User:Vanand040/sandbox

INTRODUCTION
"One major category of heat exchanger for such applications is referred to as porous media heat exchangers. The basic idea of the porous media heat exchangers is that enhanced cooling can be achieved because (i) larger surface areas available in porous particles as extended surfaces for heat transfer  (ii) mixing of fuids due to the presence of particles.  It follow the Darcy flow model.The convective heat transfer potential of flows through porous media is a relatively new topic, as the technologies of porous insulation, gas-cooled electric machinery, and nuclear reactors grew out of the contemporary concern with the cost of energy and the miniaturisation of cooling schemes."One of the major disadvantage using a porous heat exchanger, however, is the large pressure drop across the heat exchanger. In order to overcome this disadvantage,  one of the best methods is to reduce the flow velocity while keeping a higher heat transfer coefficient.

The basic problem in heat convection through porous media consists of predicting the heat transfer rate between a deferentially heated, solid impermeable surface and a fluid-saturated porous medium.We begin with the simplest wall heat transfer problem, namely, the interaction between a solid wall and the parallel flow permeating through the porous material confined by the wall.

Constant Wall Temperature
In 2D system, steady state governing equations are

$$\partial u/\partial x+\partial v/\partial y=0

$$                             (1)

from Darcy's law

$$u=-(K/\mu)\partial P/\partial x $$                           (2)

$$v=-(K/\mu)\partial P/\partial y $$                                (3)

energy equation

$$u\partial T/\partial x+v\partial T/\partial y = \boldsymbol{\alpha}{\partial^2\over\partial x^2}T $$                    (4)

where K is an empirical constant called permeability.

considering $$\rho $$ is constant and boundary layer is slender, and gravity effect is negligible.

considering uniform parallel flow

$$u= $$$$U_\infty $$                                                         $$v=0

$$

$$P(x)= -(\mu/K)U\infty  x+ constant

$$                            from (1)

Let $$\delta_t $$ be the thickness of the slender layer of length x that affects the temperature transition from $$T_0 $$ to $$T_\infty $$.

The energy equation reveals a balance between enthalpy flow in the x direction and thermal diffusion in the y direction

$$U_\infty\partial T/\partial x\sim \alpha\Delta T/\delta_t^2

$$

boundary is slender so    $$\delta_t<<x

$$

$$\delta_t/x \sim Pe_x^-.5

$$

$$Nu = hx/K \sim x/\delta_t \sim Pe_x^0.5

$$

The Peclet number is a dimensionless number used in calculations involving convective heat transfer. It is the ratio of the thermal energy convected to the fluid to the thermal energy conducted within the fluid.

$$Pe_x$$ $$= $$   Advective transport rate $$/$$ Diffusive transport rate

$$Pe_x = U_\infty x/\alpha

$$

Conclusion
The thermal boundary layer thickness $$\delta_t $$ increases as $$x^.5 $$ downstream from the point where wall heating begins. The local heat transfer coefficient decreases as $$x^-.5 $$.