toolbox 9 heat transfer in a thermal · pdf filetoolbox 9 heat transfer in a thermal apparatus...

APPLIED INDUSTRIAL ENERGY AND ENVIRONMENTAL MANAGEMENT

Z. K. Morvay, D. D. Gvozdenac

Part III:

FUNDAMENTALS FOR ANALYSIS AND CALCULATION OF ENERGY AND

ENVIRONMENTAL PERFORMANCE

1

Applied Industrial Energy and Environmental Management Zoran K. Morvay and Dusan D. Gvozdenac © John Wiley & Sons, Ltd

Toolbox 9

HEAT TRANSFER IN A THERMAL APPARATUS

DIMENSIONLESS NUMBERS

1. Referent temperatures for calculations. It is very important for proper calculation that

temperature and other fluid properties, especially pressure, are defined precisely. The temperature

definitions relevant to calculations are as follow:

t1 = temperature related to fluid at the inlet conditions, [oC]

t2 = temperature related to fluid at the outlet conditions, [oC]

tw = temperature of the wall, [oC]

tc (t∞) = temperature of the fluid flow core, [oC]

tf = temperature of the fluid calculated as: 2

ttt 21

f , [oC]

tm = temperature of the fluid calculated as: 2

ttt5.0t 21

wm , [oC]

2. The Archimedes Number (Ar) states the ratio of gravitational force and viscous force:

fs2

3s Lg

Ar (9.1)

Where:

g = gravity acceleration, [m/s2]

L = characteristic length, [m]

= dynamic viscosity, [Pa s]

f = fluid density, [kg/m3]

s = solid density, [kg/m3]

3. The Biot Number (Bi) states the ratio of internal thermal resistance and surface film resistance:

k

xhBi (9.2)

Part III – Toolbox 9:

HEAT TRANSFER IN A THERMAL APPARATUS 2

Where:

x = mid-plane distance, [m]

h = heat transfer coefficient, [W/(m2 K)]

k = fluid thermal conductivity, [W/(m K)]

4. The Colburn-Chilton Factor (jh) is used in heat transfer in general. It is equivalent to (St Pr2/3

):

3/2

p

p

h

3/2

p

p

hk

c

mc

hjor

k

c

wc

hj (9.3)

Where:


cp = isobaric specific heat, [kJ/(kg K)]

= fluid density, [kg/m3]

w = fluid velocity, [m/s]



m = fluid mass velocity, [kg/(m2 s)]

5. The Euler Number (Eu) states the ratio of friction head and velocity head:

2w

pEu (9.4)

Where:

p = pressure drop, [Pa]


w = velocity, [m/s]

6. The Fanning Friction Factor (f) states the ratio of sheer stress at pipe and conduit wall as a

number of velocity heads:

L

p

w2

dfor

Lw2

pdf

22 (9.5)

Where:

d = characteristic length, [m]

p = pressure drop, [Pa]


w = velocity, [m/s]

L = length, [m]

p/ L = pressure drop per unit length, [Pa/m]

7. The Fourier Number (Fo) is used in unsteady heat transfer calculations. It characterizes the speed

of temperature field changes during unsteady heat transfer:

22p L

FoorLc

kFo (9.6)



Where:


= thermal diffusivity, [m2/s]

= time, [s]




8. The Froude Number (Fr) states the ratio of inertial force and gravity force:

Lg

wFror

La

wFr

22

(9.7)

Where:

a = acceleration, [m/s2]

g = gravitational acceleration, [m/s2]


w = velocity, [m/s]

9. The Galileo Number (Ga) states the ratio of gravity force and viscous force:

2

23dgGa (9.8)

Where:





10. The Grätz Number (Gz) states the ratio of thermal capacity and convective heat transfer:

p

p

cwd

k

d

LGzor

Lk

cMGz (9.9)

Where:

M = mass flow rate, [kg/s]

d = diameter, [m]






11. The Grashof Number (Gr) states the ratio of buoyancy force and viscous force:

2

3

2

23 TgLGror

TgLGr (9.10)

Where:






= coefficient of volume expansion of fluid, [1/K]

T = temperature difference, [K]


= kinetic viscosity, [m2/s]

12. The Lewis Number (Le) is used in combined heat and mass transfer calculations:

DLeor

cD

kLe

p

(9.11)

Where:





= thermal diffusivity, [m2/s]

D = diffusivity, [m2/s]

13. The Mass Transfer Factor (jm) is used in mass transfer calculations:

3/2

cm

Dw

kj (9.12)

Where:


kc = diffusion rate, [m/s]




14. The Nusselt Number (Nu) states the ratio of total heat transfer and conductive heat transfer:

k

dhNu (9.13)

Where:




15. The Peclet Number (Pe) states the ratio of bulk heat transfer and conductive heat transfer:

a

wdPeor

k

cmdPeor

k

cwdPe

pp (9.14)

Where:









a = thermal diffusivity, [m2/s]

16. The Prandtl Number (Pr) states the ratio of momentum diffusivity and thermal diffusivity:

k

cPr

p (9.15)

Where:




17. The Reynolds Number (Re) states the ratio of inertial force and viscous force:

mdReor

wdRe (9.16)

Where:






18. The Schmidt Number (Sc) states the ratio of kinetic viscosity and molecular diffusivity:

DSc (9.17)

Where:

= kinetic viscosity, [m2/s]


19. The Sherwood Number (Sh) states the ratio of mass diffusivity and molecular diffusivity:

D

LkSh c (9.18)

Where:


kc = diffusion rate, [m/s]




20. The Stanton Number (St) states the ratio of heat transfer and thermal capacity of fluid:

mc

hStor

wc

hSt

pp

(9.19)

Where:






21. The Weber Number (We) states the ratio of inertial force and surface tension force:

22 md

Weorwd

We (9.20)

Where:





= surface tension, [N/m]

CORRELATIONS FOR CONVECTIVE HEAT TRANSFER

1. Forced Convection Heat Transfer. Forced convection implies a process in which heat is

transferred from a flowing fluid to a solid surface which represents one of its boundaries in space. A

device producing pressure difference causes the flow. In addition to geometrical factors, thermo

physical properties and the ratio between fluid temperature and solid surface, the speed of undisturbed

fluid core can be considered as a condition for producing uniform solutions of general differential

equations for energy conservation and momentum equations.

2. Forced Convection Flow over a Flat Plate

Laminar flow (Ref < 5×105 and 0.6 < Prf < 10). The geometrical dimension is the length of

the flat plate in flow direction. Wall temperature is constant.

n

w

f33.0f

5.0ff

Pr

PrPrRe664.0Nu (9.21)

n = 0.25 for fluid heating and n = 0.19 for fluid cooling.



Laminar flow (Ref < 5×105 and 0.6 < Prf < 10). The geometrical dimension is the length of the

flat plate in flow direction. Heat flow is constant.

n

w

f33.0f

5.0ff

Pr



Fully developed turbulent flow (Ref > 5×105 and 0.7 < Prf < 200). The geometrical dimension

is the length of the flat plate in flow direction.

Liquids: 25.0

w

c43.0c

8.0cc

Pr


Gases: 78.0

mmm PrRe057.0Nu (9.24)

3. Forced Convection Flow inside Circular Tubes and Ducts

Laminar viscous and gravitational flow within straight smooth pipes (Ref < 2300)

n

w

f1.0f

43.0f

33.0flf

Pr

PrGrPrRe15.0Nu (9.25)


Table 9.1: φl for Laminar Flow

l/d 1 2 5 10 15 20 30 40 50

φl 1.98 1.70 1.44 1.28 1.18 1.13 1.05 1.02 1.00

Transient flow within straight smooth pipes (2300 < Ref < 10,000)

25.0

w

f43.0fof

Pr

PrPrKNu (9.26)

Table 9.2: Ko for Transient Flow

Re 2100 2300 2500 3000 3500 4000 5000 6000 7000 8000 9000 10 000

Ko 1.9 3.3 4.4 6.0 10.0 12.2 15.5 19.5 24.0 27.0 30.0 33.0

Turbulent flow in straight smooth pipe or duct (10,000 < Ref < 5,000,000 and 0.6 < Prf <

