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International Journal of Thermal Sciences 100 (2016) 248e254

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

A compact closed-form Nusselt formula for laminar longitudinal flowbetween rectangular/square arrays of parallel cylinders with unequalrow temperatures

Hamidreza Sadeghifar a, b, c, *, Ned Djilali b, d, Majid Bahrami a

a Laboratory for Alternative Energy Conversion (LAEC), School of Mechatronic Systems Engineering, Simon Fraser University, Surrey V3T 0A3, BC, Canadab Institute for Integrated Energy Systems and Energy Systems and Transport Phenomena Lab (ESTP), University of Victoria, Victoria V8W 3P6, BC, Canadac Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, BC V6T 1Z3, Canadad Department of Mechanical Engineering, University of Victoria, Victoria V8W 3P6, BC, Canada

a r t i c l e i n f o

Article history:Received 6 April 2015Received in revised form29 September 2015Accepted 4 October 2015Available online 11 November 2015

Keywords:Nusselt numberHeat transfer coefficientLaminarParallel cylindersRectangular arrayUnequal cylinder temperatures

* Corresponding author. Laboratory for AlternativSchool of Mechatronic Systems Engineering, Simon0A3, BC, Canada. Tel.: þ1 (778) 782 8587.

E-mail addresses: sadeghif@sfu.ca, hsf@chhamidreza.sadeghifar@ubc.ca (H. Sadeghifar).

http://dx.doi.org/10.1016/j.ijthermalsci.2015.10.0041290-0729/© 2015 Elsevier Masson SAS. All rights re

a b s t r a c t

Axial flows over cylinders are frequently encountered in practice, e.g. in tubular heat exchangers andreactors. Using the Integral method, closed-form relationships are developed for heat transfer co-efficients or Nusselt number inside a fluid flowing axially between a rectangular/square array of parallelcylinders with unequal temperatures. The model considers the temperature variations of cylinders fromone row to another while assuming the same temperature for all the cylinders in each row. The modelcould well capture several sets of numerical data, which can be regarded as excellent in light of thesimplicity and comprehensiveness of the model. The compact and accurate formulae developed in thiswork can be readily employed, and also implemented into any software or tools, for the estimation of Nuin tubular heat exchangers, fins systems, porous media and composite manufacturing.

© 2015 Elsevier Masson SAS. All rights reserved.

1. Introduction

Heat transfer through parallel cylinders or tube assemblies is aproblem of considerable interest in a variety of industrial thermalapplications such as multi-tubular heat exchangers, fins, porousmedia and rod-bank generators, to name a few [1e3]. A general,easy-to-use and still accurate model that can predict the heattransfer coefficient under different operating conditions is essentialfor the modeling and design of such systems.

Considerable attempts have been made to study the heattransfer of a longitudinal flow between parallel cylinders. However,almost all of these studies are confined to either numerical solu-tions or asymptotic models developed for two limited cases: squareor triangular array of cylinders having the same temperatures.

e Energy Conversion (LAEC),Fraser University, Surrey V3T

be.ubc.ca, hsf@mail.ubc.ca,

served.

Table 1 summarizes all the studies performed on laminar-flow heattransfer to a fluid flowing axially between parallel cylinders.

To the authors' knowledge, and as shown in Table 1, the litera-ture lacks a model for estimating the heat transfer coefficient of alaminar flow inside a rectangular array of cylinders. Especially, nomodel or data is available for axial flow of a fluid between parallelcylinders with unequal row temperatures, which is indeed a morerealistic case in comparison to the case of equal temperatures of thecylinders. The aim of this study is to develop a general compactanalytic model for predicting the heat transfer coefficient of alongitudinal fluid flow passing through a rectangular array of par-allel cylinders with unequal row temperatures.

It should be noted that the determination of the exact temper-ature profile is not the final aim of this study. Herewe are interestedin finding a closed-form analytic relation for the prediction of theheat transfer coefficient. As a result, the integral method can beuseful, as it usually leads to compact, simple and sufficiently ac-curate relations, especially for estimating wall fluxes and averageprofiles [7e14]. In turn, we use the integral method as a powerfultechnique for obtaining approximate still reasonable solutions torather complex problems with remarkable ease. The basic idea is

Table 1Review on the works tailored for modeling the heat transfer by laminar axial flow between parallel cylinders: There is no model for the case of unequal cylinders temperaturesand/or for rectangular arrays of cylinders.

