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New semi-empirical equation of state for nuclear materials
ELSE.VIER
Journal of Nuclear Materials 211 (1994) 175-176
Letter to the Editors
New semi-empirical equation of state for nuclear materials G. Dharmadurai
’
HOPEL Research Home, 62 /4A, Poobalrayerpuram 4th street, Tuticorin 628 001, Tamil Nadu, India
Received 24 January 1994; accepted 19 May 1994
Abstract A simple semi-empirical equation of state with a single unknown behaviour of nuclear materials at high temperatures and densities.
1. Introduction
In an earlier article [l] it was proposed that the hardsphere pressure term of the Dieterici equation [2] could be refined to obtain a simple and convenient two-constant equation of state for nuclear materials of relevance to fast reactor safety analysis [3]. In this note, the empirical relation [4] between the molar volume of a material in the solid state V, and its critical volume V, is invoked to eliminate one unknown constant of the refined Dieterici equation. The resulting semi-empirical equation of state with a single unknown constant is found to predict critical pressures of nuclear materials in sensible accord with more sophisticated methods [5-71 like principle of corresponding states and significant structure theory (SST).
constant
is presented
of the Dieterici equation
The Dieterici equation represented by the following relation between pressure p, molar volume I, and temperature T is deemed [2] to be the best all-round analytical two-constant equation of state:
RT ’ = 1/(1-
b/4v)4
exp
The conditions )@/al/ at the critical temperature volume V,,
1T = 0 and [a*p/W*], = 0 T, would yield the critical
0022-3115/94/$07.00
Z, =p,I/,/RT,
Z,,
= 81/16e3 = 0.252,
while the constant
a becomes (4)
where pc is the critical pressure. It is imperative to note that the above value of Z, lies within 10% of experimental values of Z, for most substances [1,2,4] and remarkably agrees with the range of Z, = 0.24 to 0.27 for UO, and UC predicted by the traditional approach employing the principle of corresponding states [5,6].
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0022-3115(94)00307-A
(3)
and the critical compressibility
a = 3RTcVc,
’ Formerly: Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, Tamil Nadu, India.
the
Here R is the gas constant, b is the covolume equal to four times the hard-core volume of the molecules and the constant a is a measure of the cohesion between the molecules. As Eq. (1) is found to overpredict pressures at higher densities [l], its hardsphere pressure term RT/(Vb) can be replaced by the improved term RT/V(l - b/4Vj4 suggested by Guggenheim [8] to obtain the following refined equation of state:
v, = 36/4, 2. Refinement
for predicting
In view of the interesting empirical feature [4] that the ratio (3f$‘V,) between V, and V, is a universal constant close to unity, or more exactly
v, = 3.07K:,
(5)
Eq. (3) and Eq. (5) can be combined to get a simple expression for b, b = 4.093v,. Consequently,
and pc = 240 MPa which is typically of the same order as earlier predictions [6] of pc ranging from 120 to 180 MPa.
(6) Eq. (2) may be rewritten as RT
exp(-F).
Concluding remark In summary, the present semi-empirical equation of state with a single less-known parameter T, can serve as a versatile equation of state for predicting the high temperature behaviour of nuclear materials.
(7)
’ = V(1 - l.023~/~)4 A noteworthy feature of this new semi-empirical equation of state is its striking simplicity with only a single unknown constant T, for refractory materials like UO,.
Acknowledgemeat The author gratefully acknowledges that copies of Refs. [7] and [12] sent by Dr. P. Rodriguez, Director, IGCAR and the illuminating comments of the referees have been helpful in revising this article.
3. Critical pressure As demonstrated elsewhere [9-111, sensible estimates of T, for many materials can be obtained from their normal boiling point T, in the context of the following empirical relation, Tc = 1 . 4732T’.“13 b
.
