1. Isentropic Flow

2. Examples of Isentropic Flow
Air foil flow
Diffusers
Actually there is no isentropic flow but it is assumption for ideal case of flow
Most flows are real flow

3. Definition of Isentropic Flow
There are Five thermo dynamics process
1- Constant pressure process 𝑃 𝑉 0 =𝑐𝑜𝑛𝑠.
2- Constant Volume Process 𝑃 𝑉 ∞ =𝑐𝑜𝑛𝑠.
3- Constant Temperature Process 𝑃 𝑉 1 =𝑐𝑜𝑛𝑠
4- Constant entropy 𝑃 𝑉 𝛾 =𝑐𝑜𝑛𝑠
5- The general process 𝑃 𝑉 𝑛 =𝑐𝑜𝑛𝑠
Isentropic flow : the flow which the properties of it at any location or any time could be estimated According to the isentropic process equation

4. ISENTROPIC Process Governing Equation
Goal: Relate Thermodynamics to Compressible Flow
Adiabatic Process: No heat is added or removed from system
dq = 0
Note: Temperature can still change because of changing density
Reversible Process: No friction (or other dissipative effects)
Isentropic Process: (1) Adiabatic + (2) Reversible
(1) No heat exchange + (2) no frictional losses
Relevant for compressible flows only
Provides important relationships among thermodynamic variables at two different points along a streamline
g = ratio of specific heats
g = cp/cv
gair=1.4

5. ENERGY EQUATION SUMMARY
Energy can neither be created nor destroyed; can only change physical form
Same idea as 1st law of thermodynamics
Energy equation for frictionless,
adiabatic flow (isentropic)
h = enthalpy = e+p/r = e+RT
h = cpT for an ideal gas
Also energy equation for
frictionless, adiabatic flow
Relates T and V at two different points along a streamline

6. SPEED OF SOUND Sound waves travel through air at a finite speed
Sound speed (information speed) has an important role in aerodynamics
Combine conservation of mass, Euler’s equation and isentropic relations:
Speed of sound, a, in a perfect gas depends only on temperature of gas
Mach number = flow velocity normalizes by speed of sound
If M < 1 flow is subsonic
If M = 1 flow is sonic
If M > flow is supersonic
If M < 0.3 flow may be considered incompressible

7. GOVERNING EQUATIONS STEADY AND INVISCID FLOW
Incompressible flow of fluid along a streamline or in a stream tube of varying area
Most important variables: p and V
T and r are constants throughout flow
continuity
Bernoulli
continuity
Compressible, isentropic (adiabatic and frictionless) flow along a streamline or in a stream tube of varying area
T, p, r, and V are all variables
isentropic
energy
equation of state
at any point

8. Derivation of Isentropic Flow Equations
Isentropic governing equation
𝑇 2 𝑇 1 = 𝑃 2 𝑃 𝛾−1 𝛾 = 𝜌 2 𝜌 1 𝛾−1 1
Speed of sound
𝑎 2 𝑎 = 𝛾𝑅𝑇 2 𝛾𝑅𝑇 1
𝑎= 𝛾𝑅𝑇
𝑎 2 =𝛾𝑅𝑇 2
By combining Eq 1 and Eq 2
𝑎 2 𝑎 = 𝑇 2 𝑇 1 = 𝑃 2 𝑃 𝛾−1 𝛾 = 𝜌 2 𝜌 1 𝛾−1 3

9. Applying energy equation
𝐶 𝑝 𝑇 𝑉 = 𝐶 𝑝 𝑇 𝑉
𝐶 𝑝 𝑇 𝑉 𝐶 𝑝 𝑇 1 = 𝐶 𝑝 𝑇 𝑉 𝐶 𝑝 𝑇 2
𝑇 2 𝑇 1 = 1+ 𝑉 𝐶 𝑝 𝑇 𝑉 𝐶 𝑝 𝑇 2
𝑉 2 𝛾𝑅𝑇 = 𝑉 𝛾𝑅𝑇 2 = 𝑀 2
𝑉 2 2 𝐶 𝑝 𝑇 = 𝑉 2 𝛾𝑅𝑇 𝛾𝑅 2𝐶 𝑝
𝑉 2 2 𝐶 𝑝 𝑇 = 𝑉 2 𝛾𝑅𝑇 𝛾𝑅 2𝐶 𝑝 = 𝛾−1 2 𝑀 2
𝛾𝑅 2𝐶 𝑝 = 𝛾−1 2

10. 𝑇 2 𝑇 1 = 1+ 𝑉 𝐶 𝑝 𝑇 𝑉 𝐶 𝑝 𝑇 2
𝑉 2 2 𝐶 𝑝 𝑇 = 𝑉 2 2𝛾𝑅𝑇 𝛾𝑅 𝐶 𝑝 = 𝛾−1 2 𝑀 2
𝑇 2 𝑇 1 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 2 2

11. Isentropic process Equations
𝑇 2 𝑇 1 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 2 2
𝑃 1 𝑃 2 = 𝜌 1 𝜌 𝛾 = 𝑇 1 𝑇 𝛾 𝛾−1
𝑃 1 𝑃 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾 𝛾−1
𝑃 1 𝑃 2 = 𝜌 1 𝜌 𝛾 = 𝑇 1 𝑇 𝛾 𝛾−1
𝜌 1 𝜌 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾−1

12. Compressible flow Equation Referred to Stagnation Conditions
At stagnation condition the velocity is zero
𝑉 =0
𝑀 =0
ℎ 𝑜 =constant
For Any two point on flow stream
𝜌 1 𝜌 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾−1
𝑇 2 𝑇 1 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 2 2
𝑃 1 𝑃 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾 𝛾−1
𝑃 𝑜 𝑃 1 = 1+ 𝛾−1 2 𝑀 𝛾 𝛾−1
𝜌 𝑜 𝜌 1 = 1+ 𝛾−1 2 𝑀 𝛾−1
𝑇 𝑜 𝑇 1 =1+ 𝛾−1 2 𝑀 1 2
For any point on flow stream and stagnation point

13. Compressible flow Equation Referred to Critical Conditions
At critical conditions the velocity is equal to speed of sound
𝑉 =a
𝑀 ∗ =1
For Any two point on flow stream
𝜌 1 𝜌 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾−1
𝑇 2 𝑇 1 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 2 2
𝑃 1 𝑃 2 = 1+ 𝛾−1 2 𝑀 𝛾−1 2 𝑀 𝛾 𝛾−1
𝑃 𝑜 𝑃 1 = 1+ 𝛾−1 2 𝑀 𝛾 𝛾−1
𝜌 𝑜 𝜌 1 = 1+ 𝛾−1 2 𝑀 𝛾−1
𝑇 𝑜 𝑇 1 =1+ 𝛾−1 2 𝑀 1 2
𝑃 𝑜 𝑃 ∗ = 1+ 𝛾− 𝛾 𝛾−1
𝜌 𝑜 𝜌 ∗ = 1+ 𝛾− 𝛾−1
𝑇 𝑜 𝑇 ∗ =1+ 𝛾−
𝑃 𝑜 𝑃 ∗ = 𝛾 𝛾 𝛾−1
𝑇 𝑜 𝑇 ∗ = 𝛾+1 2
𝜌 𝑜 𝜌 ∗ = 𝛾 𝛾−1
For critical point of flow stream and stagnation point