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3. 1- Basic
systems &
Msys
Physical
A systems
&¥=O
(conservation
surroundings
M=dd¥
When
:
-
can
8Q
heat
]
of
produce
a
mass
net
moment
M
about
the centre of
mass
of
the
system
causing
rotation :
# (w )
I
-
H={(r×V)8m
Where
Fluid Mechanics
control Volumes
const
-
of
Laws
angular momentum
is
added ,
is
of
work 8W
or
system
done
is
.
on
a
the
system ,
system energy
DE will
also
change accordingly
SQ -8W DE
-
-
Q
w=¥-
-
Entropy change /
&
Volume
All
of
heat addition
absolute temperature
with
¥
ds>_
T:
Mass Rate of Flow
analysis
the
this
in
volume flow
chapter discusses
Q
or
mass
flow
(imaginary )
der
Vdtdtcos A)
-
-
t
volume
t
velocity
of
Integral
=/Vin)dAdt
f¥
:
Q=f(V )dA=SsVndA
-
n
surface
Where Vn
we
the
is
in the
velocity
↳ Vn positive
outflow
↳ Vn negative
inflow
can
also
constant
multiply
P and
m'=pQ=pAV
V:
normal
volume flow
direction
by
Varying
density
p
or
to
get
mass
V:
mi=fspN )dA=fspVndA
-
n
flow
in
in
passing through
a
surface
3. 2- The
3
control
main
fixed
↳
↳
moving
↳
general
-
Bev
-
ex
Outflow
inflow :
devout =Vou+ dtoutcosooutdt
-
3
ways :
any fluid
C. V.
:
we
momentum,
enthalpy
change of
instantaneous
+
outflow
B
in
the
to
system
can
etc
.
)
-
fcgddB-mpvndtout-fcsddB-mpvndh-in-f.es#mdm'
combine
have fixed
1-
be :
inflow
dd-tffeuddB-mpdtftfcsddB-mpvcosfdtout-fgdB-mpvcosfdA.in/
=
,
out
-
out
*
transport
Reynolds
!
fcsddB-mdmi.in
dm°=pVndA
outflow
and inflow
to
a
dd-tfBsys-d-tdf.eu#mPdV)tfcsddB-mp(v.n)dA
If
,
¥ (f ,u¥mpdV )
fc.g.ddB-mPVCOSO-d.lt
:
property (energy
fcsad-nf.pl/cosO-dAin
within
-
-
be
with
us
terms
we
with pressure
Volume
let B
the
force
drag
=V hdAdt
Where
If
with
changing
Control
in
2.
(Bsys)
ship
fcvddt-mdm-f.eu#-zPdV
within
change
Flow
-
change
Leaving
ex
V. ndtdt
1.
3.
nozzle
C. V.
Fixed
cases :
B can
ex
C. V.
Theorem :
consider :
volumes to
Vin d. Aincosoindt
=
=
For
Transport
C. V.
deforming
Arbitrary
devin
Reynolds
Dimensional inlets
and
outlets
,
we
net flow vector
single
can
,
simplify
dd-tfBsys-dd-tf.eu#mdm)tLddB-m(pAV)ou+-&daB-m(PA- Yin
to :
we
can
write
Reynolds Transport
Theorem as :
3. 3
-
For
Conservation of Mass :
of
conservation
mass
:
dd-tfscvpdlltf.se/Vr-n)dA=o
If
only
Scu
one
-
For steady flow
f. Aivi ) out
RAN
We
+
&(PiAiVi/
in
-0
out
inflow
=
Pz Ask =P A. V. tPyAyVy
,
also know :
:{ miout { thin
pAV=mi
For
-
:
§(Piti Vi)in=&(PiAiVi)
outflow
inlets /outlets :
dimensional
§(
debt
-_
multidimensional inlets
and outlets :
mics __fcsp(
Vin )dA
¥ cross section
Incompressible
Q=VA
Flow
volume flow
multidimensional
Qcs
-
Pau
a
given
cross
-
section
.
:
average
velocity
is :
-_¥= Shin)dA
When Vau
Same for
through
Scs (Vin )dA
The eqn for
var
passing
density
is
multiplied by
:
fpdA
Mass flow
:
( pv )au=¥fp(V
-
n
)dAñPavVav
the
cross-sectional
area
we
get
the correct volume
flow
.
3.4- Linear
Momentum
dd-tfmvlsys-EF-ad-tf cvvpdtlt,sup(
fcsvpfvr.tn/dA8Fr--dd-t(fcuUpdV)tf
Vin)dA
One Dimensional
Lets denote
Mcs
-
-
Momentum
with M
momentum
Sse
,
Msa Viki Vni Ai )
-
-
;
For 1- Dimensional
:
Vp tin )dA
flow
-
-
mi ;Vi
:
{F- dd-tffcuvpd.tl/tKmiiVi)ou+-8(mi-Vi)in
-
Energy
if
E-
:
Energy
DE
Jt
-_e=
:
Energy
with heat and
per mass
work :
df-t-dlfd-dd-F-dd-tffcvepdtltfcsepfv.tn/dA
+
Q → heat added to
by system
tw → work done
etot-l.int
e
+
kinetic
+ e
system
potential
te
other
↳ chemical nuclear
,
↳
etot
EV
it
-
'
Three parts
Wp
Work
-
us
by
electrostatic
,
course
=
stresses
Nst Np ÑV
+
pressure :
fcspfv.tn/dA--fcg
done
Wv
,
this
work :
twpresstwvis.co
done
in
gz
of
W=Ñ shaft
Work
+
neglect
-
-
-
by
fest
shear /
-
Viscous stresses :
VDA
Q-wsnaft-ddt-fcqepdvtfcslepi-PJti.li/dA--O
@
steady
state
.
Q-wsnaft-fcsfltY-tgztppfpfri.ir/dA
Specific Volume :
Enthalpy
One
:
W
=
tp
h=utPW
inlet
+
I
outlet with
uniform velocities
Q-wsnafi-mfhz-h.lt?zNi-V.YtgfZz-Z ))
,
:
magnetic
field etc
.
Bernoulli
Velocity
and
Assumptions
→
→
=
.
no
pressure
inverse
an
relationship
:
internal
friction
incompressible
const
→
velocity
→
laminar flow
=
Émv ' tmgh
↳
have
+
PA
or
tpt.ir?tpt.gh.tP.V.---'-zpV-zY+pV-zghztPeVT
incompressible
V-i V- tzpvftpgh.tl?- tzpvitpgh.tP Pt zpv2tpgh.
:
const @
any point
↳ dynamic
pressure
z­
↳ static
pressure
z­
Eta
:
.
Equation of
4. 2- Differential
From Previous
conservation
Chapter :
J-tfcu-pd.tt/esp(ri-V)dA=O
-
l
down the
breaking
first
f-tfcv-pdt-J-tpsd.lt
term :
dt=
dssdydz
It fcv-pdt~ndf-dsrdy.dz
Right
adjustments
f)
U
=p
so
/
+
¥ ¥
;=Ut¥r ¥
Mass
-
-
net
Conservation
U
+
-
←
( Pu) dsrdydz
→
continuity equation
Tyler)t¥(w)=o
djf-tp.PT)=0
¥ Is
/
Flux :
of Mass
a)
+
:
-
a
Rate of
adjustments
plea; =p ¥2s did
,
dog
• +
miss
+
left
:
¥
U
-
¥ss¥
of Mass :