abj1 1.system, surroundings, and their interaction 2.classification of systems: identified volume,...
TRANSCRIPT
abj 1
1. System, Surroundings, and Their Interaction
2. Classification of Systems:
Identified Volume, Identified Mass, and Isolated System
Questions of Interest:
Given the identified volume (IV) of interest and the relevant fields
Q1: : How much is N contained in a moving/deforming volume ?
Q2: : How much is the time rate of change of N in a moving/deforming volume
?
Q3: : Convection Flux of N Through A Surface S: At what rate is N being
transported/convected through a moving/deforming surface ?
Q4: RTT: What is the relation between the time rates of
change of N in the coincident MV and CV?
Lecture 5.0: Convection Flux and Reynolds Transport Theorem
?)( tNV )(tV
?)(
dt
tdNV )(tV
)(tS
),(),,(),,( txVtxtx
?NF
?)()(
dt
tdNf
dt
tdN CVMV
abj 2
Motivation for The Reynolds Transport Theorem (RTT)1. Physical laws (in the form we are familiar with) are applied to an identified mass (MV).
They can be written in generic form in terms of the time rate of change of property N of an MV as
2. However, in fluid flow applications, we are often interested in what happens in a region in space,
i.e., in an identified volume or CV. Hence, we want to know the time rate of change of property N
of a CV
Thus, in order to apply the physical laws from the point of view of a CV instead, we need to find
the relation
Time
N
dt
tdNS MV
N
MV(t) of N ofchange of rate TimeMV(t)in N of
change of Source
)(
Time
Momentum
dt
tPdF
Time
Mass
dt
tdM
MV
MV
)(
)(0
dt
tdNMV )(
dt
tdNCV )(
???)()(
dt
tdNf
dt
tdN CVMV
abj 3
N in any moving/deforming volume V(t)
Time rate of change of N of V(t)
Convection flux of N through a surface A
RTT
Very Brief Summary of Important Points and Equations [1]
][)()()(
NdVtNtV
V
Time
NdV
dt
d
dt
tdN
tV
V
)(
)()(
Time
NAdVF
A
Nd
md
dQ
sfN
)( /
)(or)()(,)()(:
,)()()(
)(
)(through N ofeffluxconvectionNet
)(
/
)( of N of change of rate Time
)( of N of change of rate Time
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
tV
V
tCS
tCS
sf
tCV
CV
tMV
MV
abj 4
(Physical) System
Universe / Isolated SystemSurroundings
Interaction Mechanical interaction (force)
Thermal interaction (energy and energy transfer)
Electrical, Chemical, etc.
Fundamental Concept: System-Surroundings-Interaction
The very first task in any one problem:
Identify the system
Identify the surroundings
Identify the interactions between the system and its surroundings, e.g.,
Mechanics - Force (identify all the forces on the system by its surroundings)
Thermodynamics - Energy and Energy Transfer (identify all forms of
energy and energy transfer between the system and its
surroundings)
abj 5
Classification of Systems: Identified Volume, Identified Mass, and Isolated System
Identified Volume/Region (IV / IR)
[Control Volume (CV), Open system]
• An identified region/volume of interest.
• There can be exchange of mass and energy with its surroundings.
IV
IM
IS
Identified Mass (IM)
[Material Volume (MV), Control Mass (CM), Closed system]
• A special case of an IV.
• It is an IV that always contains the same identified mass.
• Thus, there can be exchange of energy with its surroundings, but
not mass.
Isolated System (IS)
• A special case of an MV, hence of an IV.
• It is an MV that does not exchange energy with its surroundings.
• In other words, it is an IV that does not exchange both mass and energy
with its surroundings.
abj 6
Given a surface S of interest and the relevant fields
Q3: : Convection Flux of N Through A Surface S:
At what rate is N being transported/convected through a moving/deforming
surface ?
