Question

How does the finite difference formulation of a transient heat conduction problem differ from that of a steady heat conduction problem? What does the term \(\rho A \Delta x c_{p}\left(T_{m}^{i+1}-T_{m}^{i}\right) / \Delta t\) represent in the transient finite difference formulation?

Short answer

Answer: The main difference between the finite difference formulation of transient heat conduction problems and steady heat conduction problems is the presence of a time-dependent term in transient problems. This term accounts for the change in temperature within an object over time, which is essential for solving transient problems. In contrast, steady heat conduction problems do not have this time-dependency.

Step-by-step solution

Steady vs Transient Conduction Problems

Let's first understand the difference between steady and transient heat conduction problems. A steady problem refers to a situation where the temperature distribution in an object does not change over time. On the other hand, a transient problem deals with the temperature change within an object over time.

Finite Difference Formulation for Steady Problems

The finite difference formulation for a steady heat conduction problem derives from the steady-state heat conduction equation, which can be expressed as: \(\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) = 0\). In this case, the focus is on solving for temperature distribution within an object, and since it is a steady-state problem, there is no time-dependency.

Finite Difference Formulation for Transient Problems

For a transient heat conduction problem, the finite difference formulation derives from the unsteady heat conduction equation, which is expressed as: \(\rho c_{p}\frac{\partial T}{\partial t} = \frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)\). In comparison to the steady heat conduction equation, this form includes an unsteady term, being the first term on the left hand side, which represents the change in temperature with respect to time.

Understanding the Given Mathematical Term

The given term, \(\rho A \Delta x c_{p}\left(T_{m}^{i+1}-T_{m}^{i}\right) / \Delta t\), is from the finite difference formulation of a transient heat conduction problem. It represents the energy storage term in the transient heat conduction equation. Here, \(\rho\) is the density, \(A\) is the area, \(\Delta x\) is the spatial step size, \(c_{p}\) is the specific heat capacity, \(T_{m}^{i+1}\) and \(T_{m}^{i}\) are the nodal temperatures at time steps \(i+1\) and \(i\), and \(\Delta t\) is the time step size. This term essentially signifies how the temperature changes within an object over time due to energy storage in the object. In conclusion, the main difference between the finite difference formulation of transient heat conduction problems and a steady heat conduction problems is the presence of a time-dependent term. This term is responsible for considering the temperature change within an object over time and is essential for solving transient problems.