For two-dimensional unsteady conduction, derive both the...

Name: Problem 125, Chapter 3: Basic Heat And Mass Transfer
Uploaded: 2021-09-04T20:44:46-08:00
Duration: 2 min 7 s
Channel: Chai Santi
Description: For two-dimensional unsteady conduction, derive both the explicit and implicit forms of the finite-difference approximation for a convective boundary condition at an interior corner. For the explicit case, also derive the stability criterion. (See Table 3.9, item 5.)

For two-dimensional unsteady conduction, derive both the explicit and implicit forms of the finite-difference approximation for a convective boundary condition at an interior corner. For the explicit case, also derive the stability criterion. (See Table 3.9, item 5.)

Key Concepts

Interior Corner Treatment

When dealing with domains that include interior corners, special attention must be given to the numerical treatment because the boundary conditions must be applied consistently at the junction of two boundaries. This involves careful discretization to appropriately capture the behavior of the convective boundary condition at the corner.

Two-Dimensional Unsteady Conduction

This concept deals with heat transfer problems in a domain where temperature changes with time and varies spatially in two dimensions. It involves solving the transient heat conduction equation, which is a partial differential equation accounting for temporal and spatial variations in temperature.

Finite-Difference Method

The finite-difference method approximates derivatives in a differential equation using differences between function values at discrete points on a grid. This numerical technique transforms continuous models into systems of algebraic equations, making them solvable using computational methods.

Explicit Finite-Difference Scheme

An explicit scheme calculates the future state of the system directly in terms of known quantities from previous time steps. Its simplicity is balanced by a requirement for stability conditions, often expressed as a relation between time step and spatial grid sizes, to ensure that errors do not amplify.

Implicit Finite-Difference Scheme

Implicit schemes involve solving a system of equations where the unknown future states appear on both sides of the equation. Although computationally more intensive due to the need to solve these systems, implicit formulations are generally more stable and can accommodate larger time steps.

Convective Boundary Condition

This boundary condition models heat transfer at a surface that is undergoing convection, relating the heat flux at the boundary to the temperature difference between the surface and the surrounding fluid, often via a heat transfer coefficient. It is particularly challenging to implement in variably shaped domains.

Stability Criterion

The stability criterion in numerical methods ensures that the error in the computed solution does not grow uncontrollably with successive time steps. For explicit finite-difference schemes, it often manifests as a restriction that relates the time step size to the spatial step sizes and physical parameters, ensuring convergence of the solution.

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