One Dimensional Flow of Blissful Fluid -III P M V Subbarao Professor Mechanical Engineering Department I I T Delhi 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

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

Generation of High Pressure from Supersonic velocity

An Ideal Diffuser at Design Conditions p1p1 p 2d p throat p p1p1 p*p* p 2d

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. T 0 is also sometimes referred to at Total Temperature T is sometimes referred to as Static Temperature

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 P M V Subbarao Associate Professor Mechanical Engineering Department I I T Delhi 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:

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, p 0 is 20.4 Mpa, combustor temperature, T 0 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 F vac = 2298 kNt F sl =1600 kNt I sp vac =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 At correct Mach number (for given A/A*) … Define: Expand F(M) is Taylor’s series about some arbitrary Mach number M (j)

Solve for M

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

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

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

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

Plot Flow Properties Along Nozzle Length A/A *

Mach Number

Temperature T 0 = 3300  K T throat =  K

Pressure P 0 = 20.4Mpa P throat = MPa

Nozzle at Off Design Exit Pressure p1p1 p 2d p 2a > p 2d p throat

Nozzle at Off Design Exit Pressure p1p1 p 1 > p 2a > p throat p throat p p1p1 p 2a p*p* P* 2a p 2d