1
Thermodynamics Vm235
L
t
6
1
st
Law analysis for a CV
Chap6
Lecture 6
5/15/2008
37
Both Mass & Energy can cross the boundaries of a
control volume – the mass carries with it energy.
5/15/2008
38
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2
Conservation of Mass:
±
²
³
´
±
²
³
´
inlets
all
from
C.V.
the
into
rate
flow
mass
Total
C.V.
the
in
mass
of
increase
of
Rate
.
.
∑
∑
=
i
V
C
m
dt
dm
F
ext
P T
i
i
v e
i
i
Fl ow
Control
surface
²
³
´
exits
all
from
C.V.
the
of
out
rate
flow
mass
Total
−
e
m
dm
C. V.
dt
P T
e
e
v e
e
e
Fl ow
How to calculate
m
CV
, m
i
, m
e
depends on the problem:
[
]
1
:
.
.
kg
dV
v
dV
m
write
may
we
general
In
CV
CV
V
C
∫
∫
⎟
⎠
⎞
⎜
⎝
⎛
=
=
ρ
5/15/2008
39
(
)
[
]
sec
1
:
2
3
3
3
kg
dA
V
v
dA
V
m
Likewise
m
s
m
m
kg
CS
local
CS
local
m
m
kg
µ
µ ²
µ
µ ³
´
G
G
±
µ²
µ³
´
×
×
⎟
⎠
⎞
⎜
⎝
⎛
×
⎟
⎠
⎞
⎜
⎝
⎛
∫
∫
⎟
⎠
⎞
⎜
⎝
⎛
=
=
ρ
PE
KE
U
E
E
m
elevation
of
virtue
by
usually
velocity
of
virtue
by
usually
e
temperatur
of
virtue
by
usually
+
+
=
=
+
+
G
µ
µ
²
µ
µ
³
´
µ
µ ²
µ
µ ³
´
µ
µ ²
µ
µ ³
´
state)
amic
(thermodun
energy
potential
energy
kinetic
energy
Internal
:
possesses
'
'
mass
A
Conservation of Energy (1
st
Law):
P T
i
i
v e
i
i
boundary
the
at
done
is
work
Thus
P
pressure
inlet
the
against
push
to
has
it
inlet
the
through
enters
fluid
the
as
However
e
Z
g
u
e
mass
unit
per
or
+
+
=
2
2
1
,
'.
'
,
'.
'
energy
it the
with
carries
it
C.S.
he
through t
C.V.
the
into
flows
mass
this
As
:
V
Fl ow
5/15/2008
40
Pv
mass
unit
per
work
flow
or
v
m
P
V
P
dA
P
dA
P
W
Rate
Work
Flow
V
C
the
to
input
energy
total
the
get
to
e
to
added
be
must
work
flow
This
V
C
the
of
exit
entrance
exit
entrance
=
=
=
×
=
=
=
∫
∫
±
±
G
G
±
/
/
.
.
'
'
.
.
V
V
3
(
)
G
¶
·
¸
2
2
1
)
(
)
(
/
.
.
/
,
⎞
⎛
+
+
+
+
+
=
+
=
PE
d
KE
d
dU
e
m
e
m
d
dE
Z
g
h
Pv
u
Pv
e
mass
unit
per
exit
inlet
the
through
V
C
the
leaving
entering
energy
total
Thus
V
Conservation of Energy (1
st
Law):
(
)
G
±
µ
µ ²
µ
µ ³
´
G
±
±
±
2
2
1
2
2
1
.
.
.
.
.
.
.
.
:
,
&
...
:
∑
∑
+
+
⋅
−
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
+
+
⋅
+
−
=
⎟
⎠
⎜
⎝
+
+
=
=
e
e
e
i
i
i
V
C
V
C
V
C
B
B
A
A
V
C
have
we
hence
zero
or
small
are
PE
KE
usually
Z
g
h
m
Z
g
h
m
W
Q
dt
dt
dt
dt
dt
Thus
total
h
V
V
5/15/2008
41
(
)
(
)
µ
µ ²
µ
µ ³
´
±
µ
µ²
µ
µ³
´
±
µ
µ
µ
²
µ
µ
µ
³
´
±
±
C.V.
the
of
out
flow
enthalpy
C.V.
the
into
flow
enthalpy
mass
fixed
or
system
closed
a
for
Law
First
.
.
.
.
.
.
∑
∑
⋅
−
⋅
+
−
=
e
e
total
e
i
i
total
i
V
C
V
C
V
C
h
m
h
m
W
Q
dt
dE
∑
∑
=
=
V
C
V
C
V
C
dm
dt
dE
and
dt
dm
Thus
time
with
changing
stop
conditions
when
state
Steady
±
±
.
.
.
.
0
0
;
0
,
.
:
Steady State:
∑
∑
=
−
=
=
e
i
e
i
dE
m
m
or
m
m
dt
±
±
²
³
´
²
³
´
²
³
´
exits
all
from
C.V.
the
of
out
rate
flow
mass
Total
inlets
all
from
C.V.
the
into
rate
flow
mass
Total
C.V.
the
in
mass
of
increase
of
Rate
.
.
,
5/15/2008
42
(
)
(
)
∑
∑
⋅
+
=
⋅
+
⇒
=
e
e
total
e
V
C
i
i
total
i
V
C
V
C
h
m
W
h
m
Q
dt
and
±
±
±
±
.
.
.
.
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 Fall '07
 Borgnakke
 Energy, Steady State, WC .V

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