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Effects of dissipation and temperature-dependent viscosity on the performance of plate heat exchangers
Applied Thermal Engineering 29 (2009) 3132–3139
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Effects of dissipation and temperature-dependent viscosity on the performance of plate heat exchangers Iulian Gherasim a, Nicolas Galanis a,*, Cong Tam Nguyen b a b
Génie mécanique, Université de Sherbrooke, Sherbrooke, QC, Canada J1K 2R1 Génie mécanique, Université de Moncton, Moncton, NB, Canada E1A 3E9
a r t i c l e
i n f o
Article history: Received 6 March 2009 Accepted 2 April 2009 Available online 8 April 2009 Keywords: Plate heat exchangers Effectiveness Variable properties Temperature profiles
a b s t r a c t The fluid temperature profiles in a multi-passage plate heat exchanger and its effectiveness were calculated with a model which includes dissipation, temperature-dependent viscosity and appropriate correlations for the Nusselt number and the friction coefficient. When the viscosity of the two fluids is low (e.g. water) the results are identical to the classic e–NTU relations which are obtained by neglecting dissipation and by assuming that fluid properties and heat transfer coefficients are constant. But, when one of the fluids is very viscous (e.g. glycerol) the temperatures of both fluids are significantly higher while the effectiveness can be higher or lower than the value predicted by the classic relations. In particular, for cases with a very viscous hot fluid, the effectiveness may be even higher than unity. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Thermal energy is an intermediate or final form of practically all energy conversion processes and as a result, heat transfer plays an important role in determining the efficiency of such processes. In order to improve this efficiency it is therefore important to characterize the performance of heat exchangers in the most precise way possible. Traditionally, the performance and selection of heat exchangers is based on the LMTD or e–NTU methods. Both of these have been established using the following two important simplifying assumptions: viscous dissipation is neglected and thermophysical properties are considered to be constant. The classical analytical and graphical results show that the effectiveness increases monotonically with the number of transfer units. However, Bagalalel and Sahin [1] have shown that, in the case of laminar flow of high-viscosity fluids in parallel- and counter-flow heat exchangers, when dissipation is considered the effectiveness can reach a maximum and then decrease as NTU increases. According to their calculations, the maximum value of e depends on the inlet temperatures of the two fluids and on whether their viscosity is assumed to be constant or temperature dependent. The present study examines the influence of these phenomena on the effectiveness of plate heat exchangers (PHE) for both laminar and turbulent flows. PHEs are used extensively in the food, chemical, pulp and paper industries and other applications where the temperature and pressure of the fluids are moderate or low. * Corresponding author. Tel.: +1 819 821 7144; fax: +1 819 821 7163. E-mail address: [email protected] (N. Galanis). 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.04.010
Compared to more traditional designs, such as shell and tube heat exchangers, they are more compact, more efficient, more flexible (since the heat exchange surface can be changed by modifying the number of plates) and easier to clean by taking them apart. The scientific literature comprises many studies on PHEs. These include methods of classification taking into account variables such as the number of passages, the relative direction of flow, the relative position of inlets [2], calculated e–NTU results based on the standard simplifying assumptions [3], as well as experimental studies of the temperature [4] and flow [5] fields. Studies aiming the optimisation of certain design parameters have been published by Wang and Sunden [6] and Gut and Pinto [7]. One-dimensional calculations of the temperature distributions for steady-state operation have been performed by Ribeiro and Cano Andrate [8] while transient effects due to variable inlet temperatures were studies by Das and Murugesan [9] and those due to variable mass flow rates were analysed by Dwivedi and Das [10]. Bassiouny and Martin [11,12] proposed a model for the calculation of pressure drop under non-uniform flow distribution among the passages which was validated experimentally by Rao and Das [13]. Rao et al. [14] used this model to investigate the effect of flow distribution on the thermal performance of PHEs and showed that their effectiveness decreases as the flow non-uniformity increases. CFD has also been used to study fluid flow and heat transfer in PHEs. Thus Lee et al. [15] analysed a 3D, steady-state, periodic and fully-developed flow pattern and determined an optimised geometry by maximising the thermal performance. Croce and d’Agaro [16] computed the flow and thermal fields in a cell defined by four contact points between two successive plates. By plotting the friction coefficient and the Nusselt number as functions of the Rey-
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Nomenclature A B b C cp Dp dh f h k L m N NP NTU Nu Pr p pc Q Re T V
heat transfer surface (m2) width of plates (m) distance between plates (m) heat capacity ratio (C = (mcp)min/(mcp)max) specific heat (J kg1 K1) feeder tube diameter (m) hydraulic diameter, dh = 2b/U(m) friction coefficient convection heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) length of plates (m) mass flow rate (kg s1) number of passages number of subdivisions in each passage number of heat transfer units nusselt number, Nu = h dh/k prandtl number, Pr = lcp/k pressure (Pa) wavelength, see figure 2 (m) heat transfer rate between the two fluids (W) Reynolds number, Re = 2 m/((B + b) l) temperature (K) average velocity (m s1)
W x z
dissipation rate (W) coordinate parallel to the plates (m) non-dimensional coordinate (z = x/L)
Subscripts C cold fluid H hot fluid i, j indicates a passage or plate in inlet out outlet w wall Greek symbols DTln logarithmic mean temperature difference (K) e heat exchanger effectiveness H non-dimensional temperature, H = (T TC,in)/(TH,in TC,in) l dynamic viscosity (kg m1 s1) q density (kg m3) U ratio of real heat transfer surface to that of a flat surface of equal dimensions / chevron angle (°)
nolds number they found that the transition from laminar to turbulent conditions occurs at Re 300. Kanaris et al. [17] calculated the steady-state flow field between three corrugated plates and obtained good agreement with experimental values. They found that the Nusselt number and the friction coefficient increase with the Reynolds number. The present study uses the first law of thermodynamics and appropriate correlations for the heat transfer coefficients between the plates and the fluids to determine the effects of dissipation and temperature-dependent viscosity on the effectiveness of PHEs.
sages and plates are identical, that steady-state conditions prevail, that kinetic and potential energy changes are negligible and that conduction is negligible in both the fluid and the plates. Fig. 1a and b illustrates a cold passage i and a hot passage j, respectively. It should be noted that the mass flow rate of each fluid is in general not distributed uniformly among the corresponding passages (mi1 – mi+1 and mj1 – mj+1). Taking into account the dissipation rate
2. Problem description and modelling
energy conservation for the cold and hot fluids is expressed by the following equations, respectively:
The PHE under consideration has N passages or, equivalently, N + 1 plates. The hot and cold streams are flowing in opposite directions in successive passages (counter-flow arrangement). N is assumed to be an odd number so that the two end passages contain cold fluid. It is also assumed that the dimensions of all the pas-
dW ¼ m
dp
q
¼f
dx qv 2 m f m v 2 dx ¼ dh 2 q 2dh
mi cp;i dT i ¼ hC;i ðT w;i T i Þ B dx þ hC;i ðT w;iþ1 T i Þ B dx þ dW i mj cp;j dT j ¼ hH;j ðT j T w;j Þ B dx þ hH;j ðT j T w;jþ1 Þ B dx dW j
ð1Þ
ð2aÞ ð2bÞ
These relations indicate that the temperature change of the cold fluid is higher than that caused by heat transfer from the plates
Fig. 1. Control volumes. (a) cold channel and (b) hot channel.
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while the temperature change of the hot fluid is lower than that due to heat transfer toward the plates. It should be noted that the cold fluid in the first (i = 1) and last (i = N) passages receives heat from one wall only since heat transfer to or from the environment is assumed negligible. The wall temperatures can be eliminated from Eqs. (2a) and (2b) by noting that energy conservation for each plate implies that at any position the heat flux from the hot fluid to the plate is equal to the corresponding flux from the plate to the cold fluid. Finally, by introducing the following non-dimensional variables.
hi1 hi BL hi1 þ hi mi cp;i hi hiþ1 BL ¼ hi þ hiþ1 mi cp;i
NTU i;i1 ¼
ð3aÞ
NTU i;iþ1
ð3bÞ
Di ¼
fi m2i L 2
2
2dh b B q2 cp;i ðT H;in T C;in Þ
ð3cÞ
the energy equations for the hot and cold fluids can be combined into the following single expression:
dhi ¼ ð1Þiþ1 ðNTU i;i1 ðhi1 hi Þ þ NTU i;iþ1 ðhiþ1 hi Þ þ Di Þ dz
ð4Þ
The application of Eq. (4) for the calculation of the non-dimensional temperature requires the values of the mass flow rates, the convection coefficients and the friction coefficients for each passage. The former were calculated using the model proposed by Bassiouny and Martin [11,12] which determines the fraction of the total mass flow rate in each passage as a function of the number of passages, their cross-section, the average pressure loss and the cross-section of the feeder tube. The local convection and friction coefficients were obtained from the following relations, specifically recommended for PHEs [18]:
Nu ¼ 0:3 Re0:663 Pr0:333 ðl=lm Þ0:17
ð5aÞ
f ¼ a Ren
ð5bÞ
The subscript m in Eq. (5a) indicates that the dynamic viscosity is evaluated at the average fluid temperature. The constants a, n in Eq. (5b) depend on the Reynolds number and chevron angle [18]. These simple relations can be used to analyse the following four very different cases by combining uniform and non-uniform mass flow rate distributions with constant and temperature-dependent viscosity: – For uniform mass flow rate distributions and constant viscosities the values of NuC, fC are independent of x and are identical for all the cold passages; similarly, the values of NuH, fH are independent of x and are identical for all the hot passages; however, in general, NuC – NuH and fC – fH. – For uniform mass flow rate distributions and temperaturedependent viscosities the values of NuC, fC vary with x but are identical for all the cold passages; similarly, the values of NuH, fH vary with x but are identical for all the hot passages. – For non-uniform mass flow rate distributions and constant viscosities the values of NuC, fC are independent of x but are not identical in all the cold passages; similarly, the values of NuH, fH are independent of x but are not identical in all the hot passages. – For non-uniform mass flow rate distributions and temperaturedependent viscosities the values of NuC, fC, NuH and fH vary with x and are not identical in different passages. The geometry of the plates used in the present study is the same as that used in the experimental work by Tereda et al. [19]. Specif-
Fig. 2. Configuration and geometrical features of the plates.
ically, the dimensions defined in Fig. 2 are L = 600 mm (except when otherwise specified), B = 218 mm, b = 2.9 mm, Dp = 70 mm and pc = 14 mm. The chevron angle is u = 45°. The ratio U of the real heat transfer surface to that of a flat surface of equal dimensions (=B L) was calculated from the relations by Fernandes et al. [20] and is equal to 1.05. Therefore the hydraulic diameter of the geometrically identical passages is dh = 2b/U. The two fluids used in this study are water and glycerol, the same as those used by Bagalalel and Sahin [1]. Their relevant properties are specified in Table 1. It should be noted that the viscosity of glycerol for temperatures between 20 and 80 °C is several orders of magnitude higher than the corresponding values for water. 3. Numerical solution and validation The initial step of the numerical procedure determines the distribution of the hot and cold fluid mass flow rates among the corresponding passages, either by assuming this distribution is uniform or by calculating it according to the method proposed by Bassiouny and Martin [11,12]. Then, in order to solve Eq. (4), each passage is subdivided into NP sections. The resulting system of NP N unknown temperatures is solved using a fourth order Runge–Kutta method. If the viscosities are considered to be constant, this solution is the desired temperature distribution. Otherwise, the local values of the viscosities are obtained from the correlations
Table 1 Properties of water and glycerol used in the present study. Water
Glycerol [1]
Density (kg m3) Specific heat (J kg1 K1) Conductivity (W m1 K1)
997.4 4147.8 0.6025
1260 2428 0.285
Viscosity of water (kg m1 s1)
l ¼ 107 exp
Viscosity of glycerol (kg m1 s1)
l ¼ 148
1:126460:039638T 17:29769103 T
T 52:4 293
exp 23100
1 T
1 293
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in Table 1 and a new temperature distribution is obtained by solving Eq. (4). This iterative procedure is repeated until the nondimensional temperature at every node does not change significantly between successive iterations. The results presented here are within 1010 of the corresponding values of the previous iteration. In order to test the validity of the code and the sensitivity of the results on the number of sections NP a PHE with three passages, uniform mass flow rate distributions, no dissipation and constant viscosity, was considered. For this simple case it is possible to integrate analytically Eq. (4) (see Appendix). By comparing numerical predictions with these analytical temperature values it is, therefore, possible to evaluate the influence of the number of sections NP on the accuracy of the numerical results. Table 2 shows the absolute value of the maximum difference between the corresponding analytically and numerically calculated temperatures for three values of NP and three values of the heat capacity ratio. These results were calculated for water with TH,in = 80 °C and TC,in = 30 °C. We note that the error is always less than 0.5 °C. It increases as the heat capacity ratio decreases and it decreases as NP increases. In view of these results all subsequent calculations were performed using 200 sections for each passage.
Q max ¼ ðm cp Þmin ðT H;in T C;in Þ
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ð6bÞ
The actual heat transfer Q is evaluated by summing up the first two terms in the square brackets of Eq. (4) over the NP (N 1)/2 sections of the hot passages or the NP (N + 1)/2 sections of the cold passages. In fact, these two sums (indicated by QH and QC, respectively) should be equal but may be slightly different due to numerical roundoffs. Their comparison provides an indication of the accuracy of the results. As mentioned before, when dissipation is considered, QH is greater than the corresponding enthalpy change. The opposite applies to QC which is smaller than mC cpC (TC,out TC,in). Finally, the number of transfer units is calculated from its definition
NTU ¼ U A=ðm cp Þmin
ð7aÞ
where
U A ¼ Q =DT ln
ð7bÞ
The calculation of the logarithmic mean temperature-difference requires the temperatures of the hot and cold fluids at the outlet of the PHE. These are obtained by assuming that the streams issuing from the corresponding passages mix adiabatically.
4. Results and discussion
4.1. Cases with low viscosity fluids (water)
The results include temperature distributions in multi-passage PHEs as well as e–NTU graphical relations calculated with and without dissipation for constant or temperature-dependent viscosities. The effectiveness is defines in the usual way.
Fig. 3 shows the temperature profiles in the passages of a PHE with N = 19. The hot and cold fluids are both water with an identical flow rate equal to 2 l/s. Their inlet temperatures are 80 °C and 30 °C, respectively. The corresponding Reynolds numbers are 3890 for the hot fluid and 3500 for the cold one. These results were calculated with constant viscosity, assuming a uniform distribution of the flow rates among the passages and neglecting dissipation. The two heat transfer coefficients hC and hH are therefore constant throughout the corresponding passages. The results show that in this particular case the temperature variation is almost linear. They also show that in the first and last passages (passages 1 and 19) the temperature of the cold fluid does not increase as much as in the central passages (passages 3–17) since the fluid therein receives heat from one side only. As a consequence, the temperature of the hot fluid in passages 2 and 18 is lower than in the other hot passages. The temperatures of the cold and hot fluid at the outlet of the PHE are approximately 61 °C and 49 °C. The heat exchanged between the two fluids is approximately
e ¼ Q =Q max
ð6aÞ
where
Table 2 Effect of number of sections on the difference (in °C) between numerical and analytical temperatures in a PHE (N = 3; uniform flow distribution; no dissipation; L = 5 m; flow rate of cold fluid 2 l/s; flow rate of hot fluid 0.5, 1 or 2 l/s). C
Number of sections NP 10
100
200
0.25 0.5 1
2.63 101 6.12 103 3.7 1010
8.14 106 2.51 107 2.16 1010
4.82 107 1.52 108 2.03 1010
Fig. 3. Temperature profiles in a water–water PHE with 19 channels (L = 0.6 m).
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256.4 kW (QH = 256.44 kW and QC = 256.42 kW) while the DTln is 19 °C and the effectiveness is 0.6. Fig. 4 shows the calculated values of the effectiveness for this PHE (N = 19) with water on both sides. The inlet temperatures are again 80 °C and 30 °C. The heat capacity ratio was changed by varying the flow rate of one of the two fluids while the NTU was changed by varying the length L of the plates. These results are identical with those obtained from the classic analytical expression for counter-flow heat exchangers and provide further validation of the proposed model and the adopted numerical methodology. Analogous results for this PHE with water at TH,in = 80 °C and TC,in = 30 °C taking into account dissipation and considering constant or temperature-dependent viscosity show that these effects are negligible. Thus, the assumptions used to obtain the theoretical NTU–e relations are in this case valid. Furthermore, calculations with non-uniform flow rate distributions also indicate that the resulting effect is very small in the present case. 4.2. Cases with high-viscosity fluid The above remarks regarding the effects of dissipation and temperature-dependent viscosity are, however, not true if one of the two fluids is very viscous. Fig. 5 illustrates the effects of dissipation and variable viscosity on the temperature profiles
in a PHE with N = 9 and uniform flow distribution among the passages. The hot fluid is water with a flow rate of 1.5 l/s and TH,in = 80 °C (ReH = 6056). The cold fluid is glycerol with a flow rate of 2.018 l/s and TC,in = 20 °C (ReC = 0.284). The ratio of heat capacity rates is equal to 1. By comparing the temperature profiles ‘‘with” and ‘‘without” dissipation in a particular passage we note the significant temperature increases in both the cold and hot fluids. In fact, dissipation in the very viscous cold fluid causes the temperature increase in the glycerol which as a result reduces the heat transferred from the hot fluid and therefore increases the temperature of the water at the exit from the PHE. In other words, the temperature increase in the water is not due to dissipation within the hot passages. It is interesting to note that the temperature difference between the two fluids in adjoining passages is higher over approximately 90% of the length when dissipation is accounted for. Only near the inlet of the cold fluid is this temperature difference higher when dissipation is neglected. The temperatures of the hot and cold fluid at the outlet of the PHE are, respectively, 39 °C, 61 °C without dissipation and 47.7 °C, 71 °C with dissipation. The heat exchanged between the two fluids is approximately 254.3 kW (QH = 254 kW and QC = 254.6 kW) without dissipation and 200.1 kW (QH = 200.6 kW and QC = 199.6 kW) with dissipation. Thus, in this case, dissipation causes a 27% reduction in the heat transfer rate.
Fig. 4. Effectiveness of a water–water PHE with19 channels.
Fig. 5. Temperature profiles in a water–glycerol PHE with and without dissipation (L = 5 m).
I. Gherasim et al. / Applied Thermal Engineering 29 (2009) 3132–3139
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Fig. 6. Effect of dissipation on the effectiveness of a water–glycerol PHE (C = 1).
Fig. 6 shows the calculated effectiveness of the PHE with N = 9 for a ratio of heat capacity rates equal to one and uniform flow rate distribution. The relation between e and NTU was calculated using three different approaches. When dissipation is neglected the numerical predictions are in perfect agreement with the corresponding results of the classic analytical relation between e and NTU. When dissipation is taken into consideration the effectiveness is, in the present case, reduced. This reduction is more important for high values of NTU and for the case of temperature-dependent viscosity. Thus, for example, for NTU = 4 the theoretical (neglecting dissipation) effectiveness is 0.799 while when dissipation is taken into consideration it is 0.704 for constant viscosity (reduction of 11.9%) and 0.637 for temperature-dependent viscosity (reduction of 20.4%). Fig. 7 shows analogous results for a case with glycerol as the hot fluid (flow rate = 1.35 l/s, TH,in = 80 °C, ReH = 0.236) and water as the cold fluid (flow rate = 1 l/s, TC,in = 20 °C, ReC = 3230). The ratio of heat capacity rates is equal to 1, N = 9 and the flow rate distribution is assumed to be uniform as in the case of Fig. 6. As a result, when dissipation is neglected the relationship between e and NTU is the same for these two cases and is identical to the theoretical one predicted by the classic analytical relation. However, when dissipation is considered the calculated effectiveness is in the present case higher than the theoretical one. This surprising result, totally different from that in Fig. 6, is obviously due to the fact that the high-viscosity fluid (glycerol) is in the present case the hot fluid. Thus, dissipation in the glycerol keeps its temperature high
despite the heat transferred to the water and increases the amount of heat exchanged between the two fluids. Indeed, the calculated value of this quantity is approximately 187.7 kW (QH = 187.5 kW and QC = 187.9 kW) without dissipation and 196.5 kW (QH = 196.2 kW and QC = 196.8 kW) with dissipation. Thus, in this case, dissipation causes a 4.7% increase in the heat transfer rate. Even more surprising are the results of Fig. 8 which were obtained by reducing the flow rate of the glycerol by 50% from that used to calculate those in Fig. 7 while keeping all the other conditions unchanged. Thus for Fig. 8 the hot fluid is glycerol (flow rate = 0.675 l/s, TH,in = 80 °C, ReH = 0.118), the cold fluid is water (flow rate = 1 l/s, TC,in = 20 °C, ReC = 3230), the ratio of heat capacity rates is equal to 0.5, N = 9 and the flow rate distribution is assumed to be uniform. Once again, when dissipation is neglected the relationship between e and NTU is identical to the theoretical one predicted by the classic analytical relation. When dissipation with constant viscosity is taken into consideration the effectiveness increases as in the previous case, albeit more rapidly. When dissipation and temperature-dependent viscosity are considered the effectiveness increases even more and reaches values higher than one for NTU > 3.5. At first glance, this result seems physically unrealistic and unacceptable. However, it can be explained and justified by noting that in the present case Eq. (6b) becomes
Q max ¼ ðm cp ÞH ðT H;in T C;in Þ while energy conservation for the hot fluid is
Fig. 7. Effect of dissipation on the effectiveness of a glycerol–water PHE (C = 1).
ð8aÞ
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Fig. 8. Effect of dissipation on the effectiveness of a glycerol–water PHE (C = 0.5).
Q ¼ ðm cp ÞH ðT H;in T H;out Þ þ W diss;H
ð8bÞ
files in the two cold passages are the same (h1 = h3). Taking into consideration this symmetry, energy conservation (Eq. (4)) for the hot and cold fluid are, respectively,
ð8cÞ
dh2 ¼ 2 NTU H ðh2 h1 Þ dz dh1 ¼ NTU C ðh2 h1 Þ dz
Therefore
e¼
T H;in T H;out W diss;H þ T H;in T C;in Q max
The first term on the right hand side of this last expression is the conventional expression for the effectiveness and is always smaller than one since TH,out > TC,in (in the case of Fig. 8, TH,out = 40.4 °C while TC,in = 20 °C). On the other hand, the dissipation rate can be an important fraction of Qmax and therefore, the value of the effectiveness when dissipation is taken into consideration does not have an upper bound as is the case when dissipation is neglected. Stated differently, the result e > 1, or Q > Qmax, respects both the first and second laws of thermodynamics and is therefore acceptable. 5. Conclusions A one-dimensional model of heat transfer in a multi-passage plate heat exchanger using appropriate correlations for the Nusselt number and the friction coefficient was used to illustrate the effects of dissipation and temperature-dependent viscosity on the temperature profiles of the fluids and the effectiveness of the PHE. The results show that these effects can be neglected if the viscosity of both fluids is low (such as water). In that case the present results are identical with those presented in heat transfer textbooks. On the other hand, if one of the fluids is very viscous (e.g. glycerol) neglecting dissipation leads to significantly erroneous results: the temperature of both fluids is underestimated while the effectiveness of the PHE may be over-, or under-estimated depending on whether the high-viscosity fluid is the hot or cold fluid and on the ratio of the heat capacity rates. Acknowledgements This project is part of the R&D program of the NSERC Chair in Industrial Energy Efficiency established in 2006 at Université de Sherbrooke. The authors acknowledge the support of the Natural Sciences & Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and CANMET Energy Technology Center. Appendix A For a PHE with N = 3, uniform flow distribution, no dissipation and constant viscosity the heat transfer coefficients hC in passages #1 and #3 are identical and do not vary with z. Similarly hH in passage #2 is also independent of z. Furthermore the temperature pro-
ðA:1Þ ðA:2Þ
Taking the difference of these two relations and integrating gives
h2 h1 ¼ C 1 expða zÞ
ðA:3Þ
where
a ¼ 2 NTU H NTU C
ðA:4Þ
Replacing Eq. (A.3) in Eq. (A.2) and integrating gives the general solution for h1 and the corresponding expression for h2 is obtained from Eq. (A.3). The two integration constants are evaluated by applying the boundary conditions (h1 = 0 at z = 0, h2 = 1 at z = 1) and the final expression for the temperature distributions in the cold and hot passages are, respectively:
NTU C ðexpða zÞ 1Þ NTU C expðaÞ þ a expðaÞ NTU C ðNTU C þ aÞ expða zÞ NTU C h2 ¼ NTU C expðaÞ þ a expðaÞ NTU C
h1 ¼ h3 ¼
ðA:5Þ ðA:6Þ
References [1] S.M. Bagalagel, A.Z. Sahin, Design optimization of heat exchangers with highviscosity fluids, International Journal of Energy Research 26 (2002) 867–880. [2] J.A.W. Gut, J.M. Pinto, Modeling of plate heat exchangers with generalized configuration, International Journal of Heat and Mass Transfer 46 (2003) 2571–2585. [3] S.G. Kandlikar, R.K. Shah, Multipass heat exchangers – effectiveness–NTU results and guidelines for selecting pass arrangements, ASME Journal of Heat Transfer 111 (1989) 300–313. [4] K. Okada et al., Temperature distribution in plate type heat exchangers, Kagaku Kogaku 34 (1) (1970) 93–95. [5] A. Lozano, F. Barreras, N. Fueyo, S. Santodomingo, The flow in an oil/water plate heat exchanger for the automotive industry, Applied Thermal Engineering 28 (10) (2008) 1109–1117. [6] L. Wang, B. Sunden, Optimal design of plate heat exchangers with and without pressure drop specifications, Applied Thermal Engineering 23 (2003) 295–311. [7] J.A.W. Gut, J.M. Pinto, Optimal configuration design for plate heat exchangers, International Journal of Heat and Mass Transfer 47 (2004) 4833–4848. [8] C.P. Ribeiro, M.H. Cano Andrade, An algorithm for steady-state simulation of plate heat exchangers, Journal of Food Engineering 53 (2002) 59–66. [9] S.K. Das, K. Murugesan, Transient response of multipass plate heat exchangers with axial dispersion in fluid, International Journal of Heat and Mass Transfer 43 (2000) 4327–4345. [10] A.K. Dwivedi, S.K. Das, Dynamics of plate heat exchangers subject to flow variations, International Journal of Heat and Mass Transfer 50 (2007) 2733–2743.
I. Gherasim et al. / Applied Thermal Engineering 29 (2009) 3132–3139 [11] M.K. Bassiouny, H. Martin, Flow distribution and pressure drop in plate heat exchangers – 1: U-type arrangement, Chemical Engineering Science 39 (4) (1983) 693–700. [12] M.K. Bassiouny, H. Martin, Flow distribution and pressure drop in plate heat exchangers – 2: Z-type arrangement, Chemical Engineering Science 39 (4) (1983) 701–704. [13] B.P. Rao, S.K. Das, An experimental study on the influence of flow maldistribution on the pressure drop across a plate heat exchanger, Journal of Fluids Engineering 126 (2004) 680–691. [14] B.P. Rao, K. Kumar, S.K. Das, Effect of flow distribution to the channels on the thermal performance of a plate heat exchanger, Chemical Engineering and Processing 41 (2002) 49–58. [15] K.S. Lee, W.S. Kim, J.M. Si, Optimal shape and arrangement of staggered pins in the channel of a plate heat exchanger, International Journal of Heat and Mass Transfer 44 (2001) 3223–3231.
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[16] G. Croce, P. D’Agaro, Numerical analysis of forced convection in plate and frame heat exchangers, International Journal of Numerical Methods for Heat and Fluid Flow 12 (6) (2002) 756–771. [17] A.G. Kanaris, K.A. Mouza, S.V. Paras, Flow and heat transfer prediction in a corrugated plate heat exchanger using a CFD code, Chemical Engineering Technology 29 (8) (2006) 923–930. [18] S. Kakaç, H. Liu, Heat Exchangers: Selection, Rating and Thermal Design, CRC Press, 1992. [19] F.A. Tereda, N. Srihari, B. Sunden, S.K. Das, Experimental investigation on port to channel flow maldistribution in plate heat exchangers, Heat Transfer Engineering 28 (5) (2007) 435–443. [20] C.S. Fernandes, R. Dias, J.M. Nòbrega, J.M. Maia, Laminar flow in chevron-type plate exchangers: CFD analysis of tortuosity shape factor and friction factor, Chemical Engineering and Processing 46 (2007) 825–833.
Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Effects of dissipation and temperature-dependent viscosity on the performance of plate heat exchangers Iulian Gherasim a, Nicolas Galanis a,*, Cong Tam Nguyen b a b
Génie mécanique, Université de Sherbrooke, Sherbrooke, QC, Canada J1K 2R1 Génie mécanique, Université de Moncton, Moncton, NB, Canada E1A 3E9
a r t i c l e
i n f o
Article history: Received 6 March 2009 Accepted 2 April 2009 Available online 8 April 2009 Keywords: Plate heat exchangers Effectiveness Variable properties Temperature profiles
a b s t r a c t The fluid temperature profiles in a multi-passage plate heat exchanger and its effectiveness were calculated with a model which includes dissipation, temperature-dependent viscosity and appropriate correlations for the Nusselt number and the friction coefficient. When the viscosity of the two fluids is low (e.g. water) the results are identical to the classic e–NTU relations which are obtained by neglecting dissipation and by assuming that fluid properties and heat transfer coefficients are constant. But, when one of the fluids is very viscous (e.g. glycerol) the temperatures of both fluids are significantly higher while the effectiveness can be higher or lower than the value predicted by the classic relations. In particular, for cases with a very viscous hot fluid, the effectiveness may be even higher than unity. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Thermal energy is an intermediate or final form of practically all energy conversion processes and as a result, heat transfer plays an important role in determining the efficiency of such processes. In order to improve this efficiency it is therefore important to characterize the performance of heat exchangers in the most precise way possible. Traditionally, the performance and selection of heat exchangers is based on the LMTD or e–NTU methods. Both of these have been established using the following two important simplifying assumptions: viscous dissipation is neglected and thermophysical properties are considered to be constant. The classical analytical and graphical results show that the effectiveness increases monotonically with the number of transfer units. However, Bagalalel and Sahin [1] have shown that, in the case of laminar flow of high-viscosity fluids in parallel- and counter-flow heat exchangers, when dissipation is considered the effectiveness can reach a maximum and then decrease as NTU increases. According to their calculations, the maximum value of e depends on the inlet temperatures of the two fluids and on whether their viscosity is assumed to be constant or temperature dependent. The present study examines the influence of these phenomena on the effectiveness of plate heat exchangers (PHE) for both laminar and turbulent flows. PHEs are used extensively in the food, chemical, pulp and paper industries and other applications where the temperature and pressure of the fluids are moderate or low. * Corresponding author. Tel.: +1 819 821 7144; fax: +1 819 821 7163. E-mail address: [email protected] (N. Galanis). 1359-4311/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2009.04.010
Compared to more traditional designs, such as shell and tube heat exchangers, they are more compact, more efficient, more flexible (since the heat exchange surface can be changed by modifying the number of plates) and easier to clean by taking them apart. The scientific literature comprises many studies on PHEs. These include methods of classification taking into account variables such as the number of passages, the relative direction of flow, the relative position of inlets [2], calculated e–NTU results based on the standard simplifying assumptions [3], as well as experimental studies of the temperature [4] and flow [5] fields. Studies aiming the optimisation of certain design parameters have been published by Wang and Sunden [6] and Gut and Pinto [7]. One-dimensional calculations of the temperature distributions for steady-state operation have been performed by Ribeiro and Cano Andrate [8] while transient effects due to variable inlet temperatures were studies by Das and Murugesan [9] and those due to variable mass flow rates were analysed by Dwivedi and Das [10]. Bassiouny and Martin [11,12] proposed a model for the calculation of pressure drop under non-uniform flow distribution among the passages which was validated experimentally by Rao and Das [13]. Rao et al. [14] used this model to investigate the effect of flow distribution on the thermal performance of PHEs and showed that their effectiveness decreases as the flow non-uniformity increases. CFD has also been used to study fluid flow and heat transfer in PHEs. Thus Lee et al. [15] analysed a 3D, steady-state, periodic and fully-developed flow pattern and determined an optimised geometry by maximising the thermal performance. Croce and d’Agaro [16] computed the flow and thermal fields in a cell defined by four contact points between two successive plates. By plotting the friction coefficient and the Nusselt number as functions of the Rey-
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Nomenclature A B b C cp Dp dh f h k L m N NP NTU Nu Pr p pc Q Re T V
heat transfer surface (m2) width of plates (m) distance between plates (m) heat capacity ratio (C = (mcp)min/(mcp)max) specific heat (J kg1 K1) feeder tube diameter (m) hydraulic diameter, dh = 2b/U(m) friction coefficient convection heat transfer coefficient (W m2 K1) thermal conductivity (W m1 K1) length of plates (m) mass flow rate (kg s1) number of passages number of subdivisions in each passage number of heat transfer units nusselt number, Nu = h dh/k prandtl number, Pr = lcp/k pressure (Pa) wavelength, see figure 2 (m) heat transfer rate between the two fluids (W) Reynolds number, Re = 2 m/((B + b) l) temperature (K) average velocity (m s1)
W x z
dissipation rate (W) coordinate parallel to the plates (m) non-dimensional coordinate (z = x/L)
Subscripts C cold fluid H hot fluid i, j indicates a passage or plate in inlet out outlet w wall Greek symbols DTln logarithmic mean temperature difference (K) e heat exchanger effectiveness H non-dimensional temperature, H = (T TC,in)/(TH,in TC,in) l dynamic viscosity (kg m1 s1) q density (kg m3) U ratio of real heat transfer surface to that of a flat surface of equal dimensions / chevron angle (°)
nolds number they found that the transition from laminar to turbulent conditions occurs at Re 300. Kanaris et al. [17] calculated the steady-state flow field between three corrugated plates and obtained good agreement with experimental values. They found that the Nusselt number and the friction coefficient increase with the Reynolds number. The present study uses the first law of thermodynamics and appropriate correlations for the heat transfer coefficients between the plates and the fluids to determine the effects of dissipation and temperature-dependent viscosity on the effectiveness of PHEs.
sages and plates are identical, that steady-state conditions prevail, that kinetic and potential energy changes are negligible and that conduction is negligible in both the fluid and the plates. Fig. 1a and b illustrates a cold passage i and a hot passage j, respectively. It should be noted that the mass flow rate of each fluid is in general not distributed uniformly among the corresponding passages (mi1 – mi+1 and mj1 – mj+1). Taking into account the dissipation rate
2. Problem description and modelling
energy conservation for the cold and hot fluids is expressed by the following equations, respectively:
The PHE under consideration has N passages or, equivalently, N + 1 plates. The hot and cold streams are flowing in opposite directions in successive passages (counter-flow arrangement). N is assumed to be an odd number so that the two end passages contain cold fluid. It is also assumed that the dimensions of all the pas-
dW ¼ m
dp
q
¼f
dx qv 2 m f m v 2 dx ¼ dh 2 q 2dh
mi cp;i dT i ¼ hC;i ðT w;i T i Þ B dx þ hC;i ðT w;iþ1 T i Þ B dx þ dW i mj cp;j dT j ¼ hH;j ðT j T w;j Þ B dx þ hH;j ðT j T w;jþ1 Þ B dx dW j
ð1Þ
ð2aÞ ð2bÞ
These relations indicate that the temperature change of the cold fluid is higher than that caused by heat transfer from the plates
Fig. 1. Control volumes. (a) cold channel and (b) hot channel.
