Full text of DOI:10.1080/014576301300092522

HeatTransfer Engineering, 22:17±25, 2001
°
C
Copyright 2001Taylor &Francis
0145±7632/01$12.00+.00
Application of a New
Analogy for Predicting
Heat Transfer to Cross
Rod Bundle Heat
Exchanger Surfaces
PRASANT NANDA
Department of Mechanical Engineering, Indira Gandhi Institute of Technology, Sarang, Orissa, India
SARIT KUMAR DAS
Heat Transfer and Thermal Power Lab, Department of Mechanical Engineering, Indian Institute of
Technology, Madras, India
HOLGER MARTIN
¨
¨
(
)
Institut fur Thermische Verfahrenstechnik, Universitat Karlsruhe TH , Karlsruhe, Germany
A new technique for prediction and measurement of compact heat exchanger surfaces has been
established. The method utilizes an analogy between heat transfer and  uid friction proposed by a
French scientist in 1928. The applicability of this analogy to plate heat exchanger surfaces was
examined recently. The present work proposes to use the same analogy for crossed rod bundles used
in regenerators. For this purpose, the analogy equation has been modiŽ ed to include a geometric
factor which takes care of the fraction of the pressure drop arising out of skin friction. The proposed
form of equation is found to be more general, applicable to crossed rod structures of any geometric
conŽ guration. The proposition was formulated based on standard data from the literature and also
by conducting independent experiments. The result of the transient experiment indicate excellent
adherence to the proposed analogical equation. The experimental evidence clearly indicates the
possibility of predicting heat transfer characteristics of such compact surfaces by measuring
pressure drop alone across the packed bed formed by the surface under consideration.
The use of compact heat exchangers has been on
the rise in recent times. The primary reasons contribut-
ing to the phenomenal increase in the use of compact
heat exchangers are the diversiŽ cation of heat trans-
fer studies in emerging areas of technology and the
unprecedented growth of materials and manufacturing
technology. Today, heat transfer studies are concen-
trated in areas such as aerospace, heat transfer at the
nano scale in electronic chips, biomedical and biotech-
nological applications, renewable energy resources, and
environment-friendly energy conversion processes, to
name a few. All these applications call for very efŽ cient,
Address correspondence to Dr. Sarit Kumar Das, Head, Heat Trans-
fer and Thermal Power Lab, Department of Mechanical Engineering, In-
dian Institute of Technology, Madras 600 036, India. E-mail: sarit das@
hotmail.com
17

unique, reliable, and innovative design of heat exchang-
ers ranging from a size of a few millimeters to a few
meters and handling thermal loads between a few watts
and megawatts. Incidentally, almost all the innovations
for these special applications are in the area of compact
heat exchangers. For example, the wave which appeared
in the 1980s in Europe [1] was the invasion of plate and
spiral plate heat exchangers. Even a newer generation
of shell-and-tube heat exchangers was built to achieve
higher efŽ ciency through complex  ow patterns like
that in microminiature heat exchangers [2] and heli-
cally baf ed heat exchangers [3]. The developments in
the purely compact regime, such as multistream plate-
Ž n heat exchangers [4] and micro-strip-channel heat
exchangers [2], have also made heat transfer analy-
sis complex due to their typical construction and  ow
conŽ guration.
In all these examples, one common feature that can
be noted is the very complex  ow pattern aimed at aug-
menting the heat transfer primarily by using surfaces
and obstructions which act as turbulence promoters or
vortex generators. The major challenge before a de-
signer of such heat exchangers is to determine accu-
rately the heat transfer coefŽ cient of that side of the
heat exchanger which has a complex  ow structure. It
is impossible at present to derive any analytical corre-
lation for heat transfer coefŽ cient over such compact
surfaces. Numerical solution is too expensive and time
consuming, as supported by the fact that a detailed nu-
merical study of even the most commonly used surfaces
such as herringbone plates are hardly available in the
literature. The only way seems to be experimental stud-
ies such as those presented by Kays and London [5].
