Show that the Joule coefficient yi is zero. (b) Show that the Joule—Thomson coefficient is zero. e...

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Show that the Joule coefficient yi is zero. (b) Show that the Joule—Thomson coefficient is zero. e specific internal energy of a van der Waals gas is given by u u0 + c„T — —a , uo, a constants. (a) Find an expression for 17. Show that 7) = 0 if a = XXXXXXXXXXFind an expression for the specific enthalpy h as a function of v and T. (c) Show that /J. = RTv Cp (V b) cp. (d) Calculate the isothermal compressibility K for the van der Waals gas. (e) Show that if a = b = 0, K = R1, and µ = 0.5-4 how that (a) (TT), = cp(1 — K) ah (b) (6ahv)T = P VK (c) ( aT) = P- '17) h V (PP — K). 5-5 arefully sketch a Carnot cycle for an ideal gas on (a) a u-v diagram; (b) a u-T diagram; (c) a u-h diagram; (d) a P-T diagram. 5-6 A Cannot engine is operated between two heat reservoirs at temperatCannot(a) If the engine receives 1200 kilo at 400 K i 400 K to 300 K. calories from the reservoi r cycle, how much heat does it reject to the reservoir at 300 K? if_the engine is operated as a refrigerator (i.e., in reverse) and receive at 00 K, how much heat does it del
	
Document Preview: Heat
Physics 306
Homework Assignment 5
(Heat Engines.)
1) Classical and Statistical Thermodynamics, Carter Chapter 5
– 5.3 through 5.5
– 5.14 and 5.15
2) The ?gure shows a diesel cycle approximating the behavior
ofadieselengine. Processabisanadiabaticcompression, pro-
cess bc is an expansion at constant pressure, process cd is an
adiabatic expansion, and processda is cooling at constant vol-
ume. Find the e?ciency of this cycle in terms of the volumes
V , V , V , andV .
a b c d
3)CalvinCli?sNuclearPowerPlant, locatedontheHobbesRiver, generates1GW ofpower.
In this plant liquid sodium circulates between the reactor core and a heat exchanger located
in the superheated steam that drives the turbine. Heat is transfered into the liquid sodium
in the core, and out of the liquid sodium (and into the superheated steam) in the heat
exchanger. The temperature of the superheated steam is 500K. Waste heat is dumped
o
into the river which ?ows by at 25 C. (a) What is the highest e?ciency that the plant
can have? (b) How much waste heat is dumped into the Hobbes River every second? (c)
How much heat must be generated to supply 1GW of Power? (d) Assume that new tough
environmental laws are passed (to preserve the unique wildlife of the river). Because of this
o
the plant is not allowed to heat the river by more than 0.5 C. What is the minimum ?ow
rate that the hobbes river must have (in L/sec)?

Robert · Accepted Answer

5.3) Specific internal energy of a Van der walls gas is given by 
 
v
a
T
C
u
u
v
-
+
=
0
................................................(1)
Now u0 and a are constants.
a) Now as we know joule-coefficient 'η' is given as
u
v
T
÷
ø
ö
ç
è
æ
¶
¶
=
h
Now, 
)
,
(
v
T
u
u
=
Therefore we may write
1
-
=
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
T
v
u
u
v
T
u
v
T
Now, as we know 
v
v
C
T
u
=
÷
ø
ö
ç
è
æ
¶
¶
Therefore, 
T
v
u
v
u
C
v
T
÷
ø
ö
ç
è
æ
¶
¶
-
=
÷
ø
ö
ç
è
æ
¶
¶
=
1
h
From eq. (1) We get,
2
v
a
v
u
T
=
÷
ø
ö
ç
è
æ
¶
¶
So, Joule-coefficient 
u
v
T
÷
ø
ö
ç
è
æ
¶
¶
=
h
=
2
v
C
a
v
-
   (Ans)
Now for a=0, clearly we can see η = 0 
b) Specific enthalpy 
Pv
u
h
+
=
Now 
2
)
(
v
a
b
v
RT
P
-
-
=
So, 
)
(
2
)
(
0
2
0
b
v
RTv
v
a
T
C
u
v
av
b
v
RTv
v
a
T
C
u
h
V
v
-
+
-
+
=
-
-
+
-
+
=
So, h is a function of v and T
c) We know that Joule-Thomson Co-efficient 
h
P
T
÷
ø
ö
ç
è
æ
¶
¶
=
m
Or, we can write as 
ú
û
ù
ê
ë
é
-
÷
ø
ö
ç
è
æ
¶
¶
=
÷
ø
ö
ç
è
æ
¶
¶
=
v
T
v
T
C
P
T
P
P
h
1
m
.......................................(2)
Now, we know that the van der walls eq. is 
(
)
RT
b
v
v
a
P
=
-
÷
ø
ö
ç
è
æ
+
2
....................................................(3)
or, 
2
)
(
v
a
b
v
RT
P
-
-
=
.........................................................(4)
We may write 
T
v
P
v
P
T
P
T
v
÷
ø
ö
ç
è
æ
¶
¶
÷
ø
ö
ç
è
æ
¶
¶
-
=
÷
ø
ö
ç
è
æ
¶
¶
Now from eq. 3 we have,
)
(
b
v
R
T
P
v
-
=
÷
ø
ö
ç
è
æ
¶
¶
and 
kv
v
P
T
1
-
=
÷
ø
ö
ç
è
æ
¶
¶
, where k is isothermal compressibility
Putting these values in eq.

Show that the Joule coefficient yi is zero. (b) Show that the Joule—Thomson coefficient is zero. e specific internal energy of a van der Waals gas is given by u u0 + c„T — —a , uo, a constants. (a)...

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