Question

Write down the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, and indicate what each variable represents.

Short answer

Question: Write down the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form and define each variable used in the equation. Answer: The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form is: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + \frac{q_{g}}{k}\) where: T = temperature (K), t = time (s), x = distance (m), α = thermal diffusivity (m²/s), q_g = heat generation rate per unit volume (W/m³), k = thermal conductivity (W/m·K).

Step-by-step solution

One-dimensional transient heat conduction equation

First, let's write down the general one-dimensional transient heat conduction equation, which is given by: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\) where T = temperature (K), t = time (s), x = distance (m), α = thermal diffusivity (m²/s).

Adding heat generation term

Now, we need to add the heat generation term, which is represented as: \(\frac{q_g}{k}\) where q_g = heat generation rate per unit volume (W/m³), k = thermal conductivity (W/m·K). The one-dimensional transient heat conduction equation, including the heat generation term, is then written as: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + \frac{q_g}{k}\)

Simplify the equation

Considering that the thermal conductivity (k) is constant, the equation can be rewritten as: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} + \frac{q_{g}}{k}\) This is the one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat generation in its simplest form, where: T = temperature (K), t = time (s), x = distance (m), α = thermal diffusivity (m²/s), q_g = heat generation rate per unit volume (W/m³), k = thermal conductivity (W/m·K).