A large slab of 3 cm-thick Teflon at 20^∘ C is exposed to a radiative heat flux from a high tem

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A large slab of 3 cm-thick Teflon at 20^∘ C is exposed to a...

A large slab of $3 \mathrm{~cm}$-thick Teflon at $20^{\circ} \mathrm{C}$ is exposed to a radiative heat flux from a high temperature heat source of $3500 \mathrm{~W} / \mathrm{m}^{2}$ on one side, and is cooled by an array of air jets on the other side. The jet air temperature is $10^{\circ} \mathrm{C}$, and the convective heat transfer coefficient is $35 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. For the given mesh, calculate the nodal temperatures for the first four time steps using the explicit method. Use a Fourier number of $0.2$ but show that it satisfies appropriate stability criteria. Neglect radiative heat transfer from the slab surfaces to the surroundings.

Key Concepts

Boundary Conditions in Heat Transfer

Boundary conditions define how heat interacts with the surfaces of a material. In heat transfer problems, different types of boundary conditions may be applied, such as prescribed fluxes (for instance, radiative heat flux from an external source) and convective cooling. The proper implementation of these conditions is critical for accurately modeling the temperature distribution within the material, as they govern the heat exchange at the boundaries and influence the overall transient behavior.

Fourier Number

The Fourier number is a dimensionless parameter that characterizes heat conduction processes. It is defined as the ratio of diffusive transport rate to the rate of thermal energy storage and is used in the analysis of transient heat conduction to relate time, material properties, and spatial dimensions. A given Fourier number can inform the choice of time step and ensure that the numerical simulation captures the essential physical behavior.

Stability Criteria

In the context of explicit finite difference schemes, stability criteria are conditions that must be met to ensure that the numerical solution does not diverge over time. For transient heat conduction, this usually involves a restriction on the time step relative to the spatial discretization and thermal diffusivity of the material. Such criteria, often derived from a Courant-Friedrichs-Lewy (CFL) condition, ensure that the Fourier number remains below a specified threshold to prevent numerical instability.

Transient Heat Conduction

This concept involves analyzing heat transfer in a material where the temperature changes with both time and position. Transient heat conduction is governed by partial differential equations that describe how heat diffuses through a medium, requiring both spatial and temporal discretization to track the evolution of temperatures within the material over time.

Explicit Finite Difference Method

The explicit finite difference method is a numerical approach used to solve parabolic partial differential equations, such as those in transient heat conduction. It replaces the continuous spatial and temporal derivatives with discrete approximations, solving for the temperature at each node directly at the next time step based on known values from the current time step. This method is straightforward but conditionally stable, meaning the time step must be chosen carefully to maintain stability.

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