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Journal of the Brazilian Society of MechanicalSciencesPrint version ISSN 0100-7386

J. Braz. Soc. Mech. Sci. vol.22 no.4 Rio de Janeiro 2000

http://dx.doi.org/10.1590/S0100-73862000000400007

Comparison between One-Dimensional Uncoupled andConvection-Conduction ConjugatedApproaches in Finned Surface HeatTransfer

Cláudia R. Andrade Edson L. Zaparoli Departamento de Energia – IEME. Instituto Tecnológico de Aeronáutica – ITA Pça Marechal Eduardo Gomes, 50 Vila das Acácias 12228-900 São José dos Campos. SP. Brazil zaparoli@mec.ita.cta.br

This work studies the forced convection problem in internal flow betweenconcentric annular ducts, with radial fins at the internal tube surface. The finnedsurface heat transfer is analyzed by two different approaches. In the first one, itis assumed one-dimensional heat conduction along the internal tube wall andfins, with the convection heat transfer coefficient being a known parameter,determined by an uncoupled solution. In the other way, named conjugatedapproach, the mathematical model (continuity, momentum, energy and K-eequations) applied to tube annuli problem was numerically solved using finiteelement technique in a coupled formulation. At first time, a comparison wasmade between results obtained for the conjugated problem and experimentaldata, showing good agreement. Then, the temperature profiles under these twoapproaches were compared to each other to analyze the validity of the one-dimensional classical formulation that has been utilized in the heat exchangerdesign. Keywords: Conjugated heat transfer, Nusselt number, fin, heat exchanger

Introduction

The use of extended surfaces is an important mechanism to intensify the heat transfer ratein the tube internal flow, particularly applied to the compact heat exchangers design.Usually, in the extended surface study, the fins are treated as a one-dimensional

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17/5/2014 Journal of the Brazilian Society of Mechanical Sciences - Comparison between one-dimensional uncoupled and convection-conduction conjugated ap…

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conduction problem while the interaction at the interface solid-fluid is represented by aconvection coefficient (Gardner, 1945; Brown, 1965). In this classic approach the complexinteraction between the heat transfer processes of conduction in the solid domain (internaltube wall and fins) and convection in the flow along the fins is substituted approximatelyby a uniform convection coefficient at the whole interface.

The experimental results for the convection heat transfer coefficient are extrapolated to aninfinite solid conductivity to represent a uniform temperature at solid-fluid interface, asdetailed in Stasiulevicius and Skrinka (1988). When the convection coefficient isdetermined by a theoretical solution a boundary condition of uniform temperature isimposed through the solid-fluid interface. These values of the convection coefficient showaccentuated variations and it is needed a method to obtain a medium value to be appliedto the classic fin theory. The procedure described in Zukauskas and Skrinka (1992) isadopted in this work for the uncoupled formulation. However, the application of theconstant temperature as boundary condition doesn't represent the correct interaction of theheat transfer mechanisms in the solid-fluid interface. The temperature distribution and theheat flow in the solid-fluid interface should be the result of the interaction between theconduction in the solid domain and convection in the fluid region. To do so, it should beused a procedure where the conditions of heat flux and temperature equality areintrinsically imposed by the solution method, considering the solid and the fluid as a singledomain. This coupled formulation is mentioned in the literature as conjugated problem(Davis and Gill, 1970). Several works studied the conjugated heat transfer in laminarregime as He et al. (1995) that studied the conjugated problem for the flow betweenparallel plates, using the finite difference technique to analyze the fluid region and thecontour element method in the solid domain.

There are different experimental works that describe the use of extended surfaces toenhance the heat transfer rate. Knudsen and Katz (1950) analyzed the heat exchangeprocesses and pressure drop in duct water flow, with the attachment of circular fins.Obermeier and Henne (1993) studied the effect of the fin pitch in the heat transfer rate forconcentric annular tubes configuration. The present work focuses on the turbulent flow infinned concentric annuli. The results obtained by two different approaches are compared: inthe first treatment, a numerical solution is obtained for the conjugated heat transferproblem and, under the second one, an uncoupled one-dimensional formulation, which hasbeen applied to the heat exchangers design, is used.