Author(s) & year Limitations & remarks

Array of cylinders Temperatures of cylinders Porosity Type of study

Szaniawski & Lipnicki (2008) [1] Square Equal Very high (>95%) Analytic (complex series form)Miyatake & Iwashita (1990) [2] Square Triangle Equal e NumericalAntonopoolos (1985) [4] Rectangle Equal wall heat flux e NumericalYang (1979) [5] Square Equal e NumericalSparrow et al. (1961) [6] Triangle Equal e Analytic, series form

H. Sadeghifar et al. / International Journal of Thermal Sciences 100 (2016) 248e254 249

that, from the physics of the system, we assume a general shape ofthe temperature profile. It must be noted that we are not interestedin the precise shape of the temperature profile but rather need toknow the heat flux and the average profile of the temperature overthe considered domain to calculate the heat transfer coefficient. Asmentioned earlier, estimating the average profile and the fluxvalues can be well performed by the integral method. The integralmethod has been successfully applied to several classical problemssuch as moving plate and boundary layer [7e14]. However, the useof this method to develop a heat transfer model for the fluid flowbetween parallel cylinders is a novel approach. In the followingsections, the model will be presented in a general form to be alsosuitable, with only minor changes, to other possible applicationssuch as catalytic reactors and the beds packed with cylindricalmaterials.

2. Model development

Fig. 1 shows a fluid flowing through parallel cylinders of diam-eter d and length L extended along the x-direction and spaced in arectangular array. The spacing between the cylinders centers is H inthe vertical (z) direction and is W in the horizontal (y) direction. Inthis model, contrary to available similar studies as listed in Table 1,W is not necessarily equal to H and these parameters can takedifferent values (H � W). In other words, the general case of lon-gitudinal flow through a rectangular array of parallel cylinders is

Fig. 1. Fully developed laminar flow between a rectangular array of parallel cylinders;the coordinate system (x,y,z), the fluid temperature and velocity (T∞ and u∞), thetemperatures of each row of cylinders (Twi: Temperature of all the cylinders in the ithrow) and the geometrical parameters W, L, H, and d are shown on figure (H � W).

considered. The model also assumes fully developed steady state,laminar (creeping) [4,6,14e17] incompressible flow. The physicalproperties are assumed to be constant, and dissipation, gravity andbuoyancy effects are negligible. The cylinders temperatures canchange from one row to another (Twi � Twi þ 1 where i denotes therow number) but are assumed to be the same in each row. Thetemperature and velocity of the fluid at the inlet is T∞ and u∞,respectively (Tw1 > T∞).

Fig. 2 shows the lateral and front views of a longitudinal flowbetween parallel cylinders and the spacing (d) between the upperand lower boundaries of the control volume considered:

d ¼

8>>>><>>>>:

H2

0 � y<W � d

2

H2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2

4��W2

� y�2

sW � d

2� y � W

2

9>>>>=>>>>;

(1)

The analytic expression of the velocity profile obtained bySparrow and Loeffler [18] shows that the velocity is almost uniformexcept for the area at the vicinity of the cylinders surfaces. Thispoint is also confirmed by the Fluent simulation results shown laterin the “Model verification” section. For this reason, the averagevelocity (u), which can be readily obtained from the mass flow rate[2,3], is used in the model derivation throughout this study.

The energy equation and the corresponding boundary condi-tions are (T(x ¼ 0,z) ¼ T∞):

uvTvx

¼ a

v2Tvz2

!(2)

T�x; z ¼ þd

� ¼ Tw1 (3)

T�x; z ¼ �d

� ¼ Tw2 (4)

Fig. 2. Front and lateral views of four parallel cylinders with the axial flow; the presentmodel considers the general case of rectangular (H � W) arrays of cylinders which canhave unequal row temperatures (Twi � Twi þ 1).

H. Sadeghifar et al. / International Journal of Thermal Sciences 100 (2016) 248e254250

where the average value of d (i.e., d) is used to eliminate the minor,

undesired dependency of d on y for the corners�W�d2 � y � W

2

�:

d ¼

8>><>>:

dM ¼ H2

0 � y<W � d

2

dC ¼ H2� pd

8W � d

2� y � W

2

9>>=>>; (5)

It should be noted that for the space in the range of0 < y < W � d/2, the temperature of the fluid on the upper andlower borders (edges) is generally not equal to the temperature ofthe adjacent cylinders (wall temperatures). As a result, Eqs. (3) and(4) are not accurate for this range of y and the value of the bordertemperature (subtracted from the wall temperature) must beadded to the right-hand sides of Eqs. (3) and (4), even though theborder temperature is negligible for packedmaterials. Here, at first,no border temperature assumption is made to be able to proceedwith the modeling. Later, the border temperatures are consideredto make the model as accurate as possible, especially for highlyporous materials.

In order to solve Eq. (6) with its boundary conditions (Eq. (3) and(4)) using the integral method, the model assumes a parabolicprofile for the temperature according to the physics of the problem:

Tðx; zÞ ¼ a0ðxÞ þ a1ðxÞzþ a2ðxÞz2 (6)

Solving the model's equations by using the integral method, thefinal form of the temperature profile is obtained as:

Tðx; zÞ ¼�Tw1 þ Tw2

2��Tw1 þ Tw2

2� T∞

�exp

�� 3ax

d2u

��

þ�Tw1 � Tw2

2

��zd

�þ�Tw1 þ Tw2

2� T∞

�

exp�� 3ax

d2u

��z2

d2

� (7)

It should be noted that the fluid temperature is mainly a func-tion of two variables x and z for the rectangular array of cylinderswith unequal row temperatures. The variation in the fluid tem-perature is approximated in the y-direction by dividing the spacebetween the four cylinders (the control volume) into two main andcorner blocks as shown in Fig. 3. In turn, considering these twoblocks allows accounting for the average temperature variations inthe y-direction. However, the temperature gradient at the cylinderwall is later estimated based on a similar quadratic temperatureprofile obtained for the z-direction.

Similarly to the border velocity profile defined in Refs. [14,15],we consider a border temperature profile based on the averagetemperature in the case of no border temperature (wall tempera-tures as boundary conditions) as follows (0 � y<W�d

2 (half of themain block)):

Tborðx; yÞ ¼�Tw1 þ Tw2

2

�þ�TðxÞ � Tw1 þ Tw2

2

��

12� 2yW � d

�gðεÞ

(8)

where TðxÞð ¼Z z¼þd

z¼�d

Tðx; zÞdzÞ=2dÞ is the average temperature in-

side the main block, given by:

TðxÞ ¼�Tw1 þ Tw2

2

�� 23

�Tw1 þ Tw2

2� T∞

�exp

�� 3ax

d2u

�(9)

And g(ε) shows the dependency on porosity (ε) [14,15], which isobtained from a linear interpolation between two extreme cases ofminimum (Tbor ¼ Tw) and maximum (Tbor ¼ T∞) possible porosities:

gðεÞ ¼ 1:274ε� 0:274 (10)

ε ¼ 1� pd24

WH(11)

In order to eliminate the weak, not important dependency ofTbor(x,y) on the undesired variable y, we can take an average fromthat over y to obtain TborðxÞ:

TborðxÞ ¼�Tw1 þ Tw2

2

�þ�TðxÞ � Tw1 þ Tw2

2

��gðεÞ2

�0 � y<

W � d2

(12)

Finally, one can reach the temperature profile and its average as:

TBðx; zÞ ¼

8>>><>>>:

Tðx; zÞ þ�TborðxÞ �

Tw1 þ Tw2

2

�0 � y<

W � d2

Tðx; zÞ W � d2

� y � W2

9>>>=>>>;

(13)

TBðxÞ ¼

8>>><>>>:

TðxÞ þ�TborðxÞ �

Tw1 þ Tw2

2

�0 � y<

W � d2

TðxÞ W � d2

� y � W2

9>>>=>>>;(14)

The average temperature of the entire block can be approxi-mated using:

TBðxÞ ¼

Z z¼þd

z¼�d

Z y¼W=2

y¼�W=2Tðx; zÞdydz

2dMðW � dÞ þ 2dCd(15)

The final form of the average temperature will then be:

TBðxÞ ¼

�Tw1þTw2

2

�� 2

3

�Tw1þTw2

2 � T∞

�exp

� 3ax

d2Mu

!�1þ gðεÞ

2

�

1þ dCdM

dW�d

þ

�Tw1þTw2

2

�� 2

3

�Tw1þTw2

2 � T∞

�exp

� 3ax

d2Cu

!

dMdC

W�dd þ 1

(16)

The local heat transfer coefficient can be obtained using:

h ¼�k�vTBðx;zÞ

vr

�z¼dC�

TBðxÞ � Tw1� (17)

The term�vTBðx;zÞ

vr

�z¼dC

in the above equation is mathematically

calculated as (r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ y2

p):

vTvr

¼ vTvz

vzvr

þ vTvy

vyvr

(18)

Fig. 3. 3D view of four parallel cylinder quarters considered for the modeling and the space between them considered as the control volume through which the fluid flows.

