chapter 4 thermal effectiveness of a spiral plate heat...

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CHAPTER 4

THERMAL EFFECTIVENESS OF A SPIRAL PLATE

HEAT EXCHANGER

4.1 INTRODUCTION

This chapter discusses the study of thermal effectiveness of the

spiral plate heat exchanger with liquid-liquid two-phase mixtures. Thermal

effectiveness is a measure of heat transfer efficiency of the exchanger.

It depends on the exchanger geometry, the properties of the hot and cold

fluids used, and on the range of operating conditions. Quantifying thermal

effectiveness of a heat exchanger is critical in designing the exchanger for

different applications. To determine the thermal effectiveness of a heat

exchanger, a geometric model of the exchanger has to be developed and a

model of heat transfer in the exchanger has to be developed. The model

development that follows is that of Bes and Roetzel (1993), which has been

used to study the thermal effectiveness of the spiral plate heat exchanger

involving heat transfer to liquid-liquid mixtures.

4.2 THERMAL EFFECTIVENESS

Thermal effectiveness measures the efficiency of heat transfer in

a heat exchanger. There are different definitions for the thermal

effectiveness, such as the ε-effectiveness and the P-effectiveness. Both are

56

usually studied as a function of the mean number of transfer units (NTU).

NTU measures the amount of heat to be transferred from the hot side to the

cold side for unit temperature difference in the fluid. Depending on the

length of the heat exchanger needed to accomplish this temperature

difference, the exchanger may be sized. The basic definition of thermal

effectiveness is the ratio of temperature difference accomplished in one of

the sides to the maximum span that can be accomplished. Thus, for the hot

side and the cold side, thermal effectiveness is defined as:

1c1h

1c2cc

1c1h

2h1hh

TT

TTP;

TT

TTP

−

−=

−

−= (4.1)

Thermal effectiveness is a function of NTU, heat capacity ratio,

and the flow arrangement (curvature). Further, the flow patterns in

two-phase flows can also significantly affect the thermal effectiveness of the

exchanger. It is of interest to evaluate whether a purely thermal theory for a

spiral plate heat exchanger such as the ones derived by Bes and Roetzel

(1993) and Martin (1992) is sufficient to reasonably capture the thermal

performance of the exchanger. In this chapter the thermal theory of Bes and

Roetzel (1993) is presented first, and expressions are derived for the thermal

effectiveness of the hot side and cold side. Using the experimental data and

the model equations, thermal effectiveness for hot and cold sides are

calculated and the trends analyzed. Thermal effectiveness is also calculated

from the inlet and outlet temperatures using the definition provided in

Equation 4.1. The predictions of the model are compared against

experimental data and the limitations of a purely thermal theory in

predicting the performance of a spiral plate heat exchanger with immiscible

liquid mixtures are examined.

57

4.3 THERMAL THEORY OF A SPIRAL PLATE HEAT

EXCHANGER

Bes and Roetzel (1993) have developed the thermal theory of

spiral plate heat exchangers wherein they have derived analytical

expressions for the LMTD correction factor and the temperature

effectiveness (P) of the exchanger. In what follows, the derivation of Bes

and Roetzel is presented. Direct mathematical manipulations are eschewed

as the original source is accessible.

4.3.1 Assumptions Involved

• The model applies only for the middle part of a spiral plate heat

exchanger, where heat is transferred to cold fluid from both walls.

Therefore the model works better for a spiral heat exchanger with

a large number of turns. In this study, the number of turns n = 60,

large enough for end-effects to be neglected.

• Flow is assumed to be countercurrent, and a correction factor F is

introduced for deviation from countercurrent flow.

• Flow is assumed to be steady.

• The exchanger geometry is assumed to be an Archimedean spiral.

This implies that the radius of the spiral increases, continuously

with the angle ϕ as:

( ) ,rb

r mm +ϕ−ϕπ

= (4.2)

58

where b - plate spacing, rm - minimum radius, and ϕm - angle at

which the minimum radius is achieved. Therefore, temperatures

may be expressed as a function of either radius or the angle.

• Fluids are assumed to be completely mixed in the radial and axial

directions within the flow channel. Thus in a cross-section chosen

at a fixed angle, the temperature changes step by step from

channel to channel.

• Hot fluid enters the exchanger in the centre of the apparatus and

cold fluid flows in at the outermost channel.

