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
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
907
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.
910
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 <s/d<
1
A l i g n e d slots
2 0 0 < Re < 1 4 0 0
0 . 0 7 5 < s/a < 0 . 8 8
Re < 1 6 0
Re > 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
E~
A ,-~- ~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
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