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International Journal of Heat and Mass Transfer 49 (2006) 2859–2863

Analytical function theory approach to the heat transfer problemof a cylinder in cross-flow at small Peclet numbers

Volfango Bertola a,*, Emilio Cafaro b

a University of Edinburgh, School of Engineering and Electronics, The King’s Buildings, Mayfield Road, Edinburgh EH9 3JL, United Kingdomb Politecnico di Torino, Dipartimento di Energetica, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Received 18 May 2005Available online 17 April 2006

Abstract

The theory of analytical functions is used to study the heat transfer from a uniformly heated cylinder with large length to diameterratio in cross-flow, in the limit of small Peclet numbers. The energy conservation equation is solved in Fourier’s space, and inverted bymeans of the residue theorem to obtain an analytical expression of the average Nusselt number in closed form. The result agrees withother theoretical solutions of the same problem existing in the literature.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Heat transfer between a cylinder and a fluid stream incross-flow is relevant in several practical applications.Examples are tube bundles in heat exchangers, but alsohot-wire anemometers, submarine pipelines, electric andtelecommunication cables, and many others. The heattransfer process is greatly affected by the quite complexhydrodynamic problem arising from the interactionbetween the cylinder and the flow, with the developmentof a boundary layer on the surface facing the flow and itssubsequent separation in the rear part of the cylinder [1].As it is well known, this leads to a non-uniform heat trans-fer coefficient [2], which is strongly influenced by the natureof boundary layer development.

From the standpoint of engineering calculations one ismore interested in overall average conditions, so that theaverage heat transfer coefficient on the cylinder surface isrequired. The overall heat transfer coefficient is usually cal-culated by means of empirical correlations, which expressthe average Nusselt number as a function of the Reynoldsand the Prandtl numbers of the fluid flow [3], and can be

0017-9310/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2006.02.016

* Corresponding author.E-mail address: v.bertola@ed.ac.uk (V. Bertola).

used in a given range of flow conditions. Unfortunately,uncertainties associated with the measurement of fluidvelocities, the estimation of boundary effects at the cylinderends, and the averaging of surface temperature, which var-ies both circumferentially and axially, render the experi-mental results accurate no better than 20–30%. As aconsequence, calculations relying on empirical correlationsare within the experimental uncertainty of the measure-ments on which they are based.

A case of special relevance in applications is that of uni-formly heated cylinders with a large length to diameterratio: examples are, among others, hot-wire anemometersand accidentally burning wires or cables. Following thepioneering work of King [4], this problem received muchattention by researchers from the experimental, the theo-retical and, more recently, the computational point of view.

Experimental studies generally show a large scatter ofmeasured data, both at high values of the Reynolds num-ber [4–7] and at smaller ones [8–10]. This is often a conse-quence of limitations in measurement accuracy, of theinappropriate treatment of the variation of fluid propertieswith temperature, of slip effects due to the small size ofwires and, in the range of very small Reynolds numbers(Re < 0.1), it is also a consequence of the effects of naturalconvection. Thus, analytical and numerical approaches are

Nomenclature

cp specific heat at constant pressure, J kg�1K�1

g gravity acceleration, m s�2

h convection heat transfer coefficient, W m�2 K�1

k thermal conductivity, W m�1 K�1

K0 modified Bessel function of second kindL length, mNu Nusselt’s number ðNuL ¼ hL

k ÞP pressure, PaPe Peclet’s number ðPeL ¼ qucpL

k ÞPr Prandtl’s number ðPr ¼ cpg

k ÞR cylinder radius, mt time, sT 0 temperature, KT dimensionless temperatureu fluid stream velocity, m s�1

w dimensionless heat rateW heat rate per unit length, W m�1

x dimensionless spatial coordinate parallel to theflow

X spatial coordinate parallel to the flow, my dimensionless spatial coordinate perpendicular

to the flow

Y spatial coordinate perpendicular to the flow, mz complex variableZ spatial coordinate parallel to the wire, m

Greek symbols

C Euler’s constantd(•) delta functiong viscosity, Pa sk transformed dimensionless spatial variable in

Fourier spaceq density, kg m�3

h Fourier transform of the dimensionless temper-ature

x transformed dimensionless spatial variable inFourier space

f real variable

Subscripts1 asymptotic conditions of the fluid streamf filmw wallR referred to the cylinder radius

2860 V. Bertola, E. Cafaro / International Journal of Heat and Mass Transfer 49 (2006) 2859–2863

very useful, because they allow isolating the various effectsand investigating them separately.

In particular, theoretical works addressed at first thesolution of the flow field around the cylinder at small Rey-nolds numbers [11,12], which provided the starting point tosolve the heat transfer problem [13–16]. These worksshowed that for small Peclet numbers, when diffusion dom-inates over advection, the details of the near-field becomeirrelevant and heat transfer depends on the characteristicsof the far-field. Extensive and detailed reviews on the sub-ject can be found in the recent literature [17,18].

Here, an approach based on the theory of analytical func-tions is proposed to obtain a mathematical solution inclosed form for the heat transfer coefficient of a cylinder incross-flow in the range 0 < Pe < 1. The result is completelyequivalent to other well-known theoretical solutions of thesame problem that can be found in the open literature.

2. Analysis

Let’s consider a linear heat source, such as a heatedwire, placed in a gas stream uniformly flowing in the X-direction, perpendicular to the source, with constant veloc-ity, u. The heat source is characterized by a linear heatemission density W (heat rate per unit length), and it isassumed to be aligned with the Z-axis, so that all parame-ters are Z-independent. Let’s assume that all processes aresteady and occur at constant pressure, and that the pres-ence of the heat source does not perturb streamlines, whichremain parallel to the X-axis. This assumption is equivalent

to neglecting the near-field, which is a reasonable one forsmall Peclet numbers.

The continuity equation allows one to write:

qu ¼ q1u1 ¼ const ð1Þwhere q1 and u1 are the asymptotic gas density and veloc-ity. If the gas has constant thermophysical properties, theenergy equation reduces to:

q1cpu1oT 0

oX¼ kr2T 0 þ W dðX ÞdðY Þ ð2Þ

where cp is the specific heat at constant pressure, k is thethermal conductivity coefficient, and d is the delta function.The average heat transfer coefficient between the cylinderand the gas stream, h, is given by the relation:

W ¼ 2phRðT w � T1Þ ð3Þwhere R is the radius of the wire, Tw is the temperature ofthe wire surface, and T1 is the asymptotic temperature ofthe gas.

The problem can be put into a dimensionless form byintroducing the following quantities:

x ¼ XR; y ¼ Y

R

T ¼ T 0 � T1T w � T1

w ¼ WkðT w � T1Þ

; NuR ¼hRk

PeR ¼q1u1cpR

kdðxÞ ¼ RdðX Þ; dðyÞ ¼ RdðY Þ

ð4Þ

Table 1Values of the modified Bessel function of second kind K0(f)

f K0(f) Nu = 2/K0(f)

0.00 1 00.05 3.1142 0.64220.10 2.4271 0.82400.15 2.0300 0.98520.20 1.7527 1.14110.25 1.5415 1.2974

V. Bertola, E. Cafaro / International Journal of Heat and Mass Transfer 49 (2006) 2859–2863 2861

Then, Eqs. (2) and (3) can be re-written as

oTox¼ 1

PeR

½r2T þ wdðxÞdðyÞ� ð5Þ

w ¼ 2pNuR ð6Þ

Taking the Fourier transform of the energy equation oneobtains

ikhðk;xÞ ¼ � k2 þ x2

PeR

hðk;xÞ þ w2pPeR

ð7Þ

where k and x are complex variables, and

hðk;xÞ ¼ 1

2p

Z þ1

�1dxZ þ1

�1T ðx; yÞ expð�ikx� ixyÞdy ð8Þ

Eq. (7) yields

hðk;xÞ ¼ w

2pðk2 þ ikPeR þ x2Þð9Þ

The temperature distribution in the physical space ofdimensionless coordinates x and y is obtained from Fourierinversion of Eq. (9)

