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.

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

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

Applicability of Lumped-Capacity Analysis Characteristic dimension:

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.

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

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

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

Heisler Charts

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

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

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

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

In a similar manner, Conclusion:

Heat Transfer in Multidimensional Systems

4-6 TRANSIENT NUMERICAL METHOD

For one-dimensional problem:

Boundary conditions

if Convergence condition:

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。

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

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