2 - Heat Conduction Equation

Question 1

What is the difference between temperature and heat transfer as quantities?

Answer 1

Temperature is scalar, heat transfer is a vector.

Question 2

What is the driving force for any form of heat transfer?

Answer 2

Temperature difference

Question 3

What are the three prime coordinate systems?

Answer 3

Rectangular
Cylindrical
Spherical

Question 4

Rectangular coordinate system variables

Answer 4

T(x, y, z, t)

Question 5

Cylindrical coordinate system variables

Answer 5

T(r, phi, z, t)

Question 6

Spherical coordinate system variables

Answer 6

T(r, phi, theta, t)

Question 7

Define steady heat transfer

Answer 7

Steady implies no change with time at any point in the medium.
T(x, y, z)

Question 8

Define transient heat transfer

Answer 8

Transient implies variation with time or time dependence.

T(x, y, z, t)

Question 9

How are lumped systems a special case of heat transfer?

Answer 9

There is variation with time but not with position, so the temperature of the medium changes uniformly with time.
T(t)

Question 10

When is heat transfer one-dimensional?

Answer 10

If the temperature in the medium varies in one direction only and thus heat is transferred in one direction, and the variation of temperature and thus heat transfer in other directions are negligible or zero.

Question 11

When is heat transfer two-dimensional?

Answer 11

If the temperature in a medium varies mainly in two primary directions, and the variation in temperature and thus the heat transfer in that direction is negligible.

Question 12

How can some three-dimensional problems be classified as one or two-dimensional?

Answer 12

Dependent on the relative magnitudes of heat transfer rates in different directions.
Dependent on the level of accuracy required.

Question 13

How is the rate of heat conduction in a specified direction in a one-dimensional system expressed?

Answer 13

Fourier’s Law of Heat Conduction

Question 14

Why is the temperature gradient negative when heat is conducted in the positive x-direction for Fourier’s Law of Heat Conduction?

Answer 14

Heat is conducted in the direction of decreasing temperature.

Question 15

What is the temperature gradient dT/dx?

Answer 15

The slope of the temperature curve on a T-x diagram.

Question 16

Examples of heat generation

Answer 16

Resistance heating (electrical energy being converted to heat at a rate of I^2R)
Nuclear reactions (fuel elements of nuclear reactors)
Chemical reactions (exothermic chemical reactors)

Question 17

What kind of phenomenon is heat generation?

Answer 17

Volumetric phenomenon (i.e. heat generation occurs throughout the volume of the medium).

Question 18

With what may the rate of heat generation in a medium vary?

Answer 18

Time and position.

Question 19

Define boundary conditions

Answer 19

The mathematical expressions of the thermal conditions at the boundaries.

Question 20

Define initial conditions

Answer 20

The mathematical expression for the temperature distribution of the medium initially.

Question 21

Types of boundary conditions

Answer 21

Specified temperature boundary condition
Specified heat flux boundary condition
Insulated boundary
Thermal symmetry
Convection boundary condition
Radiation boundary condition
Interface boundary conditions

Question 22

What is an insulated boundary condition?

Answer 22

(Adiabatic)
A well-insulated surface can be modelled as a surface with a specified heat flux of zero.
Therefore the first derivative of temperature with respect to the space variable (i.e. the temperature gradient) in the direction normal to the insulated surface is zero.

Question 23

What is the thermal symmetry boundary condition?

Answer 23

Occurs when the temperature distribution in one half of the plate is the same as that in the other half, so the heat transfer problem possesses thermal symmetry about the centre of the plane at x = L/2.
Therefore, the centre plane can be viewed as an insulated surface, resembling the insulated or zero heat flux boundary condition.

Question 24

What are the conditions for the boundary conditions at an interface?

Answer 24

Two bodies in contact must have the same temperature at the area of contact.
An interface cannot store any energy, and thus the heat flux on the two sides of an interface must be the same.

Question 25

Assumptions for steady one-dimensional heat conduction problems

Answer 25

Heat transfer is steady.
Heat transfer is one-dimensional.
Constant thermal conductivity.
There is no heat generation.

Question 26

What is the solution procedure for solving heat conduction problems?

Answer 26

1 - formulate the problem by obtaining the applicable differential equation in its simplest form and specifying the boundary conditions
2 - obtain the general solution of the differential equation
3 - apply the boundary conditions and determine the arbitrary constants in the general solution