Biomedical Engineering Reference
In-Depth Information
done to the surroundings by the system. Mathematically, the first law of thermodynamics can
be expressed for an open system (this concept of system or control volume comes into use
again) as
d
ð
E
,
n
Þ
¼ Q
W
þ
F
in
E
in
F
out
E
out
(3.121)
d
t
where E is energy per mole of substance,
Q
is the rate of heat or thermal energy acquired by
the system (flow in),
W
is the rate of work done by the system to the surroundings (energy
flow out), and F is the molar flow rate. The total work done to the surroundings consists of
two parts, one is due to the fluid streams flowing in and out of the system, and another is the
energy exchange with the surroundings due to the shaft work imported from or exported to
the surroundings. Thus,
2
4
X
3
2
4
X
3
N
S
N
S
W
5
in
þ
5
out
þ W
s
¼
F
j
Pv
j
F
j
Pv
j
(3.122)
j
¼1
j
¼1
W
s
Here p is pressure, v is molar volume, and
is the shaft work done to the surroundings (for
example, by a turbine). Thus,
d
P
N
S
j
2
4
X
3
2
4
X
3
n
j
E
j
N
S
N
S
¼1
¼ Q
W
s
þ
5
in
5
out
F
j
ð
E
j
þ
pv
j
Þ
F
j
ð
E
j
þ
pv
j
Þ
(3.123)
d
t
j
¼1
j
¼1
Noting that:
u
2
2
þ
E
j
¼
U
j
þ
M
j
M
j
gz
(3.124)
where U is internal energy, u is average flow velocity, M is molecular mass, g is the gravita-
tional acceleration, and z is the vertical distance from a reference point. And the enthalpy is
defined as
H
j
¼
U
j
þ
pv
j
(3.125)
Therefore:
d
nU
gz
þ
2
mu
2
þ
H
j0
þ
M
j
X
N
S
¼ Q
W
s
þ
u
0
þ
1
2
F
j0
gz
0
d
t
(3.126)
j
¼1
F
j
h
H
j
þ
gz
i
M
j
2
P
N
S
u
2
þ
j
¼1
where the subscript 0 denotes for the inlet, whereas no subscript indicating the outlet.
Note:
Q
is the net heat transfer into the system, including through electric conversion and
subtraction of the frictional thermal loss by the fluid mixture. Heat of reaction is not in either
Q
or
W
s
.
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