A Numerical Study of Heat Transfer Enhancement Mechanisms in Parallel-Plate

Fin Heat Exchangers

L. Zhang, S. Balachandar, F. M. Najjar, and D. K. Tafti

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A NUMERICAL STUDY OF HEAT TRANSFER ENHANCEMENT MECHANISMS IN PARALLEL-PLATE FIN HEAT EXCHANGERS

L. Zhang, Department of Mechanical & Industrial Engineering

S. Balachandar, Department of Theoretical and Applied Mechanics

EM. Najjar, D.K. Tafti, National Center for Supercomputing Applications

University of Illinois at Urbana-Champaign

Urbana, IL 61801, U.S.A.

ABSTRACf

The heat transfer enhancement mechanisms and the performance of parallel-plate-fin heat exchang­

ers are studied numerically by solving the unsteady two-dimensional Navier Stokes and energy equa­

tions. Different fm arrangements are considered and the effect of boundary layer restart and self-sus­

tained oscillatory mechanisms on heat transfer enhancement and overall performance have been

compared. Results of grid dependence study showed satisfactory convergence of the solution. These

computations were performed efficiently on the massively parallel connection machine (CM5).

1. INTRODUCTION

It has been known from simple theory and from empirical experimental results [1-9]. that surface

interruption can be used for enhancing heat transfer. Some examples which exploit surface interrup­

tion are the offset strip-fins and perforated-plate surfaces. The surface interruption prevents the con­

tinuous growth of the thermal boundary layer by periodically interrupting it Thus the thicker ther­

mal boundary layer in continuous plate-fins, which offer higher thermal resistance to heat transfer,

are maintained thin and their resistance to heat transfer is reduced. Previous experimental and numer­

ical studies have shown that this heat transfer enhancement mechanism is operational even at low

Reynolds numbers when the flow is steady and laminar [1,4,6]. At higher Reynolds numbers, above

criticality, the interrupted surface offers additional mechanisms of heat transfer enhancement by in­

ducing self-sustained oscillations in the flow in the form of shed vortices.

In addition to heat transfer enhancement, the surface interruption also increases the pressure drop

and thus requires higher pumping power. This is partly due to the higher skin friction associated with

the hydrodynamic boundary layer restarting and also due to the Stokes layer dissipation [9] and high­

er Reynolds stresses [10] in the unsteady regime. Thus the boundary layer restart and the self-sus­

tained oscillatory mechanisms simultaneously influence both the overall heat transfer and the pump­

ing power requirement. Therefore design optimization must take into account the impact of design

parameters on the relative importance of the different heat transfer enhancement mechanisms and

their attendant effect on pumping cost.

Numerical investigation of the geometry effect on heat transfer enhancement mechanisms in

compact heat exchangers is very limited and most past studies are limited to the steady laminar flows

[4,6]. At higher Reynolds numbers, when the flow becomes unsteady, it is important to compute the

flow in a time accurate manner and furthermore the additional length scales that appear at higher

Reynolds numbers in the form of small eddies need to be accurately resolved as well. This increases

the dynamic range of length and time scales to be computed and makes the problem computationally

demanding.

2. MATHEMATICAL FORMULATION

The three kinds of fin arrangements considered in this study are shown in Figure 1. The fIrst case

is the inline arrangement where flat-fins of thickness, t, and length, /, form a periodic pattern with

a pitch, Lx, along the flow direction, x, and a fm separation of 2H between adjacent rows along the

transverse, y, direction. Thus the basic elemental unit, indicated by the dashed line, contains a single

fm. Here we consider a large periodic array of this basic unit and Figure l(a) shows only six basic

elements of this large array. The next is the staggered arrangement, which is obtained from the inline

arrangement by shifting alternate rows of fIn elements by half a wavelength along the flow direction.

The basic elemental unit, again marked by the dashed line in Figure 1 (b), is periodically repeated

along the streamwise and transverse directions. The most common arrangement investigated in the

past is the staggered arrangement due to its relevance to offset fIns and it differs from the inline ar­

rangement in that the transverse distance between the adjacent fm elements is doubled. Figure l(c)

shows the staggered-IT arrangement, which is obtained from the inline arrangement by shifting alter­

nate colums of fms in the transverse direction by half a wavelenth and the basic unit is marked by

the dashed line. In all these three cases the heat transfer surface area per unit volume is maintained

the same and the actual sizes and lengths employed in the simulations are given in terms of H.

