Balloons need to be filled for a welcoming party. Determine the heat that has to be removed to prevent the balloon from being overheated. Also determine the time needed to fill a single balloon.

Assumptions:

• The supply line is steady
• The state inside the balloon is uniform
• Neglect the changes in kinetic energy and potential energy
• No work is involved
• The cooling system is steady

Take the Balloon as a
Control Volume

Take the balloon as a control volume. The filling process belongs to unsteady-flow 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 ^oC, Minnie can use this supply line to fill her balloons. Hence, temperature in the balloon at the end of the filling process (T₂) 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

      m₁ = 0
        
Since the balloon has only one inlet and no exit (m_e = 0), the mass balance for the control volume is

      m₂ - m₁ = m_i
      m₂ - 0 = m_i
      m₂ =   m_i

The energy balance for unsteady-flow process,

can be simplified using the initial assumptions (v = 0, z = 0, ke = 0, pe = 0) and the mass terms, giving

      Q = - m_ih_i + m₂u₂

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_PT_i 
      u₂ =c_v T₂

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, u₂ and Q (recall, m_i =  m₂) gives

     
or
    T₂ = 31 ^oC < 50 ^oC

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 ideal-gas equation of state.

      P₂V₂ = m₂ RT₂

where
      R = the gas constant, R = 2.0769 kJ/kg-K

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

      P₂ = 10V₂
      V₂ = 120 /10 = 12 m³

Hence, the mass in the balloon at the end of the filling process is

       m₂ = 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.