Download Joule–Thomson coefficients of quantum ideal

Applied Energy 70 (2001) 49–57
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Joule–Thomson coefficients of quantum
ideal-gases
Hasan Saygın*, Altuğ Şişman
Nuclear Energy Institute, Istanbul Technical University, 80626-Maslak, Istanbul, Turkey
Received 30 January 2001; accepted 3 March 2001
Abstract
The temperature drop of a gas divided by its pressure drop under constant enthalpy conditions is called the Joule–Thomson coefficient (JTC) of the gas. The JTC of an ideal gas is equal
to zero since its enthalpy depends on only temperature. On the other hand, this is only true for
classical ideal gas which obeys the classical ideal gas equation of state, pV= mRT. Under
sufficiently low-temperature or high-pressure conditions, the quantum nature of gas particles
becomes important and an ideal gas behaves like a quantum ideal gas instead of a classical one.
In such a case, enthalpy becomes dependent on both temperature and pressure. Therefore, JTC
of a quantum ideal gas is not equal to zero. In this work, the contribution of purely quantum
nature of gas particles on JTC is examined. JTCs of monatomic Bose and Fermi type quantum
ideal gases are derived. Their variations with temperature are examined for different pressure
values. It is shown that JTC of a Bose gas is always greater than zero. Minimum value of
temperature is limited by the Bose–Einstein condensation phenomena under the constant
enthalpy condition. On the other hand, it is seen that JTC of a Fermi gas is always lower than
zero and there is not any limitation on its temperature. For high temperature values, JTCs of
Bose and Fermi gases go to zero since the quantum nature of gas particles becomes negligible.
Moreover, variation of temperature versus pressure under the constant enthalpy condition is
also examined. Consequently, it is understood that the quantum nature of a Bose-type gas contributes to the positive values of JTC while the quantum nature of a Fermi type gas contributes
to the negative values of JTC. Therefore, a Bose-type gas is more suitable for cryogenic
refrigeration systems. # 2001 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction
Isenthalpic pressure variations in a real gas cause temperature variations. This effect is
well known as the Joule–Thomson effect (JTE) in the literature [1–3] and it is defined by
* Corresponding author. Tel.: +90-212-285-3948; fax: +90-212-285-3884.
E-mail address: [email protected] (H. Saygın).
0306-2619/01/$ - see front matter # 2001 Published by Elsevier Science Ltd. All rights reserved.
PII: S0306-2619(01)00018-6
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H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
Nomenclature
cip
CF
g
H
h
hp
i
kb
Li
m
mo
N
n
p
R
T
T0
x
z
JT
BJT
FJT
JT
Specific heat at constant pressure of i type (Bose or Fermi) gas (J/kgK)
Correction factor for classical ideal gas equation of state
Number of possible spin orientations of the gas particle
Enthalpy (J)
Specific enthalpy (J/kg)
Planck’s constant (Js)
Positive integer indice of the summation for Polylogarithm function
Boltzmann’s constant (J/K)
P
l i
The polylogarithm function defined as Li ðxÞ ¼ 1
l¼1 x =l
Gas mass (kg)
Rest mass of a gas particle (kg)
Total number of gas particles
Number density of gas particles (m3)
Pressure (Pa)
Gas constant (J kg1 K1)
Temperature (K)
Bose–Einstein condensation temperature (K)
Variable of polylogarithm function, Lii(x)
Fugacity defined as z=exp(/kbT)
Chemical potential of a gas particle (J)
Joule–Thomson coefficient (K/Pa)
Joule–Thomson coefficient of a monatomic ideal Bose gas (K/Pa)
Joule–Thomson coefficient of a monatomic ideal Fermi gas (K/Pa).