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while the temperature change of the hot fluid is lower than that due to heat transfer toward the plates. It should be noted that the cold fluid in the first (i = 1) and last (i = N) passages receives heat from one wall only since heat transfer to or from the environment is assumed negligible. The wall temperatures can be eliminated from Eqs. (2a) and (2b) by noting that energy conservation for each plate implies that at any position the heat flux from the hot fluid to the plate is equal to the corresponding flux from the plate to the cold fluid. Finally, by introducing the following non-dimensional variables.
hi1 hi BL hi1 þ hi mi cp;i hi hiþ1 BL ¼ hi þ hiþ1 mi cp;i
NTU i;i1 ¼
ð3aÞ
NTU i;iþ1
ð3bÞ
Di ¼
fi m2i L 2
2
2dh b B q2 cp;i ðT H;in T C;in Þ
ð3cÞ
the energy equations for the hot and cold fluids can be combined into the following single expression:
dhi ¼ ð1Þiþ1 ðNTU i;i1 ðhi1 hi Þ þ NTU i;iþ1 ðhiþ1 hi Þ þ Di Þ dz
ð4Þ
The application of Eq. (4) for the calculation of the non-dimensional temperature requires the values of the mass flow rates, the convection coefficients and the friction coefficients for each passage. The former were calculated using the model proposed by Bassiouny and Martin [11,12] which determines the fraction of the total mass flow rate in each passage as a function of the number of passages, their cross-section, the average pressure loss and the cross-section of the feeder tube. The local convection and friction coefficients were obtained from the following relations, specifically recommended for PHEs [18]:
Nu ¼ 0:3 Re0:663 Pr0:333 ðl=lm Þ0:17
ð5aÞ
f ¼ a Ren
ð5bÞ
The subscript m in Eq. (5a) indicates that the dynamic viscosity is evaluated at the average fluid temperature. The constants a, n in Eq. (5b) depend on the Reynolds number and chevron angle [18]. These simple relations can be used to analyse the following four very different cases by combining uniform and non-uniform mass flow rate distributions with constant and temperature-dependent viscosity: – For uniform mass flow rate distributions and constant viscosities the values of NuC, fC are independent of x and are identical for all the cold passages; similarly, the values of NuH, fH are independent of x and are identical for all the hot passages; however, in general, NuC – NuH and fC – fH. – For uniform mass flow rate distributions and temperaturedependent viscosities the values of NuC, fC vary with x but are identical for all the cold passages; similarly, the values of NuH, fH vary with x but are identical for all the hot passages. – For non-uniform mass flow rate distributions and constant viscosities the values of NuC, fC are independent of x but are not identical in all the cold passages; similarly, the values of NuH, fH are independent of x but are not identical in all the hot passages. – For non-uniform mass flow rate distributions and temperaturedependent viscosities the values of NuC, fC, NuH and fH vary with x and are not identical in different passages. The geometry of the plates used in the present study is the same as that used in the experimental work by Tereda et al. [19]. Specif-
Fig. 2. Configuration and geometrical features of the plates.
ically, the dimensions defined in Fig. 2 are L = 600 mm (except when otherwise specified), B = 218 mm, b = 2.9 mm, Dp = 70 mm and pc = 14 mm. The chevron angle is u = 45°. The ratio U of the real heat transfer surface to that of a flat surface of equal dimensions (=B L) was calculated from the relations by Fernandes et al. [20] and is equal to 1.05. Therefore the hydraulic diameter of the geometrically identical passages is dh = 2b/U. The two fluids used in this study are water and glycerol, the same as those used by Bagalalel and Sahin [1]. Their relevant properties are specified in Table 1. It should be noted that the viscosity of glycerol for temperatures between 20 and 80 °C is several orders of magnitude higher than the corresponding values for water. 3. Numerical solution and validation The initial step of the numerical procedure determines the distribution of the hot and cold fluid mass flow rates among the corresponding passages, either by assuming this distribution is uniform or by calculating it according to the method proposed by Bassiouny and Martin [11,12]. Then, in order to solve Eq. (4), each passage is subdivided into NP sections. The resulting system of NP N unknown temperatures is solved using a fourth order Runge–Kutta method. If the viscosities are considered to be constant, this solution is the desired temperature distribution. Otherwise, the local values of the viscosities are obtained from the correlations
Table 1 Properties of water and glycerol used in the present study. Water
Glycerol [1]
Density (kg m3) Specific heat (J kg1 K1) Conductivity (W m1 K1)
997.4 4147.8 0.6025
1260 2428 0.285
Viscosity of water (kg m1 s1)
l ¼ 107 exp
Viscosity of glycerol (kg m1 s1)
l ¼ 148
1:126460:039638T 17:29769103 T
T 52:4 293
exp 23100
1 T
1 293
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in Table 1 and a new temperature distribution is obtained by solving Eq. (4). This iterative procedure is repeated until the nondimensional temperature at every node does not change significantly between successive iterations. The results presented here are within 1010 of the corresponding values of the previous iteration. In order to test the validity of the code and the sensitivity of the results on the number of sections NP a PHE with three passages, uniform mass flow rate distributions, no dissipation and constant viscosity, was considered. For this simple case it is possible to integrate analytically Eq. (4) (see Appendix). By comparing numerical predictions with these analytical temperature values it is, therefore, possible to evaluate the influence of the number of sections NP on the accuracy of the numerical results. Table 2 shows the absolute value of the maximum difference between the corresponding analytically and numerically calculated temperatures for three values of NP and three values of the heat capacity ratio. These results were calculated for water with TH,in = 80 °C and TC,in = 30 °C. We note that the error is always less than 0.5 °C. It increases as the heat capacity ratio decreases and it decreases as NP increases. In view of these results all subsequent calculations were performed using 200 sections for each passage.
Q max ¼ ðm cp Þmin ðT H;in T C;in Þ
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ð6bÞ
The actual heat transfer Q is evaluated by summing up the first two terms in the square brackets of Eq. (4) over the NP (N 1)/2 sections of the hot passages or the NP (N + 1)/2 sections of the cold passages. In fact, these two sums (indicated by QH and QC, respectively) should be equal but may be slightly different due to numerical roundoffs. Their comparison provides an indication of the accuracy of the results. As mentioned before, when dissipation is considered, QH is greater than the corresponding enthalpy change. The opposite applies to QC which is smaller than mC cpC (TC,out TC,in). Finally, the number of transfer units is calculated from its definition
NTU ¼ U A=ðm cp Þmin
ð7aÞ
where
U A ¼ Q =DT ln
ð7bÞ
The calculation of the logarithmic mean temperature-difference requires the temperatures of the hot and cold fluids at the outlet of the PHE. These are obtained by assuming that the streams issuing from the corresponding passages mix adiabatically.
4. Results and discussion
4.1. Cases with low viscosity fluids (water)
The results include temperature distributions in multi-passage PHEs as well as e–NTU graphical relations calculated with and without dissipation for constant or temperature-dependent viscosities. The effectiveness is defines in the usual way.
