Transient Heat Conduction

Chapter 4 The Temperature is usually changing with time as well as position. T = T(x,y,z,t) for transient 3-dimensional HT. T = T(z,y,z) for steady 3-dimensional HT. In the previous lectures, we discussed the steady state heat transfer. In this chapter we discuss the heat conduction as a function of time in one dimension.

Objectives We will start with the analysis of lumped systems in which the temperature of a solid varies with time but remains uniform throughout the solid at any time. Then, we consider the variation of T with time and position for one dimensional heat conduction in walls, cylinders, and spheres.

The rate of heat convection between the body and the environment is The Total Heat Transfer is The Maximum Heat transfer is

Define the Characteristic length Define Biot Number Lumped system analysis is applicable if

In this topic, we consider the variation of temperature with time and position in one dimension. Consider a plane wall of thickness 2L, along cylinder of radius ro, and a sphere of radius ro initially at a uniform temperature Ti as shown below.

Temperature profiles

Solution of the problem 1. Analytical solution Be careful of L in Biot number

One-term approximate solution

Coefficients used in the solution

Heat transfer

2. Graphical solution Temperature at the center

Temperature at a point other than the center

Heat Transfer

Conditions of using the one-term and graphical solutions The body is initially at a uniform temperature. T and h of the environment are constant and uniform. No energy generation in the body.

Solution

Since Bi=1/45.8=0.022 < 0.1, we can use the lumped system analysis: