Numerical Finite-Difference Methods for Unsteady-State Conduction

Introduction. As discussed in previous sections of this chapter, the partial differential equations for unsteady-state conduction in various simple geometries can be solved analytically if the boundary conditions are constant at T = T, with time. Also, in the solutions the initial profile of the temperature at t = 0 is uniform at T = To. The unsteady-state charts used also have these same boundary conditions and initial condition. However, when the boundary conditions are not constant with time and/or the initial conditions are not constant with position, numerical methods must be used. 

Numerical calculation methods for unsteady-state heat conduction are similar to numerical methods for steady state discussed in Section 4.15. The solid is subdivided into sections or slabs of equal length and a fictitious node is placed at the center of each section. Then a heat balance is made for each node. This method differs from the steady-state method in that we have heat accumulation in a node for unsteady-state conduction. 

Equations for a slab. The unsteady-state equation for conduction in the x direction 

The figure shows the temperature profile at a given instant of time t s. Making a heat balance on this node or slab, the rate of heat in — the rate of heat out = the rate of heat accumulation in At s. 

Note that in Eq. (5.4-2) the temperature, +c,T at position or node n and at a new time t -h At is calculated from the three points which are known at time t, the starting time. This is called the explicit method, because the temperature at a new time can be 

---

NUMERICAL FINITE-DIFFERENCE METHODS FOR UNSTEADY-STATE CONDUCTION... 

Sec. 5.4 Numerical Finite-Difference Methods for Unsteady-State Conduction... 

---