@T
¼
@p h
ð1aÞ
or alternatively,
JT ¼ 1 @h
cp @p T
ð1bÞ
where JT is the Joule–Thomson coefficient (JTC), T is temperature, p is pressure, h
is specific enthalpy and cp is the specific heat at constant pressure. JTE is widely used
in gas liquefaction systems such as a Linde (or Hampson) system or in cryogenic
cooling systems [1–3]. At sufficiently low temperatures and pressures, JT is positive
for a typical real gas. At sufficiently high temperatures and pressures, however, JT
becomes negative. Therefore, temperature increases with increasing pressure for an
isenthalpic process, reaches a maximum point and then starts to decrease with
H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
51
increasing pressure. The temperature corresponding to this maximum point is called
the ‘‘inversion temperature’’. For a given pressure, the inversion temperature is
determined by the condition JT=0. There is a maximum inversion temperature
(MIT). At temperatures higher than MIT, JT is always negative and there is no
inversion temperature [1–3].
The JTC of an ideal gas, however, is always equal to zero and JTE is not observed
in an ideal gas [1–3]. For a classical ideal gas, this is true since enthalpy depends on
only temperature and constant enthalpy means constant temperature. At sufficiently
low temperatures or high pressures, which appear in cryogenic systems, the quantum
nature of the gas becomes important and an ideal gas should be treated as a quantum ideal gas instead of a classical ideal gas. For a quantum ideal gas, enthalpy
depends on both temperature and pressure. Therefore, the JTC of a quantum ideal
gas is not equal to zero even if it is an ideal gas. In the literature, Joule–Thomson
integral inversion curves of real quantum gases have been examined [4]. However,
JTE for quantum ideal gases has not been studied previously. In this work, JTE in
quantum ideal gases is examined and the contribution of purely quantum nature of
the gas particles on the JTC is investigated. By using the corrected equation of state
for monatomic quantum ideal gases, Joule–Thomson coefficients of Bose and Fermi
type of monatomic quantum ideal gases are derived. Their variations with temperature are examined for different pressures by choosing 4He and 3He as Bose and
Fermi monatomic gases, respectively. It is shown that the JTC of a Bose gas is
always greater than zero and the minimum value of temperature is limited by the
Bose–Einstein condensation phenomena under constant enthalpy conditions. On the
other hand, it is seen that the JTC of a Fermi gas is always lower than zero and there
is not any limitation on its temperature. For high-temperature values, the JTCs of
Bose and Fermi gases go to zero since the quantum nature of gas particles becomes
negligible. Moreover, the variation of temperature versus pressure under constant
enthalpy condition is also examined. Consequently, it is understood that the quantum nature of a Bose-type gas contributes to the positive values of JT, while the
quantum nature of a Fermi-type gas contributes to the negative values of JT.
Therefore, Bose types of gases are more suitable for cryogenic refrigeration systems.
2. Derivation of the JTC for quantum ideal gases
To derive the JTC of a quantum ideal gas, there is a need for an equation of state
for quantum ideal gases. For quantum ideal gases, the equation of state is called the
corrected equation of state and it is given for monatomic gases as follows [5–7]:
p ¼ nkb T CFð=kb TÞ
ð2Þ
where n is the number density of gas particles, kb is Boltzmann’s constant, is the
chemical potential of gas particles and CF(/kbT) is a correction factor and it is
defined for a Fermi gas as
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H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
CFF ð=kb TÞ ¼
Li5=2 ½expð=kb TÞ
Li3=2 ½expð=kb TÞ
ð3Þ
and for a Bose gas as
T > T0 ) CFB ð=kb TÞ ¼
Li5=2 ½expð=kb TÞ
Li3=2 ½expð=kb TÞ
T
T4T0 ) CFB ð=kb TÞ ¼ 0:5135
T0
ð4aÞ
3=2
ð4bÞ
P
l i
where Li(x) is the polylogarithm function defined as Lii ðxÞ ¼ 1
l¼1 x =l [8] and T0 is
the Bose–Einstein condensation temperature. T0 is defined in Refs. [9–12] and it can
easily be rearranged in terms of pressure, p, as [5,7]
T 0 ð pÞ ¼ p
2=5
=kb ð2m0 Þ
3=5
h3p
g 2:612 0:5135
!2=5
ð5Þ
where g is the number of possible spin orientations of a gas particle.
The quantity of /kbT is implicitly calculated from the following equation as a
function of ( p, T),
p ¼ ð2m0 Þ3=2 g=h3p ðkb TÞ5=2 Li5=2 ½
expð=kb TÞ
ð6Þ
where the upper signs are used for the Bose gas and lower signs are used for the
Fermi gas. Consequently, correction factors can be determined for given ( p, T)
parameters by using Eqs. (3)–(6).
