Question

Consider a converging-diverging nozzle with an exit-to-throat area of 3. The inlet reservoir pressure is 1 atm and the exit pressure is 0.5 atm.

(a) For this pressure ratio a normal shock will stand somewhere inside the divergent portion of nozzle. Calculate the location of shock (A/At =?).

(b) back pressure if the flow were isentropic throughout.

Answer 1

a) Refer the diagram below.

P₀₀= 1 atm=101.3 kPa

T₀₀= 25⁰C (assumed)=298 K

M₁=1

We adopt a trial and error method

Iteration 1

Let A₂/A_t= 2

The Mach No before the shock (from air isentropic table)

M₂= 2.2

From normal shock table

M₃= 0.5471

We assume an imaginary throat at 3 with area A₃^*

With M₃= 0.5471, A₃/A₃^*= 1.2593

A₄/A₃^*= A₃/A₃^*xA₄/A₃

But A₃= A₂= 2A_t

A₄= 3A_t

Hence

A₄/A₃^*= 1.2593x(3/2)=1.889

Hence (from isentropic table)

M₃= 0.325

Using Excel function M_exit for normal shock at diverging part with P_b/P₀₀= 0.5

M₄= 0.38>0.325

Hence we go for next iteration

Iteration -2

Let A₂/A_t= 2.5

The Mach No before the shock (from air isentropic table)

M₂= 2.444

From normal shock table

M₃= 0.5185

We assume an imaginary throat at 3 with area A₃^*

With M₃= 0.5185, A₃/A₃^*= 1.306

A₄/A₃^*= A₃/A₃^*xA₄/A₃

But A₃= A₂= 2.5A_t

A₄= 3A_t

Hence

A₄/A₃^*= 1.306x(3/2.5)=1.5672

Hence (from isentropic table)

M₃= 0.409

Using Excel function M_exit for normal shock at diverging part with P_b/P₀₀= 0.5

M₄= 0.380.409

By interpolation

b) From property of convergent divergent nozzle flow characteristics

P₄=P_b= 0.5 atm