In the uncoupled approach, a global convection coefficient is adopted as representative ofthe heat exchanges with the fluid. It is intended, therefore, to verify the validity of thisprocess of obtaining a medium convection coefficient proposed by Zukauskas and Skrinka(1992). Besides, the constant temperature hypothesis at the interface (applied to theclassic solution of the convection problem) will be tested against the real variations thatoccur along the interface solid-fluid. A CFD (Computational Fluid Dynamics) code, based onthe finite element technique, was used to solve the fluid flow and heat transfer problems.The properties of the fluid were considered constant and the k-e standard model wasemployed to treat the turbulence terms. The numerical results obtained for the Nusseltnumber were compared with the experimental data of Obermeier and Henne (1993)showing a good agreement. Besides, the influence of the fluid to wall thermal conductivityratio in the heat flux rates and in the fin temperature distribution was analyzed, comparingthe two formulations.

Results showed that the one-dimensional formulation provides good estimate for the totalheat exchange rate, but it presents accentuated differences in relation to the conjugatedapproach when the fin temperature distribution is estimated.

Problem Geometry and Mathematical Formulation

The numeric simulations were accomplished in a domain representing the experimentalapparatus specified in Obermeier and Henne (1993). A schematic representation of theexperiment is described in Fig. 1, where s indicates the fins height, constant along L(heated tube extension), and p the fins spacing.

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Nomenclature

A = Heat transfer area [m2] Bi = Biot number = ht/K [-] Cp = Specific heat [J/Kg×K]

Dh = Hydraulic diameter [m] h = convection coefficient

[W/m2×K] L = Heated tube extension[m] k = Kinetic turbulent energy[J] K = Thermal conductivity[W/m×K] Nu = Nusselt number =h×Dh/Kf [-] p = Fin pitch [m] P = Pressure [Pa] Pr = Prandtl number = mCp/k[-] Qm = Conjugate heattransfer rate [J/s] r = Radial coordinate [m] R = Thermal resistance [K/W]

Re = Reynolds number =rUeDh/m [-]

s = Fin height [m] t = Fin thickness [m] T = Temperature [K] TQ = Uncoupled heat transferrate [J/s] u = Axial velocity [m/s] Ue = Uniform entrance

velocity [m/s] v = Radial velocity [m/s] r = Axial coordinate [m] x = Radial coordinate

Greek Symbols b = Fin parameter [-]

e = Dissipation rate [J]

h = Fin efficiency [-] m = Viscosity [Pa×s]

r = Density [Kg/m3]

sk = turbulent Pr for diffusion

of k se = turbulent Pr for diffusion

of e

Subscripts c = relative to convection e = external or entrance ef = effective property eq = equivalent i = internal m = medium property max = maximum value s = relative to solid t = turbulent property w = referring to the wall

Figure 2 shows a tube annuli section where rw e ri represent the radii of the internal and

external surfaces of the inner tube, respectively, and re indicates the external tube radius.

The fin has a external radius rf and thickness t.

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In the conjugated approach the water flow between concentric tubes is simulated, underthe hypotheses of constant properties and steady state conditions. The continuity,momentum, energy and k-e equations in cylindrical coordinates are expressed by:

where Gk indicates the viscous dissipation term:

The constants of the standard k-e model are: Cm = 0.09, C1 = 1.45, C2 = 1.9, sk = 1.0, se= 1.3 and Prt = 1.0.

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In the system of equations (1) to (7), r is the fluid density, T is the temperature, u and vindicate the axial and radial velocity components, Cp the constant pressure specific heat

and mef represents the effective dynamic viscosity. The effective dynamic viscosity (mef) is

defined as the sum of the molecular (m) with the turbulent contribution (mt), indicated by:

The molecular (Pr) and turbulent Prandtl (Prt) numbers are expressed as:

where K and Kt are the molecular and turbulent thermal conductivities, respectively.

In this work, all the turbulent properties are calculated according to k-e model of Launderand Spalding (1974). In the solid-fluid interface, both boundary values of u and v are set tozero. In the heated tube extension a constant temperature is specified (Tw) while the

initial and final regions are insulated. The entrance conditions are uniform velocity,temperature, k and e profiles, as schematized in the Fig. 1. In the tube exit a value isimposed for the pressure (p = 0) and natural conditions for the other variables. In theproximity of the solid wall, the program employs the Wall Logarithmic Law to establishvalues for k and e .

