Session 10

Chairman: D. Pepper

Natural convection heat transfer from a fin array is one of the major concerns on cooling of integrated circuits & electronic devices. Many analytical or experimental studies have been reported. This paper presents an analytical study on natural convection from a vertical fin array using a spectral finite difference method. Both the flow & the temperature fields are presented.

It is assumed that the fin array is mounted vertically & the temperature on the base is constant & uniform. It is also assumed that the height of the fin is small & the fin thermal efficiency is approximately unity. In this analysis, all the length are made dimensionless with respect to the gap c between the fins. Dimentionless temperature T is defined as , where T' is the local temperature of the fluid, : the ambient temperature, : the uniform temperature on the base plate. Reference velocity U is defined so that ,where Gr : Grashof number based on and c; : kinematic viscosity. Since the heat and fluid flow field is periodic in space, only one unit cell domain corresponding to the region enclosed by two fins is considered for numerical solution. Without loss of generality, it is assumed that the region extended to semi-infinite space is expressed in a Cartesian coordinate system by . corresponds to the base plate. Let the dimensionless height of the fins be L, so that fins are denoted by ,or Due to the symmetry of the heat and fluid flow field, along the axes ( or ) which does not lead to steady-state fields of infinite extension with finite temperature. Instead, to get steady-state fields of semi-infinite extension, the following assumptions along the axes are introduced : , where is a suitable constant to be given a priori. Using the following coordinate system ( one of a conformal mapping system ) ,

or equivalently

The equation of vorticity transport under Boussinesq approximation, the relation between vorticity and the stream function, and the energy equation neglecting dissipation can be given by ( in dimensionless forms )

(1)

(2)

(3)

(4)

where , and Pr denote a dimensionless stream function, dimensionless vorticity, and a Prandtl number respectively. The boundary of the domain is expressed by

Boundary conditions are

or or

or

or

or or

or

or

Since the analyzed region is expressed by the independent variable can be transformed in such a way that . Thus the coordinate system can be replaced by A spectral finite difference scheme by a time marching method is applied. For a symmetrical solution the following form is assumed:

, (5)

To maintain numerical stability for getting a steady-state solution by a semi-implicit scheme independently of each component at a stage of time integration, terms are added to the left hand side of Eqs.(1),(2), and (3) respectively, where is a time increment for integration; is a suitably chosen constant. Treatment for mixed boundary conditions ( Dirichlet and Neumann conditions ) is the same as in Ref.2. Auxiliary conditions at are given by As an initial condition for numerical integration, the following are used: ( inside the domain ). Time integration is based on a semi-implicit scheme.

Steady-state mean Nusselt number over the fin and base plate surfaces is given by

(6)

where Steady-state streamlines and isotherms for air are shown in Figs.1 - 6 under three conditions. Figure 7 shows variation of mean Nusselt number against for air.

It was demonstrated that, with proper transformation of governing equations and boundary conditions, it is possible to solve the natural convection heat transfer from a fin array, using a spectral finite difference method. Both the flow and temperature fields were presented in terms of various parameters.

1. Y.Mochimaru, A Spectral Finite Difference Scheme and its Effectiveness, Proc. 26th Ann. Iranian Math. Conf., Vol.2, 245-254(1995).
2. Y.Mochimaru, Extension of Applicability of a Spectral Finite Difference Scheme to Mixed Boundary Conditions, 9th International Sym. Transport Phenomena in Thermal-Fluids Engineering, Vol.1, 315-320(1996).

The main factor limiting transformer life is the hottest spot temperature inside the windings. According to the Montsinger relation, life expectancy of the electric insulation around those windings is halved for every 6 K temperature rise. This makes accurate prediction of the temperature distribution inside the transformer and particularly of the hottest spot temperature essential.

Since in operating conditions only the top oil temperature is measured, correlations are needed to calculate the temperatures in the windings. The available Nusselt correlations in literature involve 2D natural convection between vertical plates, subjected to either symmetric, uniform heat flux or uniform wall temperature as can be seen in Figure 1. These simplified boundary conditions hardly correspond with the real life problem, where asymmetric, highly non-uniform heating rates are present.

This study conducts a CFD-calculation of the hot spot temperatures for the parallel plate geometry, with highly non-uniform heating and mineral oil as the working fluid. After experimental verification of the CFD-model, it compares the obtained Nusselt number with Nusselt numbers, calculated from available correlations in literature.

The application of these results lies in the power transformer industry, where better predictive models of heat transfer are needed. This study has been dedicated to foil type winding of a 25 MVA transformer, restricted to one channel. The foil type winding consists mainly of copper or aluminium foil, covered with insulation and wound cylindrically, with radial cooling channels in between. The non-uniform heat production is caused by the electric current passing through the foil and supplementary losses near top and bottom of the windings due to stray losses.

The problem considered is a 2D conjugate heat transfer problem. The laminar flow of the incompressible fluid is governed by the continuity equation and the Navier-Stokes equations extended with a buoyancy source term.

(1) (2)

Since heat transport is involved, an additional transport equation has to be solved to compute the temperature. Temperature dependence of cooling mineral oil has to be taken into account.

