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Thermophysical property correlations for the niobium-1% zirconium alloy
Journal of Nuclear North-Holland
Materials
261
173 (1990) 261-273
Thermophysical property correlations for the niobium-1 % zirconium alloy David J. Senor, J. Kelly Thomas * and K.L. Peddicord Advanced Nuclear Fuels Laboratory, Department of Nuclear Engineering, Texas A & M University, College Station, TX 77843-3133, USA Received
17 October
1989; accepted
14 May 1990
Correlations were deveioped for the temperature dependence of specific heat (c,), enthalpy (!I), entropy (s), Gibbs free energy (G), free energy function (FEF), mean coefficient of linear thermal expansion (a), density (p) and lattice parameter (u) of niobium-l% zirconium. The specific heat and thermal expansion correlations were developed as empirical fits to experimental data. The enthalpy, entropy and free energy correlations were derived from the specific heat correlation, and the density and lattice parameter correlations were derived from the thermal expansion correlation. 2.784 x 10’ ~~=0.2441+5.105~10-~(T)+
2.295~10~
kJ/kg K,
T
T2
h(T)-h(298K)=-75.01+0.2441(T)+2.553x10~s(T)2+1.213x105exp
-
2.295 x lo4 T
i s = 0.2441 ln( T) + 5.105 X lo-s(
T) -0.9947
kJ/kg
K,
G=1.2388(T)-2.552~10~~(T)~-0.2Q41(T)ln(T)+1.213xlO~exp
kJ/kg. 75.01
FEF== -1.2388+0.2441
p = 8637 -0.200(T) a=3.293+2.622~10-‘(T)
kJ/kg,
ln(T)+2.552X10-5(T)+-
1.213~10~
-
T
2.295 x lo4
T
kJ/kg K,
kg/m3, ,&.
The specific heat correlation has a relative standard deviation of 0.98% and is valid over the temperature range 298 to 2700 K. The enthalpy, entropy and free energy correlations have the same temperature range of validity as the specific heat correlation. Statistical methods were employed to determine the propagation of error from the specific heat correlation to estimate the un~rtainty associated with each of the derived correlations. The relative standard deviation of the thermal expansion correlation is 0.96% and it has a valid temprature range of 575 to 1627 K. The density and lattice parameter correlations are valid over the temperature range 297 to 1627 K. The propagation of error introduced by the derivation of the density and lattice parameter correlations was calculated in the same manner as for the derived thermodynamic correlations.
1. Introduction The niobium-l% zirconium (Nb-1Zr) rently under consideration for application
alloy is curas a cladding
* Current Address: Westinghouse Savannah River, Bldg 77341A, Room 171, Aiken, SC 29808, USA. 0022-3115/90/$03.50 0 1990 - Elsevier Science Publishers
and structural material for the SP-100 space reactor system. Although several property reviews have been conducted for Nb-1Zr and elemental niobium [l-lo], no comprehensive effort has been made to construct correlations suitable for application in fuel element modeling. A consistent and well-v~idat~ set of correlations for the relevant properties is necessary in order to
B.V. worth-HoIland)
262
D.J. Senor et al. / Thermophysical propeq
accurately model fuel element performance. The correlations should be valid for the entire temperature range of interest, including start-up and transient overpower scenarios. The temperatures encompassed by such considerations range from 300 K to the melting point of the alloy (approximately 2700 K). To model the cladding as closely as possible, only data for Nb-1Zr were used in the construction of the correlations. Niobium data were employed only as a means of validation to ensure the reliability of the Nb-1Zr data. The thermodynamic properties of fuel element cladding materials are of primary importance during reactor transients. The behavior of the cladding specific heat is of interest under these conditions and a description of the temperature dependence of enthalpy is necessary to perform energy balances between the fuel, cladding and coolant. The cladding entropy and Gibbs free energy are of interest primarily at high temperatures in an accident scenario when chemical reactions between the various components of the reactor system are possible. To determine the strains introduced by changes in cladding temperature, knowledge of the thermal expansion behavior is required. The variation of cladding density with temperature is important for the modeling of many steady-state and transient scenarios and knowledge of the temperature dependence of the lattice parameter aids in modeling the migration of atomic species through the cladding.
2. Experimental data
A literature review revealed only one data set for the specific heat of Nb-1Zr [ll]. The measurements were made using a transient technique which involved passing a large current through a hollow cylindrical specimen in short pulses and measuring the temperature rise of a water jacket. The length and magnitude of the pulse were adjusted to yield the desired specimen temperature which was measured using optical pyrometry. Test temperatures ranged from 1500 to 2700 K. The nominal zirconium content of the specimen was 1.05% by weight. Total impu~ties amounted to less than 0.17%, with tantalum comprising the majority at 0.09%. Micrographs of the specimen indicated that considerable grain growth occurred as a result of the pulse heating. The data assume a sawtooth pattern when plotted as a function of temperature, but this does not appear to have been the result of any physical phenomenon.
~orr~luhons for Nh- I 9; %r crll~
.?.L Lineur
thermul
expi~rwron
Two data sets were available in the literature for the thermal expansion of the Nb-1Zr alloy [12,13] as a function of temperature. Ewing et al. [12] tested asfrabricated materials in three separate tests with a base temperature of 297 K. Test temperatures ranged from 57.5 to 1627 K and all specimens had less than 0.20% impurities. Fisher and Achener [13] tested two specimens in an argon atmosphere. The specimens were cut at right angles to each other and represent cross sections parallel and perpendicular to the rolling direction. The amount of cold work applied to the specimens was not reported. The base temperature was 297 K. and test temperatures ranged from 465 to 1305 K. No microanalysis was reported. Bigelow et al. [74] report a single data point at a temperature of 1366 K. 2.3. Density No information was available in the literature concerning the temperature dependence of density for the Nb-1Zr alloy. Several different values for the room temperature density were reported which range from 8400 to 8660 kg/m3, but the most commonly referenced value was 8570 kg/m3 [4,15-181. Since this was the most frequently reported value and represents a median for the entire range of reported values, it was used as the reference room temperature density in the development of the density correlation. 2.4. Latlice parameter Hobson [19] measured the lattice parameter of NbIZr as a function of oxygen content at 1873 K. He reports a lattice parameter of approximately 3.302 A at 1873 K for very low oxygen impurity levels. No additional microanalysis was provided.
3. Existing correlations 3.1. Specific heat Cezairliyan constructed a third-order polynomial fit to his own specific. heat data [ll]. Although the correlation accurately follows the trend of the experimental data, it becomes unstable below 1500 K. This behavior is inconsistent with the trend of the data and is the result of employing a third-order polynomial beyond the range of data on which it was based. The primary temperatures of interest for space reactor cladding
263
D.J. Senor et al. / Thermophysicalproperty correlationsfor Nb-I % Zr alloy
material are approximately 1000 to 1500 K and the unstable nature of the Cezairhyan correlation in this range renders it unsuitable for fuel element modeling applications. For this reason, a new correlation based on the Cezairliyan data was needed that would provide stability in the temperature regime below 1.500 K. 3.2. Linear thermal expansion A correlation for the linear thermal expansion of Nb-1Zr was developed by the Thermophysical Properties Research Center (TPRC), and reported by Toulaukian and Ho [lo]. It was a second-order poly nomial in terms of AL/L, but was converted for use in this work to the mean coefficient of linear thermal expansion. They report the correlation to be applicable for annealed material, but the database employed in its formation was not explicitly defined. Touloukian and Ho report the data of both Ewing et al. [12] and Fisher and Achener [13], but the trend of the correlation does not follow that of either of the data sets and for this reason a new fit to the data was deemed necessary.
where R = universal gas constant (8.314 J/g mol K), 8 = Einstein temperature, T = absolute temperature. The specific heat at constant pressure is related to that at constant volume by the Nemst-Lindemann equation which adds a linear term to eq. (1). The resulting expression for c,, is given by [21]:
a@’expfQ/T) f ” = T*(exp(B/T)
bT
’
where a and b are empirical fit coefficients. The first term of eq. (2) describes the low temperature behavior of the material, while the linear term dominates the temperature regime above room temperature. For pure elements, the leading coefficient (a) theoretically assumes a values of 3R as indicated in eq. (1). An additional term to account for defect formation processes at extremely high temperatures can be added to eq. (2). The complete expression for specific heat of a solid over all temperatures is then given by [22]: a@
exp(@/T)
” = T’(exp( 0/T) 4. Correlation development
- 1)2
+bT+-
cEn RT2
exp
(3)
4.1. Specific heat
where = activation energy for defect formation (J/gmol K), = universal gas contant (8.314 J/g mol K), R T = absolute temperature, a, b, c = empirical coefficients. A more rigorous model of heat capacity is that proposed by Debye ]22] which allows for interaction between the atoms in a lattice. This theory describes the vibration of the entire lattice as opposed to only the uncoupled vibrations of the individual atoms as described by the Einstein theory. However, the approximation made by Einstein only has a noticeable effect at cryogenic temperatures, so for fuel element modeling applications, the Einstein theory is acceptable. Additionally, the Debye model is much more complex and requires much greater computational effort than the Einstein model. For these reasons, the Einstein model with the additional defect formation term as given in eq. (3) was used to develop the Nb-1Zr specific heat correlation. Since experimental specific heat data for Nb-1Zr were only available over the temperature range 1500 to 2700 K, no information about the low temperature specific heat behavior of the alloy was available. For this reason, the first term in eq. (3) was replaced by a E,
A variety of fits to the Cezairliyan [ll] data were studied including second- and third-order polyno~~, power law and exponential forms. None modeled the data well or provided adequate predictive capability over all temperatures of interest. Due to the lack of experimental data below 1500 K, a physically-based fit form was required to ensure stability at the lower temperatures. The specific heat of a solid at constant volume from low to moderate temperatures is approximated by the Einstein theory of specific heat. The explicit assumption involved is that the material Iattice is composed of N atoms which are free to vibrate harmonically in three dimensions about their lattice site. Since each atom is assumed to vibrate independentiy of all others in the lattice, the effective result is that of 3N harmonic oscillators. Based on these assumptions, the interna energy of 3N oscillators may be derived from statistical mechanics. By differentiating the internal energy with respect to temperature at constant volume, an expression for the specific heat is obtained [20]: 3R@12exp( 8/T) cv = T’(exp(B/T)
- 1)”
- 1)2 ’
(1)
264
D.J. Senor et al. / Thermaphysral
0.15”
200
400
Km
ml
lcoo
proper?v correhtions Ior Nb-- i S Zr u&v
.’ 1200
-‘. 1400
1600
‘*
’
1800
2ooO
3
“I. 2200
2400
’ * 2600 2800
Temperature (K) Fig. 1. Nb-IZr specific heat experimental data and correlations.
