Question

Express the general stability criterion for the explicit method of solution of transient heat conduction problems.

Short answer

Answer: The general stability criterion for the explicit method of solution for transient heat conduction problems is 0 < F ≤ 1/2, where F = (αδt)/(Δx)². F is the control variable, α is the thermal diffusivity of the material, δt is the time step, and Δx is the spatial step. The criterion determines the appropriate time and spatial steps for ensuring a stable and accurate solution.

Step-by-step solution

Understanding the Explicit Method for Transient Heat Conduction Problems

The explicit method calculates the solution for each time step based on the previous time step values. The general equation for the transient heat conduction in one dimension is given by the Fourier's heat equation: \[ \frac{\partial T(x,t)}{\partial t} = \alpha\frac{\partial^2 T(x,t)}{\partial x^2} \] Here, \(T(x,t)\) is the temperature at spatial position x and time t, and \(\alpha\) is the thermal diffusivity of the material. To solve this equation using the explicit method, a discretization scheme is adopted in both space and time. Let the spatial grid points be represented by \(x_i\) and the time grid points be represented by \(t_j\). Hence, the temperature at grid point i and time step j, \(T^{j}_i\), can be written as: \[T^{j+1}_{i} = T^j_i + \delta t\frac{\alpha(\Delta x)^2}{\delta x^2}(T^j_{i+1} - 2T^j_i + T^j_{i-1}) \]

Deriving the Stability Criterion

Stability criteria are used to determine if the explicit method will produce a stable and accurate solution for the transient heat conduction problem. The most commonly used stability criterion is the Von Neumann stability analysis. The Von Neumann stability analysis states that, if the amplification factor of the numerical method is less than or equal to 1, the method is stable: \[\left|\frac{T^{j+1}_i}{T^j_i}\right| \leq 1 \] Substitute the explicit method equation into the amplification factor expression: \[\left|\frac{T^j_i+\delta t\frac{\alpha(\Delta x)^2}{\delta x^2}(T^j_{i+1}-2T^j_i+T^j_{i-1})}{T^j_i}\right| \leq 1 \] Define the control variable \(F\) as: \[ F = \frac{\alpha\delta t}{(\Delta x)^2} \] Re-write the previous equation using the control variable F: \[\left|1 + F(T^j_{i+1}-2T^j_i+T^j_{i-1})\right| \leq 1 \]

Express the General Stability Criterion

For the explicit method of solution for transient heat conduction problems, its stability criterion will depend on the value of the control variable F. The general stability criterion can be expressed as: \[ 0 < F \leq \frac{1}{2} \] This means that for the explicit method to be stable and accurate, the value of F must be between 0 and 1/2. Otherwise, the method might result in inaccurate or unstable results. Remember that F is defined as: \[ F = \frac{\alpha\delta t}{(\Delta x)^2} \] The stability criterion is crucial in determining the appropriate time step, \(\delta t\), and spatial step, \(\Delta x\), for solving transient heat conduction problems using the explicit method.