1. CHAPTER 3 TWO-DIMENSIONAL STEADY STATE CONDUCTION
3.1 The Heat Conduction Equation
• Assume: Steady state, isotropic, stationary,
k =  = constant
(3.1)

2. • Special case: Stationary material, no energy
generation
(3.2)
• Cylindrical coordinates:
(3.3)
Eq. (3.1), (3.2) and (3.3) are special cases of

3. 3.2 Method of Solution and Limitations
(3.4)
Eq. (3.4) is homogenous, second order PDE with
variable coefficients
3.2 Method of Solution and Limitations
• Method: Separation of variables
• Basic approach: Replace PDE with sets of ODE

4. • The number of sets depends on the number of independent variables
• Limitations:
(1) Linear equations
• Examples: Eqs. (3.1)-(3.4) are linear
(2) The geometry is described by an orthogonal
coordinate system. Examples: Rectangles, cylinders, hemispheres, etc.

5. 3.3 Homogeneous Differential Equations and Boundary Conditions
• Example: Eq. (3.1): Replace T by cT and divide through by c
 NH
(a)

6.  NH Example: Boundary condition (b) Replace T by cT (c)
• Simplest 2-D problem: HDE with 3 HBC and 1 NHBC
• Separation of variables method applies to NHDE and NHBC

7. 3.4 Sturm-Liouville Boundary Value Problem: Orthogonality
• PDE is split into sets of ODE. One such sets is known as Sturm-Liouville equation if it is of the form
(3.5a)
Rewrite as
(3.5b)

8. • w(x) = weighting function, plays a special role
where
(3.6)
NOTE:
• w(x) = weighting function, plays a special role
• Equation (3.5) represents a set of n equations corresponding to n values of
• The solutions
are known as the characteristic
functions

9. • Important property of the Sturm-Liouville problem:
orthogonality
• Two functions,
and
are said to be orthogonal
in the interval a and b with respect to a weighting function w(x), if
(3.7)
• The characteristic functions of the Sturm-Liouville problem are orthogonal if:

10. (1) p(x) , q(x) and w(x) are real, and
(2) BC at x = a and x = b are homogeneous of the
form
(3.8a)
(3.8b)
(3.8c)

11. Special Case: If p(x) = 0 at x = a or x = b , these
Special Case: If p(x) = 0 at x = a or x = b , these conditions can be extended to include
(3.9a)
and
(3.9b)
(3.9a) and (3.9b) are periodic boundary conditions.
• Physical meaning of (3.8) and (3.9)

12. • Relationship between the Sturm-Liouville problem,
• Relationship between the Sturm-Liouville problem, orthogonality, and the separation of variables
method:
PDE + separation of variables
2 ODE
One of the 2 ODE =
Sturm-Liouville problem
Solution to this ODE = n f
= orthogonal functions

13. 3.5 Procedure for the Application of Separation of Variables Method
Example 3.1: Steady state 2-D conduction in
a rectangular plate
Find T (x,y)
2 HBC x y 1 . 3
Fig.
T = f(x)
T = 0
(1) Observations
• 2-D steady state
• 4 BC are needed
• 3 BC are homogeneous

14. (2) Origin and Coordinates
• Origin at the intersection of the two simplest BC
• Coordinates are parallel to the boundaries
(3) Formulation
(i) Assumptions
(1) 2-D
(2) Steady
(3) Isotropic
(4) No energy generation
(5) Constant k

15. (ii) Governing Equations
(3.2)
(iii) Independent Variable with 2 HBC: x- variable
(iv) Boundary Conditions
T = 0
2 HBC x y 1 . 3
Fig.
T = f(x)
(1) H
(2) H
(3) H

16. (4) Solution (4) non-homogeneous (i) Assumed Product Solution (a)
(a) into eq. (3.2)
(b)

17. (c)
(d)
where
is known as the separation constant

18. • The separation constant is squared
NOTE:
• The separation constant is squared
• The constant is positive or negative
• The subscript n. Many values:
are
known as eigenvalues or characteristic values
• Special case: constant = zero must be considered
• Equation (d) represents two sets of equations
(e)

19. terms (f) • The functions and or characteristic values
are known as eigenfunctions
(ii) Selecting the sign of the
terms
Select the plus sign in the equation representing the variable
with 2 HBC. Second equation takes the minus sign

20. equations (g) and (h) become
• Important case:
equations (g) and (h) become
(i)