constant independent of temperature in the development of the correlation. The linear and high temperature terms were retained and the defect activation energy was implemented as an empirical coefficient. The final form of the correlation is given by:
correlation for the relative specific enthalpy of Nb-1Zr is given by: h(T)
-
h(298 K)
= -75.01 i 0.2441r+
CP= 0.2441+ 5.105 x 10-Y -t 2.784 x 10’ 7-z
2.553 x 10-5T2 2.295 x lo4
2.295 x lo4 T
T
kJ,‘kg K,
where T is the temperature in Kelvin. The relative standard deviation d the fit with respect to the Cezairliyan data over the temperature range 1500 to 2700 K is 0.98%, which is the same as the Cezairliyan [II] correlation. The correlation is valid from room temperature to the melting point of the alloy (298 to 2700 K). The expe~menta~ data are shown in fig, I along with the Cezairliyan correlation and the curve represented by eq. (4). 4.2. Enthafpy The enthalpy correlation was developed by integrating the specific heat correlation given in eq. (4) from the base temperature to an arbitrary upper limit temperature. The base temperature was selected to be room temperature (298 K) since most energy balance calculations require relative enthalpy values. The resulting
kJ/b
(5)
where T is the temperature in Kelvin. Since eq. (5) was derived from the specific heat correlation, it has the same temperature range of validity (298 to 2700 K). At low temperatures, the exponential term in eq. (5) yields values several orders of magnitude less than those given by the other terms and is thus negligible at these temperatures. At high temperatures, the exponential term represents approximately 3% of the total value and cannot be neglected. In some instances. it is necessary to perform calculations using an absolute enthalpy value in which the enthalpy is defined to be zero at 0 K. Absolute enthalpy predictions may be obtained from eq. (5) by neglecting the constant term. The relative enthaipy correlation given by eq. (6) is shown as a function of temperature in fig. 2. 4.3. Entropy The entropy correlation was derived from the specific heat correlation using the definition of absolute entropy
D.J. Senor et al. / T~er~op~~s~cal property cor~ei~tions for Nb- 1 X 2r alloy
265
900
800 700
200
400
600
800
loo0
1200
1400
1600
1800
2ooO
2200
2400
2600
2800
Temperature (K) Fig. 2. Nb-1Zr relative specific enthalpy correlation.
1201: s(T)
=s,t
TCp(V jTo--?--_ dT',
(6)
where se is the entropy at the reference temperature T,. The se term may be set equal to zero if the base temperature is taken to be 0 K. However, the function for specific heat must be valid over the entire integral. Since the specific heat correlation developed in this work is not valid at low temperatures, the reference temperature was taken to be 298 K. No entropy or low temperature specific heat data were available in the literature for Nb-1Zr. It was therefore necessary to use niobium specific heat data to calculate a room temperature entropy for use in the development of the Nb-1Zr entropy correlation. Niobium specific heat data from 10 to 300 K 1231 were fit using eq. (2). The resulting correlation yielded a leading coefficient (a) of 0.2431 kJ/kg K and an Einstein temperature (0) of 175 K. As mentioned previously, for pure elements the leading coefficient should approach a value of 3R. For niobium, this corresponds to 0.2685 kJ/kg K. Values of the niobium Einstein temperature reported in the literature range from 188 to 210 K [24]. For both the leading coefficient and the Einstein temperature, the empirically determined values are in reasonably good agreement with the expected values. The specific heat at room temperature for niobium predicted by the fit to the Chou et al. 1231 data is 0.2717 kJ/kg K; the specific heat of Nb-1Zr given by eq. (4) is 0.2593 kJ/kg K, a
difference of only 4.6%. This is remarkably good agreement considering the Nb-1Zr correlation was developed with only high temperature data (1500 to 2700 K). For these reasons, the fit to the low temperature niobium data of 1231 was deemed suitable to provide a room temperature entropy for the development of a Nb-1Zr entropy correlation. The resulting low temperature niobium specific heat fit was integrated from 10 to 298 K to calculate a room temperature entropy for niobium as indicated in eq. (6). The lower limit of 10 K was selected for two reasons. First, the value of the integrand approaches infinity as temperature decreases, so to avoid any nonphysical nume~ca~ effects the integration was performed at a temperature above which the integrand becomes unstable. Second, any effects associated with superconducting niobium were avoided since the critical temperature of elemental niobium is appro~mately 9.5 K [22]. The room temperature entropy of niobium was calculated to be 0.4115 kJ/kg K. This value was used as se in eq. (6) along with the Nb-1Zr specific heat correlation given by eq. (4). The integraf was evaluated with a base temperature of 298 K to yield a correlation for Nb-1Zr entropy as a function of temperature. The resulting expression is given by: s(r)
= 0.2441 In(T)
+ 5.105 X 10P5(T)
- 0.9947 kJ,‘kg K,
(7) where T is the temperature in Kelvin. An additional exponential term results after performing the integra-
500
loo0
1500
2mo
2500
3000
Temperature (K) Fig. 3. Nb-1Zr absolute specific entropy correlation.
tion, but cont~butes less than one percent to the value of s(T) at all temperatures. It has therefore been neglected in eq. (7). The entropy correlation is shown as a function of temperat~e in fig. 3. The co~ela~on is valid only over the temperature range 298 to 2700 K since the specific heat correlation is valid only over this range.
4.4. Free energy Gibbs free energy (c) and the free energy function (FEF) may be derived from enthalpy and entropy. The definition of Gibbs free energy is given by [2Oj: G-h-Ts,
Fig. 4. Nb-1Zr Gibbs free energy correlation.
(8)
267
D.J. Senor et al. / Thermophysical property correlations for Nb- I % Zr alloy where G = specific Gibbs free energy, h = absolute specific enthalpy, T = absolute temperature, s = absolute specific entropy. In eq. (8), both enthalpy and entropy are defined to be zero at 0 K. Manipulation of this equation results in the following relationship defining the free energy function: FEF=
-
G-h(298 T
K)
=s-
h(T)
- h(298 K) T ’ (9)
where FEF is the free energy function. Both free energy quantities are useful in chemical reaction calculations. To derive the correlation for Gibbs free energy, eq. (5) without the constant term and eq. (7) were inserted into eq. (8). The resulting expression is given by: G(T)
= 1.2388(T) - 0.2441(T)
- 2.552 x 10-5(T)2 ln( T) 2.295 x lo4 T
kJ/kg,
(10) where T is the temperature in Kelvin. To develop the free energy function correlation for Nb-lZr, eqs. (5) and (7) were inserted into eq. (9). The
500
1000
resulting
expression
is given by:
FEF = - 1.2388 + 0.2441 ln( T) 75 .Ol + 2.552 x 10-5(T) + 7 _ 1.213 x 10s exp - 2.295 x lo4 T T i
kJ/kg
K (11)
where T is the temperature in Kelvin. Both the Gibbs free energy and free energy function correlations are valid over the temperature range 298 to 2700 K and are shown in Figs. 4 and 5, respectively. 4.5. Linear thermal expansion The two data sets available in the literature for the thermal expansion of Nb-1Zr exhibit completely different trends. The Fisher and Achener [13] data set is concave down, while the Ewing et al. (121 data exhibit a concave upward trend. Although both experiments were performed as a component of a larger project, much greater emphasis on detail was given to the Ewing et al. data. For example, no specimen characterization was reported for the Fisher and Achener tests. The Ewing et al. data are a compilation of three separate tests and the results are very consistent. Two of the tests were performed with quartz measuring rods and one with a sapphire measuring rod. Additionally, Ewing et al. report that some of their previous thermal expansion test results on Nb-1Zr were rejected because they were in
1500
2m
Temperature (K) Fig. 5. Nb-1Zr free energy function correlation.
2500
3ooo
268
D.J. Senor et al. / Thermophysical
property
correlatrons
for Nh- I B %r ~IIO,,
8.5
7.0 -
*
Ewing et al. (1965)
q
Fisher and Achener
(1965)
*-_- TPRC Correlation 0 6.5 400
600
800
IOCQ
1200
Temperature Fig. 6. Nb-1Zr
mean coefficient
of linear thermal
error; this again supports Ewing et al. since they were able to detect errors in and reject portions of their data. The data point from Bigelow et al. [14] was not considered for inclusion in the correlation since it appears to be an evaluation of the TPRC correlation at 1366 K. Based on these criteria, only the Ewing et al. data were employed to form the correlation. A variety of fit forms were examined, but a secondorder polynomial was selected as the best fit to the Ewing et al. data. Since the data only extend from 575 to 1627 K, there was some concern about the stability of the polynomial outside this range. However, the minimum of the polynomiat occurs far below the lowest temperature of interest and at high temperatures the curve is also well behaved. The correlation for the mean coefficient of linear thermal expansion for Nb-1Zr is given by: ty, = 6.62 + 3.64 x W4( + 2.75 x 10-‘(T)’
7-) 10-“/K,
(12)
where T is the temperature in Kelvin. The relative standard deviation of the correlation is 0.96% with respect to the Ewing et al. [12] data. Fig. 6 shows all the available data along with the TPRC correlation and the curve produced by eq. (12).
