Page 156 - Modelling in Transport Phenomena A Conceptual Approach

136 CHAPTER 5. RATE OF GENERATION

customary to combine these two forces in a single term and express the rate of
momentum generation per unit volume as

(5.1- 15)

where P is the modified pressure’ defined by

P=P+pgh (5.1-16)

in which h is the distance measured in the direction opposite to gravity from any
chosen reference plane.

5.1.3.1 Physical interpretation of the modified pressure
Consider a stagnant liquid in a storage tank open to the atmosphere. Let z be the
distance measured from the surface of the liquid in the direction of gravity. The
hydrostatic pressure distribution within the fluid is given by

P = Patm + pgz (5.1- 17)

For this case the modified pressure is defined as
P = P - pgz (5.1-18)

Substitution of Eq. (5.1-18) into Q. (5.1-17) gives
P = Patm = constant (5.1-19)

The simplicity of defining the modified pressure comes from the fact that it is
always constant under static conditions whereas the hydrostatic pressure varies as
a function of position. Suppose that you measure a pressure difference over a length
L of a pipe. It is difficult to estimate whether this pressure difference comes from
a flow situation or hydrostatic distribution. However, any variation in P implies a
flow. Another distinct advantage of defining modified pressure is that the difference
in P is independent of the orientation of the pipe as shown in Table 5.1.

5.2 RATE OF GENERATION IN ENERGY

TRANSPORT

Let us consider the following paradox: “One of the most important problems that
the world faces today is energy shortage. According to the first law of thermodynam-
ics, energy is converted from one fonn to another and transfend ji-om one system
to another but its total is conserved. If energy is conserved, then there should be
no energy shortage.”

The term P is also called equivalent pressure, dynamic pressure and piezometric pressure.