The numerical simulations will assume periodicity of the velocity and temperature fIelds along

both the streamwise and transverse directions, over one basic elemental unit and therefore the actual

computation geometry will be limited to this elemental periodic unit Thus, in an attempt to model

the flow and heat transfer in a large periodic array of fm elements, the present computation ignores

the entrance and exit effects. Furthermore, the possibility of subharmonic effects along both the

streamwise and transverse directions is ignored. These effects can be taken into account by employ-

ing a larger computational domain, which includes multiple elemental units, Mx and My respectively,

along the x and y directions and assuming periodicity of the flow and temperature fields over this

extended domain. At low Reynolds numbers to be considered in this study these subharmonic modes

are not energetic and therefore for the sake of computational efficiency here we choose Mx=My = 1.

The governing equations solved in two-dimensions for the non-dimensional velocity, u, and tem­

perature, T, fields are the Navier-Stokes equations along with the incompressibility condition and

the energy equation, as shown below:

au + u.Vu = e - Vp + _1_V2u at x Re'l'

inD (1)

aT v 1 v2 at + U· T = Re'l' Pr T inD (2)

V·u = 0 inD (3)

Where D denotes the computational domain, indicated by the dashed line in Figure 1. for each of

the three cases. In the above equations, the length and pressure scales are given by the half distance

between adjacent fin rows along the transverse direction, H, and the applied pressure difference over

a unit non-dimensional length along the streamwise direction, M. The corresponding velocity and

time scales are then given by, u*=(M/e)l/2 and t=(fi2e/Ap)l!2, where e is the density of the fluid.

The temperature has been nondimensionalized by q"H/k, where q" is the specified constant heat

flux on fm surfaces and k is the thermal conductivity of the fluid. The friction Reynolds number, Rec,

is given by Hu* Iv and Pr is the Prandtl number. Furthermore, to enable periodicity of the flow field

along the streamwise direction, the non-dimensional pressure gradient has been split into an imposed

constant mean pressure gradient given by the unit vector, ex, and a fluctuating part, p, which can be

considered periodic along x and y. Thus, in the present computations the streamwise pressure gradi­

ent is maintained a constant and therefore the flow rate, Q, fluctuates over time, but for all the cases

considered the flow rate fluctuation is less than 1 % of its mean value.

Under constant heat flux boundary condition, a modified temperature field, e, can be defined

as: e(x,y, t) = T(x,y, t) - rx, where r is the mean temperature gradient along the flow direction.

From a balance of the total rate of heat input along the fin surface, r can be computed from the fol­

lowing expression: r = ~ /(QRecPr), where ~ is the perimeter of the fin surface in thex-y plane.

The modified temperature, e, can then be considered as the perturbation away from the linear tem­

perature variation and can be considered to be periodic along both x and y directions. Since the tem­

poral fluctuations in the flow rate are small, the corresponding fluctuations in r are also negligible

in magnitude. On the surface of the fm, no-slip and no-penetration conditions are imposed on the

velocity field. The corresponding boundary condition for the modified temperature is given by

(ve)·;' = 1 - re~·;' on aDfin (4)

where n is the outward normal to the fm swiace denoted by aD fin.

The numerical approach followed here is the direct simulation where the governing equations

are solved faithfully with all the relevant length and time scales adequately resolved and no models

are employed. A second-order accurate Harlow-Welch scheme is employed with a control-volume

formulation on a staggered grid with central difference approximations for the convection terms. The

equations are integrated explicitly in time until a steady or periodic state is reached. For the inline

fin arrangement, the periodic domain with one fm element is resolved with a grid of 128 X 32 points,

while in the staggered arrangements, the periodic domain with two fin elements is discretized with

256 X 64 grid points. A detailed description of the numerical methodology can be found in reference

[11].

3. RESULTS AND DISCUSSION

Before the presentation of the results the following quantities will be defmed first. Although the

computations were performed with Hand (API e) 1/2 as the length and velocity scales, in the results

to be presented the Reynolds number, Re, is defined based on the hydraulic diameter, Dh, as:

VD h Re=-­v and D - 4Am

h - A/Lx (5)

where Am is the minimum flow cross-section area, V is the average velocity at this section and A is

the heat transfer swiace area. Local heat transfer effectiveness will be expressed in terms of the

instantaneous local Nusselt number based on hydraulic diameter, defined as:

() D.IH Nu s,t = (}IJ ) _ () f )

J's, t ref's, t (6)

where s measures the length along the periphery of the fm and the local reference temperature, () ref'

is defined by taking into consideration the recirculating zones as [6]:

f 8Iuldy () re/s,t) - f

luldy (7)

Following the above definition, the instantaneous global Nusselt number, <Nu>, can be expressed

through an integration of heat flux and temperature difference around the fm swiace, as:

< Nu > = (Dh/H) <5'

f [(}/s,t) - (}re/s,t)]dl aD,

(8)

The overall Nusselt number, <Nu>, is then defmed as the average of the above over time. Further­

more we define the modified Colbumj-factor, and friction factor,/, as:

and J = LJP (Dh)

1QV24L (9)

In Figure 2 the performance of the three fin arrangements is plotted in terms of the j and J factors

against the Reynolds number based on the hydraulic diameter. Also plotted are the j and J factors

resulting from a fully developed channel flow between continuous uninterrupted flat plates with

identical transverse fin spacing as that of the inline arrangement. A comparison of theses various

performance quantities will provide deeper understanding into the net effect of the various heat

transfer enhancement mechanisms. Based on Figure 2 it is clear that the inline and staggered-II ar­

rangements have about the same performance suggesting that each column of fin has little impact

on the flow and heat transfer of the following column of fin, provided that they are well separated

by one fin length or more. This result seems to hold over the entire Reynolds number range investi­

gated. The inline and staggered-II geometries result in better heat transfer than the staggered arrange­

ment, but this increase is accompanied by a corresponding increase in the friction factor as well. All

three fin arrangements are seen to result in higher heat transfer and higher friction factor than the

corresponding continuous parallel plate case. The difference is due to the combined effect of bound­

ary layer restart mechanism and the self-sustained flow oscillations (vortex shedding).

In the results presented above it is difficult to separate the importance of boundary layer restart

mechanism and the impact of vortex shedding. Although heat transfer enhancement associated with

the onset of flow unsteadiness has been observed qualitatively in many experimental studies [7], the

separation and quantification of the individual effects has not been explored. Here we utilize the flex­

ibility of the present numerical approach to separate these individual effects. For all the three geome­

tries over the range of Reynolds numbers the flow and heat transfer were computed with appropriate

symmetries imposed on the velocity and temperature fields about the wake centerline. This symme­

trization of the flow removes all asymmetry in the wake associated with the shedding process and

thus the flow is made steady. We shall refer to these simulations as symmetrized or steady solutions.

Differences in the performance of the symmetrized and the non-symmetrized cases is solely due to

the shedding process. On the other hand, the difference between the symmetrized case and the contin­

uous flat plate arises mainly from the periodic restart of the hydrodynamic and thermal boundary

layers. Geometry effects such as those arising from the finite thickness of the fin also contribute to

this difference. For example, in the symmetrized case, in addition to the restart of the boundary lay­

ers, standing recirculating zones can also be seen in the wake, thus affecting the flow field and the

overall heat transfer.

The j and Jfactors and their ratio for the symmetrized steady and the non-symmetrized cases are

presented in Figure 3 for both the inline and the staggered geometries. In the symmetrized cases, the

j and J factors are seen to follow a power law of the form:

j = 1O.5*Re-O·879 and J = 22.5*Re-O·824 for Inline Geometry

j = 6.0*Re-O·843 and f = 6.2*Re-O·743 for Staggered Geometry

as shown by the linear behaviour in the log-log plot. This when compared with the theoretical result

for the continuous flat plate, given by:

j = 9.5*Re-1 and f= 24.0*Re-1 for Continuous Flat Plate

clearly shows the expected result that the effects of boundary layer restart and the finite fm thickness

are to increase both heat transfer and friction factor within the range of Reynolds numbers simulated.

The effect of self-sustained oscillation is to further increase the heat transfer and friction factor above

the power law behavior. This deviation from the power law is seen to occur above a Reynolds number

of around 350 in the inline arrangement, and at around 650 in the staggered arrangement. Above the

critical Reynolds number for the onset of self-sustained oscillations, the flow and temperature fields

are observed to oscillate at a single frequency corresponding to the main shedding frequency. Figure

4 shows the instantaneous velocity vector and its corresponding local Nusselt number along the fm

surface for the staggered arrangement at Reynolds number of 1245. The impact of the flow oscilla­

tion in terms of vortices travelling on the fm surface can be seen in the local Nusselt number as a

sharp increase in its local value.

The detailed effect of varying the Reynolds number on the temperature and flow fields will be

considered here. Figure 5 shows the velocity variations of the velocity signal u at.x=O, y=O as a func­

tion of time, t* (non-dimensionalized by mean flow bulk velocity and fm thickness), for the inline

geometry at three different Reynolds numbers: Re=546.3, 1407.3 and 2191.2. At Re=546.3 and

1407.3, flow exhibits a single dominant frequency. The corresponding frequency spectra shows a

single dominant frequency with its higher harmonics. At the highest Reynolds number, the frequen­

cy spectrum shows a lot more activity signifying departure from a smooth laminar flow.

