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Numerical studies of a tube-in-tube helically coiled heat exchanger

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Chemical Engineering and Processing 47 (2008) 2287–2295
Numerical studies of a tube-in-tube helically coiled heat exchanger
Vimal Kumar, Burhanuddin Faizee, Monisha Mridha, K.D.P. Nigam
∗
Department of Chemical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-110016, India
Received 6 August 2007; received in revised form 29 December 2007; accepted 2 January 2008Available online 11 January 2008
Abstract
In the present study a tube-in-tube helically coiled (TTHC) heat exchanger has been numerically modeled for ﬂuid ﬂow and heat transfercharacteristics for different ﬂuid ﬂow rates in the inner as well as outer tube. The three-dimensional governing equations for mass, momentum andheat transfer have been solved using a control volume ﬁnite difference method (CVFDM). The renormalization group (RNG)
k
–
ε
model is used tomodel the turbulent ﬂow and heat transfer in the TTHC heat exchanger. The ﬂuid considered in the inner tube is compressed air at higher pressureand cooling water in the outer tube at ambient conditions. The inner tube pressure is varied from 10 to 30bars. The Reynolds numbers for the innertube ranged from 20,000 to 70,000. The mass ﬂow rate in the outer tube is varied from 200 to 600kg/h. The outer tube is ﬁtted with semicircularplates to support the inner tube and also to provide high turbulence in the annulus region. The overall heat transfer coefﬁcients are calculated forboth parallel and counter ﬂow conﬁgurations. The Nusselt number and friction factor values in the inner and outer tubes are compared with theexperimental data reported in the literature. New empirical correlations are developed for hydrodynamic and heat-transfer predictions in the outertube of the TTHC.© 2008 Elsevier B.V. All rights reserved.
Keywords:
Tube-in-tube helical heat exchanger; Heat transfer; Helical tube; RNG k–
1. Introduction
Helical coil heat exchangers are one of the most com-mon equipment found in many industrial applications rangingfrom chemical and food industries, power production, elec-tronics, environmental engineering, manufacturing industry,air-conditioning, waste heat recovery, cryogenic processes, andspace applications. Helical coils are extensively used as heatexchangers and reactors due to higher heat and mass transfercoefﬁcients, narrow residence time distributions and compactstructure. The modiﬁcation of the ﬂow in the helically coiledtubes is due to the centrifugal forces (Dean roll cells, [4,5]). The
curvature of the tube produces a secondary ﬂow ﬁeld with a cir-culatorymotion,whichcausestheﬂuidparticlestomovetowardthe core region of the tube. The secondary ﬂow enhances heattransfer rates as it reduces the temperature gradient across thecross-section of the tube. Thus there is an additional convective
Abbreviations:
CVFDM, control volume ﬁnite difference method; HVAC,heating, ventilating and air conditioning; LMTD, log-mean temperature differ-ence; TTHC, tube-in-tube helical coil.
∗
Corresponding author. Tel.: +91 11 26591020; fax: +91 11 26591020.
E-mail address:
nigamkdp@gmail.com (K.D.P. Nigam).
heat transfer mechanism, perpendicular to the main ﬂow, whichdoes not exist in conventional heat exchangers. An extensivereview of ﬂuid ﬂow and heat transfer in helical pipes has beenpresented in the literature [1–3].
