Abstract
This work studies the forced convection problem in internal flow between concentric annular ducts, with radial fins at the internal tube surface. The finned surface heat transfer is analyzed by two different approaches. In the first one, it is assumed one-dimensional heat conduction along the internal tube wall and fins, with the convection heat transfer coefficient being a known parameter, determined by an uncoupled solution. In the other way, named conjugated approach, the mathematical model (continuity, momentum, energy and K-epsilon equations) applied to tube annuli problem was numerically solved using finite element technique in a coupled formulation. At first time, a comparison was made between results obtained for the conjugated problem and experimental data, showing good agreement. Then, the temperature profiles under these two approaches were compared to each other to analyze the validity of the one-dimensional classical formulation that has been utilized in the heat exchanger design.
Conjugated heat transfer; Nusselt number; fin; heat exchanger
Comparison between One-Dimensional Uncoupled and Convection-Conduction Conjugated Approaches in Finned Surface Heat Transfer
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
This work studies the forced convection problem in internal flow between concentric annular ducts, with radial fins at the internal tube surface. The finned surface heat transfer is analyzed by two different approaches. In the first one, it is assumed one-dimensional heat conduction along the internal tube wall and fins, with the convection heat transfer coefficient being a known parameter, determined by an uncoupled solution. In the other way, named conjugated approach, the mathematical model (continuity, momentum, energy and K-
Keywords: Conjugated heat transfer, Nusselt number, fin, heat exchanger
Introduction
The use of extended surfaces is an important mechanism to intensify the heat transfer rate in 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 conduction problem while the interaction at the interface solid-fluid is represented by a convection coefficient (Gardner, 1945; Brown, 1965). In this classic approach the complex interaction between the heat transfer processes of conduction in the solid domain (internal tube wall and fins) and convection in the flow along the fins is substituted approximately by a uniform convection coefficient at the whole interface.
The experimental results for the convection heat transfer coefficient are extrapolated to an infinite solid conductivity to represent a uniform temperature at solid-fluid interface, as detailed in Stasiulevicius and Skrinka (1988). When the convection coefficient is determined by a theoretical solution a boundary condition of uniform temperature is imposed through the solid-fluid interface. These values of the convection coefficient show accentuated variations and it is needed a method to obtain a medium value to be applied to the classic fin theory. The procedure described in Zukauskas and Skrinka (1992) is adopted in this work for the uncoupled formulation. However, the application of the constant temperature as boundary condition doesn't represent the correct interaction of the heat transfer mechanisms in the solid-fluid interface. The temperature distribution and the heat flow in the solid-fluid interface should be the result of the interaction between the conduction in the solid domain and convection in the fluid region. To do so, it should be used a procedure where the conditions of heat flux and temperature equality are intrinsically imposed by the solution method, considering the solid and the fluid as a single domain. This coupled formulation is mentioned in the literature as conjugated problem (Davis and Gill, 1970). Several works studied the conjugated heat transfer in laminar regime as He et al. (1995) that studied the conjugated problem for the flow between parallel plates, using the finite difference technique to analyze the fluid region and the contour element method in the solid domain.
There are different experimental works that describe the use of extended surfaces to enhance the heat transfer rate. Knudsen and Katz (1950) analyzed the heat exchange processes 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 for concentric annular tubes configuration. The present work focuses on the turbulent flow in finned concentric annuli. The results obtained by two different approaches are compared: in the first treatment, a numerical solution is obtained for the conjugated heat transfer problem and, under the second one, an uncoupled one-dimensional formulation, which has been applied to the heat exchangers design, is used.
In the uncoupled approach, a global convection coefficient is adopted as representative of the heat exchanges with the fluid. It is intended, therefore, to verify the validity of this process of obtaining a medium convection coefficient proposed by Zukauskas and Skrinka (1992). Besides, the constant temperature hypothesis at the interface (applied to the classic solution of the convection problem) will be tested against the real variations that occur along the interface solid-fluid. A CFD (Computational Fluid Dynamics) code, based on the 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 was employed to treat the turbulence terms. The numerical results obtained for the Nusselt number were compared with the experimental data of Obermeier and Henne (1993) showing a good agreement. Besides, the influence of the fluid to wall thermal conductivity ratio in the heat flux rates and in the fin temperature distribution was analyzed, comparing the two formulations.
Results showed that the one-dimensional formulation provides good estimate for the total heat exchange rate, but it presents accentuated differences in relation to the conjugated approach when the fin temperature distribution is estimated.
