Effect of flow distribution to the channels on the thermal performance of a plate heat exchanger

Chemical Engineering and Processing 41 (2002) 49 – 58 www.elsevier.com/locate/cep

Effect of flow distribution to the channels on the thermal performance of a plate heat exchanger B. Prabhakara Rao, P. Krishna Kumar, Sarit K. Das * Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology-Madras, Chennai-600 036, India Received 2 August 2000; received in revised form 27 December 2000; accepted 27 December 2000

Abstract Plate heat exchangers are making their presence felt in the power and process industry in the recent past. Hence, it has become necessary to model their temperature response accurately. The traditional way of modelling a plate heat exchanger with equal flow in all the channels is unrealistic, and previous studies indicate that considerable differences remain in the flow rates in different channels. The present work brings out the effect of flow maldistribution from channel to channel comprehensively. This poses a serious question about the usual method of analysis of the experimental heat-transfer data of the plate heat exchanger. Unlike previous studies, the present study indicates the importance of considering the heat-transfer coefficient inside the channels as a function of flow rate through that particular channel. This eliminates the contradictory proposition of unequal flow rates but an equal heat-transfer coefficient. A wide range of parametric study have been presented, which brings out effects such as those of the heat-capacity rate ratio, flow configuration, number of channels and correlation of heat transfer. The analysis presented here suggests a better method of heat-transfer data analysis for plate heat exchangers. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Flow distribution; Plate heat exchanger; Thermal performance

1. Introduction Plate heat exchangers have started playing increasingly important roles in the power and process industry during the last two decades. Although originally invented for hygienic industries such as brewing, dairy and food processing due to its ease of cleaning and maintenance, plate heat exchangers have other advantages such as compactness, ability to recover heat with a very small temperature difference, flexibility, lesser tendency of fouling and less susceptibility to flow-induced vibration. With these advantages, along with advances in material technology in the form of new temperature- and pressure-resistant materials for gasket or graphite plates, it is now possible to use this class of heat exchangers appropriately for the power and chemical process industry. Today, plate heat exchangers are * Corresponding author. Present address: Institut fu¨r Thermodynamik, Fachbereich Maschinenbau, Universita¨t der Bundeswehr Hamburg, D-22039 Hamburg, Germany. E-mail address: sarit – [email protected] (S.K. Das).

used in critical areas such as in the secondary circuit of a nuclear power plant or in the evaporation and condensation duty in an OTEC (Ocean Thermal Energy Conversion) plant. The application in such critical areas as well as in modern process industry demands that an accurate thermal model of the heat exchanger along with reliable heat-transfer data be available for plant simulation, for initiating safety measures and for deciding on the optimal flow configuration (i.e. number of plates and pass arrangement). The thermal simulation of plate heat exchangers has been presented by a large number of investigators [1– 4]. All these simulations have been carried out with the assumption of plug (one-dimensional) flow inside channels and an equal distribution of fluid into the channels from the port. However, it is a well-known fact [5] that the flow maldistribution within the plate and particularly from channel to channel plays a major role in contributing to the actual exchanger performance deviating from the idealised one-dimensional flow equally distributed amongst the channels.

0255-2701/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 5 - 2 7 0 1 ( 0 1 ) 0 0 1 0 5 - 2

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

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This points to one of the major areas of difficulty in the analysis of plate heat exchangers, namely the heat-transfer data of the plates. While efforts are concentrated on obtaining more universal data and correlations, little attention has been paid to how these data have been arrived at. A number of efforts have been made in the past to obtain reliable heat-transfer correlations for plates such as those by Marriott [6,7], Bond [8], Focke et al. [9], Savostin and Tikhonov [10] along with the comprehensive review by Raju and Bansal [11], but it was correctly pointed out by Shah and Focke [12] that the available data are rather divergent and hardly give any unified correlation of practical use. The excellent effort made recently by Martin [13] in this regard, though the best so far, has shown a deviation of 920% from correlation and up to 30% of deviation in specific cases. The recent sophisticated experiments [14] could only suggest correlations in specific geometries. To the experience of the present author (S.K.D.), it is interesting to note that the heat-transfer data of the plate heat exchanger not only differ from author to author but vary even where experiments are done by the same investigator in the same experimental set-up and with identical plates but different number of channels, as was done in Ref. [15]. This brings to focus the method by which the data reduction is done to arrive at the available correlations. All the references mentioned above recorded inlet and outlet temperatures and flow rates through experiments. Thus, considering the exchanger to be a pure counter flow exchanger, the overall heat-transfer coefficient was evaluated from: U=

(m; cp)DT1 (m ¯ cp)2DT2 = ADTlm ADTlm

(1)

From this, the heat-transfer coefficient of one side is evaluated with the help of a Wilson plot [16] by assuming a heat-transfer correlation of the type Nu = CRenPrr

(2)

It is implicit that in this approach, the flow is equally distributed to all the channels equally, and as a consequence, the heat-transfer coefficient in each channel is equal. In other words, the effect of flow maldistribution from channel to channel that remains in reality during the experiment is incorporated into the value of the heat-transfer coefficient. This is the reason why the data are affected when the number of channels is changed or the flow configuration is shifted from the ‘U’ type to the ‘Z’ type because a different flow distribution is obtained. Thus, the wide deviation in the plate heat exchanger is less due to ‘accuracy of experiment’ and more due to the assumption of a uniform flow distribution in the data reduction model. It may be mentioned here that heattransfer data for plate heat exchanger devoid of any

flow distribution effect can be obtained (and indeed has been the case sometimes in the past) with only two channels, i.e. one channel on each side. However, there are few such data mainly due to the problem of large losses and inaccuracies associated with the measurement of small flow rates. From the above discussion, it is clear that a proper heat-transfer model of a plate heat exchanger based on the unequal flow distribution in channels is required not only to analyse the heat exchangers but also to yield proper heat-transfer data from experiments on heat exchangers with multiple channels that will be stripped of the influence of the number of channels or flow configuration used in the experiment. The effect of an unequal distribution of fluid inside the channels was analysed using the numerical technique by Datta and Majumdar [17] and in more detail using the analytical technique by Bassiouny and Martin [18,19]. The latter work is of immense importance because of the fact that it brought out the difference between ‘U’- and ‘Z’-type plate exchangers and expressed the distribution in the channels in the form of closed form equation using the general flow channelling and unification concept of Bajura and Jones [20]. In the doctoral dissertation of Bassiouny [5], even though the flow distribution from channel to channels was explained very well, this distribution was not used for the thermal analysis, which prevented the effect of this distribution on the thermal performance of the plate heat exchanger from being assessed. The only work in this direction seems to be from Yang and Wang [21] and Singhaleta [22]. However, it must be observed that none of the abovementioned works considers the change of heat-transfer coefficient in the channels due to the change in flow rates amongst themselves. Thus, these models work under the unrealistic assumption of ‘variable flow rates in channels but an equal heattransfer coefficient’. This is the main inspiration of the present work. In the present work, the plate heat exchanger is thermally modelled with unequal flow in channels taking the distribution as suggested by Baussiouny and Martin [17,18]. While doing this, the heattransfer coefficients inside the channels are assumed to be variable with a general correlation of the type of Eq. (2). Results are presented for a wide range of values of n. The parametric study of both U- and Z-type heat exchangers is presented not only to bring out the effect of flow maldistribution but also to indicate the necessity to consider flow dependence of heat-transfer coefficient in the computation of the thermal performance of the plate heat exchanger as well as constructing a theoretical model for the reduction of experimental data.

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

For the plate:

2. Mathematical model To model the heat-transfer process between channels in a plate heat exchanger, the following assumptions are made. 1. The thermophysical properties of the fluids are considered to be independent of temperature and pressure. 2. The flow cross-sectional area of each channel has been taken to be the same. 3. Heat transfer is assumed to take place only between the channels and not between the channel and the ports, or through the seals and gaskets. 4. The heat exchanger is assumed to be insulated from the surroundings. 5. The flow distribution inside the channel is taken to be uniform giving a ‘plug flow’ of fluid inside each channel. 6. The flow maldistribution from channel to channel has been taken into account through the Baussiouny model [5,17,18], but additionally, the heat-transfer coefficient is considered to be a function of flow, as given by Eq. (2). 7. The plates are considered to be thin enough to neglect any axial conduction within. With these assumptions considering a small control volume of fluid inside the channel and a control volume of solid plate, as shown in Fig. 1, the energy balance over these control volumes taking the above assumptions into consideration yields the following fluid and plate equations. For fluid 1: (m ¯ Cp)1

dTi hi A hA (Twi −Ti ) + i (Twi + 1 −Ti ) = 2L 2L dx dTi =(− 1)i − 1 dx ×



hi − 1A hi A (Ti − 1 − Twi )+ (Ti − Twi )= 0 2L 2L

(4)

These equations can be non-dimensionalised as: dti =(− 1)i − 1NTU(twi + twi + 1 − 2ti ) dX ti − 1 − twi = −

(5)

hi

(t − twi ) hi − 1 i

(6)

Substituting twi from Eq. (6) to Eq. (5)gives dti =(− 1)i − 1NTU dX









hi − 1 hi hi ti − 1 + + − 2 ti hi − 1 + hi hi + hi − 1 hi + hi + 1

+

n

hi + 1 t hi + hi + 1 i + 1

where NTU =

hA 2(m ¯ Cp)uniform

r6(i )=

uuniform ui

rh(i )=

hi huniform

x T− T1,in X= ;t= T2,in − T1,in L In order to enable us to make comparisons with the case of uniform flow in all the channels, this equation may be written as

For fluid 2: (m ¯ Cp)2

51

dti =(− 1)i − 1NTUr6(i )rh(i ) dX

n

hi A hA (T −Ti ) + i (Twi + 1 −Ti ) 2L wi 2L

× (3) +

 



hi − 1 t hi − 1 + hi i − 1



n

hi hi + 1 hi + − 2 ti + t hi + hi + 1 hi + hi − 1 h i + hi + 1 i + 1

(8) Considering Eq. (2) for flow dependence of heattransfer coefficient and constant fluid properties, rh(i ) can be expressed as rh(i )=

Fig. 1. Control volume of fluid inside the channel.

hi huniform

=



ui

uuniform



n

(9)

It should be mentioned here that the dependence on the Prandtl number does not change h since property variation with temperature is not considered in the present model. Thus, Eq. (8) is reduced to the following form for i= 2, 3, …, N− 1

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

52

dti =(−1)i − 1NTUrh(i )r6(i ) dX

 



n i−1 n i−1

u

u

+u ni

ti − 1 +

 

 n

u ni u ni u ni +1 + n −2 ti + n t n n u i +u ni + 1 i + 1 u +u i + 1 u i +u i − 1 n i

However, the first and last channel transfer heat only to one fluid stream; hence, the equations for these two channels can be written as: dt1 =NTUr6(1)rh(1) dX



 

n

u n1 u n2 −1 t1 + n t2 n u +u 2 u 1 +u n2 n 1

(11) dtN = NTUr6(N)rh(N) dX ×



u

n N−1 n N−1 N

u

+u



tN − 1 +



n

uN −1 tN uN − 1 −uN

(12) Eqs. (10)– (12) are the governing differential equations for which the boundary conditions can be set as at X=0, ti = 0 for i = 1,3,5, … N at X= 1, ti =1 for i =2,4,6,…N − 1

"

(13)

The total number of channels is assumed to be odd, which is often the practical case in applications in order to minimize any heat loss from end channels by circulating cold fluid in both of them (i =1 and N). This completes the mathematical model, which is a set of coupled differential equations. For solution, it requires an exact distribution of the flow from channel to channel, which is described below.

