thermally developing flow in finned double-pipe heat exchanger

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids 2011; 65:1145–1159Published online 10 February 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2232

Thermally developing flow in finned double-pipe heat exchanger

Mazhar Iqbal1,∗,† and K. S. Syed2

1College of E&ME, (DBS&H), National University of Science and Technology, Islamabad, Pakistan2CASPAM, Bahauddin Zakariya University, Multan, Pakistan

SUMMARY

A numerical solution of the convective heat transfer in the thermal entry region of the finned double-pipeis carried out for the case of hydro-dynamically fully developed flow when subjected to uniform walltemperature boundary condition. Adaptive axial grid size is used in order to cater for the variation of largesolution gradients in the axial direction. It has been observed that the thermal entrance region is highlyeffective and there is a substantial enhancement in the heat transfer coefficient. A maximum of 76.4877%increase has been observed in the thermal entrance region as compared with the fully developed regionfor 24 fins and H∗ =0.6 when R=0.25, whereas for R=0.5 the maximum increase is 75.0308% forthe same number of fins of same height. It has been observed that no geometry consistently performbetter throughout the entrance region. However, the geometries that have optimal performance in the fullydeveloped region perform better in the developing region on average terms. Results show that the Nusseltnumber and the thermal entrance length are dependent upon various geometrical parameters such as ratioof radii of the inner and the outer pipe, fin height and the number of fins. The limiting case results matchwell with the literature results. This validates our numerical procedure and computer code. Copyright q2010 John Wiley & Sons, Ltd.

Received 17 October 2008; Revised 18 September 2009; Accepted 8 November 2009

KEY WORDS: heat transfer; Nusselt number; entrance length; entrance region; fully developed flow;thermally developing flow

INTRODUCTION

Convective heat transfer occurs in a variety of engineering problems such as space heating, air-conditioning, power production, chemical processing, and many other engineering phenomena,and is of particular importance where viscous fluids are heated or cooled. Since the heat transfer inthese types of fluids is generally low, therefore, there is a need for augmentation. One of the mosteffective methods of enhancing the convective heat transfer is the use of extended surfaces. Thelongitudinal finned tubes have been a widely used geometry. There is literature evidence indicativeof the fact that a significant level of heat transfer enhancement may be achieved through thistype of finned tube as compared with the plane tubes. Masliyah and Nandakumar [1] studied theheat transfer characteristics for laminar forced convection of fully developed flow in an internallyfinned circular tube with axially uniform heat flux and peripherally uniform temperature usingfinite element method. They have reported that the influence of the fin height is pronounced forits large values. Sparrow and Charmchi [2] studied the heat transfer in the externally finned tubeand reported a substantial heat transfer enhancement as compared with smooth tube. Zeitoun and

∗Correspondence to: Mazhar Iqbal, College of E&ME, (DBS&H), National University of Science and Technology,Islamabad, Pakistan.

†E-mail: [email protected]

Copyright q 2010 John Wiley & Sons, Ltd.

1146 M. IQBAL AND K. S. SYED

Hegazy [3] carried out the analysis of fully developed convective heat transfer in the internallyfinned tube with uniform outside wall temperature. They reported that the heat transfer in aninternally finned tube is mainly caused by the surfaces of the fins, especially when heights of thefins are long. The contribution of the pipe surface is very small when compared with that of the finsurface. Hu and Chang [4] studied the laminar forced convection problem subject to uniform axialand peripheral heat fluxes and reported an optimum configuration for heat transfer enhancement.

Double-pipe configurations have also been a well-probed geometry. Yu et al. [5] carried outthe experimental study in the annulus. They have used wave-like longitudinal fins in the annulus.Simulation of the turbulent flow in the annulus sector duct has been carried out by Li et al. [6]. Theyhave reported significant enhancement in the heat transfer results. In most of the studies quotedabove, fins are taken as longitudinal. Other fin shapes have also been used in the studies [7, 8].Both studies show the heat transfer enhancement as compared with the smooth tube. Geometriesother than the circular pipe have also been investigated by researchers [9, 10]. In the study of afinned annulus with external circular fins on the inner tube, Agrawal and Sengupta [11] showedthat the use of fins may not be justified because of a substantial rise in the pressure drop at Prandtlnumbers less than 2. The effects of the duct shapes on the heat transfer results have been studiedby Emin Erdogan and Erdem Imark [12]. They considered circular, semicircular, rectangular andparallel plate ducts and have shown that the Nusselt number depends upon the shape of the duct.A comprehensive review of the literature related to duct heat transfer is given in [13].

The investigations [1–11] have been carried out in the fully developed region. Researchershave shown that the heat transfer performance in the developing region (entrance region) is muchbetter than that in the fully developed regions. The studies carried out in [14, 15] provide theevidence. Rustam and Soliman [15] considered the thermally developing flow in the entranceregion of a tube with longitudinal internal fins. They reported that the rate of decrease of the localNusselt number was identical for all the finned geometries near the inlet. However, as the flowmarched downstream they observed sharp change in this rate for the large number of mediumheight fins. They have attributed this behaviour to complex velocity distribution and the irregularshape of the wall–fluid interface where the heat transfer takes place. Prakash and Liu [16] carriedout a numerical study of the forced convection in the internally finned tube. They reported thatheat transfer coefficient in the entrance region is large, which asymptotically approaches to fullydeveloped value at a large axial distance. The fins considered in [15, 16] are longitudinal withzero-thickness. Yao [17, 18] analytically carried out the study of free and forced convection inthe entrance region of a heated straight pipe. Padmanabhan [19] studied the entry flow in heatedcurved pipes. He observed that buoyancy disturbs the secondary motion induced by the curvature.Lin et al. [20] considered the laminar flow in the entry region of the annular sector duct. Theysummarized that the thermal entrance length decreases with the increase in the apex angle andthe ratio of radii. Present study is aimed to predict, numerically, the heat transfer characteristicsin thermally developing region through the finned double-pipe (FDP) subject to thermal boundarycondition of uniform wall temperature at the inner pipe and the fins. Axial heat conduction in fluidis relatively important only if the values of the wall conductance parameter and the Peclet numberare low [21, 22]. Nesreddine et al. [23] have mentioned that the axial diffusion of heat can beneglected for Pe>10 when the wall heat flux is uniform and for Pe>50 when the wall temperatureis uniform.

MATHEMATICAL FORMULATION

The heat transfer system comprises two concentric pipes with longitudinal fins, of zero-thickness,distributed uniformly around the outer surface of the inner pipe. The cold fluid is flowing in thefinned annular region. The fluid is viscous, incompressible, Newtonian and has constant properties.Viscous dissipation is negligible. All the body forces are negligible. The only driving force isthe pressure gradient in the axial direction. The flow is assumed to be steady, laminar, hydro-dynamically fully developed and thermally developing. The assumption of hydro-dynamically fully

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids 2011; 65:1145–1159DOI: 10.1002/fld

THERMALLY DEVELOPING FLOW IN FDP HEAT EXCHANGER 1147

Figure 1. Cross-section of the finned double pipe.

Figure 2. The computational domain.

developed flow allows the velocity profile to be taken as fully developed that remains invariant in theflow direction, whereas the temperature profile develops and varies in the axial direction. The axialconduction in wall and fluid is neglected. An adiabatic thermal condition is imposed at the outerpipe. A cross-section of the geometry under consideration is shown in Figure 1. The geometricalsymmetry requires the problem to be solved in the computational domain shown in Figure 2.

In the wake of the assumption of hydro-dynamically fully developed flow, we can calculate thefully developed velocity profile separately. The fully developed velocity profile u is obtained bysolving the momentum equation given as follows:

�2u�r2

+ 1

r

�u�r

+ 1

r2�2u

��2= 1

�

dp

dz(1)

The boundary conditions are

u=0 at r =ri, 0��

u=0 at r =ro, 0��



u=0 at �=�, ri�r�r1

�u�r

=0 at �=0, ri�r�ro

�u�r

=0 at �=�, r1�r�ro

The energy equation governing the thermally developing flow, neglecting axial conduction, isgiven by

�2T�r2

+ 1

r

�T�r

+ 1

r2�2T

��2= 1

�udT

dz(2)

where � is the thermal diffusivity. The energy equation (2) is solved on the computational domain,shown in Figure 2, subject to the initial and boundary conditions given as follows:

T =Te at z=0

T =Tw at r =ri, 0��

T =Tw at �=�, ri�r�r1

�T��

=0 at �=0, ri�r�ro

�T��

=0 at �=�, r1�r�ro

�T�r

=0 at r =ro, 0��

The above equations are made dimensionless by the following transformations:

�= T −TeTw−Te

, R= r

r0, R= ri

ro, u∗ = u

umax, z∗ = z

DhRePr

where

umax=− 1

4�

dp

dzr2o {1− R2+2R2 ln R}, R=

√1− R2

2 ln(1/R)

and Dh is the hydraulic diameter.The local Nusselt number averaged over the circumference at any axial location is defined

as [16]

Nu(z)=dQ(z)

dzPh(Tw−Tb)

Dh

�(3)

where Ph is the wetted perimeter, � is the thermal conductivity, Q(z) is the heat transfer up to thedistance z and is given by the relation Q(z)= mcp(Tb−Te). In this relation cp is the specific heatcapacity of fluid.

We may define thermal entrance length, L th, to be the distance from the thermal inlet to theaxial location where the local Nusselt number defined by Equation (3) becomes 1.05 times itsvalue for the fully developed flow [24].



NUMERICAL PROCEDURE

The dimensionless equations governing the thermally developing flow are solved by finite differencemethod using a fully implicit backward time central space discretization scheme [25]. The schemeis unconditionally stable and easy to implement. The assumption of hydro-dynamically fullydeveloped flow implies that there will be no flow reversal in the axial direction that allows theusage of BTCS discretization scheme conveniently. The method also allows taking large marchingstep size. Resulting differential algebraic equations were solved using Successive Over Relaxation(SOR) method with the over relaxation parameter determined by the method described in [26].Solution was computed for a number of geometries using different mesh sizes and based on theresults it was decided to use a grid of 41×21 (radial × angular) grid points in the cross-streamplane. Both the developing and the fully developed results were obtained using the same gridlayout.

It has been observed that heat transfer rate is very high in the vicinity of inlet plane, whichbecomes gradual as the fully developed region is approached. The rate at which this transitionoccurs is different for different compositions of the finned annulus. Therefore, taking same axialstep size for all geometries in the marching solution as has been done by Rustum and Soliman [15]and Prakash and Liu [16] is not a very good option. In the finned geometries, the axial step sizeshould be adaptive to the solution gradients. We have taken the initial axial step size as 1.0E−7and controlled the increment in it with the help of a solution monitor defined as

‖�z�z‖2It was also observed that axial temperature gradient is higher in the case of smaller ratio of radii.Therefore, the maximum step size allowed for a particular geometry was taken as 1.0E−2× R.

The estimation of thermal entrance length, L th, requires the velocity and the temperature distri-butions to be known for the fully developed flow. These fully developed flow results may beobtained either by carrying out the marching of solution sufficiently far downstream or by directlysolving the governing equations of the fully developed flow. In this study, we have followed thefirst approach for obtaining the fully developed temperature profile. Therefore, the solution of theenergy equation is marched till the axial variation in the local Nusselt number becomes 1.0E−7.The value of Nusselt number at this stage was taken as the fully developed value. The fully devel-oped velocity profile is calculated by solving Equation (1) discretized using central differencesand resulting equations solved by the SOR method.

VALIDATION

To validate our numerical procedure and the computer code developed for the thermally developingregion, Nu and L th have been computed for the finless annulus with different ratios of radii.Figure 3 shows the development of Nusselt number for different geometries in the thermal entranceregion of the finless annulus. The markers on the Nu curves are the numerical values of the Nusseltnumber given in Table 3.39 of [24]. It can be seen that the agreement is excellent. For furtherstrengthening the validation of our results, we have also computed fully developed Nusselt numberand compared it with two sources of literature results. Table I gives the comparison of the presentfully developed Nusselt number and thermal entrance length calculated for the finless annulus withthe corresponding results present in Tables 3.39 and 3.41 of [24]. Again the agreement is excellent.

