Performance of a Counterflow Heat Exchanger with Heat Loss Through

Cryogenics 39 (1999) 43–52

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 X Dimensionless axial co-ordinate in the exchanger defined by x/L A Surface area of heat transfer (m2) 2 Ac Cross-sectional area of the walls (m ) C Heat capacity rate of fluids defined by the Greek product of m and C (W/K) p ␪ Dimensionless temperature, defined as (T– c Specific heat capacity at constant pressure p T )/(T–T ) (J/kg K) h,in c,in ⑀ Effectiveness of the heat exchanger h Heat transfer coefficient (W/m2 K) ␭ Dimensionless axial conduction parameter, i Heat exchanger ineffectiveness ϭ l-⑀ defined as (kA /LC ) k Thermal conductivity of the wall material c min ␮ Ratio of heat capacity rates (C /C ) (W/m K) h min ␯ Ratio of heat capacity rates (C /C ) L Heat exchanger length (m) c h ␹ Dimensionless heat leak at the cold end of m Mass flow rate (kg/s) the heat exchanger ntu Number of transfer units for individual ⌬q Heat transfer rate (W) fluid sides NTU Overall number of transfer units rj Roots of characteristic equation Subscripts T Temperature (K) c Cold fluid W Width of a counterflow heat exchanger h Hot fluid (m) in Inlet x Axial co-ordinate (m) min Fluid of lower heat capacity rate max Fluid of higher heat capacity rate * Corresponding author. Tel: ϩ 44-235-1315; fax: ϩ 44-235-0509; out Outlet e-mail: [email protected] w Wall

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. 2). 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 ϫ 70 ϫ 1 mm. In any MMR, a temperature gradient of about 225 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. Longitudi- nal 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, Fig. 1. Schematic of a microminiature refrigerator. 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 refriger- MMR because of the significant parasitic heat conduc- ator heat exchangers. tion through the walls. The minimum effectiveness The Coefficient of Performance (COP) attainable in a necessary for a microminiature refrigerator to function, J-T cryocooler is strongly dependent on the effectiveness cannot, however, be evaluated using conventional heat of the heat exchanger used and the operating pressures. exchanger effectiveness expressions because of the heat It can be shown that no refrigeration can be realized in conduction between the cold end of the heat exchanger most refrigerators if the effectiveness of the heat and the evaporator. In order to estimate the performance exchanger is less than 85% [6]. The minimum heat of microminiature refrigerators accurately, the relation- exchanger effectiveness necessary for the Linde cycle ship between the heat conducted at the cold end of the process to be feasible will be even higher in the case of exchanger, the convective heat transfer between the S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52 45

equations for the hot ﬂuid, cold ﬂuid and the wall are obtained by energy balance as follows:

d␪ Hot fluid: h ϩ ntu (␪ Ϫ ␪ ) ϭ 0 (1) dx h h w

d␪ Cold fluid: c ϩ ntu (␪ Ϫ ␪ ) ϭ 0 (2) dx c w c

d␪ d␪ ␭ d 2␪ Wall: Ϫ h ϩ ␯ c ϩ w ϭ 0 (3) dx dx ␮ dx2

␪ ␪ ␪ In the above expressions, h, c and w represent the dimensionless temperature of the hot fluid, cold fluid and the separating wall respectively: Fig. 2. Linde cycle refrigerator. t Ϫ t x C C ␪ ϭ c,in X ϭ ␯ ϭ c ␮ ϭ h (4) t Ϫ t L C C streams and the longitudinal heat conduction along the h,in c,in h min wall should be clearly understood. kA hA hA ␭ ϭ c ntu ϭ ͩ ͪ ntu ϭ ͩ ͪ The main aim of this paper is to derive the expressions C h C c C relating the parasitic heat conducted at the cold end of min h c the heat exchanger, the NTUs and axial conductance of where ntuh and ntuc are the number of heat transfer units the wall. In the following sections, the governing equa- of the hot and cold fluid streams, respectively; ␯ is the tions of a counterflow heat exchanger which is losing heat capacity rate ratio and ␭ the longitudinal heat con- heat through the wall at the cold end are solved, and duction parameter defined as follows: closed form expression derived for the heat conducted at the cold end, and the effectiveness of the exchanger ␪ ϭ ␪ ϭ d w ϭ in terms of number of transfer units ‘NTU’ and axial x 0: h 1, 0 (5) conduction parameter ‘␭’ in the case of balanced flow dx (␯ ϭ 1) heat exchangers. For unbalanced flow (␯ Ͻ 1) ϭ ␪ ϭ ␪ ϭ ␪1 exchangers, the governing equations have been solved x 1: c 0, w w (6) numerically and the results are presented in graphical form. The performance of heat exchangers with heat loss Assuming that the wall is insulated at the hot end, and at the cold end is compared with conventional heat conducting at the cold end, the boundary conditions can exchangers with insulated ends. be expressed as follows:

x ϭ 1:␪c ϭ 0, ␪w ϭ 0 (7) 2. Mathematical model The heat loss to the surrounding area from the cold ␪ ϭ Let us consider a counterﬂow heat exchanger in which end will be maximum when w 0. The above condition longitudinal heat conduction is non-negligible and the therefore represents the maximum degradation possible exchanger subjected to heat loss at the cold end (Fig. due to combined longitudinal conduction and heat loss 3). Let the length of exchanger be ‘L’. The governing 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 Fig. 3. Heat exchanger model. considered. On the other hand, closed form general sol- 46 S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52

␪ utions can be obtained when w approaches zero (i.e. for the exit temperature of the hot and cold fluid streams the thermal resistance between the exchanger and the are presented here. The dimensionless heat loss at the evaporator is assumed to be negligible). The boundary cold end can be expressed in terms of the number of condition expressed in Eq. (7) is therefore preferable transfer units and the longitudinal heat conductance as over the more general boundary condition expressed in follows: Eq. (6). In the following sections the governing equations are solved with the boundary condition given by Ϫ dtw Eq. (7). kAc ␪ ␹ ϭ dx ϭϪ␭ d w Ϫ (13) Cmin(th,in tc,in) dx ؍ 3. Balanced flow exchanger (Cmin Cmax) The heat loss at the cold end of the heat exchanger can be expressed in dimensionless form as follows: The governing equations, Eqs. (1)–(3) have been solved along with the boundary conditions (Eqs. (5)–(7)) ␹ (14) for a balanced flow condition (Ch ϭ Cc), to obtain closed form expressions for the exit temperature of the hot and ␭ͩ Ϫ 1 ͪ Ϫ ͩ Ϫ 1 ͪ 1 r a1 1 r e 3 e 2 cold fluid streams as follows: ϭ

1 a1 ntuh −r 1 ntuh −r Ϫ ͩ ϩ ͪ ϩ ͫ Ϫ 2ͬ Ϫ ͫ Ϫ 3ͬ (1 a1) 1 e e NTU r2 ntuh Ϫ r2 r3 ntuh Ϫ r3 ␪ h,out (8)

r Ϫ Ϫ (e 3 a3) (1 a1) −r Ϫ ␭e 3ͫ1 Ϫ ͬ r Ϫ NTU (e 2 a2) 4. Effectiveness of the heat exchanger ϭ

1 a1 ntuh −r 1 ntuh −r Ϫ ͩ ϩ ͪ ϩ ͩ Ϫ 2ͪ Ϫ ͩ Ϫ 3ͪ (1 a1) 1 Ϫ e Ϫ e NTU r2 ntuh r2 r3 ntuh r3 The effectiveness of any heat exchanger is defined as the ratio of actual heat transfer to the maximum possible ␪ ϭ c,out 1 (9) heat transfer. Because of the heat loss at the cold end, the total heat transferred by the hot fluid stream is not 1 (a Ϫ 1)ͩ ϩ ␭ͪ 1 NTU equal to the total heat transferred to the cold fluid stream. ϩ