It goes without saying that such experiments are not
only time consuming but also need to be backed up by
proper analytical modeling if they are to be determined
originally proposed for thermally developing laminar
 ow over a short heated length. It was pointed out that
the same analogy can be used in the turbulent regime
and the corresponding equation was named the gener-
alized Le´veˆque equation [11]:
³
´
=
1 3
dh
2
( )
1
:
Nu D 0 404 f Re Pr
Lc
where dh is the hydraulic diameter and Lc is the charac-
teristic length, which was taken as the distance between
two crossing points in a plate heat exchanger [11] and
is a function of the wavelength and the chevron angle
of the corrugation pattern.
Preliminary calculations have shown that the same
approach is applicable for packed beds of particles, for
tube bundles, and possibly for many other systems with
periodic  ow structures.
The present work is aimed at a similar approach of
using the Le´veˆque analogy for cross rod bundles. Some
fraction of the total pressure drop should be used for
evaluating the heat transfer coefŽ cient, because only
the skin friction contribution of the pressure drop can
be related to heat transfer and not the form drag con-
tribution. It has been found that this fraction of
p
came out to be 0.5 for plate heat exchangers and tube
bundles.
In the present comparison with data for cross rod
bundles from the literature and from experiments of the
Ž rst and second authors, we Ž nd that the fraction of p,
to be used in the modiŽ ed Le´veˆque equation, depends
on a geometric parameter, the lateral pitch ratio, and can
be easily correlated. The dependence of the fraction of
p to be used in the Le´veˆque equation on lateral pitch
is physically explained while discussing the results.
–
by transient tests like those described in [6 8]. Such
tests are particularly unavoidable for surfaces used for
regenerative heat exchangers, and similar tests are gain-
ing popularity for transfer types of heat exchangers [9]
as well.
To date, the manufacturers of compact heat exchang-
ers have carried out design practices based on test data
which are proprietory in nature. This makes it extremely
difŽ cult to suggest a generalized correlation for com-
pact surfaces used in heat exchangers. A theoretical
approach in this direction was made by the third au-
The Le´veˆque Analogy
Transport of heat and momentum are similar in na-
ture. The classical studies concerning the transfer of
heat and momentum in fully developed temperature
and velocity proŽ les in turbulent  ow were presented
by Osborne Reynolds, Ludwig Prandtl, Theodore von
Ka´rma´n, and many other investigators. These analogy
models, starting with the simplest proportional corre-
lation given by Reynolds for turbulent  ow over a  at
plate, relate the friction factor to the heat transfer coefŽ -
cient. The Reynolds analogy proposes a linear relation-
ship between heat and momentum transfer coefŽ cients
in the form
(
)
thor of this article Holger Martin , initially for chevron
plates. In predicting  ow friction characteristics, he
has taken a logical approach in analyzing the differ-
ent  ow patterns and has developed a theoretical cor-
relation for  uid friction based on experimental data
and physical reasoning. For predicting heat transfer
characteristic, he used an old study by Le´veˆque [10],
Nu
f
( )
2
St D
D
Re Pr
8
18
heattransfer engineering
vol. 22 no. 3 2001

This analogy corresponds to a Prandtl number of nearly
one and hence can be applied without much error to
ideal gases. For  uids differing widely from ideal gas,
Ludwig Prandtl and many others have developed more
leaves no space between the different layers of matrix
material. Using the  ow friction characteristic for these
(
geometries from Kays and London Figures 10-98 to
)
10-100 of [5] , the theoretical prediction is made ac-
¨
(
)
reŽ ned forms of analogies see also Schlunder [12] .
It can be observed readily that the Le´veˆque equation,
in reality, is in the form of an analogy, too, where apart
from the friction factor a geometric scale factor is also
incorporated. The reason for inclusion of this factor is
that the Le´veˆque equation was originally derived for a
thermal boundary layer in a hydrodynamically devel-
oped laminar  ow in a duct. This is also the reason
that even after being completely aware of this theory
for more than half a century, researchers working with
practical heat transfer equipment did not consider it to
be very important, as the  ow in almost all equipment is
cording to Le´veˆque’s equation. In this equation, the
length Lc is taken as the longitudinal pitch between
repeated  ow structure as shown in Figure 1. How-
ever, for closely packed stacking, this length reduces
=
to Lc D d 2, where d is the rod diameter.
–
(
A selected number of results are plotted Figures 2
)
5 to bring out the most important features of the anal-
ogy. In all cases, the experimental data for  ow friction
(
are taken from the literature Figures 10-98 to 10-100
)
of [5] and after application of the Le´veˆque analogy,
they are compared with the heat transfer data for identi-
cal geometries from the same reference [5]. Originally,
calculation was done and plots were made for all the
geometries of Figures 10-98, 10-99, and 10-100 of [5].
However, it was interesting to observe that the Ž nal form
¨
turbulent in nature. Schlunder [12] expressed his opin-
ion in an article entitled “The Historical Development
and Present State of ScientiŽ c Theory of Heat Transfer”
(
)
( )
in German about the Le´veˆque equation which was
of the Leveque analogy suggested [Eq. 3 , given later]
(
does not depend on the arrangement i.e., inline, stag-
translated in [11] as follows:
)
gered, or random . Due to limited space here we have
–
presented in Figures 2 5 some selected values of results
. . . and there might probably be one single case of turbulent
heat transfer only for which an equation can be derived from
the classicaldifferentialequations for viscous  ow which are
bound to no modeling concept whatsoever in the mechanism
of turbulent  ow and thereforemaybe takenasrigorous in the
classical sense. It is only valid, however, for extremely short
heated lengths of tube with a developed turbulent  ow, and
therefore it is more of an academic than of practical value.
for four different piches—Figures 2 and 3 are for inline
arrangement 1I and 3I , Figures 4 and 5 are staggered
arrangement 5S and 7S as given in Table 10-10 and
Figures 10-98 and 10-99 of Kays and London [5].
Figures 2 5 show the comparison of measured heat
transfer coefŽ cients against the theoretical prediction
(
)
(
)
–
( )
using the generalized Le´veˆque equation [Eq. 1 ]. On
–
)
It took the rigorous exercise of analyzing available
experimental data on chevron plates from the literature
in support of his proposition and found that they in fact
do agree with the analogy.
(
the same Figures 2 5 is also shown the predicted heat
transfer characteristic from the modiŽ ed Le´veˆque equa-
tion to be presented later .
(
)
It can be observed from the curves that the value of
=
St Pr2 3 calculated from the generalized Le´veˆque equa-
tion deviates from the experimental result almost by a
constant factor over the entire range of the Reynolds
number. This virtually means that for each geometry
there remains a constant multiplicity factor to match
APPLICATION OF THE ANALOGY FOR WIRE
MESH AND CROSS ROD BUNDLES USED AS
REGENERATOR PACKINGS
It is quite logical from the earlier studies [10, 11] that
the analogy can be applied to heat transfer problems
where small repetitive heated lengths are involved. The
reason why the analogy has been found to be successful
for plate heat exchangers is due to the zigzag wavy  ow
path in chevron plates. In compact heat exchangers,
similar  ow structures are also encountered, and hence
the logical conclusion that the same can be applied to
the said surfaces is quite obvious.
As an example, the crossed rod bundle used for re-
generative heat exchangers can be studied. The differ-
ent kinds of crossed rod bundles used for the analy-
sis are oriented staggered, inline, and randomly packed
with the assumption that rods are packed tightly, which
Figure 1 Schematic of wire mesh cross sectionfor inline arrange-
ment.
heattransfer engineering
vol. 22 no. 3 2001
19

Figure 6 Variation of multiplicity factor for pressure drop with
dimensionless transverse pitch.
Figure 2 Comparison of original and modiŽ ed Le´veˆque equation
) (
(
with experimental data from 1I of Figure 10-98 in [5] Xt
D
)
4:675 .
with the experimental result. This is the factor which
takes careof the fraction ofthe total pressure drop result-
ing from the skin friction alone. As a consequence, this
multiplicity factor is found to be fairly independent of
(
the arrangement of the wire mesh i.e., inline, staggered,
)
or random . Figure 6 shows the variation of this mul-
tiplicity factor with the transverse pitch, Xt . It should
be mentioned here that the multiplicity factor indicated
=
1 3, the
(
)
here is for the Nusselt number, since Nu
p
fraction of pressure drop to be used, is the cube of this
multiplicity factor.