Table II shows a comparison of the present fully developed Nusslet number with that of anotherliterature source [27]. The results are bit higher in the present case as compared with those of[27]. This is because we have marched our results down from higher values in the developingregion to the lower values in the fully developed region, whereas the results given in [27] havebeen computed by solving the model representing fully developed flow. It also indicates that wehave stopped the marching process in our computation a little bit earlier before the fully developedregion starts.



Figure 3. Comparison developing Nu in finless annulus with literature.

Table I. Comparison of finless annulus results with literature.

Nu Entrance length

R Present Literature Present Literature

0.05 17.519 17.460 0.024383 0.024290.1 11.572 11.560 0.025711 0.025580.25 7.373 7.371 0.027730 0.27200.5 5.742 5.738 0.029100 0.02829

Table II. Comparison of fully developed Nusselt number with literature.

R=0.25 R=0.5

N H∗ Present Syed [27] Present Syed [27]6 0.2 3.4679 3.4869 3.9573 3.9226

0.4 2.9579 3.0287 3.4779 3.48160.6 3.3480 3.2743 3.4368 3.34680.8 3.4246 3.3642 3.0928 3.03451 2.8202 2.8189 2.5937 2.5929

12 0.2 2.0946 2.0841 2.9566 2.94120.4 1.5331 1.5589 2.4907 2.54430.6 1.7749 1.7397 2.9877 2.94150.8 2.8522 2.8553 3.5156 3.47781 2.9151 2.9128 3.0446 3.0441

18 0.2 1.3950 1.3671 2.2987 2.27690.4 0.9124 0.9145 1.7677 1.79880.6 0.9873 0.9565 2.1451 2.11340.8 1.8949 1.9279 3.4339 3.43911 2.8234 2.8208 3.3516 3.3509

24 0.2 0.9951 0.9619 1.8349 1.80390.4 0.6039 0.5977 1.2891 1.29990.6 0.6191 0.5924 1.4899 1.4590.8 1.2107 1.2357 2.8968 2.94281 2.7210 2.7172 3.4978 3.4988



RESULTS AND DISCUSSION

These investigations are carried out in the finned annulus for a combination of geometrical param-eters such as, the number of fins, N , the fin height relative to the annulus, H∗, and the ratio ofradii of inner and outer pipes, R. The numerical computations are made for the following rangeof these parameters:

R=0.25,0.5, H∗ =0.2,0.4,0.6,0.8,1.0, N =6,12,18,24

The local Nusselt number and the thermal entrance length, for the complete range of values ofthe parameters considered above, are computed for the thermally developing flow using energyequation (2). For a given geometry of the finned annulus, the solution is marched far enough in theaxial direction until the local Nusselt number reached 1.05 times its corresponding fully developedvalue. The magnitude of Z∗ at this location is taken as the thermal entrance length.

We now present Nu in the thermally developing region in tabular and graphical forms andinvestigate the influence of the parameter R on Nu from the tabular results and that of H∗ and Nfrom graphical results. Tables III–VI provide numerical values of the circumferentially averagedNusselt number in the thermally developing region corresponding to different values of the finheight and number of fins. As a general trend, Nu attains very high values near the inlet planeand decreases monotonically along the axial direction. This phenomenon is expected becausethe development of thermal boundary layer in the axial direction has a negative impact on theheat transfer at solid–fluid interface, thus reducing the heat transfer coefficient. A comparison ofTables III–IV with Tables V–VI shows that the values of Nu are higher for R=0.25, as comparedwith its values for R=0.5, in the vicinity of the inlet plane. However, as the thermal boundary layerdevelops and the thermally developed region is approached the trend reverses and the geometriescorresponding to R=0.5 become thermally more efficient as compared with those for whichR=0.25 except for few cases. These observations indicate that in the vicinity of the inlet planegeometries corresponding to smaller R are better, whereas in the vicinity of fully developed regionthose for larger R are better with few exceptions. The same trend can be traced in the results ofSyed [27] given in Table II. This behaviour can be explained in view of the actual length of thefins and the mean velocity given in Table VII. The physical length of the fins is more for thegeometries corresponding to R=0.25 as compared with those for R=0.5. Since the fins are 100%efficient, therefore, in the case of geometries corresponding to R=0.25, the heat is transferred toa larger part of the fluid and in the presence of large temperature gradients near the inlet planethe heat transfer rate is higher. As the flow marches along the axial direction, thermal boundarylayer starts developing on the inner pipe and along the fins. The narrow inter-fin gap near theinner-pipe surface, for the case of smaller value of R, is quickly filled with the thermal boundarylayer thus reducing the efficiency of inner-pipe surface and the lower part of the fins. This situationprevails throughout in the fully developed region. Moreover, since the only mode of heat transferconsidered here is forced convection, the mean velocity is the driving agent for heat transfer. Thefact that the mean velocity is consistently higher for larger R than that for the smaller one indicatesbetter performance of larger R in the fully developed region. Thus, we conclude that when wall–finassembly is assumed to be 100% efficient, smaller ratio of radii gives better performance in thevicinity of inlet plane whereas near the fully developed region, the larger one is better.

Now we study the effect of H∗ on Nu in the entrance region. Figures 4 and 5 show the Nucurves in the entrance region for R=0.25, 0.5, respectively when N =18, on the logarithmic scale.Nu is very high for smaller fin height, and as the fin height increases Nu decreases sharply nearthe inlet. Nu curves cross each other in a complicated fashion within a very brief interval and thenthe fins of height 0.6 and higher become dominant as far as the heat transfer rate is concerned.Nu decreases at almost the same rate before and after the crossing interval and then the decreasebecomes gradual and Nu asymptotically approaches its fully developed value. This means as theflow marches along the axial direction, geometries corresponding to different fin heights gain



Table III. The variation of Nu in the entrance region for R=0.25 and N =6.

H∗

Z∗ 0.2 0.4 0.6 0.8 1.0

1.0E−07 199.3667 173.6153 117.9218 99.1489 67.16935.0E−07 144.2187 135.1721 104.0907 91.2001 64.87651.1E−06 105.1447 104.2794 90.5554 82.4307 61.73945.1E−06 57.6799 60.8352 63.1517 62.0579 52.42401.1E−05 44.2423 46.1075 49.8241 50.2781 45.57455.4E−05 25.0981 26.1940 29.1462 29.1923 28.19821.1E−04 19.8226 20.6845 23.0531 23.1482 21.57755.4E−04 11.4078 11.8280 13.2303 13.3035 12.03831.1E−03 9.1652 9.4386 10.5989 10.6859 9.60125.0E−03 5.6396 5.6055 6.4117 6.5380 5.71071.0E−02 4.6808 4.4879 5.2084 5.3398 4.59355.0E−02 3.5508 3.0899 3.6043 3.7203 3.1134∞ 3.4679 2.9579 3.3480 3.4246 2.8202

Table IV. The variation of Nu in the entrance region for R=0.25 and N =12.

H∗

Z∗ 0.2 0.4 0.6 0.8 1.0

1.0E−07 229.5827 227.7043 213.7034 193.3295 85.46695.0E−07 118.1900 122.6737 132.1682 138.9741 79.91981.1E−06 83.1097 85.8884 95.2577 105.6470 73.51615.1E−06 47.5323 50.0725 56.8782 61.8287 57.94151.1E−05 36.0050 38.3724 44.4501 47.8272 48.31755.4E−05 19.7577 21.6943 26.1886 28.5209 28.11631.1E−04 15.2357 16.9418 20.7033 22.7892 21.77275.4E−04 8.2142 9.2060 11.7967 13.1709 12.43571.1E−03 6.3124 7.0729 9.3673 10.5803 9.94535.0E−03 3.6723 3.5665 5.2466 6.4054 5.98451.0E−02 3.0171 2.6409 3.9449 5.1617 4.83245.0E−02 2.1964 1.6700 2.0827 3.4025 3.2966∞ 2.0946 1.5331 1.7749 2.8522 2.9151

Table V. The variation of Nu in the entrance region for R=0.5 and N =6.

H∗

Z∗ 0.2 0.4 0.6 0.8 1.0

1.0E−07 178.5653 159.3895 70.2418 63.4591 55.73595.0E−07 144.4353 128.8753 68.7892 62.4814 55.06191.1E−06 110.2351 100.7697 66.4366 60.8466 53.90615.1E−06 63.5450 62.9904 57.4626 54.1125 48.78201.1E−05 49.2642 49.3187 49.8764 47.9453 43.68565.4E−05 28.0901 27.8825 30.3626 30.0933 28.01181.1E−04 22.4021 22.0232 23.3992 23.1504 21.71575.4E−04 12.9412 12.6806 13.2357 13.0061 11.87951.1E−03 10.4185 10.1255 10.5967 10.4326 9.45715.3E−03 6.3004 5.9801 6.2896 6.1744 5.47321.0E−02 5.2679 4.9114 5.1626 5.0189 4.41225.0E−02 4.0305 3.5915 3.6512 3.3803 2.9072∞ 3.9573 3.4779 3.4368 3.0928 2.5937



Table VI. The variation of Nu in the entrance region for R=0.5 and N =12.

H∗

Z∗ 0.2 0.4 0.6 0.8 1.0

1.0E−07 206.6035 200.6661 148.7769 135.1183 64.04335.0E−07 141.0727 142.3032 118.9388 115.9848 62.98111.1E−06 100.4523 102.2874 93.3398 95.8673 61.21725.1E−06 55.6513 57.0679 57.6089 58.9109 53.93921.1E−05 42.8316 43.9120 46.2148 46.0760 47.16595.4E−05 24.2441 25.1640 28.5056 28.6981 28.01451.1E−04 19.1075 19.8221 22.5399 23.0144 21.27295.4E−04 10.7918 11.2148 12.7629 13.0959 12.04761.1E−03 8.5850 8.8885 10.2026 10.5067 9.60755.3E−03 5.0414 5.0646 6.0121 6.3286 5.64131.0E−02 4.1566 4.0432 4.9089 5.2398 4.61065.0E−02 3.0497 2.6542 3.3303 3.8068 3.2821∞ 2.9566 2.4907 2.9877 3.5156 3.0446

Table VII. Actual height l∗ of fins and the mean velocity u for different geometries.

R=0.25 R=0.5

N H∗ l∗ u l∗ u

6 0.2 0.15 0.513 0.1 0.5920.4 0.3 0.361 0.2 0.4920.6 0.45 0.268 0.3 0.4250.8 0.6 0.232 0.4 0.3991 0.75 0.227 0.5 0.396

12 0.2 0.15 0.446 0.1 0.5320.4 0.3 0.260 0.2 0.3650.6 0.45 0.147 0.3 0.2590.8 0.6 0.099 0.4 0.2161 0.75 0.092 0.5 0.211

18 0.2 0.15 0.417 0.1 0.4870.4 0.3 0.225 0.2 0.2940.6 0.45 0.108 0.3 0.1770.8 0.6 0.057 0.4 0.1291 0.75 0.049 0.5 0.122

24 0.2 0.15 0.403 0.1 0.4570.4 0.3 0.208 0.2 0.2550.6 0.45 0.090 0.3 0.1350.8 0.6 0.038 0.4 0.0851 0.75 0.030 0.5 0.078

and lose efficiency at various locations in the entrance region resulting into unwieldy crossing ofNu curves.

We, now, attempt to justify this behaviour of Nu physically. In general, Nu falls due to thegrowth of thermal boundary layer. Since the rate of growth may not remain uniform throughoutthe entrance region, therefore, the rate of fall of Nu is not constant. The efficiency of the shorterfin at the inlet may have been due to the reason that for this case the whole surface of the finand that of the inner pipe is efficient due to zero-thickness of the thermal boundary layer and theadditional length of a higher fin is not, on the average, as efficient as the whole heated surfacecorresponding to the shorter fin geometry. However, swift development of the thermal boundarylayer causes rapid deterioration in the performance of shorter fin geometry. Since the average



Figure 4. Local Nusselt number in the developing region for R=0.25, N =18.

Figure 5. Local Nusselt number in the developing region for R=0.5, N =18.

heat transfer coefficient at a given cross-section not only depends on the thickness of the thermalboundary layer but also on the interaction of the thermal boundary layer with the velocity field,the non-monotonic dependence of the friction factor on the fin height as reported by Syed [27]causes non-monotonic dependence of Nu on it. This leads to the conclusion that for a given valueof R and N , there is no fin height that consistently performs better than the others throughout theentrance region. However, Tables III–VI indicate that as the fully developed region is approached,geometries that are optimal in the fully developed region (see Table II) start dominating. Thisreflects the consistency of the behaviour of present results with the literature results of the fullydeveloped flow [27].

