4: A wall 0.12 m thick having thermal diffusivity of 15 10-6 m2/s is initially at temperature of 85oC.Suddenly one face temperature lowered to a temperature of 20 oC while other face is perfectly insulated.Using explicit finite difference method, write descretization equation for space and time steps of 30 mmand 300 s respectively.
Question
4: A wall 0.12 m thick having thermal diffusivity of 15 10-6 m2/s is initially at temperature of 85oC.Suddenly one face temperature lowered to a temperature of 20 oC while other face is perfectly insulated.Using explicit finite difference method, write descretization equation for space and time steps of 30 mmand 300 s respectively.
Solution
The explicit finite difference method is a numerical method used to solve heat transfer problems and other partial differential equations. It involves discretizing the time and space derivatives in the heat conduction equation, which is a parabolic partial differential equation.
The heat conduction equation is given by:
∂T/∂t = α * ∂²T/∂x²
where: T = temperature t = time α = thermal diffusivity x = distance
The discretization of the above equation using the explicit finite difference method gives:
(T_i^n+1 - T_i^n) / Δt = α * ((T_i+1^n - 2T_i^n + T_i-1^n) / Δx²)
where: T_i^n = temperature at location i and time n Δt = time step Δx = space step
Given: α = 15 * 10^-6 m²/s Δx = 30 mm = 0.03 m Δt = 300 s
Substituting these values into the discretization equation gives:
(T_i^n+1 - T_i^n) / 300 = 15 * 10^-6 * ((T_i+1^n - 2T_i^n + T_i-1^n) / 0.03²)
This is the discretization equation for the given problem. The boundary conditions are that one face of the wall is at a temperature of 20°C and the other face is perfectly insulated. The initial condition is that the wall is at a temperature of 85°C. These conditions can be used to solve the discretization equation for the temperature distribution in the wall as a function of time.
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