Three additional dimensionless parameters are also used:

where Dh is the hydraulic diameter defined as:

and h indicates the convection coefficient.

The following numeric values were used for the fluid properties ( water at 333 K ): Pr =

3.0, rf = 985.47 Kg/m3, Cpf = 4.183 J/Kg×K, Kf = 0.651 W/(m×K), mf = 4.07 10-4 Kg/(m×s),

and Re = 2.6x104.

In the conjugated formulation the problem is studied as a single domain, including thesolid and the fluid. The system of equations (1) to (7) and the boundary conditionsdetermine the solution for the convection heat transfer problem. The conduction in thesolid is simulated using null values to u and v in the energy equation (4). The conditions ofadherence of the fluid and no temperature jump at the solid wall as well as heat flowequalities at the interface are intrinsic to the model.

Solution Methodology

The numerical solution of the problem was obtained by a CFD code based on a finiteelement method. The governing equations were solved using a segregated formulation

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(where the fields of u, v, P, T, k and e are separately calculated) as detailed in Rice andSchnipke (1986). The domain was discretised with a structured mesh, refined in the areasthat occurs accentuated gradients, allowing to capture boundary layer effects (thermal andhydrodynamic). The systems of algebraic equations were calculated by iterative methods asTDMA (Tri-diagonal Matrix Algorithm) and PCG (Preconditioned Conjugate Gradient).

The numerical simulations were accomplished using a Pentium-Pro microcomputer (200 MHzand 128 MB of memory RAM), and were composed of three stages:

1) - Pre-processing: represented by the problem geometry statement, mesh generation andboundary conditions application, that is, the domain representation in terms of finiteelements; 2) - Numerical analysis: in this stage, the numerical solution algorithm for theconfiguration established in the stage 1 is applied; 3) - Post-processing: once the numeric solution was concluded (obeying the imposedconvergence criterion), the visualization of the obtained results is performed. The programincludes a graphic interface to visualize basic variables as pressure, velocity, temperatureand derived quantities as stream function and heat flow.

Uncoupled One-Dimensional Formulation Equations

In this formulation the conduction and forced convection problems (schematized in Fig. 1)are analyzed by an uncoupled way. The convection study is developed in a domainincluding only the fluid and constant temperature as boundary condition at the interfacewith the solid. The convection coefficient varies strongly along the tube and fins surfacesbut the one-dimensional approach supposes that the heat exchange is represented by amedium and uniform convection coefficient at the solid-fluid interface. In the procedurepresented in Zukauskas and Skrinka (1992) this coefficient is obtained by the followingexpression:

where Qm is the total heat transfer rate, Twm denotes the average temperature between

the external surface of the inner tube and the fins root, Tfluid is the arithmetic average of

the fluid bulk temperatures at the entrance and the exit of the heated tube extension (L)and A represents the effective heat transfer area:

where As and Af indicates the unfinned and finned areas respectively, considering the fin

efficiency (h) as detailed in Krauss and Bar-Cohen (1995).

As the efficiency of the fins depend on the convection coefficient, h = h (h), an iterativeprocedure is needed to determine this parameter. In Zukauskas and Skrinka (1992) thevalue of h was calculated with a unique iteration. In the present work, several iterationswere accomplished until reaching the solution convergence. Those authors suggest that theconvection coefficient obtained according to Eq. (14) is valid for different fin materials,since the conditions of similarity of the problem are maintained. However, Stasiuleviciusand Skrinka (1988) proposed a procedure of extrapolation for the experimental curve of theconvection coefficient, obtained with different finite values of fin thermal conductivity, toinfinite solid thermal conductivity.

In Eq. (14) the total heat transfer rate (Qm) is expressed by:

where the fluid bulk temperature at the entrance ( x=0 ) and exit ( x=L ) of the heated

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tube extension is defined as:

and u and T values are determined by the conjugated approach.

In this work, a one-dimensional steady state and constant properties model is used tostudy tube wall and fins conduction problem. Then, a thermal circuit (resistance model) canrepresent the heat transfer for each cell showed in the Fig.3 (where the abbreviated termsare indicated in the nomenclature table).

The wall resistance for the cell portion with fins ( Rpf ) and without fins ( Rps ) are obtained

by the following expressions:

where t indicates the thickness of the fins and p the spacing among them.

