- Email: [email protected]
Transient response of perforated plate matrix heat exchangers
PII: S0011-2275(98)00117-9
Cryogenics 38 (1998) 1243–1249 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/99/$ - see front matter
Transient response of perforated plate matrix heat exchangers P. Ramesh and G. Venkatarathnam* Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India
Received 27 July 1998 Perforated plate matrix heat exchangers are used in a number of applications such as helium liquefiers, Joule–Thompson cryocoolers operating with pure fluids and mixtures, etc. The time taken for cool down of cryocoolers is very critical in many applications, for example, those used in mobile applications (e.g. missiles). In this paper we study the effect of different parameters on the transient response of perforated plate matrix heat exchangers. 1999 Elsevier Science Ltd. All rights reserved Keywords: heat exchangers; cool down; transient response
Nomenclature b c cp d G H h K n NTU ntuf ntup p s T Th,i Tc,i W
height of the separating wall (m) specific heat of matrix (J/kgK) specific heat capacity at constant pressure (J/kgK) diameter of plate perforations (m) fluid mass velocity in header (kg/m2s) length of the matrix heat exchanger (m) overall heat transfer coefficient (w/m2K) thermal conductivity (w/m K) number of plates overall number of heat transfer units defined by Equation (23) per side NTU (fluid) defined by Equation (17) dimensionless plate conductance defined by Equation (17) plate porosity thickness of spacer (m) temperature (K) temperature of the hot fluid at the inlet temperature of the cold fluid at the inlet width of the plate (m)
Greek ␣
effective thermal diffusivity (m2/s) dimensionless heat flow rate defined by Equation (19) plate conduction parameter defined by
s p ⌫
Equation (17) lateral coordinate overall axial conduction parameter defined by Equation (24) axial conduction parameter defined by Equation (19) lateral conduction parameter defined by Equation (19) ratio of heat capacity rates defined by Equation (18) dimensionless fluid temperature defined by Equation (18) density (kg/m3 ) dimensionless plate temperature (K) defined by Equation (18) time (sec) dimensionless time defined by Equation (20) dimensionless parameter, defined by Equation (21)
Subscripts c f h in j p 1,2 s x
cold fluid fluid hot fluid inlet jth plate plate channel numbers plate separator interface external boundary
Cryogenics 1998 Volume 38, Number 12 1243
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam Perforated plate matrix heat exchangers, invented nearly 50 years ago by McMohan et al.1, are finding increasing application in helium liquefiers, sorption refrigerators, quick cool down Joule–Thompson (J-T) refrigerators operating with pure fluid such as nitrogen and Kleemenko coolers operating with gas mixtures. A matrix heat exchanger (MHE), shown schematically in Figure 1, essentially consists of a stack of high thermal conductivity (copper or aluminium) perforated plates or wire screens, alternating with low thermal conductivity spacers (plastic, stainless steel etc). The packet of alternate high and low thermal conductivity materials is bonded together to form leak free passages for the streams exchanging heat between one another. The gaps in between the plates ensure uniform flow distribution (by continuous reheadering) and create turbulence which enhances heat transfer. High heat transfer coefficient and high surface area density (up to 6000 m−1 ) are obtained due to small flow passages (typically 0.3–1 mm in diameter). The spacers, being of low conductivity material, also help in reducing axial conduction and consequent deterioration of performance. The literature on the fabrication, modelling, heat transfer and flow friction in MHE was reviewed by Venkatarathnam and Sarangi2. They also studied the steady state performance of MHE3 and showed that the attainable effectiveness is a strong function of the number of finite plate-spacer pairs, lateral thermal resistance in the plates, longitudinal heat conduction and the finite heat transfer coefficients. Their analysis clearly showed that the effectiveness expressions used for normal heat exchangers cannot be used to evaluate the performance of a MHE. Venkatarathnam4 recently derived closed form expressions for the effectiveness of a MHE in terms of the above resistances. A simple method was also presented by Venkatarathnam5 for the optimum sizing of a MHE very recently. Rodriguez and Mills6 showed that the effect of finite plate-spacer pairs is important even in reducing the heat
transfer data obtained by single blow transient methods. They showed that the errors will be large when commonly used methods are used to reduce the single blow transient tests data for determining the convective heat transfer coefficients in a MHE because of the finite number of platespacer pairs. The transient response of cryocoolers such as quick cool down J-T coolers based on matrix tube heat exchangers and Kleemenko cryocoolers operating with MHE, particularly those used as cryoprobes, is a strong function of the number of plates and other thermal resistances, and the results available for normal heat exchangers cannot be extended to MHE directly. The main aim of this paper is to study the parameters that affect the cool down time of perforated plate MHE.
Mathematical model Let us consider a matrix heat exchanger consisting of n perforated plates each of thickness l and separated by spacer of thickness s. The plates have circular perforations of diameter d, the plate porosity (fraction of open area in the plates) being p. Two rectangular slots of dimension W ⫻ Hi are cut in the spacers, Hi (i ⫽ 1,2) being the dimension of the ith channel in the heat flow direction. The two channels are separated by a wall of height b and the overall length of the heat exchanger is L. These dimensions are illustrated in Figure 1. Before the temporal perturbation is introduced ( ⫽ 0), the heat exchanger has a specified temperature field i,j,. At time ( ⫽ 0) the warm and the cold fluid streams are introduced at temperatures Th,i and Tc,i, respectively, and continue to flow at these temperatures for > 0. The objective of transient analysis is to predict the matrix and fluid temperature fields at temperature > 0. A special case of the transient analysis is the cool down process. In this problem, before the temporal perturbation is introduced ( ⫽ 0), the entire heat exchanger is at a uniform temperature Th,i, which is the inlet temperature of the warm fluid stream. At time ( ⫽ 0), the cold fluid stream is introduced at a lower temperature Tc,i. In this section, the governing equations and solution algorithms have been presented for the special case of the cool down process, which is of special interest to cryogenic refrigerator designers. The analysis is based on the following assumptions: • the thermal capacity of the gas is negligible when compared with that of the heat exchanger core; • all the axial (longitudinal) heat conduction is assumed to take place along the separating wall only; • thermophysical properties of both the matrix and the fluid are independent of temperature; • the perforated plates are isothermal in the fluid flow direction (as the plate thickness is generally less than 1.0 mm and Biot number less than 0.01)3; and • the time taken by a fluid particle to travel from one end of the heat exchanger to the other (residence time of fluid) is small and can be neglected when compared with the total cool down time.