• Overall heat transfer coefficient U is constant throughout the

exchanger.

• There are no heat losses to the environment.

• Distance studs are not taken into account.

4.3.2 Thermal Theory

In this model, the mean temperature difference between the fluids

on both sides of a partition wall is tracked as a function of position in the

exchanger. To construct an Archimedean spiral plate exchanger, a double

slit is wound in a spiral fashion. Therefore, each “radius” corresponds to two

channels of rectangular cross sections. The ‘inner’ spiral is called the main

spiral, and hot fluid flows through it. The second spiral (made from the

‘outer’ slit) constitutes the side spiral, and the cold fluid flows

countercurrently through it. Thus, for each radius, two temperature

differences may be defined, as shown for an elementary wedge in

59

Figure 4.1. Both temperature differences are positive in the spiral plate heat

exchanger. Further, these differences of temperature depend upon each

other; however, for convenience they will be separately derived as two

different quantities and will later be combined with each other.

Figure 4.1 Temperature differences in an elementary wedge

of a spiral plate heat exchanger

4.3.3 Energy Balance Equations

Let the heat capacity of the hot fluid be Ch. For the part of the

channel between the radii r – 1 and r, the hot fluid flows in the counter

clockwise direction. Since ϕ is also measured counterclockwise, the flow

direction of the hot fluid is the positive direction of ϕ. This hot fluid loses

heat to the cold fluids in the channels between r − 1 and r − 2, and r and

r + 1. Denoting the channel between r – 1 and r as j, the above statement

may be rewritten as: the hot fluid in j loses heat to the cold fluid in j – 1 and

j + 1. Let the temperature of the hot fluid be denoted Th and that of the cold

fluid be denoted TC. In order to indicate the temperature dependence on the

60

position of the fluid in the exchanger, the temperature of the hot fluid in j is

denoted as ( ))21(rTh − , and that of the cold fluid in j + 1 is denoted as

( ))21(rTh + . Thus, the rate of change of heat transfer from the hot fluid in j

to the cold fluid in j + 1 with respect to the angle ϕ is given by:

+−

−=∆=+

2

1rT

2

1rThrwThAq ch1j,j (4.3)

Here the heat transfer area is given by A = (height of the

exchanger) × (arc length of the curved channel). This arc length is given at a

wedge by: ϕ= r1 Thus, the rate of change of heat transfer with respect to ϕ

results in the above expression, as a first approximation. Similar expressions

may be written for rates of heat transfer at other locations in the exchanger.

Performing an energy balance on the hot fluid at j whose temperature is

Th(r – ½), one obtains the equation

1j,j1j,j

hc

h qqd

2

1rd

C +− +=ϕ

−

− (4.4)

A similar energy balance for the cold fluid in j + 1 gives

2j,1j1j,j

c

c qqd

2

1rdT

C +++ +=ϕ

+

− (4.5)

It is convenient to change slightly the notation of temperatures.

The function of temperature

−

2

1rTh

and

+

2

1rTc

will be further denoted

as Th(r) and Tc(r), respectively. Instead of expressing the rate of change

61

with respect to the angle ϕ, one may convert this into the change with

respect to r. From Equation 4.2, we have drb

dπ

=ϕ .Thus, in the new

notations, Equations 4.4 and 4.5 become:

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

−−−+

−=

π−

2rTrT1r

rTrTrkWb

dr

rdTC

ch

chhh

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

−++

+−=

π−

rT2rT1r

rTrTrkWb

dr

rdTC

ch

chcc (4.6)

The boundary conditions are obtained from the inlet and outlet

temperatures of the hot and cold fluids, respectively in the centre of the

exchanger. These two equations contain two kinds of temperature

differences (see Figure 4.1):

1. Local temperature difference between the fluids flowing on both

sides of the main spiral: )r(T)r(T)r( ch −=∆

2. Local temperature difference between the fluids flowing on both

sides of the side spirals: )1r(T)1r(T)r( ch −−+=δ

Defining a mean heat capacity of the hot and cold fluids as

chCCC = , cross-sectional mean number of transfer units (NTU) as

C/hA2 cπ , the heat capacity ratio as c

h

C

CR = , and rearranging, the two

Equations in 4.6 may be recast as:

62

0)1r()1r()r(rdr

)r(dTR2 h =−ψδ−+∆ψ+

0)1r()1r()r(rdr

)r(dTR2 c =+ψδ++∆ψ+ (4.7)

To obtain the energy balance of the main and side spirals in terms

of the above defined local temperature differences, the derivatives have to be

converted to those in ∆(r) and δ(r). This involves algebraic manipulation of

the two Equations in 4.7, and expanding terms in Taylor’s series to second

derivative (Bes and Roetzel 1993). It turns out to be convenient to define a

new independent variable x which depends on r, as

rR

1R

2x

+ψ= (4.8)

The simplified final form of the two equations of ∆ and δ as a

function of x is:

( ) ( ) ( )[ ] ( )[ ]0

dx

xxdxxx

dx

xd=

δ−δ+∆µ−

∆

( ) ( ) ( )[ ] ( )[ ]0

dx

xxdxxx

dx

xd=

∆+δ+∆µ−

δ

(4.9)

The boundary conditions are: .)x( inlet);()x( ii δ=δ∆=∆

The Equations 4.9 are rearranged and solved by subtracting both equations

and integrating:

[ ] )(x)x()x(x)x()x( iiiii δ+∆−δ−∆=δ+∆−δ−∆ (4.10)

= constant

63

Since there are two variables ∆ and δ, another equation is needed

to find the final solution. This is obtained for the sum of the two variables.

Adding both Equations in 4.9 and replacing ( ) ( )xx δ−∆ using Equation 4.10

gives

( ))]x()x([x2

dx

)]x()x()[x1(d 2

δ+∆µ−δ+∆+

= [ ])(x iiiii δ+∆−δ−∆− (4.11)

Here an additional assumption is made that the radius rmin is small

enough to allow neglect of the term with ii rx ψ= in the last equation. This is

true for large n.

The solution of Equation 4.11 is readily obtainable. The final

function ( ) ( )xx δ+∆ is

( ) ( ) ( )( ) µ−

+δ+∆=δ+∆

12

iii

x1

Gxx (4.12)

where, µ−+= 12

ii )x1(G is a constant computed for the radius ri at

the inlet to the exchanger.

4.3.4 Effectiveness of Spiral Plate Heat Exchanger

The effectiveness Ph for spiral plate heat exchanger can be

calculated according to the general formula

i,ci,h

o,hi,h

hTT

TTP

−

−= (4.13)

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This may also expressed as:

[ ][ ]NTU)1R(expR1

NTU)1R(exp1Ph

×−×−

×−−= (4.14)

Recasting the solution Equation 4.12 to obtain an expression for

thermal effectiveness gives

( )( ) ( )2

i

2

i

2

i

2

o

2

i

2

o

h

x1

x)1R(

x1

x1lnexpR1

x1

x1lnexp1

P

+

−+

+

+µ−

+

+µ−

= (4.15)

Here the term µ is given by

( ) ( )

( ) 2

h

2

CNNTU1R

R1RR1R2

−=

+−ψ

=µ (4.16)

where,

( ) ( )

( )A

ANTUNTUCN

AANTUR1RCN

cch

c

π+≅

π+=

(4.17)

This number is characteristic for spiral plate heat exchanger and

therefore Bes and Roetzel (1993) recognize CN as the dimensionless

criterion number for the countercurrent spiral heat exchanger.

The second term of exponent with ln function contains an

independent variable which can be expressed as follows:

65

( ) ( ) ( ) ( )2

i

2

i

2

o

2

i

2

o x1xx1x1x1 +−+=++

and

+

−+

=+

−2

i

2

2

i

2

i

2

o

x1

1n

11

CNx1

xx

For realistic heat capacity ratios (R) and small inlet radius, the

last term in the denominator ( ) ( )2

i

2

i x1x1R +− is negligible.

It is easy to notice that Equation 4.15 has the same structure as

Equation 4.14. The LMTD correction factor is obtained as

( )2

2

CN

CN1n

111ln

F

−++

= (4.18)

Therefore, for the hot side, thermal effectiveness, correcting

for deviation from countercurrent flow, is given by

( )[ ]( )[ ]FNTU1RexpR1

FNTU1Rexp1Ph

××−×−

××−−= (4.19)

Thermal effectiveness of the cold side is obtained from the

expression known in literature (Ramesh, K Shah 2006)

RPC

CPP h

c

hhc == (4.20)

66

4.3.5 Calculation from Experimental Data

From experimental data, the overall heat transfer coefficient may

be calculated as follows

( )LMTDTFUAQ ∆= (4.21)

Calculation of the heat transfer coefficient from Equation 4.19

requires the knowledge of LMTD correction factor whereas Equations 4.17

and 4.18 indicate that calculating F requires the knowledge of heat transfer

coefficient. Therefore an iterative procedure was setup and executed in

Microsoft Excel® to estimate the LMTD correction factor F, heat transfer

coefficient U, the number of transfer units NTU, and the temperature

effectiveness P of the exchanger for both the hot and the cold sides.

