Problem 11 - Concepts and Applications of Finite Element Analysis

Question

A uniform bar is modeled by two identical two-node elements, as shown. Node 1 is maintained at temperature \(T_{1}=0^{\circ} \mathrm{C}\). The bar is surrounded by fluid of temperature \(T_{\mathrm{f} \mathrm{f}}=200^{\circ} \mathrm{C}\), which transfers heat across its cylindrical surface of area S. In units listed in Section 12.1, data are: \(A=300\left(10^{-6}\right), h=600, k=200\), and the surface area of the entire bar is \(S=0.020\). (a) Formulate element matrices. (b) Assemble element matrices, impose \(T_{1}=0^{\circ} \mathrm{C}\), and solve for \(T_{2}\) and \(T_{3}\). (c) Repeat part (b), but alter the FE model so that the left element is half as long as the right element. The overall length remains \(0.300 \mathrm{~m}\). (d) What conclusion might be drawn by comparing the results of parts (b) and (c)?

Step-by-Step Solution

Answer

The element matrices are given by the heat transfer equation. By solving the system of equations with imposed boundary conditions, we can find the temperatures \(T_2\) and \(T_3\). Changing the element lengths affects the solution and usually, it is observed that smaller elements yield more accurate results.

Step 1: Formulate Element Matrices

To formulate the element matrices, one needs to write down the heat transfer equation, given by: \[ -kA\frac{dT}{dx}+hS(T-T_{f})=0 \] where \( T \) is temperature, \( A \) is the area, \( k \) is the thermal conductivity, \( h \) is the convective heat transfer coefficient, \( S \) is the surface area, \( T_f \) is the fluid temperature. For a 2-noded 1D element, the element matrices are \( [K] \) and \( [F] \), where the stiffness matrix \( [K] \) is \( \frac{kA}{L} \), and the loading vector \( [F] \) is \(-hS(T-T_f) \)

Step 2: Assemble Element Matrices and Calculate Temperatures

For parts (b) and (c), we assemble the element matrices. Matrix assembly involves the addition of individual element matrices to form a global system matrix. After assembly, the boundary conditions, in this case \(T_1 = 0^{\circ}C \), are imposed. Now, solve the resulting system of equations using any suitable method, for example by inverting the system matrix and multiplying it with the loading vector, this will yield the unknown temperatures \(T_2\) and \(T_3\).

Step 3: Altering the Element Lengths

For part (c), the FE model is altered so that the left element is half as long as the right element. We need to recalculate the element matrices with the new element length, assemble them and calculate \(T_2\) and \(T_3\) again.

Step 4: Comparing Results

In part (d) it is necessary to analyze and compare the results obtained from steps 2 and 3 and note the effect of changing the lengths of the elements. Usually, the accuracy of the results is related to the size of the elements, where smaller elements tend to give more accurate results.

Key Concepts

Understanding Heat Transfer

Heat transfer is a process through which thermal energy is exchanged between physical systems. In this exercise, we are examining how heat moves across a uniform bar that is immersed in a fluid of a different temperature. There are three modes of heat transfer:

• Conduction: Transfer of heat through a solid material due to temperature difference.
• Convection: Transfer of heat by the movement of fluid. The fluid here is at a temperature of 200°C.
• Radiation: Transfer of heat through electromagnetic waves, which is not the focus in this exercise.

Here, conduction occurs within the bar, while convection happens on its surface with the surrounding fluid. The rate at which heat is transferred depends on factors such as the material's thermal conductivity and the surface conditions. Understanding these principles helps in calculating the temperature at different nodes of the bar using finite element analysis.

Element Matrices in Finite Element Analysis

Element matrices are fundamental in finite element analysis, as they help in constructing the global system to solve physical problems numerically.

• Stiffness Matrix \, \([K]\): Represents the resistance of the system to deformation due to thermal loads. In our case, it is calculated as \( \frac{kA}{L} \), where \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, and \( L \) is the length of the element.
• Load Vector \, \([F]\): Represents external forces or influences. For the thermal problem at hand, it is given by \( -hS(T-T_f) \), representing heat flux due to the convective boundary condition.

To solve the problem of finding the unknown temperatures, these matrices for each element are assembled into a larger global matrix that reflects the entire system. This matrix is then used to derive the element temperatures by imposing boundary conditions and solving the system.

Thermal Conductivity and Its Role

Thermal conductivity is a material property that measures a material's ability to conduct heat. It is denoted by \( k \) and is expressed in units of \( W/(m·K) \). In the exercise, a value of 200 is used, indicating the bar's ability to transfer heat effectively.Thermal conductivity directly influences the element stiffness matrix, \( \frac{kA}{L} \), which showcases how readily heat can be distributed within an element. A higher thermal conductivity value leads to a stiffer matrix, meaning the element conducts heat more efficiently, impacting the temperature distribution within the finite element model. Understanding the role of thermal conductivity is critical in problems involving heat transfer, as it affects how temperatures at different nodes in the model are computed.

The Role of Numerical Methods in Solving Systems

Numerical methods are essential for solving equations when an analytical solution is not feasible, especially in complex systems like those involving heat transfer. In problems like this one, numerical methods allow us to approximate the temperatures at the nodes of the bar.Once the element matrices are assembled to form the global system matrix, solving for temperatures \( T_2 \) and \( T_3 \) involves numerical techniques. Some common methods include:

• Matrix Inversion: Useful for small systems, involves inverting the system matrix and multiplying it by the load vector.
• Iterative Solvers: More suitable for large systems, methods like Gauss-Seidel or Conjugate Gradient minimize computational resources.

These methods help to manage computational complexity and provide a practical approach to find solutions within acceptable error margins, ensuring that the results align with the physical reality of heat transfer in the modeled system.