Now we study the effect of N on local Nusselt number, Nu. Figures 6 and 7 show the variationof Nu, in the entrance region for H∗ =0.4. In Figure 6, for R=0.25, the value of Nu is higher nearthe inlet for greater number of fins. Nu curves cross each other for a very brief interval and the



Figure 6. Local Nusselt number in the developing region for R=0.25, H∗ =0.4.

Figure 7. Local Nusselt number in the developing region for R=0.5, H∗ =0.4.

direct relation between Nu and N transforms into an inverse relation and the curves correspondingto smaller number of fins become dominant. As the fully developed region is approached, the orderof performance of different values of N starts matching with that reported for the fully developedflow in the literature [27]. Figure 7 indicates that Nu curves observe almost identical behaviourfor R=0.5. However, the transition interval shifts forward from the inlet plane as R increases. Inboth the figures Nu approaches its fully developed value at a sharp rate of decrease in the beginningand after the brief interval, in which curves cross each other, the decrease becomes gradual.

We have noted that at the inlet geometry with lager N is better whereas away from the inletsmaller N gives better performance and near the fully developed region, conditions of fullydeveloped flow prevail. The fact that the lager number fin is efficient at the inlet may be due tothe increased heated surface area with zero-thickness of the thermal boundary layer. This leads tothe conclusion that for a given value of R and H∗, there is no N that consistently performs betterthan the others throughout the entrance region.



Table VIII. Comparison of Nu in the thermally developing region with thefully developed Nu given by Syed [27].

R=0.25 R=0.5

N H∗ Nu Nu [27] Inc (%) Nu Nu [27] Inc (%)

6 0.2 4.6783 3.4869 34.1693 5.3714 3.9226 36.93430.4 4.1628 3.0287 37.4465 4.7869 3.4816 37.49090.6 4.5607 3.2743 39.2890 4.6811 3.3468 39.86750.8 4.5881 3.3642 36.3797 4.2673 3.0345 40.62611 3.8104 2.8189 35.1724 3.5954 2.5929 38.6621

12 0.2 2.8505 2.0841 36.7739 4.0720 2.9412 38.44780.4 2.2742 1.5589 45.8819 3.5785 2.5443 40.64620.6 2.8077 1.7397 61.3918 4.0774 2.9415 38.61550.8 3.8215 2.8553 33.8374 4.5812 3.4778 31.72641 3.8089 2.9128 30.7648 4.0423 3.0441 32.7926

18 0.2 1.5027 1.3671 9.9188 3.1952 2.2769 40.33140.4 1.3255 0.9145 44.9463 2.6859 1.7988 49.31660.6 1.6826 0.9565 75.9162 3.2638 2.1134 54.43140.8 2.8137 1.9279 45.9461 4.3536 3.4391 26.59181 3.6631 2.8208 29.8589 4.2560 3.3509 27.0117

24 0.2 1.3365 0.9619 38.9389 2.5539 1.8039 41.57600.4 0.8576 0.5977 43.4842 1.9832 1.2999 52.56220.6 1.0455 0.5924 76.4877 2.5537 1.4590 75.03080.8 2.0642 1.2357 67.0478 3.8845 2.9428 32.00031 3.5215 2.7172 29.6000 4.3425 3.4988 24.1148

The complex response of Nu to the variation of parameters may be attributed to the fact thatfor various configurations of the finned annulus, high-velocity region changes its position alongthe radial direction. As various parts of the heated surface (inner pipe and fin) have varyingperformance from the point of view of convective heat transfer, therefore, it matters how mucharea of high-performance part of the heated surface is swept by the high-velocity zone.

