One Dimensional Flow of Blissful Fluid -III B.GOVINDHARAJAN J.PARTHIBAN Always Start with simplest Inventions……..

Differential Form of Momentum Equation One dimensional steady inviscid flow : The relation between pressure and velocity is continuous.

Differential Form of Energy Equation One dimensional steady inviscid Adiabatic flow : The relation between enthalpy and velocity is continuous.

Summary

Subsonic Nozzle Subsonic Diffuser dA < 0 & M <1 So, du > 0 & dp <0 dA > 0 & M <1 So, du < 0 & dp>0

Supersonic Diffuser Supersonic Nozzle dA < 0 & M >1 So, du < 0 & dp >0 dA > 0 & M >1 So, du >0 & dp<0

Generation of High Pressure from Supersonic velocity

An Ideal Diffuser at Design Conditions pthroat p2d p1 p1 p2d p p*

Generation of Supersonic Velocity from Rest

RAMJET Engine

Capacity of A Cross Section : An implicit Model Mass flow rate through any cross section of area A With a condition that sonic velocity occurs at throat !

Stagnation Temperature for the Adiabatic Flow of a Calorically Perfect Gas • Consider an adiabatic flow field with a local gas Temperature T(x), pressure p(x), and a velocity V(x) • Since the Flow is adiabatic

Introduce an obstruction in the inviscid flow field : • This obstruction generates a location y, within this flow field where the gas velocity is reduced to zero. • Since the Flow is adiabatic

Holds anywhere within an adiabatic Flow field

In general for an adiabatic Flow Field the Stagnation Temperature is defined by the relationship Stagnation Temperature is constant throughout an adiabatic flow field. • T0 is also sometimes referred to at Total Temperature • T is sometimes referred to as Static Temperature

• Similarly Stagnation density for isentropic flow field is • Stagnation temperature is a measure of the Kinetic Energy of the flow Field. • Largely responsible for the high Level of heating that occurs on high speed aircraft or reentering space Vehicles … “stagnation” (total) pressure : Constant throughout Isentropic flow field. • Similarly Stagnation density for isentropic flow field is 1

Stagnation Properties of Isentropic Flow 1

What was Stagnation Temperature At Columbia Breakup Loss Of Signal at: 61.2 km altitude ~18.0 Mach Number T∞ ~ 243 K

Ideal gas Variable Properties Real gas

Capacity of A Cross Section Mass flow rate through any cross section of area A With a condition that sonic velocity occurs at throat !

Calorically perfect gas:

Specific Mass flow Rate Mass flow rate per unit area of cross section:

Design of Converging Diverging Nozzles B.GOVINDHARAJAN J.PARTHIBAN From the Beginning to the Peak or Vice Versa….

Quasi-One-Dimensional Flow

Distinction Between True 1-D Flow and Quasi 1-D Flow • In “true” 1-D flow Cross sectional area is strictly constant • In quasi-1-D flow, cross section varies as a Function of the longitudinal coordinate, x • Flow Properties are assumed constant across any cross-section • Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusers Where the cross sectional area is large when compared to length

Specific Mass flow Rate Mass flow rate per unit area of cross section:

Maximum Capacity of A Nozzle • Consider a discontinuity at throat “choked-flow” Nozzle … (I.e. M=1 at Throat) • Then comparing the massflow /unit area at throat to some other station.

Take the ratio of the above:

For a known value of Mach number, it is easy to calculate area ratio. Design Analysis For a known value of Mach number, it is easy to calculate area ratio. Throat area sizing is the first step in the design. If one needs to know the Mach number distribution for a given geometric design! Find the roots of the non-linear equation.

Typical Design Procedure The Space Shuttle Main Engines burn LOX/LH2 for propellants with A ratio of LOX:LH2 =6:1 The Combustor Pressure, p0 is 20.4 Mpa, combustor temperature, T0 is 3300 K. Decide throat diameter based on the requirement of thrust. What propellant mass flow rate is required for choked flow in the Nozzle? Assume no heat transfer through Nozzle no frictional losses. Combustion product is water vapor.

Space Shuttle Main Engines

Specifications of SSME • Specific Impulse is a commonly used measure of performance For Rocket Engines,and for steady state-engine operation is defined As: • At 100% Throttle a SSSME has the Following performance characteristics Fvac = 2298 kNt Fsl = 1600 kNt Ispvac = 450 sec.

SEA Level Performance One needs to know the Mach number distribution for a given geometric design! Find the roots of the non-linear equation.

Numerical Solution for Mach Number Caluculation • Use “Newton’s Method” to extract numerical solution • Define: • At correct Mach number (for given A/A*) … • Expand F(M) is Taylor’s series about some arbitrary Mach number M(j)

• Solve for M

• From Earlier Definition , thus Still exact expression • if M(j) is chosen to be “close” to M And we can truncate after the first order terms with “little” Loss of accuracy

• First Order approximation of solution for M “Hat” indicates that solution is no longer exact • However; one would anticipate that “estimate is closer than original guess”

• If we substitute back into the approximate expression • And we would anticipate that “refined estimate” …. Iteration 1

• Abstracting to a “jth” iteration Iterate until convergence j={0,1,….} • Drop from loop when

Plot Flow Properties Along Nozzle Length • A/A*

• Mach Number

• Temperature T0 = 3300K Tthroat = 2933.3 K

• Pressure P0 = 20.4Mpa Pthroat = 11.32 MPa

Nozzle at Off Design Exit Pressure pthroat p2d p1 p2a > p2d

Nozzle at Off Design Exit Pressure pthroat p1 p1 > p2a > pthroat p1 p2a p P*2a p* p2d