Boundary conditions definition for heat equation PDE Toolbox
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From: Marcin on 19 May 2010 06:47 Hi, I'm struggling with heat transfer problem using PDE toolbox. I need to use FEM method, and figured out that PDE will be the easiest way. I've got the following situation. Rectangular plate, where both left and right borders don't exchange heat (adiabatic), bottom border got constant temperature. So far boundary conditions are quite simple to define. I've got a problem with the top border. Instead of temperature I need to define heat stream. In other words I need to assume some heat stream, let's say producing 10W, the whole heat is acquired by the plate. The bottom border got constant temperature of 300K. I need to calculate what will be the value of temperature on the top border. Is it possible to define under PDE? How to specify such boundary condition? Thanks, Marcin
From: Bruno Luong on 19 May 2010 07:15 "Marcin " <nygma(a)epf.pl> wrote in message <ht0fj7$rre$1(a)fred.mathworks.com>... > Hi, > > I'm struggling with heat transfer problem using PDE toolbox. I need to use FEM method, and figured out that PDE will be the easiest way. I've got the following situation. Rectangular plate, where both left and right borders don't exchange heat (adiabatic), bottom border got constant temperature. So far boundary conditions are quite simple to define. I've got a problem with the top border. Instead of temperature I need to define heat stream. In other words I need to assume some heat stream, let's say producing 10W, the whole heat is acquired by the plate. The bottom border got constant temperature of 300K. I need to calculate what will be the value of temperature on the top border. Is it possible to define under PDE? How to specify such boundary condition? Heat stream is modeled by Neumann boundary condition: du/dt = something. Bruni
From: Torsten Hennig on 19 May 2010 03:12 > Hi, > > I'm struggling with heat transfer problem using PDE > toolbox. I need to use FEM method, and figured out > that PDE will be the easiest way. I've got the > following situation. Rectangular plate, where both > left and right borders don't exchange heat > (adiabatic), bottom border got constant temperature. > So far boundary conditions are quite simple to > define. I've got a problem with the top border. > Instead of temperature I need to define heat stream. > In other words I need to assume some heat stream, > let's say producing 10W, the whole heat is acquired > by the plate. The bottom border got constant > temperature of 300K. I need to calculate what will be > the value of temperature on the top border. Is it > possible to define under PDE? How to specify such > boundary condition? > > Thanks, > Marcin PDE: -div(c*grad(u)) = 0 Top border: Boundary condition: Generalized Neumann (with q=0) : n*(c*grad(u)) = 10 W / (area of top border of the plate) Best wishes Torsten.
From: Marcin on 19 May 2010 11:35 Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote in message <620032725.185902.1274267607997.JavaMail.root(a)gallium.mathforum.org>... > > PDE: > -div(c*grad(u)) = 0 > > Top border: > Boundary condition: Generalized Neumann (with q=0) : > n*(c*grad(u)) = 10 W / (area of top border of the plate) Many thanks Torsten. The equation from my post is fairly easy to resolve in analytic way. I wanted to use this example to verify if I configured pde toolbox properly. I did some exercises but I was stupid enough to not notice that I've got different units. Heat conduction coefficient was in W/mK but the heat stream density was in W/mm2. That's why the result was not correct. Now everything is working fine. Can I have another question? Let's assume the example as in my initial post, but instead of constant temperature on the bottom border, we have heat convection with surrounding gas. How the boundary condition should look in such case. I believe that in such case I should define heat transfer coefficient (q) and assign g = 0. Am I right? Thanks, Marcin
From: Torsten Hennig on 19 May 2010 22:52
> Torsten Hennig <Torsten.Hennig(a)umsicht.fhg.de> wrote > in message > <620032725.185902.1274267607997.JavaMail.root(a)gallium. > mathforum.org>... > > > > > PDE: > > -div(c*grad(u)) = 0 > > > > Top border: > > Boundary condition: Generalized Neumann (with q=0) > : > > n*(c*grad(u)) = 10 W / (area of top border of the > plate) > > Many thanks Torsten. The equation from my post is > fairly easy to resolve in analytic way. I wanted to > use this example to verify if I configured pde > toolbox properly. I did some exercises but I was > stupid enough to not notice that I've got different > units. Heat conduction coefficient was in W/mK but > the heat stream density was in W/mm2. That's why the > result was not correct. Now everything is working > fine. > > Can I have another question? Let's assume the example > as in my initial post, but instead of constant > temperature on the bottom border, we have heat > convection with surrounding gas. How the boundary > condition should look in such case. I believe that in > such case I should define heat transfer coefficient > (q) and assign g = 0. Am I right? > > Thanks, > Marcin In this case, the boundary condition is usually given by -n*(c*grad(u))=alpha*(u-u_inf) with heat transfer coefficient alpha and u_inf as the temperature of the surrounding gas. Rearranging gives n*(c*grad(u))+alpha*u = alpha*u_inf, thus q = alpha, g = alpha*u_inf. Best wishes Torsten. |