## Abstract

The ‘power integral method’ of calculating heat transfer across a convecting porous layer is extended to flux and porous boundaries. Convection starts at lower Rayleigh numbers for constant flux than for isothermal impervious boundaries and the flux is much greater. At higher Rayleigh numbers, as more of the higher modes contribute to the flux, the type of boundary has less influence on the heat transfer across the layer. For constant flux boundaries, simplified equations are developed to determine critical values for the second and higher modes and these values can be related simply to those for isothermal impervious boundaries.

## Key words

Natural convection heat transfer boundaries modes critical Nusselt power integral method## Nomenclature

*A*amplitude of disturbance

*a*wavenumber

*a*_{c,n}critical wave number of

*n*th mode of natural convection- \(a_p = \frac{{k_p }}{{(\rho c)_p }}\)
thermal diffusivity for porous layer

*B*_{b},*B*_{t}Biot numbers for bottom and top of porous layer, defined in (30)

- \(D \equiv \frac{\partial }{{\partial \xi }}\)
constants of integration in (14)

*E*_{n}factor in the flux of the

*n*th mode of natural convection, defined in (28)- \(F = (D^2 - a^2 ){\text{W}}\)
normalised bidimensional periodic function

**g**,*g*vector and scalar gravitational accelerations

*H*height of porous layer

*h*_{b},*h*_{t}heat transfer coefficients at bottom and top of porous layer

*k*_{p}effective thermal conductivity in porous layer

- Nu
Nusselt number

- \(p,\bar p\)
pressure and mean pressure on horizontal plane

*q*constant of integration

- \(T,\overline T\)
temperature and mean temperature on horizontal plane

*T*_{b},*T*_{t}mean temperatures at lower and upper boundaries \(\Delta T = T_b - T_t\)

*t*time

*u, v, w*velocity components in

*x, y, z*coordinates respectively**V**velocity

*W(ξ)*variation in the vertical direction of the vertical component of the velocity disturbance

*x, y, z*co-ordinates with

*z*vertical*Z(ξ)*variation of the temperature disturbance in the vertical direction

## Greek

*α*volumetric thermal expansion coefficient of fluid

*β*initial uniform temperature gradient

- γ
\(\surd \{ a(a + \lambda )\}\)

*δ*\(\surd \{ a(a - \lambda )\}\)

*δ*_{1}\(\surd \{ a(\lambda - a)\}\)

- \(\eta ,\zeta ,\xi\)
normalised co-ordinates with ξ vertical

*θ*temperature disturbance

*κ*permeability of porous layer

- λ
square root of Rayleigh number

- λ
_{c,n} square root of critical Rayleigh number of

*n*th mode of natural convection*μ*viscosity of fluid

*ν*kinematic viscosity of fluid

*ϱ*density of fluid

*(ϱc)φ*thermal capacity of fluid

*(ϱc)*_{p}thermal capacity of porous layer including voids

- Φ
*n* thermal flux due to

*n*th mode of natural convection*φ*void volume fraction of porous layer

*ω*pressure variation over horizontal plane

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