Figure 1 Schematic of perforated matrix heat exchangers
*Corresponding author. Tel.: ⫹ 44-235-1315; fax: ⫹ 44-235-0509 E-mail address: [email protected] (G. Venkatarathnam)
1244
Cryogenics 1998 Volume 38, Number 12
Governing equations The governing equations are same as that for the steady state performance presented elsewhere2, except for one time
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam dependent term, and can be expressed in dimensionless form as follows.
Stream 1. 1,j ⫹ 1 ⫽ 1,j ⫹ exp( ⫺ ntu,f,1 )( 1,j ⫺ 1,j ) ∂1,j 1 ⫺ exp( ⫺ ntu,f,1 ) ∂21,j ⫹ ( 1,j ⫺ 1,j ) ⫽ ∂ 1ntu,f,1 ∂21
(1) (2)
In the above expression s,j is the average wall temperature and equal to ( 1s,j ⫹ 2s,j )/2. represent the dimensionless plate temperature, the dimensionless fluid temperature, the heat capacity rate ratio, s the per plate axial heat conduction parameter, p the lateral conduction parameter and the dimensionless heat flow rate and the dimensionless lateral distance parameter. ntu,f is the per side NTU and ntu,p the dimensionless plate conductance. is the ratio of dimensionless plate conductance to per side NTU. These parameters are defined as follows:
Boundary conditions:
1,1( , ) ⫽ 1; for ⱖ 0 and 0 ⱕ 1 ⱕ 1
(3)
∂1,j ( ⫽ 1) ∂1 1
(4)
1,j ⫽ 1ntu,f,1
∂1,j ⫽ 0 at 1 ⫽ 0 ∂1
(5)
Initial condition:
1,j (0,1 ) ⫽ 1
(6)
ntu,f ⫽
i ⫽
⌫
∂2,j 1 ⫺ exp( ⫺ ntu,f,2 ) ⫹ ( 2,j ⫺ 2,j ⫹ 1 ) ∂ 2ntu,f,2
⫽
∂22,j ∂22
(7) (8)
Boundary conditions:
2,n ⫹ 1( , ) ⫽ 0; for ⱖ and 0 ⱕ 2 ⱕ 1
(m ˙ cp ) h (m ˙ cp ) c
p ⫽
KslW KplW q˙ ; ;⫽ bHGcp s sHGcp (m ˙ cp )min(th,in ⫺ tc,in )
∂2,j ⫽ 0 at 2 ⫽ 1 ∂2
2,j ⫽ 2ntu,f,2
∂2,j ( ⫽ 0) ∂2 2
(10)
(11)
Initial condition:
2,j (0,2 ) ⫽ 1
(12)
Separating wall. 1,j ⫹ 1,j
2,j ⫽ 2p( 1s,j ⫺ 2s,j ); 1 ⱕ j ⱕ n
2,j ⫺ ⫽ s(2s,j ⫺ s,j ⫺ 1 ⫺ s,j ⫹ 1 );
(13)
(14)
2ⱕjⱕn⫺1
1,1 ⫺
2,1 ⫽ s( s,1 ⫺ s,2 )
(15)
1,n ⫺
2,n ⫽ s( s,n ⫺ s,n ⫺ 1 )
(16)
(17)
(18)
(19)
is the dimensionless time coordinate defined as follows: ⫽
冉冊 冉 ␣ H2
⫽
1
冊
Kp (1 ⫺ p)cH2 1
(20)
where ␣ is effective thermal diffusivity of the perforated plates (m2/s) and is the time coordinate in seconds. and c are the density and specific heat rate of the plate material. ⌫ is a dimensionless parameter denoting the ratio of the thermal time constants of the two streams. ⌫⫽
(9)
y t ⫺ t2,in t ⫺ th,in ;⫽ ;⫽ ; Hi t1,in ⫺ t2,in th,in ⫺ tc,in
⫽
Stream 2. 2,j ⫽ 2,j ⫹ exp( ⫺ ntu,f,2 )( 2,j ⫹ 1 ⫺ 2,j )
hA Kpl ntu,p ntu,p ⫽ ⫽ 2 m (GcpH )i ntu,f ˙ cp
␣1H22 ␣2H21
(21)
In a well-built MHE, the thickness of the separator is small and the separating wall constitutes only a small fraction of the total mass. The thermal capacitance of the wall has been neglected in the derivation of Equations (13)– (16), which helps in reducing the complexity of the equation set. Further, considering that heat flow by axial heat conduction is small compared to that in the lateral direction, the temperature profile across the wall can be considered linear in the heat flow direction even under transient conditions7. Equations (1)–(16) describe the behaviour of the heat exchanger under cool down conditions. The variable 1,j, 2,j, 1,j and 2,j are strongly coupled and the set of equations have to be solved simultaneously.
Simplification of the governing equations The four dependent variables are functions of the three independent coordinates j, i and . An exact solution of the governing equations (in both and domains) will require a large amount of computer time. In order to reduce the computer time and to make the solutions more tractable, an assumption is made on the temperature profiles in the plates. It is assumed that the temperature profile, irrespective of the plate number j and time , can be represented by a parabola:
⫽ A ⫹ B ⫹ C2
(22)
Cryogenics 1998 Volume 38, Number 12 1245
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 2 Heat exchanger model
A, B and C being functions of the plate number j and the time coordinate . In the steady state, this assumption leads to a constant difference between the plate and fluid temperatures, independent of the lateral coordinate . It may be recalled that many workers have used this constant temperature difference assumption, first proposed by Fleming8, to reduce their experimental data. Very recently Venkatarathnam4 used the same assumption to derive closed form expressions for the effectiveness of a MHE. Equation (22) is used to reduce the dimensionality of the problem which reduces the number of equations by a large factor. In this process, the information on the temperature profile along the plate is somewhat altered, but that along the flow direction is retained. It is the later which largely determines the effectiveness of the heat exchanger and the cool down time. The constants A, B and C can now be expressed in terms of the boundary temperatures (at i ⫽ 0 and i ⫽ 1) 1x,j, 1s,j, 2x,j and 2s,j. These boundary temperatures have been illustrated in Figure 2. The subscript x refers to the external boundary, while s refers to the plate separator interface of the respective channel. The solution of the governing equations becomes quite straight forward with the above assumptions. In this work the finite difference method was employed to solve the set of algebraic and partial differential equations.
Results and discussion The governing equations for the cool down of MHE have been solved for a range of parameters to bring out the dependence of the cool down time on different parameters. Table 1 shows the variables and the range over which the different parameters have been varied. The governing equations have been solved using an implicit scheme with a dimensionless time step of 0.01. The temperatures of the fluid and plate have been determined for all the plates at the interface between (a) the flow channels and the surroundings, and (b) the flow channels and the separating wall. The overall NTU of the heat exchanger is defined as follows:
Figure 3 Variation of dimensionless temperature of the fluid at the heat exchanger exit with dimensionless time and at ntu,f ⫽ 0.2, 0.4 and 0.6. In all the cases, n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0 and p ⫽ ⬁
冉
冊
1 1 1 1 ⫽ ⫹ ⫹ NTU n ntu,f,1 p ntu,f,2
(23)
Similarly the overall axial heat conduction parameter is defined as follows:
⫽
s n
(24)
Figure 3 shows the variation of hot and cold fluid exit temperatures as a function of dimensionless time for different values of ‘per side NTU’ (ntuf ) for a matrix heat exchanger with 50 plates. It can be observed from Figure 3 that the hot and cold fluid (ntuf ⫽ 0.2) attain a near steady state at dimensionless time ⬍ 50, when ntuf is 0.2 or the overall NTU is 5, whereas the hot and cold fluid exit temperatures are somewhat far away from their steady state values even at a dimensionless time ⫽ 100 for ntuf of 0.4 or 0.6 (or a corresponding overall NTU of 10 and 15, respectively). It is therefore evident that the time required for cool down increases with an increase in the NTU of the exchanger. Figure 4 shows the variation of the plate temperature at different locations (see Figure 2) of plate number one. To simplify the analysis, the lateral thermal resistance of the
Table 1 Different parameters and the range over which they were varied S1 Resistance no.
Variable
Range
1 2 3 4 5 6
ntu,f n ⌫ s p
0.4…1.0 50…200 0.8…1.2 0.6…1.0 0.0…0.05 0.8…⬁
Per plate NTU Number of plates Ratio of themal constants Heat capacity ratio Axial conduction parameter Lateral conduction parameter
1246
Cryogenics 1998 Volume 38, Number 12
Figure 4 Variation of temperature of the first plate (at the warm end) shown as a function of dimensionless time for ntu,f ⫽ 0.2 0.4 and 0.6. In all cases n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 5 Variation of temperature of the last plate (at the cold end) shown as a function of dimensionless time for ntu,f ⫽ 0.2, 0.4 and 0.6. In all the cases n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 7 Dimensionless plate temperature shown as a function of dimensionless time for a heat exchanger with 100 plate-spacer pairs. In all cases n ⫽ 100, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
plate has been assumed to be zero for the separating wall. It can be observed from Figure 4 that the variation of plate temperature with ntuf is similar to that of the fluid. The plates do not reach steady state even after a dimensionless time of 200 when the ntuf is 0.6. It can also be observed from Figure 4 that the time taken for the plates to reach steady state is somewhat larger than that required by the fluid. The difference in the temperature of the plate in the lateral ( ) direction is due to the finite thermal resistance of the perforated plate. The plate conduction parameter ( ) has been assumed to be 1.0 in the above analysis, which corresponds to a fin efficiency of 75%, which is close to that observed in practical heat exchangers. Figure 5 shows the variation of temperature of the 50th plate at different locations in the lateral direction. It can be observed from Figure 5 that the time taken for the plate temperatures to reach steady state is much larger when ntuf is 0.6 than at 0.2. The final temperatures attained by the plates are dependent on the overall NTU of the heat exchanger. It can be observed from Figures 4 and 5 that the plates near the cold end of the heat exchanger reach steady state faster than those that are close to the warm end of the heat exchanger. Figure 6 shows the variation of fluid temperature at the heat exchanger exit for a MHE with 100 plate-spacer pairs. It can be seen that the time taken by the hot fluid to reach
