Flow Through Nozzle Homework Help, Assignment Help, Online Tutor, Tutoring

Submit Assignment

* *

Payment Online Assignment Help

Metallurgical Engineering

Homework Help

sfsf

Flow Through Nozzle

Isentropic flow through varying area duct

Let’s consider the varying area duct as shown in Fig. 1. Areas at different stations are mentioned in the same figure. The minimum cross-sectional area of this duct is called as throat if local Mach number of the same cross-section is 1. We can find out the area of throat under this constraint for known inlet or outlet area of the duct. We know that mass flow rate at the throat is,

Where, ρ*, A* and u* are geometric and flow properties at the throat.

For the steady flow, mass flow rate at any cross-section having geometric and flow properties as ρ, A and u will be equal to the mass flow rate of the throat. Hence,

But we know that,

Hence the area relation can be written as,

Hence, If we know Mach number M at any cross section and corresponding area A then we can calculate the are of the throat for the duct. From this expression it is also clear that the Mach number at any cross-section upstream or downstream of the throat is not dependant on the nature of variation of cross-sectional area of the duct in the stream wise direction.

Nozzle flow

Consider the convergent divergent duct shown in Fig. 1. Left end of the duct corresponds to the stagnation or total conditions (T₀, P₀ and ρ₀ ) due to its connection to the reservoir while right end of the duct is open the atmospheric pressure Pe. If initially exit pressure (Pe) is same as the reservoir pressure (P₀) then there will not be any flow through the duct. If we decrease the exit pressure by small amount then flow takes place through the duct.

Here convergent portion acts as nozzle where pressure decreases and Mach number increases in the stream wise direction while divergent portion acts as diffuser which leads to increase in pressure and Mach number along the length of the nozzle. Variation of pressure and Mach number for this condition is shown in Fig. 2 a and Fig. 2 b respectively by tag 1.

Further decrease in pressure at the exit of the duct shifts the pressure and Mach number curves as shown in Fig. 2 tagged by 2. Mass flow continues to increase with decreasing the exit pressure from conditions from 1 to 2. Condition 3 in this figure represents first critical condition or a particular value of exit pressure at which Mach number at the minimum cross-section of the duct becomes 1 or sonic. From Fig. 2 a it is clear that convergent portion continues to act as nozzle while divergent portion acts as diffuser. Pressure at the throat where Mach number has reached 1 attains the reference star value which is equal to 0.528 times the reservoir pressure for isentropic air flow. Further decrease in exit pressure beyond the first critical pressure (corresponding to situation 3), does not change the role of convergent portion as the nozzle. The pressure and Mach number in the convergent portion also remain unchanged with further decrease in exit pressure. Once the sonic state is achieved at the minimum cross section, mass flow rate through the duct attains saturation. Hence duct or the nozzle is said to be chocked for any pressure value lower than the first critical condition. Typical mass flow rate variation for air flow with change in exit pressure is shown in Fig. 3.

Variation of pressure and Mach number for a typical exit pressure just below first critical conditions is shown in Fig. 4 a and b respectively. As discussed earlier, for this situation also pressure decreases and Mach number increases in the convergent portion of the duct. Thus Mach number attains value 1 at the end of convergent section or at the throat. Fluid continues to expand in the initial part of the divergent portion which corresponds to decrease in pressure and increase in Mach number in the supersonic regime in that part of the duct. However, if fluid continues to expand in the rest part of the duct then pressure of the fluid is expected to reach a value at the exit which is much lower than the exit pressure (as shown by isentropic expansion in Fig. 4 a and b). Therefore, a normal shock gets created after initial expansion in the divergent portion to increase the pressure and decrease the Mach number to subsonic value (Fig. 5). Hence rest of the portion of divergent duct acts as diffuser to increase the pressure in the direction of flow to reach the exit pressure value smoothly.

For the further decrease in exit pressure for the same reservoir condition, the portion of divergent part acting as nozzle increases, in turn the normal shock moves towards exit of the duct. For a particular value of exit pressure normal shock stands at the exit of the convergent divergent duct. Decrease in the exit pressure beyond this condition provides oblique shock pattern originating from the edge of the duct to rise pressure in order to attain the exit pressure conditions. Corresponding condition is shown in Fig. 6.

For this exit pressure condition, the flow inside the duct is isentropic. However fluid attains the pressure at the exit of the duct which is lower than the exit pressure, hence it has to pass through the oblique shock and attain the pressure as that of exit pressure by the non-isentropic process. Hence such condition of the duct is called as 'over expanded nozzle' and it is shown in Fig. 6. Decrease in exit pressure beyond this over expanded condition, decreases the strength of oblique shock and hence the amount of pressure rise. Hence at a particular value of exit pressure fluid pressure at the exit of the duct becomes exactly equal to the exit pressure and flow becomes completely isentropic for the duct. In this condition both convergent and divergent portions of the duct act as nozzle to expand the flow smoothly, hence duct is called as convergent divergent nozzle. Expansion of the flow in the convergent divergent nozzle is mentioned as 'isentropic expansion' in Figs 4 a and b. Further decrease in exit pressure beyond the isentropic condition corresponds to more fluid pressure at the exit in comparison with the ambient pressure. Hence expansion fan gets originated from the edge of the nozzle to decrease the pressure smoothly to reach the ambient condition isentropic ally. Nozzle flow for such a situation is termed as 'under expanded nozzle flow'. Corresponding flow pattern is shown in Fig. 7.