These results show that the original generalized
Le´veˆque equation gives considerable deviation from
the actual measured heat transfer coefŽ cient as given
by Kays and London [5]. It is also important to note
Figure 3 Comparison of original and modiŽ ed Le´veˆque equation
) (
(
with experimental data from 3I of Figure 10-98 in [5] Xt
D
)
3:356 .
–
(
)
Figures 2 5 that the amount of deviation is neither ran-
dom nor constant, but is found to vary with dimension-
less transverse pitch. For an accurate prediction of heat
transfer coefŽ cient from the analogy, only a fraction of
the pressure drop resulting from skin friction should
be taken into consideration. Figure 7 shows such an
exercise, in which it is found that 30% of the total pres-
sure drop predicts the heat transfer accurately. However,
the approach is empirical in nature. Since the deviation
also varies with Xt, it is difŽ cult to ascertain what frac-
tion of the pressure drop is due to the skin friction that
should to be taken for the calculation of the heat transfer
coefŽ cient.
Figure 4 Comparison of original and modiŽ ed Le´veˆque equation
) (
(
with experimental data from 5S of Figure 10-99 in [5] Xt
D
)
2:417 .
Figure 7 Comparison of experimental observation from [5] with
original Le´veˆque equation using 30% of the pressure drop.
Figure 5 Comparison of original and modiŽ ed Le´veˆque equation
) (
(
)
with data from 7S of Figure 10-99 in [5] Xt D 1:571 .
20 heattransfer engineering
vol. 22 no. 3 2001

EXPERIMENTAL VALIDATION OF MODIFIED
LEVEQUE ANALOGY
Therefore, looking at the trends of the deviation, we
suggest that the nondimensional transverse pitch should
appear in the analogy correlation. This will take care of
the task of elimination of the form drag creeping into
Further strong foundation of the proposed modiŽ ed
Le´veˆque analogy has been laid by conducting indepen-
dent experiments to measure the pressure drop and heat
transfer coefŽ cient during the same test. These tests
also indicate the practical application of the analogy.
It is found that the pressure drop measurement is good
enough to predict the heat transfer characteristics, thus
avoiding more complex transient tests, which are not
only difŽ cult but for which the accuracy is critically
dependent on the theoretical model chosen for data
reduction.
(
)
the equation d being constant for all the cases . After
a series of studies it is proposed that the same may be
incorporated in the following form:
³
´
=
1 3
4rh Re2
=
Nu Pr¡1 3 D 0 44 f
( )
3
:
=
d 2 Xt
which may be called a modiŽ ed Le´veˆque analogy for
cross rod bundles. It is obvious from the nature of the
variation of the multiplicity factor that the part of total
pressure drop accounting for skin friction depends on
the transverse pitch of the wire mesh. This can be phys-
ically explained by the fact that when the pitch is small,
the entire amount of the  uid is covered by the boundary
layer and there is very little freedom for the  uid to turn
in order to increase the form drag. On the other hand,
when the pitch is large, the  uid at the center of each
crossed rod square is virtually unaffected by  uid fric-
tion and is free to turn, giving more pressure drop due
to form drag. Thus, as the transverse pitch increases,
less and less of the total pressure drop comes from skin
friction, lowering the multiplicity factor. The plots in
Experimental Setup
The schematic of the experimental setup is shown in
Figure 8. The test section is a small bed of tightly packed
wafers of stainless steel woven wire mesh screen. The
woven wire mesh screen has been chosen for the val-
idation because it simulates identical heat transfer and
 uid frictional data as crossed rod matrices as indicated
in Chapter 7 of [5]. The bed is kept inside a galva-
nized iron pipe. There are two valves, one of which
(
)
V1 serves as a bypass valve. The blower is run and
the air is heated by electrical heaters. At this stage, air
is not allowed to pass over the bed, until the air temper-
ature attains a steady-state value. During this phase of
operation, to avoid increase in temperature of the bed
due to heat conduction through the wall of the tube, the
bed is continuously cooled by compressed air. The bed
–
Figures 2 5 clearly show that the prediction of heat
transfer coefŽ cient from the modiŽ ed Le´veˆque analogy
is in excellent agreement with the experimental results.
The predictions are within a standard deviation of 5%
from the experimental results.
Figure 8 Schematic of the experimental setup.
heattransfer engineering
vol. 22 no. 3 2001
21

temperature throughout this phase of operation remains
fairly constant over the entire bed. Subsequently, valve
V2 is opened to allow the hot air to pass over the bed. The
bed temperature increases and after some time the out-
let air temperature approaches the inlet one. Flow rate is
measured by the gas  ow meter, and the pressure drop
across the bed is measured by a digital micromanome-
ter. The valves are readjusted for a different  ow rate
The Darcy friction factor f is calculated as
2
p dh
LG2
( )
5
f D
The evaluation of heat transfer coefŽ cient is based
on the simple theoretical modeling of a regenerator bed.
The underlying assumptions for this modeling are:
–
and the experiment is repeated. The temperature time
history at the inlet and outlet of the bed are continu-
1. The entire bed is insulated from the surrounding.
2. The temperatures of both  uid and matrix are uni-
form over any cross section of the bed, i.e., the ther-
mal conductivity along the radial direction in the bed
is inŽ nite.
(
ously recorded by four thermocouples two at each sec-
)
tion . Another thermocouple is placed before the bypass
valve to ensure that steady-state temperature has been
reached for bed inlet temperature. Between two experi-
ments, sufŽ cient time is given for the bed to cool down
to atmospheric temperature. Details of the test section
are given below.
3. The axial thermal conductivity of the bed and the
 uid along the  ow direction are neglected for the
(
)
>
range of Reynolds number Re 150 associated
with the experiment.
:
16 £ 16 stainless steel wire mesh
M D 0 74 kg
C D 480 J/kg K
Tube diameter
D 10 cm
4. The effect of holdup  uid heat capacity is negligible.
5. Depending on the above assumptions, the nondimen-
sional governing equation for the bed  uid system
can be given by
(
)
:
d wire diameter D 0 2 mm
(
Xt dimensionless transverse
)
:
pitch D 8 9375
:
dh D 2 075 mm
L D 4 cm
Tg
(
)
)
(
)
)
( )
6
gas
D Ntu Tm ¡ Tg
xD
Error Estimates
Tm
xD
(
(
(
( )
7
bed
D Ntu Tg ¡ Tm
In transient tests, it is important to choose the proper
thermocouple to match the allowable response lag. The
present Philips thermocoax thermocouples have a time
)
(
)
( )
8
;
gas Tg xD D 0
D Tg;i
–
constant of 40 50 ms, which is less than the measure-
ment time. The accuracy of the thermocouples is 0.01 K,
which is less than 0.5% of the temperature difference
measured during the experiment. In the present exper-
iment the error in  ow measurement is §2.5%. The
data reduction shows a sensitivity of less than 0.5%
of the temperature difference measured during the ex-
periment. It also shows a sensitivity of 3% in the heat
transfer coefŽ cient at the maximum error of  ow
measurement.
Temperature distribution of the gas and matrix for
the above conditions are given by [13]
Tg ¡ Tg;i
(
(
)
)
( )
9
;
;
D Go
D Fo
To ¡ Tg;i
Tm ¡ Tg;i
To ¡ Tg;i
(
)
10
where G and F are special functions known asAnzelius-
Schumann functions. Theyare detailed in the Appendix.
The transfer function for the single-blow packed-
Data Reduction
The most important part of a transient test is the anal-
ysis of experimental outputs. In the present data reduc-
tion, the usual deŽ nition of the Reynolds number in the
wire mesh has been used.
¯
(
)
;
bed system in the Laplace domain is sGo s . In the
present case, the experimental input is not an exact step
function. A typical recorded input temperature history
is shown in Figure 9. To obtain the output, a Ž tting of
the following nature has been used, which is also shown
in Figure 9.
4rhG
( )
4
Re D
Tg;in ¡ Tmin
¡ct
(
)
(
)
11
D a 1 ¡ e
C b
Tmax ¡ Tmin
Ç =
where G D m Ac is the mass velocity of the  uid.
22 heattransfer engineering
vol. 22 no. 3 2001

Figure 9 Fitted input temperature used for computation.