2,500)

25.0

w

f43.0f

8.0flf

Pr




Table 9.3: φl for Turbulent Flow

Ref l/d

1 2 5 10 15 20 30 40 50

10 000 1.65 1.50 1.34 1.23 1.17 1.13 1.07 1.03 1.00

20 000 1.51 1.40 1.27 1.18 1.13 1.10 1.05 1.02 1.00

50 000 1.34 1.27 1.18 1.13 1.10 1.08 1.04 1.02 1.00

100 000 1.28 1.22 1.15 1.10 1.08 1.06 1.03 1.02 1.00

1 000 000 1.14 1.11 1.08 1.05 1.04 1.03 1.02 1.01 1.00

Defining the geometrical dimension is diameter (d) for pipes and equivalent diameter (P

A4deq ; A

= flow cross sectional area and P = flow section perimeter regardless of what part of the perimeter

participates in heat transfer).

For bent tubes, the value of the heat transfer coefficient (h) has to be multiplied by the coefficient

ζ taking into account the relative curvature of the coil:

D

d54.31 (9.28)

where d is diameter of a tube and D is diameter of a coil turn.

4. Forced Convection Flow inside Concentric Annular Ducts, Turbulent Flow (Re > 2300)

Di

Do

Figure 9.1:Concentric Annular Pipe

ioh DDD

mh wDRe

k

DhNu h

Fluid properties have to be calculated for mean bulk temperature.

(a) Heat transfer occurs at the inner wall. The outlet wall is insulated.

16.0

i

o

TUBE D

D86.0

Nu

Nu (9.29)

(b) Heat transfer occurs at the outlet wall. The inner wall is insulated.

6.0

o

i

TUBE D

D14.01

Nu

Nu (9.30)

(c) Both walls participate in heat transfer and the wall temperatures are the same.



o

i

6.0

o

i

84.0

i

o

TUBE

D

D1

D

D14.01

D

D86.0

Nu

Nu (9.31)

The characteristic length for non-circular ducts is defined as follows (if not specified otherwise):

P

A4d c

e (9.32)

Where:

P = wetted perimeter of the duct (length of boundary), [m]

Ac = flow cross-sectional area, [m2]

5. Forced Convection Flow Perpendicular to Banks of Smooth Tubes

Re < 1,000

D

sL

sW

um

Figure 9.2: Square Pitch

D

sL

sW

um

Figure 9.3: Triangular Pitch

Ref < 1000 25.0

w

f36.0f

5.0ff

Pr

PrPrRe56.0Nu

Ref > 1000 25.0

w

f33.0f

65.0ff

Pr

PrPrRe26.0Nu

Ref > 1000 25.0

w

f33.0f

6.0ff

Pr

PrPrRe41.0Nu

The flow velocity is defined for the narrowest bank section.

Table 9.4: Coefficient φ (Angle of Attack)

Angle 90 80 70 60 50 40 30 20 10

φ 1 1 0.98 0.94 0.88 0.78 0.67 0.52 0.42



Figure 9.4: Shell & Tube Heat Exchanger

When equations from this paragraph are used for

shell & tube heat exchangers with baffles (Fig.

9.4) the coefficient φ is 0.6.

6. Heat Transfer in Flow around Bank of Transverse Finned Pipes (Fig. 9.5)

)8.4t/d3;000,25Re500,2(PrRet

h

t

dCNu f

4.0f

nf

14.054.0

f (9.33)

Where:

d = external diameter of the tube, [m]

D = diameter of the fin, [m]

t = pitch of the fin, [m]

h = height of the fin, [m]

Square pitch banks: C = 0.116; n = 0.72

Triangular pitch banks: C = 0.25; n = 0.65

This formula gives the heat transfer coefficient (h) which has to be multiplied by 0.73 to get the

so-called reduced heat transfer coefficient (hred). This reduced heat transfer coefficient is used in the

equation for overall heat transfer coefficient which is related to the total external heat transfer area.