Time
NAdVF
A
Nd
md
dQ
sfN
)( /
Convection Flux of N Through S
?NF
S
),(),,(),,( / txVtxtx sf
abj 7
Convection Flux of N Through A Surface S
MOTIVATION for The Expression and Quantification of Flux / Flowrate
• What is the volume flowrate of water through the cross section S of a pipe? [Volume /
Time]
• What is the mass flowrate of water through the cross section S of a pipe? [Mass / Time]
• What is the time rate of thermal energy being transported/convected with (the mass of)
water through the cross section S of a pipe? [Energy / Time]
• What is the time rate of any property N being transported/convected with the mass flow
through a surface S? [ N / Time]
SS
Hot water
Alaska pipeline
From http://www.hickerphoto.com/alaska-oil-pipeline-6765-pictures.htm
abj 8
Nomenclature
AdThe flow of mass through the moving
surface element over a period of dt
• Local fluid/mass velocity relative to a reference frame (RF) fV
A
Ad
Surface element Ad
sfV /
),(,, / txfV sf
Local value of the fields
sV
• Local surface velocity relative to RF
• Local relative velocity of fluid wrt surface sfsf VVV
/
• Extensive property NN
Mass
N
m
N:• Intensive property of N
abj 9
AAdThe flow of mass through the moving
surface element over a period of dt
Ad
Surface element Ad
dtVdl sf /
• Distance of fluid travelling over =dl dt
• Volume flowrate )( /
/)(
AdVdQ sf
dtVold
Time
Volume
• Volume outflow
dtAdV
dtAdV
AddlVold
sf
sf
)(
cos
cos)(
/
/
Volume
• Mass flowrate
dQ
sf AdVdQmd )( /
Time
Mass
md
sf AdVdQmdNd )()( /
Time
N• N flowrate
dl
dl cosVolume element )(Vold
sfV /
),(,, / txfV sf
Local value of the fields
abj 10
Q3: Convection Flux of N Through S Net Convection Efflux of N Through S
A
Ad
Surface element Ad
sfV /
Inside
Outside
Time
NAdVF
A
Nd
md
dQ
sfN
)( /
Convection Flux of N Through S
Open surface
Ad
Surface element Ad
sfV /
Closed surface
S
sfN AdVF )( /
Nothing but sum all over the closed surface.
Net Convection Efflux of N Through S
abj 11
Volume, Mass, and N Convection Flux/Flowrate Through S
A
Ad
Surface element Ad
sfV /
Inside
Outside
Mass Flowrate
Time
NAdVF
A
sfN )( /
Volume Flowrate
N Flowrate
Time
MassAdVm
A
sf
/
Time
VolumeAdVQ
A
sf
/
abj 12
Sign (+ / -) of Volume/Mass Flowrate
A
Ad
Surface element Ad
sfV /
Inside
Outside/20 Volume/Mass outflow is positive:
0
0
/
/
AdVdQmd
AdVdQ
sf
sf
/2Volume/Mass inflow is negative:
0
0
/
/
AdVdQmd
AdVdQ
sf
sf
A
Ad
Surface element Ad
sfV /
Inside
Outside
NOTE: The sign of N-flowrate depends also on the sign of .
If is a vector component, it can be positive or negative.
md
sf AdVNd )( /
abj 13
Net Convection Efflux Through A Closed Surface S
Closed surface S
Ad
sfV /
Flow
Mass Flowrate
Volume Flowrate
Time
MassAdVm
A
sf
/
Time
VolumeAdVQ
S
sf
/
0, mQ If there is a net rate of outflow,
If there is a net rate of inflow, 0, mQ
abj 14
Special Case: Uniform Properties Over The Surface
A
Ad
Surface element Ad
sfV /
Inside
Outside
AVAdVAdVQ sf
A
sf
A
sf
///
If is uniform over A:sfV /
QAVAdVAdVm sf
A
sf
A
sf
///
If are uniform over A:sfV /,
QmAVAdVAdVF sf
A
sf
A
sfN )()( ///
If are uniform over A: ,, / sfV
AAdAA
surface theof vector areaNet :
abj 15
A
Ad
Surface element Ad
sfV /
Inside
Outside
A
sf
A
sf
A
sf AdVQQAdVAdVm
/// :where,
If is uniform - but is not - over A: sfV /
A
sf
A
sf
A
sfN AdVmmAdVAdVF
/// :where,)(
If is uniform – but are not - over A: sfV /,
abj 16
Example: Evaluate the flux by using the elemental area element
Problem:
The velocity field is given by
Find the volume flowrate Q through the cross sectional surface S.
If the density of fluid is , find the mass flowrate through the same surface S.
The area-averaged velocity is defined by , find over the same surface S.A
QV :
ia
yUwutxV c
ˆ1),v,(),(2
V
m
idydzAd ˆ)(
x
y
z
wy = + a
y = - a
FlowAd
S
abj 17
Given the identified volume (IV) of interest and the relevant fields
Q1: : How much is N contained in a moving/deforming
volume ?
Q2: : How much is the time rate of change of N in a
moving/deforming volume ?)(tV
?)( tNV
)(tV
?)(
dt
tdNV
),(),,( txtx
)(tV
abj 18
The Total Amount of Property N in A Volume V(t) at time t:
Consider an infinitesimal volume dV at any time t:
• An infinitesimal volume dV [Volume]
• Mass in an infinitesimal volume dV = dm = dV [Mass]
• N contained in an infinitesimal volume dV = dN = dm = dV [N]
• N contained in a finite volume V at time t is then the sum of all dN’s corresponding to all dV’s in V
• V (t) can be any volume, material or control, depending upon the choice of the domain of integration.
