Abj1 Lecture 6.0 : Finite Control Volume Formulation of Physical Laws and C-Mass 1.Finite Control...
11
abj 1 Lecture 6.0 : Finite Control Volume Formulation of Physical Laws and C-Mass 1. Finite Control Volume Formulation of Physical Laws 2. Conservation of Mass Special Case 1: Incompressible Flows (Steady and Uniform Density Field) Special Case 2: Steady Density Field (e.g., Compressible Flows) 3. Conservation of Volume for Incompressible Liquids An illustration of the wider use of the Reynolds Transport Theorem to convert from the material volume MV viewpoint to control volume CV viewpoint
-
Upload
winifred-lambert -
Category
Documents
-
view
214 -
download
0
Transcript of Abj1 Lecture 6.0 : Finite Control Volume Formulation of Physical Laws and C-Mass 1.Finite Control...
abj 1
Lecture 6.0 : Finite Control Volume Formulation of Physical Laws and C-Mass
1. Finite Control Volume Formulation of Physical Laws
2. Conservation of Mass
Special Case 1: Incompressible Flows (Steady and Uniform Density Field)
Special Case 2: Steady Density Field (e.g., Compressible Flows)
3. Conservation of Volume for Incompressible Liquids
An illustration of the wider use of the Reynolds Transport Theorem to convert from the
material volume MV viewpoint to control volume CV viewpoint
abj 2
Very Brief Summary of Important Points and Equations [1]
)()(
/ )(,)()(
0:1,tV
V
tCS
sfCVMV dVtMAdVdt
tdM
dt
tdMMN
C-Mass:
)()(
/ )(),()()(
0:/1,tV
V
tCS
sfCVMV dVtVAdVdt
tdV
dt
tdVVN
C-Volume: (for incompressible liquids)
)(or)( is)(,)()(:
,)()()(
)(
)( through ofefflux convectionNet
)(
/
V(t) of of change of rate Time
V(t) of of change of rate Time)( of in
Change of Source
tCVtMVtVdVtN
AdVdt
tdN
dt
tdNS
tV
V
tCSN
tCSQdmd
sf
CN
CV
MN
MV
tMVN
N
Physical Laws
RTT
Finite CV formulation of physical laws:
abj 3
Finite Control Volume Formulation of Physical Laws
Physical Laws [1]:
MV(t) of N of
change of rateTime)( of in Change of Source
)(
dt
tdNS MV
tMVN
N
)( through ofEfflux convectionNet
)(
/
)( of N ofchange of rate Time
)( of N ofchange of rate Time
)()()(
tCSN
tCSQdmd
sf
tCV
CV
tMV
MV AdVdt
tdN
dt
tdN
RTT [2]:
sfV /
Ad
MV(t), MS(t)CV(t), CS(t)
Efflux through CSIncrease in CV
Finite CV Formulation of PL, [1] = [2]:
)(or)( is)(,)()(:
,)()()(
)(
)( through ofefflux convectionNet
)(
/
V(t) of of change of rate Time
V(t) of of change of rate Time)( of in
Change of Source
tCVtMVtVdVtN
AdVdt
tdN
dt
tdNS
tV
V
tCSN
tCSQdmd
sf
CN
CV
MN
MV
tMVN
N
Physical Laws
RTT
abj 4
Finite Control Volume Formulation of Physical Laws
sfV /
Ad
MV(t), MS(t)CV(t), CS(t)
Efflux through CSIncrease in CV
)( through ofefflux convectionNet
/
)( of of change of rate Time
)( of of change of rate Time
V of in Change of Source
)()()(
tCSN
CSQdmd
sf
tCVN
CV
tMVN
MV
MN
N AdVdt
tdN
dt
tdNS
)(
/)()(
0:1,tCS
sfCVMV AdVdt
tdM
dt
tdMMN
Mass
Entropy )(
/ )()()(
:,tCS md
sfCVMV AdVsdt
tdS
dt
tdS
T
QsSN
Energy )(
/ )()()(
:,tCS md
sfCVMV AdVedt
tdE
dt
tdEWQeEN
)(
/ )()()(
:,tCS md
sfCVMV AdVVdt
tPd
dt
tPdFVPN
Linear Momentum
)(
/,, )(
)()(:,
tCS md
sfcCVcMV
c AdVVrdt
tHd
dt
tHdTVrHN
Angular Momentum
abj 5
Conservation of Mass (C-Mass)
sfV /
Ad
MV(t), MS(t)CV(t), CS(t)
Efflux through CSIncrease in CV
)( through ofefflux convectionNet
/
)( of of change of rate Time
)( of of change of rate Time
V of in Change of Source
)()()(
tCSN
CSQdmd
sf
tCVN
CV
tMVN
MV
MN
N AdVdt
tdN
dt
tdNS
)(
/)()(
0:1,tCS
sfCVMV AdVdt
tdM
dt
tdMMN
C-Mass
abj 6
Special Case 1: Incompressible Flow (Steady and Uniform Density Field)