There is considerable amount of work reported in the lit-erature on heat transfer in coiled tubes; however, very lessattention has been paid to study the outer heat transfer coef-ﬁcient. Figueiredo and Raimunda [6], Haraburda [7], Prasad et
al.[8]andPatiletal.[9]havediscussedthedesignprocedurefor
coil-in-shell heat exchangers considering helical coiled tubes asa bank of straight tubes for calculating outer heat transfer coef-ﬁcients. In the coil-in-shell heat exchangers poor circulation isobserved in shell regions near the coil which could be avoidedby using a tube-in-tube helical coil (TTHC) conﬁguration. Atube-in-tube helical coil heat exchanger requires the knowledgeoftheheattransferratesforthetwoﬂowingﬂuid,i.e.,theﬂowinthehelicaltubeaswellasinthehelicalannulus.Karahalios[10]and Petrakis and Karahalios [11–13] reported the ﬂuid ﬂow and
heat transfer in a curved pipe with a solid core. They showedthat the size of the core affects the ﬂow in the annulus withﬂowsapproachingparabolicforlargecores[13].Karahalios[10]
studiedtheheattransferinacurvedannuluswithaconstanttem-perature gradient on both inner and outer walls of the annulus as
0255-2701/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.cep.2008.01.001
2288
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295
the thermal boundary conditions. All the above reported studiesfor helical coils were conﬁned with one of two major boundaryconditions, constant wall heat ﬂux or constant wall temperature[2,14]. However in industrial applications of heat exchangerswhere one is interested in ﬂuid-to-ﬂuid heat exchanger the useof constant wall temperature or constant wall ﬂux conditionsdoes not appear to be physically realistic. This complicates thedesignoftube-in-tubehelicalcoilheatexchangers,whereeitherthe heating or cooling is supplied by a secondary ﬂuid, with thetwo ﬂuids separated by the wall of the coil.Garimella et al. [15] reported average heat transfer coefﬁ-cients of laminar and transition ﬂows for forced convection heattransferincoiledannularducts.Twodifferentcoildiametersandtwo annular radius ratios were used in the experiment. Hot andcold waters were used as working ﬂuids. They reported that theheat transfer coefﬁcients obtained from the coiled annular ductswere higher than those obtained from a straight annulus, espe-cially in the laminar region. Xin et al. [16] studied the effects of coil geometries and the ﬂow rates of air and water on pressuredrop in both annular vertical and horizontal helicoidal pipes.Experiments were performed for the superﬁcial water Reynoldsnumberfrom210to23,000andsuperﬁcialairReynoldsnumberfrom 30 to 30,000. Their results showed that the transition fromlaminartoturbulentﬂowcoversawideReynoldsnumberrange.Rennie and Raghavan [17] experimentally reported the heattransfer in a coil-in-coil heat exchanger comprised of one loop.This conﬁguration results in secondary ﬂows in both the innertube and in the annulus, as both sections are curved and subjecttocentrifugalforces.Theﬂowsofboththeﬂuidswereco-currentand counter-current. They also reported that increasing the tubeor annulus Dean numbers resulted in an increase in the over-all heat transfer coefﬁcient. The heat transfer characteristicsof a double-pipe helical heat exchanger for both parallel andcounter ﬂow using water in the inner as well as outer tubes wasnumerically studied by Rennie and Raghavan [18]. Overall heattransfer coefﬁcients were calculated for inner Dean numbers inthe range of 38–350 for the boundary conditions of constantwall temperature and constant heat ﬂux. The results showed anincreasing overall heat transfer coefﬁcients as the inner Deannumber increased; however, ﬂow conditions in the annulus hada stronger inﬂuence on the overall heat transfer coefﬁcient. Fur-thermore, increasing the size of the inner tube resulted in lowerthermalresistancesintheannulus,thoughthethermalresistancein the inner tube remained fairly constant.From the literature it can be seen that no study has beenreported which considers the ﬂuid at high pressures and temper-ature, though in most of the industrial applications the processstreams are available at higher pressure and temperature (e.g.,chemical reactions; heating, ventilating and air conditioning(HVAC)systems;andheatexchangers).Thereforeinthepresentwork the performance of a tube-in-tube helical heat exchangerfor a compressed air–water counter-current and co-current ﬂowsystem is studied numerically over a wide range of Reynoldsnumber considering turbulent ﬂow regime. The hydrodynamicsand heat transfer of compressed air ﬂowing in the inner tube of tube-in-tube helical coil heat exchanger with high pressure (i.e.,10–30bars) are being reported the ﬁrst time, which has not been
Fig. 1. Tube-in-tube helical heat exchanger.
considered in the previous literature. Fluid-to-ﬂuid heat transferhas been studied using physically realistic boundary conditions.The effect of the ﬂuid ﬂow on the heat transfer and hydrody-namics have been studied in the tube as well as in the annulus.In the present work bafﬂes have been introduced in the annulusareaofthetube-in-tubehelicalheatexchanger.Allthecomputa-tions have been carried out on a SUN Fire V880 computer in theChemical Reaction Engineering Laboratory at Indian Instituteof Technology, Delhi.
