sks6

Matrix heat exchangers and their application in cryogenic systems G. Venkatarathnam and Sunil Sarangi Process Equipment and Design Laboratory, Cryogenic Engineering Centre, Indian Institute of Technology, Kharagpur - 721 302, India Received 7 December 1989

Published in Cryogenics, 1990, 30(11) 907-918

The necessity of high effectiveness in a small volume has led to the development of perforated plate matrix heat exchangers (MHE) for cryogenic applications. Although the basic principles have remained the same, the techniques of fabrication and bonding have changed considerably during the last four decades. With the introduction of all metal construction, these exchangers are finding increasing use in cryogenic refrigerators. The mechanism of heat transfer in a matrix heat exchanger is complex. Convection in three different surfaces and conduction in two different directions are coupled together in determining the temperature profiles. While early analyses were based on simple empirical correlations and approximate analytical solutions, they have given way to accurate numerical models. This paper traces the chronological development of the MHE and different methods of fabrication, heat transfer and fluid flow characteristics and design and simulation procedures.

Keywords: heat transfer; physical properties; reviews

Nomenclature A At A" a b C C f G H h L l m n n Nu Ntu P p p Pr q R r

Heat transfer area (m 2) Cylindrical surface area of holes (m 2) Heat transfer area per unit volume of plate (m-t) Width of slots (rectangular perforations) (m) Width of separator (m) Heat capacity rate (W K -t) Constant in heat transfer correlation Specific heat at constant pressure (J kg -~ K -t) Diameter of holes (cylindrical perforations) (m) Fanning friction factor Fluid mass velocity (kg m -2 s -l) Plate height (fin height) (m) Heat transfer coefficient (W m -2 K -1) Overall length of exchanger (m) Plate thickness (m) Fin parameter (m -t) Number of plates Index in heat transfer coefficient Nusselt number Number of heat transfer units Overall porosity Plate porosity Intermediate parameter Prandtl number Intermediate parameter Heat transfer resistance Radius of circular plate (m)

O011 - 2 2 7 5 / 9 0 / 1 1 0 9 0 7 ©

Re

s St U W

Reynolds number Spacer thickness Stanton number Overall heat transfer coefficient (W m -2 K -l) Exchanger width (m)

Greek letters

c~ /3 ~f~.

q~

exp(-Ntu/n) Heat transfer surface area per unit volume (m -t) Heat exchanger effectiveness Fin effectiveness Axial conduction parameter Intermediate parameter Heat capacity rate ratio (Cmin/Cmax) Plate conduction parameter Drag coefficient

Subscripts

e eq i i p P s w 1,2

Effective Equivalent Channel number Surface type Plate (perforated portion) Plate (unperforated portion) Spacer Wall (separator) Channel number

- 12

1990 Butterworth-Heinemann

Ltd

Cryogenics 1990 Vol 30 November

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Review

Heat exchangers are among the most vital components of any cryogenic refrigeration/liquefaction system. Unlike their counterparts in most other chemical process industries, cryogenic heat exchangers need very high effectiveness. The performance of refrigerators, liquefiers and separation units is strongly dependent on the effectiveness of the heat exchangers used. In fact, if the effectiveness of the heat exchanger is below a certain critical value, most cryogenic processes would cease to function 1. In addition, the low values of attainable coefficient of performance (COP) and the resulting high cost of refrigeration make it economically sensible to use more effective, albeit more expensive, heat transfer equipment. At low performance levels, heat exchanger effectiveness is largely determined by the heat transfer film coefficients. As the effectiveness increases beyond (say) 90 %, other sources of irreversibility become controlling factors. Among them are2: 1 2 3 4 5

axial conduction3'4; flow maldistribution5'6; wall thermal resistance4; heat exchange with the surroundingsT'8; and number of lateral heat transfer paths 9.

Conventional cryogenic heat exchangers such as the coiled tube heat exchangers of Hampson 1°'1t and Collins ~2'13 types and brazed aluminium plate fin exchangers ~4 have largely been able to circumvent these difficulties. Apart from having a high effectiveness, cryogenic heat exchangers also need to be very compact, i.e. they must accommodate a large amount of surface area in a small volume. This helps in controlling heat exchange with the surroundings by reducing exposed surface area. Besides, a small mass means a smaller cooling load and a faster cooling time for refrigerators. This requirement is particularly important for small refrigerators operating at liquid helium temperature. Several types of compact cryogenic heat exchanger have been reviewed by Dilevskaya ~5. The necessity of attaining high effectiveness and high degree of compactness together in one unit led to the invention of matrix heat exchangers (MHEs) by McMahon et al. 16 in 1950. Although McMahon et al. had originally expected their invention to be used for large scale air separation that has not happened, largely due to the availability of reliable brazed aluminium plate fin exchangers. Subsequent developments in matrix heat exchangers have largely been directed towards use in helium liquefiers ~7-25 and, more recently, towards low power cryorefrigerators 26-3° based on Claude' and reverse Brayton cycles. Bova et al. 31 also developed all metal MHEs for use in air separation units. An excellent review of MHE literature has been presented in Russian by Zablotskaya 32.

Structure of matrix heat exchanger A matrix heat exchanger, shown schematically in Figures 1 and 2, essentially consists of a stack of high thermal conductivity (copper or aluminium) perforated plates or wire screens, alternating with low thermal conductivity spacers (plastics, stainless steel). The packet of alternate

908

Cryogenics 1990 Vol 30 November

~//

/~/" S.S. spacer

L

/°¢ o

ader

~ ~ream-2

stream-1

Figure 1 Schematic of perforated plate matrix heat exchanger

Figure 2

Perforated

plate matrix heat exchanger and its

components

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 1619 ' . The small flow passages (typically 0.3 - 1.0 mm in diameter) ensure a high heat transfer coefficient and high surface area density (up to 6000 m -1 (References 19 and 25)). The spacers, being of low thermal conductivity material, also help in reducing axial conduction and consequent deterioration of performance. There is no unique geometry of the heat transfer plates. Circular and rectangular shapes have been extensively used. To reduce the fin height and thereby to achieve more effective contact between the fluid streams, McMahon et al. 16 used multiple passages for both the streams. The individual passages of the two streams alternated with each other and were connected together in the headers. This left open the possibility of flow maldistribution. Fleming 19'34 circumvented this problem by designing spacers in such a way that all the passages of the same

Review

stream were internally connected, providing continuous reheadering. Circular geometry, first patented by Garwin 35 in 1966 has been adopted by various investigators 26"27'36'37. Matsuda et al. 27 have adopted Fleming's technique to reduce fin height in circular plates. Their spacers are designed in such a way that the flow cross sections are in the form of two intertwined spirals providing intimate contact between the two fluid streams. With the use of modern bonding techniques, particularly in all-metal exchangers, it is possible to provide leak free passages at high pressures. Sotnikov et a l . 37 have reported an exchanger which was successfully tested at a pressure of 115 MPa. Still, to maintain comparable fluid velocities in both the streams, the cross-sectional area made available to any particular stream should be proportional to its volume flow rate. In most cryogenic systems the mass flow rates are nearly the same, which means that the cross-sectional area of any stream should be inversely proportional to its density or pressure. If the ratio of operating pressures is very high the above approach is not very satisfactory because the cross-sectional area and hence the heat transfer area provided to the high pressure stream are small. This adversely affects the overall heat transfer performance of the exchanger. The matrix tube heat exchanger, first proposed by Bon Mardion and Claudet 38 circumvents this problem by passing the high pressure stream in a tube coiled around a stack of perforated plates carrying the low pressure fluid. The assembly is encased in a thin walled tube of low conductivity. Although Bon Mardion and Claudet designed this exchanger for ease of fabrication and used it for liquidto-gas heat exchange, this design is ideal for a gas-to-gas heat exchanger with large pressure ratio. This class of exchangers have recently been studied by Longsworth 39.