• Since NV(t) depends upon , , and the domain V (t),
• After the volume integration (with domain variable with time t), is a function of t alone, .
in the same field, if the MV(t) and CV(t) coincide,
V(t), S (t)
x
y
z
dV,dm = dV,
dN = dm= dV
Evaluated at Fixed Time t
)()()( tNtNtN CVMVV
Q1: Property N in A Volume V(t) for A Given Field
dm
)(dV
)(overSum
)(
)(
tV
tV
V tN
dN
Dimension [N]
VN )(tNV
),( tx
abj 19
Time
NdV
dt
d
dt
tdN
tV
dN
dm
V
)(
)()(
Q2: Time Rate of Change of
After the function is found,
the time rate of change of N within the volume V(t) as we follow the volume can be found from
the time derivative
)(tNV
t = t t = t +t
V(t), S(t) V(t+t), S(t+t)
)(tV
dN
dm
dV )()(tNV
)(tNV
][N
abj 20
Example: Evaluation of Property N in A Volume V(t)
Intensive Extensive Mass Integral Volume Integral Time Rate of Change in V(t)
Property N
Mass 1
Linear
Momentum
Angular
Momentum
Energy e
Entropy s
M
N
V
Vr
N
M
VMP
VMrH
MeE
MsS
ssystem mas
ηdm
ssystem mas
dm
ssystem mas
dmV
ssystem mas
dmVr )(
ssystem mas
edm
ssystem mas
sdm
)(tV
dV
)(tV
dV
)(tV
dVV
)(
)(tV
dVVr
)(tV
dVe
)(tV
dVs
)(tV
dVdt
d
)(tV
dVdt
d
)(tV
dVVdt
d
)(
)(tV
dVVrdt
d
)(tV
dVedt
d
)(tV
dVsdt
d
abj 21
Q4: Reynolds Transport Theorem (RTT):
What is the relation between the time rates of change of N in the
coincident MV and CV?
)( through ofEfflux ConvectionNet
)(
/
)(in in Increase
)(
)(
)(in in Increase
)(
)(
)(
)( through ofEfflux ConvectionNet
)(
/
V(t) of in Increase)( of in Increase
)()()(
)(or)()(,)()(:
,)()()(
tCSN
tCS
sf
tCVN
tN
tCV
tMVN
tN
tMV
tV dm
V
tCSN
tCS
Nd
md
dQ
sf
CN
CV
tMVN
MV
AdVdVdt
ddV
dt
d
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
CVMV
abj 22
Motivation for The Reynolds Transport Theorem (RTT)1. Physical laws (in the form we are familiar with) are applied to an identified mass (MV).
They can be written in generic form in terms of the time rate of change of property N of an MV as
2. However, in fluid flow applications, we are often interested in what happens in a region in space,
i.e., in an identified volume or CV. Hence, we want to know the time rate of change of property N
of a CV
Thus, in order to apply the physical laws from the point of view of a CV instead, we need to find
the relation
Time
N
dt
tdNS MV
N
MV(t) of N ofchange of rate TimeMV(t)in N of
change of Source
)(
Time
Momentum
dt
tPdF
Time
Mass
dt
tdM
MV
MV
)(
)(0
dt
tdNMV )(
dt
tdNCV )(
???)()(
dt
tdNf
dt
tdN CVMV
abj 23
The Reynolds Transport Theorem (RTT) Problem Formulation and Notation
t = t
MV(t), MS(t)CV(t), CS(t),
Coincident MV and CV at time t
t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
III III
Due to the motion/deformation of both volumes,
MV and CV at a later time t+dt.
• MV is a moving/deforming material volume, MV (t).
• CV is a moving/deforming identified/control volume, CV (t).
At an instant t :
Coincident MV and CV : At any time t, we can identify the coincident MV and CV.
At a later instant t+dt :
Region III: Part of the identified and interest MV is moving out of the identified CV .
Region I: Part of a new MV – which is not the one of interest at present - is moving
into the identified CV.
III – Identified MV moving out.
I – New MV moving in.
abj 24
The Reynolds Transport Theorem (RTT) Problem Formulation and Notation
t = t
MV(t), MS(t)CV(t), CS(t),
Coincident MV and CV at time t
t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
III III
MV and CV at a later time t+dt.
III – Identified MV moving out.