4.4 RESULTS AND DISCUSSION

The log mean temperature difference correction factors are very

close to unity for all octane-water, kerosene-water, dodecane-water,

diesel-water and nitrobenzene-water systems. The calculation data is

presented along with those of heat transfer coefficients in Chapter 5, in

Tables 5.3 to 5.28. The value of F being close to unity indicates that the flow

in the exchanger is predominantly countercurrent. Figures 4.2 through 4.34

show the thermal effectiveness of the spiral plate heat exchanger for the hot

and cold sides. It is seen that, as predicted by the model of Bes and Roetzel

(1993), the thermal effectiveness of both the hot and the cold sides increase

with increase in NTU. A typical plot of P-NTU of a spiral heat exchanger

for constant heat capacity ratio R and LMTD correction factor F is one of

asymptotic increase of P with respect to NTU, with the asymptote depending

67

on the heat capacity ratio and F. The experiments performed, however, have

been conducted for fixed hot fluid flow rates and varying cold fluid flow

rates. This implies that, as the cold fluid flow rate increases, its heat capacity

increases. Therefore, the heat capacity ratio R decreases continuously, even

as the NTU increases and the effectiveness increases. This preliminary study

has investigated the NTU range of practical relevance and the values of

thermal effectiveness range for the cold fluid, and the hot fluid is listed in

the Table 4.1.

Table 4.1 Ranges of NTU, thermal effectiveness of hot and cold fluid

System NTU Ph Pc

Min Max Min Max Min Max

Octane - Water 1.86 4.31 0.07 0.29 0.83 0.97

Kerosene - Water 0.28 3.93 0.07 0.59 0.23 0.97

Dodecane -Water 0.44 3.93 0.07 0.66 0.34 0.97

Diesel - Water 2.30 6.83 0.07 0.22 0.89 1.00

Nitrobenzene - Water 0.49 3.93 0.07 0.49 0.38 0.97

Thermal effectiveness may also be calculated from experimental

data on the inlet and outlet temperatures. These values, depicted in Figures

4.2 to 4.34 by the subscript exp, are also plotted as a function of NTU.

Further, the predictions of Bes and Roetzel are compared against the

experimental data. The following are the main observations:

• For the hot side, effectiveness increases monotonically with

NTU, and the model is able to capture this behavior qualitatively

as well as quantitatively. This is attributed to the turbulent flow

68

regime in the hot side, where the Reynolds Number is greater

than the critical Reynolds Number of the spiral exchanger.

• However, at low NTUs, the model overpredicts the effectiveness,

whereas at high NTUs, the model underpredicts the effectiveness.

The errors in these over/underpredictions are almost similar.

This brings out the limitation in using a purely thermal model to

predict heat transfer in heat exchangers.

• The results in the cold side are seemingly counterintuitive and

qualitatively different from experimental predictions. It is seen

that, while the experiments predict a decrease of the efficiency

with NTU, the model predicts the opposite. This is true for pure

cold fluids (water or organics) as well as the immiscible mixtures.

This discrepancy is explained with the water-water system as

follows: For a given quantity of heat transferred from the hot side

to the cold side, the more the heat capacity (i.e., the more the flow

rate), the smaller the temperature difference achieved.

This explains why thermal efficiency is poorer when heat transfer

coefficients are high. This behavior is not captured by the model.

Actually, in the hot side, the model performs admirably. Here, the

heat transfer coefficients (hh) decrease with increasing cold fluid

flow rate, and the thermal efficiency correspondingly increases.

