fluid mechanics and applications mecn 3110
DESCRIPTION
Fluid Mechanics and Applications MECN 3110. Inter American University of Puerto Rico Professor: Dr. Omar E. Meza Castillo. Chapter 3. Integral Relations for a Control Volume. Course Objectives. To define volume flow rate, weight flow rate, and mass flow rate and their units. - PowerPoint PPT PresentationTRANSCRIPT
Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
LectureLecture
33Fluid Mechanics and Applications
MECN 3110
Inter American University of Puerto RicoProfessor: Dr. Omar E. Meza Castillo
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
Integral Relations for a Control Volume
Chapter 3
2
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
To define volume flow rate, weight flow rate, and mass flow rate and their units.
To understand the Reynolds Transport Theorem.
To apply Conservation of Mass Equation Linear Momentum Equation Energy Equation
Frictionless Flow: The Bernoulli Equation
Th
erm
al S
yst
em
s D
esi
gn
U
niv
ers
idad
del Tu
rab
oTh
erm
al S
yst
em
s D
esi
gn
U
niv
ers
idad
del Tu
rab
o
3
Course Objectives
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Introduction
All the laws of mechanics are written for a system, which is defined as an arbitrary quantity of mass of fixed identity. Everything external to this system is denoted by the term surrounding, and the system is separated fro its surrounding by its boundaries.
A control volume is defined as a specific region in the space for study.
System
Control Volume
4
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Volume and Mass Rate of Flow
All the analyses in this chapter involve evaluation of the volume flow Q or mass flow m passing through a surface (imaginary) defined in the flow.
5
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Volume and Mass Rate of Flow
The integral dV /dt is the total volume rate of flow Q through the surface S.
Volume flow can be multiplied by density to obtain the mass flow m. If density varies over the surface, it must be part of the surface integral
If density is constant, it comes out of the integral and a direct proportionality results:
6
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Volume and Mass Rate of Flow
The quantity of fluid flowing in a system per unit time can be expressed by the following three different terms: The volume flow rate is the volume of fluid flowing
past a section per unit time
where A is the area of the section and ν is the average velocity of flow
The weight flow rate is the weight of fluid flowing past a section per unit time
where ɣ is the specific weight
s/ms/m*mAvQ 32
s/Ns/m*m/NQW 33
7
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Volume and Mass Rate of Flow
The mass flow rate is the mass of fluid flowing through a section per unit time
where ρ is the density
s/kgs/m*m/kgQm 33
8
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on The Reynolds Transport Theorem
To convert a system analysis to a control-volume analysis is needed the Reynolds transport theorem.
Arbitrary Fixed Control Volume
Fixed Control Volume
B is any property of the fluid and β is an intensive property
CS
9
Compact form of the Reynolds Transport
Theorem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
10
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
11
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
12
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
13
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Conservation of Mass
For conservation of mass B is m (mass) and β is 1.
If the volume control has only a number of the one-dimensional inlets and outlets, we can write
14
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Conservation of Mass
Other special cases occur. Suppose that flow within the control volume is steady, then
This states that in steady flow the mass floes entering and leaving the control volume must balance exactly. For steady flow
15
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Conservation of Mass
The quantity ρVA is called mass flow m with units of kg/s or slugs/s
In general, the steady-flow mass conservation relation can be written as
16
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Conservation of Mass
Incompressible Flow: The variation of density can be considered negligible.
If the inlets and outlet are one-dimensional, we have
Where Q=VA is called the volume flow passing through the given cross section.
17
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
18
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
19
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
20
Solution
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
21
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
22
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
23
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on The Linear Momentum Equation
For linear momentum equation for a deformable control-volume.
For a fixed control-volume, the relative velocity Vr=V
If the volume control has only a number of the one-dimensional inlets and outlets, we can write
24
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
25
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
26
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Solution
If The components x and z of the linear momentum
equation are:
27
kwiuV
AX
CSCV
x FdAn.Vududt
dF
AZ
CSCV
z FdAn.Vwdwdt
dF
0
0
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Solution
28
AXFAVCosVAVV 211111
AZFAVSinVAV 211110
Writing the previous equations in the scalar form:
Using the conservation of mass V1A1=V2A2 or A1=A2, since V1=V2.
SinVAF
CosVACosVAVAF
AX
AX
211
211
211
211 1
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Solution
29
Replacing the values:
lbSin.
Sinsftft.ftslugs.F
lbCos.sft.slugsCos.
Cossftft.ftslugs.F
AZ
AX
6411
10060941
1641116411
110060941
223
2
223
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
30
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
31
Solution
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
32
Solution
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Energy Equation
As the final basic law, we apply the Reynolds transport theorem to the first law of thermodynamics. The dummy variable B becomes energy E, and the energy per unit mass is β=dE/dm=e.
Positive Q denotes heat added to the system and positive W denotes work done by the system
33
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Energy Equation
The Steady Flow Energy Equation
If
34
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Energy Equation
The Steady Flow Energy Equation
Where hf the friction loss is always positive, the pump always add energy (increase the left-hand side) hpump and the turbine extracts energy from the flow hturbine.
35
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
36
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
37
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
38
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
39
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
40
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
41
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on Frictionless Flow: The Bernoulli Equation
Closely to the steady flow energy equation is a relation between pressure, velocity, and elevation in a frictionless flow, now called the Bernoulli Equation.
For an unsteady frictionless flow
For steady frictionless flow
42
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
43
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
44
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
45
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
46
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
47
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
48
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
49
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
50
Problem
Chapter 3Fluid
Mech
anic
s and A
pplic
ati
ons
Inte
r -
Bayam
on
Due, Wednesday, March ??, 2011
Omar E. Meza Castillo Ph.D.
Homework3 Webpage
51