steady state is nearly independent of the overall NTU of the heat exchanger when ntuf is lower than 0.6. The time taken for the cold fluid to reach steady state, however, is dependent on the overall NTU of the heat exchanger. Figure 7 shows the variation of plate temperature at different plate locations for ntuf of 0.2 and the total platespacer pairs being 100. The plate temperatures follow the same trend as that in Figure 4 for the case of 50 plates. Figure 8 shows the variation of fluid exit temperatures for a heat exchanger with an overall NTU of 10 and 20 for heat exchangers with 50, 100 and 200 plates, respectively. It can be seen that the transient response of the heat exchanger is nearly independent of the number of plates when the overall NTU is either 10 or 20. Figure 9 shows the variation of temperature for the last plate (at the cold end) with dimensionless time. It can be seen that time taken by the heat exchanger with 200 plates is marginally greater than that with 50 plates. It is therefore evident that for all practical purposes, the response of plate temperature is nearly same for all heat exchangers with plates greater than 100 plates. Figure 10 shows the variation of cold and hot fluid exit temperatures with dimensionless time for a heat exchanger with 50 plates with an overall NTU of 20, as a function of axial heat conduction parameter. It can be observed that the
Figure 6 Dimensionless fluid exit temperature shown against the dimensionless time for a heat exchanger with 100 platespacer pairs. In all cases n ⫽ 100, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 8 Effect of overall number of transfer units (NTU) and the number of plate-spacer pairs on the response of a MHE. The results for 50, 100 and 200 plates are very close, and the curves are indistinguishable. In all the cases: ⌫ ⫽ 1.0, s ⫽ 0.0, ⫽ 1.0 and p ⫽ ⬁
Cryogenics 1998 Volume 38, Number 12 1247
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 9 Effect of number of plates on the variation of temperature of the last plate (at the cold end) with time. In all the cases ⌫ ⫽ 1.0, ⫽ 1.0, 1.0, s ⫽ 0.0, overall NTU ⫽ 20 and p ⫽⬁
Figure 10 Effect of longitudinal (axial) heat conduction parameter on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, overall NTU ⫽ 20 and p ⫽ ⬁
higher the axial conduction, the lower will be the cool down time. Figure 11 shows the corresponding plate temperature variation for the plate at the cold end. It can be seen that the effect of axial heat conduction on the cool down time is same for the plates and the fluids.
Figure 11 Effect of longitudinal (axial) heat conduction parameter on the variation of temperature of the plate at the cold end with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, overall NTU ⫽ 20 and p ⫽ ⬁
1248
Cryogenics 1998 Volume 38, Number 12
Figure 12 Effect of lateral heat conduction parameter on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, s ⫽ 0.02 and p ⫽ ⬁
Figure 12 shows the variation of the hot and cold fluid exit temperatures with dimensionless time as a function of the lateral heat conduction parameter N, for an overall NTU of 10 and of 0.5 and 3.0, respectively. It can be observed that the effectiveness increases with an increase in or a decrease in the lateral thermal resistance of the plate. The cool down time also increases with an increase in or the overall effectiveness of the heat exchanger. Figure 13 shows the variation of fluid exit temperatures with dimensionless time for different ratios of the thermal time constants and an overall axial conduction parameter of 0.04. It can be observed that the overall time taken is nearly independent of the ratio of thermal constants, and is largely controlled only by the thermal resistance offered by the plate and fluid streams. The difference between the time–temperature profiles will be even closer for heat exchangers with much smaller axial heat conduction parameters. Figure 14 shows the variation of fluid exit temperatures with dimensionless time for different heat capacity rate ratios and the same overall NTU. It can be seen that the variation of hot fluid exit temperature is nearly the same during the initial cool down period, when the fluid change is nearly 80%, for different heat capacity rate ratios. On the other hand, the temperature change of the hot fluid at the exit is strongly influenced by the heat capacity rate ratios. A similar trend is observed with the plates in different channels as shown in Figure 15.
Figure 13 Effect of ratios of thermal time constants on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⫽ 1.0, ⫽ 1.0, ntu,f ⫽ 0.2 s ⫽ 0.04 and p ⫽ ⬁
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 14 Effect of heat capacity rate ratio and overall NTU on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 15 Effect of heat capacity rate ratio and overall NTU on the variation of cold end plate temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Conclusions 1.
2.
3. 4.
The overall NTU or the effectiveness of the heat exchanger is the most important parameter that controls the cool down of MHEs. The higher the heat exchanger effectiveness, the higher will be the cool down time. Thus, for mobile applications with a requirement of fast cool down, a choice has to be made between fast cool down (low effectiveness) or low cryogen consumption (high effectiveness). The cool down time decreases due to irreversibilities such as axial heat conduction, plate thermal resistance, etc. The cool down time is indirectly proportional to and . The cool down time is only marginally dependent on the ratio of time constants (heat capacities) of the two flow channels. The cool down time is nearly independent of the number of plates when the number of plates is greater than 50.
References 1. McMohan, H. O., Bowen, R. J. and Bleyle, G. A., A perforated plate heat exchanger. Trans. ASME, 1950, 72, 623–632. 2. Venkatarathnam, G. and Sarangi, S., Matrix heat exchangers and their application in cryogenic systems. Cryogenics, 1990, 30, 907–912. 3. Venkatarathnam, G. and Sarangi, S., Analysis of Matrix heat exchangers performance. ASME J. Heat Transfer, 1991, 113, 830– 837. 4. Venkatarathnam, G., Effectiveness–Ntu relationship in perforated plate matrix heat exchangers. Cryogenics, 1996, 36, 235–241. 5. Venkatarathnam, G., A straight forward method for the sizing of perforated matrix heat exchangers. Advances in Cryogenic Engineering, 1998, 43, 0. 6. Rodriguez, J. I. and Mills, A. F., Analysis of single-blow transient testing of perforated plate heat exchangers. Int. J. Heat Mass Transfer, 1990, 33, 1969–1976. 7. Venkatarathnam, G., Matrix heat exchanger. PhD dissertation, Indian Institute of Technology, Kharagpur, 1990. 8. Fleming, R. B., A compact perforated plate heat exchanger. Adv. Cryog. Eng., 1969, 14, 197–204.
Cryogenics 1998 Volume 38, Number 12 1249
Cryogenics 38 (1998) 1243–1249 1999 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0011-2275/99/$ - see front matter
Transient response of perforated plate matrix heat exchangers P. Ramesh and G. Venkatarathnam* Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India
Received 27 July 1998 Perforated plate matrix heat exchangers are used in a number of applications such as helium liquefiers, Joule–Thompson cryocoolers operating with pure fluids and mixtures, etc. The time taken for cool down of cryocoolers is very critical in many applications, for example, those used in mobile applications (e.g. missiles). In this paper we study the effect of different parameters on the transient response of perforated plate matrix heat exchangers. 1999 Elsevier Science Ltd. All rights reserved Keywords: heat exchangers; cool down; transient response
Nomenclature b c cp d G H h K n NTU ntuf ntup p s T Th,i Tc,i W
height of the separating wall (m) specific heat of matrix (J/kgK) specific heat capacity at constant pressure (J/kgK) diameter of plate perforations (m) fluid mass velocity in header (kg/m2s) length of the matrix heat exchanger (m) overall heat transfer coefficient (w/m2K) thermal conductivity (w/m K) number of plates overall number of heat transfer units defined by Equation (23) per side NTU (fluid) defined by Equation (17) dimensionless plate conductance defined by Equation (17) plate porosity thickness of spacer (m) temperature (K) temperature of the hot fluid at the inlet temperature of the cold fluid at the inlet width of the plate (m)
Greek ␣
effective thermal diffusivity (m2/s) dimensionless heat flow rate defined by Equation (19) plate conduction parameter defined by
s p ⌫
Equation (17) lateral coordinate overall axial conduction parameter defined by Equation (24) axial conduction parameter defined by Equation (19) lateral conduction parameter defined by Equation (19) ratio of heat capacity rates defined by Equation (18) dimensionless fluid temperature defined by Equation (18) density (kg/m3 ) dimensionless plate temperature (K) defined by Equation (18) time (sec) dimensionless time defined by Equation (20) dimensionless parameter, defined by Equation (21)
Subscripts c f h in j p 1,2 s x
cold fluid fluid hot fluid inlet jth plate plate channel numbers plate separator interface external boundary
Cryogenics 1998 Volume 38, Number 12 1243
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam Perforated plate matrix heat exchangers, invented nearly 50 years ago by McMohan et al.1, are finding increasing application in helium liquefiers, sorption refrigerators, quick cool down Joule–Thompson (J-T) refrigerators operating with pure fluid such as nitrogen and Kleemenko coolers operating with gas mixtures. A matrix heat exchanger (MHE), shown schematically in Figure 1, essentially consists of a stack of high thermal conductivity (copper or aluminium) perforated plates or wire screens, alternating with low thermal conductivity spacers (plastic, stainless steel etc). The packet of alternate high and low thermal conductivity materials is bonded together to form leak free passages for the streams exchanging heat between one another. The gaps in between the plates ensure uniform flow distribution (by continuous reheadering) and create turbulence which enhances heat transfer. High heat transfer coefficient and high surface area density (up to 6000 m−1 ) are obtained due to small flow passages (typically 0.3–1 mm in diameter). The spacers, being of low conductivity material, also help in reducing axial conduction and consequent deterioration of performance. The literature on the fabrication, modelling, heat transfer and flow friction in MHE was reviewed by Venkatarathnam and Sarangi2. They also studied the steady state performance of MHE3 and showed that the attainable effectiveness is a strong function of the number of finite plate-spacer pairs, lateral thermal resistance in the plates, longitudinal heat conduction and the finite heat transfer coefficients. Their analysis clearly showed that the effectiveness expressions used for normal heat exchangers cannot be used to evaluate the performance of a MHE. Venkatarathnam4 recently derived closed form expressions for the effectiveness of a MHE in terms of the above resistances. A simple method was also presented by Venkatarathnam5 for the optimum sizing of a MHE very recently. Rodriguez and Mills6 showed that the effect of finite plate-spacer pairs is important even in reducing the heat
transfer data obtained by single blow transient methods. They showed that the errors will be large when commonly used methods are used to reduce the single blow transient tests data for determining the convective heat transfer coefficients in a MHE because of the finite number of platespacer pairs. The transient response of cryocoolers such as quick cool down J-T coolers based on matrix tube heat exchangers and Kleemenko cryocoolers operating with MHE, particularly those used as cryoprobes, is a strong function of the number of plates and other thermal resistances, and the results available for normal heat exchangers cannot be extended to MHE directly. The main aim of this paper is to study the parameters that affect the cool down time of perforated plate MHE.
Mathematical model Let us consider a matrix heat exchanger consisting of n perforated plates each of thickness l and separated by spacer of thickness s. The plates have circular perforations of diameter d, the plate porosity (fraction of open area in the plates) being p. Two rectangular slots of dimension W ⫻ Hi are cut in the spacers, Hi (i ⫽ 1,2) being the dimension of the ith channel in the heat flow direction. The two channels are separated by a wall of height b and the overall length of the heat exchanger is L. These dimensions are illustrated in Figure 1. Before the temporal perturbation is introduced ( ⫽ 0), the heat exchanger has a specified temperature field i,j,. At time ( ⫽ 0) the warm and the cold fluid streams are introduced at temperatures Th,i and Tc,i, respectively, and continue to flow at these temperatures for > 0. The objective of transient analysis is to predict the matrix and fluid temperature fields at temperature > 0. A special case of the transient analysis is the cool down process. In this problem, before the temporal perturbation is introduced ( ⫽ 0), the entire heat exchanger is at a uniform temperature Th,i, which is the inlet temperature of the warm fluid stream. At time ( ⫽ 0), the cold fluid stream is introduced at a lower temperature Tc,i. In this section, the governing equations and solution algorithms have been presented for the special case of the cool down process, which is of special interest to cryogenic refrigerator designers. The analysis is based on the following assumptions: • the thermal capacity of the gas is negligible when compared with that of the heat exchanger core; • all the axial (longitudinal) heat conduction is assumed to take place along the separating wall only; • thermophysical properties of both the matrix and the fluid are independent of temperature; • the perforated plates are isothermal in the fluid flow direction (as the plate thickness is generally less than 1.0 mm and Biot number less than 0.01)3; and • the time taken by a fluid particle to travel from one end of the heat exchanger to the other (residence time of fluid) is small and can be neglected when compared with the total cool down time.