1400
1600
1800
(K)
expansion
experimental
data and correlations.
expansion of the alloy and its density at room temperature. The density correlation was derived from the definition of the mean coefficient of linear thermal expansion. The explicit assumptions involved were an initial perfectly cubic unit cell and isotropic expansion with temperature. The base temperature was taken to be the same as that of the data used in the formation of the thermal expansion correlation (297 K). The density at this temperature was taken to be 8570 kg/m3 as discussed previously. After simplifying the expression for the density at any temperature, the following equation results:
P(7’) = (1 +
a;;!297))’ wm3+
(13)
where T is the temperature in Kelvin and a is the thermal expansion correlation given by eq. (12). Although eq. (13) is not extremely complicated, it does require computation of the thermal expansion correlation To facilitate use in fuel modeling applications a simpler expression was developed by fitting a series of values generated by eq. (13). Of the fit forms examined. a linear correlation provided the best agreement with the trend of eq. (13), provided a simple fit form, and guaranteed stability over the valid temperature range. The resulting Nb-1Zr density correlation is given by:
4.6. Density
p = 8637 - 0.200( 7’) kg/m3,
A relationship for the temperature dependence of density for Nb-1Zr was derived based on the thermal
where T is the temperature in Kelvin. Since the density correlation employs an experimental room temperature density, the lower limit of validity is 297 K. However.
(14)
269
D. J. Senor et al. / Thermophysical property correlations for Nb- I 5%Zr alloy
800
loo0
1200
Temperature (K) Fig. 7. Nb-1Zr
density
the thermal expansion correlation is valid only to 1627 K, this upper limit applies to the density correlation as well. Fig. 7 shows the density correlation as a function of temperature over this range.
since
4.7. Lattice parameter The available data for the lattice parameter of Nb1Zr do not offer any insight into the temperature dependence of the property. For this reason, a correlation
correlation.
for the lattice parameter of Nb-1Zr was derived from the density correlation. The correlation was derived following the same assumptions made for the derivation of the density correlation. In a manner similar to that used for the density correlation, a simple fit was applied to the resulting expression to yield a more manageable form. The Nb-1Zr lattice parameter correlation is given by: a = 3.293 + 2.622 X 10M5(T)
A,
(15)
3.3x
3.325
3.275
3.250 400
600
800
Ical
1200
1400
1600
Temperature (K) Fig. 8. Nb-1Zr lattice parameter experimental data and correlation.
1800
2ooo
270
D.J. Senor et al.
property correlations for N&l
/ Thermophysical
where T is the temperature in Kelvin. Eq. (12) has the same temperature range of validity as the density correlation (297 to 1627 K). Fig. 8 shows the lattice parameter correlation as a function of temperature along with the low oxygen content Hobson [19] data point at 1873 K for comparison.
5. Correlation validation 5. I. Specific heut Comparison of the Nb-1Zr correlation to elemental niobium data for specific heat provides evidence that the trend of the correlation developed in this work below 1500 K is reasonable. The niobium data [25-281 are shown with the Nb-1Zr correlation in fig. 9. The niobium data exhibit a linear trend below approximately 1800 K. Since little difference would be expected in the specific heat behavior of niobium and Nb-lZr, the similarity between the correlation and the niobium specific heat data is encouraging. The Nb-1Zr specific heat correlation may therefore be used with confidence from room temperature to 2700 K. 5.2. Enthalpy Since no experimental enthalpy data were available, it was necessary to determine the error propagated by deriving the enthalpy correlation from the specific heat
%, Zr crll~v
correlation. To determine the propagation of error involved in deriving a correlation, the following expression may be used [29]:
(16)
cl,. =
where u,, = absolute standard deviation of derived correlation, .r = dependent variable, /, = independent variables (total of n). 0, = absolute standard deviation of independent variables. For the enthalpy correlation, the only independent variable involved was the specific heat correlation itself. By inserting the appropriate values and assuming that the specific heat correlation varies slowly over its valid temperature range, it may be shown that the relative standard deviation of the enthalpy correlation is approximately the same as that for the specific heat correlation (0.98%). 5.3. Entropy The uncertainty associated with the derivation of the entropy correlation from the specific heat correlation was determined in a fashion similar to that described for enthalpy. For entropy, however, there were two independent variables (sa and cp( T)) which had the potential to propagate error into the derived correlation,
L
0.4 s g 8 0.3 -
0
400
800
1200
IMX)
2cQo
24co
2800
Temperature (K) Fig. 9. Niobium
specific heat experimental
data and Nb-1Zr
correlation
developed
in this work.
211
D.J. Senor et al. / Thermophysical property correlations for Nb-I % Zr alroy
[32,34,35] clearly exhibit a concave upward trend and one [30] displays the opposite behavior. The other data sets either exhibit no clear trend or are fairly linear. The general trend and magnitude of the data agree reasonably well with the Nb-1Zr correlation developed in this work. This behavior provides support for the selection of the Ewing et al. [12] data as the sole basis for construction of the correlation. The Nb-1Zr thermal expansion correlation may be used with confidence over the temperature range 575 to 1627 K. However, the comparison to the niobium data indicates that the correlation should provide reasonable predictions over all temperatures of interest in space reactor fuel element modeling.
