Page 224 - Modelling in Transport Phenomena A Conceptual Approach

204 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

The other initial condition can be obtained from Eq. (5) as

at t =O dT, (9)
- =d,(To-T1)
dt
The solution of Eq. (6) subject to the initial conditions defined by Eqs. (8) and (9)
is
43
T, = TI - -(TI - To) 11 - exp (- $t)] (10)
4
The use of Eq. (10) in Eq. (5) gives the fluid temperature in the form
TI -To
+
[d,
Tf = Tl - - df exp (- dt)] (11)
d
b) Under steady conditions, i.e., t + 00, Eqs. (10) and (11) reduce to

Comment: Note that the final steady-state temperature, T,, can simply be ob-
tained by the application of the first law of thermodynamics. Taking the sphere and
the fluid together as a system, we get
BD;
AU = -p,Cps(TW - Ti) + mfCpf(T, - To) = 0 (13)
6
Noting that

Equation (13) reduces to

Solution of Eq. (15) results in Eq. (12).

Example 7.8 A spherical steel tank of volume 0.5m3 initially contains air at
?bar and 5OOC. A relief valve is opened and air is allowed to escape at a constant
flow rate of 12 mol/ min.
a) If the tank is well insulated, estimate the temperature and pressure of air within
the tank after 5 minutes.
b) If heating coils are placed in the tank to maintain the air temperature constant
at 5OoC, estimate the pressure of air and the amount of heat transferred after 5
minutes.
Air may be assumed an ideal gas with a constant cp of 29 J/ mol. K.