H. Sadeghifar et al. / International Journal of Thermal Sciences 100 (2016) 248e254 251

where the term vT/vz can be obtained from Eq. (7) and the terms vz/vr and vy/vr are:

vzvr

¼ zffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ y2

p (19)

vyvr

¼ yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ y2

p (20)

The variation of T along the y-directionwas approximated in thepresent model by defining two main and corner blocks. In order toaccurately estimate the T-y slope at the cylinder surface with Tw1,according to the physics of the problem and the analytic temper-ature profile obtained for the z-direction, a quadratic temperatureprofile similar to Eq. (7) is considered for the y-direction:

h ¼ �Dk

dCdM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2C þ Y2

C

q0BB@dMdCð1� FÞ

2A exp

� 3ax

d2Cu

!þ B

!þ d

2CF

2A exp

� 3ax

d2Mu

!þ B

!þ 2dMdCðTw1 � T∞Þexp

� 3ax

Y2Cu

!

dMðW � dÞ C � 4

3A exp

� 3ax

d2Mu

!�1þ gðεÞ

2

�!þ dCd

C � 4

3A exp

� 3ax

d2Cu

!!� Tw1D

1CCA (23)

Tðx; yÞ ¼"Tw1 � ðTw1 � T∞Þexp

� 3ax

Y2Cu

!þ ðTw1 � T∞Þ

exp

� 3ax

Y2Cu

! y2

Y2C

!# (21)

where YC ¼ W2 � pd

8 since�H2 � d

2

�� z � H

2. It should be noted that

the second term of the polynomial temperature has disappeared asthe temperatures of the cylinders are the same in each row.

The derivation of temperature with respect to y at the cylindersurface will then be:

vTvy

¼ ðTw1 � T∞Þexp

� 3ax

Y2Cu

! 2y

Y2C

!(22)

All the above derivations in Eq. (18) are evaluated at z ¼ dC andy ¼ YC.

Knowing�vTðx;zÞ

vr

�z¼dC

from the above calculations and TBtðxÞfrom Eq. (16), the local heat transfer coefficient is reached as aclosed-form relationship:

where constants A, B, C, D and F are given as below:

A ¼�Tw1 þ Tw2

2� T∞

�(24)

H. Sadeghifar et al. / International Journal of Thermal Sciences 100 (2016) 248e254252

B ¼�Tw1 � Tw2

2

�(25)

C ¼ Tw1 þ Tw2 (26)

D ¼ 2dMðW � dÞ þ 2dCd (27)

F ¼ W � dð Þ= Wεð Þ (28)

The Nusselt numbers can be obtained using:

Nu ¼ hdk

(29)

- Special case of equal cylinders temperatures: Twi ¼ Twiþ1 ¼ Tw

Assuming the same temperature for all the cylinders, the localheat transfer coefficient reduces to:

h ¼ �2DkðTw � T∞ÞdCdM

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2C þ Y2

C

q0BB@

dMdCð1� FÞexp

� 3axd2Cu

!þ�d2CF�exp

� 3ax

d2Mu

!þ dMdCexp

� 3ax

Y2Cu

!

2dMðW � dÞ Tw � 2

3 ðTw � T∞Þexp

� 3axd2Mu

!�1þ gðεÞ

2

�!þ 2dCd

Tw � 2

3 ðTw � T∞Þexp

� 3axd2Cu

!!� TwD

1CCA

(30)

3. Model verification

Using Fluent Ansys, the axial flow of air with the velocities of 0.5and 1 m/s is simulated for several rectangular array of cylinderswith different row temperatures as shown in Fig. 4. The modelpredictions are compared to the results (at x ¼ d, 7d and 14d) ofthese simulations in Fig. 5a and b for the model verification for thetwo velocities. All the fluid properties used for the model

Fig. 4. Temperature (K) and velocity (m/s) contours for the air flowing axially between foTw2 ¼ 342 K, u ¼ 1 m/s and T∞ ¼ 298 K. Due to the symmetry in the y-direction, half of th

verification are listed in Table 2. Fig. 5 shows that the model canwell capture the numerical results of the Nu in the fully-developedregion (see Fig. 4) with the maximum and average relative errors of34 and 16%, respectively. It should be noted that the model error atx ¼ d (the first numerical data shown in Fig. 5a) is 60%, which ismuch higher than the errors of 27 and 17% obtained for the othertwo points of x¼ L/2 and x¼ L (u¼1.0m/s,H¼ 5d), respectively. Thereason for this can be attributed to the fact that the point x ¼ d iswithin the entrance (non-fully developed) region of the flow. Forthe fully developed region, the agreement between the closed-formcompact formulae of the present study and the numerical data canbe regarded as excellent, especially in light of the simplicity andcomprehensiveness of the present model. Similar discussions canbe made for the other cases considered in Fig. 5.