The overall heat transfer coefficient is as follows:

w

in

ex

inred

RA

A

h

1

h

1

1K

(9.34)

Where:

Aex = total external surface and finned tube per unit length

Ain = internal tube surface

h2 = heat transfer coefficient of the fluid inside the tube, W/(m2 K)

wr = sum of thermal resistance of the wall and existing scaling

D

d t

h

δ

Figure 9.5: Transverse Finned Tube

7. Heat Transfer in Stirring Liquids with Agitator. The characteristic correlation for determination

of the heat transfer coefficient in an apparatus with a coil (Fig. 6) or jacket (Fig. 7) and an agitator can

be calculated by the equation:



1

k

w

mnm

mmm SPrReCNu (9.35)

Where

d

DS;

dnRe;

k

DhNu

m

2m

mm

m

D = inside diameter of vessel, m

d = diameter of the circle covered by the agitator, m

n = speed of the agitator, rpm

The values of the remaining physical properties have to be taken at the average liquid temperature

in the vessel.

Figure 9.6: Coil Type

D

d

Figure 9.7: Jacket Type

The mean values of exponents for turbulent flow are m = 0.67, n = 0.33, k = 0.14. For high

viscosity fluids (non-Newtonian) these values are: m = 0.50, n = 0.24 - 0.33, k = 0.14 – 0.24.

The constant C depends mostly on the type of agitator and its geometrical characteristics.

a. Coil Type of Vessel:

C = 0.87 – 1.0

b. Jacket Type of Vessel:

C = 0.73 – 0.74 (Ref > 400)

(turbine agitator with side bumper)

C = 0.54 – 0.64

(propeller agitator without side bumper)

C = 0.38 – 0.55

(frame agitator)



C = 0.36 - 0.40

(agitator with flat paddle)

C = 0.40 - 0.53

(turbine closed type agitator with angled blades)

8. Natural Convection. The flowing field in natural convection is determined exclusively by

temperature field. Fluid motion occurs as a consequence of the different densities of certain

macroscopic fractions arising from the difference in their temperatures. Heat transfer correlations do

not contain velocity in this case. As a condition for the same solution of general differential equations

including boundary conditions on a solid surface, there is also the shape and a size of body, the

distribution of temperature along its surface, a temperature difference between a surface and

undisturbed fluid, the thermal and physical properties of fluid and gravitational acceleration.

In the majority of cases, the thermal and physical properties of fluid refer to the middle fluid

temperature calculated as 2

ttt fw

m , [oC]. When defining the Nusselt number, the coefficient of

heat transfer (h) is considered to be the middle one for the total heat transfer surface area.

Heat transfer outside horizontal tubes (cylinders). The characteristic length is the tube’s

(cylinder) outside diameter. The equations are valid for diameters of less than 200 mm and for

gases and liquids.

25.0

mmm PrGr53.0Nu 9m 10Pr)Gr(

(9.36) 3/1

mmm PrGr10.0Nu 9m 10Pr)Gr(

Vertical flat plate and cylindrical surfaces. The characteristic length is the height of the

plate or cylinder which is valid for gases and liquids.

45.0Num 3m 10Pr)Gr(

(9.37)

8/1mmm PrGr18.1Nu 500Pr)Gr(10 m

3

25.0mmm PrGr54.0Nu )10(Pr102Pr)Gr(500 m

7m

25.0mmm PrGr65.0Nu )10(Pr102Pr)Gr(500 m

7m

3/1mmm PrGr129.0Nu 7

m 102Pr)Gr(

The defining dimension is the diameter for the horizontal tube and the height for the vertical

surface.



Horizontal plate – heat transfer surface facing up

(a)tf

tw > tf

tw

(b)

tw

tf

tw < tf

nmmm PrGrCNu (9.38)

mPrGr C n

105 - 2·10

7 fw tt 0.54 0.25

2·107 - 3·10

10 fw tt 0.14 1/3

105 - 2·10

7 fw tt 0.25 0.25

If the plate is rectangular, the characteristic length (L) is the shorter side (if L < 0.6 m). If L > 0.6

m, increased plate dimensions do not influence the heat transfer coefficient.