After evaluating eq. (16) with the appropriate values and converting to relative standard deviation, the error associated with the entropy correlation was calculated to be approximately 1.70%. The bulk of the increase in uncertainty over the specific heat correlation was introduced by the error associated with the calculation of the room temperature entropy from the fit to the low temperature niobium data. 5.4. Free energy The errors associated with the Gibbs free energy and free energy function correlations were determined using eq. (16). In both cases, there were two independent variables associated with the derivation of the correlations. Gibbs free energy is dependent on absolute enthalpy and entropy and the free energy function is dependent on relative enthalpy and absolute entropy. For Gibbs free energy, the relative standard deviation was calculated to be 1.70% essentially the same as the uncertainty for the entropy correlation. The corresponding relative standard deviation of the free energy function correlation was calculated to be 2.20%.
5.6. Density The error associated with the density correlation was calculated using eq. (16). The independent variables were taken to be the room temperature density and the thermal expansion correlation. However, since no reasonable estimate of the error in the room temperature density was available, any propagation of error due to this term was neglected. After evaluating the remaining terms and converting to relative standard deviation, the error associated with the thermal expansion correlation was shown to be on the order of lo-*%, indicating that virtually all the error associated with the density correlation is due to the accuracy of the room temperature density incorporated in the fit. This is a consequence of
5.5. Linear thermal expansion A comparison was made between the predictions offered by the Nb-1Zr thermal expansion correlation and thermal expansion data for niobium [30-371 as shown in fig. 10. Of the niobium data sets, three
10
9
8 Edwards et al. (1951) Tottle (1957) Heal (1958) Amonenko et al. (1964) Conway et al. (1965) Conway and Losekamp (1966) Vasyutinskiy et al. (1966) Johnson et al. (1968) 4’ 0
I
I
500
1000
I 1500
I . 2000
This Work 2500
3@30
Temperature (K) Fig. 10. Niobium thermal expansion experimental data and Nb-1Zr correlation developed in this work
272
D.J. Senar ef al. /
Therm~physicaf propertycorrelations for N/w/ %i.Zr alloy
the fact that a describes the expansion of the unit cell. The expansion (&AT) of a unit cell is on the order of 0.1% for AT on the order of 103. This represents a very small change in the cell dimensions compared to its initial size, and the error in this change is negligible compared to the error in the base dimensions. The reliability of the room temperature density employed in the density correlation was judged to be high since it was the most often quoted of the several values. Thus, the Nb-1Zr density correlation may be used with a reasonable degree of confidence within its valid temperature range. 5.7. Lattice parameler The lattice parameter measured by Hobson 1191 at 1873 K for Nb-1Zr with very low, levels of oxygen impurities is approximately 3.302 A, a difference of only 1.21% from the value predicted by the Nb-1Zr lattice parameter correlation. It should be emphasized, however, that the thermal expansion, and thus the density and lattice parameter correlations are strictly valid to only 1627 K. All three correlations have stable forms, however, and should predict reasonable values outside the validated temperature range. Thus, the agreement between the Hobson data point and the correlation prediction is en~o~a~ng. By inserting the appropriate values into eq. (16), it can be shown that the relative standard deviation of the lattice parameter correlation is approximately the same as the density correlation from which it was derived. There is little propagation of error in the derivation, and the lattice parameter correlation may be employed with the same level of confidence as the density correlation.
6. Summary Comparisons with niobium data indicate that the Nb-1Zr correlation developed in this work offers reasonable predictions for the specific heat of Nb-1Zr. The correlation is based on a physical fit form suggested by the Einstein theory of specific heat and is stable over the entire temperature range of interest (298 to 2700 IQ. The relative standard deviation of the correlation with respect to the experimental data is 0.98%. The Nb-1Zr enthalpy correlation was derived based on the Nb-1Zr specific heat correlation. Since the specific heat correlation has a physical fit form with ernpi~~ly determined coefficients, the enthalpy corre-
lation also has a physical basis, lending to its credibility despite the lack of experimental data to verify its predictive capability. The enthalpy correlation is valid over the temperature range 298 to 2700 K and has an approximate relative standard deviation of 1.0% based on the propagation of errors introduced in its derivation. A correlation for the entropy of Nb-1Zr was derived from the Nb-1Zr specific heat correlation. The entropy correlation is stable and yield predictions consistent with physical expectations over all temperatures of interest (298 to 2700 K). A room temperature entropy was calculated based on a fit to low temperature niobium data using the Einstein formulation. The relative standard deviation of the Nb-1Zr entropy correlation was calculated to be appro~mately 1.70%. based on the propagation of errors from the Nb-1Zr specific heat correlation and the fit to the low temperature niobium data used in its formation. Correlations for Gibbs free energy and the free energy function of Nb-1Zr were developed based on the Nb-1Zr enthalpy and entropy correlations. The correlations are stable and yield predictions which agree with physical expectations over all temperatures of interest. The free energy correlations are both valid over the temperature range 298 to 2700 K. The estimated relative standard deviations of the fits are 1.70% for Gibbs free energy and 2.20% for the free energy function. The behavior of the Nb-IZr thermal expansion correlation agrees with physical expectations and is a good empirical fit to the most reliable data with a relative standard deviation of 0.96%. The correlation is valid over the temperature range 575 to 1627 K. However, the correlation is numerically stable and yields reasonable predictions over the entire range of temperatures for space nuclear reactor applications as indicated by a comparison with niobium data at high temperatures. The accuracy of the Nb-1Zr density correlation is dependent on two factors, the standard deviation of the Nb-IZr thermal expansion correlation and the error associated with the romm temperature density value. The dominant factor in the error of the density correlation is that associated with the room temperature density, for which no estimate could be made from the availabie data. However, the level of reliability associated with the room temperature density was high, so the density correlation may be used with a reasonable degree of confidence over the temperature range 297 to 1627 K. A correlation for the lattice parameter of Nb-1Zr was derived from the Nb-1Zr density correlation. The primary component of the error associated with the lattice parameter correlation is that due to the room
D.J. Senor et al. / Therma~hys~cal property eorreiat~~~s for Nb- I % Zr alloy
temperature density value used in the formation of the density correfation. The correiation is strictly valid over 297-1627 K, but is stable and should offer reasonable predictions outside this range of temperatures based on its good agreement with a data point at 1873 K.
