A plate-glass slab with initial temperature T0 is suddenly...

A plate-glass slab with initial temperature $T_{0}$ is suddenly exposed to a gas and surroundings at temperature $T_{e}$. The slab loses heat by forced convection and radiation. The convective heat transfer coefficient can be assumed constant, and radiation can be calculated for a small gray body in large, nearly black surroundings. (i) Give the equations required to solve for the temperature response using the explicit finite-difference method. Discuss the appropriate stability criterion.

A plate-glass slab with initial temperature $T_{0}$ is suddenly exposed to a gas and surroundings at temperature $T_{e}$. The slab loses heat by forced convection and radiation. The convective heat transfer coefficient can be assumed constant, and radiation can be calculated for a small gray body in large, nearly black surroundings.
(i) Give the equations required to solve for the temperature response using the explicit finite-difference method. Discuss the appropriate stability criterion.

Key Concepts

Heat Conduction Equation

This concept involves the formulation of the energy balance in a solid that leads to a partial differential equation representing heat conduction. For a one?dimensional medium, the differential equation typically appears as dT/dt = ? d²T/dx², where T is temperature, t is time, x is the spatial coordinate, and ? is the thermal diffusivity. This equation forms the basis for simulating the structural temperature response as it models how heat diffuses within the object over time.

Explicit Finite-Difference Method

This numerical method is used to approximate the solution of partial differential equations by discretizing both time and space into finite steps. In the context of transient heat conduction, it replaces the time derivative with a forward difference and the spatial second derivative with a central difference. An update equation is constructed at each node, which calculates the future temperature based on the current temperatures at that node and its neighbors. This method is simple to implement but requires careful selection of time and space step sizes to ensure that the solution is stable and accurate.

Boundary Conditions for Convection and Radiation

At the surfaces of the object, the boundary conditions incorporate convective and radiative heat losses. The convective heat transfer can be modeled using Newton’s law of cooling, which relates the surface heat flux to the difference between the surface temperature and the ambient temperature via a constant heat transfer coefficient. Radiation is handled by applying the Stefan-Boltzmann law for a small gray body in an environment that approximates a large black body, often linearized for small temperature differences. These boundary conditions are essential for correctly simulating the energy exchange at the surface between the solid and its surroundings.

Stability Criterion

When using the explicit finite-difference method, there is a critical requirement for numerical stability known as the Courant–Friedrichs–Lewy (CFL) condition. For the heat conduction equation, the time step ?t must be chosen such that ?t ? (?x)²/(2?), where ?x is the spatial step and ? is the thermal diffusivity. This criterion ensures that the numerical scheme remains stable, preventing the amplification of errors as the simulation progresses over time.

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