3. Solution procedure The system of a first-order ordinary differential Eqs. (10) – (12) can be solved analytically using eigenvalues. The most important requirement for this solution is a proper distribution of fluid in the channels from the port. Bassiouny and Martin [18,19] presented the flow channelling formulation for normal geometries where the volumetric flow rate decreases along the flow direction in the entrance port as 6c =

mcoshm(1−z) sinhm

Fig. 2. Flow lengths before entering into and after exit from channels in plate heat exchangers: (a) ‘U’-type; (b) ‘Z’-type.

these configurations, as given by Bassiouny and Martin [18,19], are reproduced in the Table 1. Experiments with flow channelling [5] in the distributor-collectorproblem revealed values from −1.1 to 3.61 for the parameter m, corresponding to a range of a/a* values used in the experiment there. Technical plate heat exchangers have a/a*= 1. The parameter m 2 is proportional to the square of the ratio (n*ac/a) and also

Table 1 Flow distribution correlations from Bassiouny and Martin [18,19]a m2 U-type arrangement Positive 0

(14)

for the case of U-type plate heat exchangers. Here, the distribution parameter, m, given in the above expression is dependent on the exchanger geometry. The ‘U’and ‘Z’-type heat exchangers, as shown in Fig. 2, have significantly different flow distributions and hence different values of m. The corresponding equations to

Z-type arrangement Positive 0

w,w*

 a* a

wc

sinhm(1−z) sinhm

m

1−z

1

sinhmz sinhm

m

1−z

1

cosh(1−z) sinhm

coshmz sinhm

a This table gives the expressions for dimensionless volume flow rate, wc for different channels designated by the respective coordinate (z) along the port as a function of ‘m’ and flow arrangement (‘U’- or ‘Z’-type)

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58 Table 2 Flow distribution parameter, m, from Bassiouny [5]



m2

0.123 0.170 0.678 1.474 5.898 8.163

−1.1 −1.0 0.25 2.72 3.61 3.42

a* a

T( = U( B( (x)D( .

where i1,i2…iN are the eigenvalues of the matrix A( , and U( is the matrix containing the eigen vectors of A( .

where T( =[T1.T2.T3…TN ]

is the coefficient matrix given by u n1 −1 u +u n2 u n2 u +u n2 n 1

n

For i= 2, 3,…N −1 Aii − 1 =(− 1)i − 1NTUrw(i )rh(i )

n





n i−1 n i−1

u

u

For i= N



n

+u ni

n

u ni u ni + −2 u ni +u ni + 1 u ni +u ni − 1

Ai,i + 1 = (−1)i − 1NTUrw(i )rh(i )

Individual temperatures can be given by N

Ti = % dj uij e ij x

(17)

j=1

Using Eq. (17) for the boundary conditions (given by Eq. (13)), a matrix equation of the following form can be derived from which the coefficient matrix D( = (d1, d2…dN )T can be evaluated W( D( = S( .

(18)

Here, S( = (0, 1, 0, 1…up to N terms)T The coefficient matrix, W( , is generated by putting X=0 and X= 1, respectively, in the Eq. (17) to form Wij = u for i = 1, 3, 5…N =uij e ij for i= 2, 4, 6…N − 1.

4. Results and discussion

Ai,i =( −1)i − 1NTUrw(i )rh(i ) ×

U( =[uij ]i= 1, 2…N j= 1, 2…N

The system of linear equations given by Eq. (18) can be solved by any standard matrix solver. However, it fails to give a proper solution if the eigenvalues of the matrix A( are not distinct. In the case of multiple eigenvalues, a small number are added to make them distinct in such a way that there is no significant change in the solution.

T

A12 =NTUrw(1)rh(1)

(16)

B( (x)= diag(e i1x, e i2x ,e i3x…e iN x)

(15)

n 1

n

B( is the diagonal matrix given by

dT( = A( T( dX

 

u

u nn −1 + u nN

n N−1

all other elements of A( being zero. The solution to this matrix equation is given by

inversely proportional to the friction factor of the channel flow. Since the friction factor strongly depends on the chevron angle, and (less strongly) on the Reynolds number, it means that ‘m 2’ would not be exactly constant for each parallel channel, as assumed in the present analytical solution. In principle, m 2 can vary under these conditions from 0 to very large positive values of several hundred, increasing with the square of the number of parallel channels [23]. However, the present approach can be regarded as a first step towards understanding the flow-distribution effect, which can be extended in future to include the effect of variable ‘m’. Table 2 gives a typical set of values of m for different geometrical parameters. With these velocity distributions, the set of differential equations (Eqs. (10)– (12)) can be cast in a matrix differential equation of the form

A11 =NTUrw(1)rh(1)