Fig. 3 shows the temperature profiles in the passages of a PHE with N = 19. The hot and cold fluids are both water with an identical flow rate equal to 2 l/s. Their inlet temperatures are 80 °C and 30 °C, respectively. The corresponding Reynolds numbers are 3890 for the hot fluid and 3500 for the cold one. These results were calculated with constant viscosity, assuming a uniform distribution of the flow rates among the passages and neglecting dissipation. The two heat transfer coefficients hC and hH are therefore constant throughout the corresponding passages. The results show that in this particular case the temperature variation is almost linear. They also show that in the first and last passages (passages 1 and 19) the temperature of the cold fluid does not increase as much as in the central passages (passages 3–17) since the fluid therein receives heat from one side only. As a consequence, the temperature of the hot fluid in passages 2 and 18 is lower than in the other hot passages. The temperatures of the cold and hot fluid at the outlet of the PHE are approximately 61 °C and 49 °C. The heat exchanged between the two fluids is approximately
e ¼ Q =Q max
ð6aÞ
where
Table 2 Effect of number of sections on the difference (in °C) between numerical and analytical temperatures in a PHE (N = 3; uniform flow distribution; no dissipation; L = 5 m; flow rate of cold fluid 2 l/s; flow rate of hot fluid 0.5, 1 or 2 l/s). C
Number of sections NP 10
100
200
0.25 0.5 1
2.63 101 6.12 103 3.7 1010
8.14 106 2.51 107 2.16 1010
4.82 107 1.52 108 2.03 1010
Fig. 3. Temperature profiles in a water–water PHE with 19 channels (L = 0.6 m).
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256.4 kW (QH = 256.44 kW and QC = 256.42 kW) while the DTln is 19 °C and the effectiveness is 0.6. Fig. 4 shows the calculated values of the effectiveness for this PHE (N = 19) with water on both sides. The inlet temperatures are again 80 °C and 30 °C. The heat capacity ratio was changed by varying the flow rate of one of the two fluids while the NTU was changed by varying the length L of the plates. These results are identical with those obtained from the classic analytical expression for counter-flow heat exchangers and provide further validation of the proposed model and the adopted numerical methodology. Analogous results for this PHE with water at TH,in = 80 °C and TC,in = 30 °C taking into account dissipation and considering constant or temperature-dependent viscosity show that these effects are negligible. Thus, the assumptions used to obtain the theoretical NTU–e relations are in this case valid. Furthermore, calculations with non-uniform flow rate distributions also indicate that the resulting effect is very small in the present case. 4.2. Cases with high-viscosity fluid The above remarks regarding the effects of dissipation and temperature-dependent viscosity are, however, not true if one of the two fluids is very viscous. Fig. 5 illustrates the effects of dissipation and variable viscosity on the temperature profiles
in a PHE with N = 9 and uniform flow distribution among the passages. The hot fluid is water with a flow rate of 1.5 l/s and TH,in = 80 °C (ReH = 6056). The cold fluid is glycerol with a flow rate of 2.018 l/s and TC,in = 20 °C (ReC = 0.284). The ratio of heat capacity rates is equal to 1. By comparing the temperature profiles ‘‘with” and ‘‘without” dissipation in a particular passage we note the significant temperature increases in both the cold and hot fluids. In fact, dissipation in the very viscous cold fluid causes the temperature increase in the glycerol which as a result reduces the heat transferred from the hot fluid and therefore increases the temperature of the water at the exit from the PHE. In other words, the temperature increase in the water is not due to dissipation within the hot passages. It is interesting to note that the temperature difference between the two fluids in adjoining passages is higher over approximately 90% of the length when dissipation is accounted for. Only near the inlet of the cold fluid is this temperature difference higher when dissipation is neglected. The temperatures of the hot and cold fluid at the outlet of the PHE are, respectively, 39 °C, 61 °C without dissipation and 47.7 °C, 71 °C with dissipation. The heat exchanged between the two fluids is approximately 254.3 kW (QH = 254 kW and QC = 254.6 kW) without dissipation and 200.1 kW (QH = 200.6 kW and QC = 199.6 kW) with dissipation. Thus, in this case, dissipation causes a 27% reduction in the heat transfer rate.
Fig. 4. Effectiveness of a water–water PHE with19 channels.
Fig. 5. Temperature profiles in a water–glycerol PHE with and without dissipation (L = 5 m).
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Fig. 6. Effect of dissipation on the effectiveness of a water–glycerol PHE (C = 1).
Fig. 6 shows the calculated effectiveness of the PHE with N = 9 for a ratio of heat capacity rates equal to one and uniform flow rate distribution. The relation between e and NTU was calculated using three different approaches. When dissipation is neglected the numerical predictions are in perfect agreement with the corresponding results of the classic analytical relation between e and NTU. When dissipation is taken into consideration the effectiveness is, in the present case, reduced. This reduction is more important for high values of NTU and for the case of temperature-dependent viscosity. Thus, for example, for NTU = 4 the theoretical (neglecting dissipation) effectiveness is 0.799 while when dissipation is taken into consideration it is 0.704 for constant viscosity (reduction of 11.9%) and 0.637 for temperature-dependent viscosity (reduction of 20.4%). Fig. 7 shows analogous results for a case with glycerol as the hot fluid (flow rate = 1.35 l/s, TH,in = 80 °C, ReH = 0.236) and water as the cold fluid (flow rate = 1 l/s, TC,in = 20 °C, ReC = 3230). The ratio of heat capacity rates is equal to 1, N = 9 and the flow rate distribution is assumed to be uniform as in the case of Fig. 6. As a result, when dissipation is neglected the relationship between e and NTU is the same for these two cases and is identical to the theoretical one predicted by the classic analytical relation. However, when dissipation is considered the calculated effectiveness is in the present case higher than the theoretical one. This surprising result, totally different from that in Fig. 6, is obviously due to the fact that the high-viscosity fluid (glycerol) is in the present case the hot fluid. Thus, dissipation in the glycerol keeps its temperature high
despite the heat transferred to the water and increases the amount of heat exchanged between the two fluids. Indeed, the calculated value of this quantity is approximately 187.7 kW (QH = 187.5 kW and QC = 187.9 kW) without dissipation and 196.5 kW (QH = 196.2 kW and QC = 196.8 kW) with dissipation. Thus, in this case, dissipation causes a 4.7% increase in the heat transfer rate. Even more surprising are the results of Fig. 8 which were obtained by reducing the flow rate of the glycerol by 50% from that used to calculate those in Fig. 7 while keeping all the other conditions unchanged. Thus for Fig. 8 the hot fluid is glycerol (flow rate = 0.675 l/s, TH,in = 80 °C, ReH = 0.118), the cold fluid is water (flow rate = 1 l/s, TC,in = 20 °C, ReC = 3230), the ratio of heat capacity rates is equal to 0.5, N = 9 and the flow rate distribution is assumed to be uniform. Once again, when dissipation is neglected the relationship between e and NTU is identical to the theoretical one predicted by the classic analytical relation. When dissipation with constant viscosity is taken into consideration the effectiveness increases as in the previous case, albeit more rapidly. When dissipation and temperature-dependent viscosity are considered the effectiveness increases even more and reaches values higher than one for NTU > 3.5. At first glance, this result seems physically unrealistic and unacceptable. However, it can be explained and justified by noting that in the present case Eq. (6b) becomes
Q max ¼ ðm cp ÞH ðT H;in T C;in Þ while energy conservation for the hot fluid is
Fig. 7. Effect of dissipation on the effectiveness of a glycerol–water PHE (C = 1).
