An initially deflated, flat balloon is connected by a valve to a 12 m3 storage t

Question

An initially deflated, flat balloon is connected by a valve to a 12 m3 storage tank containing helium gas, 2 MPa and ambient temperature, 20°C. The valve is opened and the balloon is inflated at constant pressure, Po = 100 kPa, equal to ambient pressure, until it becomes spherical at D1 = 1 m. If the balloon is larger than this, the balloon material is stretched giving a pressure inside as
P = P0 + C (1 - D1/D) D1/D
The balloon is inflated to a final diameter of 4 m, at which point the pressure inside is 400 kPa. The temperature remains constant at 20°C. What is the maximum pressure inside the balloon at any time during this inflation process? What is the pressure inside the helium storage tank at this time?

Explanation / Answer

At the end of the process we have D = 4 m so we can get the constant C as P = 400 = P0 + C ( 1 – 14 ) 14 = 100 + C × 3/16 => C = 1600 The pressure is: P = 100 + 1600 ( 1 – X –1) X –1; X = D / D1 Differentiate to find max: dPdD = C ( - X –2 + 2 X –3 ) / D1 = 0 => - X –2 + 2 X –3 = 0 => X = 2 at max P => D = 2D1 = 2 m; V = p6 D3 = 4.18 m3 Pmax = 100 + 1600 ( 1 - 12 ) 12 = 500 kPa Helium is ideal gas A.5: m = PVRT = 500 × 4.1892.0771 × 293.15 = 3.44 kg mTANK, 1 = PVRT = 2000 × 122.0771 × 293.15 = 39.416 kg mTANK, 2 = 39.416 – 3.44 = 35.976 kg PT2 = mTANK, 2 RT/V = ( mTANK, 1 / mTANK, 2 ) × P1 = 1825.5 kPa

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