Fluidization of Bulk Solids

Ing. (grad.)

Manfred Heyde

Flow and exchange

behaviour

in theory and experiment

_

3

The published measuring results for the different phenomena, that occur when dealing with bulk ma-terials, were combined by using uniform and in-termeshing physical basics. The developed algo-rithms allow for example estimations for the mate-rial discharge from silos, for the design of pneu-matic conveying installations, and for the heat transfer and the catalytic gas phase reaction in flu-idized beds. These algorithms apply mostly up to industrial dimensions. In addition in many cases the boundaries have become clear which in labora-tory scale experiments should not be exceeded. For the direct use of the calculation methods in the pro-cedural practice is the Windows software FLUIDI with integrated material database intended. It is written in VISUAL BASIC and runs even under Windows 7 (32 bit). An English version of the pro-gram can be downloaded as a ZIP file from the Internet: https://skydrive.live.com/#cid=0239E78E2F136DDE&id=239E78E2F136DDE%21936

For the discharge from silos becomes clear that the amount of material is influenced by the ratio of outlet diameter and particle diameter. This effect is also noticeable on other occasions by restricted mobility of bulk solid beds. The calculation ap-proach using measured values from small-scale ex-periments is completely confirmed by the results from large-scale plants.

For the pneumatic conveying is from the opera-tional data of different industrial conveyor systems a characteristic field been developed, which in-cludes in particular the dense phase conveying. As dimensionless parameters serve the well-known Reynolds number together with a factor that comes from a physical model for the turbulent pipe flow.

Short-term contact and the height-dependent seg-regation determine the size of heat transfer at ex-changer installations in fluidized beds. In catalytic gas-phase reactions the segregation effect is re-sponsible for the decrease of the reaction turnover.

1. Properties of pure substances and mixtures

1.1 Phase boundaries .............................................................. 5 1.2 Thermal equations of state................................................ 7 1.3 Viscosity........................................................................... 8 1.4 Thermal conductivity and heat capacity............................ 9 1.5 Moisture.......................................................................... 10 1.6 Flammable mixtures ....................................................... 11 2. Characteristics and properties of bulk materials 2.1 Particle size .................................................................... 13 2.2 Particle size distribution ................................................. 13

2.3 Void fraction, particle shape and density ....................... 14 2.4 Adhesive forces .............................................................. 15 2.5 Moisture ......................................................................... 15 2.6 Thermal conductivity and heat capacity ......................... 16 3. Mechanisms of heat transfer 3.1 Conduction ..................................................................... 17 3.2 Convection and short-term contact ................................. 17 3.3 Radiation ........................................................................ 19

4. Dimensionless numbers for flow and

Transport processes ........................................................... 20 5. Flow of bulk solids 5.1 Material flow in silos .................................................... 21 5.1.1 Silo design............................................................... 21 5.1.2 Discharge rate ......................................................... 22 5.1.3 Measures against outflow restrictions ..................... 23 5.2 Mechanical moving of bulk materials ............................ 23 5.2.1 Heat transfer in thin layer contact devices............... 24 6. Single phase flow through pipes 6.1 Continuity and energy conservation ............................... 26 6.2 Flow pressure loss .......................................................... 26 6.3 Heat transfer................................................................... 28 6.4 Turbulence structure and exchange behaviour................ 28 6.4.1 Pressure loss............................................................ 28 6.4.2 Speed profile ........................................................... 29 6.4.3 Heat transfer............................................................ 29 7. Pneumatic conveying of bulk materials 7.1 Conveying conditions ..................................................... 31 7.2 Pressure loss................................................................... 32 7.2.1 Fine-grain solid, horizontal ..................................... 32 7.2.2 Material acceleration............................................... 33 7.3 Heat transfer at the tube wall.......................................... 34 7.3.1 Conveying vertical upward...................................... 34 7.3.2 Cyclone separator.................................................... 35 8. Flow through packed beds of bulk materials 8.1 Pressure loss................................................................... 37 8.1.1 Flow around a single particle .................................. 37 8.1.2 Flow through bulk materials.................................... 38 8.2 Heat transfer................................................................... 39 8.2.1 Exchange at the particle surface.............................. 39 8.2.2 Exchange at the tube wall........................................ 40 9. Fluidized bed 9.1 Minimum fluidization velocity ....................................... 41 9.2 Appearance..................................................................... 42 9.3 Expansion behaviour ...................................................... 42 9.3.1 Homogeneous expansion and change to inhomogeneous state ............................................... 43 9.3.2 Inhomogeneous expansion....................................... 44 9.3.3 Bubbles rise............................................................. 45 9.4 Material entrainment ...................................................... 46 9.5 Penetration by gas jets.................................................... 47 9.6 Heat transfer on internals ............................................... 47

9.6.1 Influence of bulk solid movement and expansion ... 48 9.6.2 Comparison of measurements and calculations ....... 49 9.6.3 Influence of fluidized bed dimensions..................... 52 9.7 Heat and mass transfer between gas and particles.......... 52 9.8 Catalytic gas phase reactions.......................................... 52 9.8.1 Reactor model for reactions of 1. order ................... 53 9.8.2 Application of the reactor model ............................. 53 9.8.2.1 Laboratory scale ............................................. 53 9.8.2.2 Semi-technical scale....................................... 55 9.8.2.3 Measurements on large-scale reactor ............. 55 9.9 Operation as thermal dryer ............................................. 56 9.9.1 Heat requirement ..................................................... 56 9.9.2 Influence of the gas distribution plate construction. 56 9.9.3 Multi-stage fluidized beds ....................................... 58 9.9.4 Heating surfaces internals ....................................... 60 9.9.5 Spray fluidized bed.................................................. 60 10. Solids removal by cyclone separators............................65

4

Symbols

A Surface

Ar Archimedes number

a thermal conductivity

c specific heat capacity

c unreacted proportion

c' unreacted proportion in fluidized bed reactor

D Apparatus and pipe diameter

d Diameter

F Force

f Cross sectional area

Fr Froude number

Ga Galilei number

g Acceleration due to gravity

h Height

h Enthalpy

∆∆∆∆h Enthalpy difference

k Reaction rate constant

L Pipe length

l Length

M Mass %M Molar mass

Nr Number of reaction units

Nu Nusselt number

n,Z Number

Pr Prandtl number

pD Vapour pressure

p Pressure

∆∆∆∆p Pressure loss

Q Heat quantity

q related heat quantity

R Residue

R Radius of curvature %R molar gas constant

Re Reynolds number

T absolute temperature

t Time

u Velocity

V Volume

v specific volume

x Moisture content

x Coordinate in the direction of flow

y Coordinate transverse to the flow direction

αααα Heat transfer coefficient

ββββ Material transition coefficient

γγγγ Accommodation coefficient

δδδδ Diffusion coefficient

ηηηη Dynamic Viscosity

θ Angle

ϑ Temperature

∆ϑ Temperature difference

ΛΛΛΛ mean free path length of gas molecules

λ λ λ λ thermal conductivity

µ solid/gas rate

νννν kinematic viscosity

ξξξξ Resistance coefficient

ρρρρ Density

ττττ Shear stress

ΦΦΦΦ Sphericity

ϕϕϕϕ relative humidity

ϕϕϕϕ bubbles-flowed proportion

ψψψψ Porosity, void fraction

5

1 Properties of pure substances

and mixtures

1.1 Phase boundaries

In apparatuses and equipment of process engineering exist the materials to be processed depending on the prevailing conditions, in solid, liquid or gasous state. If several substances with the same aggregate state are situated together in a space, one speaks of single-phase systems. With two aggregate states side by side, we have a two-phase systems, and a multiphase systems, if all three exist side by side.

In chemical engineering, multiphase systems are used, to bring gaseous and liquid media with each other or with solids in contact. On the other hand, substances can be transformed from one phase to another, in order to separate them from each other. During the thermal drying, for example, the material moisture is being vaporized by supply of heat, and thus removed.

Whether a solid at a certain pressure during heating passes over into the liquid state or directly into the gaseous state, is dependent on the locations of the various equilibrium curves. These vapor pressure curves represent the pressure exerted by a steam in thermal equilibrium with its liquid or solid phase. The curve of the vapor pressure pD over the liquid phase is

the boiling line, over the solid phase the sublimation line. In addition there is the fusion line, which represents the transition from the solid to the liquid state.

Fig. 1.1 is a general pD-T-diagram that shows the courses of boiling line, fusion line and sublimation line. The three curves meet up in a point that is known as the triple point. In this point are solid, liquid and gaseous phase in thermodynamical equilibrium. The boiling line ends in the so-called

critical point, where the boiling line and the condensation line meet up each other. At temperatures higher than the critical temperature, there is no sharply defined boundary between gas phase and liquid phase.

The sublimation pressure curve that ends at the triple point, separates the regions of vaporous and solid phase. Except helium has every substance, that itself

not already decomposes at lower temperatures, a triple point. Its position is decisive for the sublimation ability of a substance. Is the vapor pressure at the triple point larger than 1 bar, sublimates the substance at atmospheric pressure (simple sublimation). However, the vapour pressure is usually significantly lower, so that the simple sublimation is relative rare. For its technically sublimability an organic substance must have at not too high temperatures (to avoid decomposition) below the fusion point a sufficiently high vapour pressure (over 1.5 mbar).

In the processes of thermal drying and heating, the sublimation ability of solids is sometimes unpleasantly

noticeable. As long as the partial pressure of the solid in the surrounding drying gas is lower than the saturation vapour pressure at the temperature of the system, the solid passes directly over into the vaporous state. The vapor de-sublimates subsequently on colder parts of the system and forms relative solid wall layers, which can influence the function and safety of the respective facility. Pipes and apparatus walls must in such cases be heated in addition. Mathematical representations for the vapor pressure of pure substances, are derived from the Clausius-Clapeyron vapor pressure equation and deliver simple vapor pressure relationships for small temperature

Fig. 1.1:

In the general state diagram

of a substance meet boiling

pressure, melt pressure and

sublimation printing curve at

a point

Fig. 1.2:

Vapor pressure curves of some solvents in the plot log pD

against 1/T: a water, b methyl alcohol, c ethyl alcohol, d

n-propyl alcohol, e carbon tetrachloride, f benzene, g

toluene, h o-xylene

6

ranges, in which the evaporation enthalpies themselves not change significantly:

log pT

D = +A

B (1-1)

In the plot of log pD against 1/T the vapour pressure curve should form a straight line. Fig. 1.2 shows the vapor pressure curves of some solvents in the temperature range from 0 to 100 °C.

The Antoine equation provides better results for larger temperature ranges:

log pT

D =+

+A

CB (1-2)

The following correlation for the saturation vapor pressure has been found on the basis of an extended investigation of existing measurement values for a large number of substances by Patel, Schorr, Sha, and Yaws [1.1] to be the best:

log logpT

T T TD = + + ⋅ + ⋅ + ⋅AB

C D E 2 (1-3)

In many cases, this correlation is valid from the triple

point to the critical point. Table 1.1 indicates the correlation coefficients A, B, C, D and E, with whose help the saturation vapour pressure of various solvents can be calculated in Torr using the temperature T in K. The validity ranges are also given.

The heat that is required for the vaporization of a certain quantity of liquid can be estimated after the Trouton rule. Experience teaches namely, that the molar entropy change for the evaporation at boiling point for many substances is 80 to 110 kJ /(kmol K).

With the approach to the critical point, the heat of vaporization becomes smaller and disappears at the critical point. The heat of vaporization ∆hv can be calculated for any temperature T with good accuracy by the correlation of Watson [1.2]:

∆ ∆h hT T

T T

n

v vkr

kr

= ⋅−−

1

1 (1-4)

The correlation coefficients ∆hv1, T1 and n were specified by Thakore, Miller and Yaws [1.3] for many substances. They are listet in Table 1.2 that also indicates the critical temperature of the substances.

The phase transformation heat changes itself at the triple point abruptly, because vaporization heat and fusion heat add up themselves to the sublimation heat. The fusion enthalpy ∆hs can be calculated according to the equation of Clausius-Clapeyron using the change of the specific volume from vs to vl :

( )∆hs vl vs Tdp

dT

D= − ⋅ ⋅ (1-5)

Liquid A B C D⋅⋅⋅⋅103 E⋅⋅⋅⋅106 Range in °°°°C

Water 16.373 -2818.6 -1.6908 -5.7546 4.0073 0 to 347.2

Methyl alcohol -42.629 -1186.2 23.279 -35.082 17.578 -67.4 to 240

Ethyl alcohol -10.967 -2212.6 10.298 -21.061 10.748 -45 to 243

n-Propyl alcohol -338.31 5127.5 148.80 -175.79 74.666 0 to 263.7

Carbon tetrachlorid 50.612 -3135.7 -16.313 7.8036 - -69.7 to 283.2

Benzene 51.204 -3245.7 -16.403 7.54 - 7.6 to 289.4

Toluene 115.21 -4918.1 -43.467 38.548 -13.496 -60 to 320.6

Table 1.1

Correlation coefficients to calculate the vapour pressure curve of some

solvents according to Eq. (1-3)

Liquid ∆∆∆∆hv1 T1 Tkr n Range

kcal/kg °°°°C °°°°C - °°°°C

Water 538.7 100 374.2 0.38 0 to 347.2

Methyl alcohol 260.1 64.7 239.4 0.4 -97.6 to 239.4

Ethyl alcohol 202.6 78.3 243.1 0.4 -114.1 to 243.1

n-Propyl alcohol 162.3 97.2 263.6 0.4 -126.2 to 263.6

Carbon tetrachlorid 46.55 76.7 283.2 0.38 -22.9 to 283.2

Benzene 94.1 80.1 288.94 0.38 5.53 to 288.94

Toluene 86.1 110.6 318.8 0.38 -95.0 to 318.8

Table 1.2

Correlation coefficients of Eq. (1-4) for calculating the heat of

vaporization of some solvents

Liquid A B Tkr ρρρρ Range

°°°°C at 25°°°°C °°°°C

Water 0.3471 0.2740 374.2 1.0 0 to 374.2

Methyl alcohol 0.2928 0.2760 239.4 0.79 -97.6 to 239.4

Ethyl alcohol 0.2903 0.2765 243.1 0.79 -114.1 to 243.1

n-Propyl alcohol 0.2915 0.2758 263.6 0.80 -126.2 to 263.6

Carbon tetrachlorid 0.5591 0.2736 283.2 1.58 -22.9 to 283.2

Benzene 0.3051 0.2714 288.94 0.87 5.53 to 288.94

Toluene 0.2883 0.2624 318.8 0.86 -95.0 to 318.8

Table 1.3

Correlation coefficients of Eq. (1-6) to calculate the density of

liquids in the saturation state

7

The values of the molar entropy of fusion ∆h M Ts ⋅ % / are according to [1.4] approximately:

♦ 9.2 kJ/(kmol⋅K) for metals,

♦ 22 bis 29 kJ/(kmol⋅K) for organic compounds,

♦ 38 bis 58 kJ/(kmol⋅K) for anorganic compounds.

For the density ρl of liquids in the saturation state as a function of the temperature T, have Sha und Yaws [1.5] the following correlation introduced.

( )ρlTr= ⋅ − −

A B1 2 7/

(1-6)

Calculated values for ρl in g/cm3 using the respective constants A and B are listet in Table 1.3. Tr = T/Tkr is the reduced temperature.

1.2 Thermal eqation of state

For each of the three phases of gas, liquid and solid, the thermal equations of state

p = p (v,T)

or

v = v (p,T)

specify the relationship between the thermal state variables. These functions are very complex and not yet known exactly for the three phases.

Gases behave at low pressures and enough distance from the boiling state almost exactly after the thermal equation of state for so-called ideal gases:

p v T⋅ = ⋅R (1-7)

R is the gas constant that can be calculated by using the universal gas constant %R and the molar mass %M .

Kkmol

kJ 8.317=R

~mit M

~/R

~R

⋅= (1-8)

In addition, one should know that the volumes of a kilomole of any ideal gases at the same temperature and pressure are equal. At 0 °C und 1013.25 mbar, the molar volume is v~ = 22.414 m3/kmol.

Looking at the behavior of gases at higher pressures or in the vicinity of the boiling state, deviations from the ideal gas are observed. Fig. 1.3 shows some isotherms of CO2; they extend to the region, in which condensation occurs. To recognize is the increasing deviation from the expected hyperbolic curve shape for ideal gases at low temperatures above the critical point.

The ideal gas law is therefore a limit law. Gases, whose behavior differ, are designated as non-ideal or real gases. However, in first approximation many gases can, at least in certain ranges of pressure and temperature, be treated as ideal gases.

For the description of the temperature- and pressure-dependent behavior of liquids and solids is often being used the following in T und p linear approach:

( ) ( )[ ]v T p v T T p p( , ) = ⋅ + ⋅ − − ⋅ −0 0 0 0 01 β κ (1-9)

In Eq. (1-9) means ß the volume change that is caused due to a temperature change at constant pressure (volume coefficient of expansion):

β∂

∂=

1

v

v

Tp

(1-10)

and κ is the change in volume that is caused by the change of pressure at constant temperature (isothermal compressibility coefficient):

Fig. 1.3

Deviation from the expected hyperbolic curve shape of

the isotherms for CO2 at low temperatures above the

critical point (according to [1.6])

8

1

Tp

v

v

−=

∂∂

κ (1-11)

at the by index 0 marked reference state.

For solids, the coefficient of volume expansion is being calculated by using the linear expansion coefficient α that is experimentally easy to determine:

α∂

∂= −

1

l

l

Tp

(1-12)

l in Eq. (1-12) is a characteristic length of the investigated solid. Because the volume V is proportional to l3, follows that ß = 3α. Values for α, ß and κ can be found in many table works.

Liquids and solids are changing their volume under the influence of pressure and temperature very little; v is thus constant. Gases, however, show elastic behaviour already at little pressure and temperature change. The compressibility of gases plays for example a role in pneumatic conveying. Due to the flow pressure loss decreases the gas pressure and thus the density of the gas in conveying direction. Therefore, the gas velocity and thus the pressure loss in non-widened pipes rise significantly.

The incompressible behavior of liquids plays, for example, in the testing of pressure vessels with water an important role. If once a weld really tears, reduces a very small volume increase the pressure to zero. Would air be used for the pressure build-up, could the work capacity that is contained in the large density change of the air, the vessel tear explosively apart.

For liquids is the influence of temperature on the pressure change to note, if the specific volume is kept constant. In such cases rises the pressure steeply. Therefore, may vessels for the storage and transport of liquids, which are not equipped with a pressure relief valve, never be completely filled, to avoid the dangerous condition of v=const.

1.3 Viscosity

If a real fluid is subjected a change in shape, the internal friction of the fluid causes resistance forces. In case of moving a plate on a fluid layer (Fig. 1.4), applies for the resistance force F related to the plate surface A:

F

A

du

dy= = −η⋅τ (1-13)

τ is the shear stress, du/dy the velocity gradient und η the dynamic viskosity, a measure of the size of the internal friction. With higher temperatures, the dynamic viscosity of gases increases, while that of liquids decreases. The ratio of dynamic viscosity η and fluid density ρ is called kinematic viskosität ν: ν = η/ρ.

The relationship between dynamic viscosity ηg of gases and the temperature T can be described by a polynomial approach:

ηg T T= + ⋅ + ⋅A B C 2 (1-14)

The correlation constants A, B, and C, determined by Miller, Schorr and Yaws [1.1], are listed for some technically important gases in Table 1.4. Inserting the temperature T in K in Eq. (1-14), results the dynamic viskosity ηg in µP.

The kinetic theory of gases describes the relationship between the viscosity and the molecular properties. On this basis, the mean free path length Λ of gas molecules can be expressed with the gelp of the viskosity ηg.

Λ = ⋅⋅

⋅ ⋅⋅

16

5

~

~R

2 M

T

p

g

π

η (1-15)

For liquids, the viscosity as a function of the temperature must be correlated with a slightly more complicated equation after Miller, Gordon, Schorr and Yaws [1.1]:

log ηlT

T T= + + ⋅ + ⋅AB

C D 2 (1-16)

The correlations constants for different liquids are contained in Table 1.5. Inserting of A, B, C und D as well as T in K into Eq. (1-16) results the viscosity ηl of the liquid in the state of saturation in cP.

The fact that according to Eq. (1-13) the shear force per unit of surface area is proportional to the negative local velocity gradient, calls one the Newtonian flow law. Liquids, which behave in this way, are so-called Newtonian fluids.

Within the procedural practice one often has to work with materials - pastes or liquids - which behave differently as to be expected after the Newton's law of flow. So it can happen that a supposedly solid material,

Fig. 1.4

A moving plate, which slides

over a layer of real fluid,

induces a shear stress ττττ

9

that is grinded in a mill, becomes suddenly liquid. The flow characteristics of such substances is in contrast to the Newtonian behavior dependent of the size of the shear rate. There are structural viscosity (pseudo plasticity), Dilatancy, Bingham behaviour and elasto-viscous behaviour.

In case of pseudoplastic behaviour the viscosity decreases during increasing shear rate, however, it remains constant at constant shear rate. With changing shear rate, the viscosity changes also.

Dilatant materials behave exactly opposed. The viscosity is being increased with increasing shear rate. This property is in practice relatively rare.

The characteristic of a Bingham liquid is, that the flowing starts only at a certain minimal shear stress. From then on, the course of the flow curve can be both, pseudoplastic and dilatant.

A fourth group of non-Newtonian fluids is called elastoviskos. The behaviour of these substances is a combination from elastic solids and liquids. During the stirring of such substances there is for example no trombe

(funnel). Instead of this, the fluid climbs on the stirring element upwards. In the vicinity of a radially pumping agitator can also a reversal of flow occur.

The relationship between shear stress τxy and velocity gradient -dux/dy for Newtonian, dilatant und Bingham liquids shows Fig. 1.5.

In addition to the dependence of the viscosity from the shear rate, exists in non-Newtonian behavior often still a time-dependence. Thereby a distinction is between thixotropic and rheopex behavior. Thixotropy is being observed during the stirring of a substance in a container over a longer period of time. Despite constant shear rate, the viscosity decreases more and more. After switching off of the agitator the viscosity increases back to its original value. The opposite behavior,

namely the increase of the viscosity with increasing duration of mixing, is known as rheopex.

In practice show the majority of the non-Newtonian substances structural viscosity and thixotropic behavior.

1.4 Thermal conductivity and heat capacity

Properties with significant influence on the heat transfer during flow processes are the thermal conductivity and the heat absorption ability of the substances. Mathematically, the temperature-dependent material behaviour is likewise be described with the help of polynomial-approaches. The following equation describes the thermal conductivity λg of gas at pressures around 1 bar after Miller, Sha and Yaws [1.1] as function of the temperature T.

3

D2

CBA TTTg ⋅+⋅+⋅+=λ (1-17)

For the thermal conductivity λl of liquids is one term less necessary [1.5]:

λ l T T= + ⋅ + ⋅A B C 2 (1-18)

In Table 1.6 are the correlation constants A, B, C und D for some important technical gases and solvents listet. In the Eq. (1-17) and (1-18) must the temperature

Gas A B⋅⋅⋅⋅102 C⋅⋅⋅⋅106 ηηηηg at 25°°°°C Range

µµµµP °°°°C

Hydrogen 21.87 22.20 -37.51 84.7 -160 to 1200

Nitrogen 30.43 49.89 -109.3 169.5 -160 to 1200

Carbon dioxide 25.45 45.49 -86.49 153.4 -100 to 1400

Sulphur dioxide -3.793 46.45 -72.76 128.2 -100 to 1400

Ammonia -9.372 38.99 -44.05 103 -200 to 1200

Table 1.4

Correlation coefficients of Eq. (1-14) for the calculation of the

dynamic viscosity of some technically important gases at low

pressures

Liquid A B C⋅⋅⋅⋅102 D⋅⋅⋅⋅106 Range in°°°°C

Water -10.73 1828 1.966 -14.66 0 to 374.2

Methyl alcohol -17.09 2096 4.738 -48.93 -40 to 239.4

Ethyl alcohol -2.697 700.9 0.2682 -4.917 -105 to 243.1

n-Propyl alcohol -5.333 1158 0.8722 -9.699 -72 to 263.6

Tetrachlorkohlenstoff -5.658 994.5 1.016 -8.733 -20 to 283.2

Benzene 2.003 64.66 -1.105 9.648 5.53 to 288.94

Toluene -2.553 559.1 0.1987 -1.954 -40 to 318.8

Table 1.5

Correlation coefficients of Eq. (1-16) for the calculation of the

dynamic viscosity of some liquids in the saturation state

Fig. 1.5

Relationship between shear stress ττττyx and speed gradient

-du/dy for Newtonian, Bingham, pseudoplastic and

dilatant liquids

10

T be used in K. The thermal conductivity is then calculated in µcal/(s⋅cm⋅K).

Measurements of the specific heat capacity cp of gases and liquids were correlated with the help of a polynomial-approach [1.3 und 1.5]:

c T T Tp g l,= + ⋅ + ⋅ + ⋅A B C D2 3 (1-19)

In Table 1.7 are the correlation constants A, B, C and D listed. In Eq. (1-19) must the temperature T be used in K. The molare heat capacity cpg for gases at low

pressures gets is then calculated in in kcal/(kmol K); For liquids in the state of saturation gets one the specific heat capacity in kcal/(kg K).

The amount of heat &Q , that is necessary to heat a substance to a certain temperature, is being expressed in thermodynamics as follows: &Q h c Tp= = ⋅∫∆ d (1-20)

Insertion of Eq. (1-19) into Eq. (1-20) and the integration lead to following expression for the amount of heat &Q :

&Q T T T T

T

T

= ⋅ + ⋅ + ⋅ + ⋅AB C D

2 3 4

2 3 4

1

2

(1-21)

T1 and T2 are the starting and final tem-peratures of the substance.

1.5 Moisture

Gases can be mixed up to certain limits with steams, i.e., with substances, which are in the examined temperature range con-densable. The most important example of such gas vapour mixtures is the humid air, which for air conditioning and in the drying technology as well as in meteorology plays a major role.

Expediently are the state variables of gas-vapour mixtures not related on the mixture, but on the dry gas, because in facilities its amount is constant, in complete contrast to the total amount of moist air. The moisture content x is being specified as the mass ratio of steam to dry gas:

xM

M

D

g

= (1-22)

Because for ideal gases the mole ratio equals the ratio of steam partial pressure to

gas partial pressure is, can for the specific moisture content x be written:

xM

M

p

p p

D

g

D

D

= ⋅−

~

~ (1-23)

How far moist gases are away from the saturation, is expressed by the relative humidity ϕ. ϕ is the ratio be-tween the actual steam partial pressure pD and the satu-ration pressure pS at the mixture temperature, or the ra-

Liquid / Gas A B⋅⋅⋅⋅102 C⋅⋅⋅⋅104 D⋅⋅⋅⋅108 Range in °°°°C

Water -916.62 1254.73 -152.12

Methyl alcohol 770.13 -114.28 2.79 -97.6 to 210

Ethyl alcohol 628.0 -91.88 5.28 -114.1 to 190

n-Propyl alcohol 1442.74 -8.04 -5.29 -126.2 to 220

Carbon tetrachlorid 383.95 -45.45 -0.24 -22.9 to 224

Benzene 424.26 1.14 -9.03 5.53 to 260

Toluene 485.1 -53.84 -0.59 -95 to 308

Hydrogen 19.34 159.74 -9.93 37.29 -160 to 1200

Nitrogen 0.9359 23.44 -1.21 3.591 -160 to 1200

Carbon dioxide -17.23 19.14 0.1308 -2.514 -90 to 1400

Sulphur dioxide -19.31 15.15 -0.33 0.55 0.0 to 1400

Ammonia 0.91 12.87 2.93 -8.68 0.0 to 1400

Table 1.6

Correlation coefficients of the Eq. (1-17) and (1-18) for calculating

the thermal conductivity of gases at low pressures and liquids in the

saturation state

Liquid / Gas A B⋅⋅⋅⋅103 C⋅⋅⋅⋅106 D⋅⋅⋅⋅109 Range in°°°°C

Water 0.6741 2.825 -8.371 8.601 0 to 350

Methyl alcohol 0.8382 -3.231 8.296 -0.1689 -97.6 to 220

Ethyl alcohol -0.3499 9.559 -37.86 54.59 -114.1 to 180

n-Propyl alcohol -0.2761 8.573 -34.2 49.85 -126.2 to 200

Carbon tetrachlorid 0.01228 2.058 -7.04 8.610 -22.9 to 260

Benzene -1.481 15.46 -43.70 44.09 5.53 to 250

Toluene -0.1461 4.584 -13.46 14.25 -95 to 310

Hydrogen 6.88 -0.022 0.21 0.13 25 to 1227

Nitrogen 7.07 -1.32 3.31 -1.26 25 to 1227

Carbon dioxide 5.14 15.4 -9.94 2.42 25 to 1227

Sulphur dioxide 5.85 15.4 -11.1 2.91 25 to 1227

Ammonia 6.07 8.23 -0.16 -0.66 25 to 1227

Table 1.7

Correlation coefficients of the Eq. (1-19) for calculating the specific

heat capacity of gases at low pressures and liquids in the saturation

state

11

tio between the amount of steam in the volume unit of moist air ρD and the largest possible value of ρS for equal total pressure and equal temperature:

ϕρ

ρ= =

p

p

D

S

D

S

(1-24)

For the density ρ of moist air below the saturation state applies for a certain temperature the following equation:

ρ =⋅

⋅+

+

p

R T

x

M

Mx

D D

g

1~

~

(1-25)

State changes of humid gases can be pursued in sim-ple and clear way in the Mollier-h-x diagram. The en-thalpy of the amount of (1 + x) kg of humid gas con-sists of the enthalpies of dry gas and steam:

( )h c h c xx p g v p D1+ = ⋅ + + ⋅ ⋅ϑ ϑ∆ (1-26)

By this equation can for certain temperatures ϑ and various steam contents x the enthalpy-values be deter-mined. The isotherms are straight lines in the h-x-chart.

By inserting the saturation value of the vapour partial pressure pD at a certain temperature into Eq. (1-23) and the use of the calculated value for x can the enthalpy of the saturated steam-gas mixture be determined. With the help of for various temperatures calculated values, is it possible to plot the saturation curve ϕ = 1.

Due to the poor legibility of the diagram, has Mollier, as is known, an oblique-angled presentation proposed,

in which the x-axis is tilted, until the iso-therme for ϑ = 0 °C is horizontal. This is the case, when at the coordinate x = 1 the evaporation enthalpy ∆hv is displayed downwards. Fig. 1.6 shows the Mollier-h-x diagram for moist air for a tempera-ture range of 0 to 90 °C, at a pressure of 1 bar.

For the determination of the curve of equal relative humidity, Eq. (1-23) can be written also:

xM

M

p

p p

D

g

S

S

= ⋅⋅

− ⋅

~

~ϕ

ϕ (1-27)

1.6 Flammable mixtures

Just like technical gases and heating oil in the mixture with air can be burned, many other substances can form flammable mixtures with air. For the operating of facilities are mixtures dangerous, in which the flame front itself after an ignition explosively spreads. One speaks of an explosion, if the combustion is connected with a clear pressure increase. Such a situation can occur, if in a closed space a large mass of fuel is burned and the pressure noticably goes up, or if

the fuel is combusting so quickly, that the surrounding gas due to its mass inertia, causes a large pressure increase.

The explosion becomes a detonation, if the progress - for example, through turbulence influences - itself as far as speeded up, that the mixture will no longer being

Gas / Vapour Explosion limits in air Ignition Ignition group

(1013 mbar, 20°°°°C) Temp. in dependence

lower upper of the ignition

Vol.-% °°°°C temperature

Hydrogen 4.0 75.6 560 G1

Carbon monoxid 12.5 74 605 G1

Hydrogen sulfide 4.3 45.5 270 G3

Ammonia 15.0 28.0 630 G1

Methyl alcohol 5.5 31/44 455 G1

Ethyl alcohol 3.5 15.0 425 G2

n-Propyl alcohol 2.1 13.5

Benzene 1.2 8.0 555 G1

Chlorbenzene 1.3/1.5 7/11 (590) G1

Toluene 1.2 7.0 535 G1

o-Xylene 1.0 6.0/7.6 465 G1

Table 1.8

Key figures of some combustible gases and vapours

Fig. 1.6

Mollier-h-x diagram for moist air at temperatures from 0 to 90 °C and

atmospheric pressure [1.9]

12

ignited by the heat of the flame front, but by the resulting blast. The pressure wave moves ahead with sound velocity, in case technical fuels in air thus with approximately 1000 m/s.

The boundary between deflagration and explosion is not precisely defined. In the parlance, deflagration is a weak explosion, which causes only minor damages.

Also dusts can together with air form explosive mixtures. The lower explosion limit lies between 15 and 45 g/m3.

The first requirement for the presence of an explosive mixtures is the concentration of the combustible portion in the air, which must lie in the range between lower and upper explosion limit. Moreover needs a surface, that touches the ignitable mixture, at least a temperature as large as the ignition temperature of the mixture. In addition, is a certain minimum energy necessary, to ignite the mixture. The minimum energies for dust/air mixtures are two or three powers of ten larger than those of gas(steam)/air mixtures. The size of the ignition energy influences the explosion limits likewise. The explosion intensity is also a function of the vessel dimensions. While the explosion pressure in vessels with larger dimensions remains the same, changes the maximum temporal pressure rise (dp/dt)max with the volume V of the vessel ("cubic law")

dp

dtV konst G

⋅ = =

max

/1 3 K (1-28)

KG in bar⋅m⋅s-1 is under otherwise equal conditions a material-specific constant. The cubic law is valid also for dust-air mixtures. Dusts are characterized by using Kst-values. Dust explosions are at least as fierce - if not more fierce - as gas explosions.

In Table 1.8 are listed the safety technical key figures for some inflammable gases and vapours. Tables with comprehensive information were published by Nabert and Schön [1.7]. Regarding the intensity of dust explosions can be key figures found in the VDI-Richtlinie 2263 [1.8].

Literature of chapter 1.

[1.1] Yaws, C.L.; J.W. Miller; P.N. Sha; R.R. Schorr und P.M. Patel: Corre-lation constants for chemical compounds. Chemical Engineering No-vember 22(1976), S.153-162.

[1.2] Watson: Ind. Eng. Chem. 35(1943)398, zitiert in [1.4].

[1.3] Yaws, C.L.; R.W. Borreson; C.E. Gorin II; L.D. Hood, J.W.Miller; G.R. Schorr und S.B. Thakore: Correlation constants for chemical compounds. Chemical Engineering August (1976)16, S.79-87.[1.4] Perry, R.H.; und C.H. Chilton: Chemical engineers handbook, 5th ed. New York, Tokyo: McGraw Hill/Kogakusha, 1973.

[1.5] Yaws, C.L.; J.J. McGinley; P.N. Sha; J.W. Miller und G.R. Schorr: Correlation constants for liquids. Chemical Engineering October (1976)25, S.127-135.

[1.6] Eastman, E.D.; und G.K. Rollefson: Physical Chemistry. New York: McGraw Hill Book Co., 1947.

[1.7] Nabert, K.; und G. Schön: Sicherheitstechnische Kennzahlen brenn-barer Gase und Dämpfe. 2. Aufl., Braunschweig: Deutscher Eich-verlag.

[1.8] VDI-Richtlinie 2263: Verhütung von Staubbränden und Staub-explosionen.

[1.9] Buchholz, E.: Das i-x-Diagramm von Mollier und seine Anwendung bei der Bedienung von Luftbehandlungsanlagen. Energie 6(1964), S.316-323.

13

2 Characteristics and Properties of

bulk materials

The behaviour of bulk materials during the discharge from silos, during the pneumatic conveyance, or in flu-idized beds, is decisively influenced by certain recur-ring material characteristics. The individual particles are of different shape and size, often show a porous structure, and are often randomly arranged within the bulk solid. Under certain circumstances friction and adhesion forces are acting between the particles.

These properties affect the movement and the ex-change processes in technical devices and facilities. For a mathematical description of the applicable physical laws, must bulk materials therefore physically meaning-ful and clearly be marked. Process selection and quality evaluation requiring likewise clear specific data.

2.1 Particle size

The particle diameter dp in this context means not the geometric dimension of the particles, but describes the dimensions of particles of defined form with the same properties. It is named equivalent diameter. For exam-ple, the terminal velocity equivalent diameter is the di-ameter of a sphere with the same terminal velocity as it the particle has. In practice, one often uses the sieve mesh width of a sieving as a not very meaningful, but quickly and easily determinable equivalent diameter. Other equivalent diameters are:

♦ geometric diameter: diameter of a sphere of equal volume or equal surface,

♦ Equivalent diameter of the particle projection: di-ameter of the circle with equal area or equal circum-ference,

♦ flow-mechanical diameter: diameter of the sphere with equal flow resistance or equal sinking speed,

♦ Diameter of the sphere with equal scattered light in-tensity,

♦ Diameter of the sphere with equal electrical resis-tance change.

It is not possible to determine particle diameter of very different sizes by the same way. The analysis method is being chosen depending on particle size and other properties of the bulk material. In addition, per-fect results are anyway just to expect if the investigated sample is representative of the bulk material.

During the sieve analysis, the bulk material by test sieves of different mesh size is decomposed into vari-ous classes of particles. Without any additional means, particle sizes above 40 to 60 µm can in this way be seized. The screening is improving, by sucking and elu-triating the particles through the sieve. In this way par-ticle sizes smaller than 5 µm can be measured.

The sedimentation analysis works, using a suspension. From the determined sinking velocities during the sedimentation of the particles, the particle diameters of individual classes are calculated. In the gravitational field with this method particle diameters between 2 and 50 microns can be measured, in the centrifugal field such with 0.01 to 3 microns in diameter.

Wind sifting within gravity or centrifugal fields is suitable for dry bulk materials. During the wind sifting, the particles with a smaller sinking velocity than that of the upward airflow are discharged by the fluid. In the gravity field, particle sizes between 5 and 60 µm, in the centrifugal field particle sizes down to 1 µm can be measured by this method.

In the light microscope can particles with sizes be-tween 1 and 150 µm be analysed.

In the so-called coulter counter, particles, which are suspended in an electrically conductive fluid, are sucked through a current-carrying capillary bore. This changes the electrical resistance of the liquid approxi-mately proportional to the volume of a single particle.

In the scattered light analyser, dust loaded gas is blown into an illuminated volume. There, each particle scatters a to its size corresponding light proportion, which is photometrically registered.

2.2 Particle size distribution

Usually, the particles in a bulk material have no uni-form size, but there is a more or less broad range in which the dimensions vary.

Most of the usual analysis methods provide informa-tion about the quantity of particles of a specific size in a bulk solid. These quantities can be plotted in a chart as proportion of the number, mass, volume, or surface dependent on the investigated particle size intervals

(called classes). It is common, to depict the proportion of the total amount as relative value, as so-called fre-quency, based on the investigated class width. This ap-proach results in a step diagram as shown in Fig. 2.1, in

Fig 2.1

Relative occurrence of particle-classes in a bulk solid

14

which in case of a sufficient number of particle classes a steady course can be plotted.

This characteristic line can also as cumulative fre-quency curve be plotted, that indicates, which propor-tion of the particles greater or less than a certain parti-cle diameter dp is. Following the sieve analysis, these

graphics are designated as backlog and passage cumula-tive curves. The particles, which are larger than the mesh size, are forming the backlog, and those, which are smaller, the passage. Fig. 2.2 shows an example.

In order to describe the measured particle size distri-butions of a bulk material mathematically, different equations were proposed. The equations are, however, all of empirical nature, and describe the actual grain sizes only in limited ranges accurately.

In addition to the arithmetic and logarithmic normal distribution, as well as the power-law distribution, should the RRSB distribution according to Rosin, Rammler and Sperling [2.1] be mentioned, after which the backlog cumulative curves of many bulk solids fol-low the Eq. (2-1).

Rd

d

p

p

n

= −

exp'

(2-1)

In Eq. (2-1) means n an evenness coefficient, and ′dp

is the particle size at the value R = 0.3679. In a coordi-nate system, consisting out of a double logarithmic or-dinate and a logarithmic abscissa, the RRSB backlog cumulative curve is a straight line. In practical cases are, however, those lines usually slightly curved. With a polynomial function is it possible to reproduce their course. Transformation of analysis results into the RRSB coordinate system and subsequent determination of the polynomial function constitutes the mathematical basis for the numerical evaluation of separating and grinding processes.

The various nets for the representation of the particle size distribution are standardized according to DIN 66141 (basics), DIN 66143 (power-law distribution), DIN 66144 (logarithmic normal distribution), and DIN 66145 (RRSB distribution).

For calculations, must the particle size distribution of a bulk material to be expressed by a measure for the average particle size. This so-called mean particle di-ameter dpm

is defined as follows:

1

d

x

dpm

i

pi

= Σ (2-2)

xi is the weight proportion of a particle class, and d pi

is the arithmetic mean value of the equivalent diameter of a particle class. With the particle diameter dp is in the following always a medium value meant.

The published results of studies on bulk solids were mostly won with the help of sieved particle fractions. For most of the discovered laws one must assume therefore, that their validity only for particle size distri-butions is guaranteed, in which the ratio between the largest and the smallest particle size is not larger than 4. Despite this restriction, belong the majority of prob-lems in practice to this scope. A greater proportion of fine particles or discontinuous particle size distribu-tions have, however, a sustainable influence on the mo-tion and exchange behaviour of bulk materials.

2.3 Void fraction, particle shape and density

Like the dimensions of the individual particles of a bulk solid, so also differ the shapes from each other. Particle and bulk solid properties are dependent of this shape. To take this into account, were shape criteria be-ing established, which be mostly used in connection to the particle size. So makes for example the sphericity

Particle type Ψp Average particle size in µm

20 50 70 100 200 300

Angular sand 0.67 - 0.60 0.59 0.58 0.54 0.50

Rounded sand 0.86 - 0.56 0.52 0.48 0.44 0.42

Rounded sand

mixture

- - - 0.42 0.42 0.41 -

Carbon and glass - 0.72 0.67 0.64 0.62 0.57 0.56

Anthracite 0.63 - 0.62 0.61 0.60 0.56 0.53

Activated carbon - 0.74 0.72 0.71 0.69 - -

Catalyst 0.58 - - - 0.58 0.56 0.55

Abrasive - - 0.61 0.59 0.56 0.48 -

Table 2.1

Experimental void fractions of some bulk solids [2.2]

Fig. 2.2

Cumulative curves of particles of a bulk material: a pas-

sage, b backlog

15

Φp the comparison between the surface of a particle and a sphere of equal volume:

particle single a of surface

volumeequal with sphere a of surface=Φ p (2-3)

The ratio of the free volume between the particles of a packed bed and its total volume is referred to as poros-ity or void fraction Ψ0:

Ψ00

p 0

V

V V=

+ (2-4)

V0 is the volume of gas, and Vp is the volume of the particle mass.

Ψ0 depends on both the sphericity and the particle size distribution, and it can be calculated only under very restricted conditions. For practical use, the void frac-tion is therefore experimentally determined or on the basis of known examples being estimated. It rises ac-cording to Table 2.1 (after [2.2]) with decreasing parti-cle size and sphericity.

According to the definition, the void fraction Ψo is an average value, in order to characterise the overall bulk solid. The local void fraction ′Ψo is however very dif-ferent. Independent of the shape and size of the parti-cles reaches the void fraction due to the point contacts of the bulk solid at the apparatus wall the value of 1. It becomes with increasing distance from the wall smaller

and reaches at a distance of half particle diameter a minimum value. Then it rises again, but not to the value 1, because the second row of the particles lays itself in the gussets of the first one. The arrangement of each of the following rows of particles is more and more left to chance. From a certain distance of the wall is the distri-bution of the particles only pure random.

Fig. 2.3 shows experimentally determined porosities for spheres of equal size [2.3]. It is to recognise that the size of the local pore volume according to the transition from the well-ordered bulk material to the randomly

distributed bulk material is swinging around a mean with steadily decreasing amplitude. Upon reaching the pure random distribution this oscillation is completely subsided.

The change of the void fraction Ψ0 of the total bulk, which is caused by the wall influence, decreases with increasing ratio between vessel diameter and particle diameter.

An also important influencing factor is the density ρp of the particles. It must however be taken into account, that the particle density ρp and the actual solids density ρs in case of a porous structure are possibly not equal. In practice, the particle density is therefore also desig-nated as apparent density.

The bulk solid density ρSch is the measure for the den-sity of the whole bulk solid volume. Because of the gas proportion, which is considered as weightless, its value is reduced:

ρ ρSch p 0= ⋅ −( )1 Ψ (2-5)

If the void fraction ψo of the bulk solid is known, the

particle density ρp can be calculated with the help of the bulk solid density.

2.4 Adhesive forces

The different mechanisms of adhesion can cause in moving bulk materials disabilities. During drying for example, meets the average particle size often not the actual particle diameter, because agglomerates may be formed. So it comes to deviations in the flow and mov-ing behaviour of bulk goods.

The main adhesion mechanisms occur due to liquid bridges, van-der-Waals and electrostatic interactions, as well as solid-state bridges. Liquid bridges between two solids surfaces cause always an attraction due to the surface tension of the liquid. Van-der-Waals forces make themselves only noticeable in case of very small particles. Electrostatic forces arise due to surplus charges and can act both attractive and repulsive. Ex-cess charges form themselves for example because of frequent collisions of the particles against walls or with each other, if one of the contact partners has the charac-teristics of an insulator. Solid-state bridges can for ex-ample arise during sintering, melting and crystallizing.

Fig. 2.3

Position dependency of the void fraction

′Ψo in a bulk solid with sphere-shaped par-

ticles [2.3]

16

2.5 Moisture

Liquid-solid mixtures can exist as a real solution, col-loidal solution, fixed colloids (gels), suspensions or as crystalline solids with dispersed fluid inclusions. De-pending on the type of mixture, are the liquids in dif-ferent ways bound to the solids.

During the crystallization of solutions are many of the separated substances not fluid-free, but they incorpo-rate liquid molecules into their crystal lattice. So, we have it to do with a molar binding.

One speaks of adsorption, if the liquid molecules are bound due to van-der-Waals forces. The idea in this re-gard is, that several layers of liquid molecules them-

selves settle with decreasing binding energy from layer to layer on the free surface. Significant quantities of liquids can be adsorbed in this way only of gels with their very large specific inner and outer surface. There-fore, this type of binding plays a role particularly in case of small liquid contents.

With increasing number of molecular layers of liq-uids, the binding energy becomes smaller. When reach-ing the value zero, no more moisture can adsorptively be absorbed. However, there are colloidal materials, which consist of a broad mixture of various molecular

weights. The low-molecular materials are still partially soluble in the liquid. In contrast, the high-molecular materials form an insoluble skeleton. This skeleton houses the soluble portions, which cannot pass through the cell walls. If, however, the low-molecular liquids can pass the cell walls, an osmotic process arises. The amount of liquid that can be taken by such a Material in

this way is under circumstances a multiple of the ad-sorbed liquid. One speaks of osmotically bound or structural moisture. On free solids surfaces and in cap-illaries the moisture is mechanically bound. As is known, the surface tension of the liquid is the cause of this binding.

All known kinds of the binding, which cause a lower-ing of the vapour pressure above the surface of the moisture, for mechanically bound moisture this is only the case in the capillaries of the fine-pored materials, are summarized under the term sorption. By thermal drying of these so-called hygroscopic solids, sorbed liquid can be removed only up to the point, when the vapour pressure inside the substance equals the partial pressure in the surroundings. It is almost impossible to distinguish the binding types strictly, so that sorption isotherms, thus the curves of the liquid content X in the material in dependence of the relative air humidity ϕ are experimentally determined. Fig. 2.4 shows the sorp-tion isotherms of some plastics; the sorbed substance is water vapour.

2.6 Thermal conductivity and heat capacity

The thermal conductivity of solids can only be deter-mined with the help of experiments. Thereby was ob-serve, that electrical conductors much a higher thermal conductivity have, than electrical insulators.

Material ϑϑϑϑ ρρρρ c λλλλ

°°°°C kg/m3 kJ/(kg K) W/(m K)

Aluminium 99 75 20 2700 0.896 229

Lead, pure 0 11340 0.128 35.1

Bronze 20 8800 0.377 61.7

V2A-Steel 20 8000 0.477 15

Copper 20 8300 0.419 372

Concrete 20 2200 0.879 1.28

Ice 0 917 1.93 2.2

Soil, coarse 20 2040 1.84 0.52

Glass 20 2480 0.7/0.93 1.16

Granite 20 2750 0.75 2.9

Coal 20 1300 1.26 0.26

Table 2.2

Density ρρρρ, thermal conductivity λλλλ and specific heat ca-

pacity c for some solids

Fig 2.4

Sorption isotherms of plastics (sorbed

substance is water vapour): a polysty-

rene granules, b polyethylene powder,

c polyethylene granules

17

The thermal conductivity of metals is thereby the higher, the greater the value of the electrical conductivity is. A connection to other properties of the substances is not recognize. The temperature dependence of the thermal conductivity of non-metals is various. An influence of the density is however in such a way noticeable, that the thermal conductivity with increasing density becomes greater. Table 2.2 shows the published values of density, thermal conductivity and specific heat capacity of some solids.

The effective thermal conductivity of non flowed through bulk solids is not a simple material constant any more, but is influenced by the molecular thermal conductivity of the gas in the overall connected gas volumes. In addition play the following influences a role: the conductivity in the gas-filled gussets between the particles, the thermal conductivity of the particles, the contact between the particles, and the radiation exchange between the particle surfaces. These dependencies can no longer be represented by means of simple calculation approaches.

How much the gas proportion influences the thermal conductivity of a bulk solid, can one recognize by the characteristics of porous insulating materials. The insulating effect results almost completely from the poor thermal conductivity of the air in the pores. The solid material sceleton - the solid material conducts heat much better - is only used to prevent the convective heat transfer due to free moving air.

Schrifttum zum Abschnitt 2.

[2.1] Rammler, E.: Zur Auswertung von Körnungsanalysen in Körnungsnet-zen. Freiberger Forschungsheft, Reihe A4, 1952.

[2.2] Leva, M.: Fluidization. New York: McGraw-Hill Book Company, 1959.

[2.3] Ridgeway, K; und K.J. Tarbuck: Voidage fluctuations in random packed beds of spheres adjacent to a containing wall. Chem. Eng. Sci. 23(1969), S.1147-1155.

3 Mechanisms of heat transfer

3.1 Heat conduction

The temperature fields during the heating and cooling of materials are usually temporally variable. Calcula-tion basis for the thermal conduction is the Fourier equation that describes the relationship between the spatial and temporal change of temperature.

∇ = ⋅2 1ϑ

∂ ϑ

∂a t

(3-1)

The thermal diffusivity a = λ/(c⋅ρ) of a substance is a property just as λ, c und ρ, for example with the dimen-sion m2/s.

For the amount of heat, which passes through a sur-face area of 1 m2, applies the following equation [3.1]:

&q c t= − ⋅ ⋅2

0πλ ρ ϑ (3-2)

The heat flow grows thus with t . Die physical term λ ρc is a pure material property that is known as heat

penetration coefficient, but could illustratively be named heat storage capacity.

The comparison between the thermal diffusivity and the heat penetration coefficient shows the different ef-fects of the thermal capacity per volume unit in regard on the temperature field and on the stored amount of heat. Accordingly the material for a heated wall must be chosen under consideration of the respective neces-sities. In some cases should be the propagation of initial temperatures as slowly as possibly, and in other cases should be the heat flow into the wall in equal time in-tervals as small as possible. For example in case of a fire-retardant wall the value would be kept small, and in case of the isolation of a discontinuous-powered fur-nace the value of λ ρc would be kept small.

3.2 Convection and short-term contact

The size of convective heat transport is usually being described with the heat transfer coefficient α. The de-fining equation for α is:

α =⋅

&Q

A ∆ϑ (3-3)

∆ϑ is the temperature difference between the surface and the along flowing fluid, A is the heat exchange sur-face, and &Q is the per unit time exchanged quantity of

18

heat. The size of the heat transfer coefficient depends on the flow conditions, which are being influenced by the substance properties and the geometrical condi-tions.

If the wall is touched by bulk solid, applies for the heat transfer coefficient in case of not too short contact times t and constant wall temperature the following re-lationship [3.2]:

απ

λ ρ= ⋅

2 ( )c

t

Sch (3-4)

According to Eq. (3-4) is the heat transfer coefficient α proportional to the square root of the heat penetration coefficient (λcρ)Sch of the bulk solid and inversely proportional to the square root of the contact time t.

Forever shorter contact, the heat transfer reaches fi-nally a maximum value and remains constant despite further reduced contact time. For the limit of vanishing short contact applies after Schlünder [3.2, 3.3] for the heat transfer coefficient αp between a spherical indi-vidual particle and a wall:

lim lnmax,t

p p

g

p p

p

rdd

d

→= =

⋅ ⋅+

⋅

⋅+

−

+0

4 21

21 1α α

λ

α

σ

σα

( )3 5−

λg signifies the thermal conductivity of the gas be-tween the particles, and dp is the particle diameter. σ γ γ= ⋅ −2 2Λ ( ) / can be determined, using the mean

free path of the gas molecules Λ after Eq. (1-15).

γ is the so called accommodation coefficient. This co-efficient takes into account the imperfection of the en-ergy exchange during the collisions of the gas mole-cules against the wall and against the particle surface.

Fig. 3.1 depicts for different temperatures values for γ in dependency from the molecular mass

~Mg of the

gases. In addition, the influence of the gas-solid pair-ings is to recognize.

Fig. 3.2 shows by Eq. (3-5) for air as gas without tak-ing into account the proportion of radiation αrd com-puted values for the maximum heat transfer coefficients between a single particle and a wall during short-term contact as function of the particle diameter dp. Parame-ter is the average temperature ϑ ϑ ϑm w p= +( ) / 2 be-

tween wall and particle. To recognize is the significant increase of the heat transfer coefficient αmax,p with decreasing particle size and increasing temperature. Thereby outweighs the influence of the particle diame-ter by far.

Eq. (3-5) applies to the heat transfer between a spheri-cal single particles and a wall. During the heat transfer between a bulk solid and a wall the void fraction of the packed bed must be considered. Because the empty

Fig. 3.3

Heat transfer coefficient αααα between bulk solid and wall as

function of the contact time t

Fig. 3.2

Maximum heat transfer coefficient ααααmax,p between wall

and particles

Fig. 3.1

The accommodation coefficient γγγγ as a function of the mo-

lecular mass ~M g of the gas; parameter is the mean tem-

perature between wall and particles

19

space provides a very small amount of heat transfers, the effective transfer surface is smaller than the actual one. The missing proportion of the heating or cooling area corresponds with the average void fraction of the bulk material, so that for the maximum heat transfer be-tween the bed and wall applies:

( )α α αmax max,= ⋅ − +p rd1 0Ψ (3-6)

For a through a pipe sliding bulk solid, called Moving Bed, has Ernst [3.8 and 3.9] the heat transfer coeffi-cients measured between the bulk solid and a heated ring surface at very short contact times. Table 3.1 gives a comparison between the measured maximum values for three solids with different mean particle diameters and the according to Eq. (3-6) calculated values. The accordance is remarkably good, so that the void frac-tion also in other cases as a measure of the missing proportion of the actual heat transfer surface can be used.

dp ΨΨΨΨ0 ααααmax in W/m2 K

µm - calculated measured

150 0.48 1774 1740

400 0.42 894 904

600 0.42 639 696

Table 3.1

Comparison of measured and with Eq. (3-6) calculated

heat transfer coefficients ααααmax for bulk solids, which

slides through a pipe

The general relationship between the heat transfer coef-ficient for a bulk material (packed bed) and its contact time on a wall is once more in Fig. 3.3 depicted, whereby the radiation component was neglected.

While for the heat transfer in the validity range of the t -law according to Eq. (3-4), the material properties

of the bulk material and the contact time are relevant, is the heat transfer in the validity range of the αmax-law according to Eq. (3-5) and (3-6) only dependent of the particle diameter, the bulk solid void fraction and the properties of the interspace gas.

3.3 Radiation

During the exchange of heat between two bodies (for example, wall and bulk solid), the amount of heat that is exchanged as a result of the radiation effect can often not be neglected. The frequently occurring diatomic gases N2 and O2 do not hamper this exchange because they are diatherm.

Because radiation values in practice be rarely known or are determinable, one uses as basis for the mathe-matical description the model of the complete radiation exchange between themselves enclosing grey bodies or

flat surfaces. The exchanged amount of heat then is written:

&Q A CT T

rd = ⋅ ⋅

−

1 121

42

4

100 100 (3-7)

with

( )CC A A C Crd

121 1 2 2

1

1 1 1=

+ ⋅ −/ / / / (3-8)

For technical surfaces one can use for the radiation coefficient C sufficiently accurate a value of 4.6 W/(m2 K4). For the equivalent heat transfer coefficient αrd one uses for the radiation in case of not too great tempera-ture differences the well-known relationship:

αrdmC

T= ⋅ ⋅

0 04100

12

3

, (3-9)

Tm is the mean temperature between T1 and T2 . Fig. 3.4 shows the radiant heat transfer rates, calculated ac-cording to Eq. (3-9) for temperature differences below 200 °C; For C12 has been used a value of 4.6 W/(m2 K4).

It is noteworthy, and it is often misjudged, that the heat transfer by radiation at room temperature lies in the order of magnitude of the heat transfer for free con-vection. The propor-tion of radiation must thus in certain cases to be consid-ered in heat transfer calculations.

Another character-istic of the radiant heat exchange that is particularly noticea-bly at high tempera-tures, as they prevail in reactors concerns the increase of the equivalent heat trans-fer coefficient with decreasing tempera-ture differences be-tween the heat ex-changing surfaces. Recognizable become the functional interrelations in the complete equation for αrd:

( ) ( )αrd

c T T

T T=

⋅ −

−

12 14

24

1 2

100 100/ / (3-10)

Fig. 3.4

Equivalent heat transfer coeffi-

cient ααααrd for the radiation ex-

change between completely en-

closed surfaces of approximately

equal size

20

Literature of chapter 3.

[3.1] Gröber, Erk und Grigull: Grundgesetze der Wärmeübertragung. Berlin, Heidelberg, New York: Springer-Verlag, 1981.

[3.2] Schlünder, E.U:: Wärmeübergang an bewegten Kugelschüttungen bei kurzfristigem Kontakt. Chem.-Ing.-Tech. 43(1971)11, S.651-654.

[3.3] Wunschmann, J.; und E. U. Schlünder: vt "verfahrenstechnik" 9(1975)10, S.501-505.

[3.4] Zehner, P.: VDI-Forschungsheft 558. Düsseldorf: VDI-Verlag, 1973.

[3.5] Reiter, F.W.; J. Camposilvan und R. Nehren: Akkomodationskoef-fizienten von Edelgasen an Pt im Temperaturbereich von 80 bis 450 K. Wärme- und Stoffübertragung 5(1972)2, S.116-120.

[3.6] Eckert, E.R.G.: McGraw-Hill Inc., 1959.

[3.7] Ebert, H.: Physikalisches Taschenbuch. Wiesbaden: Vieweg Verlags-gesellschaft, 1976.

[3.8] Ernst, R.: Der Mechanismus des Wärmeübergangs an Wär-meaustauschern in Fließbetten (Wirbelschichten). Chem.-Ing.-Tech. 31(1959)3, S.166-173.

[3.9] Ernst, R.: Wärmeübergang an Wärmeaustauschern im Moving Bed. Chem.-Ing.-Tech. 32(1960)1, S.17-22.

4 Dimensionless numbers for flow

and transport processes

For the scaling of devices and summarising relation-ships between physical conditions there is the problem that a variety of possible geometries and conditions have to be set in relation to each other. Because the dif-ferential equations for flow and transport processes generally cannot be integrated, model theories are used. When using such relationships, the question arises, un-der which circumstances two conditions are equivalent to each other.

For the comparability of flow and transport processes must first the geometric similarity be met. For this pur-pose often is used the ratio of two lengths, and the equality of these ratios in two different versions fulfils then the similarity conditions. For the pipe flow for ex-ample one uses the relationship between the roughness k of the pipe wall and the pipe diameter D.

Flowing fluids are being influenced by forces, which attack at the fluid elements. Similarity of processes is given, if certain relationships of forces are equal. The ratio between inertia force and resistance force is the so-called Reynolds number Re:

Re =⋅u l

ν (4-1)

u is the flow velocity, l is a characteristic length, and ν is the kinematic viscosity.

The ratio of inertia force and gravity force is called Froude number Fr:

Fr =⋅

u

l g

2

(4-2)

u and l have the same meaning as in the Reynolds number, g is the acceleration due to gravity.

A further dimensionless number, which can be derived for the here interesting incompressible flow without free surface, is the Euler number Eu, which sets the pressure differential p p1 2− and the product of fluid

density ρf and the square of velocity in relation:

Eu =−

⋅

p p

u f

1 22 ρ

(4-3)

The combinations of dimensionless numbers provide new ones, for example, the Galilei number Ga:

GaFr

= =⋅Re2 3

2

l g

ν (4-4)

To take into account the buoyancy, one further dimensionless number was formed, the Archimedes number:

Ar Ga= ⋅−

=⋅ ⋅

⋅

ρ ρ

ρ ν ρ

s f

f f

l g3

2

∆ρ (4-5)

For the description of mass and heat transfer processes, the use of dimensionless transfer coefficients has become customary, especially the Nusselt number Nu and the Sherwood number Sh:

Nul

=⋅α

λ (heat) Sh

l=

⋅β

δ (mass) (4-6)

These dimensionless numbers depend on the Reynolds number, and are also influenced by some others for the capture of the ratio between momentum transfer and heat transfer, for example the Prandtl number Pr and the Schmidt number Sc:

Pr =ν

a (heat) Sc

a=

ν (mass) (4-7)

The Prandtl number, and the Schmidt number are based on physical characteristics, just as the ratio of the two, the Lewis number Le:

LeSc

= =Pr

a

δ (4-8)

21

5 Flow behavior of bulk solids

For moved bulk solids has one to distinguish between cohesive and non-cohesive behavior. A non-cohesive material like dry sand takes the form of a circular cone when it is being heaped on a horizontal level. Depending on the particle density and the internal friction arises always the same rise angle between the surface line of the cone and the horizontal level. The angle of repose is thus reproducible.

Cohesive material sets its deformation an additional resistance against. The size of this resistance is dependent of the compaction state of the material. For example expires moist compacted sand not from the form, even if it is standing on its head. Experience has shown, that the angle of repose of such materials is little reproducible.

The internal friction of non-cohesion bulk materials - only the flow behavior of such materials is the subject of all further considerations - is affecting all processes, in which relative movements in the bulk material occur. Quantitatively can this influence, for example be considered in the relationship for the discharge of bulk solids from silos.

5.1 Material flow in silos

Due to the gravity flows bulk solid by itself out of bunkers and silos and can be passed through pipes to their destination. The designing engineer has to do it with two issues. A silo must be constructed in a way, that the whole contained material runs evenly out. The outlet cross section must be large enough to prevent, that stable material bridges, which hinder the flow of the bulk material, occur.

For the given dimensions of the silo then arises also yet the question, how the per time unit outflowing material mass be influenced by the material properties. If the necessary geometrical conditions, for example, from lack of space can not be met, or if bulk materials with very different properties be stored in a bunker, a

smooth and trouble-free operation is often only with the help of additional discharge aids guaranteed.

5.1.1 Silo design

Two different modes of flow can be observed if a bulk solid is discharged from a silo: mass flow and funnel flow (Fig. 5a). In case of mass flow, the whole contents of the silo are in motion at discharge. Mass flow is only possible, if the hopper walls are sufficiently steep and/or smooth, and the bulk solid is discharged across the whole outlet opening.

If a hopper wall is too flat or too rough, funnel flow will appear. In case of funnel flow (Fig. 5b), only that bulk solid is in motion first, which is placed in the area more or less above the outlet. The bulk solid adjacent to the hopper walls remains at rest and is called „dead" or „stagnant" zone.

Two steps are necessary for the design of mass flow silos for bulk material: The calculation of the required hopper slope which ensures mass flow, and the determination of the minimum outlet size to prevent arching. These parameters are measured in dependency on the consolidation stress with shear testers, e.g. with the Jenike shear tester or a ring shear tester. The shear cell of the shear tester introduced by Jenike [5.1] consists of a closed ring at the bottom, a ring of

the same diameter (so-called upper ring) lying above the bottom ring (Fig. 5.2), and a lid. The sample of bulk solid is poured into the shear cell. The lid is loaded centrally with a normal force N. In addition, a bracket is fixed to the lid. The upper part of the shear cell is displaced horizontally against the fixed bottom ring. The hopper slope required for mass flow and the minimum outlet

size to prevent arching can be calculated with the measured values using Jenike’s theory [5.5].

5.1.2 Discharge rate

Equations for the discharge rate from a silo were established so far only empirically and almost exclusively for non-cohesive bulk materials. Mostly describe these equations only the respective trials with good accuracy. In silos with symmetrical funnels is the mass flow of non-cohesive bulk material, whose effective friction

Fig. 5.1

During discharge of bulk materials

from silos can occur funnel flow (a)

and mass flow (b)

Fig. 5.2

Jenike shear tester for

determining the friction

inside the bulk solid (a) and

between bulk solid and wall

(b)

22

angle corresponds to its angle of repose, being influenced by numerous parameters and dependencies [5.6].

8.25.2 to AAs ddM ≅&

05.0 to hhM s ≅&

&M s p≅ ρ

118.0 to −−≅ pps ddM&

5.0gM s ≅&

dA is the outlet diameter, h is the height of the filling, ρp is the particle density, dp is the particle diameter, g is the acceleration due to gravity. In addition there is the influence of the opening angle θ in radian measure, that can be assumed as follows [5.7]:

36.0 −≅ θsM&

In addition there is the effects of the inner friction coefficient µ of the bulk material. µ is being equated to the tangens of the angle of repose ß:

µ β= tan (5-1)

The influence of the bed height h on the outflow can be neglected. The silo bottom pressure is changing itself only at very small filling heights, and the outflow becomes then non-stationary [5.6]. In an analysis of the measurements from three studies [8.5 to 10.5] could the very useful Eq. (5-2) for the calculation of the mass flow be developed [5.11].

36.0

5.25.0

θµ

ρ

⋅

⋅⋅⋅

= Ap

p

As

dg

d

dkM& (5-2)

As the in Fig. 5.4 depicted evaluation resulted, exists between the proportionality factor k and the ratio of outlet diameter to particle diameter approximately an exponential relationship. The graph indicates also the

limits: at values of dA/dp less than 5 to 10 is the outflow of bulk material no longer possible.

( ) 13.0/1.0 pA dd

p

A ed

dk

⋅−−⋅=

(5-3)

The for the evaluation of the investigation results used data, were determined in experiments with glass, sand, lead, clay, resin and coal. The ranges of the individual parameters are stated in Table 5.1. The particle sizes of the investigated bulk solids range from 150 to 4000 microns.

With the help of the measurements by Taubman [5.12] could be proven, that Eq. (5-2) together with Eq. (5-3) also for much larger particle sizes can be applied. The results from experiments using gravel with particle sizes from 3 to 65 mm are reproduced in Fig. 5.5. The measurments for three outlet diameters show good agreement with the calculated values.

literature sign bulk sol. dp dA θθθθ µµµµ

mm mm grd grd

[5.8]

sand

glass

sand

550

1100

2540

10

to

50

180 34

25

36

[5.9]

sand 160

to

910

6,6

to

11,9

60

to

180

35

to

42

5.10]

lead,

clay,

sand,

glas,

fertilizer,

coal

790

to

4000

10

to

58

180 24

to

39

Table 5.1

Range of variation of the individual parameters

in the investigations, evaluated in Fig. 5.4

Fig 5.4

Evaluation of eq. (5-2) with the help of

measured values for cohesionless

solids [5.8 bis 5.10]

Fig. 5.5

Comparison of the calculation according to eq. (5-2) with

the measured values on gravel [5.12]

23

In Eq. (5-2) is recognizable, that for the outflow of bulk materials from vessels (silos) especially the Froude number, formed using the outlet diameter dA as characteristic length, plays a role:

Fru

g d

M

g dA

s

p A

=⋅

=⋅ ⋅

2 2

2 5

&

ρ (5-4)

5.1.3 Measures against outflow restrictions

During the storing of bulk materials in silos are being mainly discharge devices and discharge aids used as additional facilities. Discharge devices serve the dosing of outflowing bulk material and are thus at the same time discharge and feeding device. In contrast to that, discharge aids have the task to support the material flow from the silo against the occurring disabilities. Some discharge devices can also be used as discharge aids..

The flow of the bulk material is always then hindered, if the dimensions of the silo are not being attuned to the bulk material properties. Various reasons may be responsible. Either must the silo be installed without any knowledge of the properties of the material, or the silo could due to the spatial conditions not be designed according to the known material properties. Very different bulk materials are often being stored in one and the same silo. Then could in case of bad flow behavior of the material, discharge devices be needed, in order to protect the bin against bridge formation and blocking, and in order to change from a current funnel flow to the mass flow, if necessary.

The flow behavior of bulk material in the bunker can be changed fundamentally in two ways. One can influence the material properties, or uses constructive measures, to create a favourable flow profile of the material in the bunker.

A way to affect the properties of the bulk materials, is adding dispersants, also known as lubricants. Such lubricants are magnesium oxide, aerosil, urea, or diatomaceous earth with particle sizes down to 10-5 mm. The small particles distribute themselves among the actual particles of the bulk material and prevent in this way the cohesion effect.

Another way is the fluidization of the bulk solid by blowing air, in order to loosen the packed bed up. This measure is tantamount to the reducing the effective friction angle of the bulk material. To execute this, appropriate air distribution elements in the outlet section or at the outlet cone must be installed.

The measures described above, however, are often not suitable. On the one hand change the added dispersants

the quality and composition of the bulk material, on the other hand can the loosened bulk material, for example, only poorly be bagged, because the bulk density be negatively affected through the additional air in the material.

Tapping with hammers and poking around with lances are for example measures, which the properties of the bulk materials not influencing, and at the same time the sliding of material on the wall of the outlet funnels ensure. Much more convenient and without great personnel effort can Vibrators be operated, which be installed outside the silo wall and be moved by a vibrator. Also can in silos inserted cushions be pulsatingly inflated by compressed air.

With the material and the surface quality of the outlet funnel can the wall friction angle between bulk solid and hopper wall, and thus the critical dimensions of the silo be influenced. Also with the help of plastic coatings of the walls can one achieve something.

While measures, taking effect on the wall, are only be useful, if mass flow occurs, can built-in discharge devices in the silo also be used in case of funnel flow. Also with bunker vibrators of different shapes and arrangement, which are being moved from outside, and with the help of agitators, can the flow of the bulk material be stimulated. By stationary devices and with a suitable shape of the hopper can the flow profile in the bunker be likewise favourably influenced. Discharge devices, which work at the same time as discharge aids, are for example flat floors with rotating arm or bottoms, which are elastically connected with an vibrating outlet.

5.2 Mechanical movement of bulk materials

In case of mechanically moved bulk materials, for example in screw conveyors, mixers and dryers, there are apart from the flow properties of the bulk materials additional parameters like the geometric conditions of devices and mechanical equipments as well as the manner of energy input. Therefore, it is in regard to the required drive powers and attainable mixing times or heat transfer coefficients, accordingly difficult to get information by calculations. Basic researches, in particular taking into account the aspects of scale-up, are only in insufficient extent available, so that for the planning appropriate documentation is missing. In practice is the plant engineer therefore dependent on the experiences of the respective device manufacturer.

The respective movement behavior of mechanically moving bulk materials becomes for example indirectly recognizable through the interpretation of heat transfer measurements during the heating of bulk materials in

24

horizontal thin film dryers, which were published by Klocke [5.13].

5.2.1 Heat transfer in horizontal thin film dryers

Thin layer contact apparatuses with vertical heating surfaces have long been known for the vaporisation of solutions with low viscosity and for the rectification or distillation of fluid mixtures. Its rotating internals create mechanically a thin liquid film along the inner surface of a heated cylinder, and regenerate this film constantly. In this way high heat and material transfer can be achieved.

The thin film principle can also be used, for the cooling, heating and drying of pasty materials, powders and granules, as well as in order to perform reactions, where solids and liquids are involved. Fig. 5.6 shows the scheme of such apparatus, that is equipped with four blade rows, looking like paddles. The effective surfaces of the blade rows overlap themselves, and the residence time of the material can by the change of the angle of attack be adjusted. To increase the so-called hold-up, it has proven expedient, to arrange the heating surfaces horizontally. In addition, the design of the rotor was modified accordingly to the granulate form of the products, so that the material is distributed as evenly as possible along the wall in order to improve the heat exchange.

In Experiments in two facilities with diameters of 210 and 250 mm as well as heating surfaces of 1.2 and 1.5 m2, five bulk materials were heated. In all experiments was the orientation of the blades identical: "transport in flow direction" in the entry zone, "transport against the flow direction" in the discharge zone and "neutral position" in the remaining area. Would be more "transport against the flow" chosen, so that the dwell time would increase, the wall coverage could be certainly improved, on the other hand would the drive power likewise grow considerably. Under consideration of the heat transfer on the one hand and the necessary drive power on the other hand, the largely neutral position of the blades has itself proven, as far as in case the use of drying gas not an additional conveying effect occurs.

Fig. 5.7 depicts the impact of the short-term contact during the heat transfer between the bulk material and the surface, as well as the variation of rotor speed. Represented is the size of the heat transfer coefficient α as function of the contact time t. The contact time is in thin film contact devices the time between two rearrangements at the wall. Its size depends on the rotor speed n and the number Z of blade rows:

tZ n

=⋅

60 (5-5)

The relationship between the Froude number Fr, for which the diameter D of the device is used, and the contact time t looks like follow:

Frg

D

t Z=

⋅⋅

⋅

2 2π / (5-6)

Due to the circulation of the product ring at the wall of the thin film device, is the actual contact time of the product greater than the value of t, which is being calculated by Eq. (5-5). This fact is here ignored.

The measurement results for the heat transfer coefficient, related on the whole heated area, which were determined at constant throughput and variable rotor speed, can be interpreted according to Fig 5.7 as follows. In the region of large contact times (low rotation speeds), grows the material coverage significantly with growing rotation speed, and the heat transfer rises steeply. After the operating point PK1 is reached, the increase of heat transfer corresponds to the

Fig. 5.7

Heat transfer coefficient αααα in dependency of the contact

time t in a thin film dryer during heating of various bulk

materials

Fig 5.6

Horizontal thin film dryer

with four blade rows (picture

credits: BSH)

25

t - law for short-term contact; the material coverage remains seemingly constant.

The optimum operating point is being reached at PK2 because by the further increase of the rotation speed (shorter contact times) the heat transfer is no longer improved. Increasing frictional resistance of the bulk material on the wall or changed conditions for the force transmission between blades and bulk solid seem to prevent a further reduction of the contact time between the bulk solid and the wall.

Additional conclusions are possible, if one tries to summarize the measurement values with a single curve. Fig. 5.8 shows such a presentation, in which in addition to the heat transfer coefficient α and the modified contact time, the influence of the heat penetration coefficient (λcρ)Sch, the mass flow rate &Ms and the diameter dp of the particles is becoming recognisable.

As Fig. 5.8 shows, grows the heat transfer coefficient with increasing bulk material throughput steeply -

namely with &M . Due to the larger quantity of bulk material is obviously also a larger material volume for the coverage of the apparatus wall present.

As expected, leads a greater heat penetration coefficient of the bulk material likewise to an increased heat transfer. The numeric value of 0.33 for the

exponent, is already in other apparative constellations been determined.

The effect of the particle diameter that influences the contact time inversely proportional with the third root , corresponds to the behavior of bulk solids during the outflow from silos. There, smaller particle sizes causing larger material discharges.

Obviously, eases the larger number of particles per volume unit its relocation, and less resistance is being opposed to the shape change. In thin film contact apparatuses has this mechanism an opposite effect. In case of smaller particles, the frictional connection between the blades and the bulk solid is being deteriorated, so that higher rotor speeds are required, to achieve similar heat transfer rates.

Literature of chapter 5.

[5.1] Jenike, A.W.: Gravity flow of bulk solids. Engineering Experiment Station Bulletin 108. University of Utah, 1961.

[5.2] Molerus, O.: Fluid-Feststoffströmungen. Berlin, Heidelberg, New York: Springer-Verlag, 1982.

[5.3] Schwedes, J.: Entwicklung der Schüttguttechnik seit 1974. Aufberei-tungstechnik 23(1982)8, S.403-410.

[5.4] Jenike, A.W.: Das Fließen und Lagern schwerfließender Schüttgüter - Ein Überblick. Aufbereitungstechnik 23(1982)8, S.411-421.

[5.5] Rumpf, H.: Mechanische Verfahrenstechnik. München, Wien: Carl Hanser Verlag, 1975.

[5.6] Brauer, H.: Grundlagen der Einphasen- und Mehrphasenströmung. Aarau, Frankfurt a. M.: Verlag Sauerländer, 1971.

[5.7] Riedel, K.: Der Ausfluß von Schüttgütern aus Bunkern. Studienarbeit am Lehrstuhl für Thermodynamik und Verfahrenstechnik der TU Berlin, 1965.

[5.8] Brown, R.L.; und J.C. Richards: Exploratory study of the flow of granules through apertures. Transactions of Institution of Chemical Engineers 37(1959)2, S.108-119.

[5.9] Rose, H.E.; und T. Tanaka: Rate of discharge of granular materials from bins and hopper. Engineer 208(1959)5413, S.465-469.

[5.10] Franklin, F.C.; und L.N. Johanson: Flow of granular materials through a circular orifice. Chemical Engineering Science 4(1955)3, S.465-469.

[5.11] Heyde, M.: Merkmale und Fließverhalten von Schüttgutmassen. Maschinenmarkt 89(1983)46, S.1047-1050.

[5.12] Taubmann, H.: Technologie der Schüttgüter. Aufbereitungstechnik 23(1982)2, S.77-83.

[5.13] Klocke, H.-J.: Wärmeübergang im Dünnschichtkontakttrocknern - ein Beitrag zur Vorausberechnung und Übertragung von Versuchswerten auf Betriebsverhältnisse. Vortrag im GVC-Fachausschuß Trocknung-stechnik, 10./11.4.1975

Fig. 5.8

Summarizing presentation of the heat transfer in thin

film contact dryers

26

6 Single-phase flow through pipes

One speaks of flow, if movements of liquids and gases are being caused by the total pressure difference. In contrast, is by partial pressure differences caused mo-tion known as diffusion.

The one-dimensional representation of a flow process is based on the assumption of the mass flow through a given cross-section. The relationship for the flow rate of a liquid with its constant density is accordingly:

uM

f=

⋅

&

ρ (6-1)

If the pressure differences are small, the rules of the hydraulic transport apply also to gases. In case of larger pressure differences, the gas density changes itself in dependence of pressure and temperature. Processes of this kind are dealt with in the gas dynamics.

6.1 Continuity and energy conservation

In the case of the one-dimensional steady state pipe flow, applies for the mass flow &M in the axial direction at the points 1 and 2:

&M u f u f= ⋅ ⋅ = ⋅ ⋅1 1 1 2 2 2ρ ρ (6-2)

The mass flow &M thus remains constant; the Eq. (6-2) is therefore referred to as continuity equation. If the pipe cross section remains equal (f1 = f2), the current density changes neither:

&M

fu u u= ⋅ = ⋅ρ ρ1 1 2 2 oder = const. (6-3)

In case of an incompressible fluid, the flow velocity u remains thus equal.

The integration of the general Euler Equation for the one-dimensional, unsteady case at constant density, results in the Bernulli Equation:

1

2

1

212

1 1 22

2 2⋅ ⋅ + + ⋅ ⋅ = ⋅ ⋅ + + ⋅ ⋅ +ρ ρ ρ ρ νu p g z u p g z p∆

(6-4)

The Bernoulli equation connects the dynamic pressure, the initial gage pressure, and the static pressure with each other. During the frictionless pipe flow remains the sum of the three pressures along the pipe constant. In the case of the viscous pipe flow, this

sum is being reduced by the amount of the pressure loss.

Dividing Eq. (6-4) by the fluid density ρ leads to a form, in which the sum of energies per unit mass is represented, namely the kinetic energy, the pressure energy and the potential energy.

1

2

1

212 1

1 22 2

2⋅ + + ⋅ = ⋅ + + ⋅ +up

g z up

g zp

ρ ρ ρν∆

(6-5)

Along the flow path, a part of the energy is transformed into heat and must be registered as a loss.

If one relates each term of the the Eq. (6-5) to the acceleration due to gravity g, results a equation with altitude values, of which hν the friction head loss is.

u

g

p

gz

u

g

p

gz h hges

12

11

22

222 2

+⋅

+ = +⋅

+ + =ρ ρ ν (6-6)

This form is especially suitable for the graphical representation of the flow process; at every point of the flow, the sum of velocity head, pressure altitude, geodetic height and friction head loss is equal to the constant total height hges.

That additional energy is needed for the acceleration of the fluid on its flow velocity, is often forgotten in practice. In case of small frictional pressure losses, the dynamic component of the total pressure loss gets more and more weight. If this proportion would neglected in specifying the blower performance, this would have unpleasant consequences for the operation. Also, pressure loss measurements on facility parts should be performed at places with equal flow cross section, so that the measurement results are directly comparable.

6.2 Flow pressure loss

Basically, one must distinguish between the laminar and turbulent flow. In laminar state, the fluid layers move side by side without any exchange of fluid elements . In this form of flow, the shear stress at the boundary between two fluid layers alone by the viscosity of the fluid is determined. In contrast to this, the wall friction generates during the turbulent flow "vortex bales", which wander to the pipe axis. The pressure loss is thereby being caused by the acceleration and deceleration of the vortex bales.

The relation between inertia and resistance force in a pipe flow decides whether the flow is turbulent or laminar. The significant Reynolds Number is being formed with the pipe diameter D and the mean velocity u:

27

Re =⋅ ⋅u D ρ

η (6-7)

In turbulent pipe flows are velocity fluctuations and highly disordered motion typical, which be not suppressed by the viscous forces. Flows in pipes at Reynolds numbers larger than 5000 are typically (but not necessarily) turbulent.

The pressure loss ∆p of a fluid, which flows along a pipe section with diameter D and length L, can be calculated with the help the general resistance law:

∆pu L

D= ⋅

⋅⋅ξ

ρ2

2 (6-8)

The resistance factor ξ for the theoretically describable laminar flow can for Reynolds numbers smaller than ca 5000 be calculated after following equation:

ξ =64

Re (6-9)

For the turbulent flow, the resistance factor ξ is only with the help of measurements to determine. Because the conditions at the pipe wall have the greatest

importance, is in particular the influence of the wall roughness noticeable.

Fig. 6.1 shows the resistance coefficients for the laminar and turbulent flow in pipes. For different roughnesses of the pipe wall there are separate graph curves, depending on the ratio of the diameter D to the mean pipe roughness factor k.

The turbulent flow in technically smooth pipes depends only on the Reynolds number. Between Re =

2300 and Re = 105 applies for the resistance coefficient the empirical equation of Blasius [6.1]:

Re

3164,025.0

=ξ (6-10)

Up to arbitrarily high Reynolds numbers can, based on the theory of Prandtl and Karman, be an implicit form of ξ spedified:

1

4

1

2

51.2

Relog

=

⋅ ξξ (6-11)

This function represents in Fig. 6.1 the left boundary of the turbulent range, which is referred to as hydraulically smooth.

Additional pressure losses are being caused by pipe bends and by internals. Especially the shedding of the

flow leads to large pressure losses, because the cinetic energy, which is needed for the acceleration of the fluid, can no longer be recovered.

Special cases are shown in Fig. 6.2: the sudden constriction of the flow cross section from f to f0 and the sudden extention from f0 auf f.

For the pressure loss applies for suddenly constricted pipes

∆pf

fu

e

= −

⋅ ⋅0

2

021

2

ρ (6-12)

and for suddenly extended pipes

∆pf

fu= −

⋅ ⋅1

20

2

02ρ

(6-13)

The ratio f f u ue e/ /0 0= is named contraction coefficient µ. fe is the narrowest flow cross section. For sharp edged openings apply the following contraction coefficient [6.2]:

61.02

=+

=π

πµ (6-14)

Fig. 6.2

Suddenly constricted tube (left) and

suddenly exstended tube (right)

Fig. 6.1

Resistance coefficient ξξξξ as a function of the Reynolds

number for various ratios of D/k

28

6.3 Heat transfer

For laminar pipe flow, the heat transfer conditions can be theoretically described. For a long pipe, in which velocity and temperature profiles are fully develloped, one obtains for the Nusselt number, by using the pipe diameter D, constant values:

66.3=⋅

=λ

α DNu (6-15)

In case of a very short pipe, it results:

PrRe664.0 3/12/1 ⋅⋅=⋅

=λ

α DNu (6-16)

In case of turbulent pipe flow, approaches must be used, in which the correlations between the dimensionless parameters have empirically been determined. According to Hausen [6.3] applies for Re greater than 2300 the following equation:

+⋅⋅−⋅=⋅

=3/2

42.04/3 1Pr)180(Re037.0L

DDNu

λα (6-17)

One should realize, that upon heating a substance in a heated tube, the temperature difference between the wall and the materials, which are flowing on the inside and the outside, is inversely proportional to the size of the heat transfer coefficient. During heating of crude oil before the distillation is, for example, water vapor being blown into the pipes in order to generate additional velocity and turbulence in the pipe. In this way, the wall temperature on the oil side and the temperature of the oil flow are being adjusted to each other, to avoid deposits of cracked oil on the tube wall.

