Page 27 - Modelling in Transport Phenomena A Conceptual Approach

8 CHAPTER 1. INTRODUCTION

Example 1.4 Energy generation rate per unit volume as a result of an electric
cumnt passing through a rectangular plate of cross-sectional area A and thickness
L is given by
XX
8 = %osin (7)
where !R is in W/m3. Calculate the total energy generation rate within the plate.
Solution

Since Y? is dependent on position, energy generation rate is calculated by integration
of Y? over the volume of the plate, ie.,

Energy generation rate = A %lo 1' sin (F) dx

1.3.3 Rate of Accumulation Term

The rate of accumulation of any quantity cp is the time rate of change of that
particular quantity within the volume of the system. Let p be the mass density
and 8 be the quantity per unit mass. Thus,

Total quantity of cp = /// p @ dV (1.3-4)
V
and the rate of accumulation is given by

Accumulation rate = d dt (/I/ p @ dV) (1.3-5)

If 8 is independent of position, then Eq. (1.3-5) simplifies to
d
Accumulation rate = - (m +) (1.3-6)
dt
where m is the total mass within the system.
The accumulation rate may be either positive or negative depending on whether
the quantity is increasing or decreasing with time within the volume of the system.

1.4 SIMPLIFICATION OF THE RATE
EQUATION

In this section, the general rate equation given by Eq. (1.1-1) will be simpliied for
two special cases: (i) steady-state transport without generation, (ii) steady-state
transport with generation.