The cold-side thermal efficiency is simply the product of heat

capacity ratio (R) and the hot-side thermal efficiency. As the

cold-side flow rate increases, R decreases, which should pull

down the value of cold-side efficiency. This happens, but

insufficiently, resulting in the preservation of the same trend in

the cold side as seen in hot side. This is best attributed to the fact

69

that the cold side has laminar flow while the hot side has

turbulent flow. Therefore, the reduction in R is insufficient to

reverse the trend. In other words, this is a limitation on the use of

a purely thermal model in the prediction of heat transfer in

laminar convection processes.

• An analysis of the thermal effectiveness in the case of liquid

mixtures reveals further insights. While the mixtures follow the

same trend as described above, the discrepancies fall under two

categories.

• In the case of water, octane and diesel, the model consistently

overpredicts the effectiveness. In all these systems, the Prandtl

Number is of the order of 5 to 7, which implies that heat transfer

is through both conduction and convection, though convection is

significant.

• In the next chapter, where a linear mixing rule is developed to

predict heat transfer to immiscible liquids, it is shown that

mixtures of these fluids obey what approaches stratified flow

pattern. Whether or not the flow pattern is stratified (or slug or

annular or three-layer), the non-negligible role of conduction

played in the system serves to lessen the importance of flow

patterns in the heat transfer behavior. This observation is further

validated by the results in the next chapter.

• In the case of kerosene, dodecane, and nitrobenzene, the model

underpredicts effectiveness in systems with higher organics

concentration, whereas as the amount of water increases, the

70

behavior tends to overprediction. For these systems, the Prandtl

Numbers are of the order of 17 to 26. In such systems, heat

transfer due to conduction is negligible. Here, the effects of flow

should be very significant in the laminar regime. As expected,

errors are much higher, and using weighted-average properties for

the mixtures tend to underpredict the actual heat transfer

effectiveness.

• In the next chapter, mixtures of these fluids are approximated to

flow in a oil-in-water or water-in-oil dispersion, and a linear

mixing rule is applied to predict their heat transfer coefficients.

This is to better capture the effect of flow on the heat transfer.

Even in the absence of a direct dependence on velocity, the heat

transfer coefficient predictions are reasonable, within an order of

magnitude, further validating the above observations.

• Bes and Roetzel model is able to predict that the thermal

effectiveness for kerosene, dodecane and nitrobenzene would be

lower than that of water, octane and diesel. However, the

magnitudes are significantly different, as explained above.

• The flow arrangement in the exchanger is not stream symmetric.

Hot fluid has much poorer thermal effectiveness than the cold

fluid. This is to be expected, because hot fluid flows much faster

(in turbulent regime) compared to the cold fluid (in laminar

regime).

• In the cold side, thermal effectiveness is highest for water, and

lowest for nitrobenzene system. Thermal efficiencies showed no

71

discernible trend respect to the amount of organic phase present

in the cold fluid. However, thermal effectiveness was higher for

lower-molecular weight organic phases such as octane and

kerosene, than the higher-molecular weight organic phases such

as dodecane and diesel. Polarity in the substance (as in

nitrobenzene) tended to decrease thermal effectiveness. The most

significant effect is that of the Prandtl Number, which effectively

determines whether heat transfer is by convection alone or if

conduction is to play an important role.

• Unlike the predictions of Bes and Roetzel (1992) and Burmeister

(2006) in their simulations, no maxima are observed for the

effectiveness for any composition of the two-phase mixture

within the range of Dean Number and Prandtl Number studied.

Burmeister has studied only the case of equal capacitances, and

Bes and Roetzel have examined cases of constant heat capacity

ratios. In the present work, on the one hand, NTU values are

smaller than those for which the maximum effectiveness is

observed by Burmeister. Also, in our measurements, the heat

capacity ratio decreases with increasing NTU.

• Further experiments are needed to study the quantitative

dependence of thermal effectiveness on carbon number and

composition of the organic phase, as well as the full functional

form of the dependence of thermal effectiveness on NTU for

liquid-liquid two-phase mixtures.