1 a1 ntuh −r 1 ntuh −r Therefore the usual definition of heat exchanger effec- Ϫ ͩ ϩ ͪ ϩ ͩ Ϫ 2ͪ Ϫ ͩ Ϫ 3ͪ (1 a1) 1 Ϫ e Ϫ e NTU r2 ntuh r2 r3 ntuh r3 tiveness cannot be used. The effectiveness of the heat exchanger can be defined separately in terms of the heat where r2, r3 are the eigen roots of the governing equa- transfer to the hot and cold fluid streams as follows: tions and are defined as follows: Ϫ ⑀ ϭ q˙hot ϭ Ch(th,out th,in) Ϫ h (15) ntuh ntuc Ϫ r ϭ (10) q˙max Cmin(th,in tc,in) 2 2 q˙ C (t Ϫ t ) ntu ϩ ntu 2 ntu ϩ ntu ⑀ ϭ cold ϭ c c,out c,in (16) Ϫ ͩ h cͪ ϩ h c c q˙ C (t Ϫ t ) Ί 2 ␭ max min h,in c,in The above expressions can be expressed in terms of Ϫ ϭ ntuh ntuc dimensionless fluid temperatures for a balanced flow r3 (11) 2 heat exchanger as follows: ϩ 2 ϩ ntuh ntuc ntuh ntuc ⑀ ϭ Ϫ ␪ ϭ ϩ ͩ ͪ ϩ h 1 h,out (Ch Cc) (17) Ί 2 ␭ ⑀ ϭ ␪ ϭ c c,out (Ch Cc) (18) In the above expressions a1, a2 and a3 are defined as follows: The two effectiveness expressions are related to the dimensionless heat transfer at the cold end as follows: − 1 Ϫ a e r3 ϭ ͩ 3 ͪ ϭ ntuh a1 − a2 (12) Ϫ r2 ϩ ⑀ ϭ ⑀ ϩ ␹ 1 a2e ntuh r2 h c (19) ntu ϭ →ϱ ϭ h 4.1. Limiting case (Cc Ch and NTU ) a3 ϩ ntuh r3 When the number of transfer units approaches infinity, Because of space limitations only the final expressions the expressions for the temperature of the hot and cold S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52 47

ﬂuid streams at the heat exchanger exit, and the heat B ϭ (26) conducted at the cold end wall reduces to the following −r − −r simple expressions: e 1 e r2 e 3 111 1 1 − − − ␪ ϭ 0; ␪ ϭ ; ⑀ ϭ 1; ⑀ ϭ ; (20) ΂ r1 r2 r3 ΃ h,out c,out ␭ ϩ h c ϩ ␭ e e e 1 1 (1 Ϫ a r ) (1 Ϫ a r ) (1 Ϫ a r ) 1 1 r 2 2 r 3 3 r ␭ 1 2 3 ␹ ϭ ϭ ;(Ch Cc) 1 ϩ ␭ C ϭ (27)

− − − ␯ Ϫ r1 ␯ Ϫ r2 ␯ Ϫ r3 When the axial heat conduction through the wall is a1( e ) a2( e ) a3( e ) negligible Eqs. (15) and (16) reduce to the following 111 standard expression for counterﬂow heat exchangers [7]. − − − ΂ r1 r2 r3 ΃ e e e (1 Ϫ a r ) (1 Ϫ a r ) (1 Ϫ a r ) NTU 1 1 r 2 2 r 3 3 r ⑀ ϭ ⑀ ϭ (21) 1 2 3 h c NTU ϩ 1 D ϭ (28)

Ͻ a a a 5. Unbalanced flow exchanger (Cmin Cmax) 1 2 3 111 The solution of the governing equations of an unbal- − − − ΂ e r1 e r2 e r3 ΃ anced heat exchanger is not as straight forward as in the (1 Ϫ a1r1) (1 Ϫ a2r2) (1 Ϫ a3r3) case of a balanced flow heat exchanger and the govern- r1 r2 r3 ing equations need to be solved numerically to obtain the heat exchanger performance. Following Kroeger [1], In the above expressions r1, r2, and r3 are the roots it has been assumed that the ‘per side’ ntu is the same of the characteristic equation outlined below: for both the fluid channels. ␭ ntu 3 ϩ Ϫ ␯ Ϫ ␭ Ϫ ϩ ␯ ϭ ϭ r r ntuͩ 1ͪ ( (29) ntuh ntuc ntu (22) ␮ ␮ Ϫ 1)ntu2 ϭ 0 The above assumption greatly simplifies the procedure and makes the solutions tractable. The numerical sol- The effectiveness of the heat exchanger can be ution of the governing equations with ntuh not the same as ntu indicated that the maximum difference between expressed in terms of the hot and cold fluid stream exit c temperatures as follows: the heat conducted when ntuh was not equal to ntuc and that when ntu ϭ ntu is limited to about 3% in the range h c ⑀ ϭ ␮ Ϫ ␪ ⑀ ϭ ␮␯␪ Ͻ Ͻ h (1 h,out) h c,out (30) 0.8 ntuh/ntuc 1.25. With the above assumption, the exit temperature of the hot and cold fluid streams can ␮ ϭ ␮ ϭ be expressed as follows: where 1 when hot fluid is the Cmin fluid and ␯ 1/ , when cold fluid is the Cmin fluid. In both the cases ␯ A ␭ B is defined as follows: ␪ ϭ (␯ Ϫ 1) Ϫ (23) h,out C ␮ C | | | | (m˙ C ) C ␯ ϭ p c ϭ c (31) (m˙ Cp)h Ch D ␪ ϭ (24) c,out | C | The dimensionless heat conducted through the wall at the cold end can be expressed in terms of the fluid exit temperatures as follows: where the matrices A, B, C and D are defined as follows: ␹ ϭ ⑀ Ϫ ⑀ ϭ ␮ Ϫ ␪ Ϫ ␮␯␪ h c (1 h,out) c,out (32) −r −r −r a1e 1 a2e 2 a3e 3 ␯Þ →ϱ 111 5.1. Limiting case ( 1 and ntu ) A ϭ (25) − − − ΂ r1 r2 r3 ΃ e e e When the number of heat transfer units approaches

