Page 494 - Modelling in Transport Phenomena A Conceptual Approach

474 CHAPTER 11. UNSTEADY MICROSCOPIC BALANCES WITH GEN.

Figure 11.1 Unsteady-state flow in a circular pipe.

The conservation statement for momentum is expressed as
Rate of Rate of Forces acting
( momentum in ) - ( momentum out )+( on a system
= ( Rate of momentum ) (11.1-4)
accumulation

The pressure in the pipe depends on z. Therefore, it is necessary to consider only
the z-component of the equation of motion. For a cylindrical differential volume
element of thickness AT and length Az, as shown in Figure 11.1, Eq. (11.1-4) is
expressed as

(rzal, + Trzl,. 2srAz) - [~zz[,+A, STTAT + sTZlr+Ar ~T(T + AT)Az]
a
+
+ (PI, - P1z+Az)2s~Ar 2srArAzpg = - (2srArAzpv,) (11.1-5)
at
Dividing Eq. (11.1-5) by 2sArAz and taking the limit as AT -+ 0 and Az + 0
gives

+ lim ( AzzIz - sZzlz+Az ) +pg (11.1-6)
Az-0 Az

(11.1-7)