Performance of a counterflow heat exchanger with heat loss through the wall at the cold end

Cryogenics 39 (1999) 43 5 Performance of a counterflow heat exchanger with heat loss through the wall at the cold end S. Pradeep Narayanan, G. Venkatarathnam * Department of Mechanical Engineering, Indian Institute of Technology, Chennai 600 036, India Received 7 July 1998; accepted 9 November 1998 Abstract The performance of high effectiveness heat exchangers used in cryogenic systems is strongly controlled by irreversibilities such as longitudinal heat conduction and heat leak from ambient. In all heat exchanger analyses, it is assumed that no heat is lost through the heat exchanger walls. In the case of small J-T refrigerators such as microminiature refrigerators, the heat exchanger cold end is almost directly connected to the evaporator, which may result in a large amount of heat loss through the heat exchanger wall at the cold end. The rate of heat loss through the wall at the cold end is also strongly dependent on the longitudinal thermal resistance of the wall. In this paper, we present the relationship between the effectiveness of a heat exchanger losing heat at the cold end and other resistances such as number of transfer units (NTU), longitudinal thermal resistance etc. The performance of such heat exchangers under different operating conditions is also discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Compact heat exchangers; Heat leak; High effectiveness; Longitudinal heat conduction Nomenclature A Surface area of heat transfer (m ) A c Cross-sectional area of the walls (m ) C Heat capacity rate of fluids defined by the product of m and C p (W/K) c p Specific heat capacity at constant pressure (J/kg K) h Heat transfer coefficient (W/m K) i Heat exchanger ineffectiveness l- k Thermal conductivity of the wall material (W/m K) L Heat exchanger length (m) m Mass flow rate (kg/s) ntu Number of transfer units for individual fluid sides NTU Overall number of transfer units r j Roots of characteristic equation T Temperature (K) W Width of a counterflow heat exchanger (m) x Axial co-ordinate (m) * Corresponding author. Tel: 44-35-1315; fax: 44-35-0509; e-mail: gvenkat@acer.iitm.ernet.in X Dimensionless axial co-ordinate in the exchanger defined by x/l Greek Dimensionless temperature, defined as (T T h,in )/(T T c,in ) Effectiveness of the heat exchanger Dimensionless axial conduction parameter, defined as (ka c /LC min ) Ratio of heat capacity rates (C h /C min ) Ratio of heat capacity rates (C c /C h ) Dimensionless heat leak at the cold end of the heat exchanger q Heat transfer rate (W) Subscripts c h in min max out w Cold fluid Hot fluid Inlet Fluid of lower heat capacity rate Fluid of higher heat capacity rate Outlet Wall 0011-75/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S0011-75(98)0013-4

44 S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 1. Introduction The performance of low effectiveness heat exchangers is largely dependent on the finite heat transfer area (heat transfer coefficients), and can be predicted fairly accurately using conventional -NTU or Log mean temperature difference (LMTD) methods. The performance of high effectiveness heat exchangers such as those used in cryogenic systems, on the other hand, is dependent on other irreversibilities such as longitudinal (axial) heat condition through the walls and heat leak from the ambient. Different authors [1 3] have studied the effect of these two irreversibilities on the performance of heat exchangers and have presented the relationship between attainable effectiveness and the traditional number of transfer units (NTU), the lateral conduction resistance and the rate of heat leak into the heat exchangers. A common assumption made by all the authors analysing the performance of heat exchangers is that the walls of the heat exchanger are insulated at either end, i.e. there is no heat transfer by conduction at the two ends of the heat exchanger. Such an assumption is, however, not valid in the case of a microminiature cryogenic refrigerator. Fig. 1 shows the schematic of a microminiature refrigerator, which operates on the Linde cycle (Fig. ). In the microminiature refrigerator (MMR) the heat exchanger, expansion capillary and the evaporator are all etched on a thin sheet of glass or stainless steel [4,5]. The typical size of a microminiature refrigerator is 30 70 1 mm. In any MMR, a temperature gradient of about 5 K is sustained between the two ends of the refrigerator within a short distance (typically 70 100 mm). Unlike a normal heat exchanger, the cross-sectional area of the walls of the heat exchanger is much larger than the flow cross-section in a MMR. Longitudinal heat conduction through the walls will therefore be much higher in MMR heat exchangers compared to other type of cryogenic heat exchangers. Because of the small distance between the cold end of the heat exchanger and the evaporator, a significant part of the heat conducted through the wall will be transferred to the evaporator. The loss of refrigeration due to the parasitic heat conduction through the walls can be quite significant. Hence, the assumption of insulated cold end (i.e. no heat transfer between the heat exchanger walls and the evaporator) cannot be applied in the case of microminiature refrigerator heat exchangers. The Coefficient of Performance (COP) attainable in a J-T cryocooler is strongly dependent on the effectiveness of the heat exchanger used and the operating pressures. It can be shown that no refrigeration can be realized in most refrigerators if the effectiveness of the heat exchanger is less than 85% [6]. The minimum heat exchanger effectiveness necessary for the Linde cycle process to be feasible will be even higher in the case of Fig. 1. Schematic of a microminiature refrigerator. MMR because of the significant parasitic heat conduction through the walls. The minimum effectiveness necessary for a microminiature refrigerator to function, cannot, however, be evaluated using conventional heat exchanger effectiveness expressions because of the heat conduction between the cold end of the heat exchanger and the evaporator. In order to estimate the performance of microminiature refrigerators accurately, the relationship between the heat conducted at the cold end of the exchanger, the convective heat transfer between the

S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 45 equations for the hot fluid, cold fluid and the wall are obtained by energy balance as follows: Hot fluid: d h dx ntu h( h w ) 0 (1) Cold fluid: d c dx ntu c( w c ) 0 () Wall: d h dx d c dx d w 0 (3) dx Fig.. Linde cycle refrigerator. In the above expressions, h, c and w represent the dimensionless temperature of the hot fluid, cold fluid and the separating wall respectively: streams and the longitudinal heat conduction along the wall should be clearly understood. The main aim of this paper is to derive the expressions relating the parasitic heat conducted at the cold end of the heat exchanger, the NTUs and axial conductance of the wall. In the following sections, the governing equations of a counterflow heat exchanger which is losing heat through the wall at the cold end are solved, and closed form expression derived for the heat conducted at the cold end, and the effectiveness of the exchanger in terms of number of transfer units NTU and axial conduction parameter in the case of balanced flow ( 1) heat exchangers. For unbalanced flow ( 1) exchangers, the governing equations have been solved numerically and the results are presented in graphical form. The performance of heat exchangers with heat loss at the cold end is compared with conventional heat exchangers with insulated ends.. Mathematical model Let us consider a counterflow heat exchanger in which longitudinal heat conduction is non-negligible and the exchanger subjected to heat loss at the cold end (Fig. 3). Let the length of exchanger be L. The governing Fig. 3. Heat exchanger model. t t c,in t h,in t c,in ka c C min X x L ntu h ha Ch C c C h ntu c ha Cc C h C min (4) where ntu h and ntu c are the number of heat transfer units of the hot and cold fluid streams, respectively; is the heat capacity rate ratio and the longitudinal heat conduction parameter defined as follows: x 0: h 1, d w dx 0 (5) x 1: c 0, w 1 w (6) Assuming that the wall is insulated at the hot end, and conducting at the cold end, the boundary conditions can be expressed as follows: x 1: c 0, w 0 (7) The heat loss to the surrounding area from the cold end will be maximum when w 0. The above condition therefore represents the maximum degradation possible due to combined longitudinal conduction and heat loss to surroundings at the heat exchanger cold end. In a real situation, the wall temperature at the cold end is decided by the heat transfer resistance between the heat exchanger and the surroundings (evaporator). In order to estimate the true performance, the governing equations, however, need to be solved together with that for the heat transfer between the heat exchanger and the evaporator to estimate the true performance of the heat exchanger and the amount of heat loss through the wall at the cold end of the heat exchanger. General solutions can not be obtained when the heat transfer resistance between the wall and ambient at the cold end is also considered. On the other hand, closed form general sol-

46 S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 utions can be obtained when w approaches zero (i.e. the thermal resistance between the exchanger and the evaporator is assumed to be negligible). The boundary condition expressed in Eq. (7) is therefore preferable over the more general boundary condition expressed in Eq. (6). In the following sections the governing equations are solved with the boundary condition given by Eq. (7). 3. Balanced flow exchanger (C min C max ) The governing equations, Eqs. (1) (3) have been solved along with the boundary conditions (Eqs. (5) (7)) for a balanced flow condition (C h C c ), to obtain closed form expressions for the exit temperature of the hot and cold fluid streams as follows: h,out (8) (1 a 1 ) NTU (1 a 1 )1 NTU 1 a 1 r e r 31 (er 3 a 3 ) ntu h ntu h r e r (e r a ) 1 ntu r 3 h ntu h r 3 e r 3 c,out 1 (9) (a 1 1) 1 NTU (1 a 1 )1 NTU 1 a 1 ntu r h e ntu h r r 1 r 3 ntu h ntu h r 3 e r 3 where r, r 3 are the eigen roots of the governing equations and are defined as follows: r ntu h ntu c ntu h ntu c ntu h ntu c r 3 ntu h ntu c ntu h ntu c ntu h ntu c (10) (11) In the above expressions a 1, a and a 3 are defined as follows: a 1 1 a 3e r 3 a 1 a e r ntu h (1) ntu h r a 3 ntu h ntu h r 3 Because of space limitations only the final expressions for the exit temperature of the hot and cold fluid streams are presented here. The dimensionless heat loss at the cold end can be expressed in terms of the number of transfer units and the longitudinal heat conductance as follows: dt w ka c dx C min (t h,in t c,in ) d w dx (13) The heat loss at the cold end of the heat exchanger can be expressed in dimensionless form as follows: (14) 1 1 e r 3 (1 a 1 )1 NTU 1 a 1 r a 11 1 1 r 3 ntu h ntu h r e r e r 4. Effectiveness of the heat exchanger ntu h ntu h r 3 e r 3 The effectiveness of any heat exchanger is defined as the ratio of actual heat transfer to the maximum possible heat transfer. Because of the heat loss at the cold end, the total heat transferred by the hot fluid stream is not equal to the total heat transferred to the cold fluid stream. Therefore the usual definition of heat exchanger effectiveness cannot be used. The effectiveness of the heat exchanger can be defined separately in terms of the heat transfer to the hot and cold fluid streams as follows: h q hot C h(t h,out t h,in) q max C min (t h,in t c,in ) c q cold C c(t c,out t c,in) q max C min (t h,in t c,in ) (15) (16) The above expressions can be expressed in terms of dimensionless fluid temperatures for a balanced flow heat exchanger as follows: h 1 h,out (C h C c ) (17) c c,out (C h C c ) (18) The two effectiveness expressions are related to the dimensionless heat transfer at the cold end as follows: h c (19) 4.1. Limiting case (C c C h and NTU ) When the number of transfer units approaches infinity, the expressions for the temperature of the hot and cold

S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 47 fluid streams at the heat exchanger exit, and the heat conducted at the cold end wall reduces to the following simple expressions: h,out 0; c,out 1 1 ; h 1; c 1 1 ; (0) 1 ;(C h C c ) When the axial heat conduction through the wall is negligible Eqs. (15) and (16) reduce to the following standard expression for counterflow heat exchangers [7]. h c NTU NTU 1 5. Unbalanced flow exchanger (C min C max ) (1) The solution of the governing equations of an unbalanced heat exchanger is not as straight forward as in the case of a balanced flow heat exchanger and the governing equations need to be solved numerically to obtain the heat exchanger performance. Following Kroeger [1], it has been assumed that the per side ntu is the same for both the fluid channels. ntu h ntu c ntu () The above assumption greatly simplifies the procedure and makes the solutions tractable. The numerical solution of the governing equations with ntu h not the same as ntu c indicated that the maximum difference between the heat conducted when ntu h was not equal to ntu c and that when ntu h ntu c is limited to about 3% in the range 0.8 ntu h /ntu c 1.5. With the above assumption, the exit temperature of the hot and cold fluid streams can be expressed as follows: h,out ( 1) A C B C (3) c,out D C (4) where the matrices A, B, C and D are defined as follows: A a1e r1 ae r a3e r3 1 1 1 e r 1 r 1 e r r e r3 r 3 (5) B (6) (1 a 1 r 1 ) e r 1 e r 1 e r e r 3 1 1 1 r 1 (1 a r ) e r r (1 a 3 r 3 ) e r3 r 3 C (7) a1( e r1) a( e r) a3( e r3) 1 1 1 (1 a 1 r 1 ) e r 1 (1 a r r ) e r (1 a 1 r 3 r 3 ) e r3 r 3 D (8) (1 a 1 r 1 ) e r 1 a 1 a a 3 1 1 1 r 1 (1 a r ) e r r (1 a 3 r 3 ) e r3 r 3 In the above expressions r 1, r, and r 3 are the roots of the characteristic equation outlined below: r3 r ntu ntu 1 ( (9) 1)ntu 0 The effectiveness of the heat exchanger can be expressed in terms of the hot and cold fluid stream exit temperatures as follows: h (1 h,out ) h c,out (30) where 1 when hot fluid is the C min fluid and 1/, when cold fluid is the C min fluid. In both the cases is defined as follows: (ṁc p) c (ṁc p ) h C c C h (31) The dimensionless heat conducted through the wall at the cold end can be expressed in terms of the fluid exit temperatures as follows: h c (1 h,out ) c,out (3) 5.1. Limiting case (1 and ntu ) When the number of heat transfer units approaches infinity, the expressions for the exit temperature of hot

48 S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 and cold fluid streams and the heat conducted through the wall at cold end reduce to the following expressions: h,out 0 c,out er 1 e r 1 (1 ) 1 e r (33) where r ( 1) (34) 6. Results and discussion The temperature profiles vary linearly with distance in a balanced flow ( 1) heat exchanger without any irreversibilities. The temperature profiles, however, get distorted due to longitudinal heat conduction and heat loss through the ends. Fig. 4 shows the temperature profile of the hot and cold fluid in a balanced flow (C h C c ) heat exchanger with an NTU of 0 and of 0.04, with heat transfer through the wall at the cold end of the heat exchanger. Fig. 5 shows that in an equivalent but fully insulated (normal) heat exchanger with the same design parameters. Because of the longitudinal heat conduction through the walls and the heat loss at the cold end, a balanced flow heat exchanger shows a profile similar to that of an unbalanced flow heat exchanger with hot fluid being the C min fluid. Because of the heat loss through the wall at the cold end, the hot fluid will cool more than that in a fully insulated heat exchanger, while the cold fluid heats less than that in a fully insulated heat exchanger. Figs. 6 0 show the variation of the hot fluid and the cold fluid outlet temperatures, as well as the dimensionless heat loss at the cold end as a function of the design (conventional) NTU, axial conduction parameter [8] and different heat capacity rate ratios. The heat trans- Fig. 5. Temperature profile of the hot and cold fluid streams in a normal heat exchanger with non-negligible longitudinal heat conduction. Fig. 6. Relation of hot fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow heat exchanger losing heat at the cold end. Fig. 4. Temperature profile of the hot and cold fluid streams in a normal heat exchanger with non-negligible longitudinal heat conduction and heat transfer through the wall at cold end. Fig. 7. Relation of hot fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.8) heat exchanger losing heat at the cold end.

S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 49 Fig. 8. Relation of hot fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.8) heat exchanger losing heat at the cold end. Fig. 11. Relation of cold fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow heat exchanger losing heat at the cold end. Fig. 9. Relation of hot fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.6) heat exchanger losing heat at the cold end. Fig. 1. Relation of cold fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.8) heat exchanger losing heat at the cold end. Fig. 10. Relation of hot fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.6) heat exchanger losing heat at the cold end. ferred from the hot stream increases with an increase in NTU, as in a normal heat exchanger, resulting in a decrease in the temperature of the hot fluid at its outlet with an increase in NTU as shown in Figs. 6 10 at different heat capacity rate ratios. In normal heat exchangers with insulated ends, the hot fluid outlet temperature will increase with an increase in axial conduction parameter. However, when the wall at the cold end is directly connected to a cold sink, the hot fluid outlet temperature will decrease with an increase in axial conduction parameter as shown in Figs. 6 10. For example, h,out decreases by about 0% for an increase in from 0.01 to 0.08 for an NTU of 5 and 1. On the other hand, the cold fluid outlet temperature decreases with an increase in axial conduction parameter as in conventional heat exchangers at all heat capacity rate ratios (Figs. 11 15). At very large NTU, the hot

50 S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 Fig. 13. Relation of cold fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.8) heat exchanger losing heat at the cold end. Fig. 16. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow heat exchanger losing heat at the cold end. Fig. 14. Relation of cold fluid exit temperature to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.6) heat exchanger losing heat at the cold end. Fig. 17. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.8) heat exchanger losing heat at the cold end. Fig. 15. Relation of dimensionless heat loss at cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.6) heat exchanger losing heat at the cold end. Fig. 18. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.8) heat exchanger losing heat at the cold end.

S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 51 Fig. 19. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow (C h /C c 0.6) heat exchanger losing heat at the cold end. Figs. 16 0 show the variation of heat leak at the cold end with NTU and for different heat capacity rate ratios. When the hot fluid is the C min fluid, or the flow is balanced (C c C h ), the heat leak at the cold end decreases with an increase in NTU as shown in Figs. 16, 17 and 19. When the cold fluid is the C min fluid, on the other hand, the heat leak increases within NTU as shown in Figs. 18 and 0 (see Eqs. (0) and (33)). It can also be seen that the heat loss at the cold end will be much smaller when the hot fluid is the C min fluid for the same operating parameters. It can be seen that the heat loss can be lower by up to 300% for NTUs of 6 or 7, and 0.6, when the hot fluid is the C min fluid, than when the cold fluid is the C min fluid. It should be remembered that the specific heat of pure fluids is, in general, higher at higher pressures. Therefore the cold fluid is the C min fluid when pure fluids are used in MMR refrigerators. The properties of some of the mixed refrigerants such as nitrogen hydrocarbon mixtures can be such that ( H/ T) p of the cold fluid stream is higher than that of the hot fluid stream over most of the heat exchanger length. The use of such mixtures in MMRs will therefore result in much lower parasitic heat conduction to the evaporator. 7. Conclusions Fig. 0. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter () for a balanced flow (C c /C h 0.6) heat exchanger losing heat at the cold end. fluid temperature at outlet will approach that of the cold fluid at the inlet and the cold fluid temperature at the outlet will attain the value given by Eqs. (0) and (33) for balanced and unbalanced flows, respectively, as shown in Figs. 6 15. The liquid yield in a J-T liquefier will increase if the hot fluid (high pressure fluid) cools to a lower temperature before it is expanded in the J-T valve. The hot fluid outlet temperature will be lower in heat exchangers with heat leak at the cold end than in heat exchangers with insulated ends. Therefore, it may appear that the yield should increase because of the heat conduction at the cold end of the heat exchanger. However, the increase in yield is more than offset by the parasitic heat that enters the dewar (phase separator) from the heat exchanger leading to an overall reduction in the liquid yield or refrigerating capacity because of the heat transfer to surrounding at the cold end of the heat exchanger. The hot fluid will cool to a much lower temperature in a heat exchanger with heat loss at the cold than in an equivalent heat exchanger with insulated ends. Similarly, the cold fluid will heat to a temperature lower than that in a heat exchanger with insulated ends. For any given, the hot fluid exit temperature of the heat exchanger with conductive heat leak is always less than the hot fluid exit temperature of the heat exchanger with insulated ends. The difference can vary between 0 and 30% depending on NTU, and. The exit temperature of the cold fluid is, however, lower than that in an equivalent heat exchanger with insulated ends. The hot fluid exit temperature approaches zero as NTU of the heat exchanger approaches infinity, irrespective of the heat capacity rate ratios, while the outlet temperature of cold fluid and the heat leak at the cold end of the wall remains finite and a function of and. For a balanced flow and for an unbalanced flow with C h C min, the heat leak at the cold end wall () decreases with an increase in NTU, while for an unbalanced flow with C c C min, increases with an increase in NTU. Since the minimum fluid is the cold fluid (C c C min ) in MMR heat exchangers operating with nitrogen or argon, the degradation of perform-

5 S. Pradeep Narayanan, G. Venkatarathnam / Cryogenics 39 (1999) 43 5 ance due to heat leak increases with an increase in NTU of MMR exchangers. The performance of microminiature refrigerators will be better when operating mixed refrigerants which show a higher heat capacity rate in the cold fluid stream are used, compared to that operating with pure fluids, everything else being the same. The closed form expressions and limiting solutions derived give a new insight into MMR type heat exchanger performance, and should prove to be useful in the design and rating of heat exchangers employed in small J-T cryocoolers. References [] Barron RF. Effect of heat transfer from ambient on cryogenic heat exchanger performance. Advances in Cryogenic Engineering 1988;33:65 7. [3] Chowdhury K, Sarangi S. Performance of cryogenic heat exchangers with heat leak from the surroundings. Advances in Cryogenic Engineering 1988;33:73 80. [4] Little WA. Microminiature refrigerator. Review of Scientific Instruments 1984;55:661 80. [5] Mikulin EU. The miniature Joule Thomsen refrigerator. ICEC Supplement, Cryogenics 199;3:17 19. [6] Barron RF. Cryogenic systems, nd edn. Oxford University Press, UK, 1985. [7] Kays WM, London AL. Compact heat exchangers, nd edn. McGraw Hill, New York, USA, 1964. [8] Pradeep Narayanan S. Analysis of performance of microminiature refrigerator heat exchangers. MS Thesis, Indian Institute of Technology, Chennai, 1998. [1] Kroger PG. Performance deterioration in high effectiveness heat exchangers due to axial conduction effects. Advances in Cryogenic Engineering 1967;1:363 7.