It is observed that there is no geometry that performs consistently better than others throughoutthe entrance region. Therefore, we need some parameter to work as a performance indicator tomeasure the performance of different geometries in the entrance region. We use Nu for this purposeand define it as

Nu= 1

Z

∫ Z

0Nu(Z)dz

This parameter will give us the Nusselt number averaged over the entire entrance length and willalso help us to gauge the overall effect of the thermal entrance region. A comparison of Nu withfully developed Nu results reported by Syed [27] is given in Table VIII. It is evident from thetable that the geometries that give optimal performance on an average in the entrance region matchthe optimal geometries in the fully developed region. Therefore, we can conclude that althoughthere is no single geometry that has consistently better performance in the entrance region yetthe optimal trend that prevails in the fully developed region is, on the average, unaffected in theentrance region. It can be observed from the table that there is a substantial enhancement in theheat transfer coefficient in the thermal entrance region. For R=0.25 the maximum increase is76.4877% for 24 fins and H∗ =0.6, whereas for R=0.5 the maximum increase is 75.0308% forthe same number of fins of same height. It can also be observed that the thermal entrance regionis highly effective for fin heights 0.4�H∗�0.6.

Now we discuss the thermal entrance length, L th, and the influence of different geometricparameters on it. Figure 8 shows the effect of number of fins, N and H∗ on L th for ratio ofradii R=0.25. We note that for a given fin height, thermal entrance length increases with the



Figure 8. L th for R=0.25 plotted against fin height.

Figure 9. L th for R=0.5 plotted against fin height.

increase in the number of fins, attains a maximum and then falls. Same effect of these parameterson L th can be traced in Figure 9 for R=0.5. This behaviour indicates the existence of an optimalfin height for a given number of fins for which L th is maximum. Table IX gives the numericalvalues of L th for both ratios of radii. Table shows that the magnitude of L th is larger for a smallerratio R=0.25 as compared with its values for the larger ratio R=0.5 for any choice of valuesof the other parameters except for the two cases when N =6 and H∗ =0.8 and 1. This inverserelation between L th and R is physically quite valid. As smaller values of R correspond to alarger annular gap that requires the thermal boundary layer to take longer axial distance for gettingfully developed thus increasing the thermal entrance length [20]. It is evident from the table thatgeometries corresponding to H∗ =0.8 have optimal thermal entrance length when 24 fins aretaken. This happens for both values of R. The overall response of L th against variations in thegeometry of the annulus provides evidence of the existence of optimal geometry for which thethermal entrance length is optimal.



Table IX. Estimate of thermal entrance length L th.

R=0.25 R=0.5

N N

H∗ 6 12 18 24 6 12 18 24

0.2 0.03800 0.04950 0.06425 0.08150 0.03584 0.04234 0.04934 0.057340.4 0.04800 0.06475 0.09200 0.12725 0.04284 0.05634 0.06534 0.077840.6 0.06100 0.08350 0.10425 0.13975 0.05584 0.07584 0.09284 0.096840.8 0.06675 0.11300 0.15575 0.17775 0.06734 0.06934 0.11334 0.151841.0 0.07500 0.09650 0.11900 0.14000 0.07784 0.06634 0.07984 0.10134

CONCLUSION

In this paper, we have studied the heat transfer in the thermal entrance region of the circularfinned annulus subject to constant wall temperature boundary conditions. It has been observedthat there is no geometry that consistently performs better everywhere in the thermal entranceregion. However, the geometries found optimal on the average in the entire entrance region matchwith those optimal in the fully developed region. As the fully developed region is approached, thebehaviour of Nu matches the literature results of fully developed flow. This reflects the consistencyof the marching solution with the developed region. Furthermore, it has been observed that thermalentrance length has inverse relation with the ratio of radii.

NOMENCLATURE

Dh hydraulic diameter of finned geometryH∗ normalized fin heightL th thermal entrance lengthNu Nusselt numberdp/dz pressure gradient in finned geometryPr Prandtl numberPe Peclet NumberR dimensionless radial coordinateRe Reynolds numberri outer radius of inner pipero inner radius of outer pipeR dimensionless ratio of radiiR dimensionless radial coordinate of mean velocityT temperatureTb bulk mean fluid temperatureTe entrance temperature of the fluidTw wall temperatureu axial velocity componentu∗ dimensionless axial velocity componentumax maximum axial fluid speed at a cross-section for fully developed laminar flowm volumetric flow rate for finned geometryZ∗ dimensionless axial coordinate� thermal diffusivity� dimensionless temperature� fluid dynamic viscosity (absolute)



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thermally developing flow in finned double-pipe heat exchanger

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