r1 r2 r3 inﬁnity, the expressions for the exit temperature of hot 48 S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52 and cold ﬂuid streams and the heat conducted through the wall at cold end reduce to the following expressions:

er Ϫ 1 ␮(1 Ϫ ␯) ␪ ϭ 0 ␪ ϭ ␹ ϭ (33) h,out c,out ␯er Ϫ 1 1 Ϫ ␯er where

␮ ␯ Ϫ ϭ ( 1) r ␭ (34)

6. Results and discussion

The temperature profiles vary linearly with distance Fig. 5. Temperature profile of the hot and cold fluid streams in a in a balanced flow (␯ ϭ 1) heat exchanger without any normal heat exchanger with non-negligible longitudinal heat conduc- irreversibilities. The temperature profiles, however, get tion. distorted due to longitudinal heat conduction and heat loss through the ends. Fig. 4 shows the temperature pro- ϭ file of the hot and cold fluid in a balanced flow (Ch Cc) heat exchanger with an NTU of 20 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 Cmin 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. 6–20 show the variation of the hot fluid and the Fig. 6. Relation of hot fluid exit temperature to the NTU and longi- cold fluid outlet temperatures, as well as the dimen- tudinal heat conduction parameter (␭) for a balanced flow heat sionless heat loss at the cold end as a function of the exchanger losing heat at the cold end. design (conventional) NTU, axial conduction parameter [8] and different heat capacity rate ratios. The heat trans-

Fig. 4. Temperature profile of the hot and cold fluid streams in a Fig. 7. Relation of hot fluid exit temperature to the NTU and longi- ␭ ϭ normal heat exchanger with non-negligible longitudinal heat conduc- tudinal heat conduction parameter ( ) for a balanced flow (Ch/Cc tion and heat transfer through the wall at cold end. 0.8) heat exchanger losing heat at the cold end. S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52 49

Fig. 8. Relation of hot fluid exit temperature to the NTU and longi- Fig. 11. Relation of cold fluid exit temperature to the NTU and longi- ␭ ϭ tudinal heat conduction parameter ( ) for a balanced flow (Cc/Ch tudinal heat conduction parameter (␭) for a balanced flow heat 0.8) heat exchanger losing heat at the cold end. exchanger losing heat at the cold end.

Fig. 9. Relation of hot fluid exit temperature to the NTU and longi- Fig. 12. Relation of cold fluid exit temperature to the NTU and longitudinal heat conduction parameter (␭) for a balanced flow (C /C ϭ h c tudinal heat conduction parameter (␭) for a balanced flow (C /C ϭ 0.6) heat exchanger losing heat at the cold end. h c 0.8) 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 20% for an increase in ␭ from 0.01 to 0.08 for an NTU of 5 and ␯ ϭ 1. On the other hand, the cold fluid outlet temperature decreases with an increase in axial conduction parameter Fig. 10. Relation of hot fluid exit temperature to the NTU and longi- ␭ tudinal heat conduction parameter (␭) for a balanced flow (Cc/Ch ϭ as in conventional heat exchangers at all heat capacity 0.6) heat exchanger losing heat at the cold end. rate ratios (Figs. 11–15). At very large NTU, the hot 50 S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52

Fig. 13. Relation of cold fluid exit temperature to the NTU and longi- Fig. 16. Relation of dimensionless heat loss at the cold end to the tudinal heat conduction parameter (␭) for a balanced flow (Cc/Ch ϭ NTU and longitudinal heat conduction parameter (␭) for a balanced 0.8) heat exchanger losing heat at the cold end. flow heat exchanger losing heat at the cold end.

