An air tank of volume 1.5 m3 is initially at 800 kPa and 208C. At t 5 0, it begins exhausting through a converging nozzle to sea-level conditions. The throat area is 0.75 cm2 . Estimate (a) the initial mass fl ow in kg/s, (b) the time required to blow down to 500 kPa, and (c) the time at which the nozzle ceases being choked.

(a) m = 0.141 kg/s

(b) t = 47.343 s

(c) t = 143.745 s

Explanation:

Given that:

The volume of air in the tank V = 1.5 m³

The initial temperature in the tank is supposed to be 20° C and not 208 C;

So  

The initial pressure in the tank

The throat area = 0.75 cm²

To find the initial mass flow in kg/s.

Lets first recall that:

Provided that , then the flow is choked.

Then;

From what we see above, it is obvious that the ratio is lesser than 0.528, therefore, the flow is choked.

Now, for a choked nozzle, the initial mass flow rate is determined by using the formula:

where;

where;

T = 244.16 K

During a critical condition when Mach No. is equal to one;

V = 313.214  m/s

Thus, the initial mass flow rate

m = 6.03 × 0.75 × 10⁻⁴ × 313.214

m = 0.141 kg/s

(b)

The mass balance formula for the control volume surrounding the tank can be expressed as:

When the air mass flow rate is:

Thus; replacing in the previous equation; we have:

Taking the differential of both sides  from 0 → t

So, when the pressure P = 500 kPa, the time required is:

t = 47.343 s

(c)

Let us recall that:

The choking on the nozzle occurred when

From ; the time required for is:

By solving:

t = 143.745 s

grav potential energy

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