Question

How can we use the one-term approximate solutions when the surface temperature of the geometry is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient?

Short answer

Question: Explain how to use one-term approximate solutions for heat conduction problems when the surface temperature is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient. Answer: To use one-term approximate solutions for heat conduction problems with specified surface temperature, follow these steps: 1. Understand the one-term approximate solution method, which uses a single term of the Fourier series representation to estimate the exact solution. 2. Identify the given boundary condition, which is the surface temperature (T_s). 3. Modify the original boundary condition by relating T_s with the temperature of the surrounding medium (T_infty) and the convection heat transfer coefficient (h) using Newton's law of cooling. 4. Apply the one-term approximate solution method by selecting the dominant term in the Fourier series representation based on the geometry and boundary conditions. 5. Solve for the temperature field (T) in the geometry using the chosen dominant term. 6. Check the accuracy of the solution by comparing it to the exact solution or performing more detailed numerical simulations.

Step-by-step solution

Understand the one-term approximate solution method

One-term approximate solution method is used to solve heat conduction problems by approximating the exact solution of the problem using a single term of its Fourier series representation. This single term is usually the dominant term in the series, and its coefficients are determined based on the geometry and boundary conditions of the problem.

Identify the given boundary condition

In this problem, the surface temperature of the geometry is specified. That means, T_s = T_surface, where T_s is the surface temperature and T_surface is the given temperature value for the surface of the geometry.

Modify the original boundary condition

In the original problem statement, the temperature of the surrounding medium (T_infty) and the convection heat transfer coefficient (h) are given. To suit our problem, we need to relate the surface temperature (T_s) with T_infty and h. Using Newton's law of cooling, we have: q = h * (T_s - T_infty), where q is the heat transfer rate per unit area.

Apply the one-term approximate solution method

Now, we can apply the one-term approximate solution method to the modified problem by selecting the dominant term in the Fourier series representation of the solution. The selection of this term depends on the geometry and boundary conditions of the problem.

Solve for the temperature field

Following the one-term approximate solution method, we can now solve for the temperature field (T) in the geometry. This will be an approximation, but if the dominant term is chosen correctly, it will be close to the exact solution.

Check the accuracy of the solution

To ensure that our solution is accurate, we can compare the resulting temperature field (T) to the exact solution (if available) or perform more detailed numerical simulations to validate our one-term approximation. In conclusion, by modifying the boundary condition and applying the one-term approximate solution method, we can solve for the temperature field of a geometry when the surface temperature is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient.