Page 226 - Modelling in Transport Phenomena A Conceptual Approach

206 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

On the other hand, the right-hand side of Eq. (4) is expressed in terms of temper-
ature as
- cv 7
-- dfisys - ays
dt
Hence, substitution of Eqs. (5) and (6) into Eq. (4) gives

- 12RTsys = (no - 12t) CV -
- dTsys
dt (7)
For an ideal gas
&
&=Cv+R * -- (8)
-7-1
R
where
CP 29
y=y= = 1.4 (9)
cv 29-8.314
Note that Eq. (7) is a separable equation. Substitution of Eq. (8) into Eq. (7) and
rearrangement yielcls

Integration gives
Tsys =To ( no )
no - 12t '--I

The variation of pressure as a jimction of time can be estimated by using the ideal
gas law, i.e.,
ns ysR Ts ys
psys = (12)
V
Substitution of Eqs. (2) and (11) into Eq. (12) gives

Since RToIV = Po/no, Eq. (13) duces to
no - 12t '
psys =Po ( no ) (14)

Substitution of the numerical values into Eqs. (11) and (14) gives Tsys and Psys,
respectively, after 5 minutes as

Tsars = (50 + 273) [ 130.3 - (12)(5)] 1'4-1 = 252.4K
130.3