1. CV is stationary and non-deforming
2. Incompressible flow (steady and uniform density field)
)(
/)()(
0:1,tCS
sfCVMV AdVdt
tdM
dt
tdMMN
C-Mass
CS
sf
CS
sfMV
AdV
AdVdt
tdM
/
/
0
)(0 Net mass efflux through CS = 0
Net volume efflux through CS = 0
C-Mass
0
time.]offunction not are and [Both:
timeoffunction anot is andconstant,)(
)(
flow ibleIncompress
deformingnonandstationaryis
)(
CVCV
CVCVCVCVCV
CV
CV
CV
CV
tCV
CV
V
VMdt
Vd
dVdt
d
dVdt
d
dVdt
d
dt
tdM
Unsteady Term:
abj 7
Special Case 2: Steady Density Field (e.g., Compressible Flow)
1. CV is stationary and non-deforming
2. Steady density field (need not be uniform)
)(
/)()(
0:1,tCS
sfCVMV AdVdt
tdM
dt
tdMMN
C-Mass
CS
sfMV AdVdt
tdM /
)(0 Net mass efflux through CS = 0
C-Mass
0
)0(
)(
steady
deformingnonandstationaryis
)(
CV
CV
CV
tCV
CV
dV
dVt
dVdt
d
dt
tdM
Unsteady Term:
Note that – unlike the previous incompressible flow case - in this case the density at various parts of (CV and) CS may not be
equal. Hence, we have only the net mass efflux – and not necessarily net volume efflux - vanish.
Examples are in the case of steady (density), compressible flows where the density field is steady but not uniform.
abj 8
Conservation of Volume for Incompressible Liquids
An illustration of The Wider Use of The Reynolds Transport Theorem
to convert from the material volume MV viewpoint to control volume CV viewpoint
abj 9
Conservation of Volume for Incompressible LiquidsIllustration of The Wider Use of The Reynolds Transport Theorem
The RTT and Finite CV Formulation of Physical Law
are not applicable only to main “physical laws.”
They are applicable to any material volume MV so long as we have the relation for
the time rate of change of the property N of the material volume
The Conservation of Volume for incompressible liquids below illustrates this point.
Physical Laws
RTT
)( through ofefflux convectionNet
)(
/
V(t) of of change of rate Time
V(t) of of change of rate Time)( of in
Change of Source
)()()(
tCSN
tCSQdmd
sf
CN
CV
MN
MV
tMVN
N AdVdt
tdN
dt
tdNS
V(t) of of
change of rate Time)( of in Change of Source
)(
MN
MV
tMVN
N dt
tdNS
abj 10
Conservation of Volume (for incompressible liquids)
For incompressible liquids (not incompressible flow), its volume as we follow the
material volume does not change with time. Thus, we have
Physical Laws:
RTT:
Thus, we can formulate the conservation of volume for incompressible liquids as
1,,
)(0 VN
dt
tdVMV
)()(
/
)()(
/
)(),()()(
)()(,)()()(
tV
V
tCS
sfCVMV
tV
V
tCS
sfCVMV
dVtVAdVdt
tdV
dt
tdV
dVtNAdVdt
tdN
dt
tdN
)()(
/ )(),()()(
0tV
V
tCS
sfCVMV dVtVAdVdt
tdV
dt
tdV C-Volume:
(for incompressible liquids)
abj 11
Special Case :Incompressible Liquids + Stationary and Non-Deforming CV
1. Stationary and non-deforming CV
2. Incompressible liquids
)()(
/ )(),()()(
0tV
V
tCS
sfCVMV dVtVAdVdt
tdV
dt
tdV C-Volume:
(for incompressible liquids)
)necessary. is assumption r)or whateve (of steady"" no that Note
. with timechangenot does CV theof Volume(0
)(
deformingnonandstationaryis
)(
dt
dV
dVdt
d
dVdt
d
dt
tdV
CV
CV
CV
tCV
CVUnsteady Term:
CS
sfMV AdVdt
tdV)(
)(0 /
C-Volume:
(for incompressible liquids)
Net volume efflux through CS = 0
Note that neither steady nor uniform density field assumption is necessary so long as
1. CV is stationary and non-deforming
2. It is an incompressible liquids. That is, if we follow any one material volume (even though the density field inside the coincident CV may not be steady or uniform), its volume does not change.
