ME 512 Conduction
Module 3 – Section 3.2:
Spatial Transient Conduction
Instructor: Dr. Bladimir Ramos-Alvarado
Associate Professor
Department of Mechanical Engineering
The Pennsylvania State University
Fall 2025
Lecture 29
10/31/2025
Summary
Announcements:
- HW#5 is due on Monday.
Previously, we solved the following 1-D transient problem:
Governing equation:
𝜕 2 𝑇 1 𝜕𝑇
=
𝜕𝑥 2 𝛼 𝜕𝑡
𝜕𝑇
ቤ
=0
𝜕𝑥 𝑥=0
𝑘
𝜕𝑇
ቤ
= 𝑞”𝑠
𝜕𝑥 𝑥=𝐿
Boundary and initial conditions:
𝜕𝑇
𝜕𝑇
ቤ
=0
𝑘 ቤ
= 𝑞"𝑠
𝜕𝑥 𝑥=0
𝜕𝑥
𝑇 𝑥, 0 = 𝑇𝑖
𝑇 𝑥, 0 = 𝑇𝑖
𝑥=𝐿
L
We started with a variable change 𝜃 𝑥, 𝑡 = 𝑇 𝑥, 𝑡 − 𝑇𝑖
And suggested a superposition solution as: 𝜃 𝑥, 𝑡 = 𝜃𝐻 𝑥, 𝑡 + 𝜎 𝑥 + 𝜏 𝑡
We reached the following solution:
∞
𝑞𝑠 "𝐿
𝑞𝑠 "𝐿 2
𝜃 𝑥,ҧ 𝐹𝑜 =
𝐹𝑜 +
𝑥ҧ + 𝐶0 + ෍ 𝐶𝑛 exp −𝜆2𝑛 𝐹𝑜 cos 𝜆𝑛 𝑥ҧ
𝑘
2𝑘
𝑛=1
1-D Transient Problems with Steady-state solution, cont’d
1-D Transient Problems with Steady-state solution, cont’d
Matlab plotting code
COMSOL Simulation
Multidimensional transient problems: Product solution
Multidimensional transient problems in rectangular coordinates (Cartesian) can be
expressed in terms of the product of two one-dimensional solutions.
▪ Limitations: the initial condition must be expressed as the product of two/three
functions as, 𝑇 𝑥, 𝑦, 𝑧, 0 = 𝐹 𝑥, 𝑦, 𝑧 = 𝐹1 𝑥 𝐹2 𝑦 𝐹3 𝑧 . Clearly, the uniform
temperature initial condition satisfies this condition
▪ All boundary conditions must be homogeneous.
Let us consider a simple rectangular system defined as 0 ≤ 𝑥 ≤ 𝐿 and 0 ≤ 𝑦 ≤ 𝑊, where
For 0 ≤ 𝑥 ≤ 𝐿
𝜕 2 𝜃 𝜕 2 𝜃 1 𝜕𝜃
+
=
𝜕 2 𝜃1 1 𝜕𝜃1
𝜃1 𝑥, 0 = 𝐹1 𝑥
𝜕𝑥 2 𝜕𝑦 2 𝛼 𝜕𝑡
=
𝜕𝑥 2
𝛼 𝜕𝑡
𝜕𝜃1
𝜃 𝑥, 𝑦, 0 = 𝐹1 𝑥 𝐹2 𝑦
−𝑘
ቤ
= −ℎ1 𝜃1 0, 𝑡
𝜕𝑥 𝑥=0
𝜕𝜃
−𝑘 ቤ
= −ℎ1 𝜃 0, 𝑦, 𝑡
𝜕𝜃1
𝜕𝑥 𝑥=0
−𝑘
ቤ
= ℎ2 𝜃1 𝐿, 𝑡
𝜕𝑥 𝑥=𝐿
𝜕𝜃
−𝑘 ቤ
= ℎ2 𝜃 𝐿, 𝑦, 𝑡
For 0 ≤ 𝑦 ≤ 𝑊
𝜕𝑥 𝑥=𝐿
𝜃2 𝑦, 0 = 𝐹2 𝑦
𝜕 2 𝜃2 1 𝜕𝜃2
=
𝜕𝜃
2
𝜕𝜃2
𝜕𝑦
𝛼 𝜕𝑡
−𝑘 ቤ
= −ℎ3 𝜃 𝑥, 0, 𝑡
−𝑘
ቤ
= −ℎ3 𝜃 0, 𝑡
𝜕𝑦 𝑦=0
𝜕𝑦 𝑦=0
𝜕𝜃
𝜕𝜃2
−𝑘 ቤ
= ℎ4 𝜃 𝑥, 𝑊, 𝑡
−𝑘