I – New MV moving in.
dt
tNdttNdttNdt
tNdttN
dt
tdNdt
tNdttNdttNdt
tNdttN
dt
tdN
CVIII
CVCVCV
MVIIIII
MVMVMV
dt
tCVdN
dt
tMVdN
)()]()([:
)()()(:
)()]()([:
)()()(:
)()(
Obviously
???)()(
dt
tdNf
dt
tdN CVMVQ3:
abj 25
The Reynolds Transport Theorem (RTT) Derivation
t = t
MV(t), MS(t)CV(t), CS(t), t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
I II III
III – Identified MV moving out.
I – New MV moving in.
)(
/
//
/
/
)()(
)()()(
)()(
identifiedthenotbutnewaofofInflow)()(:
identifiedtheofofOutflow)()(:
;)()(
)]()([)]()([
)()(,)()(
)()()()()()(
tCS
sfCVMV
A
sf
A
sf
A
sfI
A
sfIII
IIII
dttN
III
dttN
IIIII
CVMVCVMV
CVCVMVMVCVMV
AdVdt
tdN
dt
tdN
AdVAdV
MVNAdVdtdttN
MVNAdVdtdttN
dt
dttNdttNdt
dttNdttNdttNdttN
tNtNdt
dttNdttNdt
tNdttN
dt
tNdttN
dt
tdN
dt
tdN
inflowoutflow
inflow
outflow
CVMV
For simplicity, we evaluate the difference
abj 26
The Reynolds Transport Theorem (RTT)
t = t
MV(t), MS(t)CV(t), CS(t), t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
I II III
III – Identified MV moving out.
I – New MV moving in.
)(or)()(,)()(:
,)()()(
)()()(
)(
)(through N ofeffluxconvectionNet
)(
/
)( of N of change of rate Time
)( of N of change of rate Time
)(
/
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
AdVdt
tdN
dt
tdN
tV
V
tCS
tCS
sf
tCV
CV
tMV
MV
tCS
sfCVMV
Reynolds Transport Theorem (RTT)
Unsteady/Temporal Term Net Convection Efflux Term
abj 27
Note on RTT
t = t
MV(t), MS(t)CV(t), CS(t), t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
I II III
III – Identified MV moving out.
I – New MV moving in.
1. Instantaneously coincide MV(t) and CV(t). [Coincident MV(t) and CV(t)]
2. In the form given in the previous slide, it is applicable to moving/deforming CV(t). [CV is a function
of time; hence, CV(t).]
3. As demonstrated in the RTT and the diagram (Region I, II, and III),
differ by the amount of the net convection efflux of N through
CS(t).
4. is the local relative velocity of fluid wrt the moving CS(t).
sfsf VVV
/
dt
tdN
dt
tdN CVMV )(and
)(
abj 28
Interpretation of RTT
t = t
MV(t), MS(t)CV(t), CS(t), t = t + dt
MV(t+ dt), MS(t+d t)CV (t+d t), CS (t+d t)
I II III
III – Identified MV moving out.
I – New MV moving in.
Reynolds Transport Theorem (RTT)
)(or)()(,)()(:
,)()()(
)(
)(through N ofeffluxconvectionNet
)(
/
)( of N of change of rate Time
)( of N of change of rate Time
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
tV
V
tCS
tCS
sf
tCV
CV
tMV
MV
Increase in MV = Increase in CV + Efflux Through CS
= Increase in CV + [Outflow – Inflow]
(See the diagram and Region I, II, III for better understanding.)
abj 29
In principle, in order to evaluate the unsteady term , we must
• first find the volume integral , then
• later take time derivative .
In other words, the order of differentiation and integration is important.
The Evaluation of The Unsteady Term
Reynolds Transport Theorem (RTT)
)(or)()(,)()(:
,)()()(
)(
)(
/
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
tV
V
tCS
sfCVMV
Unsteady Term
)(
)()(
tCV
CV dVdt
d
dt
tdN
dt
tdNCV )(
)(
)()(tCV
CV dVtN
)(
)()(
tCV
CV dVdt
d
dt
tdN
abj 30
1. When the whole volume integral , i.e, the total amount of NCV, is not a
function of time, regardless of the stationarity of the CV or the steadiness of and .
A container filled with water is moving.
• In this case, even though
• the CV is moving, CV(t),
• the density field as described by the coordinate system fixed to earth is not
steady (at one time, one point has the density of water, the next instant the point has the density of
air),
but since , (total mass in
the container remains constant with respect to time).
