## Q. 3.14

Converging–Diverging Nozzle Flow

Consider ideal gas flow in a converging-diverging nozzle ($\rm A_{throat} =10\, cm^2$ and $\rm A_{exit} =40\, cm^2$), fed by a reservoir ($\rm T_0$ =20°C, $\rm p_0$ = 500 kPa absolute). Determine the nozzle exit pressures such that M=1 in $\rm A_{throat} ≡ A^*$ . Specifically, a varying receiver pressure, $\rm p_r$, can produce different mass flow rates and throat conditions (see Sketch)

Sketch:

Note: Receiver (or back) pressure pr can be regulated via a valve and vacuum pump

## Verified Solution

Clearly, when $\rm p_r = p_0$ (see Curve A), no flow can occur. If $\rm p_r$ is slightly lower than $\rm p_0$, subsonic flow occurs (see Curve B). Lowering $\rm p_r$ further results in a pressure distribution $\rm p(x)/p_0$ (see Curve C) where M = 1 in the throat. Now, a second particular $\rm p_r$ – value generates again subsonic flow with M = 1 in the throat, i.e., Curve D, because there are two solutions to Eq. (3.57) for $\rm M_{exit}$ or for that matter to Eq. (3.62).

$\rm\frac{p_0}{p} =\left\lgroup1+\frac{k-1}{2}M^2 \right\rgroup ^{\frac{k}{k-1} }$                                (3.57)

$\rm\frac{A^*}{A} =M\left[\frac{k+1}{2+(k-1)M^2} \right] ^{\frac{k+1}{2(1-k)} }$                              (3.62)

Given $\rm A_{exit} / A_{throat} ≡ A / A^*= 40 /10 =4$ and using Eq. (3.62), two $\rm M_e$ – values can be obtained via trial-and-error, i.e., $\rm M_e ≈ 0.147$  and $\rm M_e ≈ 2.94$ . Employing Eq. (3.57), the corresponding exit pressures are

$\rm p_e\equiv p_r\hat =p_c=0.985p_0~\text{and}~p_D=0.0298p_0$

or with $\rm p_0$ = 500 kPa

$\rm p_c=492.5\,kPa~\text{and}~p_D=15\,kPa$

Graph:

• To generate Curve C, the pressure drop is only $\rm p_0 − p_c$ =7.5 kPa
• The associated exit temperature conditions are $\rm T_c$ =292 K and $\rm T_D$ = 107 K
$\rm v_c=M_c\sqrt{kRT_c}=50\,m/s$
$\rm v_D=M_D\sqrt{kRT_D}=610\,m/s$
• Any outlet pressure in the range of $\rm p_c < p_r < p_D$ , generating pressure distributions $\rm p(x)/p_0$ between Curves C and D, produce shock waves inside or outside the diverging part of the nozzle and the flow is generally non-isentropic.