(8)
These estimates were found to concur with earlier SST predictions [ll]. The most recent vapour pressure equation obtained [7] by applying SST for UO, yields T, = 3850 K so that Eq. (8) gives T, =L7500 K. With V, = 27.9 cm3/mol [7], Eq. (5) would yield V, - 86 cm3/mol, resulting in a value of pc = (Z,RTc)/Vc = 180 MPa, rather compatible with pc = 158 MPa predicted by Fischer [7]. Moreover, it can be readily seen that if the wide scatter [7,12] in the estimates of T, for UO, ranging from 6000 to 10600 K is considered, the resulting estimates of pc would range from 140 to 240 MPa in sensible accord with earlier predictions f5-71, ranging from 100 to 200 MPa. For UC having V, = 26 cm3/mol [3], T, estimated from Eq. (8) turns out to be around 9000 K [ll]. Hence, the above approach will yield V, = 80 cm3/mol
References [I] G. Dharmadurai, J. Nucl. Mater. 110 (1982) 256. 121 J.O. Hirschfelder,
CF. Curt& and R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1967) p. 2.50. [3] H.H. Hummel and D. Okrent, Reactivity Coefficients in Large Fast Power Reactors (American Nuclear Society, 1970). [4] R. Mayer and B. Wolfe, Trans. Am. Nucl. Sot. ‘7 (1964) 111. [S] SK. Kapil, J. Nucl. Mater. 60 (1976) 158. [6] G. Dharmadurai, Atomkernenergie-Kerntechnik 43 (1983) 259. (7) E.A. Fischer, Nucl. Sci. Eng. 101 (1989) 97, and references therein. [8] E.A. Guggeheim, Molec. Phys. 9 (1965) 1993. 193D.A. Gates and G. Thodos, Am. Inst. Chem. Eng. J. 6 (1960) 50. IlO] R.W. Ohse and H. Von Tip~lskir~h, High Temp.-High Press. 9 (1977) 367. [ll] G. Dharmadurai, Nucl. Eng. Des. 73 (1982) 287. 1121A.M. Azad and S. Ganesan, J. Nucl. Mater. 139 (1986) 91.
Journal of Nuclear Materials 211 (1994) 175-176
Letter to the Editors
New semi-empirical equation of state for nuclear materials G. Dharmadurai
’
HOPEL Research Home, 62 /4A, Poobalrayerpuram 4th street, Tuticorin 628 001, Tamil Nadu, India
Received 24 January 1994; accepted 19 May 1994
Abstract A simple semi-empirical equation of state with a single unknown behaviour of nuclear materials at high temperatures and densities.
1. Introduction
In an earlier article [l] it was proposed that the hardsphere pressure term of the Dieterici equation [2] could be refined to obtain a simple and convenient two-constant equation of state for nuclear materials of relevance to fast reactor safety analysis [3]. In this note, the empirical relation [4] between the molar volume of a material in the solid state V, and its critical volume V, is invoked to eliminate one unknown constant of the refined Dieterici equation. The resulting semi-empirical equation of state with a single unknown constant is found to predict critical pressures of nuclear materials in sensible accord with more sophisticated methods [5-71 like principle of corresponding states and significant structure theory (SST).
constant
is presented
of the Dieterici equation
The Dieterici equation represented by the following relation between pressure p, molar volume I, and temperature T is deemed [2] to be the best all-round analytical two-constant equation of state:
RT ’ = 1/(1-
b/4v)4
exp
The conditions )@/al/ at the critical temperature volume V,,
1T = 0 and [a*p/W*], = 0 T, would yield the critical
0022-3115/94/$07.00
Z, =p,I/,/RT,
Z,,
= 81/16e3 = 0.252,
while the constant
a becomes (4)
where pc is the critical pressure. It is imperative to note that the above value of Z, lies within 10% of experimental values of Z, for most substances [1,2,4] and remarkably agrees with the range of Z, = 0.24 to 0.27 for UO, and UC predicted by the traditional approach employing the principle of corresponding states [5,6].