Acknowledgements Research sponsored by the US Department of Energy under contract DE-ACO2-86NE3796~.A~ and by the University Programs Division of the Oak Ridge Associated Universities under the Nuclear Engineering and Health Physics Fellowship Program.
References [l] Aerospace Structural Materials Handbook, US Air Force, ASD-TR-64-741 Vol. II-A (1963). [2] E.S. Bartlett and J.A. Houck, Physical and Mechanical Properties of Columbium and Columbium-Base Alloys, Battelle Memorial Institute, DMIC-125 (1960). 133 Columbium and Tantalum Alloys Technical Information, Wah Chang Corporation, Alloy Data Sheet (1966). [4] D.G. Goldberg, G. Dicker and S.A. Worcester, Nucl. Eng. Des. 22 (1972) 95. (51 Niobium, Metallwerk Plansee, Alloy Data Sheet (1988) [6] F.F. Schmidt and H.R. Ogden, The Engineering Properties of Columbium and Columbium Alloys, Battelle Memorial Institute, DMIC-188 (1963). 171 N.S. Stoloff, Survey of Elevated Temperature Properties of ~lumbium, Pratt and Whitney Aircraft-CANEL, PWAC-237 (1958). [8] R. Syre, Niobium, Molybdenum, Tantalum and Tungsten, North Atlantic Treaty Organization, NATO AGARDograph 50 (1961). 191 T.E. Tietz and J.W. Wilson, Behavior and Properties of Refractory Metals (Stanford University, Stanford, CA, 1965). [lo] Eds. Y.S. Touloukian and C.Y. Ho, Thermophysical Properties of Matter (PIenum, New York, 1972). A. Cezairliyan, Simultaneous Measurement of Specific Heat, Electrical Resistivity, and Hemispherical Total Emittance of Niobium-l wtSg Zirconium Alloy in the Range 1500 to 2700 K by a Transient (Subsecond) Teehnique, National Bureau OFStandards, AFOSR-TR-73-1462 (1972). CT. Ewing, J.P. Stone, J.R. Spann, E.W. Steinkuller, D.D. Williams and R.R. Miller, High-Temperature Properties of Potassium, Naval Research Laboratory, NRL-6233 (1965). 114 C.R. Fisher and P.Y. Achener, Alkali Metals Evaluation Program: Quaterly Progress Report, October l-December 31, Aerojet-General Nucleon&, AGN-8131 (1965).
273
1141 CC. Bigelow, H.C. Gray, D.E. Robinson, G.H. Rowe and M.E. Shank, Discussion of Columbium-Zirconium Alloy Properties Relative to the PWAR-1lC Reactor Structural Design, Pratt & Whitney Aircraft-CANEL, CNLM-2487 (1960).
