How Does Airflow Through a Porous Wall Depend on Temperature Gradient?

[Cipherflak]

Homework Statement

What we know is that we have two impermeable walls with a porous material inside (the porous material gives a simply restraining force proportional to the movement itself). we have a stationaly flow, so that dw/dt=du/dt=0. u and w are the velocities in the x (horizontal) and z (vertical) directions, respectively. This is a 2-d problem, so we don't bother with the y-axis.[PLAIN]http://img135.imageshack.us/img135/4820/wallo.png

The temperature is a simple linear function of x, where Ty is the temperature of the outside wall and Ti is the temperature of the inside, like shown on the picture.

So what I'm trying to find is an expression of the vertical velocity w as a function of x, that is, w(x). A hint is given that ∂P/∂x=0 when P is pressure. Also, I need an expression for P.

Homework Equations

it is given that (stationary solution and Darcy's law for flow through porous material).
0= - ∂P/∂z - ρg - μ/kw

the temperature as a function of x is,
T(x)=1/2(Ti+Ty+(Ti-Ty)/L * x)

the density ρ of the fluid is
ρ=ρ0(1-αT) , where

as a boundary condition or whatever, we know that
w=0 @ x=0.

The Attempt at a Solution

to find and expression for w, it seems logical to solve the first equation for w and get
w=-k/μ(∂P/∂z+ρg), and then subtitute for ρ and then for T(x). But then I get this large awkward equation, and i have not used the hint that ∂P/∂x=0.

perhaps one should consider the continuity equation too, ∂u/∂x=-∂w/z ?

any insightful ideas? :)

 

[mark726]

Dear Forum Member,

Thank you for sharing your thoughts on this problem. I would approach this problem by first understanding the physical principles involved. From the given information, we know that we have a stationary flow through a porous material, with two impermeable walls and a linear temperature gradient along the x-axis. This suggests that we are dealing with a convection problem, where the flow is driven by the temperature difference between the two walls.

To find an expression for w(x), we can start by using Darcy's law, which relates the fluid velocity to the pressure gradient. In this case, the pressure gradient is given by ∂P/∂z, which includes the effects of gravity and the porous material. We can also incorporate the continuity equation, which tells us that the horizontal velocity u(x) is related to the vertical velocity w(x) through the partial derivative of w with respect to z.

Next, we can use the given expression for the temperature T(x) to calculate the density of the fluid ρ. This will allow us to simplify the expression for w(x) and eliminate any unnecessary terms. We can also use the hint that ∂P/∂x=0 to simplify the final expression for w(x).

Overall, our approach should be to use the given equations and information to derive an expression for w(x) that takes into account the effects of gravity, the porous material, and the temperature gradient. It may also be helpful to plot the solution and analyze it to gain a better understanding of the physical behavior of the system.

I hope this helps in your solution. Good luck!