)(
)()(
tCV
CV dVdt
d
dt
tdN
)(
)()(tCV
CV dVtN
1] Example of when the unsteady term vanishes
t t + dt
),( tx
)(Constant)( tftM CV
x
y
0)(
dt
tdM CV
abj 31
)(
)()(
tCV
CV dVdt
d
dt
tdN2] Example of when the unsteady term
vanishes
0
)0(
)(
)()(
time)offunction anot i.e., steady, are and both if(0)(
:
time)offunction anot i.e., steady, is if()(
:
time)offunction anot i.e., steady, is if()(
:
time)offunction anot i.e., deforming,-non and stationary is CV if()(
)(:
)()(
steady are and both
deforming-non and stationary is
)(
)(
)(
CV
CV
CV
tCV
CV
CVtCV
tCV
CV
dV
dVt
dVdt
d
dt
tdN
t
tt
tt
dVt
dVdt
d
dVdt
d
dt
tdN
1. CV is stationary and non-deforming
2. and are steady.
abj 32
)(
)()(
tCV
CV dVdt
d
dt
tdN3] Example of the evaluation of the unsteady term
when some fields are uniform over the CV
dt
Vd
dVdt
d
dVdt
d
dt
tdN
dt
Md
dVdt
d
dVdt
d
dt
tdN
VdVdV
dVdV
MdVdV
dVdt
d
dt
tdN
CVCVCV
CV
CVCV
CV
CV
CVCV
CV
CV
CV
CV
CVCVCV
CV
CVCV
CV
CV
CV
CV
CVCV
CV
CV
CV
CV
CV
)(
)(
)()(
)(
)(
)()(
CV)over uniform are and both if()()(:
CV)over uniform is if()()(:
CV)over uniform is if()()(:
)()(
CVover uniform are and
CVover uniform is
1. CV is stationary and non-deforming
2. is uniform over CV.
2. and are uniform over CV.
abj 33
The Evaluations of The Convection Efflux Term 1] Example of the evaluation of the convection flux term when some fields are uniform over the surface A of interest
Reynolds Transport Theorem (RTT)
)(or)()(,)()(:
,)()()(
)(
)(
/
tCVtMVtVdVtN
Time
NAdV
dt
tdN
dt
tdN
tV
V
tCS
sfCVMV
Convection Flux Term ...)()()(
2121
//
...)(
/ A
sf
A
sf
AAtCS
sf AdVAdVAdV
A
sfAAA
A
sfA
A
A
sf AdVmmAdVAdV )(:)signed(.)()( //
overuniformis
/
A
sfAAAA
A
sfAA
A
A
sf AdVQQAdVAdV )(:)signed(.)()( //
overuniformareand
/
1. CV is stationary and non-deforming (A is stationary and non-deforming)
)()( ,/,/
overuniformareand,,
/
/
AVAdVAdV AsfAA
A
AsfAA
AV
A
sf
sf
abj 34
Example 2: Finding The Time Rate of Change of Property N of an MV By The Use of A Coincident CV and The RTT
Problem: Flow Through A Diffuser
An incompressible flow of water (density ) with steady velocity field passes through a conical diffuser
at the volume flowrate Q. Assume that the velocity is axial and uniform at each cross section.
1. Use the RTT and the coincident stationary and non-deforming control volume CV that includes
only the fluid stream in the diffuser (as shown above) to find the time rate of change of
1. Kinetic energy (scalar field)
2. x-linear momentum (component of a vector field)
of the coincident material volume MV(t).
Given that V2 < V1 , is the kinetic energy of the coincident material volume MV(t) increasing or
decreasing?
According to Newton’s second law, should there be any net force in the x direction acting on the
MV(t) , or equivalently CV(t) ?
)(, 111 QVA )(, 222 QVA
2//,2/ 22 VkemNmVKEN
xxx VmNmVPN /,
abj 35
Example 3: Finding The Time Rate of Change of Property N of an MV By The Use of A Coincident CV and The RTT
Problem: Given that the velocity field is steady and the flow is incompressible
1. state whether or not the time rate of change of the linear momenta Px and Py of the material
volume MV(t) that instantaneously coincides with the stationary and non-deforming
control volume CV shown below vanishes;
2. if not, state also
- whether they are positive or negative, and
- whether there should be the corresponding net force (Fx and Fy ) acting on the
MV/CV, and
- whether the corresponding net force is positive or negative.
abj 36
x
y
V1V2 = V1
(a) (yes/no) If not, positive or negative
Net Fx on CV? (yes/no) If yes, Fx positive or negative
(b) (yes/no) If not, positive or negative
Net Fy on CV? (yes/no) If yes, Fy positive or negative
?0, dt
dP xMV
dt
dP xMV ,
?0, dt
dP yMV
dt
dP yMV ,
V1V2 > V1
V1
V2 = V1
V1
V2 = V1
V1
V2 = V1