For the inner tube area without fins (Fig. 3) the convection resistance (Rc) is calculated by:

The radial fin of rectangular profile is analyzed supposing one-dimensional conduction,insulated fin tip and temperature prescribed at the fin root. The solution is obtained interms of the Bessel functions (Krauss and Bar-Cohen, 1995) with the fin thermal resistance(Rf) expressed by:

with b1 = bri and b2 = brf ; I0 and I1 are the first type modified Bessel functions; K0 and

K1 are the second ones.

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In the circuit schematized in Fig. 3, an equivalent resistance (Req) can be defined as:

and the total heat transfer rate (TQ) is given by:

where Tw is the solid-fluid interface temperature and Tfluid is the fluid average temperature

between the entrance and exit of the heated tube.

For the case described, the Biot number (Eq. 11) expresses the relationship between theconduction and convection thermal resistance, adopting a uniform convection coefficientsurrounding the fin. Thus, low values of Biot (Bi < 0.1) indicate that the convectionresistance controls the heat transfer process, and the one-dimensional calculations arevalid in the classical uncoupled approach.

To compare the conjugated treatment and the uncoupled formulation, the followingprocedure was adopted:

(a) the convection coefficient was determined using Eqs. (14), (15) and (16), where thenecessary data to calculate the Qm and Tm values were obtained by the conjugated

formulation using a very large solid thermal conductivity, which implies in a uniformsolid/fluid interface temperature;

(b) for finite thermal conductivity the total heat transfer rate was calculated by twoapproaches: in the uncoupled treatment TQ was obtained by Eqs. 18 to 20, with the hvalue determined in (a), and to determine the Qm value in the conjugated heat transfer

formulation the Eq. 16 was used.

Results and Discussion

A comparison between numerical results for the Nusselt number (Nu) and the experimentaldata of Obermeier and Henne (1993) was established to evaluate the numerical solution.The numerical results were obtained by the conjugated approach. These Nu values are alsopresented in Table 1, and they show good agreement with the experimental results wherethe differences are less than 5%.

The total heat transfer rate was calculated by the uncoupled and conjugated formulationsfor 1 < Ks/Kf < 500 range. The results are shown in Fig. 4 and is verified that don't occur

significant differences between the one-dimensional and the conjugated approaches in thestudied interval of the solid to fluid conductivity ratio. This indicates that the uncoupledtreatment, which uses a global and uniform convection coefficient, furnishes a goodmethod to estimate the total heat transfer rate.

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However, the fin temperature distribution determined by the uncoupled formulation (wherethe convection coefficient is determined by the above described method) shows largedifferences when compared with the conjugated approach results. Figures 5 and 6 presentthe temperature profiles for two values of thermal conductivity ratio: Ks/Kf = 1 and Ks/Kf =

500.

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When the Ks/Kf = 1 (Fig. 5) the fin efficiency is low and the fin tip temperature ( r = rf ) is

almost the same as the fluid entrance temperature (Te = 280 K). The uncoupled one-

dimensional formulation underestimates the values of the fin temperature profile at all finextension ( ri < r < rf ).

When Ks/Kf = 500 the heat transfer rate increases due to attachment of fins with high

thermal conductivity in the tube wall. This fact implicates in larger exit temperatures. It isnoticed that for Ks/Kf = 500 the differences among the temperatures profiles obtained by

the two approaches are larger than those for the Ks/Kf = 1, as shown in Fig. 6.

The tube wall and fin temperatures calculated by the uncoupled method are much smallerthan the conjugated approach one. This happens because the wall temperature profile forKs/Kf = 500 is two-dimensional and presents a depression near the fin root, as shown in

Fig. 7. For this case, the one-dimensional formulation provides lower results for the fintemperature because it considers only the heat flux radial component in the wall andignores the axial conduction to the fin root.

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The fin temperature distribution for Ks/Kf = 500 ( Fig. 7 ) is almost one-dimensional,

except near the fin tip, while the fin temperature distribution for Ks/Kf = 1 (Fig. 8) shows

radial and axial variations. This agrees with the fin Biot number values presented in Table2. According to a two-dimensional conduction analysis, for the fin domain only, thetemperature only varies with the radial coordinates when the Biot number is smaller than0.1, showing that the one-dimensional approach is valid to calculate the heat transfer inthe fin region.

