Page 237 - Bird R.B. Transport phenomena

§7.8 Derivation of the Macroscopic Mechanical Energy Balance 221

The type of motion that the manometer liquid exhibits depends on the value of the dis-
criminant in Eq. 7.7-22:
2 2
(a) If (6fjb/pR ) > (6g/L), the system is over damped, and the liquid moves slowly to its final
position.
2 2
(b) If (в/л/pR ) < (6g/L), the system is under damped, and the liquid oscillates about its
final position, the oscillations becoming smaller and smaller.
2 2
(c) If (в/л/pR ) = (6g/L), the system is critically damped, and the liquid moves to its final
position in the most rapid monotone fashion.
The tube radius for critical damping is then

(*£)'" .7.7-25»

If the tube radius R is greater than R cr, an oscillatory motion occurs.

§7.8 DERIVATION OF THE MACROSCOPIC
1
MECHANICAL ENERGY BALANCE
In Eq. 7.4-2 the macroscopic mechanical energy balance was presented without proof. In
this section we show how the equation is obtained by integrating the equation of change
for mechanical energy (Eq. 3.3-2) over the entire volume of the flow system of Fig. 7.0-1.
We begin by doing the formal integration:

2
/ j t (W + РФ) dV = - j (V • {\pv + Ф)у) dV- j (V • pv) dV - j (V • [т • v]) dV
Р
V(t) V(t) V(t) 1/(0
p(V • v) dV + | (T:VV) dV (7.8-1)
) V(t)
Next we apply the 3-dimensional Leibniz formula (Eq. A.5-5) to the left side and the
Gauss divergence theorem (Eq. A.5-2) to terms 1, 2, and 3 on the right side.

j j (W + p&)dV=-j(n- (W + Ф)(у - v)) dS - j (n • pv) dS
t Р s
V(t) S(t) Sit)
- j (n • [T • v]) dS + J p(V • v) dV + | (T:VV) dV (7.8-2)
S(t) V(t) V(t)
The term containing v , the velocity of the surface of the system, arises from the applica-
s
tion of the Leibniz formula. The surface S(t) consists of four parts:
• the fixed surface S (on which both v and v are zero)
f
s
• the moving surfaces S (on which v = v with both nonzero)
m
s
• the cross section of the entry port Si (where v = 0)
s
• the cross section of the exit port S (where v = 0)
2 s
Presently each of the surface integrals will be split into four parts corresponding to these
four surfaces.
We now interpret the terms in Eq. 7.8-2 and, in the process, introduce several as-
sumptions; these assumptions have already been mentioned in §§7.1 to 7.4, but now the
reasons for them will be made clear.

1
R. B. Bird, Korean J. Chem. Eng., 15,105-123 (1998), §3.