Section 2.2: Examples

Equations in Engineering Problems

The following example show how the different categories of equation arise in various situations.

Example 1: Simple Mixing Process

See Figure below

Consider the behaviour of the process over a period of time when the two flowrates do not change, and any disturbances to the process resulting from previous changes in the flows have settled out.

total outflow rate = total inflow rate

F1 + F2 - F = 0

Notes

Example 2: Tank Filling

See Figure below

Liquid flows at a rate F kg/s into a tank. The mass of material in the tank initially, at say time t=0 is M0 kg. There is no outflow from the tank.

rate of accumulation = net rate of inflow - net rate of outflow

With initial condition: M(t) = M0 at t=0

Notes

Example 3: Heat Transfer in a Slab

See Figure below

Heat flows at a constant steady rate Q watts across a uniform slab. Each side of the slab is held at constant temperature. The temperature of the slab, of cross sectional area A and thermal conductivity k, varies across its thickness x and is denoted by T(x).

Notes

Example 4: Time Varying Convection by Plug Flow in a Pipe

This is described by a p.d.e. whose independent variables are time and distance along the pipe. Consider the situation where the temperature T is changing with distance x down a pipe of total length L where the velocity of flow is u.

This is described by the convective flow terms of the Navier-Stokes equation:

Although this is a complicated looking equation it turns out to have a simple solution. The temperature of the fluid leaving the pipe at time t is exactly the same as the temperature entering the pipe at a previous time (t - L/u) since L/u is the residence time for the fluid in the pipe.

More Complicated Situations

Summary and Further Points


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