2 - Heat Conduction Equation Flashcards
What is the difference between temperature and heat transfer as quantities?
Temperature is scalar, heat transfer is a vector.
What is the driving force for any form of heat transfer?
Temperature difference
What are the three prime coordinate systems?
Rectangular
Cylindrical
Spherical
Rectangular coordinate system variables
T(x, y, z, t)
Cylindrical coordinate system variables
T(r, phi, z, t)
Spherical coordinate system variables
T(r, phi, theta, t)
Define steady heat transfer
Steady implies no change with time at any point in the medium.
T(x, y, z)
Define transient heat transfer
Transient implies variation with time or time dependence.
T(x, y, z, t)
How are lumped systems a special case of heat transfer?
There is variation with time but not with position, so the temperature of the medium changes uniformly with time.
T(t)
When is heat transfer one-dimensional?
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.
When is heat transfer two-dimensional?
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.
How can some three-dimensional problems be classified as one or two-dimensional?
Dependent on the relative magnitudes of heat transfer rates in different directions.
Dependent on the level of accuracy required.
How is the rate of heat conduction in a specified direction in a one-dimensional system expressed?
Fourier’s Law of Heat Conduction
Why is the temperature gradient negative when heat is conducted in the positive x-direction for Fourier’s Law of Heat Conduction?
Heat is conducted in the direction of decreasing temperature.
What is the temperature gradient dT/dx?
The slope of the temperature curve on a T-x diagram.
Examples of heat generation
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)
What kind of phenomenon is heat generation?
Volumetric phenomenon (i.e. heat generation occurs throughout the volume of the medium).
With what may the rate of heat generation in a medium vary?
Time and position.
Define boundary conditions
The mathematical expressions of the thermal conditions at the boundaries.
Define initial conditions
The mathematical expression for the temperature distribution of the medium initially.
Types of boundary conditions
Specified temperature boundary condition Specified heat flux boundary condition Insulated boundary Thermal symmetry Convection boundary condition Radiation boundary condition Interface boundary conditions
What is an insulated boundary condition?
(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.
What is the thermal symmetry boundary condition?
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.
What are the conditions for the boundary conditions at an interface?
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.
Assumptions for steady one-dimensional heat conduction problems
Heat transfer is steady.
Heat transfer is one-dimensional.
Constant thermal conductivity.
There is no heat generation.
What is the solution procedure for solving heat conduction problems?
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