Figure 11 Comparison of present experimental heat transfer data
with modiŽ ed Le´veˆque equation.
The output in the Laplace domain is obtained by mul-
tiplying the Laplace transform of the above function
by the transfer function. The resulting equation is in-
versed back numerically to the time domain based on
Crump’s [14] algorithm using fast Fourier transform.
CONCLUSION
The present work brings out a new look at an anal-
ogy which remained unused for more than half a cen-
tury. The suggested form of analogy, which can be
termed a modiŽ ed Le´veˆque analogy, compares excel-
lently with the experimental data available in the litera-
ture for crossed rod bundles. This fact has been utilized
to suggest a method for the prediction of heat transfer
coefŽ cient based on pressure drop measurement, and
the proposition has been founded by conducting ex-
periments on wire mesh regenerators. It is interesting
to note that a unique equation can be utilized for both
crossed rod and wire mesh matrices without any neces-
sity to change the constants and the exponent involved in
the equation. This makes the present form of analogy
more general and directly usable for similar compact
surfaces. It is further suggested that experiments can
be conducted with compact surfaces with similar  ow
structures which are of practical importance for heat
transfer engineers.
–
The temperature time response is compared with the
(
)
experimental response Figure 10 for the determina-
tion of the heat transfer coefŽ cient.
Results of the Experiment
Even though the analysis of data from Kays and
London [5] proved conclusively the applicability of the
Le´veˆque analogy to crossed rod bundles, it was impor-
tant to ensure that this match was not by chance and
all similar  ow structures can be modeled through the
same analogy. Moreover, the experiment would demon-
strate how the heat transfer behavior of an unknown
compact surface can be determined directly from the
analogy concept. To do this, the prediction based on
(
)
pressure drop data has been plotted Figure 11 . The
Ž gure shows the experimental results for heat trans-
fer measurements along with the prediction from the
NOMENCLATURE
(
)
Le´veˆque analogy modiŽ ed based on measured fric-
tion factor. It demonstrates excellent agreement in the
entire range of Re from 300 to 650. The deviation is
less than 5%, which is even smaller than the accuracy
of standard experimental data.
A
Ac
C
Cp
d
dh
f
G
total heat transfer area of the matrix, m2
free  ow area of the matrix, m2
speciŽ c heat of the solid/matrix, J/kg K
speciŽ c heat of the  uid, J/kg K
diameter of the wire/crossed rod, m
hydraulic diameter, m
(
)
Darcy skin friction factor
Ç =
mass velocity in the bed m Ac, kg/m s
2
(
(
)
)
(
)
;
Gn
Gn s
h
a special function given in Appendix
¯
(
)
;
;
transformed Gn
vective heat transfer coefŽ cient, W/m2
con
K
k
L
Lc
mÇ
M
thermal conductivity of the  uid, W/m K
length of the regenerator bed, m
characteristic length, m
mass  ow rate of the  uid, kg/s
total mass of the matrix solid, kg
Figure 10 Experimental output temperature and the computa-
( )
(
)
tional result from the model used given by Eqs. 9 and 10 .
heattransfer engineering
vol. 22 no. 3 2001
23

(
)
=
k
(
[6] Mullisen,R. S., and Loehrke, R. I., A TransientHeatExchanger
Evaluation Testfor ArbitraryFluid InletTemperatureVariation
and Longitudinal Core Conduction, J. Heat Transfer, vol. 108,
Nu
Ntu
PL
PT
Pr
rh
Nusselt number D hdh
)
=
number of transfer units D h A mÇCp
longitudinal pitch, m
–
pp. 370 376, 1986.
transverse pitch, m
[7] Heggs, P. J., and Burns D., Single-Blow Experimental Pre-
diction of Heat Transfer CoefŽ cients—A Comparison of Four
Commonly Used Techniques, Exp. Thermal Fluid Sci., vol. 1,
(
)
k
=
Prandtl number D Cp
(
)
=
hydraulic radius D dh 4 , m
–
pp. 243 251, 1988.