6.4 Turbulence and exchange behavior

Generally valid equations for the heat transfer in the pipe or broader insights into the turbulence dependent pneumatic conveying process are only to obtain with the help of detailed considerations. Prandtl made the following picture: during turbulent flow arise "fluid bales" which have an own mobility, and which themselves over a certain distance in the longitudinal and transverse directions as cohesive entities move. Such a fluid bale covers a distance relative to the surrounding fluid, proportional to its diameter, which is referred to as mixing length, before it itself with the surrounding fluid mixes and its individuality looses. If such a fluid bale is due to a transverse movement being moved from its original location to a neighboring volumen, then its velocity is compared to its new surroundings higher or lower. The displacement effect of the fluid bales causes fluctuations, which are of the

same magnitude as the speed differences. These considerations are the basis for Prandtl's Mixing Length Hypothesis and the approach for the turbulent shear stress τturb [6.4] :

τ ρ∂∂

∂∂turb l

u

y

u

y= ⋅ ⋅2

(6-18)

In the usual approaches, the analogy to Newton's law of viscosity is searched by summing up the proportions of the molecular and the turbulent shear stresses. In contrast, here is assumed, that the turbulent shear stress itself proportionally to the velocity gradient and the viscosity of the flowing Fluid alters:

τ η∂

∂turb ku

y= ⋅ ⋅

(6-19)

Thus, one can insert the Eq. (6-19) into the Eq. (6-18), and one gets an expression for the turbulent shear stress, in which the gradient of the flow velocity is no longer included:

τη

ρturb

k

l=

⋅

⋅

2 2

2 (6-20)

6.4.1 Pressure loss

The relationship between the wall shear stress in the pipe and the pressure loss is as follows:

∆p

L Dw= ⋅

4τ (6-21)

The dimension of the mixing length can be introduced into Eq. (6-21) with the help of the shear stress in accordance to Eq. (6-20). The mixing length relates to the flow state along the pipe wall and ha the symbol lw.

∆p kL

l Dk k

w

= ′ ⋅⋅

⋅ ⋅′ = ⋅

η

ρ

2

224 mit (6-22)

Fig. 6.3

Ratio between pipe diameter and

wall-related mixing length lw, plottet

against the Reynolds number Re

29

Is the proportionality factor k´ chosen such, that the wall-related mixing length at a Reynolds number of 5000 is equal to the tube diameter, then give the pressure loss measurements by Nusselt [6.5] the function in Fig. 6.3. Plotted is the ratio between the pipe diameter D and the wall-related mixing length lw against the Reynolds number Re /= ⋅u D ν . The proportionality factor has the value k´=4.5 105.

The functional relationship in Fig. 6.3 can be mathematically expressed as follows:

Re1064.4 9.04 ⋅⋅= −

wl

D (6-23)

Insertion of Eq. (6-23) into Eq. (6-22), gives the following equation for the pressure loss ∆p in a hydraulically smooth pipe:

Re097,03

28.1

D

Lp

⋅⋅

⋅⋅=∆ρη

(6-24)

The validity of Eq. (6-24) can easy be verified by comparison with the usual formulation:

3

28.12 Re097,0

2 D

Lu

D

L

⋅⋅

⋅⋅=⋅⋅⋅ρηρ

ξ

This comparison gives for the friction coefficient ξ:

Re194.0 2.0−⋅=ξ (6-25)

The graph for the friction ξ according to Eq. (6-25) has a slightly flatter course than that after Blasius and averages virtually the boundary curve for the hydraulically smooth pipe in the Range of Re = 2300 an Re = 107.

6.4.2 Velocity profile

Reflections regarding the velocity profile, which itself in the pipe developes, confine themselves usually to the area, in which the turbulent exchange authoritative is. The area of the pipe wall with its laminar rules is left disregarded.

For the zone of turbulent mixing, the Eq. (6-18) and (6-19) can be equated. The result is an expression for the velocity gradient ∂ ∂ u y/ :

∂∂

η

ρ

u

y

k

l=

⋅

⋅ 2 (6-26)

Because the mixing length l inside the pipe is depending on the wall-related mixing length lw, the searched law for the velocity distribution must be precisely built up like Prandtl's "law-of-the-wall" [6.6]. This requirement can be fulfilled by the choice of an appropriate relationship between the mixing length l in the distance y from the wall and the wall-related mixing length lw:

l

l

y

lw w

2

2=

⋅=

κκ proportionality factor (6-27)

Insertion into Eq. (6-26) gives for the velocity gradient:

∂∂ κ

ηρ

u

y

k

l yw

= ⋅⋅

⋅ ⋅1

(6-28)

Designates one the expression k lw⋅ ⋅η ρ/ ( ) , which has

the dimension of a velocity, with uτ, provides the integration of the Eq. (6-28) the well-known expression [6.6]:

u u y C= ⋅ ⋅ +τ κ( ln )1

(6-29)

With the ratios u/uτ and y/lw Eq. (6-29) becomes

u

uf

y

l

y

lC

w wτ κ=

= ⋅ +

1ln (6-30)

By twice inserting the maximum speed in the middle of the tube (calculated using the velocity ratios, which are to be found in the Dubbel [6.7]), was for κ a value of 0.44 and for C a value of 20.5 determined. Eq. (6-30) is now:

5.20ln44.0

1

+⋅⋅=

wl

yuu τ (6-31)

6.4.3 Heat transfer

The physical ideas about the heat transfer in the pipe flow consider the conditions in the boundary zone as crucial for the amount of heat, that is between pipe wall and fluid exchanged. The heat transport through turbulent exchange surpasses the molecular Exchange till close to the wall by far, so that the temperature gradient in this zone is small. In contrast to that, are in the adhering wall layer due to the missing exchange very steep temperature gradients present. On the boundary of these two zones exist a speed u´ and a temperature T´.

30

This model concept can be used for deriving a relationship for the dimensionless Nusselt number Nu [6.6]:

( )[ ]Nu

D D

u u

w=⋅

=⋅

⋅ ⋅ + − ⋅ ′

α

λ

τ

η σ σ1 (6-32)

σ=⋅

1

m Pr

Eq. (6-32) contains the unknown velocity u´ on the boundary of the two zones, and also the unknown ratio m A Aq= / τ between the apparent thermal conductivity

Aq and the apparent viscosity Aτ. Eq. (6-30) now offers the opportunity, to connect u´ with the conditions on the pipe wall. For this, the wall-related mixing length lw is used in place of y.

5.20337

1ln1

⋅⋅

⋅=+⋅⋅=′

wlCuu

ρ

η

κτ (6-33)

After inserting the Eq. (6-23) for lw in the above equation, the expression for u´ is:

Re2.3 9.0

Du

⋅⋅⋅=′

ρη

(6-34)

Using the expression u D= ⋅Re /ν for the mean flow velocity in the pipe, the relationship between u´ and u is dependent on the Reynolds number:

Re2.3 1.0−⋅=′

u

u (6-35)

Taking into account that Eq. (6-20) with l = lw describes the wall shear stress, can Eq. (6-32) be rephrased:

Re0076.0

1.0Re

Pr2.3

1.0Re1

8.0

−+⋅⋅

−−

⋅=

⋅=

m

DNu

λα

(6-36)

The calculation results from above equation, are for the Prandtl number 1 equal to the published measurement values, if m = 6.25 is used. In case of other Prandtl numbers there are deviations, which can be corrected by the factor Pr0.4 For the accordance in case of small Nusselt numbers, must still the laminar proportion (Nu≈5) be taken into account, so that after a small rephrasing generally applies:

05.0PrRe05.0

PrRe0076.05

1.0

4.19.0

−+⋅⋅⋅

+=⋅

=λ

α DNu (6-37)

Fig. 6.4 shows the accordance between the Eq. (6-37) and the measurement results from a large number of researches, which have been compiled by Churchill [6.8]. It is at all cases fully developed turbulence in smooth pipes at moderate temperature differences. The equation captures not only the area, where the measured values in the double logarithmic system straight run, but also the curved curves at Prandtl numbers below Pr = 0.72.

Literature of chapter 6.

[6.1] Blasius, H.: Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüs-sigkeiten. VDI-Z. 56(1912).

[6.2] Mersmann, A.: Thermische Verfahrenstechnik. Berlin, Heidelberg, New York: Springer-Verlag, 1980.

[6.3] Hausen, H.: Neue Gleichung für die Wärmeübertragung bei freier oder erzwungener Strömung. Allg. Wärmetech. 9(1959), S.75-79.

[6.4] Prandtl, L.: Z. angew. Math. Mech. 5(1925), S.136-139.

[6.5] Nusselt, W.: VDI-Forschungsheft Nr. 89, 1910.

[6.6] Prandtl, L.; K. Oswatitsch und K. Wieghardt: Strömungslehre, 7. Au-flage. Braunschweig: Vieweg-Verlag, 1969.

[6.7] Dubbel: Taschenbuch für den Maschinenbau. Bd. 1, 13. Auflage. Ber-lin, Heidelberg, New York: Springer-Verlag, 1970.

[6.8] Churchill, W.S.: Ind. Eng. Chem., Fundam. 16(1977)1, S.109-116.

Fig. 6.4

Comparison of measurement results with the

calculation after Eq. (6-37)

31

7 Pneumatic conveying of

bulk materials

Through pipelines are most commonly liquids and gases sent, but also powders and bulk materials, whereby in case of pneumatic conveying mostly by blowers or fans supplied air or compressed air is used. The pneumatic conveyance utilizes the capability of flowing gases to transport under certain preconditions solids with their higher specific weight. This technique is however only usable for relatively short distances of up to some hundred meters.

Despite some disadvantages, which are associated with pneumatic conveying of bulk materials, there are some reasons, which make the difference of this way of bulk material transport especially in small and medium-sized internal conveying systems. It is often the only viable alternative, for example in case of tight spaces in mining or in case of subsequent automation of transport processes, if not enough space for a mechanical solu-tion is available. Weber [7.1] listed the pros and cons.

The operation of pneumatic conveying systems de-mands a considerable amount of devices, which func-tionality is owed the several decades of experience of the manufacturers.

7.1 Conveying states

During the pneumatic conveying of bulk material through a tube, the material is, depending on the re-spective flow conditions, in very different ways across the cross section distributed. This includes: uniform distribution over the pipe cross-section, subsets as strands on the bottom of the pipe, as well as bales, dunes and plugs. The different flow states Cambric [7.2] has summarized in a statechart diagram: therein is the pressure loss at constant material throughputs plot-ted against the gas velocity in the empty pipe cross sec-tion (Fig. 7.1).

At large velocities and respective large pressure losses, the state of dilute phase conveying is present, in which the solid particles pass the conveying line float-

ing and jumping. With decreasing gas velocities, the pressure loss graph passes a minimum, at which parts of the loose material be deposited on the bottom of the pipe and be conveyed as strands. At even smaller flow rates, the strands cumulate to dunes or fill as bales and plugs partially the entire cross section of the pipe. In case of the further decrease of the gas velocity the pres-sure loss rises again.

To note is that in the region of the so called dense phase on the left, the required blower performance de-creases despite the increase in the pressure loss, be-cause the effect of the decreasing air flow is much lar-ger, at least up to certain gas velocities. In addition, the considerably more gentle transport due to the lower mechanical stress on the pipe walls and the particles plays a role. For this reason one is today strived, indus-

Advantages Disadvantages

Simplicity relatively high energy demand

Customizability Wear

low space need Abrasion

easy routing Danger of clogging

Possibility of branching relatively low flexibility

Controllability

can be automated Restriction as to the suitability of

conveyed materials

Availability

low environmental impact

Low inflation rate possibly complicated particle

processing

Serviceability

Integrability

linkable to processes possibly costly dust separation

Table 7.1

Advantages and disadvantages of pneumatic conveying

installations

Fig. 7.1

Diagram of pneumatic conveyance of pourable bulk ma-

terials with particle sizes larger than 500 µm

32

trial conveying installations under dense phase condi-tions to operate.

After Krambrock [7.2] can most bulk materials with particle sizes over 50 µm be in simple, smooth pipe-lines reliable conveyed under dense conditions. In re-cent years, it has been found, that particularly bulk ma-terials with narrow particle size distributions and aver-age particle sizes between 0.5 and 5 mm can be con-veyed without major problems even if the gas velocity, related on the free cross-section, is smaller than the sinking velocity of the individual particles.

In the diagram are two characteristic conveying condi-tions recognizable. In case of dilute phase conveying, is after passing through the minimum of the pressure loss the point reached, where the bulk material possibly clogs the pipeline. Two expressions are being com-monly used for this boundary.

"Choking velocity" means the minimum velocity that is required to maintain solids in a vertical conveying line in the dilute-phase mode (its value is influenced by the particle's terminal velocity).

"Saltation velocity" means the minimum velocity that is re-quired to maintain solids in a horizontal conveying line in the dilute-phase mode. The value of this is 3 to 6 times the "Choking velocity".

Mostly, however, this condition is not the end of the pneumatic conveying, but the start of unsta-ble conditions, which with the help of a high blower power can be bridged. Further lowering of the air velocity, however, en-ables often again a stable opera-tion at constant pressure loss. Fi-nally however the conveying comes to a standstill. This condi-tion is represented by the left boundary of the statechart.

Not all bulk materials can be conveyed in dense phase without any problems, so that especially for fine, adherent and plug-forming materials additional measures be needed - for exam-ple the use of secondary air.

7.2 Pressure loss

In pneumatic conveying lines occurring pressure loss are caused by the opposite to the flow acting resis-tances. Wall friction and pipe bends cause the pressure loss of the pure airflow. The additional losses are caused mainly by the acceleration of the bulk material, the wall friction and the material weight.

Usually, one tries to capture the individual proportions of the pressure loss with the help of resistance coeffi-cients, as they are used for the calculation of the pure gas flow. Really satisfying results, which take in par-ticular the dense phase state into account, however, are up until now not available.

7.2.1 Fine-grain material in horizontal pipes

Muschelknautz and Krambrock [7.3] as well as Muschelknautz and Wojahn [7.4] published the condi-tions for the conveyance of fine-grained bulk materials through horizontal pipelines. These data sets are sum-marized in table 7.2. More measurement results were published by: Bohnet [7.5], Krambrock [7.2], Matsu-

No. Material dp50

µm

L

m

h

m

D

mm

&Ms

t/h

&Mg

Nm3/h

∆∆∆∆p

bar

u

m/s

µ

-

1 Cement raw flour 60 146 - 36 1.4 11 2 3.15 102

2 PAN powder 75 150 - 36 1.2 19 1.4 3.95 53

3 Soda 100 200 30 100 35 1050 2.5 18 28

4 Filler (SiO2) 10 30 - 65 4 100 0.3 7.3 33

5 Moist sand 200 120 - 100 19.5 300 3.5 3.75 54

6 Filler (SiO2) 15 58 - 70 1.4 30 0.14 2.1 39

7 Filler (SiO2) 15 10 - 70 1 29 0.023 2.25 29

8 Organic material 275 37 1.5 66 4.1 63 0.7 3.6 54

9 Soda 75 25 - 65 20 200 0.3 13 83

10 Fly ash 15 1200 15 200 50 4000 3.5 13 10

11 PVC powder 60 115 20 70 7 240 0.65 13.5 24

12 Rock salt (hose) 700 26 2 50 5.8 110 0.8 11 44

13 Organic material 150 6,5 - 20 0.4 8 0.17 6.5 41

14 Organic material 80 25 10 70 17 300 1.2 13 47

15 Clay 45 130 15 100 8 300 2.5 5 22

16 Plastic powder 40 300 - 100 10 310 1.5 6.8 27

17 Rubber chips 25000 45 6 125 1 1100 0.04 27 0,8

18 Plastic granules 2000 30 12 65 1 200 0.14 18 4

19 Kontakt (catalyst) 1200 - 16 100 9 460 0.1 17.6 16

20 Abbrand (burnup) 120 92 20 95 4 300 0.8 12.5 11

Table 7.2

Data of industrial pneumatic conveying installations

33

moto [7.6], Molerus [7.7] and Siegel [7.8].

Pneumatic transport of solids mainly takes place by transversal movements in the gas flow, and is therefore closely linked to the laws of turbulent pipe flow. This fact can be accounted with the help of parameters, which characterize the turbulent flow condition. In ad-dition to the Reynolds number, which is being formed using the pipe diameter, comes yet the ratio between the diameter D and the wall-related mixing length lw in question. This ratio can be specified according to the Eq. (6-23) and (6-24) as a function of the pressure loss ∆p, the pipe length L, the gas density ρg, the pipe di-ameter D and the dynamic viscosity η:

D

l

p

L

D

w

g= ⋅ ⋅

⋅0 0015

3

2,∆ ρ

η (7-1)

In Fig. 7.2, the values of D/lw, calculated using the data from the above-mentioned publications, are plotted against the Reynolds number. With the help of the sin-gle points and graphs has been constructed a general

statechart similar to that of Kram-brock [7.2]. This diagram shows that industrial conveying installa-tions are mostly be operated in dense phase mode. The importance of the turbulent flow condition is underlined by the fact, that the dia-gram is located in the area above the critical Reynolds number of 2300. Regarding the operating conditions should be noted, that two conveying installations with relatively large throughputs be-cause of not recognizable reasons be operated not in the dense phase region, but on the boundary of the dilute phase region.

With the help of the plotted val-ues and graphs, it is possible, to estimate the lines of equal material throughput. However, the pressure losses, which were published by Molerus, are too low, because the particles of the used bulk material are very large in size. The rolling particles offer only low resistance against the gas flow. Moreover, one has to be aware that in most researches only the pressure losses for the non-accelerated flow in a test assembly were measured. Un-der such conditions occurring pres-

sure gradients are not directly comparable with those of industrial convey-ing installations.

The left bound-ary of the general statechart in Fig. 7.2, which repre-sents the maxi-mum material throughputs, in-cludes the well-known, by Fig. 7.3 illustrated fact, that in case of constant pressure loss, the maximum amount of mate-rial with increasing pipe length is becoming smaller [7.2].

Fig. 7.3

Maximum material throughput &Ms of PE granules dependent on

the pipe diameter D and the pipe

length L

Fig. 7.2

General statechart for the pneumatic conveying of pourable bulk materials, con-

structed on the basis of characteristic parameters for the turbulent pipe flow,

using measured values of various researchers and data of industrial facilities.

34

7.2.2 Material acceleration at the beginning

and after pipe bends

After the product was fed into the pipeline, and after pipe bends, the bulk solid is accelerated until reaching its final velocity. The required energy is taken from the airflow, so that an additional pressure loss occurs. In short conveying lines, this pressure loss can be the main proportion.

Reference values for the size of these additional pres-sure losses can be found in the literature. After the feeding the material in horizontal pipelines requires the acceleration of 1 kg of material per kg of air, depending on the particle diameter, the following pressure losses: 250 N/m2 (dp = 8000 µm) or 550 N/m2 (dp = 1000 µm) [7.8]. These values apply to a gas velocity of 20 m/s and a particle density of 1000 kg/m3.

In case of the straight up directed conveying of bulk solids are about 100 N/m2 for the acceleration of 1 kg of solids per kg of air necessary [6]. This value was de-termined at a gas velocity of 23 m/s and for a particle density of 2600 kg/m3.

The required pressure losses for the acceleration of the bulk solids in-crease itself in gen-eral with decreasing settling velocities of the particles. The reason lies in the fact that smaller or lighter particles at constant mass flow have higher end speeds. During the flow in pipe bends, the bulk solid is by the cen-trifugal forces being pressed outward and

moves along the pipe wall. This results in increased friction between the particles and the tube wall, so that the velocity of the material decreases. In the subsequent straight pipeline, the solid is then again accelerated to the speed of the steady state. In this way, the pressure loss ∆pKr is generated.

After Schuchart [7.10] the additional pressure loss ∆pKr for horizontal deflections is dependent on the ra-tio 2R/D between pipe bend radius R and tube diameter D. The diagram in Fig. 7.4 depicts the related pressure loss for the pipe bend in relation to the related pressure loss in a straight pipe for air with and without solids loading.

For the frequently used ratio of 2R/D=10 is the pres-sure drop for the pure air flow by a factor of 2, and for the bulk solid conveying by a factor of 15 greater than

in a straight pipe. However, it remains unclear, on what basis the specified ratios be applied in the estimation of pressure losses.

7.3 Heat transfer at the pipe wall

Also regarding the heat transfer at the pipe wall, the presence of the bulk solid is during the pneumatic con-veyance noticeable. Depending on the flow conditions, the heat transfer coefficients are higher or lower, com-pared to the pure airflow.

7.3.1 Conveying vertically upward

During the vertical steady-state pneumatic convey-ance, in addition to the pipe diameter, especially the particles size of the bulk material and the gas velocity have influence on the surface-related heat transfer coef-ficient. Fig. 7.5 shows the differently orientated meas-urement graphs of the heat transfer coefficients plotted over the solid/gas-ratio µ [7.11]. The gas velocity in these investigations has been kept constant.

For sand with densities of 2600 (left chart), low mate-rial loads cause obviously an impulse loss, and the re-duced turbulence leads to reduced heat transfer. The

Fig. 7.4

Dependence of the pressure loss

ratio on the diameter ratio 2R / D

in horizontal pipe bends [7.10]

Fig. 7.6

By the solids fraction µ caused additional heat transfer

αααα-ααααg during the stationary vertical pneumatic convey-

ance

35

particles-wall-contact is not able to compensate this deficit. Only at higher material loadings, the value for the pure gas flow is reached again and with increasing loading also outbid. With decreasing particle size α rises faster and the rise takes place sooner.

The measurement values by Stockburger [7.12] in the right diagram, for which particles of smaller density were used, show a slightly different characteristic. Because of the lighter particles, a reduction of the heat transfer under the initial value is not noticable. The

gradient of the heat transfer coefficient in dependance on the solid/gas rate, reduces itself with higher gas velocities. The particle size dp was constant during these investigations, and the particle density was 1000 kg/m3.

As already shown [7.13 and 7.14], must be assumed, that the contact times between particles and wall be located in the amax region, so that the contact time itself plays no role for the size of the heat transfer between wall and particles. Thus, can the influence of the particle size be considered by the use of the maximum value for a single particle according to Eq. (3-5). This tool can be used for the summarizing presentation of the by the flow conditions influenced changes of the heat transfer coefficient.

In Fig. 7.6 is the heat transfer coefficient α-αg , that is caused only by the material proportion, plotted over an expression, that apart from the maximum heat transfer coefficient for a single particle αmax,p , takes the influences of the solid/gas ratio µ, the Froude number Fr as well as the density ratio ρp/ρg takes into account. The Froude number is being formed using the difference between the gas velocity u and the sinking velocity ut of a single particle as well as the pipe

diameter D ; g means the acceleration due to gravity. In the evaluated measurements from five researches [7.11, 7.12, 7.15 bis 7.17] vary the particle diameters between 30 and 570 µm and the tube diameters between 17 and 50 mm.

The influence of the particle size on the increase of the heat transfer becomes in the diagram as parameter visible. Really significant improvements of the heat transfer are after that only be expected in case of particle sizes below 80 µm. The reason for this is mostly the low average volume concentration of the solid in case of the dilute phase conveying. The solids concentration on the pipe wall is therefore also small.

Particle diameters of 50 µm show according to the diagram the most favourable results. For a material load of 10 kg/kg, that at a particle density of 2600 kg/m3 corresponds to a mean volume concentration of 0.4 percent, the proportion on the maximum value of heat transfer by a packed bed lies according to Eq. (3-6) at about 4 percent. Are the particle diameters greater than 200 µm, this percentage drops below 1 percent.

7.3.2 Cyclone separator

Increased solid concentrations therefore improve the surface-related heat exchange, for example in cyclone separators, in which the material, due to the acting centrifugal forces, preferably is moved along the wall.

Fig. 7.7 shows measurements in a cyclone separator with a diameter of 100 mm and an overall height of 330 mm, which were published by Székely and Carr [7.18]. The particle sizes range between 150 and 1200 µm, and the densities of the solids between 2600 and 8800 kg/m3. Plotted is the ratio and the measured heat transfer coefficients to the maximum value αmax,p·(1-ψ0) for the packed bed (moving bed), according to Eq. (3-6), against a non-dimensionless expression, using the

Fig. 7.5

Influence of the solids loading µ, the particle diameter dp

as well as the gas velocity u on the heat transfer coefficien

αααα at the tube wall

Fig. 7.7

The ratio between the measured value ααααexp and the

maximum heat transfer coefficient ( )αmax, p ⋅ −1 0Ψ for a

moving bed in a cyclone separator, plotted against a non-

dimensionless expression with the particle diameter dp,

the particle density ρρρρp and the solid flow &Ms

36

material flow &Ms , the particle density ρp and the particle size dp .

In the tested cyclone separator, the surface occupancy reaches values up to a maximum of 40 percent, although the average volume concentration is only about 0.2 percent. These are much higher values than during pneumatic conveyance. In larger cyclone separators, may the occupancies, however, be lower. In horizontal thin film contact dryers, in which the materials be also forcibly guided along the wall, are values known of up to 10 percent.

Literature of chapter 7.

[7.1] Weber, M.: Grundlagen der hyddraulischen und pneumatischen Rohrförderung. VDI-Berichte, Nr. 371(1980), S.23-29.

[7.2] Krambrock, W.: Dichtstromförderung. Chem.-Ing.-Tech. 54(1982)9, S.793-803

[7.3] Muschelknautz, E.; und W. Krambrock: Vereinfachte Berechnung horizontaler pneumatischer Förderleitungen bei hoher Gutbeladung mit feinkörnigen Partikeln. Chem.-Ing.-Tech. 41(1969)21, S.1164-1172.

[7.4] Muschelknautz, E.; und H. Wojahn: VDI-Wärmeatlas 1973, Kap Lh

[7.5] Bohnet, M.: Fortschritte bei der Auslegung pneumatischer Förder-anlagen. Chem.-Ing.-Tech. 55(1983)7, S.524-539.

[7.6] Matsumoto, S.; M. Hara, S. Saito und S. Maeda: Minimum transport velocity for horizontal pneumatic conveying. J. Chem. Engng. Jpn. 7(1974)6, S.425-430.

[7.7] Molerus, O.; und K.-E. Wirth: Die Stopfgrenze der horizontalen pneumatischen Förderung. vt "verfahrenstechnik" 15(1981)9, S.641-645.

[7.8] Siegel, W.: VDI-Forschungsheft 538: Experimentelle Untersuchung zur pneumatischen Förderung körniger Stoffe in waagerechten Rohren und Überprüfung der Ähnlichkeitsgesetze. Düsseldorf: VDI-Verlag, 1970.

[7.9] Kerker, L.: Druckverlust und Partikelgeschwindigkeit bei der verti-kalen Gas-Feststoff-Strömung. vt "verfahrenstechnik" 11(1977)9, S.549-559.

[7.10] Schuchart, P.: Widerstandsgesetz beim Transport in Rohrkrümmern. Chem.-Ing.-Tech. 40(1968)21/22, S.1060-1067.

[7.11] Brötz, W.; J.W. Hiby und K.G. Müller: Wärmeübergang auf eine Flugstaubströmung im senkrechten Rohr. Chem.-Ing.-Tech. 30(1958)3, S.138-143.

[7.12] Stockburger, D.: Der Wärmeaustausch zwischen einer Rohrwand und einem turbulent strömenden Gas-Feststoff-Gemisch (Flugstaub). VDI-Forschungsheft 518, Düsseldorf: VDI-Verlag, 1966.

[7.13] Heyde, M.: Der Wärmeübergang an der Rohrwand und das Druckver-lustminimum bei der pneumatischen Förderung. Chem.-Ing.-Tech. 51(1979)11, S.1138-1139, MS 746/79.

[7.14] Heyde, M.: Heat transfer and minimum pressure drop in pneumatic conveying. Ger. Chem. Engng. 3(1980)3, S.203-209.

[7.15] Jepson, G.; A. Poll und W. Smith: Heat transfer from gas to wall in a gas/solids trasport line. Trans. Instn. Chem. Engrs. 41(1963), S.207-211.

[7.16] Farbar, L.; und C.A. Depew: Heat transfer effects to gas-solids mix-tures using solid spherical particles of uniform size. I&EC Funda-mentals 2(1963)2, S.130-135.

[7.17] Farbar, L.; und M.J. Morley: Heat transfer of flowing gas-solids mix-tures in a circular tube. Ind. and Engng. Chem. 49(1957)7, S.1143-1150.

[7.18] Szekely, J.; und R. Carr: Heat transfer in a cyclone. Chem. Engng. Sci. 21(1966)12, S.1119-1132.

37

8 Flow through packed beds

of bulk materials

For considerations regarding the pressure loss in flowed-through packed beds, information on the spe-cific flow velocity are necessary. The difficulty thereby is, that the precise determination of the local velocities in the bed is not possi-ble. Measurable how-ever is the velocity dis-tribution behind a packed bed. Fig. 8.1 shows the qualitative results of such meas-urement.

Thereby is to recognise that the originally pis-ton-shaped velocity pro-file looks behind the bed different, showing a maximum near to the wall. Direct conclusions on the flow velocity inside the packed bed, can from such measurements not be de-rived. But it stands to reason, that inside the bed the flow velocity close to the wall likewise is larger, be-cause the void fraction near the wall increases, so that in case of the constant static pressure at each level, the flow velocity also rises. The preferred flow of the fluid in the vicinity of the wall is known as "Randgaengig-keit" (wall effect).

For quantitative considerations on the single-phase flow in packed beds, is used the mean velocity, related on the empty cross-section.

8.1 Pressure loss

8.1.1 Flow around a single particle

A part of the flow behaviour in packed beds is closely with the conditions connected, which occur at the flow around a single particle.

On a particle, that itself moves along a straight-line path at constant velocity, acts a resistance force Fν. Normally, Fν is related on the largest cross-sectional area fp of the particle, and is expressed as the propor-tion ξ of the dynamic pressure (ρf /2) urel2, using the relative velocity urel . The definition equation is there-fore:

∆pF

fu

p

frel= = ⋅ ⋅ν ξ

ρ

22 (8-1)

ξ is also known as resistance coefficient, and its size depends on the Reynolds number Rerel that is formed with the relative velocity. In Fig. 8.2 is the resistance

coefficient for a single sphere plotted as function of the Reynolds number [8.1].

For very small Reynolds numbers applies the Stokes law of resistance:

ξ =24

Rerel

(8-2)

In the range of mean Reynolds numbers (Re ≈ 103 to 105) is the resistance coefficient constant at 0.4. In case

of Reynolds numbers greater than 5⋅105 drops the value to 0.1.

The balance of forces for a sphere, that moves itself vertically, is expressed as follows:

( )F F Fg Aν = − − (8-3)

Fν means the resistance force according to Eq. (8-1), and Fg-FA is the by the buoyancy force FA reduced inertia force Fg:

( )F F g Vg A f pp f

f

− = ⋅ ⋅ ⋅−

ρ

ρ ρ

ρ (8-4)

By substituting Eq. (8-4) and (8-2) in the Eq. (8-3) re-sults in:

( ) ( )ξρ π

ρ ρπ

Re = ⋅ ⋅ ⋅ = ⋅ − ⋅ ⋅f

t p p f pu d g d2 4 6

2 2 3 (8-5)

The sinking velocity ut is tantamount to urel. Eq. (8-5) can also be written in a dimensionless form:

Ga Ar⋅−

= = ⋅ ⋅ρ ρ

ρξp f

ft

3

4(Re) Re (8-6)

The approximation equations for ξ(Re) necessitate for the determination of the sinking velocity often an itera-tive calculation [8.2]. For the practice can the method be recommended, that was proposed by Martin [8.3]. Accordingly ξ can in the whole interesting Ret-range

Fig. 8.1

Velocity distribution in a gas

stream after the flow through

a packed bed Fig. 8.2:

Resistance coefficient ξξξξ for a flow-around single sphere

as a function of Rerel

38

from 0 to 105 be correlated by using the following ap-proximate equation:

ξ = ⋅ +

1

3

721

2

Ret

(8-7)

In this way can Eq. (8-6) be resolved explicitly for the sinking velocity:

Re /tt pu d

=⋅

= ⋅ + −

ν18 1 9 1

2Ar (8-8)

8.1.2 Flow through bulk materials

The flow through bulk materials can be laminar or turbulent, just like in the pipe flow. In case of the lami-nar flow, it seems to be appropriate, to deduce the ap-plicable laws on the basis of the conditions at the flow around spherical particles. Because of the many branches and curvature of the flow channels in a bulk solid, the various individual streams overlap so that no solid flow profile is formed. Because of this fact, the laws of the laminar pipe flow, which are in particular connected with the pronounced flow profile, are not suitable as a basis.

The pressure loss, related on the largest cross-sectional area of a single particle is according to Eq. (8-1):

∆pF

fu up

p

ff= = ⋅ ⋅ = ⋅ ⋅ν ρ

ρ24

2

122 2

Re Re (8-9)

For the adding up of individual pressure losses in a bulk solid, can Eq. (8-9) only be used, while taking into account the fact, that in disordered bulk materials, in which the particles each other overlap, the effective cross section per particle length (diameter) in the flow direction fp´ is larger than the maximum cross sectional area of a single particle. As a simple correction, the void fraction comes in question, so that the following formula applies:

′ = ⋅f fp p1

0Ψ (8-10)

For a packed bed the Eq. (8-9) therefore can be writ-ten:

∆ ′ =′

= ⋅ ⋅pF

fup

pf

ν ρ12 2

Re (8-11)

For the force Fν results thus:

F u ff pν ρ= ⋅ ⋅ ⋅ ⋅12 12

0Re Ψ (8-12)

The number of particles in the entire packed bed is calculated from the ratio between the volume of the en-tire material V⋅(1-ψ0) and the volume of the single par-ticle Vp:

( ) ( ) ( )n

V

V dV

dp

p p p

=⋅ −

=⋅ −

⋅⋅ =

⋅ −

⋅⋅ ⋅

1 6 1 6 10 03

03

Ψ Ψ Ψ

π πf h0 (8-13)

The entire cross-sectional area f can be also expressed with the help of the free cross-sectional area fψ :

f f= Ψ Ψ/ 0

Eq. (8-13) can therefore be written:

nd

f hpp

=⋅

⋅−

⋅ ⋅6 1

30

00

π

Ψ

Ψ Ψ (8-14)

The entire pressure drop in a laminar flowed-through packed bed is now the sum of all individual forces Fν according to Eq. (8-12), multiplied with the number of particles np according to Eq. (8-14), related on the empty cross section fψ of the packed bed, which is also contained in Eq. (8-14).

∆Ψ

Ψ

ΨΨp

n F

fu

d

dhlam

p

f

p

p

=⋅

= ⋅ ⋅ ⋅⋅

⋅ ⋅⋅

⋅−

⋅ν

ρπ

π

12

4

1 6 122

03

0

00Re

(8-15)

In Eq. (8-15) is used the mean velocity u´ in the free cross section fψ. With the help of the velocity u in the empty cross section becomes u´=u/ψo. After a respec-tive extension, the equation for the laminar pressure loss ∆plam looks then like this:

1

Ga

Re18

81.9 30

00 Ψ

Ψ−⋅⋅⋅⋅⋅=

∆gh

pf

lam ρ (8-16)

The division of ∆p by 9.81 becomes necessary, be-cause the dimension of the right side due to the exten-sions is altered. The dimensionless form of Eq. (8-16) looks as follows:

Ga

Re6.176

1 0

30

0

⋅=Ψ−

Ψ⋅

⋅⋅

∆=∆

gh

pK

f

lam

lamp ρ (8-17)

39

Except the factor, Eq. (8-17) equals the theoretical re-sistance law after Brauer [8.1]. Brauer has empirically a factor of 160 determined for the pressure loss in the laminar flow through disordered bulk solids with spherical particles.

For the description of the turbulent flow condition in packed beds, one can orientate themselves at the laws of the turbulent pipe flow. In case of the flow through channels with non-circular cross sections, is being used a so-called hydraulic diameter Dh as characteristic di-mension. Dh is for bulk solids defined as ratio between the empty volume of the packed bed and the entire par-ticle surface. For the use of the for circular cross sec-tions valid laws, the diameter D must be replaced by 4⋅Dh. Because of the - compared to tubes - not only in one flow direction orientated bulk solid surfaces, the factor 4 is however meaningless and is being not con-sidered here. The defining equation for the hydraulic diameter Dh of a packed bed is therefore as follows:

DV V

A

V

Adh

s

Sch

p

pp=

−= ⋅

−= ⋅ ⋅

−⋅

Ψ

Ψ

Ψ

Ψ0

0

0

01

1

6 1 (8-18)

The pressure loss Eq. (6-8) with the resistance law ac-cording to Eq. (6-10) results together with the hydraulic diameter for the turbulent pressure ∆pturb in the turbu-lent flow:

2

Re194.0

0

0

16

1

022.0

pd

f

turb

hup

⋅Ψ−

Ψ⋅

− ⋅⋅⋅⋅=∆ρ

(8-19)

The extension of Eq. (8-19) as in case of the laminar pressure loss leads to the following form:

1

Ga

ReeR58.0

81.9 30

02

2.00 Ψ

Ψ−⋅⋅′⋅⋅⋅⋅=

∆ −ghp

fturb ρ (8-20)

For Re´ must instead of dp the hydraulic diameter Dh and instead of u the actual velocity u´ = u/ψ0 be used, so that for the pressure loss in the turbulent flow through bulk solids the following relationship is valid:

( )

Ga

Re1.8

1

8.1

2.1

0

30

0

⋅=Ψ−

Ψ⋅

⋅⋅∆

=∆gh

pK

f

turb

turbp ρ (8-21)

The pressure loss equations apply strictly speaking only for an average void fraction. At small ratios between the diameter of the apparatus and the average particle diameters, wall effects are noticeable. Fig. 8.3 clarifies the functional relationships in Eq. (8-17) and (8-21). The plotted measurement values originate from four in-

vestigations with air at pressures up to 5 bar and water [8.4 to 8.7]. The theoretical relationships and in par-ticular the transition from the laminar to the turbulent flow regime are being confirmed by the measurement results.

8.2 Heat transfer

8.2.1 Exchange at the particle surface

For the summary presentation of the measured values of mass transfer coefficients and heat transfer coeffi-cient in flowed-through bulk solids, the same dimen-sionless numbers Ga and Re/Ga must be considered, which are important for the calculation of the pressure loss [8.8]. Fig. 8.4 shows the correlation between the Nusselt number combined with the void fraction and the quotient of the Reynolds number and the Galilei number. In the double logarithmic presentation, Galilei number and Prandtl number appear linked with each other as parameter.

There are three flow regions with different gradients for the curves of equal parameter values. In addition varies the exponent n of the Prandtl number from re-gion to region. n takes values of n = 1, n = 2/3 and n = 1/3.

The presentation was developed with the help of measurement results from eleven studies [8.5, 8.9 to 8.18], which are plotted in Fig. 8.4 in addition. The bulk solids were arranged in form of packed beds. The tests were always carried out only in one of the three

Fig. 8.3

Pressure loss in the flow through bulk solids with spheri-

cal particles: comparison of theory with measurement re-

sults

40

flow regions, so that the transition from one region to the other remains unclear. The experimental conditions for the mostly disordered bulk solids are varying widely. The used fluids were: H2O, air, CO2, H2 und He. The values for Pr and Sc varied between 0.68 und 1650, for Ga between 10-3 and 3⋅109, for Re between 10-3 and 2.5⋅104, and for ψ0 between 0.26 and 0.48.

8.2.2 Exchange at the pipe wall

Measurements of the heat transfer coefficient at the tube wall have been performed, for example, by Leva [8.19] and Kling [8.20]. The flow conditions corresponded to the uppermost portion of the diagram in Fig. 8.4. The measured heat transfer coefficients have thereby approximately the magnitude, which can be determined for the particle surface with the help of the dimensionless representation. There is also no reason why this should not be so. Representations, which are at diameter ratios orientated [8.21], take into account, if it is no real wall effect, indirectly the influence of the pressure, that cannot be captured by means of the Reynolds number alone.

Literature of chapter 8.

[8.1] Brauer, H.: Grundlagen der Einphasen- und Mehrphasenströmung. Aarau und Frankfurt a.M.: Verlag Sauerländer, 1971.

[8.2] Kürten, H.; J. Raasch und H. Rumpf: Beschleunigung eines kugelför-migen Feststoffteilchens im Strömungsfeld konstanter Geschwindig-keit. Chem.-Ing.-Tech. 38(1966)7, S.941-948.

[8.3] Subramanian, D.; H. Martin und E.U. Schlünder: Stoffübertragung zwischen Gas und Feststoff in Wirbelschichten. vt "verfahrenstech-nik" 11(1977)12, S.748-750.

[8.4] Kling, G.: Das Wärmeleitvermögen eines Kugelhaufwerkes. Diss. TH München, 1937.

[8.5] Glaser, M.B.; und G. Thodos: Heat and momentum transfer in the flow of gases through packes beds. Amer. Chem. Engng. J. 4(1958)1, S.63-68.

[8.6] Krischer, O.: Vorgänge der Stoffbewegung durch Haufwerke und porige Güter bei Diffusion, Molekularbewegung sowie laminarer und turbulenzartiger Strömung. Chem.-Ing.-Technik 34(1962)3, S.154-162.

[8.7] Wilhelm, R.H.; und M. Kwauk: Fluidization of solid particles. Chem. Engng. Prozess 44(1948)3, S.201-218.

[8.8] Heyde, M.: Empirische Darstellung des Wärme- und Stoffaustausches in durchströmten ruhenden Kugelschüttungen. Chem.-Ing.-Technik 52(1980)8, S.654-655.

[8.9] Löf, C.O.G.; und R.W. Hawley: Unsteady-state heat transfer between air and loose solids. Ind. Engng. Chem. 40(1948), S.1061-1070.

[8.10] Grootenhuis, R.C.A.; und A.O. Saunders: Heat transfer to air passing through heatet porous metals. Proc. Instn. Mech. Engrs. (1951), S.363-366.

[8.11] Eichhorn,J.; und R.R. White: Particle-to-fluid heat transf. in fixed and fluidi. beds. Chem.Engng.Prog. Symp. Ser. 48(1952), S.11-18.

[8.12] Kunii, D.; und J.M. Smith: Heat-transfer characteristics of porous rocks. AIChEJ 7(1961)1, S.29-34.

[8.13] Donnadieu, G.: Transmission de la chaleur dans les milieux granulai-res. Revue Inst. Fr. Petrole 16(1961), S.1330.

[8.14] Mimura, R.: Studies on heat transfer in packed beds. Graduate thesis, University of Tokyo, 1963.

[8.15] Rowe, P.N.; und K.T. Claxton: Heat and mass transfer from a single sphere to fluid flowing through an array. Trans. Instn. Chem. Engrs. 43(1965), S.T321-T331.

[8.16] Bhattacharyya, D.; und D.C.T. Pei: Heat transfer in fixed gas-solid systems. Chem. Engng. Sci. 30(1975), S.293-300

[8.17] Thoenes, D.; und H. Kramer: Mass transfer from spheres in various regular packings to a flowing fluid. Chem. Engng. Sci. 8(1958)3/4, S.271-283.

[8.18] Littman, H.; R.G. Barile und A.H. Pulsifer: Gas-particle heat transfer coefficients in packed beds at low Reynolds numbers. I/EC Funda-mentals 7(1968)4, S.554-561.

[8.19] Leva, M.: Heat transfer to gases through packed tubes. Ind. Eng. Chem.

[8.20] Kling, G.: Versuche über den Wärmeaustausch in Rohren mit kugli-gen und zylindrischen Füllungen. Chem.-Ing.-Tech. 31(1959)11, S.705-710.

[8.21] Schumacher, R.: Wärmeübergang an Gasen in Füllkörper- und Kon-taktrohren. Chem.-Ing.-Tech. 32(1960)9, S.594-597.

Fig 8.4

With the help of

measurements from

various researchers

developed dimension-

less representation of

the relationship be-

tween Nu, Re, Pr and

Ga for the heat trans-

fer at the particle sur-

face in flowed-through

packed beds

41

9 Fluidized bed Flowed-through packed beds remain with increasing

gas velocity so long in rest until the so-called minimum fluidization velocity is reached. Fig. 9.1 depicts the pressure loss ∆p that occurs in the bed, as function of gas velocity u in the empty cross-section of the bed. In the fixed bed rises the pressure loss with increasing gas velocity, and remains after having reached the mini-mum fluidization, despite further increase of the gas ve-locity constant. Over the whole state of fluidization, the pressure loss corresponds to the weight of the bulk ma-terial.

Determining the minimum fluidization velocity umf is complicated by the fact, that the function profile of the pressure loss in case of increasing gas velocity often does not have a pronounced kink, but a transition re-gion. Therefore, the point of the minimum fluidization velocity is defined as the intersection of the extended function curves of the fixed bed and the fluidized bed.

The fluidized bulk material has virtually no inner friction angle, and its undulating movements are reminiscent of a liquid. In bulk solids with very broad

particle size distributions, smaller particles are being earlier fluidized than large ones, and the full fluidization of the bulk solids is being only at fluid velocities achieved, which are higher than those, defined by the intersection of the characteristic curves

9.1 Minimum fluidization velocity

At a flowed-through bulk solid arises at the fluidiza-tion point a balance between the resistance force ∆p⋅fSch that acts in the bulk solid, and the oppositely directed mass force Vs⋅(ρp-ρf)⋅g. With the void fraction at the fluidization point according to Eq. (2-4) one ob-tains from the balance of forces the following relation-ship for the pressure loss ∆pmf:

( ) ( )∆ Ψp g hmf p f= ⋅ ⋅ − ⋅ −0 01 ρ ρ (9-1)

One can also write:

( )∆

Ψp

g h

mf

f

p f

f⋅ ⋅= − ⋅

−

001

ρ

ρ ρ

ρ (9-2)

Replacing the left side of Eq. (9-2) by the pressure loss Eq. (8-17) for the laminar flowed-through bulk solid, leads to the following form:

( ) 011

Ga

Re5.160

30

0

f

fpmf

ρ

ρρ −Ψ−

Ψ

Ψ−⋅=⋅⋅ (9-3)

Out of this arises for the Reynolds number of the minimum fluidization velocity for the laminar flow:

30

30

5.160

1

5.160

1Re Ψ⋅⋅=Ψ⋅

−⋅⋅= ArGa

f

fp

lammf ρ

ρρ (9-4)

Analogous to this, delivers the equilibrium of forces together with the pressure loss Eq. (8-21) for a turbulent flowed-through bulk solid, the following relationship for the minimum fluidisation velocity:

( )

1Ga12.0Re

2.0

0

30

Ψ−

Ψ⋅

−⋅⋅=

f

fp

turbmf ρ

ρρ (9-5)

or in other notation:

Fig. 9.1

Pressure loss ∆∆∆∆p and behaviour of flowed-through bulk

solids as function of the fluid velocity u

Fig. 9.2

Minimum fluidization velocity for bulk materials: com-

parison between theory and measurements

42

( )

1Ar31.0Re

11.0

0

67.100.56

Ψ−

Ψ⋅⋅=

turbmf (9-6)

Fig. 9.2 depicts the functions according to the Eq.s (9-4) and (9-6),whereby the term Remf⋅(1-ψ0)/ψ03 was plotted against the Archimedes number Ar.

Measurement results from experiments on bulk solids [1.9 to 3.9], flowed through by water and air, confirm the theoretically deduced relationships.

The width of the transition region between laminar and turbulent flow as well as the spread of the measurement results are not insignificant. This fact and the circumstance that in industrial practice often rough estimations without detailed knowledge of the characteristic data of a bulk material must be carried out, justifies the use of empirical calculation approaches. For a void fraction at the fluidization point of ψ0 = 0.45 reproduces the following equation the measurement results with good accuracy [9.4]:

Ar5.22+1400

ArRe =mf (9-7)

9.2 Appearance

As known from many experiments, there are two forms of fluidized bed [9.5]. If the bulk solid is fluidized by a liquid, the concentration of particles is generally in the entire fluidized bed volume temporally and spatially constant. This state is called as homogeneous. In contrast, most gas-solid fluidized beds

show gas bubbles, which rise in the surrounding suspension virtually solids-free up - the fluidized bed is inhomogeneous. Experiments show, however, that also gas-solid fluidized beds in case of conditions just above the minimum fluidization velocity can be homogeneous, to turn at a higher gas velocity into the inhomogeneous state.

The growth of the bubbles with increasing height above the gas distribution goes hand in hand with an increase in the bubbles ascent rate. Due to the "continuity of flow", the proportion of the cross sectional area, that is flowed-through of bubbles, reduces itself with increas-ing height above the gas distribution. Because of that, fluidized beds have in case of equal gas throughput with increasing filling height smaller volume-related turnovers in catalytic gas-phase reactions, as well as smaller volume-related heat transfer coefficients in de-pendence on the installation height of heat transfer sur-faces.

9.3 Expansion behaviour

One distinguishes between void fraction ψb that oc-curs during the expansion of a fluidized bed, and the void fraction ψ0 in the suspension phase. With Vp as volume of the material within the fluidized bed, with V0 as the gas volume in the state of minimum fluidization, and with Vb as expansion volume, apply using the usual definitions for the void fraction, the following equa-tions:

Ψ Ψ00

0 0=

+=

+ +V

V V

V

V V Vp

b

p b

und b

For the entire void fraction ψ applies:

Ψ =+

+ +V V

V V V

b

p b

0

0

The connection between these three variables pro-duces the following relationship:

( ) ( ) ( )1 1 10− = − ⋅ −Ψ Ψ Ψb (9-8)

The void fraction ψ0 is relatively easy to measure. The expansion void fraction ψb, in contrast, is being influenced by the bulk solid properties, the flow condi-tions and the fluidized bed geometry.

Fig 9.3

In inhomogeneous fluidized are rising

gas bubbles up, which grow due to

coalescence

43

9.3.1 Change from homogeneous to inhomogene-

ous expansion

In Fig. 9.4 plotted measurement values from studies indicate for the homogeneous expansion of water-fluidized beds, that for the minimum fluidization veloc-ity in a turbulent flow region approximately Eq. (9-6) applies, if one replaces the void fraction of the packed

bed ψ0 through the void fraction ψ of the expanded flu-idized bed:

( )

1Ar31.0Re

11.0

67.156.0

Ψ−

Ψ⋅⋅= (9-9)

The few measurements from the analysed study [9.1] that lie in the laminar region, reveal unfortunately no uniform overall picture.

Several researches provide values of gas velocities for the change from homogene-ous to inhomogeneous fluidization [9.1, 6.9 to 8.9]. Especially in one study [9.6] have been the gas pressure and thus the ratio between the particle density and the gas density systematically varied. An in-fluence of the gas distributor plate on the behaviour of the fluidised bed has been largely excluded by use of porous plates.

Fig. 9.5 shows very clearly the influence of the density ratio ρp/ρf on the position of the transition point. In case of density ra-tios above 1000, the fluidization state be-comes already in the vicinity of the mini-mum fluidization velocity inhomogeneous. In contrast, remains a water-fluidized bed with a density ratio of 1.6 up until fluidi-zation numbers of u/umf ≈14 homogenous. The increasing bed expansion with de-creasing density ratio makes the behaviour of liquid-fluidized beds plausible, which remain throughout the whole range of flu-idization homogeneous.

Fig. 9.4

Homogeneous expansion of liquid-fluidized beds: Com-

parison of calculation with measured values

Fig. 9.5

Boundaries between homogeneous and inhomogeneous

fluidization state according to measurement results from

four studies

Lit. dp ρρρρp Fluid p Ar (ρρρρp-ρρρρf)/ρρρρf Remb Remb

- µµµµm kg/m3 - bar - - - Gl.(9-10)

[9.6] 125 1185 Air 1 86 650 0.16 0.19

2 162 475 0.33 0.32

4 322 237 0.66 0.63

8 650 118 1.4 1.27

14 1135 68 2.5 2.20

186 1185 Air 1 266 650 0.3 0.39

2 532 475 0.65 0.68

4 1066 237 1.33 1.37

8 2124 118 2.67 2.73

14 3740 68 4.68 4.79

121 240 Air 1 15 192 0.09 0.09

2 40 96 0.18 0.22

4 65 48 0.38 0.39

[9.1] 282 2600 Water 1 371 1,6 3.50 3.96

[9.7] 39 3186 Air 1 6 2600 0.05 0.02

58 21 2600 0.093 0.045

Table 9.1

Boundaries between homogeneous and inhomogeneous fluidization

state: Comparison of measured and by Eq. (9-10) calculated values

for Remb

44

The connection in Fig. 9.5 can empirically be captured with the following equation:

ArAr1.0Re

35.0−

−⋅⋅⋅=

f

fp

mb ρ

ρρ (9-10)

Table 9.1 contains the comparison between calcula-tion results and measured values. One investigation [9.7] shows thereby greater deviations that under cir-cumstances can be explained with the different diame-ter of the test apparatus. However, this problem should be here not pursued.

Because of the high fluidization numbers u/umf com-mon in practice, technical gas-solid fluidized beds at atmospheric pressure are virtually always inhomogene-ous. In addition, the used gas-distributions are usually equipped with discrete holes, so that the gas jets hinder the formation of a homogeneous state in principle.

9.3.2 Inhomogeneous expansion

As already mentioned, rises the gas in gas-solid fluid-ized beds, which are operated at large ratios ρp/ρg be-tween particle density and gas density, as bubbles up. The proportion ϕ of the cross sectional area, that is flowed-through of bubbles, reduces itself thereby with increasing height above the gas distribution. This proc-ess can be described as follows [9.9 and 9.10]:

ϕ( )hh

h h=

⋅

+≥ =

+

+

220 h 20 cm, h cm+ (9-11)

h+ represents the fluidized bed level, in which no in-fluence of the height is noticeable, while h is the height of the expanded fluidized bed. In Fig. 9.6 ϕ is shown as a function of the fluidized bed height h.

Between the height h of a fluidized bed and the height h0 of the resting bulk solid exists the following connec-tion:

h

h0

01

1=

−

−

Ψ

Ψ (9-12)

It follows from Eq. (9-8):

hh

h hb

=−

≤ +0

1 Ψ für (9-13)

Regarding greater heights, it should be noted that the void fraction decreases with increasing altitude. The ratio between the average void fraction in the volume between the bed levels h and h+ and the void fraction in the volume up to the level h+, is indicated by the factor ϕ(h). On this basis one can formulate for the average

void fraction Ψb in the entire bed volume:

( ) ( ) ( )h h h h hb b b− ⋅ = − ⋅ + − ⋅ ⋅+ +0 0Ψ Ψ Ψ ϕ( )

This results in:

( )[ ]ΨΨ

bb

hh h h h= ⋅ + − ⋅+ + ϕ( ) (9-14)

Thus applies to larger fluidized bed heights:

hh

h hb

=−

≥0

1 Ψ + (9-15)

Values for Ψb were measured in a large number of published studies. For evaluating these measurement values, they must be related on the bed height h+. There, the void fraction is independent of the bed

Fig. 9.6

Of bubbles flowed-through fluidized bed proportion ϕϕϕϕ as

a function h of the expanded fluidized bed height h

Fig. 9.7

Expansion void fraction ψψψψb, related on the height h+ and

therefore independent of the height, connected with flu-

idization number and Galileo number

45

height. In this way can the corresponding void fractions ψb be determined for all measured values. The rela-tionship for the conversion results from the Eq. (9-14) and (9-15):

( ) ( )Ψ

Ψb

b h

h h h h

h h

h h h h=

⋅

+ − ⋅=

−

+ − ⋅+ + + +ϕ ϕ( ) ( )

0 (9-16)

Measurement values from eight studies [9.1, 9.7, 9.8, 9.11 to 9.15] have been evaluated, in which various fluidized bed systems were investigated, but only at ambient temperature and only with air as fluidizing gas. The evaluation of these data showed, that a relationship between the calculated void fraction ψb of the expanded fluidized bed, the fluidization number u/umf and the Galileo number Ga exists. For constant Galileo numbers arise themselves namely within a series of measurements for the expression

( )( )( ) 75.01/

1ln

−

Ψ−−

mf

b

uu

constant numeric numbers. The complete analysis is depicted in Fig. 9.7.

The straight line, that in the double-logarithmic coordinate system can be approximately laid through the measurement points, delivers the expansion law for the by the height unaffected region of the fluidized bed:

1Ga12.0exp1

75.0

29.0

−⋅⋅−=Ψ−

mf

bu

u (9-17)

The use of porous gas distributor or perforated plates does not seem to produce any significant differences [9.15].