Chronological development Matrix heat exchangers were first introduced in 1949 by McMahon et al. 16. Their design consisted of a series of perforated aluminium plates separated by thin die cut neoprene gaskets which also served to separate the fluid streams. The whole pack, together with a cast aluminium header at each end was held together by means of steel tie rods. The neoprene spacers almost eliminated axial conduction and provided gas tight seals even at liquid air temperatures. The construction was extremely simple and repairs were easy. Full scale commercial units were intended to be fabricated for use in production of liquid oxygen, but subsequent reports are not available. Garwin's design 35 which came sixteen years after that of McMahon et al. contained several new features. He proposed a cylindrical geometry which permitted encasement of the entire assembly in a thin walled tube to eliminate leakage of gas to the surroundings. He also proposed the use of resilient members (springs) at both ends to compensate for differential contraction between the heat exchanger core and the support structure on cooling. In spite of the obvious advantages in easy assembly and repair, heat exchangers made with demountable elastomeric spacers (e.g., neoprene L6and nylon 35) were I not very successful. They probably developed sealing problems at low temperatures, particularly after repeated thermal cycling. This led to the use of epoxy resin adhesives and non-elastomeric polymer spacers to bond

perforated plates or wire screens. The heat exchanger developed by Vonk 17'18was made of a stack of alternate layers of copper wire screens and resin impregnated paper. The stack was placed in a jig and cured at a specified temperature under a given load. This exchanger was used in the Stirling cycle helium liquefier made by Philips ~8. The helium liquefier development program at General Electric Corporation, USA led to the development of matrix heat exchangers made of perforated aluminium plates and plastic spacers 19,34 Although perforated plates are inferior to wire screens in terms of heat transfer and pressure drop performance, they were probably chosen because of their better bonding properties. The exact specifications of the plastics and the adhesives used have not been published in open literature. During the last decade, plastic spacers and adhesive bonding techniques have given way to all metal construction. First patented by Fleming in 1 9 6 7 33 , these exchangers use copper (k = 400 W m -I K -l) perforated plates and stainless steel (k = 12 W m -I K -l) spacers bonded together by soldering, brazing or diffusion (solid state) welding 4°. The all metal matrix heat exchangers offer distinct advantages over their adhesive bonded predecessors: 1 thermal stresses developed during bonding and during operation are small because of close coefficients of expansion of copper and stainless steel; 2 the probability of forming hermetic seals between two metals is much higher than that between a metal and a plastic; and 3 metals are less likely than plastics to show deterioration in ductility on ageing. All metal exchangers are, generally speaking, poorer in thermal performance than metal-plastic combinations. This is largely due to higher axial conduction through stainless steel spacers ( k = 12 W m -l K-~), than through plastics (k < 1 W m -1 K-l). Besides, all metal exchangers cannot use wire screens, which have better thermal-hydraulic characteristics than perforated plates. While adhesive bonded exchangers can use aluminium plates, all metal exchangers must use copper, which makes the exchanger much heavier. (Aluminium does not bond properly to stainless steel or other low conductivity metals.) But Bova et al. 31 have shown that under proper design and operating conditions, all metal MHEs are on the level of metal-polymer exchangers and better than them with respect to strength. All metal matrix heat exchangers have been extensively studied in the USSR, particularly at the Odessa Institute of Technology for the Refrigeration Industry in collaboration with NPO-Kislorodmash 22'3~'32'37 and at the N.E. Bauman Moscow Technical College (MVTV)jointly with the Scientific Research Institute for Chemical Engineering41-44. Diffusion bonded MHEs, reported by Sotnikov et al. 37 of the Odessa Institute, was hermetic even at very high pressures. Along with development work, Mikulin and co-workers at M V T V 41-44 and Orlov, Shevyakova and co-workers at the Balashnik Scientific Industrial Association of Cryogenic Machine Construction 23'45'46 have done extensive studies on heat transfer and flow friction characteristics of all metal MHEs. These heat exchangers were largely designed for use in helium liquefiers, although some were proposed

Cryogenics 1990 Vol 30 November

909

Review to be used in air separation plants 31. Diffusion bonded heat exchangers similar to those described by Zablatoskaya 2~'32 have been developed at the Hitachi Corporation, Japan 28'29, for use in a small Claude' cycle cryorefrigerator based on micro-turbines. Harada et al. 28 report that the matrix heat exchanger at the coldest stage of the refrigerator was a factor of four smaller than the brazed aluminium plate fin exchanger that it replaced. While the matrix heat exchangers made of copper perforated plates and stainless steel spacers remain the most successful, other types of matrix heat exchanger have also been reported in the literature. Steyert and Stone 47'48 constructed a tube-in-tube heat exchanger where disks of oxygen annealed copper wire screens were diffusion bonded to the inner tube in both channels.

Fabrication of MHEs Many different techniques have been used in the last four decades for the manufacture of matrix heat exhangers. These techniques have mostly represented the state of art in bonding of perforated plates and spacers. The fabrication of MHEs consists mainly of two steps: fabrication of plates, spacers and headers; and bonding.

Plates, spacers and headers Perforated plate matrices can be fabricated by many different methods. The early models of McMahon et al. ~6 and Garwin 35 used commercially available punched plates which are widely used in the manufacture of sieves and fdters. The presence of perforations in the wall region which is bonded to the spacers adversely affects bonding strength, leak tightness and heat conduction between the streams. Therefore, all subsequent designs have used custom-fabricated plates, in which perforations are avoided in the walls and the borders where the plates are bonded to the spacers and the headers. Perforated plates and spacers can be fabricated by one of the following techniques: mechanical methods - drilling (of plates), milling (of spacers) and punching; chemical methods - photochemical milling, electroforming. While drilling of plates using a numerically controlled machine can be a very useful technique in the development stage, it is not suitable for mass production. Punched plates fabricated using custom designed dies are cheap, particularly in mass production. This is viable if plate thickness is > 1 mm. Punching usually leaves rough edges which affect bonding and may alter heat transfer and flow friction characteristics. Subsequent finishing operation is sometimes useful in removing the burrs created in punching. Photochemical milling (PCM) 49 has been the method chosen by most workers in this field for making both plates and spacers. In this process a pattern showing the desired profile is drawn on paper to a convenient scale. The pattern is photographed to make a mask of the exact size on transparent plastic. Copper and stainless steel sheets, coated with photoresist, are exposed from an appropriate light source through the mask. Exposed sheets are developed and etched in chemical baths. Hates and spacers in desired shapes are produced to high dimensional accuracy at low cost. This technology is similar to the manufacturing of printed circuit boards (PCB). However, it is often necessary to expose the sheets from both sides.