Figure 1 Schematic of perforated matrix heat exchangers
*Corresponding author. Tel.: ⫹ 44-235-1315; fax: ⫹ 44-235-0509 E-mail address: [email protected] (G. Venkatarathnam)
1244
Cryogenics 1998 Volume 38, Number 12
Governing equations The governing equations are same as that for the steady state performance presented elsewhere2, except for one time
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam dependent term, and can be expressed in dimensionless form as follows.
Stream 1. 1,j ⫹ 1 ⫽ 1,j ⫹ exp( ⫺ ntu,f,1 )( 1,j ⫺ 1,j ) ∂1,j 1 ⫺ exp( ⫺ ntu,f,1 ) ∂21,j ⫹ ( 1,j ⫺ 1,j ) ⫽ ∂ 1ntu,f,1 ∂21
(1) (2)
In the above expression s,j is the average wall temperature and equal to ( 1s,j ⫹ 2s,j )/2. represent the dimensionless plate temperature, the dimensionless fluid temperature, the heat capacity rate ratio, s the per plate axial heat conduction parameter, p the lateral conduction parameter and the dimensionless heat flow rate and the dimensionless lateral distance parameter. ntu,f is the per side NTU and ntu,p the dimensionless plate conductance. is the ratio of dimensionless plate conductance to per side NTU. These parameters are defined as follows:
Boundary conditions:
1,1( , ) ⫽ 1; for ⱖ 0 and 0 ⱕ 1 ⱕ 1
(3)
∂1,j ( ⫽ 1) ∂1 1
(4)
1,j ⫽ 1ntu,f,1
∂1,j ⫽ 0 at 1 ⫽ 0 ∂1
(5)
Initial condition:
1,j (0,1 ) ⫽ 1
(6)
ntu,f ⫽
i ⫽
⌫
∂2,j 1 ⫺ exp( ⫺ ntu,f,2 ) ⫹ ( 2,j ⫺ 2,j ⫹ 1 ) ∂ 2ntu,f,2
⫽
∂22,j ∂22
(7) (8)
Boundary conditions:
2,n ⫹ 1( , ) ⫽ 0; for ⱖ and 0 ⱕ 2 ⱕ 1
(m ˙ cp ) h (m ˙ cp ) c
p ⫽
KslW KplW q˙ ; ;⫽ bHGcp s sHGcp (m ˙ cp )min(th,in ⫺ tc,in )
∂2,j ⫽ 0 at 2 ⫽ 1 ∂2
2,j ⫽ 2ntu,f,2
∂2,j ( ⫽ 0) ∂2 2
(10)
(11)
Initial condition:
2,j (0,2 ) ⫽ 1
(12)
Separating wall. 1,j ⫹ 1,j
2,j ⫽ 2p( 1s,j ⫺ 2s,j ); 1 ⱕ j ⱕ n
2,j ⫺ ⫽ s(2s,j ⫺ s,j ⫺ 1 ⫺ s,j ⫹ 1 );
(13)
(14)
2ⱕjⱕn⫺1
1,1 ⫺
2,1 ⫽ s( s,1 ⫺ s,2 )
(15)
1,n ⫺
2,n ⫽ s( s,n ⫺ s,n ⫺ 1 )
(16)
(17)
(18)
(19)
is the dimensionless time coordinate defined as follows: ⫽
冉冊 冉 ␣ H2
⫽
1
冊
Kp (1 ⫺ p)cH2 1
(20)
where ␣ is effective thermal diffusivity of the perforated plates (m2/s) and is the time coordinate in seconds. and c are the density and specific heat rate of the plate material. ⌫ is a dimensionless parameter denoting the ratio of the thermal time constants of the two streams. ⌫⫽
(9)
y t ⫺ t2,in t ⫺ th,in ;⫽ ;⫽ ; Hi t1,in ⫺ t2,in th,in ⫺ tc,in
⫽
Stream 2. 2,j ⫽ 2,j ⫹ exp( ⫺ ntu,f,2 )( 2,j ⫹ 1 ⫺ 2,j )
hA Kpl ntu,p ntu,p ⫽ ⫽ 2 m (GcpH )i ntu,f ˙ cp
␣1H22 ␣2H21
(21)
In a well-built MHE, the thickness of the separator is small and the separating wall constitutes only a small fraction of the total mass. The thermal capacitance of the wall has been neglected in the derivation of Equations (13)– (16), which helps in reducing the complexity of the equation set. Further, considering that heat flow by axial heat conduction is small compared to that in the lateral direction, the temperature profile across the wall can be considered linear in the heat flow direction even under transient conditions7. Equations (1)–(16) describe the behaviour of the heat exchanger under cool down conditions. The variable 1,j, 2,j, 1,j and 2,j are strongly coupled and the set of equations have to be solved simultaneously.
Simplification of the governing equations The four dependent variables are functions of the three independent coordinates j, i and . An exact solution of the governing equations (in both and domains) will require a large amount of computer time. In order to reduce the computer time and to make the solutions more tractable, an assumption is made on the temperature profiles in the plates. It is assumed that the temperature profile, irrespective of the plate number j and time , can be represented by a parabola:
⫽ A ⫹ B ⫹ C2
(22)
Cryogenics 1998 Volume 38, Number 12 1245
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 2 Heat exchanger model
A, B and C being functions of the plate number j and the time coordinate . In the steady state, this assumption leads to a constant difference between the plate and fluid temperatures, independent of the lateral coordinate . It may be recalled that many workers have used this constant temperature difference assumption, first proposed by Fleming8, to reduce their experimental data. Very recently Venkatarathnam4 used the same assumption to derive closed form expressions for the effectiveness of a MHE. Equation (22) is used to reduce the dimensionality of the problem which reduces the number of equations by a large factor. In this process, the information on the temperature profile along the plate is somewhat altered, but that along the flow direction is retained. It is the later which largely determines the effectiveness of the heat exchanger and the cool down time. The constants A, B and C can now be expressed in terms of the boundary temperatures (at i ⫽ 0 and i ⫽ 1) 1x,j, 1s,j, 2x,j and 2s,j. These boundary temperatures have been illustrated in Figure 2. The subscript x refers to the external boundary, while s refers to the plate separator interface of the respective channel. The solution of the governing equations becomes quite straight forward with the above assumptions. In this work the finite difference method was employed to solve the set of algebraic and partial differential equations.
Results and discussion The governing equations for the cool down of MHE have been solved for a range of parameters to bring out the dependence of the cool down time on different parameters. Table 1 shows the variables and the range over which the different parameters have been varied. The governing equations have been solved using an implicit scheme with a dimensionless time step of 0.01. The temperatures of the fluid and plate have been determined for all the plates at the interface between (a) the flow channels and the surroundings, and (b) the flow channels and the separating wall. The overall NTU of the heat exchanger is defined as follows:
Figure 3 Variation of dimensionless temperature of the fluid at the heat exchanger exit with dimensionless time and at ntu,f ⫽ 0.2, 0.4 and 0.6. In all the cases, n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0 and p ⫽ ⬁
冉
冊
1 1 1 1 ⫽ ⫹ ⫹ NTU n ntu,f,1 p ntu,f,2
(23)
Similarly the overall axial heat conduction parameter is defined as follows:
⫽
s n
(24)
Figure 3 shows the variation of hot and cold fluid exit temperatures as a function of dimensionless time for different values of ‘per side NTU’ (ntuf ) for a matrix heat exchanger with 50 plates. It can be observed from Figure 3 that the hot and cold fluid (ntuf ⫽ 0.2) attain a near steady state at dimensionless time ⬍ 50, when ntuf is 0.2 or the overall NTU is 5, whereas the hot and cold fluid exit temperatures are somewhat far away from their steady state values even at a dimensionless time ⫽ 100 for ntuf of 0.4 or 0.6 (or a corresponding overall NTU of 10 and 15, respectively). It is therefore evident that the time required for cool down increases with an increase in the NTU of the exchanger. Figure 4 shows the variation of the plate temperature at different locations (see Figure 2) of plate number one. To simplify the analysis, the lateral thermal resistance of the
Table 1 Different parameters and the range over which they were varied S1 Resistance no.
Variable
Range
1 2 3 4 5 6
ntu,f n ⌫ s p
0.4…1.0 50…200 0.8…1.2 0.6…1.0 0.0…0.05 0.8…⬁
Per plate NTU Number of plates Ratio of themal constants Heat capacity ratio Axial conduction parameter Lateral conduction parameter
1246
Cryogenics 1998 Volume 38, Number 12
Figure 4 Variation of temperature of the first plate (at the warm end) shown as a function of dimensionless time for ntu,f ⫽ 0.2 0.4 and 0.6. In all cases n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 5 Variation of temperature of the last plate (at the cold end) shown as a function of dimensionless time for ntu,f ⫽ 0.2, 0.4 and 0.6. In all the cases n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 7 Dimensionless plate temperature shown as a function of dimensionless time for a heat exchanger with 100 plate-spacer pairs. In all cases n ⫽ 100, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
plate has been assumed to be zero for the separating wall. It can be observed from Figure 4 that the variation of plate temperature with ntuf is similar to that of the fluid. The plates do not reach steady state even after a dimensionless time of 200 when the ntuf is 0.6. It can also be observed from Figure 4 that the time taken for the plates to reach steady state is somewhat larger than that required by the fluid. The difference in the temperature of the plate in the lateral ( ) direction is due to the finite thermal resistance of the perforated plate. The plate conduction parameter ( ) has been assumed to be 1.0 in the above analysis, which corresponds to a fin efficiency of 75%, which is close to that observed in practical heat exchangers. Figure 5 shows the variation of temperature of the 50th plate at different locations in the lateral direction. It can be observed from Figure 5 that the time taken for the plate temperatures to reach steady state is much larger when ntuf is 0.6 than at 0.2. The final temperatures attained by the plates are dependent on the overall NTU of the heat exchanger. It can be observed from Figures 4 and 5 that the plates near the cold end of the heat exchanger reach steady state faster than those that are close to the warm end of the heat exchanger. Figure 6 shows the variation of fluid temperature at the heat exchanger exit for a MHE with 100 plate-spacer pairs. It can be seen that the time taken by the hot fluid to reach
steady state is nearly independent of the overall NTU of the heat exchanger when ntuf is lower than 0.6. The time taken for the cold fluid to reach steady state, however, is dependent on the overall NTU of the heat exchanger. Figure 7 shows the variation of plate temperature at different plate locations for ntuf of 0.2 and the total platespacer pairs being 100. The plate temperatures follow the same trend as that in Figure 4 for the case of 50 plates. Figure 8 shows the variation of fluid exit temperatures for a heat exchanger with an overall NTU of 10 and 20 for heat exchangers with 50, 100 and 200 plates, respectively. It can be seen that the transient response of the heat exchanger is nearly independent of the number of plates when the overall NTU is either 10 or 20. Figure 9 shows the variation of temperature for the last plate (at the cold end) with dimensionless time. It can be seen that time taken by the heat exchanger with 200 plates is marginally greater than that with 50 plates. It is therefore evident that for all practical purposes, the response of plate temperature is nearly same for all heat exchangers with plates greater than 100 plates. Figure 10 shows the variation of cold and hot fluid exit temperatures with dimensionless time for a heat exchanger with 50 plates with an overall NTU of 20, as a function of axial heat conduction parameter. It can be observed that the
Figure 6 Dimensionless fluid exit temperature shown against the dimensionless time for a heat exchanger with 100 platespacer pairs. In all cases n ⫽ 100, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 8 Effect of overall number of transfer units (NTU) and the number of plate-spacer pairs on the response of a MHE. The results for 50, 100 and 200 plates are very close, and the curves are indistinguishable. In all the cases: ⌫ ⫽ 1.0, s ⫽ 0.0, ⫽ 1.0 and p ⫽ ⬁
Cryogenics 1998 Volume 38, Number 12 1247
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 9 Effect of number of plates on the variation of temperature of the last plate (at the cold end) with time. In all the cases ⌫ ⫽ 1.0, ⫽ 1.0, 1.0, s ⫽ 0.0, overall NTU ⫽ 20 and p ⫽⬁
Figure 10 Effect of longitudinal (axial) heat conduction parameter on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, overall NTU ⫽ 20 and p ⫽ ⬁
higher the axial conduction, the lower will be the cool down time. Figure 11 shows the corresponding plate temperature variation for the plate at the cold end. It can be seen that the effect of axial heat conduction on the cool down time is same for the plates and the fluids.
Figure 11 Effect of longitudinal (axial) heat conduction parameter on the variation of temperature of the plate at the cold end with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, ⫽ 1.0, overall NTU ⫽ 20 and p ⫽ ⬁
1248
Cryogenics 1998 Volume 38, Number 12
Figure 12 Effect of lateral heat conduction parameter on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, s ⫽ 0.02 and p ⫽ ⬁
Figure 12 shows the variation of the hot and cold fluid exit temperatures with dimensionless time as a function of the lateral heat conduction parameter N, for an overall NTU of 10 and of 0.5 and 3.0, respectively. It can be observed that the effectiveness increases with an increase in or a decrease in the lateral thermal resistance of the plate. The cool down time also increases with an increase in or the overall effectiveness of the heat exchanger. Figure 13 shows the variation of fluid exit temperatures with dimensionless time for different ratios of the thermal time constants and an overall axial conduction parameter of 0.04. It can be observed that the overall time taken is nearly independent of the ratio of thermal constants, and is largely controlled only by the thermal resistance offered by the plate and fluid streams. The difference between the time–temperature profiles will be even closer for heat exchangers with much smaller axial heat conduction parameters. Figure 14 shows the variation of fluid exit temperatures with dimensionless time for different heat capacity rate ratios and the same overall NTU. It can be seen that the variation of hot fluid exit temperature is nearly the same during the initial cool down period, when the fluid change is nearly 80%, for different heat capacity rate ratios. On the other hand, the temperature change of the hot fluid at the exit is strongly influenced by the heat capacity rate ratios. A similar trend is observed with the plates in different channels as shown in Figure 15.
Figure 13 Effect of ratios of thermal time constants on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⫽ 1.0, ⫽ 1.0, ntu,f ⫽ 0.2 s ⫽ 0.04 and p ⫽ ⬁
Perforated plate matrix heat exchangers: P. Ramesh and G. Venkatarathnam
Figure 14 Effect of heat capacity rate ratio and overall NTU on the variation of fluid exit temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Figure 15 Effect of heat capacity rate ratio and overall NTU on the variation of cold end plate temperature with time. In all the cases: n ⫽ 50, ⌫ ⫽ 1.0, ⫽ 1.0, s ⫽ 0.0 and p ⫽ ⬁
Conclusions 1.
2.
3. 4.
The overall NTU or the effectiveness of the heat exchanger is the most important parameter that controls the cool down of MHEs. The higher the heat exchanger effectiveness, the higher will be the cool down time. Thus, for mobile applications with a requirement of fast cool down, a choice has to be made between fast cool down (low effectiveness) or low cryogen consumption (high effectiveness). The cool down time decreases due to irreversibilities such as axial heat conduction, plate thermal resistance, etc. The cool down time is indirectly proportional to and . The cool down time is only marginally dependent on the ratio of time constants (heat capacities) of the two flow channels. The cool down time is nearly independent of the number of plates when the number of plates is greater than 50.
References 1. McMohan, H. O., Bowen, R. J. and Bleyle, G. A., A perforated plate heat exchanger. Trans. ASME, 1950, 72, 623–632. 2. Venkatarathnam, G. and Sarangi, S., Matrix heat exchangers and their application in cryogenic systems. Cryogenics, 1990, 30, 907–912. 3. Venkatarathnam, G. and Sarangi, S., Analysis of Matrix heat exchangers performance. ASME J. Heat Transfer, 1991, 113, 830– 837. 4. Venkatarathnam, G., Effectiveness–Ntu relationship in perforated plate matrix heat exchangers. Cryogenics, 1996, 36, 235–241. 5. Venkatarathnam, G., A straight forward method for the sizing of perforated matrix heat exchangers. Advances in Cryogenic Engineering, 1998, 43, 0. 6. Rodriguez, J. I. and Mills, A. F., Analysis of single-blow transient testing of perforated plate heat exchangers. Int. J. Heat Mass Transfer, 1990, 33, 1969–1976. 7. Venkatarathnam, G., Matrix heat exchanger. PhD dissertation, Indian Institute of Technology, Kharagpur, 1990. 8. Fleming, R. B., A compact perforated plate heat exchanger. Adv. Cryog. Eng., 1969, 14, 197–204.
Cryogenics 1998 Volume 38, Number 12 1249