(
)
=
Re
s
Reynolds number D Gdh
[8] Stang, J. H., and Bush, J. E., The Periodic Method for Test-
transformed time variable in frequency
domain
ing Compact Heat Exchanger Surfaces, ASME J. Eng. Power,
–
vol. 96A, no. 2, pp. 87 94, 1974.
(
)
=
St
t
Stanton number D h GCp
[9] Roetzel, W., Luo, X., and Xuan, Y., Measurements of Heat
Transfer CoefŽ cient and Axial Dispersion CoefŽ cient Using
Temperature Oscillations, Exp. Thermal Fluid Sci., vol. 7,
time coordinate, s
Tm
Tg
Tg;i
Tg;in
To
temperature of the matrix, K
–
pp. 345 353, 1993.
(
)
temperature of the  uid gas , K
gas inlet temperature to the packed bed, K
instantaneous inlet temperature, K
ambient temperature, K
[10] Le´veˆque, A., Les jois de la transmission de chaleur par con-
–
vection, Ann. Mines. Ser., vol. 12, no. 13, pp. 201 415, 1928.
[11] Martin,H., A TheoreticalApproach toPredictthe Performance
of Chevron-Type Plate Heat Exchanger, Chem. Eng. Process-
–
Tmin
minimum of the inlet temperature transient,
K
maximum of the inlet temperature transient,
K
instantaneous outlet temperature, K
distance coordinate along the axis, m
ing, vol. 35, pp. 301 310, 1996.
¨
[12] Schlunder, E. U., Analogy between Heat and Momentum
Transfer, Chem. Eng. Processing, vol. 37, pp. 103 107, 1998.
[13] Das, S. K., and Sahoo, R. K., Thermodynamic Optimization
of Regenerators, Cryogenics, vol. 31, pp. 862 868, 1991.
[14] Crump, K. S., Numerical Inversion of Laplace Transforms Us-
–
Tmax
–
Tg;out
x
xD
Xt
ing a Fourier Series Approximation, J. Assoc. Commer. Ma-
chinery, vol. 23, pp. 89 96, 1976.
–
(
)
(
=
dimensionless distance D x L
)
=
d
dimensionless transverse pitch D PT
(
z
dimensionless time D real time/residence
)
APPENDIX
time
p
pressure drop, N/m2
(
)
Special Functions
dimensionless charging time D Ntu
(
=
dimensionless charging time D mÇCpt
(
)
;
)
The special function Gn X Y known as the
MC
(
Anzelius-Schumann function is given by
dimensionless inlet temperature [D Tg;in
¡
in
) (
=
)
]
Tmin Tmax ¡ Tmin
dimensionless outlet temperature [D Tg;out
) (
1
)
Y (
nCrC1
X
(
out
¡XCY
(
)
;
Gn X Y D e
)
=
¡ Tmin Tmax ¡ Tmin
]
(
)
n C r C 1 !
rD0
viscosity of the  uid, kg/ms
density, kg/m3
r
h
i
p
X
(
)
n C r ¡ p ! x
dimensionless distance D xD Ntu
for n ¸ 0
(
)
r ¡ p !
p!
pD0
REFERENCES
The Fn function can be expressed as
[1] Samdani, G., Fouhly, K., and Moore, S., Heat Exchanger: The
¡
¢
Fn D e¡Y
eY Gn
–
Next Wave, Chem. Eng., June, pp. 30 37, 1993.
Y
[2] Das, S. K., Advances in Heat Exchanger Technology, the
New Route, Proc. Workshop on Heat Exchangers, IIT Madras,
17 August 1996.
[3] Krat, D., Stehlik, P., Van der Prolog, H. J., and Bashir I. Master,
Helical Baf es in Shell and Tube Heat Exchanger, Part I: Ex-
However, it is easier to evaluate Fn from the recurrence
relations,
–
perimental VeriŽ cation, Heat Transfer Eng., vol. 17, pp. 93
100, 1996.
(
)
(
)
(
)
;
;
;
Fn X Y D Gn X Y C Gn¡1 X Y
[4] Prasad, B. S. V., The Thermal Design and Rating of Multi-
;
For n D ¡1 Gn can be expressed as
(
)
stream Plate-Fin Heat Exchangers, in R. K. Shah ed. , Com-
pact Heat Exchangers for the Process Industries, pp. 79 100,
–
³
´ ³
´
1
Xr
Y r
X
Begell House, New York, 1997.