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Cryogenics 1990 Vol 30 November

PCM offers several advantages over mechanical methods: perforations and passages in complex shapes can be fabricated easily; the time required for producing a new type of plate and spacer is small; and the surfaces are burrfree, chemically active and well suited for diffusion bonding process 4°. The only disadvantage of this process is the possibility of undercuts in the perforations, caused by lateral etching, which distort the perforations. The problem is less severe if the plate is exposed from both sides. PCM is not suitable for machining plates of thickness >0.5 mm. This method has also been used for the fabrication of compact plate heat exchangers 5°'51. Electroforming is as yet another modern manufacturing technique well suited to fabrication of plates and spacers. In this method a negative of the desired shape is prepared as the master punch, which is used as the cathode in an electrolytic bath, the plate to be perforated serving as the anode. Selected sites on the anode dissolve rapidly on application of a suitable d.c. voltage. This technique is expensive but fast, accurate and suitable for complex shapes. Mikulin et al. 44 have used electroforming for manufacturing MHEs. Since plate-to-plate gaps cause continual reheadering throughout the exchanger, the design of headers to avoid jet effects is not as critical in MHEs as in other types of heat exchanger 19. The fabrication of headers for most rectangular cross sections can be done using a conventional milling machine. The manufacture of headers with annular, spiral or other complex flow cross sections, however, requires special attachments. They can also be done with numerically controlled milling machines or by electrodischarge machining.

Bonding The most critical process in the fabrication of MHEs is bonding of plates and spacers to form hermetic flow passages. The successes and failures of MHE development are closely related to those of the bonding techniques. The different types of bonding method used can be generally classified into three categories: 1 demountable assembly using pressure devices; 2 adhesive bonding of metal-plastic exchangers; and 3 soldering, brazing and diffusion welding of all metal MHEs. The early models of MHEs used demountable structures. McMahon et al. ~6used steel tie rods to bond punched aluminium plates and neoprene gaskets. Garwin 35 obtained a patent for fabricating MHEs using resilient members (springs) which allow automatic compensation of differential contraction between the matrix and its support structure. Ostbo 36 has patented a matrix exchanger with demountable seals for use in liquid-to-liquid and liquid-to-gas heat exchangers at near ambient temperatures. Although demountable assembly had obvious advantages such as flexibility, reusability and low cost, the elastomeric gaskets pose serious problems. They tend to harden at low temperatures and develop cracks on temperature cycling. The real breakthrough in MHE technology came with the introduction of adhesive bonding of metal-plastic heat exchangers by Vonk ~7 in 1968. During the last two

Review

decades several other workers have used this method for both wire screen-resin and perforated plate-plastic spacer heat exchangers. The most important among them are References 19, 20, 25 and 27. Detailed information on the types of adhesive and joining techniques have largely remained proprietary and not available in open literature. Anakshin et al. 25 have found that epoxy silicone based adhesives similar to type VT-200 are the most suitable for bonding MHEs. The use of other Russian formulations such as resin YP614T and hardner YP5-139T and the cold hardening epoxide glue based on resin EDA-20 have been reported by Zablotskaya 32. Matsuda 52 recommends use of AF163 adhesive sheet, available from 3M corporation USA, to bond aluminium plates and Kapton (DuPont) spacers. The process requires a special technique and skill. Special fixtures are required to cut the adhesive sheets to the required shape, to assemble the matrices and to cure them at a specified temperature and pressure. Although poorer in thermal performance than metalplastic units, all metal MHEs have slowly replaced them in many applications because they are easier to fabricate using one of the standard metal joining techniques. The common lead-tin solder has been successfully used by Zablotskaya 32. The exchangers were hermetic at pressure of 2 - 4 MPa even after repeated temperature cycling. However, some of the perforations were found to be blocked by excess solder. Soldering is a simple process and no expensive equipment is required. Low temperatures ( < 200°C) avoid distortion of the plates and spacers as well as residual stresses caused by differential contraction on cooling from the bonding temperature. The main disadvantage is the poor overall strength compared with other methods of bonding. The quantity of solder to be applied is critical. If the amount of solder is too small the seal is poor; if the amount is too high excess solder blocks the perforations. Silver brazing is a high temperature joining process. An alloy of Ag - C u - Z n - Cd, melting between 600 and 700°C, is used instead of the P b - S n alloy discussed earlier. The joints are strong, ductile and usually leakfree. This method of bonding was patented by Fleming in 196933 . Recently it has been successfully used to construct experimental MHEs at the Indian Institute of Technology, Kharagpur 53, A special fixture has been designed to ensure uniform heating under an applied load. The most successful method of bonding c o p p e r stainless steel MHEs, by far, has been the diffusion welding process. An excellent monograph has been written on this subject by Kazakov 4°. Diffusion bonding of stainless steel perforated plates has also been discussed by Voronin et al. 54. Alternate layers of copper perforated plates and stainless steel spacers with a header at each end are assembled and placed between the anvils of a vacuum hot press. In the vacuum chamber (p < l0 -2 Pa) the assembly is subjected to an axial pressure between 1 and 20 MPa and a temperature between 900 and 1000°C for a few minutes. Because of high temperature and low oxygen pressure in the environment, the surface oxides dissociate, creating an active metallic surface. The axial pressure ensures intimate contact between the surfaces. At high temperature, the atoms of copper and steel diffuse into each other making a near perfect bond between the two sheets. An interlayer of

nickel has been found to aid in the bonding of copper and stainless steel 4°. The process is extremely clean and fast. The joint is as strong as the parent metals and peffecty leak tight. Since no filler metal is used and no liquefaction takes place, there is no possibility of blockage of the passages by squeezing out of filler metal. The process is, however, expensive and requires specialized equipment and skilled personnel.

Heat transfer and flow friction The heat transfer process in a matrix heat exchanger is complex. The total heat transfer resistance between the two fluid streams is composed of the following parts: 1 convective heat transfer between the hot fluid stream and the perforated plates; 2 conduction along the perforated plates in the first channel; 3 conduction across the separator; 4 conduction along the perforated plates in the second channel; and 5 convective heat transfer between the perforated plate and cold fluid stream. This description appears similar to that of a conventional heat exchanger with fins on both sides. In fact, for the purpose of analysis, the plates have been treated as fins by some authors t9. The peculiarities of the matrix exchanger manifest themselves when attempts are made to find the heat transfer correlations, the fin efflciencies and the role of the separator. Although an MHE can be made either of wire screens or of perforated plates, the review in this section is largely limited to the latter. This is because heat transfer and flow friction in wire mesh matrices have been extensively studied 55-58 in connection with their use in regenerators. An excellent review of wire mesh exchangers has already been made by Zablotskaya 32. The heat transfer and flow friction data of Coppage and London 55 can be used for most design purposes.

Convective heat transfer

Convective heat transfer between the fluid stream and a perforated plate takes place on three surfaces: the tubular surface of the perforations; the front face of the plates; and the back face of the plates. In a perforated plate exchanger the flow cross section changes continuously between that of the perforations and that of the spacer, the former being substantially smaller than the latter. Consequently, the fluid undergoes alternate expansion and contraction as it flows through the exchanger. Because of the interruption of the boundary layer at every plate, the flow through the tubular portion can be considered as developing flow with associated high heat transfer coefficient. The length to diameter ratio is usually between 0.3 and 4.0. The heat transfer correlations for developing flow are given by London et al. 59.60. The leading face of a plate is subjected to arrays of impinging jets emanating from the plate before it. Heat transfer due to jets impinging on a blind surface has been studied in detail by several authors 61-64. Hollworth and