21. (iii) Solutions to the ODE
(j)
(iii) Solutions to the ODE
(k)
(l)
(m)
(n)

22. NOTE: Each product is a solution
The complete solution becomes
(q)
(iv) Applying Boundary Conditions

23. NOTE: Each product solution must satisfy the BC BC (1)
Therefore
(r)
(s)
Applying (r) to solution (k)

24. Therefore
Similarly, (r) and (m) give
B.C. (2)
Therefore

25. Applying (k)
Therefore
(t)
Thus
(u)
Similarly, BC 2 and (m) give

26. Therefore
With
the solution corresponding to
vanishes.
BC (3)

27. Which gives
Results so far:
therefore
and
Temperature solution:
(3.10)

28. Remaining constant
B.C. (4)
(3.11)
To determine
from (3.11) we apply orthogonality.
(v) Orthogonality
• Use eq. (7) to determine

29. • The function
in eq. (3.11) is the
solution to (g)
• If (g) is a Sturm-Liouville equation with 2 HBC,
orthogonality can be applied to eq. (3.11)
Return to (g)
(g)
Compare with eq. (3.5a)
(3.5a)

30. and
Eq. (3.6) gives
and
• BC (1) and (2) are homogeneous of type (3.8a).
• The characteristic functions
are orthogonal with respect to to
Multiply eq. (3.11) by
integrate from

31. (3.12)
Interchange the integration and summation, and use
Introduce orthogonality (3.7)
(3.7)

32. Apply (3.7) to (3.12)
Solve for
(3.13)
(5) Checking

33. The solution is
(3.10)
Dimensional check
Limiting check
Boundary conditions
Differential equations
(6) Comments
Role of the NHBC

34. 3.6 Cartesian Coordinates: Examples
Example 3.3: Moving Plate with Surface
Convection y x U T h , L 3 .
Fig.
HBC 2
Semi-infinite plate leaves a furnace at
Outside furnace the plate exchanges heat by convection.
Determine the temperature distribution in the plate.

35. Solution (1) Observations • Symmetry • At plate is at • NHBC
(2) Origin and Coordinates
(3) Formulation
(i) Assumptions
(1) 2-D
(2) Steady state

36. (3) Constant
and
(4) Uniform velocity
(5) Uniform
and
(ii) Governing Equations
Define
(a)
(b)

37. (iii) Independent variable with 2 HBC: y- variable
(iv) Boundary Conditions y x U T h , L
Fig. 3.3
HBC 2 o
(1)
(2)
(3)
(4)

38. (4) Solution terms (i) Assumed Product Solution (c) (d)
(ii) Selecting the sign of the
terms

39. (e)
(f)
For
(g)
(h)

40. (iii) Solutions to the ODE
(j)
(k)
And
(l)

41. Complete solution:
(m)
(iv) Application of Boundary Conditions
BC (1)
BC (2)
(n)

42. BC (3)
With
(3.17)
BC (4)
(o)

43. (v) Orthogonality Characteristic Functions:
are solutions to equation (f).
Comparing (f) with eq. (3.5a) shows that it is a Sturm-
and
Liouville equation with
Thus eq. (3.6) gives
and
(p)
2 HBC at
and
are orthogonal

44. to
Multiply both sides of (o) by
integrate
from
and apply orthogonality
(q)

45. (5) Checking
Dimensional check
Limiting check: If
(6) Comments
For a stationary plate,

46. 3.7 Cylindrical Coordinates: Examples
Example 3.5: Radial and Axial Conduction in a Cylinder o r z T h , L
2 HBC w
Fig. 3.5
Two solid cylinders are pressed co axially with a force F and rotated in opposite directions.
Coefficient of friction is
Convection at the outer surfaces.

47. Solution Find the interface temperature. (1) Observations • Symmetryz T h , L
2 HBC w
Fig. 3.5
• Symmetry
• Interface frictional heat
= tangential force x velocity
• Transformation of B.C.,
(2) Origin and Coordinates

48. (3) Formulation
(i) Assumptions
(1) Steady
(2) 2-D
(3) Constant
and
(4) Uniform interface pressure
(5) Uniform
and
(6) Radius of rod holding cylinders is small compared to cylinder radius
(ii) Governing Equations

49. (3.17)
(iii) Independent variable with 2 HBC: r- variable
(iv) Boundary conditions
(1) or
finite