0 1994 Elsevier Science B.V. All rights reserved
SSDI 0022-3115(94)00307-A
(3)
and the critical compressibility
a = 3RTcVc,
’ Formerly: Indira Gandhi Centre for Atomic Research, Kalpakkam 603 102, Tamil Nadu, India.
the
Here R is the gas constant, b is the covolume equal to four times the hard-core volume of the molecules and the constant a is a measure of the cohesion between the molecules. As Eq. (1) is found to overpredict pressures at higher densities [l], its hardsphere pressure term RT/(Vb) can be replaced by the improved term RT/V(l - b/4Vj4 suggested by Guggenheim [8] to obtain the following refined equation of state:
v, = 36/4, 2. Refinement
for predicting
In view of the interesting empirical feature [4] that the ratio (3f$‘V,) between V, and V, is a universal constant close to unity, or more exactly
v, = 3.07K:,
(5)
Eq. (3) and Eq. (5) can be combined to get a simple expression for b, b = 4.093v,. Consequently,
and pc = 240 MPa which is typically of the same order as earlier predictions [6] of pc ranging from 120 to 180 MPa.
(6) Eq. (2) may be rewritten as RT
exp(-F).
Concluding remark In summary, the present semi-empirical equation of state with a single less-known parameter T, can serve as a versatile equation of state for predicting the high temperature behaviour of nuclear materials.
(7)
’ = V(1 - l.023~/~)4 A noteworthy feature of this new semi-empirical equation of state is its striking simplicity with only a single unknown constant T, for refractory materials like UO,.
Acknowledgemeat The author gratefully acknowledges that copies of Refs. [7] and [12] sent by Dr. P. Rodriguez, Director, IGCAR and the illuminating comments of the referees have been helpful in revising this article.
3. Critical pressure As demonstrated elsewhere [9-111, sensible estimates of T, for many materials can be obtained from their normal boiling point T, in the context of the following empirical relation, Tc = 1 . 4732T’.“13 b
.
(8)
These estimates were found to concur with earlier SST predictions [ll]. The most recent vapour pressure equation obtained [7] by applying SST for UO, yields T, = 3850 K so that Eq. (8) gives T, =L7500 K. With V, = 27.9 cm3/mol [7], Eq. (5) would yield V, - 86 cm3/mol, resulting in a value of pc = (Z,RTc)/Vc = 180 MPa, rather compatible with pc = 158 MPa predicted by Fischer [7]. Moreover, it can be readily seen that if the wide scatter [7,12] in the estimates of T, for UO, ranging from 6000 to 10600 K is considered, the resulting estimates of pc would range from 140 to 240 MPa in sensible accord with earlier predictions f5-71, ranging from 100 to 200 MPa. For UC having V, = 26 cm3/mol [3], T, estimated from Eq. (8) turns out to be around 9000 K [ll]. Hence, the above approach will yield V, = 80 cm3/mol
References [I] G. Dharmadurai, J. Nucl. Mater. 110 (1982) 256. 121 J.O. Hirschfelder,
CF. Curt& and R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1967) p. 2.50. [3] H.H. Hummel and D. Okrent, Reactivity Coefficients in Large Fast Power Reactors (American Nuclear Society, 1970). [4] R. Mayer and B. Wolfe, Trans. Am. Nucl. Sot. ‘7 (1964) 111. [S] SK. Kapil, J. Nucl. Mater. 60 (1976) 158. [6] G. Dharmadurai, Atomkernenergie-Kerntechnik 43 (1983) 259. (7) E.A. Fischer, Nucl. Sci. Eng. 101 (1989) 97, and references therein. [8] E.A. Guggeheim, Molec. Phys. 9 (1965) 1993. 193D.A. Gates and G. Thodos, Am. Inst. Chem. Eng. J. 6 (1960) 50. IlO] R.W. Ohse and H. Von Tip~lskir~h, High Temp.-High Press. 9 (1977) 367. [ll] G. Dharmadurai, Nucl. Eng. Des. 73 (1982) 287. 1121A.M. Azad and S. Ganesan, J. Nucl. Mater. 139 (1986) 91.