The numerical data of Ref. [2] provided for the special case ofsquare (W ¼ H) array of cylinders with the same temperatures(Twi¼ Twiþ1¼ Tw) can also be used for the verification of the presentmodel. Fig. 6 shows that the model results are in reasonableagreement with the numerical data of Ref. [2] for pitch-to-diameter(PD ¼ W/d ¼ H/d) ratios of 2 and 4. For PD ¼ 1.5, the model couldprovide rough, still reasonable, estimations of the Nu. The reason

for such rough Nu estimation (at very low PD ratios) by the modelcan be attributed to the fact that for the packed arrangements(PD < 1.5), the velocity profile cannot be approximated to theaverage velocity. For such cases, the flowmay be approximated to a(internal) flow inside a duct or channel.

It should be noted that the analytic model of Ref. [1] developedfor the specific case of the square (W ¼ H) array of thin cylinders(very high porosities) with equal temperatures (Twi ¼ Twiþ1 ¼ Tw)cannot be used for comparison with the present model. The Nuresults reported in Ref. [1] shows some inconsistency with therealistic trends of Nu with porosity. For instance, with increasing

ur cylinders (cylindrical quarters): L ¼ 14d, W ¼ 2d, H ¼ 5d, d ¼ 8 mm, Tw1 ¼ 350 K,e space (0 � y � W/2) has been shown.

Fig. 5. Comparison of the present model (solid curves) with Fluent Ansys simulations(data points) for the two velocities of: (a) u ¼ 1.0 m/s and (b) u ¼ 0.5 m/s. The otherparameters are d ¼ 8 mm, L ¼ 14d, W ¼ 2d, H ¼ 3d, 5d and 10d, Tw1 ¼ 350 K,Tw2 ¼ 342 K, and T∞ ¼ 298 K.

Fig. 6. Comparison of the present model with the numerical results of Ref. [2] for aspecial case: d ¼ 1 mm, L ¼ 100d, W ¼ H ¼ 1.5d, 2d and 4d (square array of cylinderswith the same temperature of Tw ¼ 350 K), u ¼ 1 m/s and T∞ ¼ 298 K.

H. Sadeghifar et al. / International Journal of Thermal Sciences 100 (2016) 248e254 253

porosity, the model of Ref. [1] predicts a continuous increase in Nu,which is not consistent with the numerical data of Ref. [2].

4. Summary and conclusion

A compact analytic model was developed for predicting the heattransfer coefficient or Nusselt number for longitudinal laminar flow

Table 2Air thermo-physical properties used for the model validation.

Property Value Unit

k 0.024 W/m Km 1.7 � 10�5 Kg/m sr 1.2 Kg/m3

a 1.99 � 10�5 m2/s

passing through parallel cylinders. The model accounts for thesalient geometric parameters, the fluid thermo-physical propertiesand the operating conditions. Contrary to all the studies conductedon the longitudinal flow between parallel cylinders, the presentmodel considers the general case of rectangular arrays of parallelcylinders whose temperatures can vary from row to row. For suchgeneral complex case, the model can accurately predict the Nu forany porosity of the tube bank and for any spacings between thecylinders, except for the narrow range of 1 � W/d, H/d < 1.5 wherethe model gives approximate estimates of the Nu. The closed-formsimple formulae presented in this study can be readily used for avariety of heat exchangers and rod-generators where longitudinalflows through parallel tubes/pipes are encountered.

References

[1] Andrzej Szaniawski, Zygmunt Lipnicki, Heat transfer to longitudinal laminarflow between thin cylinders, Int. J. Heat Mass Transf. 51 (2008) 3504e3513.

[2] O. Miyatake, H. Iwashita, Laminar-flow heat transfer to a fluid flowing axiallybetween cylinders with uniform surface temperature, Int. J. Heat Mass Transf.33 (1990) 417e425.