50. (4) Solution (2) (3) (4) (i) Assumed Product Solution r z T h , LT h , L
2 HBC w
Fig. 3.5
(3)
(4)
(4) Solution
(i) Assumed Product Solution

51. (a)
(b)
(c)
(ii) Selecting the sign of the
r-variable has 2 HBC

52. (d)
(e)
For
(f)
(g)

53. (iii) Solutions to the ODE
(d) is a Bessel equation:
Since
and
is real
(h)
Solutions to (e), (f) and (g):
(i)

54. (iv) Application of Boundary Conditions
(j)
(k)
Complete solution
(l)
(iv) Application of Boundary Conditions
BC (1)
and
Note:

55. BC (2)
(m)
(n)
BC (2)

56. Since
BC (3)
(3.20)

57. (v) Orthogonality BC (4) (o) is a solution to (d) with 2HBC in and
Comparing (d) with eq. (3.5a) shows that it is a Sturm-
Liouville equation with
Eq. (3.6):
and
(p)

58. and
are orthogonal with respect to
Applying orthogonality, eq. (3.7), gives
(q)
Interface temperature: set
in eq. (3.20)

59. (r)
(5) Checking
Dimensional check: Units of
Limiting check: If or
then
(6) Comments
is not always equal to unity.

60. 3.8 Integrals of Bessel Functions
(3.21)

61. Table 3.1 Normalizing integrals for solid cylinders [3]
Boundary Condition at
(3.8a):
(3.8b):
(3.8c):

62. 3.9 Non-homogeneous Differential Equations
Example 3.6: Cylinder with Energy Generation L a T o r q z
Fig. 3.6
Solid cylinder generates
heat at a rate
One end is at
while the other is insulated.
Cylindrical surface is at
Find the steady state temperature distribution.

63. Solution (1) Observations • Energy generation leads to NHDE • Defineq z
Fig. 3.6
• Energy generation leads to
NHDE
• Define
to make
BC at surface homogeneous
• Use cylindrical coordinates
(2) Origin and Coordinates

64. (3) Formulation
(i) Assumptions
(1) Steady state
(2) 2-D
(3) Constant
and
(ii) Governing Equations
Define

65. (iii) Independent variable with 2 HBC
(3.22)
(iii) Independent variable with 2 HBC
(iv) Boundary conditions L a T o r q z
Fig. 3.6
(1)
finite
(2)
(3)
(4)

66. (4) Solution Let: (a) Substitute into eq. (3.22) (b) Split (b): Let
(c)

67. NOTE: • (c) is a HPDE for Therefore (d) • (d) is a NHODE for the
• Guideline for splitting PDE and BC:
should be governed by HPDE and three HBC.
Let
take care of the NH terms in PDE and BC

68. BC (1)
finite
(c-1)
BC (2)
(c-2)
BC (3)
Let
(c-3)

69. Thus
(d-1)
BC (4)
Let
(c-4)
Thus
(d-2)
Solution to (d)

70. (e)
(i) Assumed Product Solution
(f)
(f) into (c), separating variables
(g)
(h)

71. (ii) Selecting the sign of the
terms
(i)
(j)
For
(k)

72. (l)
(iii) Solutions to the ODE
(m)
(j) is a Bessel equation with
(n)

73. Solution to (k) and (l)
(o)
(p)
Complete Solution:
(q)
(iv) Application of Boundary Conditions
BC (c-1) to (n) and (p)

74. BC (c-4) to (m) and (o)
BC (c-3) to (m) and (o)
Equation for or
(r)
With
The solutions to
become

75. (s)
BC (d-1) and (d-2)
and
(t)
BC (c-2)
(u)

76. (v) Orthogonality are solutions to equation (i). and
Comparing (i) with eq. (3.5a) shows that it is a Sturm-
Liouville equation with
Eq. (3.6) gives
and
and
2 HBC at
are orthogonal
with respect to

77. Evaluating the integrals and solving for

78. Solution to
(w)
(5) Checking
Dimensional check: Units of
and of in
solution (s) are in °C
Limiting check:
and

79. 3.10 Non-homogeneous Boundary Conditions
Method of Superposition:
• Decompose problem
Example:
Problem with 4 NHBC is decomposed
into 4 problems each having one NHBC ) ( y g x f L W o q T h ,
3.7
Fig. DE

80. = + Solution: 3 2 4 ) ( y g x f L W q ¢ T h , x y L W T h , ) ( f g ¥= x y L W T h , 4 + ) ( f 3 g 2