Cryogenics 1990 Vol 30 November

911

Review

Dagan ~4 have studied arrays of impinging jets with spent fluid removal through vent holes on the target surface. The heat transfer rates obtained with air jets impinging on blind surfaces are an order of magnitude higher than those ~enerally associated with gaseous heat transfer media °'. The correlations expressing average Nu as a function of Re, Pr and geometrical factors that have been given in these references are generally applicable to higher Reynolds numbers and lower plate porosities, but can be extrapolated to conditions appropriate to MHEs. Heat transfer rate in the back face of the plate is also high due to flow separation and resulting turbulence. Heat transfer coefficients on the downstream face of an abrupt enlargement have been studied by Sparrow and O'Brien6Yat Reynolds numbers higher than usual in MHEs. For design work, as well as for comparison with experimental results, an average heat transfer coefficient may be defined as

(1) the index i referring to the three different surfaces: the tubular surfaces of the holes and the front and back faces of the plates. Similarly, the flow friction characteristics of MHEs are characterized by an average friction factor f A large amount of experimental data has been collected over the years. Most authors have derived empirical correlations for Nusselt number and friction factor versus Reynolds number. The general approach has been to find a relation of the form Nu = C Re"

(2)

where C and n are functions of geometric parameters and Re is usually based on ~6'3°'44-46 flow velocity in the perforation and the perforation diameter as the characteristic dimension, although s o m e a u t h o r s 25'3L32 have preferred average flow rate and hydraulic diameter. Orlov et al.45'46 have shown that the heat transfer coefficient and friction factor do not vary significantly with spacer thickness and hence the correlations based on average flow rate and hydraulic diameter do not reflect the true hydrodynamics of the flow. McMahon et al. ]6 used the conventional heat exchanger relation q = UAIATLm

(3)

to determine the overall heat transfer coefficient U. For the heat transfer area A~ they took only the tubular portion of the holes, but admit that a weighted combination of hole surface area and that of the front and back faces would have been a more realistic choice. The average heat transfer coefficient h~ was determined by the formula 1/U = 2/(,/n.h0 + bA]/k~4.

(4)

where: b = width of the separator; kw = thermal conductivity of the fin material; Aw = mean area of cross section in one set of channels; and ~/nn = fin effectiveness.

912

Cryogenics 1990 Vol 30 November

For fin effectiveness they used the standard hyperbolic tangent formula 66, but subsequent workers have universally used a different relation derived by Fleming twenty years later 19. Fleming's relation

(5)

7hi. = (1 + m2H2/3)-1

is based on the assumption that the temperature difference between the fluid and the matrix, rather than the fluid temperature itself, remains constant throughout the length of the fin. This assumption is generally valid for all plates except the first few and has been experimentally verified by Mikulin et al. 43. In Equation (5) m 2 = hA"/kp;

h = heat transfer coefficient; kp = effective thermal conductivity of the plates in the transverse direction; H = plate height (fin height); and A " = heat transfer area per unit volume of the plate. Based on the same assumptions, Mikulin et al. 43 derived the following relation for fins of circular geometry ~/fi. = (1 + m2r2/6) -l

(6)

The most extensive experimental studies on heat transfer in matrix heat exchangers are due to Mikulin and coworkers at N.E. Bauman Technical Institute, MOSCOW41-44. References 41 and 42 deal with wire screen heat exchangers and the other two deal with perforated plate units. For wire gauge exchangers they obtained the relation 4~ Nu = 1.21Re°47(L/de) -(°SRe-°2')

L/de < 200

(7)

Nu = 0.05Re °'85

L/de > 200

(8)

L being the overall length of the exchanger and de the mean hydraulic diameter. The apparent dependence of Nu on L/de at low L/de values is probably due to axial conduction effects in a short exchanger. These heat transfer coefficients are generally lower than those of Coppage and London 55, particularly at high L/de. Heat transfer coefficients in perforated plate heat exchangers have been measured by Mikulin et al. 44, Shevyakova and co-workers 45'46 and Ornatskii et al. 30 under different conditions. A comparison of their results is given in Table 1 and Figure 3. The results obtained by different authors differ both quantitatively and qualitatively. Strictly speaking, a comparison of the different correlations is justified only when experiments are conducted with identical combinations of geometric parameters 46. The heat transfer and flow friction in MHE are strongly dependent on the geometric parameters, namely, shape and size of the perforation, hole to hole spacing (plate porosity), spacer and plate thicknesses and alignment of perforations between adjacent plates. The differences between the observations of different workers may be ascribed to the differences in the geometric parameters. In their experiments, Orlov e t al. 45 used condensing steam in the second channel, creating an isothermal boundary condition. The effects of conduction resistance along the perforated plates were accounted for by using Fleming , s 19 fin effectiveness relation. Flem-

Review

Table 1

Heat transfer correlations for perforated plate M H E

Correlation

Limitations

Source

h = a Re n

8 0 0 < Re < 4 0 0 0

16

0.09 < p < 0.35

30

Shifted holes 7 0 < Re < 2 1 0 0 Shifted slots 3 0 < Re < 1 6 0 0 Aligned holes 2 0 0 < Re < 1 0 0 0 0.11 < s / d < 1 Aligned slots 2 0 0 < Re < 1 4 0 0 0 . 0 7 5 < s/a < 0.88 p = 0.3246 3 0 0 < Re < 3 0 0 0 3 0 0 < Re < 3 0 0 0 0.3 < p < 0.6

44

0 . 8 8 < n < 1.14 N u = C Re ~

C = 4 . 9 3 p - 0.11 n = 0.77 1.12p N u = 0 . 2 Re T M Nu = 0 . 2 2 Re °69 Nu = 0 . 0 6 5 R e 0 7 4 ( s / d ) °'21

N u = 0 . 0 4 5 R e ° 8 7 ( s / a ) °'5°

S t Pr 2/3 = 1.2 Re -062 St Pr 2/3 = C Re n

C = 0 . 0 0 0 3 6 [ ( 1 - p ) p - 0 . 2 ] -2.07 n = - 4 . 3 6 x 1 0 - 2 x p 2.34 Ntu per plate = b Re m b = 2.1-3.7 m = 0.37-0.57

'

I

~

I

'

'

I

'

I

~

c

~

......

I

i7

c ~3

2.0

45 46

67

been observed 44-46. On the other hand, in exchangers with aligned holes, Mikulin et al. 44 observed that, as the spacer thickness is increased, the heat transfer coefficient increases and finally equals that of an isolated plate. Their observation coincides with that of Fleming 33, who recommends a higher spacer thickness for plates with aligned holes. This is in contrast to the early results of McMahon et al. ~6, who observed that the effect of spacer thickness is first to increase and then to decrease the heat transfer coefficient. It must be borne in mind that while Mikulin et al.44 conducted experiments in the Reynolds number range 200-2000, the experiments of McMahon et al. 16 were for Re between 800 and 4000. At fixed spacer thickness, the effect of plate porosity is to reduce the heat transfer coefficient 44.46' '67 . The early observation of McMahon et al. ~6, suggesting higher heat transfer coefficient with increased perforation diameter d, has not been corroborated by subsequent workers. Flow

friction

Like heat transfer coefficient, friction factor correlations have been determined by many authors. A summary of available correlations is given in Table 2. The drag coefficient ~(=pAp/nG2), used by most of the authors, is related to the standard Fanning friction factor by the relation

E t~ I-0

s r~ %"~

44

0.09 < p < 0.245

5.0

0.5

44

Re < 100

ing's expression (Equation (5)) is based on the assumption that the temperature difference between a plate and the fluid remains constant over the whole plate. This condition, expected to be valid in a counterflow exchanger, is not valid for the experiments of Orlov et al. It is difficult to estimate the resulting error in the heat transfer coefficients. The effect of spacer thickness has been studied by several authors. In randomly arranged and shifted plate exchangers no significant effect of spacer thickness has ~0"0