The evaluation of the Eq. (9-17) with the help of a model for the heat exchange between fluidized bed and internals showed, that the factor -0.12 in front of the Galileo number applies only for air as the fluidizing gas. For other gases with other temperatures and pressures and other materials, the ratio between material density and gas density must be considered, so that for ψb in general applies:

−⋅⋅

⋅−−=Ψ

75.0

29.0

54.0

1Ga002.0exp1mfg

p

bu

u

ρ

ρ

(9-18)

With the help of Eq.s (9-11), (9-14), (9-15) and (9-18) can the expanded fluidized bed height h iteratively be calculated. After the determination of ψb, using an estimated value for h, is being calculated the bed

proportion ϕ(h), that is flowed-through by bubbles. From this results the value for ψb. The new value for h, that is determined with the help of Eq. (9-15), serves, where appropriate, as the start value for the new loop. The iteration converges very rapidly, so that only two or three iterations are necessary.

9.3.3 Bubble rise

The growth of the bubble diameter d db b− 0 with

increasing height above the gas distribution is, according to measurement results [9.16 to 9.21] for constant particle sizes dependent on the fluidization number u/umf . In addition, there is an influence of the flow conditions at the fluidization point, that can be expressed by the Froude number

( )gdu pmfmf ⋅= /Fr 2

Within a measurement series one obtains for the expression

d d

h

b b

mf

−

⋅0

Fr

constant numeric values. Fig. 9.8 shows the converted measurement results, plotted against the Fluidisationszahl. The initial bubble diameter dbo was used according to the course of the measurements.

In the log-log plot can be two different regression lines specified, without that for that a special reason being recognizable. The slope is for both straights nearly the same . Thus, applies for the bubble growth the following law:

1Fra

7.0

0

−⋅⋅=

−

mf

mf

bb

u

u

h

dd (9-19)

For air as the fluidizing gas lie the numeric values for the factor a between 0.08 and 0.2.

Fig. 9.8

Influence of the fluidization number u/umf on the change

db-dbo of the bubble diameter with increasing height h

above the gas distribution

46

The bubble rise velocity is often described being proportional to the root of the bubble diameter [9.22]:

u g db b= ⋅ ⋅k (9-20)

With the bubble diameter db according to Eq. (9-19) and without taking into account an initial bubble diameter, this relationship is written:

1Frk

7.0

−⋅⋅⋅⋅=

mf

mfbu

uhgu (9-21)

9.4 Material entrainment

When the gas flows through the bulk material in a fluidized bed, a certain material amount is being entrained. This share should be recycled to the fluidized bed, or be replaced in some way. Especially for the downstream separator may be the knowledge of the discharged amount of bulk material of interest.

In inhomogeneous gas-solid fluidized beds, particles are hurled into the free space above the fluidized bed.

This is mainly being caused due to the rising gas bubbles, which burst when leaving the fluidized bed (Fig. 9.9). The initial velocity of the particles is dependent on the height h of the expanded fluidized bed. With increasing clearance height H above the fluidized bed, an increasing part of the discharged solid falls back into the bed, so that the actual amount of discharged material is reduced.

There are only few published studies on the material entrainment, which can be evaluated [9.23 to 9.25]. Two of these investigations range however in technically interesting areas with equipment diameters of

0.45 m and 0.6 m. Further experimental data are contained in the Table 9.2.

If one the expansion ψb of the fluidized bed considers as a relevant state variable, one can the test results accordingly to Fig. 9.10 summarize in one single diagram. It can be seen that the influence of the expanded height h on the material entrainment &M s is greater than that of the clearance height H. As indicated

at lower expansion values, the material entrainment above certain clearance heights is no longer influenced by the fluidized bed conditions and is then only still dependent on the flow conditions and the material properties.

Literature Material ρρρρp dp H h

- kg/m3 µm m m

[9.23] Catalyst 1000 67 0,87 1,0

[9.24] Catalyst 1000 58 0.3-3.0 1.05

[9.25] Catalyst 2600 120/260 0.2-0.55 0.3/0.2

Table 9.2

Experimental data of the material entrainment in fluidized

bed facilities

Unwanted material entrainment from the fluidized bed can be prevented not only by appropriately dimensioned clearance heights. Much more effective are for example flow obstructions (baffle plates) or flow diversions, which ensure, that the particles lose their upward directed proportion of kinetic energy and fall back into the bed.

Fig. 9.9

Material entrainment in

inhomogenous gas-solid

fluidized beds

Fig. 9.10

Influence of the void fraction ψ ψ ψ ψb of the expanded bed

and the height dependence of ϕϕϕϕ in gas-solid fluidized

beds in conjunction with the bed height h and the height

H of the free space above the bed on the amount of

entrained material

47

9.5 Penetration of gas jets

When gases be injected by nozzles into a fluidized bed, it forms in front of the nozzle usually a jet, which the surrounding material entrains and accelerates. The

erosive effect in this fluidized bed region, makes itself extremely negative noticable, for example, if heat exchanger tubes are not placed at a sufficient distance. But also apparatus walls and gas distribution plates can be affected. During the spray drying in fluidized beds, where the moist material is distributed through nozzles into the bed, must the mutual interference of the jets be prevented, and the jets must have a corresponding distance from the apparatus wall.

Empirical relationships for the depth of penetration of jets were publicated by Merry [9.26 and 9.27]. For the description of the

vertical and the horizontal penetration, two different equations were needed. These dimensionless equations mix the characteristics of the jet and the fluidized bed with each other and appear therefore confusing.

The jet effects and the conditions in the fluidized bed can be separated from each other using the Froude number, which is formed with the jet velocity u0 at the nozzle exit and the nozzle diameter d0. For the description of the fluidized bed remain as influencing factors the fluidisation number u/umf, the Galilei number Ga and the

density ratio (ρp-ρg)/ρg. The evaluation of measurement values from investigations on horizontal gas jets leads to the following relationship:

( )

Ga

/8.3

5.0

08.0

5.045.0

0

20

0

−⋅

⋅

⋅⋅=

g

gp

mfuu

gd

u

d

L

ρ

ρρ (9-22)

Fig. 9-11 shows the comparison of calculated and measured values.

Some values for the propagation angle of jets were likewise published by Merry [9.26]. Fig. 9.12 shows that especially the ratio of nozzle diameter to particle diameter d0/dp in certain areas plays a role. In addition, the density ratio, that goes with the fourth root, is seemingly again important.

9.6 Heat transfer between fluidized bed and heat

exchange surfaces

Gas-solid fluidized beds with immersed heating and cooling elements are being used in a variety of technical processes, such as cooling, heating and drying of bulk solids, as well as endothermic or exothermic reactions. Advantageous are the ease of use of the bulk materials in the fluidized state, the good mixing of the material and, not least, the high heat transfer coefficients between fluidized bed and heat exchange internals.

Because of the great mobility and intensive mixing of the particles in the fluidized bed, and because of the large specific surface of the material mass, is the temperature of the fluidised bed in the entire volume practically equal. Therefore, the heat transfer coefficient α, related on the heat transfer surface Αw, can be defined, using the bed temperature ϑSch :

( )αϑ ϑ

=⋅ −

&Q

Aw w Sch

(9-23)

The heat transfer to the heating or cooling surfaces takes place in three ways, which can be approximately regarded as being independent of each other, namely by Partikelkonvektion (αpc), by gas convection (αgc) and by radiation (αrd). The resulting heat transfer coefficient α is then the sum of these three components:

α α α α= + +pc gc rd (9-24)

Fig. 9.13 shows the course of the heat transfer coefficient α in dependance of the gas velocity u. In the stationary packed bed (a), the heat is being transferred only by conduction in the bulk solid and by convection as well as radiation. Therefore, the heat transfer is bad and increases only gradually with rising gas velocity.

After reaching the point umf (a/b), at which the fluidization starts, increases the heat transfer

Fig. 9.11

Comparison of penetration

depths of jets, calculated

according to Eq. (9-22), with

measured values

Fig. 9.12

Influence of the diameter ratio

do/dp and the density ratio ρρρρp/ρρρρg

on the propagation angle θθθθ of gas

jets

48

coefficient α itself rapidly, because due to the material movement, the bulk solid layer at the transfer surfaces

is frequently renewed. With increasing gas velocity (b) grow ascent speed and frequency of the bubbles and thus the mixing of the fluidized bed. At the same time disappear the temperature profiles, so that the heat exchange within the bed is no longer

inhibited. Therefore increases the heat transfer, although due to the bed expansion the mean number of particles per unit volume of the fluidized bed becomes smaller. Outweighs the effect of the bed expansion, the heat transfer can despite increasing gas velocity no longer be increased (b/c). The heat transfer reduces itself again.

9.6.1 Influence of material movement and bed

expansion

It can be assumed that also the particle-convective heat transfer between fluidized bed and internals has to do with the short-term contact between the individual particles and the wall [9.28]. For the bulk material in the fluidized bed, however, must except the void fraction ψ0 of the resting bed, in addition the expansion and movement of the fluidized bed be considered. By summarizing these influences in an alternative heat transfer void fraction Ψα , can the Eq. (3-6) for the stationary fixed bed be rewritten on the particle-convective heat transfer coefficient:

( ) ( )α α αpc = ⋅ − ⋅ −max,p 1 10Ψ Ψ (9-25)

The heat transfer void fraction Ψα is firstly determined through the material occupancy of the heat exchange surface, that is directly connected with the expansion of the fluidized bed. In addition, however, the proportion of the heat exchange surface must be given, which is in contact with a material amount, that fullfills the conditions of the maximum heat transfer, namely sufficiently short contact time and vanishing temperature profile. As a measure for this, the ascent velocity of the bubbles can be used, so that with the help of Eq.s (9-17) and (9-21) for 1-ψα can be written:

−⋅⋅−⋅

−⋅⋅=Ψ−

75.0

29.0

7.0

1Gaexp1Fr1mfmf

mfu

un

u

ukα

(9-26)

Insertion into Eq. (9-25) gives for the particle-convective heat transfer coefficient αpc:

( )

1exp

11

75.0

29.0

7.0

0pmax,

−⋅⋅−

⋅

−⋅⋅⋅Ψ−⋅=

mf

mf

mfpc

u

uGan

u

uFrkαα

(9-27)

By evaluating data from many studies with several gases at various temperatures could be determined, that for the coefficients k and n in Eq. (9-27) an additional influence through the density ratio ρp/ρg has to be considered [9.28]

3.2k

27.0−

⋅=

g

p

ρ

ρ (9-28)

und

54.0

289

1=n

⋅

g

p

ρ

ρ (9-29)

For the heat transfer by radiation, which is mainly at high temperatures significant, applies according to Eq. (3-9):

K 100

04.03

12

⋅⋅= mrd

TCα (9-30)

For the evaluation of the measurement results from the literature, a mean value of 4.5 W/(m2⋅K) was used for C12. Tm is the mean absolute temperature between wall and fluidized bed. For the minimum fluidization velocity umf was used the value calculated according to Eq. (9-7).

The proportion of the direct heat transfer between the gas and the heating or cooling surface rises with increasing gas velocity. For very large particles (dp ≥ 3mm), which require correspondingly high gas velocities for its fluidization, this proportion can play a dominant role. After Baskakov [9.29] can αgc be estimated for gas velocities above αmax with the help of the following simple relationship:

Fig. 9.13

Heat transfer coefficient αααα at heat

exchanger internals in fluidized

beds as function of the gas velocity u

49

ArPr009.0 3/1

maxgc ⋅⋅⋅=p

g

d

λα (9-31)

For gas velocities between umf and uαmax, this maximum value must still be multiplied by (u/uαmax)0.3.

9.6.2 Comparison with measured values

Investigations in laboratory fluidized beds have been carried out with submerged bodies of very different shape. Regarding the comparability of the different experimental setups, there are no knowledges, so that the determination of the coefficients k and n in Eq. (9-27), was possible only under consideration of certain effects, which were caused by the experimental setups. If one compares only the measured maximum values of the heat transfer coefficients, which are listed in Table 9.4, becomes apparent, that between the investigated bulk solids, and from author to author at otherwise the same size and density of the particles significant differences occur.

Fig. 9.14 shows, for example, measurements in systems with very different density ratios ρp/ρg [9.30]. In the experiments, a rod with fairly small diameter was hung coaxially in a laboratory fluidized bed. The tendency of calculated and measured values correlates quite well. That in case of the lead particles the measured values do not quite match with the calculated ones, could lie at the specific material properties of lead.

In Fig. 9.15, trend and magnitude of the calculated values [9.31] correspond to the measurement results, although in one case, due to the large particle size, the convective proportion of the heat transfer was relatively great. In these experiments, cooling coils were installed in a laboratory fluidized bed.

Different again are the conditions in the experiments, which were described by Karchenko [9.32]. These investigations have been carried out at partly very high gas temperatures. In a laboratory fluidized bed was above the gas distributor plate a sphere installed, which has been selected with respect to the fluidized bed dimensions extremely large. In such a case one must probably assume, that the mobility of the bulk material, that rises in center of the bed with the bubbles, is limited by the large volume of the sphere. Therefore, the material occupancy of the surface is bigger, than due to the average void fraction of the fluidized bed would be expected. After Fig. 9.16, the heat transfer coefficients are about 30 percent higher, as may be expected according to the invoice or similar measurements of other researchers. Only in case of the largest investigated particles, which already at low

Fig. 9.15

Heat transfer at fluidized bed internals as function of the

fluidization number: comparison with measurement

results

Fig. 9.16

Heat transfer at fluidized bed internals as function of the

fluidization number: comparison of measurement results

with widely varying temperatures

Fig. 9.14

Heat transfer at fluidized bed internals as function of the

fluidization number: comparison of measurement results

with widely varying density ratios

50

fluidization numbers show a relative high expansion, this effect was no longer present.

A larger material occupancy of the heat exchange surface is also to be expected, if the heat transfer surfaces are situated at the casing wall of the fluidized bed. In laboratory apparatuses, the particles move down along the wall in the direction of the gas distribution plate. They are however no longer involved in the actual material movement, so that the conditions equal a moving bed. Fig. 9.17 shows the results of such experiments [9.33]. The tendency of the calculated and measured values are being completely equal, however,

the measured heat transfer coefficients are considerably higher, and their values equal almost exactly the maximum values of a fast-moving bed of densely packed particles.

In some investigations (eg [9.34]), the behavior of bulk materials with particle sizes below of about 120 microns differs from the described laws. Partly there are no maxima of the heat transfer coefficients, and the achievable heat transfer coefficients are higher than usual.

Table 9.4 contains a number of maximum values for the heat transfer and the associated fluidization numbers which are being compared with the from the Eq. (9-25) and (9-27) derived relationships for the optimal fluidization number

Ga6851

72.0

39.0

max

−

−

⋅⋅+=

g

p

mfu

u

ρ

ρ

α

(9-32)

and the maximum heat transfer coefficient

( )

-

GaRe114

52.0

38.05.00max,max

rdgc

g

gp

mfp

ααρ

ρρ

αα

++

⋅⋅⋅Ψ−⋅⋅=

−

−

(9-33)

Fig. 9.17

Heat transfer coefficients at heating surfaces, which are

situated on the casing wall: comparison of measured

values with those, which were calculated for immersed

heating elements

Table 9.4

Comparison of measured and calculated fluidization numbers u/umf und as well as heat transfer coefficients

αααα for the state of maximum heat transfer coefficient at heat transfer installations

Experimental conditions Measurement Calculation

Lit. Gas p ϑ Material dp ρρρρp ΨΨΨΨ0 u/umf ααααmax u/umf ααααmax,p ααααmax

- - bar °C - µm kg/m3 - - W/m2K - W/m2K W/m2K

[9.35] Air 1 20 redurit 52 4000 0,6 21 435 30 7500 550

[9.36] 95 0,6 13 370 15 4760 430

[9.37] 150 0,6 7 336 9,5 3300 355

260 0,57 3,2 284 5,5 2115 299

460 0,55 2 232 3,3 1320 240

mullite 350 1000 0,5 272 3,2 1660 267

700 203 2 930 223

[9.30] Air 1 20 abrasive 65 1600 0,45 25 707 22 6300 682

dust 150 6,3 570 8,9 3300 492

river 315 2600 0,45 3,5 360 3,3 1810 356

sand 450 2,7 290 2,5 1350 300

750 1,9 255 1,8 875 238

alumina 310 2700 0,45 2,2 284 3,3 1830 356

grit 450 2 250 2,5 1345 300

750 1,7 220 1,8 875 238

lead powd. 125 11000 0,45 2,7 395 3,4 3815 528

glas 900 2600 0,40 1,7 226 1,7 750 234

H2 1 20 river sand 315 2600 0,45 3,9 1080 2,5 7880 990

Al grit 310 2700 0,45 2,8 900 2,5 7970 990

CO2 1 20 river sand 315 2600 0,45 2,7 278 3 1244 272

Al grit 310 2700 0,45 1,8 185 3 1260 272

[9.31] Air 1 40 polyamide 2240 1135 0,47 1,7 150 1,4 346 143

Al 450 2700 0,42 2,8 330 2,5 1360 306

[9.38] Air 1 50 carbon 245 1230 0,56 6,3 360 6,5 2390 329

346 0,61 5,7 310 4,7 1803 254

447 0,59 4,8 270 3,8 1460 237

560 0,61 3,7 255 3,1 1210 206

690 0,61 2,5 230 2,7 1020 189

775 0,63 2,5 220 2,5 924 172

980 0,63 2 210 2,1 760 158

280 0,50 350 5,8 2140 348

sand 224 2600 0,52 5 450 4,6 2563 369

296 3,6 406 3,6 2047 328

447 0,50 2,5 360 2,6 1460 283

561 1,8 300 2,2 1210 256

[9.32] Air 1 900 quartz 340 2600 0,43 8 730 5,9 2540 567

700 sand 7,5 660 5,5 2346 451

500 7,5 590 4,9 2220 384

300 7 510 4,2 2026 348

500 clay 420 2300 0,53 4 390 4,3 1880 315

710 3 300 2,8 1235 273

1660 1,7 210 1,7 615 223

9.6.3 Influence of fluidized bed dimensions

Measurements in apparatuses with diameters from 70 to 4500 mm and mounting heights of the heat exchanger elements to about 200 mm, do not give any evidence to a dependence between the average heat transfer and the diameter of the fluidized bed [9.9 and 9.10]. Important is seemingly not the diameter ratio of gas bubbles to fluidized beds [9:39], but a sufficient distribution of the bubbles over the entire cross

sectional area. In low laboratory flu-idized beds the lat-ter is ensured.

Like the bed expansion, is also the heat transfer affected by the with increasing height decreasing proportion of the bubbles flowed-

through cross sectional area. With increasing height, increases the proportion of the fluidized bed, in which the heat transfer only correspond to that of a slowly moving bed, so that the average heat transfer is significantly reduced.

This relationship is being confirmed by the measurement values in Fig. 9.18. The ratio between the maximum heat transfer coefficient at the bed height h and the bed height h+ gets with increasing height above the gas distribution plate constantly smaller. h+ means the height, up to which a sufficient distribution of bubbles across the cross sectional area is present.

The evaluation of the measurement gave for h+ a value of about 20 cm.

For the particle convective proportion of the maximum heat transfer, the influence of the fluidized bed height can be considered accordingly [9.9 and 9.10]:

( ) ( )α α ϕpc hu hob pc hu hobmax;

max;= ⋅ (9-34)

ϕ describes the mean bubble flowed-through proportion of the fluidized bed between the lower and upper installation height hu and hob of the heat exchanger surfaces. From Eq. (9-11) results the following relationship:

( )ϕ h hh h

h h cmu obu ob

ob u; =⋅

+≥ ≥

4 520 (9-35)

9.7 Heat and mass transfer between the fluid and

particles

Investigations of heat and mass transfer between fluid and particles were mainly carried out in extremely low fluidized beds, in which the initial bed heights were only in the centimeter range. By plotting the measurement results similar to those of fixed beds, results the diagram in Fig. 9.19.

Eight studies have been evaluated [9.43 to 9.50]. Plotting the Nusselt number Nu against the quotient Re/Ga, the expression Ga⋅Pr1/3 appears as a parameter. Measurements in systems with Galileo numbers less or equal to 4.3, can however only be classified, if the parameter value is corrected by the factor of 6. Because these facts apply to the investigations of various researchers, it might be assumed, that for the single particles at Galileo numbers of about 5, the flow conditions themselves change fundamentally . Another peculiarity in fluidized beds is the fact that, dependent on the parameter values, the Nusselt values stagnate above certain limits of Re/Ga.

The evaluated measurement results originate from experiments with air (20 und 1000°C) and H2O. The range of the experimental conditions is not very broad, so that the depicted connections apply only roughly. In particular, the impact of larger fluidized bed heights remains unclear.

9.8 Catalytic gas-phase reactions

Heterogeneous catalytic gas-phase reactions in fluidized are like the heat transfer being influenced by the expansion behaviour of the fluidized bed [9.51].

Fig. 9.18

The heat transfer between fluidized

bed and internals as function of the

fluidized bed height h

Fig. 9.19

With the help of measurement values, which have been

published by various researchers, detected connection

between the dimensionless numbers Nu, Pr, Re and Ga

for the heat and mass transfer in low fluidized beds

53

Thereby, the following idea can serve as discussion basis:

A portion of the total amount of gas flows with the minimum fluidizing velocity through the suspension phase of the fluidized bed, while the excess gas in form of bubbles ascends. Close to the gas distribution, gas bubbles and material are being mixed ideally. The same holds for larger hights in the fluidized bed, but for a decreasing amount of material. The size of the ideally mixed proportion of the fluidized bed is described by the function ϕ(h) in accordance to Eq. (9-11). Additionally must still be taken into account, that in the bubbles flowed-through and therefore ideal stirred reactor volume the expansion ψb according to Eq. (9-18) prevails. Accordingly applies in this area for the suspension fraction 1-ψb.

9.8.1 Reaction model for first order reactions

For the homogeneous catalytic reaction of first order in an ideal stirred reactor, applies for the unreacted proportion c [9.52]:

cNr

=+1

1 (9-36)

Nr is being designated as number of the reaction units. In accordance to the prevailing conditions in the fluidized bed it must Nr be given separately for the quantity of gas, which flows through the void fraction of the suspension phase, and for the quantity of gas, which ascends in the fluidized bed as bubbles. The expression for the first-mentioned gas proportion is the same as for a with minimum fluidization velocity umf flowed-through fixed bed:

Nh

urmf

00=

⋅k (9-37)

k is the reaction rate constant, related on the suspension or fixed bed volume. For the as bubbles ascending gas is being written under consideration of ϕ(h) and 1-ψb:

( ) ( )N

h h

urb

b=

⋅ ⋅ ⋅ −k ϕ 1 Ψ (9-38)

The unreacted portion c´ in the fluidized bed then is the sum of the proportions, which are calculated under use of Nro und Nrb:

′ = ⋅+

+−

⋅+

cu

u N

u u

u N

mf

r

mf

rb

1

1

1

10

(9-39)

9.8.2 Application of the reaction model

By comparing published measurement results of studies on first order reactions can the useability of the reaction model be demonstrated.

9.8.2.1 Laboratory scale up to fluidized bed

diameters of 23 cm

The catalytic low temperature oxidation was investigated by Massimilla and Johnstone [9:53]. The reaction proceeds at 250 °C in the presence of manganese and bismuth oxide, that has been applied on a Al2O3 catalyst support. The reaction gas consisted of 90 percent oxygen and 10 percent ammonia. The experiments were conducted in a cylindrical fluidized bed of 11.4 cm in diameter with various initial heights. The reaction rate constant k is a mean value, that has been determined from the measurement results [9:54]. Fig. 9.2 shows the comparison of measured and calculated values.

Investigations for the hydrogenation of

ethylene on nickel contact were carried out by Lewis, Gilliland and Glass [9:55]. The reaction proceeded at temperatures around 110 °C at ethylene surplus with 10 percent hydrogen and 90 percent ethylene, so that with respect to H2, a first-order reaction can be taken as a basis. The diameter of the fluidized bed was 5.2 cm,

Fig. 9.21

Ethylene hydrogenation at ethylene surplus: comparison

of measurement and calculation

Fig. 9.20

Catalytic NH3 oxidation:

comparison of the by Werther

evaluated measurements with the

calculation

54

was thus rather small. The fixed bed heights were around 0.4 m.

The measurement results in Fig. 9.21 show no full accordance with the calculation, but trend and magnitude are correct in any case. The differences have certainly something to do, with the very small diameter of the experimental apparatus.

The same reaction, the hydrogenation of ethylene, was also investigated by Heidel, Fetting, Schügerl and Schiemann [9.56]. However, the reaction was verified at hydrogen surplus with 70 percent hydrogen and 30 percent ethylene at temperatures of 400 K on a copper-bearing contact. The reactor had a diameter of 7.5 cm, was thus again relatively small. In separate series of measurements, the reaction kinetics

have been studied in a fixed bed reactor of 2 cm in diameter. For the chosen operating conditions was found out, that the conversion of ethylene can be described by a first-order reaction. The value of the reaction rate constant k comes also from the fixed bed experiments.

Fig. 9.22 shows that close to the minimum fluidization velocity like in the previously described 5 cm reactor a tendency to better reaction conversions prevails, than according to the calculation would be expected.

Results for the catalytic decomposition of nitrous oxide N2O at manganese oxide-bismuth-contact were published by Shen and Johnstone [9.57]. The reaction was carried out at temperatures around 400 °C in a reactor with 11.4 cm in diameter. The reaction gas

consisted of 99 percent air and 1 percent of nitrous oxide. The values for the reaction rate constant k were taken from the publikation of Werther [9.54].

As the comparison of the measurement results in Fig. 9.23 shows, there is for a fluidized bed diameter of 11.5 cm an exact accordance with the results of the first investigation. This also applies to the area near the minimum fluidization velocity and for fixed bed heights of up to 1.1 m.

In literature is the catalytic ozone decay a widely held model for first-order reaction, which proceeds at temperatures slightly above the ambient conditions. For comparison purposes, are being used here two investigations in devices with diameters of 10, 15 and 23 cm [9.52 and 9.58]. The fill heights were 0.6 m and 1 m.

In Fig. 9.24, the unreacted proportion c´ is plottet against the reaction rate constant km, that is related on the catalysator mass. In Fig. 9.25 is c´ plotted against the expression k⋅h0/u, formed by using the reaction rate constant k. The connection between km and k, that is related on the volume of the suspension phase or the fixed bed, is as follows:

Fig. 9.22

Ethylene hydrogenation at

hydrogen excess: comparison of

measurements and calculation

Fig. 9.23

Catalytic decomposition of N2O: comparison of measurements and

calculation

Fig. 9.25

Catalytic decomposition of ozone:

comparison of measurements and

calculation

Fig. 9.24

Catalytic decomposition of ozone: comparison of

measurements and calculation

55

( )k p m= − ⋅ ⋅1 0Ψ ρ k (9-40)

As on the basis of the gained findings is to be expected, there is even in these cases an extensive accordance between the measured results and the calculation.

9.8.2.2 Semi-technical scale

Greater reaction volumes than in the previously evaluated studies were investigated by Hovmand, Freedman and Davidson [9.59]. Here again, the catalytic ozone decomposition was chosen as a model reaction of first order. The fluidized bed had a diameter of 46 cm, and it was being operated with fixed bed heights of 1.3 m and 2.6 m. Apart from the commonly used porous gas distributors, were in these studies in addition two perforated plates used, which rather fullfill the industrial requirements. One of them had 14 holes with 6.4 mm diameter, the other had 230 holes with 2.7 mm diameter.

The course of the measurement values in the semi-industrial fluidized bed installations indicate a behaviour, that in laboratory apparatuses could not be observed. The achievable reaction conversion strives namely with increased reaction rate not against the 100% value. This implies, that a certain proportion of the gas from the holes of the gas distribution plate directly into the forming bubbles flows, and thereby no contact with the material has, and there also takes place no further exchanges with the surrounding suspension phase. This gas proportion leaves the fluidized bed unreacted.

Mathematically the described situation can be taken into consideration, by modifying the Eq. (9-39) with an unreacted proportion c'byp that represents the proportion of the bypass gas.

( )′ = − ′ ⋅ ⋅+

+−

⋅+

+ ′c cu

u N

u u

u Ncbyp

mf

r

mf

rbbyp1

1

1

1

10

(9-41)

For the size of c´byp shall here no general law be given. The comparisons of measurements and calculation in Fig. 9.26 show, however, that in the investigated area measured and calculated values can be matched to each other - here was c´byp used just as an adjusting factor. The values of c´byp are the greater, the smaller the fluidized bed height is. Moreover, it is striking that even the porous gas distributer plate, especially at low bed heights and higher gas velocities, causes a bypass flow.

