Chapter 3
Two-Dimensional, Steady-State
Conduction
General considerations
• Two-dimensional conduction:
– Temperature distribution is characterized by two spatial coordinates,
e.g., T (x, y).
– Heat flux vector is characterized by two directional components,
e.g., q x and q y .
• Heat transfer in a long, prismatic solid with two isothermal surfaces
and two insulated surfaces:
Note the shapes of lines of constant temperature (isotherms) and heat flow lines
(adiabats).
The Heat Equation and Methods of Solution
• Assuming steady-state, two-dimensional conduction in a rectangular domain
with constant thermal conductivity and heat generation, the heat equation is :
• Solution Methods :
 2T  2T q
 2  0
2
k
x
y
– Exact/Analytical: Separation of Variables
 Limited to simple geometries and boundary conditions.
– Approximate/Graphical (q  0) : Flux Plotting
 Of limited value for quantitative considerations but a quick aid to
establishing physical insights.
– Approximate/Numerical: Finite-Difference Method, Finite Element or Boundary
Element Method.
 Most useful approach and adaptable to any level of complexity.
Exact solution: Separation of Variables
 2 T  2T

0
x 2 y 2
Separation of variables method
Analytical Solution
The Finite-Difference Method
• An approximate method for determining temperatures at discrete
(nodal) points of the physical system.
• Procedure:
– Represent the physical system by a nodal network.
– Use the energy balance method to obtain a finite-difference
equation for each node of unknown temperature.
– Solve the resulting set of algebraic equations for the unknown
nodal temperatures.
The Nodal Network (Grid or Mesh)
• The nodal network identifies discrete points at which the temperature is
to be determined and uses an (m, n) notation to designate their location.
Derivation of the Finite-Difference Equations
- The Energy Balance Method • As a convenience that obviates the need to predetermine the direction of heat
flow, assume all heat flows are into the nodal region of interest, and express all
heat rates accordingly. Hence, the energy balance becomes:
E in  E g  0
• Consider application to an interior nodal point (one that exchanges heat by
L
conduction with four, equidistant nodal
points):
Rt ,cond

kA
Solutions Methods
• Matrix Inversion: Expression of system of N finite-difference equations for
N unknown nodal temperatures as:
 AT   C 
Coefficient
Matrix (NxN)
Solution
Solution Vector
(T1,T2, …TN)
(4.52)
Right-hand Side Vector of Constants
(C1,C2…CN)
T    A1 C 
• Gauss-Seidel Iteration: Each finite-difference equation is written in explicit
form, such that its unknown nodal temperature appears alone on the lefthand side:
N aij
C i 1 aij k
k
Ti   i   Tj   
Tj( k 1)
aii j 1 aii
j  i 1 aii
where i =1, 2,…, N and k is the level of iteration.
Iteration proceeds until satisfactory convergence is achieved for all nodes:
k
k 1
Ti   Ti   
Typical problems
Problem 3.2.1: Derive Finite-difference equations for (a) nodal point on a
diagonal
surface and (b) tip of a cutting tool.
(a) Diagonal surface
(b) Cutting tool.
Typical problems
Solution
(a) Diagonal surface
(a)
(b)
(b) Cutting tool.