# Finite Element Method Questions and Answers – Three Dimensional and Axis Symmetric Heat Transfer

This set of Finite Element Method Multiple Choice Questions & Answers (MCQs) focuses on “Three Dimensional and Axis Symmetric Heat Transfer”.

1. Which of the following is the governing equation for three dimensional heat transfer?
a) ∂ / ∂x(kx ∂T / ∂x) + ∂ / ∂y(ky ∂T / ∂y) + ∂ / ∂z(kz ∂T / ∂z) + Q = 0
b) ∂ / ∂x(kx ∂T / ∂x) + ∂ / ∂y(ky ∂T / ∂y) + ∂ / ∂z(kz ∂T / ∂z) = 0
c) ∂ / ∂x(kx ∂T / ∂x) + ∂ / ∂y(ky ∂T / ∂y) – ∂ / ∂z(kz ∂T / ∂z) + Q = 0
d) ∂ / ∂x(kx ∂T / ∂x) – ∂ / ∂y(ky ∂T / ∂y) – ∂ / ∂z(kz ∂T / ∂z) + Q = 0

Explanation: The governing equation for three dimensional heat transfer is given by –
∂ / ∂x(kx ∂T / ∂x) + ∂ / ∂y(ky ∂T / ∂y) + ∂ / ∂z(kz ∂T / ∂z) + Q = 0
where,
k = heat transfer coefficient
T = temperature
Q = Heat flux

2. Procedure for assembling global equations for of a three dimensional heat transfer model is very different from its two dimensional counterpart.
a) True
b) False

Explanation: The given statement is false. The procedure for assembling global equations for of a three dimensional heat transfer model is very similar to its two dimensional counterpart. Firstly, the element type is selected based on various geometric considerations. Following this, the model is divided into a mesh of elements by introducing nodes.

3. What are the types of boundary conditions used for three dimensional heat transfer?
a) Specified temperature only
b) Convection conditions only
c) Specified temperature, specified heat flux and convection conditions
d) Specified temperature and specified heat flux

Explanation: Incase of three dimensional heat transfer problems, the boundary conditions used are very similar to the two dimensional ones. Specified temperatures is the simplest of the three; with the temperatures being input into system equations. The other two boundary conditions come into the picture only when there are surfaces explicit to the global value under consideration.

4. Which of the following is the governing equation for non steady heat conduction?
a) ∇ * k * ∇T = ρcT°
b) ∇ * ∇T = ρc
c) ∇ * k * T = ρcT
d) ∇ * k * ∇ = ρ

Explanation: The governing equation for non steady heat conduction is given by –
∇ * k * ∇T = ρcT°
k = thermal conductivity
ρ = density
c = specific heat
T = temperature
T° = time derivative of temperature

5. What is the final simplified equation that we arrive to, upon application of Galerkin’s method and Gauss’s theorem?
a) CT° + Y = v
b) CT° + YT = v
c) CT + YT = v
d) C° + YT = v

Explanation: Once the governing equation for non steady heat conduction is subjected to Galerkin Method and Gauss’s theorem, we arrive at the final simplified equation as follows –
CT° + YT = v
where, C & Y are symmetric matrices that represent thermal capacitance and thermal admittance
T = nodal temperatures
v = vector that depends upon thermal boundary conditions

6. For which of the following thermal boundary conditions, does the sub vector v correspond to zero?
a) Conductive boundary condition
b) Convective boundary condition
c) Insulated boundary condition
d) Constant temperature boundary condition

Explanation: For an insulated boundary condition, the sub vector v corresponds to zero. This is because dT/dn = 0. Whereas, for convective boundary condition, -k(dT/dn) = h(T – 0); therefore, making the sub vector correspond to a non zero value.

7. The basic relations between rectangular and cylindrical coordinates are given by x = rsinθ, y = rcosθ.
a) True
b) False

Explanation: The given statement is false. Basic relations between rectangular and cylindrical coordinates are given by y = rsinθ, x = rcosθ. This further corresponds to r2 = x2 + y2 and tanθ = y/x. For axisymmetric problems, cylindrical coordinates are preferred for computational ease and speed.

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