Chapter 4 Unsteady-State Conduction4-1 INTRODUCTIONApplication of separation-of variables method in the determination of temperature distribution in an infinite plate subjected to sudden cooling of surfaces.Assumingresults in

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

4-2 LUMPED-HEAT-CAPACITY SYSTEMTime constant ( )Energy balance:When the time equals to time constant,

Applicability of Lumped-Capacity AnalysisCharacteristic dimension:

4-3 TRANSIENT HEAT TRANSFER IN A SEMI-INFINITE SOLIDThe 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.The problem is solved by Laplace-transform technique.Gauss error function:Constant surface temperature

At surface the heat flow isHeat flow at any x position:

Constant Heat Flux on Semi-Infinite SolidEnergy Pulse at Surface

4-4 CONVECTION BOUNDARY CONDITIONSFor a semi-infinite solid with a convection boundary conditionThe solution is:

Heisler Charts

The Biot and Fourier NumbersApplicability of the Heisler ChartsIn Lumped Heat Capacity analysis, characteristic dimension can be defined asThe time constant becomes

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

Initial and boundary conditions:For plate 1 with thickness 2L1Initial and boundary conditions:For plate 2 with thickness 2L2To 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

ifConvergence condition:

Forward and Backward DifferencesForward difference and explicit formulationBackward difference and implicit formulation

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

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