T ðx; yÞ ¼ w4p2

Z þ1

�1dxZ þ1

�1

expðikxþ ixyÞk2 þ ikPeR þ x2

dk

¼ w4p2

Z þ1

�1expðixyÞdx

Z þ1

�1

expðikxÞk2 þ ikPeR þ x2

dk

ð10Þ

Eq. (10) can be integrated once over the k-variable bymeans of the residue theorem [19]. In particular, since theintegrand function f(k) satisfies Jordan’s lemma, we havethat for x > 0Z þ1

�1

expðikxÞk2 þ ikPeR þ x2

dk ¼ 2piX

k

Resff ðzÞ; kkg ð11Þ

where kk are the poles of f(k) contained in the upper-half ofthe complex plane, that is, Im(kk) > 0, and the residue isgiven by

Resff ðzÞ; kkg ¼ ðz� kkÞf ðzÞjz¼kkð12Þ

Analogously, for x < 0 we have that

Z þ1

�1

expðikxÞk2 þ ikPeR þ x2

dk ¼ �2piX

k

Resff ðzÞ; kkg ð13Þ

where Im(kk) < 0. The function f(k) has two first-orderpoles

k1;2 ¼ �iPeR

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2

R

4þ x2

s0@

1A ð14Þ

These poles are purely imaginary, and lay in the lower-and upper-half of the complex plane, respectively, so thatEqs. (10), (11) and (13) yield

T ðx; yÞ ¼

w4p

Rþ1�1 expðixyÞ

exp xPeR

2 �

ffiffiffiffiffiffiffiffiffiffiffiffiPe2

R4 þx2

q� �� �ffiffiffiffiffiffiffiffiffiffiffiffiPe2

R4 þx2

q dx x > 0

w4p

Rþ1�1 expðixyÞ

exp xPeR

2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiPe2

R4 þx2

q� �� �ffiffiffiffiffiffiffiffiffiffiffiffiPe2

R4 þx2

q dx x < 0

8>>>>>>>>><>>>>>>>>>:

ð15ÞFor x = 0, the temperature can be expressed as

T ð0; yÞ ¼ w4p

Z þ1

�1

expðixyÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2

R

4þ x2

q dx ð16Þ

Since on the wire surface we have that T(0, ± 1) = 1, Eq.(16) can be written as

2pw¼Z þ1

0

cosðxÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2

R

4þ x2

q dx ð17Þ

Integrating the right hand side of Eq. (17) yields a mod-ified Bessel function of second kind [20], so that

2pw¼ K0

PeR

2

� �ð18Þ

According to Eqs. (18) and (4), one finds the expressionof NuR

NuR ¼1

K0PeR

2

� � ð19Þ

Referring Nusselt’s and Peclet’s dimensionless groups tothe cylinder diameter, as it is standard practice in the liter-ature, Eq. (19) can be re-written as

Nu ¼ 2

K0Pe4

� � ð20Þ

Values of the function K0(f) and of the correspondingNusselt number are reported in Table 1.

3. Discussion

The validity of Eq. (20) is not generalized, but is obvi-ously related to the assumptions made during its deriva-tion. In particular, we have that:

Fig. 1. Comparison of the proposed solution for the Nusselt number (Eq.(20)) with the solutions of Cole–Roshko (Eq. (21)) and Nakai–Okazaki(Eq. (22)).

Fig. 2. Comparison between the proposed formula (Eq. (20)) andHilpert’s correlation (Eq. (23)), for different values of the Reynoldsnumber.