The inline arrangement has higher higher heat transfer than staggered arrangement for the same

Reynolds number as shown in Figure 2. However the inline arrangement also has a higher friction

factor. In order to gain a better understanding of the significant differences in friction factor of the

two arrangements, we pursued to investigate the corresponding mean flows. We have considered

mean flow for the inline and staggered arrangements at approximately the same Reynolds number

(Re= 1407.2 for inline and Re= 1465.3 for staggered). Shown in Figure 6 are the mean flow velocity

vectors and their corresponding time averaged skin friction distributions around the top and bottom

surfaces of the fin element. From the local reversal of skin friction it can be seen that in the staggered

arrangement, a recirculating zone is present in the mean flow at the leading edges, one on top and

bottom of the fm surfaces. The recirculation bubble at Re=1465.3, extends over one third of the fm surface, resulting in the significant decrease in the skin friction contribution to the f factor over the

corresponding inline arrangement. An added explanation for the higher heat transfer and friction fac-

tor in the inline arrangement is that, since the distance between adjacent fms along the y-direction

is half of that in the staggered geometry, the thermal and hydrodynamic boundary layer are main­

tained thin.

Finally we report results from a grid independence study conducted with three different resolu­

tions of size 128x32, 256x64 and 512x128 for the inline geometry at a Re of about 2000. Figure 7

shows j and / factors for the three different grid resolutions. We observe that by doubling the grid

in each direction to 256x64, the friction factor reduces by about 9% while the j factor reduces by 6%.

Further doubling of the grid to 512x128, results in a nominal reduction of 1 % and 2% for the j and

/factors, respectively. The present results are in very good agreement with experimental results [7]

up to the Reynolds number where secondary frequencies begin to appear, after which the present

calculations overpredict thej and/factors slightly. We suspect that this is possibly due to the onset

of strong three-dimensional effects which are neglected in the present simulations. Overprediction

of the mean and rms drag and lift coefficients in two-dimensional models has also been observed

in a recent study by Mittal and Balachandar[lO]. Further details are are provided in reference [13].

4. CONCLUSION

Direct numerical simulation is a powerful tool which can be used to explore in detail the flow

and heat transfer phenomena in heat exchangers. It provides detailed information about the flow pat­

tern and temperature field over the entire computational domain. Here unsteady Navier-Stokes and

energy equations are solved in two-dimensions to simulate flow over a large array of parallel-plate

fin elements.

'Three different fin arrangements have been studied and their performance have been compared

with theoretical results where the boundary layer restart and self-sustained oscillatory mechanisms

are absent In all the arrangements, the heat transfer surface area per unit volume is maintained the

same to facilitate comparison. It is shown that both the boundary layer restart and self-sustained os­

cillatory mechanisms are beneficial for increasing the heat transfer. Furthermore, steady flows in

these geometries were computed with imposed symmetry boundary conditions and their perfor­

mance was compared with that of unsteady simulation to quantify the effect of unsteadiness in heat .

transfer enhancement. A complete grid dependence study showed satisfactory convergence of our

solution. The limitations of 2-D simulations in predicting flow and heat transfer quantities at high

Reynolds numbers has also been illustrated.

5. ACKNOWLEDGEMENTS AND REFERENCES

The authors would like to thank the Air Conditioning and Refrigeration Center (ACRC) at the Uni­

versity of lllinois at Urbana-Champaign for their support. Dr. F. N. Najjar was partially supported

by a Division of Advanced Scientific Computing, National Science Foundation Post-Doctoral Fel­

lowship. These computations were performed on the CM5 at National Center for Supercomputing

Applications.

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99 (1977) pp 180-186.

2. E.M. Sparrow, B.R Baliga and S.Y. Patankar, Journal of Heat Transfer, Transaction of ASME,

Vol. 99 (1977) pp 4-11.

3. N. Cur, and E.M. Sparrow, International Journal of Heat and Mass Transfer, Vol. 21 (1978) pp

1069-1080.

4. E.M. Sparrow, and C.H. Liu, International Journal of Heat and Mass Transfer, Vol. 22 (1979) pp

1613-1624.

5. E.M. Sparrow, and A. Hajiloo, Journal of Heat Transfer, Vol. 102 (1980) pp 426-432.

6. S.Y. Patankar and C. Prakash, International Journal of Heat and Mass Transfer, Vol., 24 (1981)

pp 51-58.

7. R.S. Mullisen, and RI. Loehrke, Journal of Heat Transfer, Transaction of ASME, Vol. 108 (1986)

pp 377-385. 8. N.K. Ghaddar, G.E. Karniadakis, and A.T. Patera, Numerical Heat Transfer, Vol. 9 (1986) pp

277-300. 9. C.H. Amon, and B.B. Mikic, Numerical Heat Transfer, Part A, Vol. 19 (1991) pp 1-19.

10. R Mittal, and S. Balachandar, Physics of Fluids, 7 (1995) pp 1841-1865.

11. D.K. Tafti, "A Study of High-Order Spatial Finite Difference Formulation for the Incompress­

ible Navier-Stokes Equations, NCSA Pre print, 031, 1993.

12. C.H.K. Williamson and A. Roshko, Z. Flugwiss. Weltrawnforch, 14, (1990) pp 36.

13. D.N. Tafti, L. Zhang, EM. Najjar and S. Balachandar, S., "A Tune-Dependent Calculation Pro­

cedure for Studying Heat Transfer in Parallel-Plate Fin Heat Exchangers on the Connection Ma­

chine-5", to be submitted to Numerical Heat Transfer, 1995.

. ...,

1 •• =14.4H .. 11=0.75 1 __ _ .................... ~ tZH

!!I .... !!!I!I!I!I!~. 1=6.4H

~ __ ~/~.,=~/4~.4~H_~~ ....

,=11.7S,"._ lZH !! .... ~!!!!! .. ,,=0.75

~ ..... ~ .. ~ ... ~ .. ~ ... ~ ................ ~~~~~~ ... . : 1=6.4H : ... --.

.... _-_._--------_._-----:

• _._ •• A •••••

IC

(al (b) (el

Figure 1. Three Fin Arrangements: (a) Inline (b) Staggered (c) Staggered-II

CurTent Simuiation-Inline

Current Simulation-Staggered

....

• • •••• A •••••

IC Current Simuiation.S·Gftgered.n 10.3 !:-r __ -'--_-'---'--'-..................... ~!'"'5r--......... - __ 10 10

Continuous Par.dlel Plates

CUrTent Simuiation-lnUne

CUrTent Simulation-Staggered

CUrTent Simuiation.Staggered.n 10.3 &,.,-----~~ ................. -......,..I,-,.---~~

10 10 Re Re

(a) (b)

Figure 2. Overall Performance: (a) Colburn j Factor vs

Reynolds Number (b) Friction Factor f vs Reynolds Number

10" 'if ! .. ... ... til ... ~jIf

10"

'S.

"-IO~

- ..... - Inline Unsymmetrized

--l(-J( - Inline Symmetrized

- ...... - Staggered Unsymmetrized

--lI-X - Staggered Symmetrized

10"1...,...---~-~-~~~~.......L,:----~~ 102 Re 10'

10~OL' ___ ~-~~-~~~I-'-:O':----~~ Re

(a) (b)

Figure 3. Impact of Vortex Shedding on Overall Perfonnance: (a) Inline (b) Staggered

15.~ 10

05 1IIIiliiiili 0.0

-0.5

:;:~

(a)

la'

10'

o 2 4 5

Fin Surface Location

Figure 4. (a) Instantaneous Flow Field for Staggered Geometry at Re=1245.4

(b) Corresponding Instantaneous Local Nusselt Number vs Fin Surface Location

4

3

2

::--1 ci ~O o II x -1 >

-2

-3

6

4

C"2 ci II :>:'-0 o II x

>-2

-4

(a)

10'

10'

10·'

10" 1oS'------~--~~~10~------~~~

f*

10'

-60 10 20 30 40 50 60 70 80 90

10

5

!... ci II

:;0 II x >

-5

-100

t*

(b)

Hi 10'

E 210'

~ 10·' a. ~10·2

g 10'" III

5-10'" ~

IL 10"

10"

10·'

50 t* 100 150 10' 10

f* (c)

Figure 5. Velocity v Probes for Inline at (x=O,y=O) and Corresponding Frequency Spectrum at: (a) Re=546.3 (b) Re=1407.3 (c) Re=2191.2

00

.5

(a)

2 I

(b)

..

Skin Friction(Top Fin)

Skin Friction(Bottom Fin)

10 12 13

Skin Friction(Top Fin)

Skin Friction(BoUom Fin)

Figure 6. TIme Averaged Mean Flow Field and Corresponding Skin Friction Distribution

on Top and Bottom Fin Surfaces: (a) Inline at Re=I407.2 (b) Staggered at Re=1465.3

~---------~--------~ x x f

.--------------~------------~ A ... j

Figure 7. Grid Dependence Study: j and f at Different Grid Sizes