AN,N = (−1)N − 1NTUrw(N)rh(N)

53

n

u ni + 1 u +u ni + 1 n i



AN,N − 1 =(− 1)N − 1NTUrw(N)rh(N)

n

u nN − 1 u nN − 1 +u nN

In the literature, it has been clearly indicated [17,18] that the flow distribution in U-type plate heat exchangers differs significantly from that of the Z type. Hence, results have been calculated for both ‘U’ and ‘Z’ configurations and a wide range of other parameters. It should be mentioned here that due to flow maldistribution, each channel has a different heat-transfer coefficient, and hence, it is difficult to define Ntu for the entire equipment. However, in order to facilitate a comparison with the uniform distribution model, NTU is defined on the basis of the heat-transfer coefficient with the hypothetical case of equal flow distribution when (m 2 = 0).

54

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

Fig. 3. Effectiveness of a ‘Z’-type PHX containing 32 channels and constant heat-transfer coefficient.

Fig. 5. Comparison of effectiveness between the ‘U’ and ‘Z’ arrangement for an even number of channels.

Figs. 3 and 4 show the effectivenesses of a ‘Z’- and ‘U’-type 32 channel plate heat exchanger, respectively. In this calculation, the simpler assumption of constant heat-transfer coefficient has been used, which means setting n= 0 in Eq. (2). It can be seen that the flow maldistribution parameter plays a major role in both types of heat exchangers. However, the difference between the two types of heat exchangers can be observed only for an even number of channels as given by Figs. 5 and 6. These two figures indicate that for an odd number of channels, not only does an equal flow distribution (m 2 =0) give the same effectiveness for ‘U’- and ‘Z’-type exchangers, but even with a large difference of flow in channels (m 2 =25), they behave in a similar way. This is contrary to practical experience, and this precisely brings out the limitation of the model of a constant heat-transfer coefficient. For a constant heat-transfer coefficient, the heat capacity rate ratio naturally plays a major role since the only effect of the variation of flow from channel to channel is reflected in the difference in heat capacity rates. Fig. 7 shows the effect of the heat

capacity rate ratio at two different NTU values (1.0 and 3.0) for a 24-channel plate exchanger. It can be seen that the effectiveness shows a minimum almost at equal heat capacity rates of the two fluids. The above results clearly indicate the necessity of using a variable heat-transfer coefficient in the form of a flow-dependent Nusselt number [i.e. n" 0 in Eq. (2)]. It must be emphasized here that the heat-transfer data of plate heat exchangers show a very wide variation. From the experimental observations for the value of the exponent of the Reynolds number, n in Eq. (2) ranges from 0.32 to 0.732. The present results clearly indicate that the problem lies in the evaluation of the heat-transfer coefficient from the experimental data. The usual technique of determination of the heat-transfer coefficient in a plate heat exchanger considers it to be a purely counter flow heat exchanger with identical flow in each channel. This actually lumps the flow maldistribution effect (i.e. effect of m 2 " 0) and the variable heat-transfer coefficient effect into a mean (the so-called ‘measured’) value of the heat-transfer coefficient. Thus, we can say:

Fig. 4. Effectiveness of a ‘U’-type PHX containing 32 channels and constant heat-transfer coefficient.

Fig. 6. Comparison of effectiveness between the ‘U’ and ‘Z’ arrangement for an odd number of channels.

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

Fig. 7. Effect of flow distribution and heat capacity ratio on the effectiveness of a PHX with 24 channels.

variation of n :variation in m 2 + variation in h from channel to channel. Now, since the experiments have been performed by different investigators with different port and plate dimensions as well as different numbers of plates, the above effects have influenced the determination of the heat-transfer coefficient considerably. This can be seen from Fig. 8, where the effect of the exponent of the Reynolds number, n, has been shown. It is interesting to note that the range of variation of effectiveness (m) is same for n as for m 2, as given in Fig. 9. This indicates very clearly that the spread in experimental data in the form of a wide range of n has very little to do with experimental accuracy: it is primarily due to flow distribution, and the two figures virtually act as a validation for the present approach since the ranges of values for

Fig. 8. Effect of index of Reynolds number (n) on the NTU-effectiveness character of a PHX.