2862 V. Bertola, E. Cafaro / International Journal of Heat and Mass Transfer 49 (2006) 2859–2863

1. The problem was treated in two dimensions, with noboundary effects considered at the two ends of the cylin-der. Thus, the cylinder aspect ratio L/R cannot be toosmall (L/R� 10).

2. The thermophysical properties of the fluid are assumedto be uniform in space, whereas the fluid temperatureis a function of the spatial coordinates. To account forthe differences due to the temperature gradient, the fluidproperties should be evaluated at the so-called mean filmtemperature Tf = (Tw + T1)/2.

3. As mentioned above, assuming that streamlines are par-allel to the X-axis, therefore neglecting their deforma-tion in the vicinity of the cylinder surface, means thatdiffusion is the dominating mass transport mechanismin the flow around the cylinder: thus, the validity ofthe present approach is limited to small values ofPeclet’s number (0 < Pe < 1).

An assessment of Eq. (20) can be obtained from thecomparison with some theoretical solutions of the heattransfer problem at small Reynolds numbers existing inthe open literature. Cole and Roshko [13] applied theOseen–Lamb solution for the flow [11,12] to the thermalenergy equation (i.e., they linearized the inertia term ofthe flow equations at infinity), for small temperature differ-ences (constant fluid properties). They found the first termof an expansion series in [ln(Pe)]�1 for the Nusselt number

Nu ¼ 2

ln 8Pe

� �� C

ð21Þ

where C � 0.5772 is the Euler constant. Later, Nakai andOkazaki [16] matched two solutions for the temperaturefield, one corresponding to pure conduction in the vicinityof the cylinder and the other to a similarity solution forconvection in the far field. They obtained the followingexpression:

Nu ¼ 223þ ln 8

3

� �� lnðPeÞ

ð22Þ

In the proposed approach, the flow field in the vicinity ofthe cylinder is deliberately neglected, because its contribu-tion vanishes in the limit Pe! 0. This allows one to obtaina solution in closed analytical form (Eq. (20)), instead oftruncated solutions. Fig. 1 shows that in spite of thatapproximation Eq. (20) is in very good agreement with bothof these analytical solutions, and in practice cannot be dis-tinguished from the Cole–Roshko solution for Pe < 0.5.

As expected, all solutions converge to a same value forPe! 0, because in this case any differences in modellingthe flow field become irrelevant. When the Peclet numbergrows, the solutions shown in Fig. 1 diverge, as the flowfield is taken into account to different degrees ofapproximation.

It is also of interest the comparison of Eq. (20) withsome empirical correlations for cylinders in cross-flow.Here, we consider the well-known correlation proposedby Hilpert [5]:

Nu ¼ CRemPr1=3 ¼ CPe1=3Ren ð23Þwhere the values of C and m (or C and n) depend on theReynolds number, and the fluid properties are evaluatedat the film temperature Tf. Fig. 2 shows that, in the range0 < Pe < 1, the proposed correlation exhibits a similartrend as long as the flow is laminar (Re < 2000).

V. Bertola, E. Cafaro / International Journal of Heat and Mass Transfer 49 (2006) 2859–2863 2863

4. Conclusions

The heat transfer problem between a uniformly heatedcylinder with large length to diameter ratio and a fluidstream in cross-flow has been solved, in the limit of smallPeclet numbers (0 < Pe < 1), using the theory of analyticalfunctions. This approach allows one to obtain an exactsolution of the problem in a closed analytical form, whichreturns the average Nusselt number as a monotonicallygrowing function of Peclet’s number.

The proposed solution is in very good agreement withother theoretical solutions of the same problem existingin the literature, which are based on the analysis of the flowfield around the cylinder. Moreover, in the range of Pecletnumbers considered, the present result exhibits the sametrend as some well-known empirical correlations for theheat transfer coefficient of cylinders in cross-flow.

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