2. Mathematical modeling of TTHC
The geometry considered and the systems of the coordinatesare illustrated in Fig. 1 (where,
d
i,inner
is the diameter of innercoiled tube;
R
c
is the coil radius;
H
is the distance between thetwo turn; and
d
o,inner
is the inner diameter of the outer tube).The TTHC geometry was developed with inner and outer diam-eters of 0.023 and 0.0508m, respectively, with coil diameterof 0.762m and with a pitch of 0.020m. The TTHC was com-prised of four full turns. As the tube side ﬂuid was under highlyturbulence ﬂow regime therefore it was assumed that the ﬂowwillgetfullydevelopedwithinfourturns.TheTTHCdetailsaregiven in Table 1 for both inner as well as for outer tubes, respec-tively. The coils were orientated in the vertical position. Inletsand outlets were located at each end of the coil. The boundaryconditions associated with the inlets speciﬁed the inlet veloci-ties in the axial direction. Coil properties were set to those of stainless steel, with a thermal conductivity of 16Wm
−
1
K
−
1
,density of 7882kg.m
−
3
and a speciﬁc heat of 502Jkg
−
1
K
−
1
.The outer coil was set to be adiabatic (representing an insulatedtube) and the inner coil was set to allow conductive heat ﬂowthrough the tube.
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295
2289Table 1Geometrical and ﬂow parameters for the inner and outer tube of TTHCInner tube Outer tubeOuter diameter (m) 0.0254 0.0508Inner diameter (m) 0.023 0.0484Coil diameter (m) 0.762 0.762Pitch (m) 0.100 0.100Number of turns 4 4Flow rate (kg/h) 40–85 200–600Pressure (bars) 10, 20 and 30 1Prandtl number 1 0.74, 7, 33 and 150
The pressure drop and heat transfer in the TTHC is studiedfor ﬁve different Reynolds number (30,000<
N
Re
<70,000) inthe inner tube. The outer tube mass ﬂow rates is varied from200 to 600kg/h. The Reynolds number in the outer tube isdifferent for pressure drop and heat transfer calculations as itdepends upon the equivalent diameter of the outer tube. Thedetailed description of the equivalent diameter calculation isreported in Kumar et al. [19]. The simulations were carried outat three different levels of pressure values, i.e., form 10–30bar.The annulus side ﬂuid properties were calculated at averagetemperature i.e., mean of inlet and outlet temperature in theannulus. Compressed air density in the inner tube was cal-culated using ideal gas law. The total number of simulationsperformed were 150 (5 inner tube ﬂow rates, 5 outer tube ﬂowrates, 3 operating pressure values and 2 ﬂow arrangement sys-tems). The output of the simulations included the inlet andoutlet velocities, mass ﬂow rates and enthalpy rates, as wellas velocity, pressure, and temperature ﬁelds at various speciﬁedcross-sections.
2.1. Governing equations
The geometry, system of coordinates and details of TTHCconsidered for the present work are discussed in our previouswork [19]. In the TTHC the cold and hot ﬂuids enter from theircorresponding inlets and heat transfer take place between thetwo ﬂuids due to conduction and forced convection. Heat trans-fer by radiation can be neglected because temperature in theTTHC studied is quite low (max. 353K). The Cartesian coor-dinate system (
x
,
y
,
z
) was used to represent ﬂow in numericalsimulation. The ﬂow was considered to be steady, and incom-pressible.At the inlet (
φ
=0
◦
), ﬂuid with turbulence intensity
I
andtemperature
T
0
enters the TTHC heat exchanger at a velocityof
u
0
. Turbulent ﬂow and heat transfer develop simultaneouslydownstreaminthetubes.Fortheturbulentﬂowandheattransfermodeling the RNG k–
model proposed by Yakhot and Orszag[20]wasusedintheTTHCbecausetheRNGmodelincludedanadditional term in its
