fluid mechanics and applications inter - bayamon lecture 3 fluid mechanics and applications mecn...
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Fluid
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s and A
pplic
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ons
Inte
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LectureLecture
33Fluid Mechanics and Applications
MECN 3110
Inter American University of Puerto Rico
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Chapter 3Fluid
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Integral Relations for a Control Volume
Chapter 3
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Chapter 3Fluid
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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
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Course Objectives
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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.
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Chapter 3Fluid
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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:
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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
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Compact form of the Reynolds Transport
Theorem
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Chapter 3Fluid
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Chapter 3Fluid
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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
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Chapter 3Fluid
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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.
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Chapter 3Fluid
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Chapter 3Fluid
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Problem
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Solution
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Chapter 3Fluid
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Problem
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Chapter 3Fluid
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Problem
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Chapter 3Fluid
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Problem
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Chapter 3Fluid
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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
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Chapter 3Fluid
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Chapter 3Fluid
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Problem
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Chapter 3Fluid
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on Solution
If The components x and z of the linear momentum
equation are:
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kwiuV
AX
CSCV
x FdAn.Vududt
dF
AZ
CSCV
z FdAn.Vwdwdt
dF
0
0
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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
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Replacing the values:
lbSin.
Sinsftft.ftslugs.F
lbCos.sft.slugsCos.
Cossftft.ftslugs.F
AZ
AX
6411
10060941
1641116411
110060941
223
2
223
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Chapter 3Fluid
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Problem
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Solution
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Solution
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Chapter 3Fluid
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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
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Chapter 3Fluid
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on Energy Equation
The Steady Flow Energy Equation
If
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Chapter 3Fluid
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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.
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Problem
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Chapter 3Fluid
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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
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Problem
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