Take the balloon as a control volume. The filling process belongs to
unsteadyflow processes since the balloon is empty initially and contains
some helium when the filling process ends. It has only one inlet and
has no exit.
The pressure in the balloon will reach the pressure in the supply line
at the end of the filling process. If the temperature in the balloon
at the end of the filling process is
less
than 50 ^{o}C,
Minnie can use this supply line to fill her balloons. Hence, temperature
in the balloon at the end of the filling process (T_{2}) needs
to be determined.
(1) Determine the temperature in the balloon when the filling process
ends.
At the beginning of the filling process, the balloon is empty. If 1 denotes the initial state and 2 denotes the final filled state, then the mass at state 1 is
m_{1} = 0
Since the balloon has only one inlet and no exit (m_{e} = 0), the mass balance
for the control volume is
m_{2}  m_{1} = m_{i}
m_{2}  0 = m_{i}
m_{2} = m_{i}
The energy balance for unsteadyflow process,
can be simplified using the initial assumptions (v = 0, z = 0, ke = 0, pe = 0) and the mass terms, giving
Q = 
m_{i}h_{i} + m_{2}u_{2}
It should be noted that work and potential energy are not really zero, but they offset each other. The work done to filling the balloon is assumed to be 100% converted into potential energy.
Next, the enthalpy and the internal energy of Helium can be expressed
as
h_{i} = c_{P}T_{i}
u_{2} =c_{v} T_{2}
Also, the total heat removed from the balloon by the cooling system during
the filling process can be expressed as
where
t = time needed to fill one balloon
= heat removed per unit time by the
cooling system
Substittuting h_{i}, u_{2} and Q (recall, m_{i} = m_{2}) gives
or
T_{2} = 31 ^{o}C
< 50 ^{o}C
It is safe to fill balloons using this supply line.
(2 ) Determine the time needed to fill a single balloon.
Helium is an
ideal gas and thus obeys the idealgas equation of state.
P_{2}V_{2} = m_{2} RT_{2}
where
R = the gas constant, R = 2.0769 kJ/kgK Since the relation between the pressure and the volume of the balloon
is given as P = 10 V, the volume of the balloon at state 2 can be determined.
That is
P_{2} = 10V_{2}
V_{2} = 120 /10 = 12 m^{3 }
Hence, the mass in the balloon at the end of the filling process is
m_{2} = 120(1,000)(12)/(2.0769(1,000)(31+273))
= 2.28 kg
The assumption states that the flow is
steady from the supply line. Hence,
=
2.28/0.01 = 228 s = 3.8 minute
It will take 3.8 minutes to fill one balloon.
