Isentropic Flow. Gas Dynamics

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Hillary Nancy Walsh
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1 Isentropic Flow

2 Agenda Introduction Derivation Stagnation properties IF in a converging and converging-diverging nozzle Application

3 Introduction Consider a gas in horizontal sealed cylinder with a piston at one end. he gas expands outwards moving the piston and performing work. he walls of the piston are insulated and no heat transfer takes place. Is this an isentropic process? q he mathematical relationships between pressure, density, and temperature are known as the isentropic flow relations.

4 Introduction Examples of isentropic flows: Jet or rocket nozzles, diffusers Airfoils Isentropic core flow But in reality there is no real flow is entirely isentropic!!

5 Derivation For isentropic flow: And: So: Applying energy eqn to get relation between & : a R a a c c p p ( ) ( ) c c p p R a

6 Derivation cont. But: And: So: a a c R R c p p

7 Derivation cont. o find relation between A & : Using relation - and - Where: A A A A R R A A ( ) ( ) K K K K K K A A K

8 Basic Equations for One-Dimensional Compressible Flow Control olume

9 Basic Equations for One-Dimensional Compressible Flow Continuity omentum

10 Basic Equations for One-Dimensional Compressible Flow First Law of hermodynamics Second Law of hermodynamics

11 Basic Equations for One-Dimensional Compressible Flow Equation of State roperty Relations

12 Isentropic Flow of an Ideal Area ariation Basic Equations for Isentropic Flow

13 Isentropic Flow of an Ideal Area ariation Isentropic Flow

14 Stagnation Conditions otal (Stagnation) conditions : A point (or points) in the flow where. Fluid element adiabatically slow down A flow impinges on a solid object

15 Stagnation Conditions (cont.) From Energy Equation and the first law of thermodynamics otal enthalpy Static enthalpy Kinetic energy (per unit mass) Steady and adiabatic flow h const (h h ) Steady, inviscid, adiabatic flow const Isentropic flow const and const (Slow down adiabatically and reversibly) For a calorically perfect gas, h C or h C

16 Stagnation condition is a condition that would exist if the flow at any point was isentropically brought to or come from rest ( ). Stagnation values: Stagnation roperties

17 Stagnation roperties cont. Examples Stagnation point is point. Stagnation point is inside the chamber.

18 Example: / / / a /a A/A* θ Isentropic Relations in abular Form ( ) * A A a a

19 itot robe easurement for Compressible Flow: Incompressible flow (Bernoulli eqn): Compressible flow: ( ) a

20 Example (Compressible pitot tube) Given: Air at u 75 fts, ercury manometer which reads a change in height of 8 inches. Find: Static pressure of air in psia Assume: Ideal gas behavior for air

21 Analysis: First consider the manometer which is governed by fluid statics. In fluid statics, there is no motion, thus there are no viscous forces or fluid inertia; one thus has a balance between surface and body forces. Consider the linear momentum equation:

23 3

24 Critical condition is a condition that would exist if the flow was isentropically accelerated or decelerated until. * *, *, *, A*, a* Critical Conditions: * * * * a a 3 4 ( ) ( ) * A A AA* < > ach number A/A* 5

25 Homeworks. Calculate the ach number of two aircraft both travelling with an airspeed of 3m/s. One is traveling at sea level (5 C); the other at an altitude of km (-6 C.).4. A perfect gas with is traveling at ach 3with astatic temperature of 5K, a static pressure of ka, and astatic density of.477kg/m 3. Determine the stagnation temperature, pressure, and density values. 3. An aircraft is flying at 8m/s at sea level where the temperature is C, density is.5kg/m3 and pressure is 3.mbar. Assuming R87 J/kgK what ach number is the aircraft flying? Air stagnates near the leading edge. Assuming isentropic compressible flow calculate the stagnation pressure. Assuming incompressible flow, use Bernoulli s equation to calculate the stagnation pressure. What is the error in assuming incompressible flow at this ach number?

Objectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation

Objectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation Objectives Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation Conservation of Mass Conservation of Mass Mass, like energy, is a conserved