72

Figure 4.2 Thermal effectiveness with number of transfer units,

when pure cold fluids are used: From experiments

Figure 4.3 Thermal effectiveness with number of transfer units, when

pure cold fluids are used: From Bes and Roetzel model

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8

Pc,e

xp

NTUc

Pure Water

Pure Octane

Pure Kerosene

Pure Dodecane

Pure Diesel

Pure Nitrobenzene

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8

Pc,p

red

NTUc

Pure Water

Pure Octane

Pure Kerosene

Pure Dodecane

Pure Diesel

Pure Nitrobenzene

73

Figure 4.4 Thermal effectiveness comparison of model

vs. experiments when pure cold fluids are used

Figure 4.5 Thermal effectiveness with number of transfer units

in hot fluid for octane-water system: From experiments

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pc,e

xp

Pc,pred

Pure Water

Pure Octane

Pure Kerosene

Pure Dodecane

Pure Diesel

Pure Nitrobenzene

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.5 1.0 1.5

Tem

per

atu

re e

ffec

tiven

ess

Ph

, exp

Number of transfer units NTUh

Hot Fluid Pure Water

20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

74


in hot fluid for octane-water system: From Bes and

Roetzel Model


vs. experiments in hot fluid for octane-water system

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.15 0.65 1.15 1.65

Tem

per

atu

re e

ffec

tiven

ess

Ph

,pre

d



20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

h,

pre

d

Temperature effectiveness Ph, exp

Pure Water

20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

75


in cold fluid for octane-water system: From experiments


in cold fluid for octane-water system: From Bes and

Roetzel Model

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5

Tem

per

atu

reef

fect

iven

ess

Pc,

exp

Number of transfer units NTUc

Cold FluidPure Water

20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

0.8

0.9

1.0

1.5 2.0 2.5 3.0 3.5 4.0

Tem

per

atu

re e

ffec

tiv

enes

s P

c,

pred


Cold Fluid

Pure Water

20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

76


vs. experiments in cold fluid for octane-water system

Figure 4.11 Thermal effectiveness with number of transfer units in

hot fluid for kerosene-water system: From experiments

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

c,

pre

d

Temperature effectiveness Pc, exp

Pure Water

20% Octane

40% Octane

60% Octane

80% Octane

Pure Octane

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.0 0.5 1.0 1.5

Tem

per

atu

re e

ffec

tiven

ess

Ph

, exp



20% Kerosene

40% Kerosene

60% Kerosene

80% Kerosene

Pure Kerosene

77


in hot fluid for kerosene-water system: From Bes and

Roetzel Model


vs. experiments in hot fluid for kerosene-water system

0.05

0.15

0.25

0.35

0.45

0.55

0.65

0.15 0.65 1.15 1.65

Tem

per

atu

re e

ffec

tiv

enes

s P

h,

pred



20 % Kerosene

40 % Kerosene60 % Kerosene

80 % Kerosene

Pure Kerosene

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

h,

pre

d


Pure Water

20% Kerosene

40% Kerosene

60% Kerosene

80% Kerosene

Pure Kerosene

78


cold fluid for kerosene-water system: From experiments


in cold fluid for kerosene-water system: From Bes and

Roetzel Model

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 2 4 6

Tem

per

atu

reef

fect

iven

ess

Pc, exp


Cold FluidPure Water

20% Kerosene

40% Kerosene

60% Kerosene

80% Kerosene

Pure Kerosene

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

Tem

per

atu

re e

ffec

tiven

ess

Pc,

pre

d


Cold Fluid

Pure Water

20 % Kerosene

40 % Kerosene

60 % Kerosene

80 % Kerosene

Pure Kerosene

79


vs. experiments in cold fluid for kerosene-water system


hot fluid for dodecane-water system: From experiments

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

c,

pred


Pure Water

20% Kerosene

40% Kerosene

60% Kerosene

80% Kerosene

Pure Kerosene

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.0 0.5 1.0 1.5 2.0

Tem

per

atu

re e

ffec

tiv

enes

s P

h,

exp



20% Dodecane

40% Dodecane

60% Dodecane

80% Dodecane

Pure Dodecane

80


in hot fluid for dodecane-water system: From Bes and

Roetzel Model


vs. experiments in hot fluid for dodecane-water system

0.0

0.2

0.4

0.6

0.8

0.0 0.5 1.0 1.5 2.0

Tem

per

atu

re e

ffec

tiven

ess

Ph

, p

red



20 % Dodecane

40 % Dodecane

60 % Dodecane

80 % Dodecane

Pure Dodecane

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

h,

pre

d


Pure Water

20% Dodecane

40% Dodecane

60% Dodecane

80% Dodecane

Pure Dodecane

81


cold fluid for dodecane-water system: From experiments


in cold fluid for dodecane-water system: From Bes

and Roetzel Model

0.2

0.4

0.6

0.8

1.0

0 2 4 6

Tem

per

atu

reef

fect

iven

ess

Pc,

exp


Cold Fluid Pure Water

20% Dodecane

40% Dodecane

60% Dodecanel

80% Dodecane

Pure Dodecane

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 2.0 4.0 6.0

Tem

per

atu

re e

ffec

tiven

ess

Pc,

pre

d



20 % Dodecane

40 % Dodecane

60 % Dodecane

80 % Dodecane

Pure Dodecane

82


vs. experiments in cold fluid for dodecane-water system


in hot fluid for diesel-water system: From experiments

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

c,

pre

d


Pure Water

20% Dodecane

40% Dodecane

60% Dodecane

80% Dodecane

Pure Dodecane

0.0

0.1

0.2

0.3

0.4

0.5

0.0 0.5 1.0 1.5

Tem

per

atu

re e

ffec

tiv

enes

s P

h,

exp



20% Diesel

40% Diesel

60% Diesel

80% Diesel

Pure Diesel

83


in hot fluid for diesel-water system: From Bes and

Roetzel Model


vs. experiments in hot fluid for diesel-water system

0.05

0.10

0.15

0.20

0.25

0.1 0.3 0.5 0.7 0.9

Tem

per

atu

re e

ffec

tiven

ess

Ph

, p

red



20% Diesel

40% Diesel

60% Diesel

80% Diesel

Pure Diesel

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

h,

pre

d


Pure Water

20% Diesel

40% Diesel

60% Diesel

80% Diesel

Pure Diesel

84


in cold fluid for diesel-water system: From experiments


in cold fluid for diesel-water system: From Bes and

Roetzel Model

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6 8

Tem

per

atu

reef

fect

iven

ess

Pc, exp



20% Diesel

40% Diesel

60% Diesel

80% Diesel

Pure Diesel

0.85

0.90

0.95

1.00

2 4 6 8

Tem

per

atu

re e

ffec

tiv

enes

s P

c,

pred



20 % Diesel40 % Diesel60 % Diesel80% DieselPure Diesel

85

Figure 4.28 Thermal effectiveness comparison of model vs.

experiments in cold fluid for diesel-water system


in hot fluid for nitrobenzene-water system: From

experiments

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

c,

pred


Pure Water

20% Diesel

40% Diesel

60% Diesel

80% Diesel

Pure Diesel

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.5 1 1.5

Tem

per

atu

reef

fect

iven

ess

Ph

, ex

p



20% Nitrobenzene

40% Nitrobenzene

60% Nitrobenzene

80% Nitrobenzene

Pure Nitrobenzene

86


in hot fluid for nitrobenzene-water system: From Bes

and Roetzel Model


vs. experiments in hot fluid for nitrobenzene-water

system

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.1 0.6 1.1 1.6

Tem

per

atu

re e

ffec

tiv

enes

s P

h,

pre

d



20 % Nitrobenzene

40 % Nitrobenzene

60 % Nitrobenzene

80 % Nitrobenzene

Pure Nitrobenzene

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

h,

pre

d


Pure Water

20% Nitrobenze

40% Nitrobenzene

60% Nitrobenzene

80% Nitrobenzene

Pure Nitrobenzene

87


in cold fluid for nitrobenzene-water system: From

experiments


in cold fluid for nitrobenzene-water system: From Bes

and Roetzel Model

0.0

0.2

0.4

0.6

0.8

1.0

0 2 4 6

Tem

per

atu

reef

fect

iven

ess

Pc,

exp



20% Nitrobenzene

40% Nitrobenzene

60% Nitrobenzene

80% Nitrobenzene

Pure Nitrobenzene

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 1.0 2.0 3.0 4.0

Tem

per

atu

re e

ffec

tiv

enes

s P

c,

pre

d


Cold Fluid

Pure Water

20 % Nitrobenzene

40 % Nitrobenzene

60 % Nitrobenzene

80 % Nitrobenzene

Pure Nitrobenzene

88


vs. experiments in cold fluid for nitrobenzene-water

system

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tem

per

atu

re

effe

ctiv

enes

s P

c,

pre

d


Pure Water

20% Nitrobenze

40% Nitrobenzene

60% Nitrobenzene

80% Nitrobenzene

Pure Nitrobenzene

chapter 4 thermal effectiveness of a spiral plate heat...

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