Problem Geometry and Mathematical Formulation
The numeric simulations were accomplished in a domain representing the experimental apparatus specified in Obermeier and Henne (1993). A schematic representation of the experiment is described in Fig. 1, where s indicates the fins height, constant along L (heated tube extension), and p the fins spacing.
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 heat transfer 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 transfer rate [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 rwe ri represent the radii of the internal and external surfaces of the inner tube, respectively, and reindicates the external tube radius. The fin has a external radius rf and thickness t.
In the conjugated approach the water flow between concentric tubes is simulated, under the 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.
In the system of equations (1) to (7), r is the fluid density, T is the temperature, u and v indicate the axial and radial velocity components, Cp the constant pressure specific heat and mefrepresents 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 Ktare the molecular and turbulent thermal conductivities, respectively.
In this work, all the turbulent properties are calculated according to k-e model of Launder and Spalding (1974). In the solid-fluid interface, both boundary values of u and v are set to zero. 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 is imposed for the pressure (p = 0) and natural conditions for the other variables. In the proximity of the solid wall, the program employs the Wall Logarithmic Law to establish values 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 the solid and the fluid. The system of equations (1) to (7) and the boundary conditions determine the solution for the convection heat transfer problem. The conduction in the solid is simulated using null values to u and v in the energy equation (4). The conditions of adherence of the fluid and no temperature jump at the solid wall as well as heat flow equalities 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 finite element method. The governing equations were solved using a segregated formulation (where the fields of u, v, P, T, k and e are separately calculated) as detailed in Rice and Schnipke (1986). The domain was discretised with a structured mesh, refined in the areas that occurs accentuated gradients, allowing to capture boundary layer effects (thermal and hydrodynamic). The systems of algebraic equations were calculated by iterative methods as TDMA (Tri-diagonal Matrix Algorithm) and PCG (Preconditioned Conjugate Gradient).
The numerical simulations were accomplished using a Pentium-Pro microcomputer (200 MHz and 128 MB of memory RAM), and were composed of three stages:
1) - Pre-processing: represented by the problem geometry statement, mesh generation and boundary conditions application, that is, the domain representation in terms of finite elements;
2) - Numerical analysis: in this stage, the numerical solution algorithm for the configuration established in the stage 1 is applied;
3) - Post-processing: once the numeric solution was concluded (obeying the imposed convergence criterion), the visualization of the obtained results is performed. The program includes a graphic interface to visualize basic variables as pressure, velocity, temperature and 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 domain including only the fluid and constant temperature as boundary condition at the interface with the solid. The convection coefficient varies strongly along the tube and fins surfaces but the one-dimensional approach supposes that the heat exchange is represented by a medium and uniform convection coefficient at the solid-fluid interface. In the procedure presented in Zukauskas and Skrinka (1992) this coefficient is obtained by the following expression:
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 Afindicates 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 iterative procedure is needed to determine this parameter. In Zukauskas and Skrinka (1992) the value of h was calculated with a unique iteration. In the present work, several iterations were accomplished until reaching the solution convergence. Those authors suggest that the convection coefficient obtained according to Eq. (14) is valid for different fin materials, since the conditions of similarity of the problem are maintained. However, Stasiulevicius and Skrinka (1988) proposed a procedure of extrapolation for the experimental curve of the convection coefficient, obtained with different finite values of fin thermal conductivity, to infinite 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 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 to study tube wall and fins conduction problem. Then, a thermal circuit (resistance model) can represent the heat transfer for each cell showed in the Fig.3 (where the abbreviated terms are 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 in terms 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.
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 the conduction and convection thermal resistance, adopting a uniform convection coefficient surrounding the fin. Thus, low values of Biot (Bi < 0.1) indicate that the convection resistance controls the heat transfer process, and the one-dimensional calculations are valid in the classical uncoupled approach.
To compare the conjugated treatment and the uncoupled formulation, the following procedure was adopted:
(a) the convection coefficient was determined using Eqs. (14), (15) and (16), where the necessary data to calculate the Qm and Tmvalues were obtained by the conjugated formulation using a very large solid thermal conductivity, which implies in a uniform solid/fluid interface temperature;
(b) for finite thermal conductivity the total heat transfer rate was calculated by two approaches: in the uncoupled treatment TQ was obtained by Eqs. 18 to 20, with the h value determined in (a), and to determine the Qmvalue 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 experimental data 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 also presented in Table 1, and they show good agreement with the experimental results where the differences are less than 5%.