[3] O. Miyatake, H. Iwashita, Laminar-flow heat transfer to a fluid flowing axiallybetween cylinders with uniform wall heat flux, Int. J. Heat Mass Transf. 34(1991) 322e327.

[4] K.A. Antonopoulos, Heat transfer in tube assemblies under conditions oflaminar axial transverse and inclined flow, Int. J. Heat Fluid Flow 6 (1985)193e204.

[5] I.W. Yang, Heat Transfer and Fluid Flow in Regular Rod Arrays with OpposingFlow, US Department of Energy, 1979.

[6] E.M. Sparrow, A.L. Loeffler, H.A. Hubbard, Heat transfer to longitudinal laminarflow between cylinders, J. Heat Transf. 83 (1961) 415e422.

[7] M.N. Ozisic, Boundary Value Problem of Heat Conduction, InternationalTextbook Company, Scranton, Pennsylvania, 1968.

[8] H. Schlichting, Boundary-layer Theory, seventh ed., McGraw-Hill, New York,1979.

[9] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, second ed., JohnWiley & Sons, NJ, 2002.

[10] F.P. Incropera, P.D. Dewitt, Fundamentals of Heat and Mass Transfer, fifth ed.,John Wiley & Sons, NJ, 2006.

[11] W. Kays, M. Crawford, B. Weigand, Convective Heat and Mass Transfer, fourthed., McGraw-Hill, NY, 2005.

[12] L.C. Burmeister, Convective Heat Transfer, second ed., Wiley-Interscience, NY,1993.

[13] A. Bejan, Convection Heat Transfer, second ed., Wiley-Interscience, 1994.[14] A. Tamayol, M. Bahrami, Analytical determination of viscous permeability of

fibrous porous media, Int. J. Heat. Mass Transf. 52 (2009) 2407e2414.[15] A.A. Kirsch, N.A. Fuchs, Studies on fibrous aerosol filters e II. Pressure drops in

systems of parallel cylinders, Ann. Occup. Hyg. 10 (1967) 23e30.

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[16] R. Wiberg, N. Lior, Heat transfer from a cylinder in axial turbulent flows, Int. J.Heat Mass Transf. 48 (2005) 1505e1517.

[17] O.R. Tutty, Flow along a long thin cylinder, J. Fluid Mech. 602 (2008) 1e37.[18] E.M. Sparrow, A.L. Loeffler Jr, Longitudinal laminar flow between cylinders

arranged in regular array, AIChE J. 5 (1959) 325e330.

Nomenclature

A, B, C, D, F: constants defined by Eqs. (24)e(28), respectivelya0, a1, and a2: coefficients in the integral methodcp: heat capacity of fluid J/Kg Kd: cylinder diameter mGz: Graetz number ¼ rucpd

2

44ðPDÞ2�p

k x -h: (local) heat transfer coefficient W/m2 KH: spacing between cylinders centers in z-direction mk: fluid thermal conductivity W/m KL: cylinder length mNu: (local) Nusselt number (¼hd

k ) -PD: pitch-to-diameter ratio (¼W/d ¼ H/d) for square array of cylinders -r: radial variable (¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2 þ y2

p) m

T: fluid temperature KT: average fluid temperature KTbor: border temperature (defined for the main block) KTw: cylinder (wall) temperature KTw1: A constant temperature KTw2: A constant temperature KTwi: temperature of all cylinders in ith row KTwiþ1: temperature of all cylinders in (iþ1)th row Ku: fluid velocity m/su: average fluid velocity m/su∞: velocity of free stream of fluid m/s

W: spacing between cylinders centers in the y-direction mx: coordinate system variable along the flow directiony: coordinate system variablez: coordinate system variable

Greek

m: fluid viscosity Kg/m sa: fluid thermal diffusivity m2/sd: half of the spacing between the upper and lower boundaries of the control volume

md: half of the average spacing between the upper and lower boundaries of the control

volume mdC : d in the corner block (¼H

2, see Eq. (5)) mdM: d in the main block (¼H

2 � pd8 , see Eq. (5)) m

ε: porosityr: fluid density Kg/m3

Y: half of the spacing between the left and right boundaries of the control volume mYC: half of the average spacing between the left and right boundaries of the control

volume in the corner block�

¼ W2 � pd

8

�) m

Subscript

∞: free streami: ith row of cylinders (i ¼ 1,2,3, …)B: entire blockbor: border temperatureC: corner part of the blockM: main or middle part of the blockw: wall