1. Chapter 4 Unsteady-State Conduction
4-1 INTRODUCTION
Assuming
results in
Application of separation-of variables method in the determination of temperature distribution in an infinite plate subjected to sudden cooling of surfaces.

2. From boundary condition [b], C1=0
From boundary condition [c] or
Final series form of the solution is

3. 4-2 LUMPED-HEAT-CAPACITY SYSTEM
Energy balance:
Time constant ( 时间常数 )
When the time equals to time constant,

4. Applicability of Lumped-Capacity Analysis
Characteristic dimension:

6. 4-3 TRANSIENT HEAT TRANSFER IN A SEMI-INFINITE SOLID
Constant surface temperature
The problem is solved by Laplace-transform technique.
Gauss error function:
The initial temperature of the semi-infinite solid is Ti, the surface is suddenly lowered to T0. Seek an expression for the T distribution in the solid as a function of time.

7. Heat flow at any x position:
At surface the heat flow is

8. Constant Heat Flux on Semi-Infinite Solid
Energy Pulse at Surface

13. 4-4 CONVECTION BOUNDARY CONDITIONS
For a semi-infinite solid with a convection boundary condition
The solution is:

16. Heisler Charts

30. The Biot and Fourier Numbers
In Lumped Heat Capacity analysis, characteristic dimension can be defined as
The time constant becomes
Applicability of the Heisler Charts

35. 4-5 MULTIDIMENSIONAL SYSTEMS
Definition:
Governing eq.
Initial and boundary conditions:
Governing eq.
Initial and boundary conditions:

36. For plate 1 with thickness 2L1
Initial and boundary conditions:
Initial and boundary conditions:
To be shown that

37. Dimensionless temperature distribution can be expressed as a product of the solutions for the two plate problems

38. In a similar manner,
Conclusion:

39. Heat Transfer in Multidimensional Systems

46. 4-6 TRANSIENT NUMERICAL METHOD

47. For one-dimensional problem:

48. Boundary conditions

49. if
Convergence condition:

50. Forward and Backward Differences
Forward difference and explicit formulation
Backward difference and implicit formulation
向前差分：将时间步长末时节点的温度用时间步长起点时周围节点的温度表示的差分方法。
向后差分：空间微分用当前时刻温度表示的差分方法。

51. 4-7 THERMAL RESISTANCE AND CAPACITY FORMULATION
Forward difference:
Stability requirement:
Consideration on round-off error

52. Backward difference:
Heat source term:
For radiation input to the node,
=net radiant energy input to the node per unit area