ENME315: Chapter 2 HCE Formulas Handout
HEAT CONDUCTION EQUATION
T(x, y, z, t)
T(r, , z, t)
T(r, , , t)
Figure: Three prime coordinate systems.
General Heat Conduction Equation (GHCE)
Coordinate system
Governing Partial Differential Equation
Rectangular
Cylindrical
Spherical
1-D Heat Conduction Equation (1-D HCE)
Large Plane Wall
(rectangular)
Case I
Constant properties
Case II
Steady
= 0 , with heat
generation 𝑒̇
Case III
Transient, no heat generation
𝑒̇
=0
Case IV
Steady, no heat generation
= 0 & 𝑒̇
=0
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ENME315: Chapter 2 HCE Formulas Handout
1-D Heat Conduction Equation (1-D HCE)
Long Cylinder
(cylindrical)
Case I
Constant properties
Case II
Steady
= 0 , with heat
generation 𝑒̇
Case III
Transient, no heat generation
𝑒̇
=0
Case IV
Steady, no heat generation
= 0 & 𝑒̇
=0
Sphere
(spherical)
Case I
Constant properties
Case II
Steady
= 0 , with heat
generation 𝑒̇
Case III
Transient, no heat generation
𝑒̇
=0
Case IV
Steady, no heat generation
= 0 & 𝑒̇
=0
Boundary Conditions
𝑞̇
𝑞̇
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ENME315: Chapter 2 HCE Formulas Handout
Steady 1-D HCE Solutions
Large Plane Wall
(rectangular)
Case IV
Steady, no heat generation
= 0 & 𝑒̇
=0
General
Solution
Example 2-10, p. 100
BC
Specified surface temperature
Unique
solution
Example 2-11, p. 102
BC
Specified heat flux and
temperature
𝑇
𝑞̇
Unique
Solution
Example 2-12, p. 104
BC
Specified heat flux and
Convection boundary
Unique
Solution
Example 2-13, p. 106
BC
Specified surface temperature
and Convection boundary
Unique
Solution
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ENME315: Chapter 2 HCE Formulas Handout
Steady 1-D HCE Solutions
Large Plane Wall
(rectangular)
Case II
Steady
= 0 with heat
generation 𝒆̇ 𝒈𝒆𝒏
General
Solution
Example 2-20, p. 119
BC
Specified surface temperature
and Convection boundary
𝑒̇
Unique
Solution
For the symmetrical boundary condition (T1=T2):
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ENME315: Chapter 2 HCE Formulas Handout
Steady 1-D HCE Solutions
Long Cylinder
(cylindrical)
Case IV
Steady, no heat generation
𝜕
= 0 & 𝑒̇
=0
𝜕𝑡
General
Solution
BC
Example 2-16, p. 111
Specified surface temperatures
Unique
Solution
Steady
= 0 , with heat generation 𝒆̇ 𝒈𝒆𝒏
Case II
General
Solution
BC
Example 2-18, p. 115
Specified surface temperatures
𝑇
Unique
Solution
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ENME315: Chapter 2 HCE Formulas Handout
Steady 1-D HCE Solutions
Sphere
(spherical)
Case IV
Steady, no heat generation
= 0 & 𝑒̇
=0
General
Solution
BC
Example 2-16, p. 111
Specified surface temperatures
Unique
Solution
Case II
Steady
= 0 , with heat
generation 𝑒̇
General
Solution
BC
Example 2-103, p. 143
Specified surface temperatures
𝑇
Unique
Solution
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