The inlet temperature was held constant at 52.1 °C. The pressure drop of 17 Pa in the cooling unit (not modelled here) was imposed as a counter pressure at the outlet. The heat losses, due to the electric current through the foil and supplementary losses at top and bottom, were imposed as a volumetric heat generation. The heat generation is highly non-uniform and exceeds the average heat flux with 30 % at the bottom and the top of the foil.

The model was implemented in the PHOENICS code, using a staggered grid arrangement. The 2D simplest algorithm was used. The hybrid difference scheme was applied. Convergence was reached at 800 sweeps, using conservation of mass and energy as convergence criterion.

A profound grid study pointed out that a minimum of 35 exponentially distributed cells have to cover the width of each cooling channel, to reach grid independent results.

A linear relaxation factor of 0.2 was used for pressure and 0.6 for temperature. For the vertical velocity a false time relaxation of 0.001 was used. Pressure and temperature were solved by the whole-field method, velocity was solved point-by-point.

Pauwels Trafo performed temperature measurements inside the windings at the presumed hot spots on a full scale transformer. To do so, Copper-Constantane thermocouples with a thin epoxy insulating layer were wound together with the foil. They were located at 9 mm from the top of the foil and mid-width of each bundle. Temperature measurements were started after nineteen hours of operation, to ensure stabilised temperatures.

Figure 2 shows the temperature field in the foil windings and the cooling channels. Cold fluid entering at the bottom is heated by the heat, released in the windings. Figure 3 presents the computed Nusselt number in one of the cooling channels. The higher Nusselt number at the entrance is due to entrance effects. In the middle section the Nusselt number is quite constant. Near the exit region the Nusselt number increases again, because of the exponential increase in the heat, released in the windings near the top. The mean discrepancy derived from the measured temperatures versus the CFD-calculated temperatures for each of the foil bundles is 2.5ºC.

Figure 4 presents the CFD-calculated wall temperatures over the full height of the foil both for uniform and non-uniform volumetric heat production. Hot spot temperatures for non-uniform heat production are up to 30% higher than for uniform heat production.

Figure 4 also shows the wall temperatures at height x=L, as predicted by several correlations, found in literature. Those correlations all largely underestimate the corresponding CFD-calculated wall temperature at that height. This urges for new correlations which should show a spatial dependency factor including entrance effects and non-uniform heat flux distributions. Research is now directed towards CFD based computations.

Comparison of the CFD-calculated wall temperatures at the top of the foil with those calculated from correlations in literature, clearly shows that available correlations in literature cannot accurately predict hot spot temperatures. Even the best fitted correlation gives an underestimation of the hot spot temperature by 20% for a uniform heat input and by 40% for a non-uniform heat input. Therefore, at this instance, CFD calculations are needed to accurately predict hot spot temperatures. Also, while available correlations in literature only allow to calculate wall temperature at a certain height, CFD renders much more elaborate information, like a full height temperature profile.

In further investigations, improved Nusselt correlations will be derived taking into account the non-uniformity of the heat flux, so that wall temperatures can be predicted more accurately over the full height of the foil.

1. Bar-Cohen, A., and Rohsenow, W. M., “Thermally Optimum Spacing of Vertical, Natural Convection Cooled, Parallel Plates”, ASME Journal of Heat Transfer, Vol. 106, p 116-123, 1984.
2. Wirtz, R. A., Stutzman, R. J., “Experiments on Free Convection Between Plates With Symmetric Heating”, ASME Journal of Heat Transfer, Vol. 104, p 501-507, 1982.
3. Churchill, S. W., Usagi, R., “A General Expression for the Correlation of Rates of Heat Transfer and Other Phenomena”, Journal of American Institute of Chemical Engineers, Vol. 18, p 1121-1138, 1972.
4. Barrow, H., Prasad A.,Sherwin K., “Heat Transfer by Free Convection in Open Ended Vertical Ducts”, Natural Convection, p 59-63.

This paper presents an overview of the methodology and assumptions of the transient interfacial area model FLOWR. The paper also describes the implementation and testing of FLOWR in a prototype version of CATHENA¹ (Canadian Algorithm for THErmalhydraulic Network Analysis), a two-phase thermalhydraulic network code.

The transient interfacial area density model developed in the present work is integrated in a prototype version of the two-phase, two-fluid, thermalhydraulic systems code CATHENA. CATHENA has been in development at the Whiteshell Laboratories of AECL (Atomic Energy of Canada Limited) since the early 1980’s. It has extensively been validated against separate effects as well as integrated experiments. CATHENA is currently used in the safety and licensing analysis of AECL designed CANDUâ (CANadian Deuterium Uranium) PHWRs (Pressurized Heavy Water Reactors). It is also used in the design, safety and licensing analysis of reactors for research and isotope production such as the IRF (Irradiation Research Facility) or MAPLE (Multipurpose Applied Physics Lattice Experimental).

1. Richards, D.J., Hanna, B.N., Hobson, N. and Ardron, K.H., A Two-Fluid Code for CANDU LOCA Analysis, Proceedings of Third International Meeting on Thermal Hydraulics, Vol.2, Newport, R.I., 1985.