The total heat transfer rate was calculated by the uncoupled and conjugated formulations for 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 the studied interval of the solid to fluid conductivity ratio. This indicates that the uncoupled treatment, which uses a global and uniform convection coefficient, furnishes a good method to estimate the total heat transfer rate.
However, the fin temperature distribution determined by the uncoupled formulation (where the convection coefficient is determined by the above described method) shows large differences when compared with the conjugated approach results. Figures 5 and 6 present the temperature profiles for two values of thermal conductivity ratio: Ks/Kf = 1 and Ks/Kf = 500.
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 fin extension ( 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 is noticed 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 smaller than the conjugated approach one. This happens because the wall temperature profile for Ks/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 fin temperature because it considers only the heat flux radial component in the wall and ignores the axial conduction to the fin root.
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 Table 2. According to a two-dimensional conduction analysis, for the fin domain only, the temperature only varies with the radial coordinates when the Biot number is smaller than 0.1, showing that the one-dimensional approach is valid to calculate the heat transfer in the fin region.
The one-dimensional conduction heat transfer analysis (for the fin domain only) is valid when 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 reproduce the conduction heat transfer to the fin root that elevates the temperature at this point, as can be seen in Fig. 6 and Fig.7. This fact implies in larger differences between the temperature 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 low thermal conductivity (insulation material). An opposite effect occurs in Fig. 7 where the heat is transferred towards the fin base.
Conclusions
Results obtained for Nusselt number by the conjugated treatment using the standard k-e were compatible with the experimental data. It was shown that the one-dimensional formulation 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-dimensional conjugated approach when the temperature distribution fins is analyzed. The one-dimensional uncoupled formulation doesnt capture the temperature depression or elevation close to the fin root while the two-dimensional conjugated approach reproduces the tube wall temperature distribution combining the radial and axial conduction effects.
Comparison between the results obtained by the two approaches showed that the classic one-dimensional formulation was valid for the heat exchangers design to determine the total heat transfer rate. However, to calculate the temperature distribution in finned tubes the 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).
Manuscript received: June1999. Technical Editor: Angela Ourívio Nieckele.
- Brown, A., 1965, "Optimum Dimensions of Uniform Annular Fins", Int. J. Heat Mass Transfer, Vol. 8, pp 655-662.
- Davis, E.J. and Gill, W.N., 1970, "The Effects of Axial Conduction In The Wall On Heat Transfer With Laminar Flow", Int. J. Heat Mass Transfer, Vol. 13, pp 459-470.
- Gardner, K.A., 1945, "Efficiency of Extended Surface", Trans. ASME, Vol. 67, pp 621-631.
- He, M., Bishop, P.J.; Kassab, A.J. e Minardi, A., 1995, "A coupled FDM/BEM Solution for the Conjugate Heat Transfer", Numerical Heat Transfer, Parte B, Vol. 28, pp 139-154.
- Krauss, A.D., and Bar-Cohen, A., 1995, "Design and Analysis of Heat Sinks", John Wiley & Sons, Inc.
- Knudsen, J.G., and Katz, D.L., 1950, "Heat Transfer and Pressure Drop In Annuli", Chem. Eng. Progress, Vol. 56, pp.490-500.
- 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.
- Obermeier, E, and Henne, R. A., 1993, "Performance Analysis of Augmented Heat Transfer Surfaces - Theoretical and Experimental Results for Tubes with Circunferencial Fins in Longitudinal Flow", Proceedings of the First International Conference on Aerospace Heat Exchanger Technology, pp. 329-345, Palo Alto, USA.
- Rice, J.G. and Schnipke, R.J., 1986, "An Equal-Order Velocity-Pressure Formulation That Does Not Exhibit Spurious Pressure Modes", Computer Methods In Applied Mechanics And Engineering, Vol.58, pp 135-149.
- Stasiulevicius, J. and Skrinka, A., 1988, "Heat Transfer of Finned Tube Bundles in Crossflow", Hemisphere Publishing Corporation.
- Zukauskas, A. and Skrinka, A., 1992, "Banks of Plain and Finned Tubes - Part B: Finned Tubes", Handbook of Heat Exchanger Design, pp 2.5.3-6 a 2.5.3-16.
Publication Dates
-
Publication in this collection
13 Sept 2001 -
Date of issue
2000
History
-
Received
June 1999
