# How can I get steady state of complicated heat transfer problem?

• yhyoon0511
yhyoon0511
Homework Statement
What needs to be obtained is the temperature distribution of thermos that have reached thermal equilibrium. The thermos consists of three parts. The 'pillar' is a hollow cylinder whose inner radius is R_in and outside radius is R_out, height h. The 'lid border' is a solid of revolution made by rotating a quadrant with a radius of R_out - R_in. The 'lid center' is a cylinder, the radius and the height is R_out - R_in. The 'lid border' is placed on the 'pillar', and 'lid center' occupies the inside of the 'lid center'. The surface of the thermos is in contact with two heat sources. First, the high heat source which has a temperature of Th covers the inside. Next, the low heat source which has a temperature or Tl covers the outside except the bottom of the thermos. And the bottom of the 'pillar' is insulated.
Relevant Equations
Heat transfer equation
Lapace equation in cylindrical coordinate
This problem is created by my cousin who is learning about 1-D heat transfer problems. He made this problem and made solve it with fourier's law of heat transfer with calculating thermal resistance by integrating 1/A dx, which presupposes a false assumption; if the distance away from the heat source is the same, the temperature will be the same. I find out it was a wrong solution, but I can't solve this problem with my knowledge. The hardest problem I've solved used the separation of variables which assuming the steady state temperature can be expressed as R(r)Z(z)(applying this solution for this problem, theta term is omitted because there is theta directional symmetry). However, with r direction boundary conditions varying through the z direction, I don't think I can get suitable answer with this problem.

yhyoon0511 said:
Homework Statement: What needs to be obtained is the temperature distribution of thermos that have reached thermal equilibrium. The thermos consists of three parts. The 'pillar' is a hollow cylinder whose inner radius is R_in and outside radius is R_out, height h. The 'lid border' is a solid of revolution made by rotating a quadrant with a radius of R_out - R_in. The 'lid center' is a cylinder, the radius and the height is R_out - R_in. The 'lid border' is placed on the 'pillar', and 'lid center' occupies the inside of the 'lid center'. The surface of the thermos is in contact with two heat sources. First, the high heat source which has a temperature of Th covers the inside. Next, the low heat source which has a temperature or Tl covers the outside except the bottom of the thermos. And the bottom of the 'pillar' is insulated.
Relevant Equations: Heat transfer equation
Lapace equation in cylindrical coordinate

This problem is created by my cousin who is learning about 1-D heat transfer problems. He made this problem and made solve it with fourier's law of heat transfer with calculating thermal resistance by integrating 1/A dx, which presupposes a false assumption; if the distance away from the heat source is the same, the temperature will be the same. I find out it was a wrong solution, but I can't solve this problem with my knowledge. The hardest problem I've solved used the separation of variables which assuming the steady state temperature can be expressed as R(r)Z(z)(applying this solution for this problem, theta term is omitted because there is theta directional symmetry). However, with r direction boundary conditions varying through the z direction, I don't think I can get suitable answer with this problem.View attachment 337565
Suppose the lid border were insulation rather than the same material as the pillars and the lid center. Could you solve it then? In your judgment, how does the heat loss through the lid border compare to the heal loss through the pillar and the lid center (qualitatively) in the actual set up?

## 1. What is a steady-state heat transfer problem?

A steady-state heat transfer problem refers to a situation where the temperature distribution in a material does not change over time. This means that the heat entering any part of the system is equal to the heat leaving it, leading to a constant temperature field.

## 2. How do I set up the boundary conditions for a steady-state heat transfer problem?

To set up boundary conditions, you need to specify the temperatures or heat fluxes at the boundaries of the domain. Common types include Dirichlet boundary conditions (fixed temperatures), Neumann boundary conditions (fixed heat flux), and Robin boundary conditions (convective heat transfer). Properly defining these conditions is crucial for solving the problem accurately.

## 3. What numerical methods can be used to solve steady-state heat transfer problems?

Numerical methods such as Finite Difference Method (FDM), Finite Element Method (FEM), and Finite Volume Method (FVM) are commonly used to solve steady-state heat transfer problems. These methods discretize the domain into smaller elements or volumes and solve the heat transfer equations iteratively.

## 4. How do I ensure convergence when solving a steady-state heat transfer problem numerically?

To ensure convergence, you need to choose appropriate discretization parameters, such as time step size and grid resolution. Additionally, using iterative solvers like Gauss-Seidel or Successive Over-Relaxation (SOR) with proper relaxation factors can help achieve convergence. Monitoring the residuals and ensuring they fall below a certain threshold is also important.

## 5. Can software tools help in solving steady-state heat transfer problems?

Yes, various software tools like ANSYS, COMSOL Multiphysics, and MATLAB can assist in solving steady-state heat transfer problems. These tools offer built-in solvers and user-friendly interfaces for setting up the geometry, boundary conditions, and material properties, making the process more efficient and accurate.

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