equation that signiﬁcantly improve theaccuracyforrapidlystrainedﬂows,suchasthoseincurvedpipes.The effect of swirl on turbulence is included in the RNG model,enhancing accuracy for swirling ﬂows. The three dimensionalgoverning equations of turbulent ﬂow and the heat transfer inthe TTHC can be written in tensor form in the master Cartesiansystem as follows:Ideal gas law:
P
=
ρRT
(1)Continuity equation:
∂u
i
∂x
i
=
0 (2)Momentum equation:
∂
(
ρu
i
u
j
)
∂x
j
=
∂∂x
j
µ
eff
∂u
i
∂x
j
+
∂u
j
∂x
i
−
23
µ
eff
∂u
k
∂x
k
−
∂p∂x
i
(3)Energy equation:
∂
(
ρu
i
c
p
T
)
∂x
i
=
∂∂x
i
α
T
µ
eff
∂T ∂x
i
+
du
i
dx
j
µ
eff
∂u
i
∂x
j
+
∂u
j
∂x
i
−
23
µ
eff
∂u
k
∂x
k
δ
ij
(4)Turbulent kinetic energy equation:
∂
(
ρu
i
k
)
∂x
i
=
∂∂x
i
α
k
µ
eff
∂k∂x
i
+
µ
t
S
2
+
G
b
−
ρε
(5)Dissipation rate of turbulent kinetic energy equation:
∂
(
ρu
i
ε
)
∂x
i
=
∂∂x
i
α
ε
µ
eff
∂ε∂x
i
+
C
1
ε
εkµ
t
S
2
−
C
2
ε
ρε
2
k
−
Ra
(6)The effective viscosity,
µ
eff
can be deﬁned as
µ
eff
=
µ
mol
1
+
C
µ
µ
mol
k
√
ε
2
(7)where
µ
mol
is the molecular viscosity. The coefﬁcients
α
T
,
α
k
and
α
ε
in Eqs. (4)–(6) are the inverse effective Prandtlnumbers for
T
,
k
, and
ε
, respectively. The values of inverseeffective Prandtl numbers,
α
T
,
α
k
and
α
ε
are computed usingthe following formula derived analytically by the RNG theory:
α
−
1
.
3929
α
0
−
1
.
3929
0
.
6321
α
+
2
.
3929
α
0
+
2
.
3929
0
.
3679
=
µ
mol
µ
eff
(8)where
α
0
is equal to 1/
N
Pr
, 1.0, and 1.0, for the computationof
α
T
,
α
k
and
α
ε
, respectively. When a non-zero gravity ﬁeldand temperature gradient are present simultaneously, the
k–
ε
model account for the generation of
k
(kinetic energy) due tobuoyancy(
G
b
inEq.(5))andthecorrespondingcontributiontothe production of
ε
(energy dissipation) in Eq. (6). The effectsof buoyancy are also included despite the fact that the effect of buoyancy is not so signiﬁcant at very high Reynolds number.The generation of turbulence due to buoyancy is given by
G
b
=
βg
i
µ
t
N
Pr,t
∂T ∂x
i
(9)where
N
Pr
,t
is the turbulent Prandtl number for energy,
g
i
is thecomponentofthegravitationalvectorinthe
i
thdirection.Inthe
2290
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295
case of the RNG k–
model,
N
Pr
,t
=1/
α
T
, and
β
, the coefﬁcientof thermal expansion, is deﬁned as
β
=
−
1/
ρ
(
∂
p
/
∂
T
)
T
. In Eq.(6), R is given by
R
=
C
µ
ρη
3
(1
−
η/η
0
)1
+
ζη
3
ε
2
k
(10)where
η
=S
k
/
ε
,
η
0
≈
4.38,
ζ
=0.012. The model constants
C
,
C
1
, and
C
2
are equal to 0.085, 1.42 and 1.68, respectively.The term
S
in Eq. (5) and Eq. (6) is the modulus of the mean
rate-of-strain tensor, deﬁned as
S
=
2
S
ij
S
ij
, where
S
ij
=
12
∂u
i
∂x
j
+
∂u
j
∂x
i
(11)Thetwo-layerbased,non-equilibriumwallfunctionwasusedfor the near-wall treatment of ﬂow in the given geometry. Thenon-equilibrium wall functions are recommended for complexﬂowsbecauseofthecapabilitytopartlyaccountfortheeffectsof pressure gradients and departure from equilibrium. The numer-ical results for turbulent ﬂow tend to be more susceptible togrid dependency than those for laminar ﬂow due to the stronginteraction of the mean ﬂow and turbulence. The distance fromthe wall at the wall-adjacent cells must be determined by con-sidering the range over which the log-law is valid. The size of wall adjacent cells can be estimated from
y
p
(
y
+
p
ν/u
τ
), where
u
≡
(
τ
w
/
ρ
)
0.5
=
u
(c
f
/2)
0.5
.Inthepresentstudy,the
y
+
p
wastakenin the range of 30–60.The wall heat ﬂux was computed using
q
w
=
ρ
C
p
u
T/T
+
,where
T
+
is obtained from
T
+
=
N
Pr
uu
∗
, y
+
≤
y
+
v
N
Pr,w
uu
∗−
N
Pr
, y
+
> y
+
v
(12)
N
Pr
=
π/
4sin(
π/
4)
A
T
κ
1
/
2
N
Pr
N
Pr,w
−
1
N
Pr,w
N
Pr
1
/
4
(13)where
N
Pr
is the molecular Prandtl number,
N
Pr
,w
, the turbu-lent wall Prandtl number (
N
Pr
,w
=1.2) and
A
T
, the Van Driestconstant (
A
T
=26).At the inlet, uniform proﬁles for all the dependent variableswere employed:
u
s
=
u
0
, T
=
T
0
, k
=
k
0
, ε
=
ε
0
.