44

/

(9)

=2fl/d

I

0.2

0'1

t

0.0

I

o.t

t

0-2

0-3 Plate

Figure 3

0'4

I 0-5

I

I 0.6

J

0"7

porosity

C o n s t a n t s C and n in t h e h e a t t r a n s f e r c o r r e l a t i o n N u = C Re n as a f u n c t i o n o f plate p o r o s i t y . 1, O r n a t s k i i e t al. 30; 2, S h e v y a k o v a and Orlov46; 3, O r l o v e t a1.45; 4, 5, M i k u l i n e t al. 44

Many workers 44'46'67have observed that the friction factor is independent of Reynolds number even at Reynolds numbers near 200. This indicates that form drag dominates over skin drag even at a Reynolds number of 200. This is probably due to the complex flow pattern in an MHE. Unlike heat transfer coefficient, friction factor has a strong dependence on spacer thickness for both aligned and shifted plates 44. Shevyakova and Orlov 46, on the other hand, observed that friction factor does not significantly

Cryogenics

1990

Vol

30

November

913

Review Table 2

Friction f a c t o r c o r r e l a t i o n s

in p e r f o r a t e d plate M H E

Correlation = ~j'[1 + O . 0 8 ( s / d ) - ° 8 ] ; ~' = [ 0 . 7 0 7 ( 1 - p)O.5 + (1 - p ) ] 2

0 . 0 7 5 < s/a < 1.1 A l i g n e d holes Re > 2 0 0 0.22 1 6 0

= 1 6 . 3 4 Re °'55~sir~ = ~sim ~sir. = ( 1 . 7 0 7 - p ) 2 / 2

depend on the spacer thickness in randomly arranged plates. While the correlations given by Mikulin et al. 44 as well as those by Shevyakova and Orlov 46 are independent of the plate thickness, Hubbell and Cain 67 observed that the friction factor is more sensitive to plate thickness-to-hole diameter ratio than to porosity. From the point of view of both convective heat transfer and pressure drop, Mikulin et al. ~ recommend optimum values: s/d = 0.3 for circular holes and s/a --- 0.5 for rectangular slots in plates with shifted holes. Similar recommendations have also been made by Ornatskii et al. 30.

Conduction along perforated plates Apart from convection, conductive heat transfer through perforated plates also has a dominant role in the overall heat transfer process. An extensive review on the conductivity of perforated plates has recently been published by Churchill 6s. The conductivity of a medium containing an array of parallel circular cylinders of different property was derived by Maxwell 69 and Lord Rayleigh 7° more than a century ago. For perforated plates used in MHEs, Maxwell's relation reduces to keq _ 1 - p kp 1 + p

(lO)

where: k~q = equivalent conductivity of perforated plate; kp = conductivity of unperforated plate: and p = plate porosity. Lord Rayleigh's 7° relations were modified by Runge 71 and recently by Perrins et al.72. The effective thermal conductivity of a plate with square and hexagonal arrays of holes, respectively, can be approximated by the relations 7z t/ 0.0305827p 4 _0.013362p8"~ -~

;

(lla)

914

Cryogenics 1 9 9 0 Vol 3 0 N o v e m b e r

44

S h i f t e d slots

= O.44(s/a) °'72

pkl-i

S h i f t e d holes

44

Re > 1 0 0

= O.78(s/d) °5

- 1-

Source

Re > 1 0 0 0.11 < s/d < 1.1

~j = ~ ' [ 1 + 0.18(s/a)-1.58] ~' = [ 0 . 7 0 7 ( 1 - p)O.5 + (1 - p ) ] 2

kp

Limitations

44

44

46

keq - 1 - 2p(- 0"75422p6 - 0.000076p 12)-1 kp \1 - ~ p t2

(1 lb) The above expressions are accurate for p < 0.6. For higher porosities, a higher order expansion is necessary for very accurate results. Perrins et al. 72 have also experimentally verified Equations (11). Keller and Sachs 73 used a finite difference method to calculate the effective thermal conductivity of perforated plates. Forp > 0.5 the effective conductivity of plates with a square array of holes has been found to be keq _ I 711-1 kp [2 - (16p/a') °5] 0.5 - 1.95

(12)

Parasnis TMhas given a set of empirical equations based on his experiments. Comprehensive results for square and hexagonal arrays of circular holes as well as square and rectangular arrangement of square holes are available in Reference 75. Their data clearly indicates that the effective thermal conductivity of a perforated plate does not differ significantly along different directions. This is particularly useful while designing circular shaped MHEs made out of commercially available punched sheets, which have square or hexagonal arrays of perforations. A theoretical analysis of the heat transfer process in a perforated fin has also been given by Kirpikov and Leifman 76. They have given an approximate relation to calculate the effective thermal conductivity of a plate with a square array of holes. Mikulin et al.43 have used this approximate relation in calculating the fin efficiency of perforated plate MHEs. Babak et al. 77 have used another approximate method for calculating the effective thermal conductivity in their numerical model.

Analytical and numerical models The first major step towards modelling of matrix heat exchangers was the work of Fleming 19 in 1969. Fleming's approach was to use standard heat exchanger relations, the perforated plate being considered as the secondary sur-

Review

face or a fin with a certain fin efficiency (Equation (5)). Fleming also considered the effect of axial conduction and expressed his results in terms of an 'apparent N~', a concept which is very useful in expressing the performance of heat exchangers. Fleming got good agreement between computed and experimental performance for the range of exchangers that he constructed. Although an MHE is made up of plates and spacers, it is often treated as being essentially uniform in the axial direction. An alternative approach, in which the MHE was treated as a discrete set of plate-spacer pairs, was adopted by Sarangi and Barclay\ They assumed that the full temperature drop in the axial direction took place over the spacers, neglecting any temperature gradient along the plates. They used the principle of superposition to suggest the following expressions for the effectiveness of a high Ntu exchanger with n plates -

1 -p" 1 -

up"

1 - q"-l

+

1 -

1 -

uZq n - l

u"

(13)

1 - u "+1

with p-

,q1 +tza2

,/z-

u

1 +unX

1-ol 2

where o~ = exp(-NtJn), is the number of heat transfer units, Nt~, per plate of each side, u = capacity rate ratio and k is an axial conduction parameter. For balanced flow operation, Equation (13) reduces to

C=

n(1 - oq)(1 - o~2) n(1 - cq)(1

n+l

-

o~2) q -

(1

-

+ OtlO(2)

n(1 + X) 1 + n(1 + 2X)

(14)

The derivation of these formulae is based on the principle that in high-effectiveness exchangers the ineffectiveness due to different sources can be added to give the total ineffectiveness. Based on Equations (13) and (14), Sarangi and Barclay 9 have also shown that the maximum attainable Ntu per plate is 1.0. Experimental results, recently reported by Hubbell and Cain 67, support this conclusion. It should be noted that Hubbell and Cain have incorrectly attempted to fit a straight line to their experimental data, getting an Ntu per plate value little greater than 1.0 at low Reynolds numbers. Reference 9 does not take into consideration the 'fin effect' of the perforated plates. An alternative approach to matrix heat exchanger design has been proposed by Kim and Qvale 21, who used a pin fin model to represent wire screen disks. The work of Lins and Elkan 2° uses the standard heat exchanger expression but considers axial conduction. Although both these papers deal with wire screen exchangers, their results are also applicable to perforated plate units. The most extensive analytical studies on matrix heat exchanger performance are by Babak and co-workers 77-82. In their initial model 79 the authors neglected axial conduction and considered the exchanger as a homogeneous

medium with fixed transverse conductivity. They define 77 an equivalent conductivity ke which includes the conductivity of the perforated plates as well as the convective film coefficient in the perforations. This enables them to represent both the matrix and the fluid by a single temperature, which is a function of the axial coordinate z and the transverse coordinate y. By doing an energy balance over a differential control volume Wdzdy, the authors derive the equation aT,