9.8.2.3 Large-scale reactor

One of the few studies at commercial reactors were published by de Vries and his co-workers

[9.14]. The large-scale reactor for the Shell process for catalytic oxidation of HCl had a diameter of 3 m and a hight of 10 m. In the experiments was the effect of a targetedly set fine grain content determined. As fine-grained proportion has been defined the bed material with particle sizes smaller than 44 microns.

Fine grain proportion in bulk

Fig. 9.26

Catalytic decomposition of ozone in semi-technical scale:

comparison of measurements and calculation according to Eq. (9-

41)

Fine-grain

content

u Convers

ion

m/s %

20 0.2 95.7

17 95.0

12 93.5

7 91.0

Calculation 88

Table 9.5

Reaction conversion in a large-

scale reactor for the Shell process

for the catalytic oxidation of HCL

depending on the fine grain

proportion (k = 0.68 1/s)

56

solids reduces the friction between the bulk of the larger particles, so that the gas bubbles coalesce slower with the height. Therefore increases the proportion of the ideal with gas commingled bulk material, and the catalytic reaction conversion increases as well. The measurement results in Table 9.5 confirm the with increasing fine grain content improved catalytic reaction. The calculation by Eq. (9-39) delivers in this case for the reaction conversion a value of 88 percent, thus exactly the magnitude, that according to the experiments with close particle size distributions is to be expected.

9.9 Operation as thermal dryer

With the notion of the fluidized bed drying, is connected for a long time the use of a single-stage convection dryer for powder, crystalline, granular, or short fibrous goods. For some time there are also multi-stage facilities and facilities with internals for heating and cooling in drying technology. A relatively recent development is the spray fluidized bed for drying and granulating solutions and suspensions.

The advantages of the fluidized bed are decisive for their wide dissemination in drying technology: high heat and mass transfer, no product residues on hot surfaces, convektive drying with longer dwell times of good, no mechanically moving parts and therefore low repair requirements, constant pressure loss even with fluctuating gas throughput, low maintenance and investment costs and good space/time yield.

The construction of a fluidized bed drying facility is shown in Fig. 9.27. The hot drying gas flows through a

gas distribution, for which plates of various kind are being used. Subsequently, the material on the plate is being fluidized by the gas. The gas transfers a part of its heat to the product, so that liquid evaporates. The steam-gas mixture contains dry particles from the

fluidized bed, which will be separated in a downstream device, such as cyclone, textile filter, or wet scrubber. Fluidized bed dryers are continuously and discontinuously operated, with fresh air, recirculated air or inert gas.

9.9.1 Heat requirement

Fluidized bed dryers are mainly suitable, if on the one hand the drying rate is depending on the heat and mass transfer between gas and moist material, and if on the other hand the moisture does not as pure surface humidity exist, so that longer dwell times are required. Due to fact, that the physical parameters, which determine the diffusion processes, usually are not known, are for the design of dryers model experiments necessary.

As lower limit can be determined the smallest gas-solids ratio & / &M Mg s for continuously operating dryers.

For a simplification, the cross flow is being neglected, what because of the considerable cross-mixing and the thus nearly isothermal conditions seems justified. Under the additional condition of non-hygroscopic material, and the assumption that the surface temperature is equal to the cooling limit temperature (first drying section), applies the following simple equation.

( )&

&

M

M

q X X

c

g

s

s s

g g g

=⋅ −

⋅ −

α ω

α ωϑ ϑ

(9-42)

During the drying of water-wetted drying-goods, the heat consumption q for 1 kg of evaporated water with surcharges for the heat loss of the apparatus lies between 3000 and 3300 kJ/kg. The gas outlet temperature is being equated in a first approximation with the cooling limit temperature. The gas inlet temperature depends on the product. To avoid sticking at the gas distribution plate, the gas temperature should preferably lie below the softening point of the dry material. Results of the Eq. (9-42) lie frequently quite near the actual gas solids ratios in dryers, because due to the neglection of the cross flow, a too high gas throughput is being calculated (fluidized beds work between the extremes of an ideal mixer and an ideal cross-flow device), and because of the fact that for reasons of economic efficiency in single-stage fluidized bed devices, only goods with a not too large "second drying section" are treated. In case of a very large "second drying section", is for the post-drying a second facility with low gas flow advisable, for example a multistage fluidized bed.

Fig. 9.27

Basic construction of a fluidized bed drying facility

57

9.9.2 Influence of the gas distribution plate

construction

The size of the cross-sectional area of a fluidized bed dryer is being calculated on the basis of the thermally necessary quantity of gas according to Eq. (9-42) and the fluidization velocity. For the required gas velocity can be assumed, that fluidized beds with expansion values of ψb ≈ 0,5 in terms of bed mobility and solid material discharge be advantageously operated. The fluidization number, that belongs to this bed expansion value, can be determined by using Eq. (9-18). For large particles, the gas velocities are by a factor of 1.5 to 3

higher, than the minimum fluidization velocity. For very small particle sizes lie the values of this factor between 15 and 30. Because of the particle size dependence of the minimum fluidization velocity, remains the actual gas velocity despite different particle sizes however in the same order of magnitude.

The products that are to be dried often have wide par-ticle size distributions and consist of agglomerates, which were formed due to moisture bridges, and which during the drying process partially disintegrate. There-fore, the fluidization velocity must so be chosen, that the agglomerated material is likewise being moved, and that the gas as little as possible fine particles carries with it. Because during the drying process the average particle size, the moisture content, as well as the parti-

cle shape constantly changing, the optimal fluidisation velocity can only experimentally be determined.

A constructive way, to reduce the proportion of parti-cles that is entrained by the gas, is the enlarging of the cross section of the fluidized bed dryer above the fluid-ized bed up to twice the size. The amount of the en-trained material of fine particles can also be influenced through the position of the gas outlet (Fig. 9.28).

For the even distribution of the gas over the cross sec-tion of the fluidized bed, is a gas distributor plate nec-essary that must meet the following requirements:

♦ Uniform gas distribution over the cross-section of the drying facility can be realised best with a large number of smaller gas inlets. Prerequisite is a minimum pressure loss at the gas distributor plate.

♦ No zones with resting material in immediate vicin-ity of the gas distributor plate, especially in cases, in which the gas inlet temperature has to be higher, than the softening temperature of the material.

♦ No trickling through of solid particles during the drying process, often also during interruptions, when the material as packed bed in the facility re-mains.

♦ Holes should not plug, and the material to be dried should form no deposits at the gas distribution plate.

♦ Minimum particle crushing by gas jets from the gas distribution plate.

Crushing and abrasion of particles is closely con-nected with the construction of the gas distribution de-vice. Particles, which are accelerated by the gas jets from the holes of the plate, cause increased friction and collision between the particles in the jet region. In addi-tion to the jet velocity, plays especially the jet diameter a role, because its influence on the jet pulse is stronger than that of the gas velocity. As is to be expected, in-creases the crushing of particles with larger nozzle di-ameters, even in case of equal fluidizing velocities.

The described relationships are being confirmed by abrasion-measurements on a fibre precursor (Fig 9.29), which were published by Stockburger [9.62]. A sin-tered metal plate caused the smallest abrasion. This kind of gas distributors, with a average pore size of just 0.035 mm, produce the most even gas distribution and the thinnest gas jets.

The hole plate with holes of 1.6 mm diameter pro-duces a greater abrasion than a fine-hole-sheet with openings of 0.35 mm diameter. And that, even though it’s opening ratio is twice as large, and therefore the ve-locity of the leaking gas only half as large.

Fig. 9.28

Influence of the gas outlet location on the amount of

product, which is entrained by the fluidizing gas

58

The gas distribution plate types, which are commonly used in the drying technology, are shown in Fig. 9.30. Only in laboratory apparatus one uses still frits plates. This kind of plate ensures due to the many small pores an even distribution of gas, but tends - especially if dust-laden drying-air is used - easily to the clogging. Generally, this type of gas distribution plate causes a very high pressure loss.

For facilities on a technical scale one often uses plates with cylindrical holes as gas distributor. The holes are usually larger than the particles. Thus, the plates tend less to the clogging, but have the disadvantage that the gas is unevenly distributed and in thick jets into the fluidized bed flows. Moreover, trickles the dry material through the gas distribution plate.

Fine hole sheets have themselves very successful proven and are most commonly used in fluidized layer dryers. The holes are triangular to half-eliptical and strong conical. The passage of the drying gas is inclined. The holes are kept very small (up to 60 µm), so that an even distribution of the gas and low abrasion of the product can be achieved. The trickling through of product during operating interruption is no problem at this type of gas distribution plate.

Trickling through of product and clogging of holes can be avoided best by using gas distributor heads or bell-shaped distributing elements. Such plate constructions are however expensive to produce, and the gas distribution is very uneven, so that the bulk solid movement between the gas distribution elements very small is, and partly also dormant product accumulations occur. In case of gas distributor heads, the product abrasion is still far greater than with perforated sheets with round holes, because the thick gas jets are in addition directed against each other. If the nozzles are directed down to the plate, erosion is to be expected. Such gas distribution plates are being rarely used in drying technology.

A special feature is the so-called spouted bed, which has no gas distribution plate, but instead is funnel-shaped [9.63]. The spouted bed is used for the drying of suspensions of fine particles on inert coarser particles as heat carrier, and for the drying of large particles in millimetre range with narrow particle size distributions, which are difficult to fluidize. The cone angle should have a size of at least 40° [9.64], while the diameter-ratio between nozzle and device should lie below of 0.35 [9.65].

To achieve, that through all holes of a gas distribution plate an equal amount of gas is flowing, must a certain pressure loss be generated by the plate. 15 to 40 percent of the pressure drop in the fluidized bed is being considered sufficient. The upper value applies to flat

Fig. 9.29

Abrasion behaviour of fibre precursor granules using dif-

ferent gas distribution plates

Fig. 9.30

Design of gas distributor floors for fluidized bed dryers

Fig. 9.31

Two-stage circular fluidized bed with disc for the material

feeding, air-broom and heated internals

59

fluidized beds, in which the influence of the approach flow conditions is particularly great. The distribution plate of a dryer usually has pressure losses between 5 and 10 mbar.

During drying of sticky and highly agglomerated filter cakes, the formation of channels can be avoided by distributing the wet material with the help of a stirrer into already dry material, or by feeding the product with the help of a rotating disk. Clumps, which are contained in the wet material, can be crushed by using built-in grinding discs.

Mainly in multistage fluidized beds, the clogging of the holes in the gas distribution plate can be a problem. This often can be prevented with the help of an air-broom. This device rotates below the plate and blows out the holes with the help of sharp air jets.

In some cases, the considerable horizontal equilibration of concentration and temperature, which occur in con-tinuously operated fluidized beds, can be of disadvan-tage. To achieve a greater average dwell time for the main proportion of the material, a larger length/width ratio is to be aimed for. In a compact dryer one can meet this requirement due to the installation of spiral-shaped baffles in a round bed. Fig. 9.31 shows a two-stage fluidized bed with disc-product feeding, air-broom and spiral-shaped internals [9.66].

9.9.3 Multi-stage fludized beds

For substances that have a low moisture content, but due to the moisture-transport from the inside of the product to its surface larger dwell times needing, can convection dryers like tower dryer or multistage fluid-ized bed be used. Through multistage fluidized beds can a better balanced dwell time be realized, which of-ten is connected with a positive effect on the quality of the dried product. With the use of this kind of dryers, the formation of channels in the product can be avoided, which in case of cohesive bulk solids in chamber dryers often occurs, so that the largest part of the bulk solid has no contact with the drying gas. Com-pared to the single-stage fluidized bed, the multi-stage requires a far lower amount of drying gas, and the dwell-time distribution is tighter. Moreover, counter current flow of product and gas is realized, so that es-pecially in the drying of hygroscopic products, heat and mass transfer are positively affected.

Fig. 9.32 shows schematically the two different types of multistage fluidized beds. In both cases, a separate stage consists of a sieve plate with a fluidized bed on it. The material moves from top to bottom, the drying gas from bottom to top.

The multistage fluidized bed with downpipe matches largely with its structure the sieve plate columns from the rectification and absorption. In this type of con-struction flows the drying gas through the sieve plate of each stage, while the material reaches the next sieve plate through an overflow pipe. With no cellular wheel sluice in the overflow pipe, the pipe must immerse into the fluidized bed on the next plate. The filling height of this pipe must be such, that the pressure loss, that is caused by the gas flow through the sieve plate and the associated material bed, is smaller than the pressure loss, that is generated by the unwanted flow through the overflow pipe. These conditions require a relatively large ground clearance. Furthermore, the start up of the facility is problematic, because the overflow tubes are not yet being filled. Such problems can however be cir-cumvented by using locks and shutoff devices, as they are specified in Perry's Chemical Engineer Hand-book [9.67]. So the tube can remain for example using a slider when starting so long closed, until a sufficiently high product column exists.

Particularly useful and reliable are rotary feeder as shut-off devices. There is no difficulty during start-up of the facility because the dryer without special meas-ures can be filled. The cell wheel can be arranged close to the gas distribution plate, and at low fill level is also the jamming of particles in the cell-wheel to get around.

Fig. 9.32

Multistage fluidized bed

60

The mode of action in a simpler type of fluidized bed, which is known as 'trickle-stage fluidized bed' (Riesel-boden-Wirbelschicht), is less problematic. As gas dis-tribution plates serve so-called Dual-flow sheets, which are equipped with such large holes, that both gas and material can pass [9.68 and 9.69]. Overflow pipes are therefore not required. For technically interesting di-mensions of such an apparatus are, however, difficul-ties are expected, because the pressure loss, that is re-quired for an even distribution of gas by the sieve plate, is difficult to reach. This applies especially to fine ma-

terials and such goods, which due to varying moisture distribution, varyingly agglomerate, so that they are particularly reliant on an even gas distribution by the plate.

The multi-stage 'trickle-stage fluidized bed' will re-main limited for this reason on the drying of same-sized particles in the millimetre range, for example plastic granules.

Multistage fluidized beds can be understood as chained ideally mixed single-stage fluidized beds. Fig. 9.33 depicts the temperature profile along a six-stage apparatus [9.70]. There are significant temperature variations between the stages. In every single stage, the temperatures are, however, constant over the height.

With increasing gas velocity and growing free cross-sectional area of the sieve plate rises but also the pro-portion of material, which is thrown from the lower flu-idized bed through the holes onto the overlying plate. This material arrives on a stage, which it already has passed through. Fig. 9.34 depicts this process quantita-tively: depicted is the axial mixing coefficient depend-ing on the gas velocity. As parameter serve the free cross section of hole plates (13 and 24 percent), as well as that of a slit plate (52 percent).

Multistage fluidized bed dryers are only used for products with large second drying sections. Its drying time is dependent on complex diffusion processes. Therefore, are for the design of the drying apparatus investigations necessary. Results from such tests are the inlet and outlet temperatures of the drying gas, the re-quired gas-solid ratio and the appropriate gas velocity,

from which the diameter of the apparatus for the re-quired material throughput is determined. Finally, the experimentally determined drying time is used, to spec-ify the number of stages and the fluidized bed height. The bed height can be adjusted with the height of the overflow pipe.

9.9.4 Heating surfaces internals

For drying of material with low softening point, the temperature difference between the incoming and the outgoing gas is relatively small, so that relatively large amounts of gas and thus large areas for fluidization are required. It is much more economic to provide a portion of the required drying heat with the help of heating sur-faces, which are inbuilt into the fluidized bed.

For internals there is a distinction between plates and bundles of round or oval tubes. Neither the shape of the

Fig 9.33

Temperature profile along a multistage fluidized bed

Fig. 9.34

Axial mixing coefficient in multistage fluidized beds

61

inbuilt bodies [9.71] nor the use of pipes and their ar-rangement (aligned, offset or crossed) [9.72] causes a significant change in the amount of the average heat transfer. The heat transfer is also being influenced hardly by the diameter of the pipe, as long as this is lar-ger than 15 mm [9.73]. Finned tubes can be used for the increase the transferred amount of heat in proportion to the volume of the pipe. Excessive obstruction of parti-cles between the ribs can be avoided by using tubes with only two longitudinal ribs, which are arranged in the direction of flow [9.71]. Due to the increased risk of incrustation in the corners between the tubes and the ribs is such a solution for dryers not fully to recom-mend and has to be carefully checked in long-term tests.

For the undisturbed fluidization in beds with dived tube bundles, some authors recommend optimum tube spacing t (pipe midst distance to diameter ratio), which from t = 2 [9.74] to t = 4...6 [9.75] vary. More sense seems to make, however the use of the ratio between clear distance of the tubes and particle diameter, to take the mutual influence into account. According to Petrie [9.76] lies this distance at a value of about 43 times the largest particle diameter. This value corresponds with the size ratio for the undisturbed bulk solid discharge from silos. In addition to the unimpeded moving of the material, the chosen tube spacing should the possibility of the cleaning ensure.

The distance between internals and gas distributor plate must be larger than the penetration depth of the gas jets, which can be estimated by Eq. (9-22). Fur-thermore, internals cause an increased pressure loss in the fluidized bed, if their resistance against the gas flow is greater than the weight loss of the fluidized bed due to the reduced volume of the bed. The increase in the pressure loss, however, is not significant. According to Neukirchen [9.77], the increase amounts in the worst case about 18 percent of the total pressure loss (gas dis-tribution, fluidized bed, internals).

9.9.5 Spray fluidized bed drying

A further development in the field of fluid bed drying is the spray fluidized bed. This technique is applied for solutions and suspensions, for which in former times mostly spray dryers and roller dryers were used.

The principle of the spray fluidized bed dryer consists therein, that via two-component nozzles, rarely via sin-gle-component nozzles, flowable product is sprayed into a fluidized bed, which contains already dried prod-uct. The moist product is during this process distributed over the surface of the individual particles. The drying of the so generated thin liquid layer takes place in the first drying section, and is therefore extremely fast. The

energy for evaporation of the liquid is delivered through the drying gas.

If the procedure is operated continuously, two, some-times three steps are required: firstly a drying and granulating stage, secondly a crushing stage, which the grain growth during the drying process counteracts and new granulation germs creates, and thirdly, if neces-sary, a classification stage, where the desired end prod-uct is separated.

The basic design of a spray fluidized bed facility is depicted in Fig. 9.35. It consists essentially of the fluid-ized bed with gas distributor plate, a nozzle for the product infeed, a device such for example a worm con-

veyor for the material withdrawal, a gas outlet and a downstream textile filter, which is mounted above the fluidized bed [9.78]. For the classifying of the desired particle size [9.78 up 9.80] is in many cases a sieve or sifter responsible, and a mill creates the necessary granulation germs.

Fundamental prerequisite for the functioning of this procedure is the combination of the three steps of granulating, crushing and grading. As expected, the particle size increases in the drying and granulating stages according to an exponential growth law [9.81]:

Fig. 9.35

Spray fluidized bed drying facility

62

d

d

M

MZ

p

p

s

s03

= ⋅ ⋅

exp

&K (9-43)

If the ratio & /M Ms s as well as the temporal increase of the layer thickness per particle surface unit are con-stant, if the particles are approximately spherical, and if the actual particle size distribution can be expressed by an equivalent particle diameter, the constant K has the experimentally confirmed value of 1 [9.62]. It is obvi-ous, that in the limiting case, the fluidized bed would consist of only a few large particles. So, a size reduc-tion and nucleation process must superimpose the growth process, to maintain the fluidization process. Crushing can be done in three ways: mechanically within or outside of the fluidized bed, or taking advan-tage of the fluidized bed abrasion.

The easiest way is the exploitation of the particle abra-sion that occurs in the fluidized bed. Apart from prod-uct properties as strength and shape of the granules, are the following influence factors important: design of the gas distribution, fluidization number as well as height of fluidized bed. More important than the gas distribu-tion are the fluidization number and the height of the fluidized bed. These effects are depicted in Fig. 9.36. With growing fluidization and bed height, the abrasion rate increases significantly [9.62]. As an additional re-quirement however, the Reynolds number, formed with the gas velocity and the particle diameter, must be lar-ger than 100 [9.61].

Finally the significant crushing during the atomisation should be mentioned. This effect is in case of two-substance nozzles especially strongly noticeable. Their effect complies almost with that of a fluidized bed jet mill [9.82].

A targeted control of the particle size is possible, us-ing a mill and a classifying device outside the fluidized bed. For instance can a screening machine be used for this task, but the installation of a classifier in the entire process is much easier. The fine-grain proportion is be-ing normally by augers or injectors recycled into the fluidized bed [9.79 and 9.80].

The nozzles can be so arranged, that they spray from above [9.78 and 9.79], from the side [9.80 and 9.84] or from below [9.85].

Orientation values about the immediate sphere of in-fluence of a two-component nozzle, while spraying horizontally into a fluidized bed, was gained during drying and granulating of fertilisers by measuring the temperature distribution in the spray zone [9.86]. The liquid loading of the nozzle lied between 45 and 240 kg/h, thus at values, which are in technical facilities common. The amount of atomising air was maintained

at a constant value of 270 kg / h, at gas pressures be-tween 1.9 and 3.4 bar. First, the temperature in the axis of the spray cone became with increasing liquid load smaller, but reached at the highest liquid load in a dis-tance of 350 to 400 mm from the nozzle the tempera-ture of the fluidized bed. The temperature profile in ra-

dial direction resembles parabolic curves with a mini-mum in the spray axis. Even at the highest loads was in a distance of 100 to 150 mm from the nozzle nearly the bed temperature reached.

Due to the small extent of the spray cone, the liquid distribution over a larger area of fluidized bed is only achievable by the moisture-laden particles migrate out of the jet region. This process leads to a significant ex-

Fig 9.36

Abrasion behaviour of fibre precursor pellets depending

on the fluidization number and the bulk height

Fig. 9.37

Mixing coefficients in fluidized beds depending

on the fluidization number and the bed height

63

change of energy inside the bed, because the volumetric heat capacity of the material is about 1000 times as large as that of the gas.

The experience shows, that the intensity of the mate-rial movement of a fluidized bed is influenced by the condition of the bed, by the fluidization number, and by the bed height. As Fig. 9.37 shows, the mixing coeffi-cient starts with the value zero at the minimum fluidiza-tion velocity and increases with growing fluidization [9.87]. The significant influence of the fluidized bed height is by the graphic likewise attested [9.88]. So, high spray rates require large bed heights and large flu-idization numbers.

The advantage of the spray fluidized bed dryer, com-pared to competing processes, lies except in the often higher economic efficiency, in the significant im-provement of the product properties. Depending on the intended use particle sizes between 1 and 6 mm can be produced. The products are free of dust and consist of compact particles with mostly higher bulk density and better flow behaviour than they can be produced in any other technique. Nevertheless, the solubility in liquids is often better, than that of dust-free products, in which the primary particles often clump together. Prerequisite for the application of the spray fluidized bed technol-ogy is, however, the granulation ability of the material to be dried. This is first and foremost a product specific property, which however can also be influenced by the process conditions, such as gas outlet temperature, moisture, and kind of atomisation [9.78].

Literature of chapter 9. [9.1] Wilhelm, R.H.; und M. Kwauk: Fluidization of solid particles. Chem.

Engng. Prozess 44(1948)3, S.201-218.

[9.2] Saxena, S.C.; und G.J. Vogel: The measurements of incipient fluidiza-tion velocities in a bed of coarse dolomite at temperature and pres-sure. Trans. IChemE 55(1977), S.184-189.

[9.3] Rowe P.N.; und C.X.R. Yacono: The bubbling behaviour of fine pow-ders when fluidized. Chem. Engng. Sci. 31(1976)12, S.1179-1192.

[9.4] Goroschenko, W.D.; R.B. Rosenbaum und O.M. Todes: Nachr. d. höh. Lehranst. d MWO d. UDSSR. Erdöl und Gas 1(1958)25.

[9.5] Leva, M.: Fluidization. New York: McGraw-Hill Book Company, 1959. [9.6] Godard, K.E.; und J.F. Richardson: Proceedings of the Tripartite Chemical Engineering Conference, Montreal, 1968.

[9.7] Davies, L.; und J.F. Richardson: Trans. Inst. Chem. Eng. 44(1966), S.T293.

[9.8] Rowe, P.N.; und C.X.R. Yacono: The bubbling behavior of fine pow-ders when fluidized. Chem. Engng. Sci. 31(1976)12, S.1179-1192.

[9.9] Heyde, M.; und H.J. Klocke: Wärmeübergang zwischen Wirbelschich-ten und Einbauten - ein Problem des Wärmeübergangs bei kurzfristi-gem Kontakt. Chem.-Ing.-Tech. 51(1979)4, S.318-319; MS 679/79.

[9.10] Heyde, M.; und H.J. Klocke: Wärmeübergang zwischen Wirbel-schicht und Wärmetauschereinbauten. vt "verfahrenstechnik" 13(1979)11, S.886-892.

[9.11] de Groot, J.H.: Proceedings of the international Symposium on Flu-idization. Eindhoven: 6.-9. Juni 1967.

[9.12] Huisung, T.H.; und C. Thodos: Expansion characteristics of gas-fluidized beds. The Canadian J of Chem. Engng. 55(1977)4, S.221-226.

[9.13] Lewis, W.K.; E.R. Gilliland und W.C. Bauer: Characteristics of fluid-ized particles. Ind. Engng. Chem. 41(1949), S.1104-1117.

[9.14] de Vries, R.J.; W.P.M. van Swaaij, C. Mantovani und A. Heijkoop: Design chriteria and performance of the commercial reactor for Shell Chlorine Process. Proc. 5th Europ. Symp. Reaction Engng., Amster-dam, 1972, S.B9.-59/69.

[9.15] Xavier, A.M., D.A. Lewis und J.F. Davidson: The expansion of bub-bling fluidized beds. Trans. Instit. Chem. Eng. 56(1978)4, S.274-280.

[9.16] Werther, J.: Effekt of gas distributor on the hydrodynamics of gas fluidized beds. Germ. Chem. Engng. 1(1978)3, S.166-174.

[9.17] Geldart, D.: The effect of particle size and size distribution an the behavior of gas-fluidized beds. Powder Technol. 6(1972), S.201-215.

[9.18] Park, W.H.; W.K. Kang, C.E. Capes und G.L. Osberg: The properties of bubbles in fluidized beds of contacting particles as measured by an electroresistivity probe. Chem. Eng. Sci. 24(1969), S.851.

[9.19] Rowe, P.N.; und D.J. Everett: Fluidized bed bubbles viewed by x-rays, Part III. Trans. Instn. Chem. Engrs. 50(1972), S.55-60.

[9.20] Whitehead, A.B.; und A.D. Young: Fluidization performance in a large-scale equipment, Part I. Proc. Intern. Symp. on Fluidization, Eindhoven, 1967, S.284.

[9.21] Fryer, C.; und O.E. Potter: AIChEJ 22(1976)1, S.88.

[9.22] Reuter, H.: Steiggeschwindigkeit von Blasen im Gas-Feststoff-Fließbett. Chem.-Ing.-Tech. 37(1965)10, S.1062-1066.

[9.23] Moroka, S.; K. Kawazuishi und Y. Kato: Holdup and flow pattern of solid particles in freeboard of gas-solid- fluidized bed with fine parti-cles. Powder Technol. 26(1980)1, S.75-82.

[9.24] Fournol, A.B.; M.A. Bergougnou und C.G. Baker: Solids entrainment in a large gas fluidized bed. The Can. J of Chem. Engng. 51(1973), S.401-404.

[9.25] Demmich, J.; und M. Bohnet: Feststoffaustrag aus Wirbelschichten. vt "verfahrenstechnik" 12(1978)7, S.430-435.

[9.26] Merry, J.M.D.: Penetration of a horizontal gas jet into a fluidized bed. Tran. Instn. Chem. Engrs 49(1971), S.189-195.

[9.27] Merry, J.M.D.: Penetration of vertical jets into fluidized beds. AIChE Journal 21(1975)3, S.507-510.

[9.28] Heyde, M.: Durchströmen, Aufwirbeln und Fördern von Schüttgütern. Fortschrittberichte der VDI-Z. Reihe 3, Nr. 66, 1982.

[9.29] Baskakov, A.P.; et al.: Heat transfer to objects immersed in fluidized beds. Powder Technol. 8(1973), S.273-282.

[9.30] Wicke, E.; und F. Fetting: Wärmeübergang in Gaswirbelschichten. Chem.-Ing.-Tech. 26(1954)6, S.301-309.

[9.31] Knuth, M.; und P.M. Weinspach: Experimentelle Untersuchungen des Wärme- und Stoffübergangs an die Partikeln einer Wirbelschicht bei der Desublimation. Chem.-Ing.-Tech. 48(1976)10, S.893; MS 413/76.

