Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer

Fluid Mechanics and Energy TransportFluid Mechanics and Energy TransportBIEN 301BIEN 301

Lecture 15Lecture 15Compressible Flows Continued – Isentropic FlowCompressible Flows Continued – Isentropic Flow

Juan M. Lopez, E.I.T.Juan M. Lopez, E.I.T.

Research ConsultantResearch ConsultantLeTourneau UniversityLeTourneau University

Adjunct LecturerAdjunct LecturerLouisiana Tech UniversityLouisiana Tech University

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

Area Changes Area Changes (White 9.4)(White 9.4)

Flow can be considered one-dimensional Flow can be considered one-dimensional when:when:• There is a no slip condition at the wallThere is a no slip condition at the wall• dh/dx << 1 and h(x) << R(x)dh/dx << 1 and h(x) << R(x)• V = V(x), reacts to changes in areaV = V(x), reacts to changes in area

The 1D approximation:The 1D approximation:• p(x)V(x)A(x) = dm/dt = constantp(x)V(x)A(x) = dm/dt = constant

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

Differential form of continuity for area-dependent Differential form of continuity for area-dependent 1D flow:1D flow:

0A

dA

V

dVd

This simple relationship helps us predict the behavior This simple relationship helps us predict the behavior of a compressible fluid in a 1D flow situation.of a compressible fluid in a 1D flow situation.

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

dV < 0dV < 0

dp > 0dp > 0

Subsonic Subsonic DiffuserDiffuser

dV > 0dV > 0

dp < 0dp < 0

Subsonic Subsonic NozzleNozzle

dV > 0dV > 0

dp < 0dp < 0

Supersonic Supersonic NozzleNozzle

dV < 0dV < 0

dp > 0dp > 0

Supersonic Supersonic DiffuserDiffuser

dA > 0dA > 0

dA < 0dA < 0

Ma < 1Ma < 1 Ma > 1Ma > 1

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

More relationshipsMore relationships

22

2

1

1

:flows ibleincompressFor

Sound) of (Speed

Equation) (Momentum0

V

dp

MaA

dA

V

dV

dadp

VdVdp

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

Ideal Gas Area ChangeIdeal Gas Area Change• The previous equation helps us to identify a maximum mass The previous equation helps us to identify a maximum mass

flow (choking condition) for compressible, isentropic flows:flow (choking condition) for compressible, isentropic flows:

kk

k

p

p

p

p

k

k

p

RT

A

m

RT

ARTAm

1

0

2

00

0

21

0

021

00max

11

2

:choking)(not function flow mass local aWith

6874.06874.0

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow

Eq’s 9.48a-dEq’s 9.48a-d Provide good relationships for the mach number at Provide good relationships for the mach number at

different area relationships.different area relationships.

We will be skipping the sections on shock We will be skipping the sections on shock waves.waves. Read them for your own edification-the material is Read them for your own edification-the material is

good, the pictures are quite interestinggood, the pictures are quite interesting

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Blood FlowBlood Flow

From these equations to blood flow is From these equations to blood flow is quite a ways. However, I wanted to be quite a ways. However, I wanted to be sure we had the foundations before we sure we had the foundations before we saw a true blood flow solution worked out.saw a true blood flow solution worked out.

Here is a full blood flow solution for Here is a full blood flow solution for pulsatile flow. Poiseuille flow is utilized.pulsatile flow. Poiseuille flow is utilized.

Examples of Comprehensive Flow Solutions

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Isentropic FlowIsentropic Flow Converging and Diverging NozzlesConverging and Diverging Nozzles

A few examplesA few examples• Example 9.8Example 9.8• Example 9.9Example 9.9• P9.63P9.63• P9.65P9.65

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Chapter 9 HomeworkChapter 9 Homework

DUE 2/22/2007DUE 2/22/2007

P9.11P9.11P9.43P9.43P9.78P9.78P9.88P9.88

EC: P9.114EC: P9.114

Due with the Ch. 7 and Ch. 8 HW.Due with the Ch. 7 and Ch. 8 HW.

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

RememberRemember

Study Session tomorrow nightStudy Session tomorrow night Project Questions – Office hours today, Project Questions – Office hours today,

tomorrow, and during the study sessiontomorrow, and during the study session Report format is required on the final Report format is required on the final

projects.projects. Projects due Thursday!Projects due Thursday!

2/6/20072/6/2007 BIEN 301 – Winter 2006-2007BIEN 301 – Winter 2006-2007

Questions?Questions?