55

Fig. 9. Comparison of effectiveness calculated by constant and variable heat-transfer coefficient models (both ‘U’- and ‘Z’-type PHX with 31 channels).

n and m 2 used are practically observed values. The subsequent results for Fig. 9 and Fig. 10 have been obtained for a value of n= 0.67, which has been chosen from the literature as the most representative value. Fig. 9 shows that the constant and variable heat-transfer coefficient models only match at m 2 = 0 (which is obvious because an equal flow rate in the channels will give an equal heat-transfer coefficient even for n"0). For all other values of m 2, the variable heat-transfer model gives a considerably different result, which increases the effectiveness with the value of m 2. The higher values of effectiveness for the variable heattransfer model can be attributed to the fact that the constant heat-transfer coefficient model brings more heat capacity to some of the channels, thus keeping the heat-transfer rate constant, which virtually indicates an increase in the wastage of available heat capacity in the form of a lowering of effectiveness. However, with the

Fig. 10. Comparison of effectiveness calculated by constant and variable heat-transfer coefficient models (‘U’- and ‘Z’-type PHX with 32 channels).

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B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

flow maldistribution and variation of heat-transfer coefficient are crucially important. The results also indicate that by an inverse solution, it is possible to calculate the values of n using experimental data where the resulting correlation will show a ‘true’ dependence of heat-transfer coefficient in the channel without incorporating a maldistribution effect within the value of n. All the above calculations have been carried out with a strict check on energy balance in the whole apparatus, and the numerical determination of eigen solutions are carried out until an accuracy of 0.1% is achieved in the energy balance.

5. Conclusion and recommendations

Fig. 11. Effectiveness of both ‘U’-type and ‘Z’-type PHX with 24 channels for various heat capacity ratios with a variable heat-transfer coefficient model.

variable heat-transfer coefficient, not only does the heat capacity of some of the channels increase, but so does its corresponding heat-transfer rate, which is reflected by the increase in effectiveness. Fig. 10 shows that the difference between the constant and variable heat-transfer coefficient model is more for a ‘U’-type exchanger than for a ‘Z’-type exchanger, which is due to the asymmetry of flow configuration of different channels in the ‘U’ type. The effect of heat capacity rate is shown in Fig. 11, which shows that the dependence is quite considerable, even though it is significantly different from the constant heat-transfer coefficient model. One more distinguishable result that indicates the importance of the evaluation process of the experimental data is presented in Fig. 12 in the form of the effect of the number of channels on the effectiveness of the heat exchanger. The first inference that can be drawn from this figure is that considering a model of a finite number of channels rather than purely counterflow (which can be assumed at N \35), experiments need to be done with fewer channels. The second observation is that the flow maldistribution characterised by m 2 is important, and this can bring about a considerable error in estimating the heat-transfer coefficient if a model given by Eq. (1) is used. Finally, it is observed that the effect of a variable heat-transfer coefficient is extremely large, and it is absolutely erroneous to consider flow maldistribution without considering the effect of a variable heat-transfer coefficient, as done by some authors [21,22]. The results clearly indicate that not only in the analysis of plate heat exchangers but also for experimental data reduction, the consideration of

A generalized model for thermal simulation of a single pass plate heat exchanger has been presented. The model considers flow variation from channel to channel in a realistic way in keeping with available results from literature. The results indicate that only a consideration of variable flow rate in channels is not sufficient as it tends to underestimate the exchanger performance. A more realistic way is to consider a variable flow rate along with the heat-transfer coefficient, which is flow-dependent. The results indicate that the effects of parameters such as the heat capacity rate ratio, flow configuration, plate geometry and number of channels are considerably different from the unrealistic paradigm of ‘unequal flow rates but equal heat-transfer coefficient’ used so far. Based on the above features, the following recommendations are made:

Fig. 12. Simultaneous effect of flow distribution and number of channels for different models on the performance of a ‘U’-type PHX with 32 channels.