[9.32] Karchenko, N.V.; und K.E. Makhorin: The rate of heat transfer be-tween a fluidized bed and an immersed body at high temperatures. Int. Chem. Engng. 4(1964)4. S.650-654.

[9.33] Botterill, J.S.M.; und M. Desai: Limiting factors in gas-fluidized bed heat transfer. Powder Technol. 6(1972), S.231-238.

[9.34] Yamazaki, R.; und G. Jimbo: Heat transfer between fludized beds and heated surfaces. J of Chem. Engng. of Japan 3(1970)1, S.44-48.

[9.35] Mersmann, A.: Bestimmung der Lockerungsgeschwindigkeit in Wir-belschichten durch Wärmeübergangsmessungen. Chem.-Ing.-Tech. 38(1966)10, S.1095-1098.

[9.36] Mersmann, A.: Zum Wärmeübergang zwischen dispersen Zweipha-sensystemen und senkrechten Heizflächen im Erdschwerefeld. vt "verfahrenstechnik" 10(1976)10, S.641-645.

[9.37] Mersmann, A.: Zum Wärmeübergang in Wirbelschichten. Chem.-Ing.-Tech. 39(1967)5/6, S.349-353.

[9.38] Kim, K.J.; D.J. Kim, S. Chun und S.S. Choo: Heat and mass transfer in fixed and fluidized bed reactors. Int. Chem. Engng. 8(1968)3, S.205-222.

64

[9.39] Werther, J.: Strömungsmechanische Grundlagen der Wirbelschicht-technik. Chem.-Ing.-Tech. 49(1977)3, S.193-202.

[9.40] Wright, S.J.; R. Hickmann und H.C. Kethely: Heat transfer in fluid-ized beds of wide size spectrum at elevated temperatures. British Chem. Engng. 15(1970)12, S.1551-1554.

[9.41] Howe, W.C.; C. Alusio, Pope, Evans und Robbins: Control variables in fluidized bed steam generation. Chem. Engng. Progr. 73(1977)7, S.69-73.

[9.42] Betriebsmessung

[9.43] Subramanian, D.; H. Martin und E.U. Schlünder: Stoffübertragung zwischen Gas und Feststoff in Wirbelschichten. vt "verfahrenstech-nik" 11(1977)12, S.748-750.

[9.44] Donnadieu, G.: Transmission de la chaleur dans les milieux granulai-res. Revue Inst. F. Pétrole 16(1961), S.1330.

[9.45] Rowe, P.N.; und K.T. Claxton: Heat and mass transfer from a single spehre to fluid through an array. Trans. Instn. Chem. Engrs. 43(1965), S.T321-331.

[9.46] Zabescheck, G.: Experimentelle Bestimmung und analytische Beschreibung der Trocknungsgeschwindigkeit rieselfähiger, kapillarporöser Güter in der Wirbelschicht. Diss. Universität Karlsruhe, 1977.

[9.47] Petrovic, V.; und G. Thodos: Effectiveness factor for mass transfer in fluidized systems. Proc. Int. Symp. on Fluidization, Eindhoven, 1967.

[9.48] Mosberger, E.: Über den Wärme- und Stoffaustausch zwischen Par-tikeln und Luft in Wirbelschichten. Diss. TH Darmstadt, 1964.

[9.49] Richardson, J.F.; und P. Ayers: Heat transfer between particles and gas in a fluidized bed. Trans. Instn. Chem. Engrs. 37(1959)6, S.314-322.

[9.50] Yanata, J.; K.E. Makhorin und A.M. Glukhomanyuk: Inverstigation and modelling of the combustion of natural gas in a fluidized bed of inert heat carrier. Int. Chem. Engng. 15(1975)1, S.68-72.

[9.51] Heyde, M.: Strömungsmechanik und katalytische Reaktionen in Wir-belschichten. Maschinenmarkt 88(1982)78, S.1596-1599.

[9.52] Orcutt, J.C.; J.F. Davidson und R.L. Pigford: Reaction time distribu-tions in fluidized catalytic reactors. Chem. Engng. Progr. Symp. Ser. 58(1962)38, S.1-15.

[9.53] Massimilla, L.; und U.H.F. Johnstone: Reaction kinetics in fluidized beds. Chem. Eng. Sci. 16(1961), S.105-112.

[9.54] Werther, J.: Mathematische Modellierung von Wirbelschichtreak-toren. Chem.-Ing.-Tech. 52(1980)2, S.106-113.

[9.55] Lewis, W.K.; E.R. Gilliland und W. Glass: Solid-catalyzed reaction in a fluidized bed. AIChE 5(1959)4, S.419-426.

[9.56] Heidel, K.; K. Schügerl, F. Fetting und G. Schiemann: Einfluß von Mischungsvorgängen auf den Umsatz bei der Äthylenhydrierung im Fließbett. Chem. Eng. Sci. 20(1965), S.557-585.

[9.57] Shen, C.Y.; und H.F. Johnstone: Gas-solid contact in fluidized beds. AIChI Journal 1(1955)3, S.349-354.

[9.58] Van Swaaij, W.P.M.; und F.J. Zuiderweg: Investigations of ozone decomposition in fluidized beds on the Basic of a two-phase model. Proc. 5th Europ. Symp. Reaction Engng., Amsterdamm, 1972, S.B9-25/36.

[9.59] Hovmand, S.; W. Freedman und J.F. Davidson: Chemical reactions in a pilot-scale fluidized bed. Trans. Instn. Chem. Engrs. 49(1971), S.149-162.

[9.60] Baskakov, A.P.; und I.V. Gubin: Khim. Prom. (russ.) 7(1968), S. 54/56.

[9.61] Reh, L.: Wirbelschichtreaktoren für nicht katalytische Reaktionen. Ullmann Enzyklopädie der techn. Chemie, 3. Weinheim: Verlag Chemie, 1973, S.433-460.

[9.62] Stockburger, D.: Fortschritte und Entwicklungstendenzen in der Trocknungstechnik bei der Trocknung formloser Güter. Chem.-Ing.-Tech. 48(1976)3, S.199-205.

[9.63] Brauer, H.: Grundlagen der Einphasen- und Mehrphasenströmung, Kapitel 9: Wirbelschicht. Aarau und Frankfurt am Main: Sauerländer Verlag, 1971.

[9.64] Hunt, C.H.; und D. Brennan: Estimation of spout diameter in an spouted bed. Austral. Chem. Engng. (1965)3, S.9-18.

[9.65] Becker, H.A.: An investigation of laws governing the spouting of coarse particles. Chem. Engng. Sci. 13(1961), S.245-262.

[9.66] Klocke, H.J.; und D. Stockburger: Neues aus der Trocknungstechnik. Chem. Ind. (1973)25, S.708-712.

[9.67] Perry's Chemical Engineer Handbook, 4th e., S. 20-46/49.

[9.68] Brauer, H.; und V. Asbeck: Druckverlust und Feststoffkonzentration in mehrstufigen Rieselboden-Wirbelschichten. vt "verfahrenstech-nik" 6(1972)7, S.230-238.

[9.69] Röben, K.; und E. Steffen: Praktische Anwendungen von Mehrstufen-Rieselboden-Wirbelschicht-Apparaten. Aufbereitungstechnik 15(1974)12, S.665-669.

[9.70] Nishinaka, M.; Sh. Morooka und Y. Kato: Longitudinal dispersion of solid particles in fluid beds with horizontal baffles. Powder Technol 9(1974)1, S.1-16.

[9.71] Natusch, H.J.; und H. Blenke: Zur Wärmeübertragung an horizon-talen Längsrippenrohren in Gas-Fließbetten. vt "verfahrenstechnik" 8(1974)10, S.287-293.

[9.72] Reh, L.: Strömungs- und Austauschverhalten von Wirbelschichten. Chem.-Ing.-Tech. 46(1974)5, S.180-189.

[9.73] Vreedenberg, H.A.: Heat transfer beetween a fluidized bed and an horizontal tube. Chem. Engng. Sci 9(1958)1, S.52-60.

[9.74] Lese, H.K.: Heat tranfer from a horizontal tube to an fluidized bed in the presende of unheated tubes. Dissertation, Lexington/USA (1969).

[9.75] Offergeld, E.: Wirbelbetten mit beheizten Einbauten in der Wirbel-schicht. vt "verfahrenstechnik" 8(1974)12, S.336-338.

[9.76] Petrie, J.C.; W.A. Freeby und J.A. Buckham: In-bed heat exchangers. Chem. Engng. Progr. 64(1968)7, S.45-51.

[9.77] Neunkirchen, B.: Gestaltung horizontaler Rohrbündel in Gas-Wirbelschichten nach wärmetechnischen Gesichtspunkten. Disserta-tion Universität Stuttgart, 1973.

[9.78] Rosch, M.; und R. Probst: Granulation in der Wirbelschicht. vt "ver-fahrenstechnik" 9(1975)2, S.59-64.

[9.79] Bean, S.L.; et al: Process for the production of hydrous granular so-dium silicate. USA Patent 3.748.103.

[9.80] Shakova, N.A.; u.a.: Investigation of the granulation of ammonium nitrate in a fluidized bed under industrial conditions. Internat. Chem. Engng. 13(1973), S.658-661.

[9.81] Shakova, N.A.; u.a.: Kinetics of granule formation in a fluidized bed. Theor. Foundations Chem. Engng. (Teor. osnovy khim technol.) 5(1971)5, S.656-661.

[9.82] Kaiser, F.: Die Fließbettstrahlmühle. Chem.-Ing.-Tech. 45(1973)10a, S.676-680.

[9.83] Fritsch, R.; u.a.: Austragsvorrichtung für Granulierwirbelapparate. DBP 2303212 (1973).

[9.84] Christmann, G.: Zerstäubungs-Wirbelbettgranulator. Chemie-anlagen + Verfahren (1973)1, S.42-43.

[9.85] de Jong, B.: Fließbettanlage fördert Flexibilität von Prozeßvorgängen. Chemie-anlagen + Verfahren (1971)8, S.49-51.

[9.86] Shakova, N.A.; und G.A.Minaev: Investigation of the temperature field in the spray zone of a granular with a fluidized bed. Internat. Chem. Engng. 13(1973)1, S.65-68.

[9.87] Mori, Y.; und K. Nakamura: Solid mixing in a fluidized bed. Kagaku Kogaku 4(1966)1, S.154-157.

[9.88] Pippel, W.; u.a.: Über die Vermischung des Feststoffes in Gas-Feststoff-Wirbelschichten. Chem. Techn. 20(1968)12, S.750-755.

65

10 Solids removal in cyclone

separators Bulk solid, which is transported by a gas stream, must

be separated from the gas. An elegant way is the separation in cyclone separators, which are because of their simplicity and reliability widely used. For this, the solids-laden gas is being fed into the centrifugal separator over tangential or spiral-shaped inlets, sometimes also with the help of guide vanes. In the cyclone, the gas moves helically downward, then in spiral orbits inward, and flows through the dip tube centrally back off (Fig. 10.1).

The separation process is being described vividly by Krambrock [10.1]. The circumferential velocity in the cyclone is constantly increasing from the outside to the inside up to the dip tube radius, and drops in the vortex core down to zero. The pressure decreases because of the curved flow with diminishing radius. In the axis of the cyclone there is a strong vacuum compared to the outlet tube. At the technically common circumferential speeds of 20 to 60 m/s and radiuses from 0.1 to 1 m, the

centrifugal acceleration is a hundred to a thousand times greater than the acceleration due to gravity, so that even small particles opposite to the inward directed flow be driven to the cyclone wall.

The boundary layer on the walls of the cyclone rotates due to the friction much more slowly than the main flow. Therefore, the there acting centrifugal forces are relatively small. Because, however, the pressure drop of the main flow is impressed on the boundary layer, it comes to strong, almost radially inward directed secondary currents on the cover and on the conical bottom. These secondary flows capture the segregated particles, and discharge them in the conical lower section of the apparatus, also when the cyclone lies or

"standing on its head". Characteristic are the product strands, which are being formed due to disturbances in the boundary layer in connection with the rotational flow. On the cyclone bottom transports the secondary flow likewise larger amounts of dust inward, which, however, during the flow around the dip tube will be again thrown to the wall of the cyclone.

Design and construction of a cyclone separator concerns first and foremost the pressure drop and the separation rate. Extensive theories of these complexes were established by Barth [10.2] and Muschelknautz [10.3]. However, regarding the separation efficiency are the circumstances in the technical practice mostly such, that the bulk material reaches the cyclone wall in unfractionated state, because the gas already in the conveyor pipe and in particular in the cyclone can carry only a certain solids portion. In case of solid/gas rates of µ = 0.1, as they are often present behind mills, dryers and Sifters, about 90 percent of the material is separated in this way [10.1]. The calculation on the basis of the fractional separation efficiency and the particle size distribution is valid only for the remaining portion of material.

For high separating performances are cyclones with large height and high inflow velocities necessary, with an adverse effect on facility and operating costs. Useful separation performances and acceptable pressure losses are achievable at inflow velocities between 7 and 20 m/s, in devices with a ratio between cyclone diameter D and dip tube diameter di of about 3, and a ratio between height hi and dip tube diameter di between 6 and 12. The pressure loss without taking into account the influence of solid material, can at least for such flow conditions and dimensions be described by a simple power law:

2

0.5 2

4.0

e

gid

ih

i

e uf

f

d

hp

i

i ⋅⋅

⋅⋅=∆

⋅ ρ (10-1)

Starting from the ratio between cyclone wall and dip tube diameter of 3, Eq. (10-1) describes exactly the conditions when changing the size of the cross-sectional area of the inlet. A confirmation of this offers Fig. 10.2, that depicts the results of studies, carried out by Gloger and Niendorf [10.4] using a model cyclone, in which the cross-sectional area of the inlet fe and thus the ratio to the cross sectional area of the dip tube fi, systematically has been varied. With the design data of a cyclone series with three different diving tube diameters, which before years was introduced by the BASF, could be verified, that the variation of the dip

Fig 10.1

Schematic representation of the speed and pressure

conditions in a cyclone separator according to Krambrock

[10.1]

66

tube cross-sectional area causes identical pressure losses compared to the calculation after the method of Muschelknautz.

Trouble-free operation of cyclone separators can be supported by specific constructive measures. Proven elements are for example collection container at the discharge of the cyclone. They prevent, that already secluded product passes into the area of the dip tube vortex. This effect can in vacuum systems be reinforced by the use of leak gas shocks. A cover cone in the collection reservoir removes the material finally permanently from the influence area of the vortex.

Many of the to be separated materials have a strong abrasive effect on the cyclone wall. Due to the high circumferential velocity in the lower area of the cyclone, this is the first affected part, later then the overlying area. A cylindrical instead of a conical shape of the cyclon can counteract this development. Low flow velocities and linings with cast basalt are likewise suitable, to hold the abrasive effect of solids in borders.

Literature of chapter 10.

[10.1] Krambrock, W.: Die Berechnung des Zyklonabscheiders und prak-tische Gesichtspunkte der Auslegung. Aufbereitungstechnik (1971)7, S.391-401.

[10.2] Barth, W.: Berechnung und Auslegung von Zyklonabscheidern auf-grund neuerer Untersuchungen. Brennstoff-Wärme-Kraft 8(1956)1, S.1-8.

[10.3] Muschelknautz, E.: Chemie-Ing.-Tech. 44(1972)1/2, S.63-71.

[10.4] Gloger, J.; und G.Niendorf: Untersuchungen an einem Modellzyklon über den Einfluß verschiedener geometrischer Parameter auf Ab-scheidegrad und Druckverlust. Chem. Techn. 22(1970)9, S.525-532.

Fig 10.2

Pressure loss in a cyclone with different inlet cross-

sectional areas depending on the gas inlet velocity:

comparison of measurements [10.4] with the calculation

according to Eq. (10-1)

1

Appendix

Industrial pneumatic conveying installations

A computer program, that includes a database, allows the quick and direct use of the here describt

algorithms for evaluation and design of processes, apparatuses and equipment. This free software is

written in Visual Basic and runs even under Windows 7 (32 bit).The german version can be

downloaded from the following link:

https://skydrive.live.com/redir.aspx?cid=0239e78e2f136dde&resid=239E78E2F136DDE!340

&authkey=s60rZ0M*8fA%24

An english version of the software can be downloaded from the following link:

https://skydrive.live.com/#cid=0239E78E2F136DDE&id=239E78E2F136DDE%21936

Literature researches, as they underlie this brochure, are despite the ubiquity of the Internet even today

only possible with the help of technical-scientific libraries. A godsend in this respect, is the for more

than 10 years existing bulk-online portal. In the forum on the subject of pneumatic conveying exists

meanwhile a real fund, that however can only with great effort be utilized. These data are the base for

the following considerations, which are based on my universal state diagram (Fig. 7.2) and my

software. The following link leads to the bulk-online portal:

http://forum.bulk-online.com/forumdisplay.php?11-Pneumatic-Conveying

2

( ) = calculated values, µ = solid /gas ratio

Table 1: Operating conditions of industrial installations

(1) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?4754-Pneumatic-Conveying-of-PP

In this installation is the measured pressure loss of 0.3 bar about 0.1 bar bigger than the calculated

value for the minimum pressure loss in dilute phase. This surplus is surely caused by the 8 (not

specified) pipe bends of the conveying line.

(2) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?3348-Conveying-of-LLDPE-Granules

These operating data fit extremely well in the scheme of the dense phase, although the particle size of

this material, in contrast to the fine-grained solids, which are commonly conveyed under such

conditions, lies in millimeter range. There were observed no general operational problems, except of

vibrations at pipe bends, which lead the flow vertically upwards. In regard to the terminal velocity of

particles with a size of 4 mm is the actual gas velocity too low. So, material can accumulate

themselves in the pipe bends and possibly develop a behaviour like a pulsating spouting bed, and thus

generate vibrations.

Nr. Product dp50

µm

L

m

h

m

D

mm

&Ms

t/h

&Mg

Nm3/h

∆∆∆∆p

bar

u

m/s

µ

-

1 PP pellets (1) (4000) 36.5 15.2 150 14 ca 2010 0.3 (30) (6)

2 LLDPE Granules (2) (mm) 100 -- 300 60 -- (0.41) (10.3) 20

3 Cement (3) -- 216 40 ca 285 85 ca 6000 1.6 -- --

4 Polycarbonat:

Powder & Pellets (4)

-- / 2500 50 + 60 15 + 20 100/125 6 / 7 ca 1700 1(0.73) (38) (3)

5 Expanded Perlite (5) 28.5 35.2 28.3 83 3,28 ca 280 0.25 -- (9.35)

6 PTA (6) 130 120 40 150 90 3000 2.66 (27.5) (24)

7 PP Pellets (7) -- 140 22 (187) 30 ca 2700 0.66 (21) (10)

8 Rapeseed (8) 2500 230 10 206 12 (2700) 0.4 (21.5) (3.6)

9 PET Virgin Chips (9) (>12700) 40 20 80 1.5 (ca 570) 0.15 (32) (2)

10 Cement (10) -- 176 -- 50 7.63 (233) 3.15 (18) (26)

11 Spent Cell Liner (11) -- 85 -- 77 2.5 (230) (0.3) (13) (9)

12 Cement (12) -- 132 55 250 120 ca 4200 1.7 (14.5) (24)

3

(3) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?7057-Pneumatic-Transport-Cement-Pipeline-Pigging

The operating data of this system offered the opportunity to determine the boundary of dense flow

more accurately, because the operating conditions are lying exactly on this line. For a gas temperature

of 125 °C and a bulk solid throughput of 85 t / h, the calculation on the basis of the improved

coordinates (no longer equal to those of the original state diagram) yields the following values: 7060

kg / h for the gas flow and 1.66 bar for the pressure loss. In this thread was also an increase to 100 t / h

solids throughput discussed; according to my calculations, however, this requires a gas flow of 7860

kg / h at a pressure loss of 1.92 bar, for which the installed compressor capacity does not suffice.

(4) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?21119-Increasing-the-Conveying-Capacity

(5) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?16367-Optimisation-of-a-Pneumatic-Conveying-Line

After the program's calculation lies the operating point of this installation on the boundary of dilute

phase, the so-called saltation line.

(6) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?7259-Conveying-of-PTA

After the calculation lies the operating point of this installation also on the boundary of dilute phase,

the so-called saltation line. The operational pressure loss is equal to the calculated value, when a

(unconfirmed) gas temperature of 90 °C is used.

4

(7) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?4797-PP-Pellet-Conveying-Problem

This operating point lies close to the so-called saltation line of the dilute phase. For this operation

conditions yields the calculation a slightly smaller gas flow rate and looks as follows:

The average gas velocity is much smaller than for the conveying of PP pellets in the installation No. 1

of Table 1.

(8) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?5655-Pneumatic-Conveying-of-Rapeseed

This is another good example, to verify the calculated results. For a bulk solid throughput of 12 t / h

yields the calculation for the condition of minimum pressure loss in dilute phase a value of 0.36 bar.

This value lies near the pressure difference of 0.4 bar of the blower, what the possible solids

throughput accordingly limits.

5

(9) Data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?4794-Why-Is-Throughput-Reduced

As usual for bulk materials with large particle diameters, the gas velocity is much higher than the

calculated value for the minimum pressure loss in dilute phase: 32 m/s instead of 14 m/s. However, the

bulk solid throughput is too small for such a gas velocity, and I am also of the opinion, that the reason

for this is the poorly functioning material feed. The fan is designed for pressures up to 0.5 bar at a gas

flow rate of 700 m3 / h and would be able, to handle a much higher bulk solid throughput. The

calculation result for the current bulk solid throughput of 1.5 t / h is as follows:

(10) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?5597-Typical-Solids-Friction-Factor-for-Cement

The calculation results for the saltation condition on the boundary of the dilute phase for a

(unconfirmed) gas temperature of 110 °C are identical to the operating data:

6

(11) For completeness, here are added the operating conditions for the conveying of used, very

abrasive cell liner:

http://forum.bulk-online.com/showthread.php?5437-Lean-Phase-Conveying-of-Abrasive-Cell-Liner

The information for this system are not complete, However, the calculated pressure loss of 30 kPa for

the saltation conditions fits quite well to the blower specification of 40 or 60 kPa.

(12) data set coming from the following bulk-online thread:

http://forum.bulk-online.com/showthread.php?3709-Determinining-Pneumatic-Conveying-Parameters

The operating point lies a little left hand from the boundary of the dense phase. The estimated values

for the parameters K and Re of the state diagram are 450 and 450,000. The calculation results for gas

flow rate and the pressure drop confirm the assumptions.

7

State diagram

The main concern of the state diagram is the universal and clear presentation of the operating

data of industrial conveying installations with their large number of parameters. For this

purpose a large number of published data from industry and research was added into the

coordinate system. They form the skeleton of the state diagram and reveal the areas of dilute

phase and dense phase conveying as well as their boundaries. There are three characteristic

conveying states, that lie in the diagram on particular lines. Two of the states are the pressure

loss minimum in dilute phase as well as the boundary to the transition area, the so-called

saltation (choking) line. The third one is the boundary of dense phase on the other side of the

transition area, the position of which was determined by the help of the operational data No. 3

in Table 1 more accurately. From the data of dense phase conveying was derived in addition

an estimation for frequently used pipe diameters as ratio to the bulk material throughput.

In order to facilitate the estimation of the sizes of conveying lines, the software helps, as

mentioned, in determining a usable pipe diameter. Furthermore, can be determined on the

basis of the design data, the relevant parameters for the three characteristic conveying states.

The respective values are being calculated by means of simple mathematical equations for

the respective straight lines. For all other possible operating points, the respective parameters

can be read from the state diagram. However, the left border line of the display, which

represents the boundary of the bulk solid movement, is only indicated, but may be precised, as

described above.

For the calculations of conditions in dilute phase uses the program its default pipe diameter

and corrects the calculation result in the case of deviations from the actual diameter. This

correction has been found to be necessary during the recalculation of the used literature data, ,

but it is surely not particularly sophisticated. For bulk solids with big particles is the gas

8

velocity in some cases bigger than the velocity at the point of minimum pressure loss in dilute

phase: for example, 38 m / s instead of 21 m / s for the installation No. 4 (2.5 mm) in Table 1.

In dense phase the operating points are distributed over the entire range. This opens up the

possibility to draw lines of constant bulk solids throughput. Originally, the maximum bulk

solid throughput was 19.5 t / h. Fortunately, the installation No. 3 offered the opportunity for

an expansion of the throughputs and the accurate determination of the position of the

boundary of the dense phase region, because this operating point lies exactly on this line. As

already described above, yields the calculation on the basis of the new coordinates (which are

no longer those of the original in the state diagram registered correspond) for a gas

temperature of 125 °C and a bulk solid flow of 85 t / h the following result: 7060 kg / h for

the gas flow and 1.66 bar for the pressure loss.

Once again here is the attention being focused on the installation No. 2 in Table 1. These

operating data fit extremely well in the scheme of the dense phase, although the particle size

of this material, in contrast to the fine-grained solids, which are commonly conveyed under

such conditions, lies in millimetre range. There were observed no general operational

problems, except of vibrations at pipe bends, which lead the flow vertically upwards. In

regard to the terminal velocity of particles with a size of 4 mm is the actual gas velocity too

low. So, material can accumulate themselves in the pipe bends and possibly develop a

behaviour like a pulsating spouting bed, and thus generate vibrations.

*) In the case of horizontal conveyor lines with vertical pipe sections for the steady conveying

state an equivalent length must be used. For this, the value for the whole vertical pipe length

is being enlarged by a factor between 1.7 and 2.0.

**) In the evaluated data of industrial installations was no evidence of additional influences,

which are dependent on the conveyed bulk solids. The only relevant parameter is the material

throughput. Also no other physical similarities could be discovered, which are connected with

the bulk solid properties in some way and possibly exist in form of hidden parameters. Even

the oft-cited solid /gas ratio turned out to be unusable. A factor, which is in this context

closest to a parameter, is the bulk solid throughput related to the cross-sectional area of the

pipe. This fact possibly shows a connection with the Froude number according to Eq. (5-4).

9

This is the scetch for the dense phase area of the state diagram, which was improved by using

the here analyzed operating data . The on the border to the transition area lying endpoints of

the lines of equal bulk solid throughput were calculated by the appropriate modified software

program.

10

Pneumatic conveying at high temperatures

Operating data for industrial pneumatic conveyor lines that are operated at high gas

temperatures are to be found at the following Internet address:

http://www.enviro-engineering.de/pdf/InjektorenBauformenVariantenUndAnwendungen.pdf

( ) = Calculated values, µ = solid / gas ratio

Table 2: Operating data of industrial installations with high gas temperatures

(1) As in other installations for the transport of bulk materials with large particles, the gas

velocity in this conveying line is by a factor of 1.6 higher than those, calculated for the

minimum pressure loss in dilute phase. The extrapolation from the calculated operating point

of minimum pressure loss gives values of 100 000 and 25 for the parameters Re and K. The

calculation results look like this.

Nr. Product dp50

µm

L

m

h

m

D

mm

&Ms

t/h

&Mg

Nm3/h

∆∆∆∆p

bar

u

m/s

µ

-

1 Ash (1) / 80 °C 0-3000 20 25 80 0.6 ca 350 (0.14) (23.6) (ca 1.5)

2 Ash (2) / 250 °C 0-1000 11 15 178 6.5 ca 750 (0.133) (16.5) 6.5

3 Cement (3) / 125 °C -- 216 40 ca 285 85 ca 6000 1.6 -- --

11

(2) The data of this conveying system describe the operating condition at a gas temperature

of 250 °C, which due to a longer cooling period of the ash under the actual operating

temperature of 450 °C lies. The measured ash throughput of 6.5 t / h is assessed as being

surprisingly high. The calculation for this condition results in a pressure loss of 0.133 bar at a

gas flow of approximately 1000 kg / h.

The following calculation results describe now the situation at the actual operating

temperature of 450 °C and at the same pressure loss. The gas flow rate lies at about 900 kg /

h, while the ash throughput at this gas temperature is reduced to a value of 3500 kg / h; this

value corresponds exactly to the design conditions of the conveying line.

(3) This is once again the installation No. 3 of Table 1 with the corresponding interpretation: The

operating data of this system offered the opportunity to determine the boundary of dense flow more

accurately, because the operating conditions are lying exactly on this line. For a gas temperature of

125 °C and a bulk solid throughput of 85 t / h, the calculation on the basis of the improved coordinates

(no longer equal to those of the original state diagram) yields the following values: 7060 kg / h for the

gas flow and 1.66 bar for the pressure loss. In this thread was also an increase to 100 t / h solids

throughput discussed; according to my calculations, however, this requires a gas flow of 7860 kg / h at

a pressure loss of 1.92 bar, for which the installed compressor capacity does not suffice.