**Statement (I) :** In an isentropic nozzle flow, discharge reaches a maximum value when the throat pressure reaches the critical value.

This question was previously asked in

ESE Mechanical 2013 Official Paper - 1

Option 2 : Both Statement (I) and Statement (II) are individually true but Statement (II) is NOT the correct explanation of Statement (I)

CT 3: Building Materials

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10 Questions
20 Marks
12 Mins

__Concept:__

A nozzle is a device that increases fluid velocity while causing pressure drop of the fluid i.e. dP > 0,

dP < 0.

The equation for a nozzle is given by

\(\frac{{dA}}{A} = \frac{{dP}}{{\rho {{\vec V}^2}}}\left( {1 - {M^2}} \right)\)

Where A is the cross-sectional area of the nozzle, P is the fluid pressure, is the fluid velocity, M is the Mach number.

Various conditions are described as below:

- Subsonic: M < 1, dP(1 – M
^{2}) < 0, dA < 0 - Sonic: M = 1, dP(1 – M
^{2}) = 0, dA = 0 - Supersonic: M > 1, dP(1 – M
^{2}) < 0, dA > 0

As depicted from the above picture, to accelerate the subsonic flow, the nozzle flow area must first decrease in the flow direction. The **flow area reaches a minimum at the point where the Mach number is unity**. To continue to accelerate the flow to supersonic conditions, the flow area must increase.

The minimum flow area is called the **throat of the nozzle**.

Now if we plot mass flow rate vs. static to stagnation pressure ratio, we get a plot like this

The plot shows that there is a value of P/P_{o} that makes the **mass flow rate a maximum**.

The pressure ratio that makes the **mass flow rate a maximum** is the same pressure ratio at which the Mach number is unity at the flow cross-sectional area. This value of pressure ratio is called **critical pressure ratio** for nozzle flow or we can say mass flow rate and hence discharge reaches a maximum value when the throat pressure reaches the critical value.

So, from the above discussion, it is clear that both the statements are individually true but statement II is not the correct explanation of statement I.