How to get rid of laplace of pressure in an equation

[Bianca123]

Hello!

I have a question about an equation.
I have an equation which is a boundary condition to a problem I have concerning fluid flow in a layered porous medium. I have a equation where my only variable is the temperature T, and I have 10 boundary conditions with it. In order to solve the system, I need to express all my boundary contitions in terms of T. I have managed it with all but i miss one, which contains Pressure.

I need to have the pressure expressed in terms of T in this equation:

∇^2 P = dT/dz.

I feel like I need some sort of conection where ∇P=aP, where a is a constant. Does anyone know how to solve it, or if such a relation exists?

 

[Chestermiller]

Can you elaborate a little more? I have experience with this this sort of thing, but I'm having trouble understanding the physical situation. First, please describe the physical situation, and then please write out the equations. Thanks.

Chet

 

[Bianca123]

I am looking at natural convection between two coaxial cylinders. Between them there are to layers of different height and permeability, which I will change in my calculations and see how this effects my solution.

Originally, after doing a permutation to the steady state solution, and after nondimensionalizing my variables, I am left with the following 3 equations:

Darcy's law;
v_i=-∇P_i + Ra_i/(b_i)² * T_i k

Mass conservation:
∇v_i=0

Energy conservation;
-v_(z,i)=∇²T_i

where
∇=d/dr + 1/r *d/dθ + 1/b_i * d/dz
b_i=h_i/h ( ratio of the heitght of the layer i, to the total height h of the cylinder)
Ra=Rayleigh number

I have managed to combine these three into the following equation:

∇⁴T_i+Ra/b_i² *∇T_i=0
I have boundary conditions at the bottom and top of my cylinder, at the sidewalls and 4 boundary conditions at the interface between the two layers: continuity in temperature, heat flux, pressure and vertical flow. I have managed to write all the boundary conditions in terms of T, except for the pressure. I have tried to take ∇ of darcy law, and replace the left hand side by 0( from mass eq). Then I have :
∇²P_i=Ra/b_i² * ∇T_i.

If there excist a relation for ∇P=aP, I would be able to solve the system. Is there any such relations?

 

[Chestermiller]

First of all, please let me ask. Is this a homework problem?

Secondly, I'm going to try to articulate the problem description in the way that I understand it.

You have an annular region between two concentric cylinders, and you have two porous plugs stacked on top of one another in the gap between the cylinders. The permeabilities and the heights of the plugs are different. I'm guessing that the temperatures of the two cylinders are different, and the temperatures are held constant, so that there is a steady state natural convection flow occurring. The bottom and top of the stack are insulated and do not permit vertical flow. You are to determine the velocity distribution and the temperature distribution in the porous media. Is this correct?

I would also like to see your equations in dimensional form. Please also state what assumptions were made so far. Also, please define the dimensionless variables.

You are aware that the system is axisymmetric, so that all derivatives with respect to theta are zero, correct?

How does the gap spacing between the cylinders compare with the vertical dimensions of the plugs?
How does the gap spacing compare with the inner radius?

Chet

 

[Bianca123]

No it is not a homework, it is my master thesis. In order to write out the system in a 8x8-matrix I need all the variables in T.

Top and bottom of the cylinders are impermeable and perfectly heat conducting. The sidewalls are impermeable and insulated. I am to see how natural convection cells appear, how many in direction of r and tetha. So I need to find a relation between a wavenumber and the critical rayleigh number. My supervisor said I should avoid an axisymmetric problem where I use that knowlegde.

My equations in dimensional form:

v_i=-K_i/μ(∇P_i + ρgk)

∇⋅v_i=0

(ρc)_m dT_i/dt + (ρc)_f v_i ⋅∇T_i=∇⋅(k_(m,i) ∇T_i)

where (ρc)_m = (1-φ)(ρc)_s +φ(ρc_p)_f (s:solid, f:fluid, c_p: spesific heat at constant pressure)The dimensionless variables with a *:

r*=r/h
Z_i *=(z-h_(i-1))/h_i
v_i=v_i'⋅h /(b_i)(α_f,i) where (α_f,i)=k_(m,i)/(ρc_p)_f
T_i*=T_i'/ΔT_i
P_i*=(P_i' )(K_i) /μ⋅b_i⋅(α_f,i)h is the total height, h_i is the height of layer i.

Did this help to understand what I am looking for?

 

[Chestermiller]

No it is not a homework, it is my master thesis. In order to write out the system in a 8x8-matrix I need all the variables in T.

Top and bottom of the cylinders are impermeable and perfectly heat conducting. The sidewalls are impermeable and insulated. I am to see how natural convection cells appear, how many in direction of r and tetha. So I need to find a relation between a wavenumber and the critical rayleigh number. My supervisor said I should avoid an axisymmetric problem where I use that knowlegde.

My equations in dimensional form:

v_i=-K_i/μ(∇P_i + ρgk)

∇⋅v_i=0

(ρc)_m dT_i/dt + (ρc)_f v_i ⋅∇T_i=∇⋅(k_(m,i) ∇T_i)

where (ρc)_m = (1-φ)(ρc)_s +φ(ρc_p)_f (s:solid, f:fluid, c_p: spesific heat at constant pressure)The dimensionless variables with a *:

r*=r/h
Z_i *=(z-h_(i-1))/h_i
v_i=v_i'⋅h /(b_i)(α_f,i) where (α_f,i)=k_(m,i)/(ρc_p)_f
T_i*=T_i'/ΔT_i
P_i*=(P_i' )(K_i) /μ⋅b_i⋅(α_f,i)h is the total height, h_i is the height of layer i.

Did this help to understand what I am looking for?

Yes. You're trying to solve a stability problem, with the cold boundary above the plugs, and the hot boundary below the plugs. Some questions:

Are you just trying to solve for the critical conditions at the onset of instability (in which case you can use a linear stability analysis), or are you trying to describe the system behavior beyond the value of the critical Rayleigh number?

Has this problem been solved for the case of a single plug, and do you have that solution available? If it has not been solved for a single plug, why are you doing two plugs?

Do you have any idea what the analytic form of the temperature and velocity disturbances look like? This should be available if the problem has been solved for a single plug.

I assume you are representing the temperature variation by a periodic function of time, and the theta variation by sines and cosines, correct? What about the variation in the axial direction?

Again, what is the radius to gap ratio?

The dimensionless radial coordinate should be normalized in terms of the gap, not h.

Why do you feel that you need to reduce this problem to an equation in a single variable?

Chet