ð8aÞ
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Fig. 8. Effect of dissipation on the effectiveness of a glycerol–water PHE (C = 0.5).
Q ¼ ðm cp ÞH ðT H;in T H;out Þ þ W diss;H
ð8bÞ
files in the two cold passages are the same (h1 = h3). Taking into consideration this symmetry, energy conservation (Eq. (4)) for the hot and cold fluid are, respectively,
ð8cÞ
dh2 ¼ 2 NTU H ðh2 h1 Þ dz dh1 ¼ NTU C ðh2 h1 Þ dz
Therefore
e¼
T H;in T H;out W diss;H þ T H;in T C;in Q max
The first term on the right hand side of this last expression is the conventional expression for the effectiveness and is always smaller than one since TH,out > TC,in (in the case of Fig. 8, TH,out = 40.4 °C while TC,in = 20 °C). On the other hand, the dissipation rate can be an important fraction of Qmax and therefore, the value of the effectiveness when dissipation is taken into consideration does not have an upper bound as is the case when dissipation is neglected. Stated differently, the result e > 1, or Q > Qmax, respects both the first and second laws of thermodynamics and is therefore acceptable. 5. Conclusions A one-dimensional model of heat transfer in a multi-passage plate heat exchanger using appropriate correlations for the Nusselt number and the friction coefficient was used to illustrate the effects of dissipation and temperature-dependent viscosity on the temperature profiles of the fluids and the effectiveness of the PHE. The results show that these effects can be neglected if the viscosity of both fluids is low (such as water). In that case the present results are identical with those presented in heat transfer textbooks. On the other hand, if one of the fluids is very viscous (e.g. glycerol) neglecting dissipation leads to significantly erroneous results: the temperature of both fluids is underestimated while the effectiveness of the PHE may be over-, or under-estimated depending on whether the high-viscosity fluid is the hot or cold fluid and on the ratio of the heat capacity rates. Acknowledgements This project is part of the R&D program of the NSERC Chair in Industrial Energy Efficiency established in 2006 at Université de Sherbrooke. The authors acknowledge the support of the Natural Sciences & Engineering Research Council of Canada, Hydro Québec, Rio Tinto Alcan and CANMET Energy Technology Center. Appendix A For a PHE with N = 3, uniform flow distribution, no dissipation and constant viscosity the heat transfer coefficients hC in passages #1 and #3 are identical and do not vary with z. Similarly hH in passage #2 is also independent of z. Furthermore the temperature pro-
ðA:1Þ ðA:2Þ
Taking the difference of these two relations and integrating gives
h2 h1 ¼ C 1 expða zÞ
ðA:3Þ
where
a ¼ 2 NTU H NTU C
ðA:4Þ
Replacing Eq. (A.3) in Eq. (A.2) and integrating gives the general solution for h1 and the corresponding expression for h2 is obtained from Eq. (A.3). The two integration constants are evaluated by applying the boundary conditions (h1 = 0 at z = 0, h2 = 1 at z = 1) and the final expression for the temperature distributions in the cold and hot passages are, respectively:
NTU C ðexpða zÞ 1Þ NTU C expðaÞ þ a expðaÞ NTU C ðNTU C þ aÞ expða zÞ NTU C h2 ¼ NTU C expðaÞ þ a expðaÞ NTU C
h1 ¼ h3 ¼
ðA:5Þ ðA:6Þ
References [1] S.M. Bagalagel, A.Z. Sahin, Design optimization of heat exchangers with highviscosity fluids, International Journal of Energy Research 26 (2002) 867–880. [2] J.A.W. Gut, J.M. Pinto, Modeling of plate heat exchangers with generalized configuration, International Journal of Heat and Mass Transfer 46 (2003) 2571–2585. [3] S.G. Kandlikar, R.K. Shah, Multipass heat exchangers – effectiveness–NTU results and guidelines for selecting pass arrangements, ASME Journal of Heat Transfer 111 (1989) 300–313. [4] K. Okada et al., Temperature distribution in plate type heat exchangers, Kagaku Kogaku 34 (1) (1970) 93–95. [5] A. Lozano, F. Barreras, N. Fueyo, S. Santodomingo, The flow in an oil/water plate heat exchanger for the automotive industry, Applied Thermal Engineering 28 (10) (2008) 1109–1117. [6] L. Wang, B. Sunden, Optimal design of plate heat exchangers with and without pressure drop specifications, Applied Thermal Engineering 23 (2003) 295–311. [7] J.A.W. Gut, J.M. Pinto, Optimal configuration design for plate heat exchangers, International Journal of Heat and Mass Transfer 47 (2004) 4833–4848. [8] C.P. Ribeiro, M.H. Cano Andrade, An algorithm for steady-state simulation of plate heat exchangers, Journal of Food Engineering 53 (2002) 59–66. [9] S.K. Das, K. Murugesan, Transient response of multipass plate heat exchangers with axial dispersion in fluid, International Journal of Heat and Mass Transfer 43 (2000) 4327–4345. [10] A.K. Dwivedi, S.K. Das, Dynamics of plate heat exchangers subject to flow variations, International Journal of Heat and Mass Transfer 50 (2007) 2733–2743.
I. Gherasim et al. / Applied Thermal Engineering 29 (2009) 3132–3139 [11] M.K. Bassiouny, H. Martin, Flow distribution and pressure drop in plate heat exchangers – 1: U-type arrangement, Chemical Engineering Science 39 (4) (1983) 693–700. [12] M.K. Bassiouny, H. Martin, Flow distribution and pressure drop in plate heat exchangers – 2: Z-type arrangement, Chemical Engineering Science 39 (4) (1983) 701–704. [13] B.P. Rao, S.K. Das, An experimental study on the influence of flow maldistribution on the pressure drop across a plate heat exchanger, Journal of Fluids Engineering 126 (2004) 680–691. [14] B.P. Rao, K. Kumar, S.K. Das, Effect of flow distribution to the channels on the thermal performance of a plate heat exchanger, Chemical Engineering and Processing 41 (2002) 49–58. [15] K.S. Lee, W.S. Kim, J.M. Si, Optimal shape and arrangement of staggered pins in the channel of a plate heat exchanger, International Journal of Heat and Mass Transfer 44 (2001) 3223–3231.
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[16] G. Croce, P. D’Agaro, Numerical analysis of forced convection in plate and frame heat exchangers, International Journal of Numerical Methods for Heat and Fluid Flow 12 (6) (2002) 756–771. [17] A.G. Kanaris, K.A. Mouza, S.V. Paras, Flow and heat transfer prediction in a corrugated plate heat exchanger using a CFD code, Chemical Engineering Technology 29 (8) (2006) 923–930. [18] S. Kakaç, H. Liu, Heat Exchangers: Selection, Rating and Thermal Design, CRC Press, 1992. [19] F.A. Tereda, N. Srihari, B. Sunden, S.K. Das, Experimental investigation on port to channel flow maldistribution in plate heat exchangers, Heat Transfer Engineering 28 (5) (2007) 435–443. [20] C.S. Fernandes, R. Dias, J.M. Nòbrega, J.M. Maia, Laminar flow in chevron-type plate exchangers: CFD analysis of tortuosity shape factor and friction factor, Chemical Engineering and Processing 46 (2007) 825–833.