Page 23 - Modelling in Transport Phenomena A Conceptual Approach

4 CHAPTER 1. INTRODUCTION

Example 1.1 A Newtonian fluid with constant viscosity, p, and density, p, is
initially at rest in a very long horizontal pipe of length L and radius R. At t = 0,
a pressure gradient [APIIL is imposed on the system and the volumetric flow rate,
Q, is expressed as

where r is the dimensionless time defined by

and A1 = 2.405, A2 = 5.520, AB = 8.654, etc. Determine the volumetric flow rate
under steady conditions.
Solution
Steady-state solutions are independent of time. To eliminate the time from the
unsteady-state solution, we have to let t --+ 00. In that case, the exponential term
approaches zero and the resulting steady-state solution is

which is known as the Hagen-Poiseuille law.
Comment: If time appears in the exponential term, then the term must have a
negative sign to assure that the solution does not blow as t ---f 00.

Example 1.2 A cylindrical tank is initially half full with water. The water is fed
into the tank from the top and it leaves the tank from the bottom. The volumetric
flow rates are different from each other. The differential equation describing the
time rate of change of the height of water in the tank is given by
dh
-=6-8&
dt
where h is the height of water in meters. Calculate the height of water in the tank
under steady conditions.
Solution

Under steady conditions dhldt must be zero. Then
0=6-S&

or,
h = 0.56m