02T,

(Gcp)i Oz = ke,i---Oyz

i = 1, 2

(15)

with appropriate boundary conditions, where G is the mass velocity in the header and Cp the fluid specific heat at constant pressure. The subscript i refers to the channel number. The pair of equations (15) have been solved analytically by separation of variables. Using a 'series solution' approach Babak et al. have expressed the performance of the exchanger in terms of four dimensionless quantities X, Y, Z and P. Many limiting cases have been explicitly worked out. This model may be termed a 'continuum model', because it does not take into account the 'discrete' nature which typifies a matrix heat exchanger. In subsequent publications 8°-82, Babak et al. have further extended this technique to many different cases and presented the results graphically 8°. By considering about 20 terms in the series and using a computer they determined the effective Ntu of the MHE with an accuracy better than 2% 82. The expressions are complex. Unlike reports on simpler analytical relations hardly any physical insight is given. In a later paper 77 Babak et al. have adopted finite difference instead of the analytical approach. However, instead of considering the whole exchanger, they have considered only one channel subjected to a constant heat flux (independent of axial coordinate). The discrete structure of the exchanger has been ignored and convective heat transfer resistance has not been considered separately. They have also considered axial conduction in the separating wall. Effective thermal conductivities of the separator in the transverse and axial directions have been estimated from those of the plate and the spacer as kt

. . . . . . . . •.

_ Ikp + sks

(16a)

s+l

kaxial=(S-~-l)/(~p -'~-~ss)

(16b)

s and 1 being the spacer and plate thicknesses, respectively. The results of this numerical model are in reasonable agreement with the experimental data of Mikulin et al. 43. A different approach has been taken by Bova et al. 31. The overall thermal resistance of the matrix heat exchanger based on the area of longitudinal section on any side is expressed by the following relation Rt,: =

[( hi,2

)7-'

A Z+ P + 7/fi,/3Ht 2

Cryogenics 1990 Vol 30 November

(17)

915

Review

where H is the height and P is the overall porosity of the fin. A = 0 and Z = 0 for perforated plates and A = 1 and Z = 2 for wire screen matrices./3 is the overall surface area density of the exchanger. R is to be computed for each stream separately. A recent investigation by Hubbell and Cain 67 gives both theoretical expressions and experimental data on MHE performance. Heat transfer data have been expressed in terms of Nt, per plate. While Sarangi and Barclay 9 have derived a limiting value of 1.0 for Nm per plate, the theoretical expressions of Hubbell and Cain predict values as high as 5.0. On the other hand, their experimental values have remained below 1.0 confirming the general applicability of the Sarangi and Barclay model. The agreement between their experimental and theoretical friction factor data is, however, much better. A complete analysis of the MHE which takes into consideration both convective and conductive resistances and axial conduction as well as the distinctively discrete nature of the MHE has recently been undertaken at the Indian 100

l

I

'

'

I'

'''1

= 0.01

5O

I '

~

...~ ~.f

/..12~::~---r-~

-- 0 +

_

20

]0

5 10

I 20

i

, I , t,tl 50 100

I 200

,

, I , L,, 500 1000

Ntu- design Figure 4 Effective Ntu as a function of design Ntu for given number of plates n, axial conduction parameter k and plate conduction parameter ~. A, Continuum approach, axial conduction included 3; B, Fleming's model with axial conduction effects19'3; C, Sarangi and Barclay formulation9; D, modified Sarangi and Barclay formulation; E, Venkatarathnam and Sarangi B3

100

'

'

'

I ....

I

n = 50

50

' D~

),= 0"01

EA ~ ,-~- ~f ~/ ,",f, ,~~

.~

~

' ,

'

I '

~~ " " ~ -

~~ -

_ -_

c

+o

z

,

10

,

+

I t,,,l

50

100 Nlu -design

500 =

Cryogenics 1990 Vol 30 November

During the last forty years the matrix heat exchanger has undergone an evolutionary change in terms of its method of construction, heat transfer analysis and design techniques. Chemical milling and diffusion bonding techniques have been perfected for the fabrication of all metal perforated plate exchangers. They are finding increasing application in high performance cryorefrigerators with considerable savings in volume and cost. Although poorer in performance, silver brazing offers a widely available and cheaper alternative to diffusion bonding. Silver brazed exchangers are expected to find application in low and medium pressure refrigerators. There is considerable difference among the different heat transfer and flow friction correlations available in the literature. This may be due to the use of incorrect data reduction procedures. In fact, although excellent MHEs have been constructed and used in several laboratories, a proper design/analysis procedure was not available until recently. With better understanding of the overall heat transfer process, r better correlations are expected to emerge. Until now, most heat transfer data have been available in the form of empirical correlations. Attempts should be made to predict heat transfer and flow friction performance from fundamental relations on different heat transfer mechanisms. Such studies will result in optimized geometries and operating conditions.

1000

Figure 5 Effective Ntu as a function of design Ntu, axial conduction parameter ~kand plate conduction parameter ~. A, D, Fleming's model w i t h axial conduction effects 19'3 w i t h ~ = 0.5 (A) and 2.5 (D); B, E, modified Sarangi and Barclay formulation w i t h ~ = 0.5 (B) and 2.5 (E); C, F, Venkatarathnam and Sarangi B3 w i t h ~ = 0.5 (C) and 2.5 (F)

916

1 At low design Nt,, Nt,,D, the effectiveness of an MHE is fairly independent of the number of plates n and the axial conduction parameter X. The effective Nt, is adequately given by Ntu.D/(1 + 1/3~b). 2 At high Ntu,D the effectiveness is limited by n and X, instead of Nt,,o or q~. If n < l/X, n is the controlling factor; otherwise the effective Ntu is determined by the axial conduction parameter X. 3 The formula suggested by Sarangi and Barclay 9 with Nt,.D replaced by Nto,D/(1 + 1/3~b) is sufficient in most cases to predict the performance of matrix heat exchangers of rectangular geometry.

Conclusion

'2

Z

)

Institute of Technology, Kharagpur 83. The model assumes that the axial temperature gradient in the (copper) plate is negligible and that the full temperature change takes place along the spacers. With this assumption the energy balance and heat transfer equations reduce to two secondorder ordinary differential equations and four algebraic equations for each plate. A stable and robust algorithm has been developed to solve the equations by an iterative scheme. Some of the results are presented in Figures 4 and 5 where the effective Ntu of a balanced flow exchanger is plotted against the design Nt, for a given number of plates n, axial conduction parameter X and plate conduction parameter ~b. The results are compared with earlier (approximate) models. The following observations can be made:

Acknowledgement This work was supported by the Department of NonConventional Energy Sources, Ministry of Energy, Government of India under the project 'Liquid hydrogen: production, storage and transfer'.

Rev~w

References 1 Barron, R. Cryogenic Systems Oxford University Press (1985) 129 2 Chowdhury, K. Some aspects of performance of cryogenic heat exchangers, Ph.D. Dissertation, Indian Institute of Technology, Kharagpur (1984) 3 Kroeger, P.G. Performance deterioration in high effectiveness heat exchangers due to axial heat conduction effects Adv Cryo Eng (1967) 12 363-372 4 Chowdhury, K. and Sarangi, S. A second law analysis of the concentric tube counterflow heat exchanger: optimization of wall conductivity lnt J Heat Mass Transf (1983) 26 783-786 5 Fleming, R.B. The effect of flow maldistribution in parallel channels of counterflow heat exchangers Adv Cryo Eng (1967) 12 352-362 6 Chowdhury, K. and Sarangi, S. The effect of flow maldistribution on multipassage Heat Exchanger performance Heat TransfEng (1985) 6 45-54 7 Barton, R. Effect of heat transfer from ambient on cryogenic heat exchanger performance Adv Cryo Eng (1984) 29 265-272 8 Chowdhury, K. and Sarangi, S. Performance of cryogenic heat exchangers with heat leak from the surroundings Adv Cryo Eng (1984) 29 273-280 9 Sarangi, S. and Barclay, J.A. An analysis of compact heat exchanger performance, in Cryogenic Processes and Equipment - 1984 (Ed Kerney, P.J. et al.) ASME, New York, USA (1984) 37-44 10 Timmerhaus, K.D. and Schoenhals, R.J. Design and selection of cryogenic heat exchangers Adv Cryo Eng (1974) 19 445-462 11 Abadzic, E.E. and Seholz, H.W. Coiled tubular heat exchangers Adv Cryo Eng (1973) 18 42-52 12 Collins, S.C. A helium cryostat Rev Sci Instrum (1947) 18 157 13 Collins, S.C. Liquid nitrogen generator Rev Sci lnstrum (1955) 26 671 14 Lenfesty, A.G. Low temperature heat exchangers, in Progress in Cryogenics Vol. 3, (Ed. Mendelssohn, K.) Academic Press, New York (1961) 25 15 Dilevskaya, A. Micro Cryogenic Heat Exchangers (in Russian) Mashinostroenie, Moscow (1978) 16 McMahon, H.O., Bowen, R.J. and Bleyle Jr., G.A. A perforated plate heat exchanger Trans ASME (1950) 72 623-632 17 Vonk, G. A new type of compact heat exchanger with high thermal efficiency Adv Cryo Eng (1968) 13 582-589 18 Information sheetno. DKT41-20, N.V. Philips Gloeilampenfabrieken, The Netherlands 19 Fleming, R.B. A compact perforated plate heat exchanger Adv Cryo Eng (1969) 14 197-204 20 Lins, R.C. and Elkan, M.A. Design and fabrication of compact heat exchanger using wire mesh surfaces Adv Cryo Eng Plenum (1975) 20 283 -299 21 Kim, J.C. and Qvale, E.B. Analytical and experimental studies of compact wire screen heat exchangers Adv Cryo Eng (1971) 16 302- 310 22 Zablotskaya, N.S., Vaselov, V.A., Bodyul, C.V. and Sharnopolskaya, E.T. Metallic matrix heat exchangers (in Russian) lzv Buzov Mashin (1978) No. 3, 82-85 23 Kovalenko, V.D., Mnshin, L.B., Orlov, V.K., Tsfasman, G.Yu. and Shevyakova, S.A. Perforated-plate heat exchangers for cryogenic helium units Khim Neff Mashin (1980) No. 7, 11-12; transl: Chem Petr Eng (1980) 406-470 24 Pron'ko, V.G., ~ v s k i i , E.U., Usanov, V.V., Krasnikova, O.K., Zhuravleva, I.N. and Chernysheva, E.A. Manufacture of high efficiency heat exchangers for cryogenic systems, Kh/m Neff Mashin (1975) No. 9, 8-10; transl: Chem Petr Eng (1976) 791-793 25 Anashkin, O.P., Keilin, V.E. and Patrikeev, V.M. Compact high efficiency perforated-plate heat exchangers Cryogenics (1976) 16 437 -439 26 Koizumi, T., Takalmshi, M., Uchida, T., Kanazawa, Y. and Suzuki, M. A small and light weight heat exchanger for on-board helium refrigerator, in Refrigeration for Cryogenic Sensors (Ed Gasser, M.) NASA Conference Publication 2287, Washington, DC, USA (1983) 179-88 27 Matsuda, T., Imamura, M., Sato, N., Ogata, H., Harada, S. and Hosiomi, N. Small cryocooler system for superconducting devices, in Cryogenic Process and Equipment - 1984 (Ed. Kerney, P.J. et al.) ASME, New York, USA (1984) 45-50 28 Horatio, S., Matsuda, T., Saito, S. and lhara, K. Small size helium refrigerator with micro turbine expanders Proc lnt Cryocooler Conf (1986) 29 Izumi, H., Harada, S., Matsubara, K. and Saito, S. Development of small size Claude cycle helium refrigerator with micro turbine expander Adv Cryo Eng (1986) 31 811