The one-dimensional conduction heat transfer analysis (for the fin domain only) is validwhen Biot number is << 1, which correspond to Ks/Kf = 500, as shown in Table 2.

However, the two-dimensional formulation in the tube wall is very important to reproducethe conduction heat transfer to the fin root that elevates the temperature at this point, ascan be seen in Fig. 6 and Fig.7. This fact implies in larger differences between thetemperature profiles provided by the two approaches in the case of Ks/Kf = 500.

When Ks/Kf = 1, an elevation of the fin root temperature is noticed since the lower

thermal conductivity fin performs as an insulation at this tube wall region, according to Fig.8.

Isotherms presented in Fig. 8 shows that heat is deviated from the fin base due to its lowthermal conductivity (insulation material). An opposite effect occurs in Fig. 7 where theheat is transferred towards the fin base.

Conclusions

Results obtained for Nusselt number by the conjugated treatment using the standard k-ewere compatible with the experimental data. It was shown that the one-dimensionalformulation provides good estimate for the total heat transfer rate in the 1< Ks/Kf < 500

range but it presents accentuated differences in comparison with the two-dimensionalconjugated approach when the temperature distribution fins is analyzed. The one-dimensional uncoupled formulation doesn’t capture the temperature depression or elevationclose to the fin root while the two-dimensional conjugated approach reproduces the tube

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wall temperature distribution combining the radial and axial conduction effects.

Comparison between the results obtained by the two approaches showed that the classicone-dimensional formulation was valid for the heat exchangers design to determine thetotal heat transfer rate. However, to calculate the temperature distribution in finned tubesthe adoption of a two-dimensional treatment provides better results.

Ackowledgements

The authors are grateful to FAPESP, which supported this work (grant No. 99/03471-5).

References

Brown, A., 1965, "Optimum Dimensions of Uniform Annular Fins", Int. J. Heat MassTransfer, Vol. 8, pp 655-662. [ Links ]

Davis, E.J. and Gill, W.N., 1970, "The Effects of Axial Conduction In The Wall On HeatTransfer With Laminar Flow", Int. J. Heat Mass Transfer, Vol. 13, pp 459-470. [ Links ]

Gardner, K.A., 1945, "Efficiency of Extended Surface", Trans. ASME, Vol. 67, pp 621-631. [ Links ]

He, M., Bishop, P.J.; Kassab, A.J. e Minardi, A., 1995, "A coupled FDM/BEM Solution for theConjugate Heat Transfer", Numerical Heat Transfer, Parte B, Vol. 28, pp 139-154. [ Links ]

Krauss, A.D., and Bar-Cohen, A., 1995, "Design and Analysis of Heat Sinks", John Wiley &Sons, Inc. [ Links ]

Knudsen, J.G., and Katz, D.L., 1950, "Heat Transfer and Pressure Drop In Annuli", Chem.Eng. Progress, Vol. 56, pp.490-500. [ Links ]

Launder, B.E., and Spalding, D.B., 1974, "The Numerical Computation of Turbulent Flows",Computer Methods In Applied Mechanics And Engineering, Vol.3, pp 269-289. [ Links ]

Obermeier, E, and Henne, R. A., 1993, "Performance Analysis of Augmented Heat TransferSurfaces - Theoretical and Experimental Results for Tubes with Circunferencial Fins inLongitudinal Flow", Proceedings of the First International Conference on Aerospace HeatExchanger Technology, pp. 329-345, Palo Alto, USA. [ Links ]

Rice, J.G. and Schnipke, R.J., 1986, "An Equal-Order Velocity-Pressure Formulation ThatDoes Not Exhibit Spurious Pressure Modes", Computer Methods In Applied Mechanics AndEngineering, Vol.58, pp 135-149. [ Links ]

Stasiulevicius, J. and Skrinka, A., 1988, "Heat Transfer of Finned Tube Bundles inCrossflow", Hemisphere Publishing Corporation. [ Links ]

Zukauskas, A. and Skrinka, A., 1992, "Banks of Plain and Finned Tubes - Part B: FinnedTubes", Handbook of Heat Exchanger Design, pp 2.5.3-6 a 2.5.3-16. [ Links ]

Manuscript received: June1999. Technical Editor: Angela Ourívio Nieckele.

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