Page 96 - Bird R.B. Transport phenomena

§3.3 The Equation of Mechanical Energy 81

§3.3 THE EQUATION OF MECHANICAL ENERGY

Mechanical energy is not conserved in a flow system, but that does not prevent us from
developing an equation of change for this quantity. In fact, during the course of this
book, we will obtain equations of change for a number of nonconserved quantities, such
as internal energy, enthalpy, and entropy. The equation of change for mechanical en-
ergy, which involves only mechanical terms, may be derived from the equation of mo-
tion of §3.2. The resulting equation is referred to in many places in the text that follows.
We take the dot product of the velocity vector v with the equation of motion in Eq.
3.2-9 and then do some rather lengthy rearranging, making use of the equation of conti-
nuity in Eq. 3.1-4. We also split up each of the terms containing p and т into two parts.
The final result is the equation of change for kinetic energy:
2
f(W) = -(V • > v) - (V • pv) - p(-V • v)
t
rate of rate of addition rate of work rate of reversible
increase of of kinetic energy done by pressure conversion of
kinetic energy by convection of surroundings kinetic energy into
per unit volume per unit volume on the fluid internal energy

- (V • (T•v]) (-T:VV) + p(v • g) (3.3-1) 1
rate of work done rate of rate of work
by viscous forces irreversible by external force
on the fluid conversion on the fluid
from kinetic to
internal energy
At this point it is not clear why we have attributed the indicated physical significance to
the terms p(V • v) and (T:VV). Their meaning cannot be properly appreciated until one
has studied the energy balance in Chapter 11. There it will be seen how these same two
terms appear with opposite sign in the equation of change for internal energy.
2
We now introduce the potential energy (per unit mass) Ф, definedby g = - V<f>. Then
the last term in Eq. 3.3-1 may be rewritten as -p(v • V<f>) = -(V -руФ) + Ф(У -pv). The
equation of continuity in Eq. 3.1-4 may now be used to replace +Ф(У • pv) by -<b(dp/dt).
The latter may be written as -<?(рФ)/^, if the potential energy is independent of the
time. This is true for the gravitational field for systems that are located on the surface of
the earth; then Ф = gh, where g is the (constant) gravitational acceleration and h is the el-
evation coordinate in the gravitational field.
With the introduction of the potential energy, Eq. 3.3-1 assumes the following form:

-(V • pv) - p(-V • v) - (V • [T • v]) - (-T:VV) (3.3-2)

This is an equation of change for kinetic-plus-potential energy. Since Eqs. 3.3-1 and 3.3-2 con-
tain only mechanical terms, they are both referred to as the equation of change for mechani-
cal energy.
The term p(V • v) may be either positive or negative depending on whether the fluid
is undergoing expansion or compression. The resulting temperature changes can be rather
large for gases in compressors, turbines, and shock tubes.

1 The interpretation under the (T:VV) term is correct only for Newtonian fluids; for viscoelastic
fluids, such as polymers, this term may include reversible conversion to elastic energy.
2 If g = —b g is a vector of magnitude g in the negative z direction, then the potential energy per
z
unit mass is Ф = gz, where z is the elevation in the gravitational field.