(14)For the inner and outer walls of the heat exchanger, heat con-duction in solid wall was considered. Ambient temperature wasconsidered across the outer wall of the TTHC heat exchanger,i.e.,
T
=300K. Turbulent kinetic energy at the inlet,
k
0
, and thedissipation rate of turbulent kinetic energy at the inlet,
ε
0
, wereestimated by
k
0
=
32(
u
0
I
)
2
ε
0
=
C
3
/
4
µ
k
3
/
2
L
(15)In Eq. (14), the turbulence intensity level,
I
, is deﬁned as
u
/
/
u
×
100%. As the turbulent eddies cannot be larger than thepipe, the turbulence characteristic length scale,
L
, is set to be0.07(
d
i
,inner
/2) in the present study. The factor of 0.07 is basedon the maximum value of the mixing length in fully developedturbulent pipe ﬂow.The computation domain (4 turns) used in this study canensure that the outﬂow condition (i.e., fully developed ﬂow andthermal assumed) can be satisﬁed at the exit plane of the curvedpipe.Therefore,attheoutlet,thediffusiontermforalldependentvariables were set to zero in the exit direction:
∂φ∂n
=
0 (16)where
φ
represents the variables
u
i
, T, k
and
ε
.
3. Numerical computation
3.1. Numerical method
The governing equations for ﬂow and heat transfer in theTTHC were solved in the master Cartesian coordinate systemwith a control volume ﬁnite difference method (CVFDM) sim-ilar to that introduced by Patankar [21]. The grid topology anddensity considered was similar as used in our previous work [19].The convection term in the governing equations was mod-eled with the bounded second-order upwind scheme and thediffusion term was computed using multi linear interpolat-ing polynomial nodes. The SIMPLEC algorithm was used toresolve the coupling between velocity and pressure [22]. Toaccelerate the convergence, the under-relaxation factor for thepressure,
p
, was 0.3; that for temperature,
T
, was 0.9; that forthe velocity component in the
i
-direction,
u
i
,
k
and
ε
was 0.7;and that for body force was 0.8. In order to predict the per-formance of the TTHC, the double precision solver was usedfor simulations. The numerical computation was consideredconverged when the residual summed over all the computa-tional nodes at the
n
th iteration was less than or equal to10
−
6
.
3.2. Calculation of heat transfer coefﬁcient
Theoverallheat-transfercoefﬁcient,
U
o
,wascalculatedfromtheinletandoutlettemperaturedataandtheﬂowrates,usingthefollowing equation:
U
o
=
qA
o
T
LMTD
(17)where
T
LMTD
is the log mean temperature difference. Theoverall heat transfer rates were based on the outer surface area,
A
o
, of the inner tube. The
T
LMTD
was calculated based on thetemperaturedifference,
T
1
and
T
2
usingfollowingequation:
T
LMTD
=
T
2
−
T
1
ln(
T
2
/T
1
) (18)Heat transfer coefﬁcients were calculated for both the innerand outer tubes. For these calculations, average bulk tempera-tures and average temperature of the coil at each cross-sectionwere used. The inner and outer heat transfer coefﬁcients are
V. Kumar et al. / Chemical Engineering and Processing 47 (2008) 2287–2295
2291
usually obtained from the overall thermal resistance consistingof three resistances in series: the convective resistance in theinner surface, the conductance resistance of the pipe wall andthe convective resistance on the outer surface by the followingequation:1
U
o
=
A
o
A
i
h
i
+
A
o
ln(
d
o
/d
i
)2
πkL
+
1
h
o
(19)where
d
o
istheouterdiameterofthetube;
d
i
istheinnerdiameterof the tube;
k
is the thermal conductivity of the wall; and
L
isthe length of the tube.Heattransfercoefﬁcientsfortheshellside,h
o
,andforthetubeside, h
i
, were calculated using traditional Wilson plot technique[17,19].Forthecalculationofinnerheattransfercoefﬁcient,themass ﬂow rate in the shell side was kept constant; and assumedthat the outer heat transfer coefﬁcient is constant. The innerheat transfer coefﬁcient was assumed to behave in the followingmanner with the ﬂuid velocity in the tube side,
u
i
:
h
i
=
Cu
ni
(20)Eq. (20) was placed into Eq. (19) and the values for the con-
stant
C
andtheexponent
n
weredeterminedthroughcurveﬁtting.Theinnerandouterheattransfercoefﬁcientscouldthenbecalcu-lated. This procedure was repeated for each outer tube ﬂow rate.Similar procedure was adopted for the calculation of annulusheat transfer coefﬁcient (by keeping tube side ﬂow rate constantand annulus side ﬂow rate varying).
4. Results and discussion
The TTHC heat exchanger was analyzed in terms of heattransfer in downstream length and heat exchanger efﬁciencybefore discussing the friction factor and fully developed heat-transfer coefﬁcient in the inner and outer tube, respectively.
4.1. Effectiveness-NTU
The effectiveness is deﬁned as the ratio of the actual amountof heat transfer to the maximum possible heat transfer for thegivenheatexchanger.Themaximumamountofheattransferred,
q
max
, is calculated by:
q
max
=
C
min
(
T
in
,
h
−
T
in
,
c
) (21)where
C
min
is the minimum between ˙
m
c
C
p,c
and ˙
m
h
C
p,h
(massﬂow rate multiplied by speciﬁc heat),
T
in,h
is the inlet tempera-ture of the hot ﬂuid, and
T
in,c
is the inlet temperature of the coldﬂuid. The effectiveness is generally plotted vs. the NTU
max
toobtaingraphsthatcontaincurvesofconstant
C
min
to
C
max
ratios.NTU
max
is deﬁned as:NTU
max
=
A
o
U
o
C
min
(22)Furthermore, the theoretical effectiveness,
ε
, for a co-currentand counter-current ﬂow heat exchangers are given by the fol-
Fig. 2. Effectiveness vs. number of transfer units in TTHC.
lowing two equations [23], respectively:
ε
=
1
−
exp(1
−
NTU(1
+
C
min
/C
max
))1
+
C
min
/C
max
,
forco-currentﬂow (23)
ε
=
1
−
exp(
−
NTU(1
−
C
min
/C
max
))1
−
C
min
/C
max
exp(
−
NTU(1
−
C
min
/C
max
))
,
forcounter-currentﬂow (24)Fig. 2 shows the plot of effectiveness vs. NTU for both co-current and counter-current ﬂow in the TTHC heat exchangerfor
m
s
=500kg/h and
P
=30bars. It can be seen from the ﬁg-ure that the simulated data ﬁts well with the model values (Eqs.(23) and (24)). In the present work the effectiveness was cal-culated considering the average heat transfer coefﬁcient whichcausesnegligibledeviationfromthetheoreticalpredictions(Eqs.(23) and (24)). Fig. 2 also shows that with increasing NTU the
effectiveness increases.
4.2. Friction factor in TTHC
TheinnerandouterFanningfrictionfactorsweredeterminedas
f
=
P
×
D
e
2
ρu
20
×
L
(25)where
L
is the length of the heat exchanger and
D
e
is the equiv-alent diameter for the inner and outer tubes and
u
0
is the inletvelocity for the inner and outer tubes, respectively. The fric-tion factor measurements in the present work were comparedwith the experimental data of Mishra and Gupta [24]. Mishraand Gupta’s experimental data were presented by the followingempirical correlation:
f
c
−
f
s
=
0
.
0075
√
λ
(26)

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