Chapter 5 Numerical Method in Heat Conduction

Question 1

What is the limitation of analytical methods

Answer 1

highly simplified problems in simple geometries or problems than can be simplified with reasomable approximations

Question 2

Issues with the numerical method

Answer 2

only get numbers dont have neat solution -> plotting data without understanding physics

Question 3

Numerical method adv

Answer 3

often oversimplify analytical soltuion to allow easier maths -> remove non linearity, therefore mathematical model more likely to be accurate
Allows quick iterations to check if variables chosen are correct and their impact
analytical solution may result in problem so complex not worth the effort to try and solve

Question 4

what is m in numerical method

Answer 4

m is the number of points on the mesh

Question 5

what do you want to happen to m to maximise accuracy

Answer 5

m-> infinity such that delta x tends to zero

Question 6

Difference between the finite difference equation and the differential equation

Answer 6

differential equation is valid at every point whereas the finite difference equation is valid onoly at discrete points (the nodes)

Question 7

What is the energy balance method

Answer 7

subdividing the medium into a sufficient number of volume elements and then applying an energy balance on each element

Question 8

what is the energy balance in the energy balance method

Answer 8

rate of heat conduction at the left surface + rate of heat conduction at the right surface + rate of heat generation inside the element = rate of change of the energy content of the element

Question 9

Energy balance in energy balance method =

Answer 9

Q.cond left + Q.con right + E.gen element = delta E element/ delta t

Question 10

what is the change in energy content of an element equal to

Answer 10

zero

Question 11

what is the numerical equivalent of a first order differential

Answer 11

Tm-1 - Tm / delta x

Question 12

what is the numerical equivalent of a second order differential

Answer 12

Tm-1 - 2Tm + Tm+1 / delta x ^2`

Question 13

What is assumed to happen between nodes in the finite difference formulation

Answer 13

variation is linear between nodes

Question 14

What are the boundary conditions most commonly encountered in pracice

Answer 14

specified temperature, specified heat flux, convection and radiation

Question 15

How is specified temperature dealt with

Answer 15

T(0)  = To = Specified value
T(L) = Tm = Specified value

Question 16

What type of boundary conditions are dealt with using an energy balance on the volume element at the boundary

Answer 16

specified heat flux, convection, radiation or combined convection and radiation

Question 17

Equation for energy balance of volume element at boundary

Answer 17

Sum of Q. + E.gen element = 0

Question 18

Specified heat flux boundary condition

Answer 18

q.o * A + kA(T1-T0)/delta x + e.gen (Adelta x/2)

Question 19

If a surface is adiabatic how is boundary condition resolved

Answer 19

treat as symmetry line, temperature gradient slop is zero and if at T0, T1 and T-1 will be equal

Question 20

How many equations will you end up with

Answer 20

m equations

Question 21

how will the equations be computed

Answer 21

in a massive matrix of the form Ax = B or for our case AT=B
Where A is a matrix of known coefficients,
T is a matrix of unknown temperatures
and B a matrix of heat generation + other known stuff

Question 22

When resolving a boundary condition what will remain no matter what

Answer 22

at boundaries the conduction term on the other side of the element and the energy generation term inside the element, will the just ad additional terms for radiation, convection etc, internal will have the second derivative and energy generation terms

Question 23

What is an interface boundary condition

Answer 23

Two different solid media are assumed to be in perfect contact, thus at the same temperature interface node, but the rate of heat transfer via convection will be different due to different, convection coefficients

Question 24

What is the mirror image approach

Answer 24

can be used for problems that possess thermal symmetry, replace the plane of symmetry by insulation and only consider only half the medium in the solution -

Question 25

What does the finite difference formulation of steady heat conduction problems usually result in

Answer 25

a system of m algebraic equation in m unknown nodal temperatures that need to be solved simultaneously

Question 26

Two types of method for solving many simultaneous equations

Answer 26

direct and iterative

Question 27

Whats the difference between a direct and iterative method

Answer 27

direct methods are based on a fixed number of well defined steps that results in solution in a systematic manner, iterative methods are based on an initial guess for the solution that is refined by iteration until a specified convergence criterion is reached

Question 28

Give an example of direct approach

Answer 28

finding the inverse of the A matrix and solving A-1*b =T

do not use this approach in exams

Question 29

What is the gauss seidal method

Answer 29

iterative method, have equations relating all the temperatures, make initial guess and sub into equations, everytime you solve one update previous results