Enthalpy of a monatomic i type quantum ideal gas can be expressed by using the
general definition of enthalpy and Eq. (2) as
5
5
Hi ¼ Nkb TCFi ð p; TÞ ¼ mRTCFi ð p; TÞ
2
2
ð7Þ
where N is the total number of gas particles, R is the gas constant and i refers to
Bose or Fermi type gas. Specific heat at constant pressure can be obtained by Eq. (7)
as follows
"
#
@h
5
@CFi ð p; TÞ
i
cp ¼
¼ R CFi ð p; TÞ þ T
ð8Þ
@T p 2
@T
p
Therefore, the JTC of a monatomic i type quantum ideal gas is determined by Eqs.
(1b), (7) and (8) as
H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
iJT
@CFi ð p; TÞ
@CFi ð p; TÞ
T
T
@p
@p
T
¼
¼
@CFi ð p; TÞ
cip =2:5R
CFi ð p; TÞ þ T
@T
p
53
ð9Þ
3. Results and discussion
In all numerical calculations, 4He and 3He gases are chosen as Bose and Fermi
monatomic gases, respectively. In Fig. 1, variations of CFB and CFF with pressure
for different temperature values are shown. It is seen that the slope of CFB is always
negative, while the slope of CFF is always positive. By considering Eq. (9), therefore,
it can be said that BJT is always positive and FJT is always negative since cBp and cFp
have positive values. Maximum value of pressure is restricted by the Bose–Einstein
condensation pressure for a given temperature: p0(T) can easily be calculated from
Eq. (5). At the pressures higher than p0(T), pressure becomes only temperature
dependent. Thus, the pressures higher than p0(T) are not possible for a given constant temperature. Since p0(T)=88 atm for T=10 K, the pressure interval of the
curve of CFB(p, 10 K) in Fig. 1 is 0–88 atm.
Fig. 1. Variations of correction factors with pressure for different temperature values.
54
H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
In Fig. 2, the variations of BJT and FJT with temperature are given for different
pressure values. It is seen that BJT and FJT go to zero at high temperatures. This is
an expected result since the quantum degeneracy becomes unimportant and the gas
behaviour approaches that of an ideal gas at high temperatures. On the other hand,
BJT and |FJT| increase with decreasing temperature. This situation results from the
increment in quantum exchange force between the identical particles. These forces
are attractive for Bose particles and repulsive for Fermi particles [12]. This is the
reason why BJT is positive and FJT is negative. Exchange force is not a real force
derived from a real potential. Its origin is the ‘‘statistical potential’’ which originates
solely from the symmetry properties of the wave functions of gas particles. Thus, it
is sometimes called ‘‘statistical forces’’ [12]. The magnitude of exchange forces
increase with increasing quantum degeneracy. Therefore, BJT and |FJT| increase with
decreasing temperature. For a given temperature, BJT increases with increasing
pressure values while |FJT| decreases for the same situation. To understand the reason for this difference in the effect of pressure on BJT and FJT, analytical expressions
for correction factors in terms of pressure and temperature are needed. These
expressions can be obtained for high temperature (or low pressure) limits. For
sufficiently high temperatures or low pressures, the term of exp(/kbT) is much
less than unity. If exp(/kbT) is designated by z (namely fugacity), Eqs. (3), (4a) and
(6) can be simplified by expanding them into series in the vicinity of z=0 and
considering z << 1. Therefore, Eqs. (3), (4a) and (6) are rewritten as the following
simple forms respectively,
Fig. 2. Variations of BJT and FJT with temperature for different pressure values.
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H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
CFF ffi 1 þ 0:17678z 0:06580z2
ð10Þ
CFB ffi 1 0:17678z 0:06580z2
ð11Þ
p ffi ð2m0 Þ3=2 g=h3p ðkb TÞ5=2 z 0:17678z2
ð12Þ
The quantity z can be expressed by Eq. (12) in terms of ( p, T) as
z¼
h3p
ð2m0 Þ
3=2
p
g ðkb TÞ5=2
ð13Þ
By using Eq. (13), Eqs. (10) and (11) are expressed in terms of (p, T) as follows
respectively,
CFF ffi 1 þ 0:17678
CFB ffi 1 0:17678
h3p
h6p
p
p2
0:06580
5=2
3 2
g ðkb TÞ
ð2m0 Þ g ðkb TÞ5
ð14Þ
h6p
p
p2
0:06580
3=2
5=2
3 2
ð2m0 Þ g ðkb TÞ
ð2m0 Þ g ðkb TÞ5
ð15Þ
ð2m0 Þ
3=2
h3p
Eqs. (14) and (15) are valid only under the condition that z << 1 (p <<(2m0)3/2
g(kbT)5/2/h3p). If Eqs. (14) and (15) are used in Eq. (9), the JTCs of ideal Fermi and
Bose gases for high temperature or low pressure limits are obtained as
"
#
h3p
h6p
1
p
E
0:1316
JT ffi 0:17678
ð16Þ
T3=2
ð2m0 Þ3 g2 k5b T4
ð2m0 Þ3=2 gk5=2
b
"
BJT
h3p
h6p
1
p
ffi 0:17678
þ
0:1316
3 2 5 T4
3=2
T
ð
2m
Þ
g
k
ð2m0 Þ3=2 gk5=2
0
b
b
#
ð17Þ
For a given temperature, by considering the pressure dependences of Eqs. (16) and
(17), it can easily be said that BJT increases with increasing pressure, while |FJT|
decreases for the same situation. This result can be seen in Fig. 2. The first term in
Eq. (16) represents the maximum value of |FJT|, while the first term in Eq. (17)
represents the minimum value of BJT for a given temperature.
Since temperature cannot be decreased below T0(p) for a given pressure, the
minimum value of temperature is restricted by T0(p) for the curves of BJT . It is seen
that BJT goes to its maximum value when T=T0(p). An analytical expression for this
maximum value of BJT can be obtained by using Eqs. (4b) and (5) in Eq. (9) and
then taking its limit
when T goes to T0(p). Therefore, the following analytical
expression for BJT max is obtained:
56
H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
BJT
Max
¼
2 T0 ðpÞ
5 T
Under constant-enthalpy conditions, temperature can be calculated for a given
pressure by numerically solving Eq. (7). In Fig. 3, the variation of gas temperature
versus pressure under constant-enthalpy conditions is given for Bose and Fermi
gases. A decrement in gas pressure leads to temperature increment in a Fermi gas
and temperature decrement in a Bose gas. This result is consistent with the signs of
JTCs of Fermi and Bose gases. This is an expected result since the internal energy
decreases in an ideal Fermi gas and increases in an ideal Bose gas with decreasing
pressure. The reason for this variation of internal energy, depending on pressure, is
the repulsive or attractive exchange forces between the particles. This exchange forces
decrease with decreasing pressure. When gas expands, therefore, repulsive forces
lead to a temperature increment in a Fermi gas and attractive forces leads to a
temperature decrement in Bose gas.
In Fig. 3, it is seen that there is a discontinuity point on the curve for hB/2.5R=10
at T=T0 due to Bose–Einstein condensation. Discontinuity points of the other
curves for Bose gas are out of range of the scales of this figure. As can be seen in the
figure, there is an upper limit for pressure of a Fermi gas for a given enthalpy. At
this upper pressure limit, enthalpy completely originates from the potential energy
caused by repulsive forces and the kinetic energy content of the enthalpy is zero.
Fig. 3. Variation of temperature versus pressure for different constant-enthalpy values.
H. Saygın, A. Şişman / Applied Energy 70 (2001) 49–57
57
Consequently, quantum degeneracy of a Fermi gas contributes to the negative
values of the JTC, while quantum degeneracy of a Bose gas contributes to the positive values of the JTC. Therefore, Bose type of gases are more suitable for cryogenic
refrigeration systems using the JTE. Furthermore, an inversion temperature does
not appear for quantum ideal-gases.
Acknowledgements
This research was supported by the Research Fund of Istanbul Technical University under contract No. 1311.
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