30 Ornatskii, A.P., Perkov, V.V., and Khud~rt~, V.M. Experimental study of perforated plate heat exchanger for micro cryogenic systems (in Russian) Promish Teplo Tekhn (1983) 5 28-33 31 Bova, V.I., Zablotskaya, N.S., V~elov, V.A. and Smnnsenkov, N.V. Development of efficient regenerators for alr-fractionating units Khim Neff Mash (1978) No, 7, 11 - 12; transl: Chem Petr Eng (1978) 309-313 32 Zablotskaya, N.S. Matrix Heat Exchangers in Cryogenic Engineering (in Russian) CentChemNeflmash, Moscow, USSR (1980) 33 Fleming, R.B. Heat exchanger of porous metal US Patent 3 433 299 (1969) 34 Coyler, D.B. and Fleming, R.B. Porous metal and plastic heat exchanger US Patent 3 477 504 (1969) 35 Garwin, R.L. Heat exchange device US Patent 3 228 460 (1966) 36 Ostbo, K.R.A. Heat exchanger US Patent 3 865 504 (1969) 37 Sotnikov, A.A., Vaselev, V.A., Bova, V.I. and Gorenshtein, I.V. Matrix heat exchangers for high pressure systems Khim Neff Mash (1985) No. 4, 27-29; transl: Chem Petr Eng (1985) 187-190 38 Ban Mardion, G. and Clandet, G. A counter flow gas-liquid helium heat exchanger with copper grid Cryogenics (1979) 19 552 39 Longsworth, R.C. Advances in Joule Thomson cryo coolers Proc Cryo Eng Conf (1989) 40 Kazakov, N.F. (Ed) Diffusion Bonding of Materials Mir Publishers, Moscow (1985) 41 Mikulin, E.I. and Shevieh, Yu. A. Experimental study of heat transfer in meshed matrices lnz Fiz Zh (1972) 22 1116-1117; transl: J Eng Phys (1972) 777-778 42 Shevich, Yu.A. and Mikulin, E.I. Heat exchange and resistance in a wire-gauge heat exchanger Khim Neff Mashin (1973) No. 6, 20-22; transl: Chem Petr Eng (1973) 522-525 43 Mikulin, E.I., Shevich, Yu.A. and Potapov, V.N. Efficiency of perforated plate array heat exchangers ghim Neff Mashin (1979) No. 5, 13-15; transl: Chem Petr Eng (1979) 351-355 44 Mikulin, E.I., Shevich, Yu.A., Potapov, V.N., Solntsev, M.Ya. and Yusova, G.M. Study of matrix-type heat exchangers made of perforated plates Khim Neff Mashin (1980) No. 9 8 - 10; transl: Chem Petr Eng (1980) 514-519 45 Orlov, V.K., Shevyakova, S.A. and Valeev, G.W. Heat exchange and hydraulic resistance in perforated-plate heat exchangers Khim Neff Mashin (1978) No. 8, 10-11; transl: Chem PetrEng (1978) 701-704 46 Shevyakova, S.A. and Orlov, V.K. Study of hydraulic resistance and heat transfer in perforated-plate heat exchangers Inz Fiz Zh (1983) 45 32-36; transl: J Eng Phys (1983) 734-737 47 Steyert, W.A. and Stone, N.A. Low flow velocity, fine-screen heat exchangers and vapor-cooled cryogenic current leads Report No. LA-7395 Uc 38 Los Alamos Scientific Laboratory, USA (1978) 48 Steyert, W.A. and Stone, N.A. An ultra compact low cost heat exchanger Cryogenics (1979) 18 627 49 Harris, W.T. Chemical Milling: The Technology of Cutting Materials by Etching Oxford University Press, UK (1976) 50 Swift, G., Migliori, A. and Wheatley, J. Construction of and measurements with an extremely compact cross-flow heat exchanger Heat TransfEng (1985) 6 39-47 51 Johnston, T. Miniaturized heat exchangers for chemical processing The Chem Eng (1984) 36-38 52 Matsuda, T. Private communication (1988) 53 Venkatarathnam, G. Matrix heat exchangers Ph.D. Dissertation Indian Institute of Technology, Kharagpur, in preparation 54 Voronin, V., Trekov, V.N. and Voronin, A.V. Using vacuum diffusion bonding for producing laminated perforated materials Welding Production (1984) 16-17 55 Coppage, I.E. and London, A.L. Heat transfer and flow friction characteristics of porous media Chem Eng Prog (1956) 52 57-63 56 Tong, L.S. and London, A.L. Heat transfer and flow friction characteristics of woven-screen and cross-rod matrices Trans ASME (1956) 78 1558-70 57 Qvale, E.B. and Smith, J.L. Jr A simple correlation for the heat transfer characteristics of a family of matrices subject to complex flow conditions Cryogenics (1969) 9 62-63 58 Bahnke, G.D. and Howard, C.P. The effect of longitudinal conduction on periodic-flow heat exchanger performance Trans ASME (1964) 86-A 105 59 Kays, W.M. and London, A.L. Compact Heat Exchangers 2nd Edn, McGraw Hill, New York, USA (1966) 60 Shah, R.K. and London, A.L. Laminar flow in ducts Adv Heat Transf (1978) 8 Suppl 1

Cryogenics 1990 Vol 30 November

917

Review 61 Gardon, R. and Akfirat, J.C. Heat transfer characteristics of impinging two-dimensional air jets, Trans ASME J Heat Transf (1966) 88 101 - 107 62 Heat Transfer Data Book (1968) General Electric Co., USA, Section 503.6, 1 - 8 63 Pawlowskii, M. and Suszek, E. Heat transfer by the perpendicular impact of an air stream on a flat surface, lnt Chem Eng (1988) 28 3 6 9 - 375 64 Hollworth, B.R. and Dagan, L. Arrays of impinging jets with spent fluid removal through vent holes on the target surface Trans ASME J Eng Power (1980) 102 994-999 65 Sparrow, E.M. and O'Brien, J.E. Heat transfer coefficients on the down stream face of an abrupt enlargement or inlet constriction in a pipe Trans ASME J Heat Transf (1980) 102 408-414 66 Eckert, E.R.G. and Drake R.M. Jr. Heat and Mass Transfer Tata McGraw Hill, New Delhi, India (1974) 67 Hubbell, R. and Cain, C.L. New heat transfer and friction factor design data for perforated heat exchangers Adv COo Eng (1986) 31 383 - 3 9 0 68 Churchill, S. Thermal conductivity of dispersions of packed beds an illustration of unexplored potential of limiting solutions for correlation, in Advances in Transport Processes Vol. 4 (Eds Mujumdar, A.S. and Masbelkar, R.A.) Wiley Eastern, New Delhi, India (1986) 394-418 69 Maxwell, J.C. A Treatise on Electricity and Magnetism Vol. 1, Clarendon Press, Oxford, UK (1873) 70 Rayleigh, Lord (Strutt, J.W.) On the influence of obstacles arranged in a rectangular order upon the properties of medium Phil Mag (1882) 481 - 502 71 Runge, I. On the electrical conductivity of metallic aggregates Z Tech Phys (1925) 6 6 1 - 6 8 72 Perrins, W.T., McKenzie, D.R. and McPhedran, R.C. Transport properties of regular arrays of cylinders Proc R Soc Lond (1979) A369 207 -225 73 Keller, H.B. and Sachs, D. Calculations on the conductivity of a medium containing cylindrical inclusions, J Appl Phys (1964) 35 537 - 5 3 8

918

Cryogenics 1990 Vol 30 November

74 Parasnis, D.S. Electrical and thermal conductivity of a medium containing arrays of spherical and cylindrical particles, Nature, Lond (1966) 211 1135-1137 75 Heat Transfer Data Book General Electric Co., USA, (1969) Section 502.2, 1 - 3 76 Kirpikov, V.A. and Leifman, I.I. Calculation of the temperature profile of a perforated fin lnz Fiz Zh (1972) 23 316-321; transl: J Eng Phys (1972) 23 1024-1027 77 Babak, T.B., Zablatoskaya, N.S., Babak, V.N., Kholpanov, L.P. and Malyusov, V.A. Applicability of the convective heat transfer equation for design of matrix heat exchangers Theor Osno Khim Tekh (1985) 19 488-494; transl: Theor Found Chem Eng (1985) 19 314-321 78 Babak, V.N., Babak, T.B., Kholpanov, L.P., Malyusov, V.A. and Zhavoronkov, N.M. Heat and mass exchange between a liquid film and a gas in countercurrent and straight through regimes on pistonlike phase movement Theor Osno Khim Tekh (1978) 12 3 - 10; transl: Theor Found Chem Eng (1978) 12 1 - 8 79 Babak, V.N., Babak, T.B., Kholpanov, L.P. and Malyusov, V.A. On the computation of a matrix heat exchanger from perforated plates Theor Osno Khim Tekh (1983) 17 347-360; transl: Theor Found Chem Eng (1984) 17 236-248 80 Babak, V.N., Babak, T.B., Kbolpanov, L.P. and Malyusov, V.A. Design of a sectional heat exchanger made of perforated plates with a highly conductive wall Theor Osno Khim Tekh (1985) 19 762 -767; transl: Theor Found Chem Eng (1985) 19 484-489 81 Babak, V.N., Babak, T.B., Kholpanov, L.P. and Malyusov, V.A. Calculation of compact matrix beat exchanger with plane-parallel channels Theor Osno Khim Tekh (1986) 20 327-331: transl: Theor Found Chem Eng (1986) 20 201-205 82 Babak, V.N., Babak, T.B., Kholpanov, L.P. and Malyusov, V.A. A procedure for calculating matrix heat exchangers from perforated plates Inz Fiz Zh (1986) 50 457-458; transl: J Eng Phys (1986) 50 330-335 83 Venkatarathnam, G. and Sarangi, S. Analysis of matrix heat exchanger performance Trans ASME J Heat Transf submitted for publication