ቤ
= ℎ4 𝜃 𝑊, 𝑡
𝜕𝑦 𝑦=𝑊
𝜕𝑦
𝑦=𝑊
Multidimensional transient problems: Product solution, cont’d
Then, the solution of the original problem is:
𝜃 𝑥, 𝑦, 𝑡 = 𝜃1 𝑥, 𝑡 𝜃2 𝑦, 𝑡
Which can be easily demonstrated by substituting the product solution into the original
governing equation,
𝜕 2 𝜃1
𝜕 2 𝜃2 1
𝜕𝜃1
𝜕𝜃2
𝜃2
+ 𝜃1
=
𝜃
+ 𝜃1
𝜕𝑥 2
𝜕𝑦 2
𝛼 2 𝜕𝑡
𝜕𝑡
𝜃2
𝜕 2 𝜃1 1 𝜕𝜃1
𝜕 2 𝜃2 1 𝜕𝜃2
−
+ 𝜃1
−
=0
𝜕𝑥 2 𝛼 𝜕𝑡
𝜕𝑦 2 𝛼 𝜕𝑡
Since the boundary conditions are homogeneous, it is easily verifiable that the product
solution also satisfies them.
Multidimensional transient problems: Product solution, cont’d
Product solution example: Solve the following problem using the product solution approach.
Note that not all boundary conditions are homogeneous and that the initial condition is a
uniform temperature value, T0 > T∞.
𝑦
𝜕𝑇
=0
𝜕𝑥
𝜕 2 𝑇 𝜕 2 𝑇 1 𝜕𝑇
+
=
𝜕𝑥 2 𝜕𝑦 2 𝛼 𝜕𝑡
ℎ, 𝑇∞
𝑇 𝑥, 𝑦, 0 = 𝑇0
In 0 ≤ 𝑥 ≤ 𝐿 and 0 ≤ 𝑦 ≤ 𝑊
ℎ, 𝑇∞
𝑇 𝑥, 𝑦, 𝑡
Now, the problem can be turned into a
homogeneous one by defining 𝜃 = 𝑇 − 𝑇∞
𝑥
𝜕𝑇
=0
𝜕𝑦
The product solution can be formulated as follows:
𝜕 2 𝜃1
𝜕𝜃1
=
𝐿2 𝜕 𝑥/𝐿 2 𝜕 𝛼𝑡
𝜕𝜃1
ቤ
=0
𝜕𝑥ҧ 𝑥=0
ҧ
𝜕 2 𝜃1
𝜕𝜃1
=
𝜕𝑥ҧ 2
𝜕𝐹𝑜1
𝜃1 𝑥,ҧ 0 = 1
𝜕𝜃1
ቤ
= −𝐵𝑖1 𝜃1 1, 𝐹𝑜1
𝜕𝑥ҧ 𝑥=1
ҧ
ℎ𝐿
𝐵𝑖1 =
𝑘
𝜕 2 𝜃2
𝜕𝜃2
=
𝑊 2 𝜕 𝑦/𝑊 2 𝜕 𝛼𝑡
𝜕𝜃2
ቤ
=0
𝜕𝑦ത 𝑦=0
ത
𝜕𝜃2
ቤ
= −𝐵𝑖2 𝜃2 1, 𝐹𝑜2
𝜕𝑦ത 𝑦=1
ത
𝜕 2 𝜃2
𝜕𝜃2
=
𝜕𝑦ത 2
𝜕𝐹𝑜2
𝜃2 𝑦,
ത 0 = 𝜃0
𝐵𝑖2 =
ℎ𝑊
𝑘
Multidimensional transient problems: Product solution, cont’d
Fortunately, we already have the solution for this problem, see Example T1:
Multidimensional transient problems: Product solution, cont’d
Multidimensional transient problems: Product solution, cont’d
1𝑚2
COMSOL solution for, 𝐵𝑖1 = 2, 𝐵𝑖2 = 1, 𝛼 =
, 𝜃𝑖 = 50𝐾, 𝑎𝑛𝑑 𝑇∞ = 300𝐾
𝑠
Temperature contours for t = 2.0 s
Temperature contours for t = 0.1 s
Multidimensional transient problems: Product solution, cont’d
Temperature profiles along x = 0
Temperature profiles along y = 0