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Chapter 5: Problem 91

Is there any limitation on the size of the time step \(\Delta t\) in the solution of transient heat conduction problems using (a) the explicit method and \((b)\) the implicit method?

Short Answer

Expert verified
Answer: In the explicit method, the time step size ∆t must follow the Courant-Friedrichs-Lewy (CFL) condition to maintain stability, which states that ∆t should not be greater than the ratio of the spatial step size squared (∆x²) to the thermal diffusivity (α): ∆t ≤ ∆x²/(4α). In the implicit method, there is no stability criterion or limit on the size of the time step ∆t, but an increased time step size may result in a loss of accuracy.

Step by step solution

01

(a) Explicit method limitation

In the explicit method, the temperature at each point in the system is evaluated based on the temperatures of its neighboring points in the previous time step. The time step size in the explicit method has a stability criterion, often referred to as the Courant-Friedrichs-Lewy (CFL) condition. The CFL condition states that the time step size ∆t should not be greater than the ratio of the spatial step size squared (\(\Delta x^2\)) to the thermal diffusivity (\(\alpha\)): $$\Delta t \leq \frac{\Delta x^2}{4 \alpha}$$ If the time step size exceeds this limit, the explicit method may become unstable, leading to erroneous results.
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(b) Implicit method limitation

In the implicit method, the temperature at each point is updated simultaneously with the temperatures of its neighboring points. The implicit method is unconditionally stable, which means that there is no stability criterion or limit on the size of the time step ∆t for this method. However, increasing the time step size too much may lead to a loss in accuracy. Therefore, it is recommended to choose an appropriate value for the time step size depending on the specific problem and desired accuracy level.

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Most popular questions from this chapter

Consider transient heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(50^{\circ} \mathrm{C}\) while the right surface (node 6) is subjected to a solar heat flux of \(600 \mathrm{~W} / \mathrm{m}^{2}\). The wall is initially at a uniform temperature of \(50^{\circ} \mathrm{C}\). Express the explicit finite difference formulation of the boundary nodes 0 and 6 for the case of no heat generation. Also, obtain the finite difference formulation for the total amount of heat transfer at the left boundary during the first three time steps.

What happens to the discretization and the round-off errors as the step size is decreased?

Consider transient one-dimensional heat conduction in a plane wall that is to be solved by the explicit method. If both sides of the wall are subjected to specified heat flux, express the stability criterion for this problem in its simplest form.

Explain why the local discretization error of the finite difference method is proportional to the square of the step size. Also explain why the global discretization error is proportional to the step size itself.

Consider the base plate of a \(800 \mathrm{~W}\) household iron having a thickness of \(L=0.6 \mathrm{~cm}\), base area of \(A=160 \mathrm{~cm}^{2}\), and thermal conductivity of \(k=20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The inner surface of the base plate is subjected to uniform heat flux generated by the resistance heaters inside. When steady operating conditions are reached, the outer surface temperature of the plate is measured to be \(85^{\circ} \mathrm{C}\). Disregarding any heat loss through the upper part of the iron and taking the nodal spacing to be \(0.2 \mathrm{~cm},(a)\) obtain the finite difference formulation for the nodes and \((b)\) determine the inner surface temperature of the plate by solving those equations. Answer: (b) \(100^{\circ} \mathrm{C}\)
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