Fig. 14. Relation of cold fluid exit temperature to the NTU and longi- Fig. 17. Relation of dimensionless heat loss at the cold end to the ␭ ϭ tudinal heat conduction parameter ( ) for a balanced flow (Ch/Cc NTU and longitudinal heat conduction parameter (␭) for a balanced 0.6) heat exchanger losing heat at the cold end. ϭ flow (Ch/Cc 0.8) heat exchanger losing heat at the cold end.

Fig. 15. Relation of dimensionless heat loss at cold end to the NTU Fig. 18. Relation of dimensionless heat loss at the cold end to the and longitudinal heat conduction parameter (␭) for a balanced ﬂow NTU and longitudinal heat conduction parameter (␭) for a balanced

(Cc/Ch ϭ 0.6) heat exchanger losing heat at the cold end. ﬂow (Cc/Ch ϭ 0.8) heat exchanger losing heat at the cold end. S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52 51

Figs. 16–20 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 Cmin fluid, or the flow ϭ ␭ is balanced (Cc Ch), 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 Cmin ﬂuid, on the other hand, the heat leak increases within NTU as shown in Figs. 18 and 20 (see Eqs. (20) 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 Cmin 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 ␯ ϭ 0.6, when the hot fluid is the Cmin fluid, than when the cold fluid is the Cmin fluid. It should be remembered that the specific heat of pure Fig. 19. Relation of dimensionless heat loss at the cold end to the NTU and longitudinal heat conduction parameter (␭) for a balanced fluids is, in general, higher at higher pressures. Therefore ϭ the cold fluid is the C fluid when pure fluids are used flow (Ch/Cc 0.6) heat exchanger losing heat at the cold end. min in MMR refrigerators. The properties of some of the mixed refrigerants such as nitrogen–hydrocarbon mix- ∂ ∂ tures 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

¼ 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 Fig. 20. Relation of dimensionless heat loss at the cold end to the ends. ␭ NTU and longitudinal heat conduction parameter ( ) for a balanced ¼ For any given ␭, the hot fluid exit temperature of the flow (C /C ϭ 0.6) heat exchanger losing heat at the cold end. c h heat exchanger with conductive heat leak is always less than the hot fluid exit temperature of the heat fluid temperature at outlet will approach that of the cold exchanger with insulated ends. The difference can fluid at the inlet and the cold fluid temperature at the vary between 20 and 30% depending on NTU, ␯ and outlet will attain the value given by Eqs. (20) and (33) ␭. The exit temperature of the cold fluid is, however, for balanced and unbalanced flows, respectively, as lower than that in an equivalent heat exchanger with shown in Figs. 6–15. insulated ends. The liquid yield in a J-T liquefier will increase if the ¼ The hot fluid exit temperature approaches zero as hot fluid (high pressure fluid) cools to a lower tempera- NTU of the heat exchanger approaches infinity, irres- ture before it is expanded in the J-T valve. The hot fluid pective of the heat capacity rate ratios, while the outlet outlet temperature will be lower in heat exchangers with temperature of cold fluid and the heat leak at the cold heat leak at the cold end than in heat exchangers with end of the wall remains finite and a function of ␭ insulated ends. Therefore, it may appear that the yield and ␯. should increase because of the heat conduction at the ¼ For a balanced flow and for an unbalanced flow with ϭ ␹ cold end of the heat exchanger. However, the increase Ch Cmin, the heat leak at the cold end wall ( ) in yield is more than offset by the parasitic heat that decreases with an increase in NTU, while for an ϭ ␹ enters the dewar (phase separator) from the heat unbalanced flow with Cc Cmin, increases with an exchanger leading to an overall reduction in the liquid increase in NTU. Since the minimum fluid is the cold ϭ yield or refrigerating capacity because of the heat trans- fluid (Cc Cmin) in MMR heat exchangers operating fer to surrounding at the cold end of the heat exchanger. with nitrogen or argon, the degradation of perform- 52 S. Pradeep Narayanan, G. Venkatarathnam /Cryogenics 39 (1999) 43–52

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