Conjugated Conduction-Free Convection Heat Transfer in an Annulus Heated at Either Constant Wall Temperature or Constant Heat Flux

In this paper, we investigate numerically the effect of thermal boundary conditions on conjugated conduction-free convection heat transfer in an annulus between two concentric cylinders using Fourier Spectral method. The inner wall of the annulus is heated and maintained at either CWT (Constant Wall Temperature) or CHF (Constant Heat Flux), while the outer wall is maintained at constant temperature. CHF case is relatively more significant for high pressure industrial applications, but it has not received much attention. This study particularly focuses the latter case (CHF). The main influencing parameters on flow and thermal fields within the annulus are: Rayleigh number Ra; thickness of inner wall Rs; radius ratio Rr and inner wall-fluid thermal conductivity ratio Kr. The study has shown that the increase in Kr increases the heat transfer rate through the annulus for heating at CWT and decreases the inner wall dimensionless temperature for heating at CHF and vice versa. It has also been proved that as the Rs increases at fixed Ra and Rr, the heat transfer rate decreases for heating at CWT and the inner wall dimensionless temperature increases for heating at CHF at Kr <1. The study has also discussed that the effect of increase in Rs for both cases of heating at Kr>1 depends on Rr. It has been shown that for certain combinations of controlling parameters there will be a value of Rr at which heat transfer rate will be minimum in the annulus in case of heating at CWT, while there will be a value of Rr at which inner wall dimensionless temperature will be maximum in case of heating at CHF.


INTRODUCTION
include thermal storage systems, solar collector-receivers, underground power transmission cables, cooling system in nuclear reactor and many others [1]. The heat transfer in such geometry occurs as a result of temperature difference created due to heating of one wall and cooling B uoyancy driven flow and associated heat transfer in an annular enclosure between two concentric/eccentric circular walls has long been investigated because of its pertinence to many practical engineering applications. These applications the other. The heating process may be done at either prescribed wall temperature or prescribed heat flux. The present work considers heating the inner wall at either CWT or CHF while outer wall is cooled and maintained at constant temperature. The case of isothermal heating (CWT) approximates the cooling of microelectronic equipment while dissipation of heat generated within an underground power transmission cables from its surface to the surrounding enclosure is a practical example of heating at CHF [2].
Kumar [22] showed that wall temperature is a function of diameter ratio and Rayleigh number. On the other hand, Castrejon and Spalding [23] made an experimental and theoretical study of transient free-convective flow between horizontal concentric cylinders. Yoo [24] discussed flow pattern of natural convection while Yoo [25] studied dual free-convective flows in a horizontal annulus with constant heat flux on inner wall. Ho, et. al. [26] conducted a numerical study of natural convection in concentric and eccentric horizontal cylindrical annuli with mixed boundary conditions. The thermal boundary conditions definitely affect the flow and thermal fields within the concentric annulus. A common practice adopted by heat transfer community, as can be inferred from previous works , is to prescribe thermal boundary condition at the fluid-wall interface.
Consequently, energy equation of fluid alone has to be solved and numerical results become unreliable for wall having high thickness or low thermal conductivity. In most of high-pressure industrial applications, thickness of wall is high and conduction in solid wall needs to be coupled with convective heat transfer in fluid (named as conjugate heat transfer).
The effect of conjugation for Newtonian fluid in a concentric annulus was considered by Kolesnikov and Bubnovich [27]. The authors investigated conjugated conduction-free-convective heat transfer in horizontal cylindrical coaxial channels subjected to CWT heating.
The case of CHF heating is more relevant to industrial applications than CWT case but it has not been widely investigated so far as per author's best knowledge. The objective of this study is to investigate conjugate heat transfer through concentric annulus, heated either isothermally or through constant heat flux with focus on the latter case (CHF). Moreover, the study aims to provide accurate numerical results by adopting numerical solution based on Fourier Spectral method. Fig. 1 illustrates the physical domain of the problem which is the annulus space between two long concentric cylinders.

PROBLEM FORMULATION AND SOLUTION
The inner cylinder has inner radius r i (heating side) and outer radius r sf (fluid side), while outer wall has radius r o . The

The Governing Equations
For Buoyancy driven flow, time dependent governing equations of mass, momentum and energy in Cartesian coordinates in terms of stream function, vorticity and temperature for the above mentioned problem can be written as: Energy equation in fluid part (r sf < r < r o ) In the solid wall of the inner cylinder (r i < r < r sf ) the energy equation can be written as: Where τ is the time, ψ' is the stream function, ζ' is the vorticity, T is the temperature, ρ is the density, v is the kinematic viscosity and F x = ρgβ (T-T o ) is upward buoyancy force F b .k s , α s and k f , α are thermal conductivity and thermal diffusivity of the solid and fluid respectively.

Initial and Boundary Conditions
At outer wall ξ = ξ o

Method of Solution and Numerical Procedure
The process of solving the governing Equations (9-12) in modified polar coordinates along with the boundary conditions Equations (13)(14)(15), is based on Fourier Spectral method [28][29][30]. In this method, the stream function, vorticity and temperature are approximated as a Fourier series expansion. This approach is related to that used by Mahfouz and Imtiaz [1], Collins and Dennis [28], Badr and Dennis [29] and Mahfouz and Badr [30]. The Where S 1n , S 2n and S 3n are all easily identifiable functions of ξ and t. The boundary conditions in term of Fourier coefficients can be expressed as: Equation (20) is multiplied by e -nξ and then integrated with respect to ξ from ξ = ξ sf to ξ = ξ o . After using Equation (24), the resulting equation can be written as: Equation (26) is used for calculating the g n on the fluid side of the inner wall while g n at outer wall can be calculated by using Equation (20) directly. Finite difference method is used for discretization of differential Equations (19)(20)(21)(22)(23)(24)(25) where all spatial derivatives are computed using central

Heat Transfer Parameters
The local Nusselt Number at the inner side of inner wall and at the solid-fluid interface is defined as: Where Similarly the local Nusselt number at the outer wall is defined as: The equivalent thermal conductivity along inner and outer wall is defined as: The average Nusselt number can be seen as a dimensionless heat transfer rate at the wall and can be defined as: Nud Nu (30) In terms of Fourier coefficients, the average Nusselt number at the two sides of inner wall can be written as: And at the outer wall as: The Nu calculated from Equation (30) is used to quantify the heat transfer rate through the concentric annulus. At the steady state, the energy conservation entails that Nu at heating side of inner wall would be equal to that of fluid side and both become equal to that of outer wall.
That is, at the steady state the In case of heating at CHF, the rate of heat transfer is known (heat flux multiplied by surface area) and for this case the most important parameter is the inner wall temperature which should be controlled in order to avoid the overheating of the annulus for industrial purposes.
The value of φ for either side of the inner wall is calculated from Equation (18) and the corresponding mean dimensionless temperature is then calculated as:

RESULTS AND DISCUSSION
The

Heating at CWT
In case of heating at CWT the main target is to study the effect of controlling parameters on heat transfer rate  On the other side, for low conductivity inner wall material at Kr = 0.25 the increase in Rs means replacing the high conductive fluid at low Rayleigh number with lower conductive material, which in turn increases the overall thermal resistance and thus decreases the heat transfer through the annulus as can be seen in Table 2.

Conjugated Conduction-Free Convection Heat Transfer in an Annulus Heated at Either Constant Wall Temperature or Constant Heat Flux
The thermal conductivity of material of inner wall plays a vital role in controlling heat transfer rate through the concentric annulus. In industrial applications, the replacement of Aluminum with Copper enhances the heat transfer through the annulus. In order to quantify the effect of Kr on heat transfer rates the results of steady Nu at Rs = 1.3 and at different values of Kr are presented in Table 3. It can be seen in Table 3  is low in case of conduction dominating flows while for convection dominating flows, this increase is high. For instance, at Ra = 10 3 , Rr = 5.0 the percentage increase in steady Nu is 81% as Kr = 0.5-100, whereas at Ra = 10 5 , Rr = 5.0 this percentage goes to 217%.

Heating at CHF
This section focuses on the effect of controlling parameters on both flow and thermal fields in case of heating inner wall at CHF. In this case the inner wall temperature and temperature of the solid-fluid interface are the most important parameters for the annulus design.
In case of heating at CWT, it was found that the steady state heat transfer rate through the annulus depends on the overall thermal resistance between the two cylinders.
Similarly, in case of heating at CHF the dimensionless temperature of the two sides of the inner wall depends on the overall thermal resistance between the two cylinders.
The overall thermal resistance dependence on the controlling parameters has been discussed in case of CWT. In case of CHF the smaller the thermal resistance, the higher the overall heat transfer coefficient is and thus the lower the temperature of the inner wall and vice versa.   temperature at the fluid side of the inner wall is less than that of the heating side. The temperature difference between heating side and fluid side depends on the inner wall thickness (Rs) and thermal conductivity ratio (Kr), the less Rs and/or the higher Kr, the smaller the temperature difference between the two inner wall sides. It can be observed from the two tables that the increase in Rs for conduction dominating cases (Ra = 10 3 ) leads to increase in inner wall temperature for Kr < 1, while at Kr >1 the inner wall temperature rises with increase in Rs only at high Rr (i.e. Rr > 2.6). The effect of increase in Rs for convection dominating cases (Ra F = 10 5 ) in the annulus results in an increase in inner wall temperature except for very narrow annulus space (Rr = 2.0) and Kr >1. The material properties of inner wall for CHF heating are investigated in Table 6 Table 6.
One further observation that can be inferred from data of