# Unit - I BASIC CONCEPTS AND ISENTROPIC FLOW IN VARIABLE AREA DUCTS

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09ME66 GAS DYNAMICS AND SPACE PROPULSION

ME 6604 - GAS DYNAMICS AND JET PROPULSION

UNIT I

BASIC CONCEPTS AND FUNDAMENTALS OF COMPRESSIBLE FLOW

T.SURESHASSISTANT PROFESSOR DEPT OF MECHANICAL ENGGKAMARAJ COLLEGE OF ENGINEERING

PART - AFUNDAMENTALS OF COMPRESSIBLE FLOW

Energy and momentum equations for compressible fluid flows, various regions of flows, reference velocities, stagnation state, velocity of sound, critical states, Mach number, critical Mach number, types of waves, Mach cone, Mach angle, effect of Mach number on compressibility.

PART B

Flow through variable area duct Isentropic flow through variable area ducts, T-s and h-s diagrams for nozzle and diffuser flows, area ratio as a function of Mach number, mass flow rate through nozzles and diffusers, effect of friction in flow through nozzles

FLOW THROUGH VARIABLE AREA DUCTS

FLOW THROUGH VARIABLE AREA DUCTSAs a gas is forced through a tube, the gas molecules are deflected by the walls of the tube. If the speed of the gas is much less than the speed of sound of the gas, thedensityof the gas remains constant and the velocity of the flow increases. However, as the speed of the flow approaches thespeed of soundwe must considercompressibility effectson the gas. The density of the gas varies from one location to the next. Considering flow through a tube, as shown in the figure, if the flow is very gradually compressed (area decreases) and then gradually expanded (area increases), the flow conditions return to their original values. We say that such a process isreversible. From a consideration of thesecond lawof thermodynamics, a reversible flow maintains a constant value ofentropy. Engineers call this type of flow anisentropicflow; a combination of the Greek word "iso" (same) and entropy.

FLOW THROUGH VARIABLE AREA DUCTS

FLOW THROUGH VARIABLE AREA DUCTSTheconservation of massis a fundamental concept of physics. Within some problem domain, the amount of mass remains constant; mass is neither created or destroyed. Themassof any object is simply the volume that the object occupies times thedensityof the object. For afluid(a liquid or agas) the density, volume, and shape of the object can all change within the domain with time and mass can move through the domain.Theconservation of mass(continuity) tells us that the mass flow ratemdotthrough a tube is a constant and equal to the product of the densityr, velocityV, and flow areaA:

Conservation of mass

Conservation of massSolid MechanicsTheconservation of massis a fundamental concept of physics along with theconservation of energyand theconservation of momentum. Within some problem domain, the amount of mass remains constant--mass is neither created nor destroyed. This seems quite obvious, as long as we are not talking about black holes or very exotic physics problems. The mass of any object can be determined by multiplying thevolumeof the object by thedensityof the object. When we move a solid object, as shown at the top of the slide, the object retains its shape, density, and volume. The mass of the object, therefore, remains a constant between state "a" and state "b."Fluid StaticsIn the center of the figure, we consider an amount of astatic fluid,liquid or gas. If we change the fluid from some state "a" to another state "b" and allow it to come to rest, we find that, unlike a solid, a fluid may change its shape. The amount of fluid, however, remains the same. We can calculate the amount of fluid by multiplying thedensitytimes the volume. Since the mass remains constant, the product of the density and volume also remains constant. (If the density remains constant, the volume also remains constant.) The shape can change, but the mass remains the same.Fluid DynamicsFinally, at the bottom of the slide, we consider the changes for a fluid that is moving through our domain. There is no accumulation or depletion of mass, so mass is conserved within the domain. Since the fluid is moving, defining the amount of mass gets a little tricky. Let's consider an amount of fluid that passes through point "a" of our domain in some amount of timet. If the fluid passes through an areaAat velocityV, we can define the volumeVolto be:Vol = A * V * t

Conservation Laws for a Real Fluid

Conservation of Mass Applied to 1 D Steady Flow

Conservation of Mass:Conservation of Mass for Stead Flow:

Integrate from inlet to exit :

One Dimensional Stead Flow

A,V rA+dA,V+dV r+dr

dl

Conservation of Momentum For A Real Fluid Flow

No body forcesOne Dimensional Steady flow

A,V rA+dA,V+dV r+dr

dl

Conservation of Energy Applied to 1 D Steady Flow

Steady flow with negligible Body Forces and no heat transfer is adiabatic real flow

For a real fluid the rate of work transfer is due to viscous stress and pressure. Neglecting the the effect of viscous dissipation.

For a total change from inlet to exit :

Using gauss divergence theorem:One dimensional flow

Summary of Real Fluid Analysis

Further Analysis of Momentum equation

Frictional Flow in A Constant Area Duct

Frictional Flow in A Constant Area Duct

twThe shear stress is defined as and average viscous stress which is always opposite to the direction of flow for the entire length dx.

Divide by rAV2

One dimensional Frictional Flow of A Perfect Gas

Sonic Equation

Differential form of above equation:

Energy equation can be modified as:

1D steady real flow through constant area duct : momentum equation

Differential Equations for Frictional Flow Through Constant Area Duct

Second Law Analysis

Fanno Line

Adiabatic flow in a constant area with friction is termed as Fanno flow.

Isentropic Nozzle and Adiabatic Duct

C Nozzle Discharge Curve

CD Nozzle + Discharge Curve

Nature of Real FlowEntropy of an irreversible adiabatic system should always increase!

MdMdpdTdV1-ve+ve+ve-ve

Compressible Real Flow

Effect of Mach number is negligible.

Pressure drop in Compressible FlowLaminar Flow Turbulent Flows

Moody Chart

Compressible Flow Through Finite Length DuctIntegrate over a length l

is a Mean friction factor over a length l .

Maximum Allowable LengthThe length of the duct required to give a Mach number of 1 with an initial Mach number MiSimilarly

Compressible Frictional Flow through Constant Area Duct

M

Frictional Flow in A Variable Area Duct

A,V rA+dA,V+dV r+dr

Constant Mach number frictional flow

Sonic Point : M=1

54Stagnation Properties

Consider a fluid flowing into a diffuser at a velocity , temperature T, pressure P, and enthalpy h, etc. Here the ordinary properties T, P, h, etc. are called the static properties; that is, they are measured relative to the flow at the flow velocity. The diffuser is sufficiently long and the exit area is sufficiently large that the fluid is brought to rest (zero velocity) at the diffuser exit while no work or heat transfer is done. The resulting state is called the stagnation state.

We apply the first law per unit mass for one entrance, one exit, and neglect the potential energies. Let the inlet state be unsubscripted and the exit or stagnation state have the subscript o.

55Since the exit velocity, work, and heat transfer are zero,

The term ho is called the stagnation enthalpy (some authors call this the total enthalpy). It is the enthalpy the fluid attains when brought to rest adiabatically while no work is done.

If, in addition, the process is also reversible, the process is isentropic, and the inlet and exit entropies are equal.

The stagnation enthalpy and entropy define the stagnation state and the isentropic stagnation pressure, Po. The actual stagnation pressure for irreversible flows will be somewhat less than the isentropic stagnation pressure as shown below.

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