(
)
¡ XCY
(
)
;
G
X Y D e
¢
[5] Kays, W. M., and London, A. L., Compact Heat Exchangers,
3rd ed., McGraw-Hill, New York, 1984.
¡1
r!
r!
rD0
24
heattransfer engineering
vol. 22 no. 3 2001

(
)
(
)
;
From the above the functions, Go X Y and Fo X Y
can be expressed as
of the editorial board of the International Journal of Heat Exchangers.
His research interests include heat exchangers, dynamic analysis of heat
exchangers, nuclear heat transfer, thermal storage, exergy analysis of heat
transfer processes, computational  uid dynamics and heat transfer, analogies
in heat transfer, and liquid metal heat transfer. He is a Humboldt Fellow for
;
q
1
p
YqC1
X
X
X
(
)
¡ XCY
(
)
–
;
Go X Y D e
2000 2001.
q C 1
p!
qD0
pD0
Holger Martin has been a professor of thermal
process engineering at the Universitaet Karlsruhe
(
)
(
)
;
;
Fo D Go X Y C G X Y
¡1
(
)
TH , Germany, since 1980. His research inter-
ests are in the Ž elds of drying, heat and mass
transfer, and heat exchangers. He holds a Dr.-
Ing. and a Dr.-Ing. habil.-degree from Karlsruhe
University. Since 1980 he has also been the sci-
Prasant Nanda is a lecturer at Indira Gandhi
(
)
Institute of Technology—Sarang, Orissa India .
He completed his master’s degree at the Heat
Transfer and Thermal Power Laboratory of the
Indian Institute of Technology—Madras in 1998.
The present work is part of his dissertation for
Master of Technology in Mechanical Engineer-
ing. His research interests include numerical sim-
ulation of heat transfer processes, heat transfer
analogy, and application of artiŽ cial neural network to heat transfer
problems.
(
)
entiŽ c director of the International Seminar IS
for Teaching and Research in Chemical Engineer-
ing and Physical Chemistry at the Universitaet Karlsruhe. Numerous of his
original papers on the mentioned research topics have been published in
professional journals. Book contributions in Advances in Heat Transfer,
Vol. 13 1977 on impinging jet  ow heat and mass transfer, in the Heat
Exchanger Design Handbook on transient conduction,  uidized beds, and
impinging jets, and in the VDI Waermeatlas on the same topics as well as on
(
)
(
plate heat exchangers. Books include Waermeuebertrager Heat Exchang-
) (
)
(
)
ers in German , 1988; Heat Exchangers in English , 1992; and, withErnst-
(
Ulrich Schluender, Einfuehrung in die Waermeuebertragung Introduction
) (
)
Sarit Kumar Das is head of the Heat Transfer
and Thermal Power Section of the Mechanical
Engineering Department at the Indian Institute
of Technology—Madras. He received his B.E.
and M.E. degrees from Jadavpur University,
to Heat Transfer in German , 1995. In 1980 he was awarded the Arnold-
Eucken-Preis from GVC-VDI in Strasbourg, and in 1984 he obtained the
French-German Alexander-von-Humboldt-Preis, which enabled him a stay
of several months as a guest researcher at the Laboratoire des Sciences du
(
)
Ge´nie Chimique LSGC at Nancy, France, which belongs to the French
(
)
( )
National ScientiŽ c Research Centre CNRS . He also has been a guest pro-
Calcutta India , in 1984 and 1987, respectively.
He received his Ph.D. degree from Sambalpur
(
)
fessor at the Indian Institute of Technology IIT in Madras. Some years ago,
he discovered the “Le´veˆque analogy,” a method to calculate heat or mass
transfer rates from pressure drop in periodic structures such as the cross-
corrugated channels of chevron-type plate heat exchangers, pached beds,
tube bundles, etc.
University in 1994. He has published one book
and more than 40 technical papers in various jour-
nals and conference proceedings. He had been a guest professor at the Uni-
versity of Federal Armed Forces, Hamburg Germany . He is a member
(
)
heattransfer engineering
vol. 22 no. 3 2001
25

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