Horizontal plate – heat transfer surface facing down

(a)tf

tw < tf

tw

(b)

tw

tf

tw > tf

mPrGr C m

105 - 2·107 fw tt 0.25 0.25

105 - 2·107 fw tt 0.54 0.25

2·107 - 3·1010 fw tt 0.14 1/3

If the plate is rectangular, the characteristic length (L) is the shorter side (if L < 0.6 m). If L > 0.6 m,

increased plate dimensions do not influence the heat transfer coefficient.

9. Heat Transfer in Thermal Radiation of Solids

Heat passing from a hotter body to a colder one by radiation is defined as follows:

42

4121radiation TTAWQ (9.39)

where:

Qradiation Heat energy transmitted by radiation, W

A Area of radiating surface, m2

W1-2 Power of radiation, W/(m2 K

4)

T1 Hotter body surface temperature, K

T2 Colder body surface temperature, K

φ Dimensionless angular coefficient



The power of radiation depends on (1) mutual arrangement and (2) emissivity of radiating

surfaces having temperatures T1 and T2.

If a body with radiating surface A1 is totally within a body with radiating surface A2 then

A = A1 and angular coefficient φ = 1 and the power of radiation is as follows:

bb12

1

1

21

W

1

W

1

A

A

W

1

1W

(9.40)

where:

W1 = ε1·Wbb Radiating power of smaller body

W2 = ε2·Wbb Radiating power of larger body

ε1 and ε2 Emissivities of the surface of the smaller and larger bodies,

respectively. The emissivity for some materials is given in the Table

below.

Wbb Radiating power of a black body, = 5.6687·10-8

W/(m2 K

4)

Material Emissivity

(ε)

Material Emissivity

(ε)

Material Emissivity

(ε)

Aluminum 0.05–0.07 Iron, galvanized 0.27 Varnish 0.80–0.98

Asbestos 0.96 Iron (steel), oxidize 0.74–0.96 Varnish, aluminum 0.4

Brick masonry 0.93 Iron (rough), oxidize 0.85 Water 0.93

Copper 0.30–0.87 Lead 0.28 Wood, dressed 0.90

Glass 0.94 Oil paint 0.78–0.96

Gypsum 0.78–0.90 Plaster 0.93

If area A2 is much bigger compared to area A1, the ratio A1/A2 in equation (9.39) above is

close to zero, and the radiation power becomes W1-2 = W1.

If A1 = A2, the radiation power is as follows:

bb11

21

W

1

W

1

W

1

1W

(9.41)

The summary coefficient of radiation and convection heat transfer is:

conrad hhh (9.42)

where

21

42

4121

21

radiationrad

TT

TTW

ATT

Qh (9.43)

and hcon (coefficient of convection heat transfer) is defined by the relevant equation.

The following approximate formula is used to calculate the heat losses of apparatus in closed

premises when the temperature of the apparatus surface is less than 150 oC:

t07.074.9h (9.44)



h Coefficient of heat transfer by radiation and convection, W/(m2 K)

Δt Temperature difference of the apparatus surface and surrounding air, K

References Dimic, M. (1973) Kriterijalne jednacine za konvektivni prelaz toplote i materije (Correlations for

Convective Heat and Mass Transfer), Novi Sad (in Serbian).

Nashchokin, V.V. (1979) Engineering Thermodynamics and Heat Transfer, Mir Publishers

Moscow.

Pavlov, K.F., Romankov, P.G., Noskov, A.A. (1976) Examples and Problems to the Course of Unit

Operations of Chemical Engineering, Mir Publishers, Moscow.

Spang, B. Correlations for Convective Heat Transfer, http://www.cheresources.com.

Thomas, L.C. (1992) Heat Transfer, Prentice Hall.

www.processassociates.com/process/dimen/dn_all.htm

http://www.cheresources.com/

http://www.processassociates.com/process/dimen/dn_all.htm

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