All Questions
Tagged with heat-transfer finite-difference
12 questions
1
vote
1
answer
251
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Coupled Partial Differential Equations
I'm trying to solve the following system of coupled differential equations, the two-temperature model for $e$ = electrons and $l$ = lattice.
$$
\rho_{e}C_{p,e}\frac{\partial T_{e}}{\partial t} = k_{e}\...
- 11
2
votes
1
answer
410
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Where am I making a mistake in solving the heat equation using the spectral method (Chebyshev's differentiation matrix)?
I would like to numerically solve the following heat equation problem:
$$ u_t = \Bigg(2{a \over l}\Bigg)^2 u_{xx} \tag 1$$
$$ x \in [ -1, 1 ] \tag 2$$
$$ u(x, 0) = 0 \tag 3$$
$$ u(1, t) = A \sin \Bigg(...
2
votes
0
answers
114
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Efficient heat diffusion implementation with varying coefficients
I have the following heat diffusion equation:
\begin{alignat}{3}
\partial_t u(t, \vec{x}) &= g(\vec{x})\Delta u(t,\vec{x}), &\quad& \vec{x} \in\Omega, \, t\in(0,\infty],\\
\partial_n u(t,\...
- 2,882
2
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0
answers
92
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How to accelerate the computing of implicit finite difference method for heat conduction between two solids
Edit on May 3rd: I have found the problem. Because the difference of between $k_1$ and $k_2$ is huge, a very small time step need to be chosen so that the right green part can "feel" the ...
- 21
1
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1
answer
2k
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How to solve heat equation in spherical coordinates with finite differences?
I have a problem dealing with heat transfer which is spherically symmetrical. I was thinking it should be possible to solve this as a 1d problem in spherical coordinates using the radius only.
...
- 111
0
votes
1
answer
214
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Incorporating heat flux into Laplace Equation
I need to find the temperature distribution of a square plate using the Laplace equation by using FDM:
$$ \frac{d^2T}{dx^2} + \frac{d^2T}{dy^2} = 0$$
But there is a heat flux entering from the top ...
- 145
1
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0
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456
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Incorporating radiation boundary condition at the edge in finite difference
I am trying to solve the 2-d heat equation on a rectangle using finite difference method. I am confused as to how to incorporate non linear radiation boundary condition at the edge.
$-k\frac{\partial ...
- 19
1
vote
1
answer
15k
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Python Finite Difference Schemes for 1D Heat Equation: How to express for loop using numpy expression
Hello all,
I've recently been introduced to Python and Numpy, and am still a beginner in applying it for numerical methods. I've been performing simple 1D diffusion computations. I suppose my ...
2
votes
1
answer
708
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Solve 3-D Heat equation with Neumann boundaries
I want to solve the Poisson PDE for heat flow in a 3-D solid cube with given dimensions $x$, $y$, and $z$:
$$\rho C\frac{\partial T}{\partial t} = k \Delta T$$
The cube is irradiated with a constant ...
- 21
1
vote
2
answers
398
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V-cycle Multigrid for 2D transient heat transfer on a square plate using finite difference
I'm currently developing a program to solve 2D transient state heat conduction on a square plate using the V-cycle multigrid. Althought my program is able to reach the steady state solution, it's ...
- 11
1
vote
2
answers
2k
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Finite Difference Grid Spacing and Scaling
I have been exploring finite differences and heat transfer using the 2D heat equation to further expand my knowledge. So far I think it is going well.
I am running into some confusion around grid ...
- 11
0
votes
1
answer
852
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Heat equation with Neumann and Dirichlet conditions on same boundary
I am looking at numerical solutions to the heat equation with Dirichlet and Neumann conditions on the same boundary. That is $u(x,t)$ satisfying
$$
u_t = u_{xx}\,, \quad x \in[0,1]\,, \quad t>0\,,...
- 531