B. Prabhakara Rao et al. / Chemical Engineering and Processing 41 (2002) 49–58

1. The analysis of a plate heat exchanger should always be carried out with an eye towards flow maldistribution and flow dependence of the heat-transfer coefficient in each channel. 2. Determination of the heat-transfer coefficient should be done by solving an inverse problem using experimental values of temperature and an usual definition of NTU to obtain the value of n, by matching with computed plots such as Fig. 8. This will eliminate the entry of the flow distribution effect into the heat-transfer data. 3. A similar model can be built up for a determination of a ‘true’ friction factor as well, which is left for future investigation.

DTlm T T1,in T2,in t

S( U U( W( w/w* x X

Acknowledgements z The authors express their sincere thanks to Professor W. Roetzel of Universita¨ t der Bundeswehr Hamburg for his suggestions regarding defining this problem. The third author (S.K.D.) also expresses his sincere thanks to Alexander von Humboldt Foundation, Germany, because the present work acted as preparation for the research fellowship granted to him. The authors express their deep gratitude to Professor H. Martin of University of Karlsruhe, Germany, for his suggestion in the form of a very helpful discussion cited in Ref. [23].

Appendix A. Nomenclature

A a/a* A( B( C cp D( h L m ¯ m n N NTU Nu Pr Re rh r ru

heat-transfer area for effective plate (m2) cross-sectional area of the intake/outlet conduit (m2) coefficient matrix, Eq. (15) diagonal matrix, Eq. (16) constant in Eq. (1) isobaric specific heat of the fluid (J kg−1 k−1) matrix resulting from boundary conditions heat-transfer coefficient (W m−2 K−1) fluid flow length in a channel (m) mass flow rate (kg s−1) flow-distribution parameter [18,19] exponent of Re number of plates number of transfer units Nusselt number Prandtl number Reynolds number ratio of velocity exponent of Pr ratio of heat-transfer coefficients

ij wc m

57

logarithmic mean temperature difference (K) temperature of fluids (K) inlet temperature of the fluid (1) (K) inlet temperature of the fluid (2) (K) non-dimensional temperature of fluids= T−T1,in T2,in−T1,in matrix of differential temperatures (K) velocity of fluid (m/s) matrix of eigenvalues of the matrix A( coefficient matrix in Eq. (18) dimensionless velocity in the intake/outlet conduit space coordinate for the fluid flow (m) non-dimensional flow path coordinate=x/ L dimensionless flow path coordinate in the port [18,19] j th eigenvalue of matrix A dimensionless volume flow rate in the channels [18,19] effectiveness of the plate heat exchanger= (m; Cp)1(T1,out−T1,in) (m; Cp)min(T2,in−T1,in)

Subscripts and superscripts c channel i th channel i uniform the case of uniform flow distribution amongst channels w plate i th plate wi 1 fluid in the odd channel 2 fluid in the even channel * exhaust port

References [1] E.L. Watson, A.A. McKillop, W.L. Dunkley, R.L. Perry, Plate heat exchanger-flow characteristics, Ind. Eng. Chem. 52 (9) (1960) 733 – 744. [2] B.W. Jackson, R.A. Troupe, Plate heat exchanger design by o-NTU method, Chem. Eng. Prog. Symp. Ser. 62 (64) (1966) 185 – 190. [3] R.A. Buonopane, R.A. Troupe, J.C. Morgan, Heat transfer design method for plate heat exchangers, Chem. Eng. Prog. 59 (7) (1963) 57 – 61. [4] J.J. Marono, J.L. Jechura, Analysis of heat transfer in plate heat exchangers, AIChE Symp. Ser. 81 (245) (1985) 116 – 121. [5] M.K. Bassiouny, Experimentelle und theoretische Untersuchungen u¨ ber Mengenstromverteilung, Druckverlust und Wa¨ rmeu¨ bergang in Plattenwa¨ rmeaustauschern, Forsch-Ber, VDI, Reihe 6, Ni 181, VDI-Verlag, Du¨ sseldorf, 1985. [6] Marriott, J., Personal communication, Alfa Laval Thermal AB, PO Box 1721, S-211 01 Lund, Sweden, 1985. [7] J. Mariott, Performance of an Alfaflex plate heat exchanger, Chem. Eng. Progress 73 (2) (1977) 73 – 78.

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