Isentropic nozzle flow

Fig: A supersonic nozzle

A nozzle for a supersonic flow must increase in area in the flow direction, and a diffuser must decrease in area, opposite to a nozzle and diffuser for a subsonic flow. So, for a supersonic flow to develop from a reservoir where the velocity is zero, the subsonic flow must first accelerate through a converging area to a throat, followed by continued acceleration through an enlarging area.

The nozzles on a rocket designed to place satellites in orbit are constructed using such converging-diverging geometry. The energy and continuity equations can take on particularly helpful forms for the steady, uniform, isentropic flow through the nozzle. Apply the energy equation with Q_ W_S 0 between the reservoir and some location in the nozzle to obtain

                                               ${\displaystyle c_{p}\cdot T_{o}={\frac {V^{2}}{2}}+c_{p}\cdot T}$

Any quantity with a zero subscript refers to a stagnation point where the velocity is zero, such as in the reservoir. Using several thermodynamic relations equations can be put in the forms:

${\displaystyle {\frac {T_{o}}{T}}=1+{\frac {k-1}{2}}M^{2}}$
${\displaystyle {\frac {p_{o}}{p}}=\left(1+{\frac {k-1}{2}}M^{2}\right)^{\left({\frac {k}{k-1}}\right)}}$
${\displaystyle {\frac {\rho _{o}}{\rho }}=\left(1+{\frac {k-1}{2}}M^{2}\right)^{\left({\frac {1}{k-1}}\right)}}$

If the above equations are applied at the throat (the critical area signified by an Asterisk (*) superscript, where M =1), the energy equation takes the forms

${\displaystyle {\frac {T^{*}}{T_{o}}}={\frac {2}{k+1}}}$ 
${\displaystyle {\frac {p^{*}}{p_{o}}}=\left({\frac {2}{k+1}}\right)^{\left({\frac {k}{k-1}}\right)}}$
${\displaystyle {\frac {\rho ^{*}}{\rho _{o}}}=\left({\frac {2}{k+1}}\right)^{\left({\frac {1}{k-1}}\right)}}$

The critical area is often referenced even though a throat does not exist. For air with k =1.4, the equations above provide

   T* = 0.833333·To
   p* = 0.528282·po
   ρ* = 0.633938·ρo

The mass flux through the nozzle is of interest and is given by:

     ${\displaystyle {\dot {m}}=\rho AV={\frac {p}{RT}}\cdot A\cdot M{\sqrt {kRT}}=p{\sqrt {k \over RT}}AM}$

With the use of Eq. (9.28), the mass flux, after applying some algebra, can be Expressed as

                 ${\displaystyle {\dot {m}}=p_{o}MA{\sqrt {\frac {k}{RT_{o}}}}\left(1+{\frac {k-1}{2}}M^{2}\right)^{\frac {k+1}{-2(k-1)}}}$

If the critical area is selected where M =1, this takes the form

                       ${\displaystyle {\dot {m}}=p_{o}A^{*}{\sqrt {\frac {k}{RT_{o}}}}\left(1+{\frac {k-1}{2}}\right)^{\frac {k+1}{-2(k-1)}}}$ 

which, when combined with previous it provides:

${\displaystyle {\frac {A}{A^{*}}}={\frac {1}{M}}\left({\frac {2+(k-1)\cdot M^{2}}{k+1}}\right)^{\frac {k+1}{2(k-1)}}}$

Fig: A Converging Nozzle

Consider a converging nozzle connecting a reservoir with a receiver. If the reservoir pressure is held constant and the receiver pressure reduced, the Mach number at the exit of the nozzle will increase until Me=1 is reached, indicated by the left curve in figure 2. After Me =1 is reached at the nozzle exit for ${\displaystyle p_{r}=0.5283p_{0}}$, the condition of choked flow occurs and the velocity throughout the nozzle cannot change with further decreases in${\displaystyle p_{r}}$. This is due to the fact that pressure changes downstream of the exit cannot travel upstream to cause changes in the flow conditions. The right curve of Fig2. represents the case when the reservoir pressure is increased and the receiver pressure is held constant. When ${\displaystyle M_{e}=1}$, the condition of choked flow also occurs; but Eq indicates that the mass flux will continue to increase as ${\displaystyle p_{0}}$ is increased. This is the case when a gas line ruptures.

Fig2: The pressure variation in the nozzle.

It is interesting that the exit pressure ${\displaystyle p_{e}}$ is able to be greater than the receiver pressure ${\displaystyle p_{r}}$. Nature allows this by providing the streamlines of a gas the ability to make a sudden change of direction at the exit and expand to a much greater area resulting in a reduction of the pressure from ${\displaystyle p_{e}}$ to ${\displaystyle p_{r}}$. The case of a converging-diverging nozzle allows a supersonic flow to occur, providing the receiver pressure is sufficiently low. This is shown in Fig. 9.6 assuming a constant reservoir pressure with a decreasing receiver pressure. If the receiver pressure is equal to the reservoir pressure, no flow occurs, represented by curve ${\displaystyle A}$. If pr is slightly less than p_0, the flow is subsonic throughout, with a minimum pressure at the throat, represented by curve B. As the pressure is reduced still further, a pressure is reached that result in M =1 at the throat with subsonic flow throughout the remainder of the nozzle. There is another receiver pressure substantially below that of curve C that also results in isentropic flow throughout the nozzle, represented by curve D; after the throat the flow is supersonic. Pressures in the receiver in between those of curve C and curve D result in non-isentropic flow (a shock wave occurs in the flow) and will be considered in the next section. If pr is below that of curve D, the exit pressure pe is greater than pr. Once again, for receiver pressures below that of curve C, the mass flux remains constant since the conditions at the throat remain unchanged. It may appear that the supersonic flow will tend to separate from the nozzle, but just the opposite is true. A supersonic flow can turn very sharp angles, since nature provides expansion fans that do not exist in subsonic flows. To avoid separation in subsonic nozzles, the expansion angle should not exceed 10°. For larger angles, vanes are used so that the angle between the vanes does not exceed 10°.

Figure : A converging-diverging nozzle with reservoir pressure fixed.