WI R.H. Bums, F.S. Shuker, Jr. and P.E. Manning, in: Industrial Applications, Ed. R.E. Smallwood (ASTM, Philadelphia, 1984) p. 50. [I61 J.B. Conway and P.N. Flagella, Creep-Rupture Data for the Refractory Metals to High Temperatures, (Gordon and Breach, New York, 1971). 1171 D. Peckner, Mater. Des. Eng. 54 (1963) 107. Creep I181 R.H. Titran and R.W. Hall, Ultra~~-Vacuum Behavior of Columbium and Tantalum Alloys of 2000 and 2200 ’ F for Time Greater than 1000 Hours, NASA-Lewis Research Center, NASA-TND-3222 (1966). 1191 D.O. Hobson, Aging Phenomena in Columbium-Base Alloys, Oak Ridge National Laboratory, ORNL-3245 (1962). WI J.H. Noggle, Physical Chemistry (Little, Brown and Company, Boston, 1985). 1211 Hj. Matzke, Science of Advanced LMFBR Fuels (NorthHolland, Amsterdam, 1986). WI C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966). 1231 C. Chou, D. White and H.L. Johnson, Phys. Rev. 109 (1958) 788. 1241J.P. Moore, R.S. Graves and R.K. Williams, High Temp.High Press. 12 (1980) 579. 1251 I.B. Fieldhouse, J.C. Hedge and J.I. Lang, Measurements of Thermal Properties, Wright Air Development Center, WADC-TR-58-274 (1958). 1261 F.M. Jaeger and W.A. Veenstra, Rec. Trav. Chim. 53 (1934) 677. 1271 G.C. Lowenthal, Austr. J. Phys. 16 (1963) 47. [28] J.B. Conway and R.A. Hein, in: Proc. 3rd ASME Symposium on Advances in Thermophysical Properties at Extreme Temperatures and Pressures (ASME, Lafayette, IN, 1965). [29) N. Tsoulfa~dis, Measurement and Detection of Radiation (Hemisphere, W~~ngton, 1983). 1301 B.M. Vasyutinskiy et al., Phys. Met. Metallogr. 21 (1966) 134. 1311J.B. Conway and A.C, Losekamp, Trans. Met. Sot. AIME 236 (1966) 702. [32] V.M. Amonenko, P.N. Vyugov and V.S. Gumenyuk, High Temp. (USSR) 2 (1964) 22. [33] J.W. Edwards. R. Speiser and H.L. Johnston, J. Appl. Phys. 22 (1951) 424. (341 P.M. Johnson, R.L. Lincoln and E.R. McClure, Development of a High-Temperature Interferometric Dilatometer Using a Laser Light Source, US Bureau of Mines, BMRI7142 (1968). 1351 J.B. Conway,
R.M. Fin& and A.C. Losekamp, Trans. Met. Sot. AIME 233 (1965) 844. [36] C.R. Tottle, J. Inst. Met. 85 (1957) 375. [37] T.J. Heal, in: Proc. 2nd UN Int. Conf. on Peaceful Uses of Atomic Energy (AGARD, Geneva, 1958).
Materials
261
173 (1990) 261-273
Thermophysical property correlations for the niobium-1 % zirconium alloy David J. Senor, J. Kelly Thomas * and K.L. Peddicord Advanced Nuclear Fuels Laboratory, Department of Nuclear Engineering, Texas A & M University, College Station, TX 77843-3133, USA Received
17 October
1989; accepted
14 May 1990
Correlations were deveioped for the temperature dependence of specific heat (c,), enthalpy (!I), entropy (s), Gibbs free energy (G), free energy function (FEF), mean coefficient of linear thermal expansion (a), density (p) and lattice parameter (u) of niobium-l% zirconium. The specific heat and thermal expansion correlations were developed as empirical fits to experimental data. The enthalpy, entropy and free energy correlations were derived from the specific heat correlation, and the density and lattice parameter correlations were derived from the thermal expansion correlation. 2.784 x 10’ ~~=0.2441+5.105~10-~(T)+
2.295~10~
kJ/kg K,
T
T2
h(T)-h(298K)=-75.01+0.2441(T)+2.553x10~s(T)2+1.213x105exp
-
2.295 x lo4 T
i s = 0.2441 ln( T) + 5.105 X lo-s(
T) -0.9947
kJ/kg
K,
G=1.2388(T)-2.552~10~~(T)~-0.2Q41(T)ln(T)+1.213xlO~exp
kJ/kg. 75.01
FEF== -1.2388+0.2441
p = 8637 -0.200(T) a=3.293+2.622~10-‘(T)
kJ/kg,
ln(T)+2.552X10-5(T)+-
1.213~10~
-
T
2.295 x lo4
T
kJ/kg K,
kg/m3, ,&.
The specific heat correlation has a relative standard deviation of 0.98% and is valid over the temperature range 298 to 2700 K. The enthalpy, entropy and free energy correlations have the same temperature range of validity as the specific heat correlation. Statistical methods were employed to determine the propagation of error from the specific heat correlation to estimate the un~rtainty associated with each of the derived correlations. The relative standard deviation of the thermal expansion correlation is 0.96% and it has a valid temprature range of 575 to 1627 K. The density and lattice parameter correlations are valid over the temperature range 297 to 1627 K. The propagation of error introduced by the derivation of the density and lattice parameter correlations was calculated in the same manner as for the derived thermodynamic correlations.
1. Introduction The niobium-l% zirconium (Nb-1Zr) rently under consideration for application
alloy is curas a cladding
* Current Address: Westinghouse Savannah River, Bldg 77341A, Room 171, Aiken, SC 29808, USA. 0022-3115/90/$03.50 0 1990 - Elsevier Science Publishers
and structural material for the SP-100 space reactor system. Although several property reviews have been conducted for Nb-1Zr and elemental niobium [l-lo], no comprehensive effort has been made to construct correlations suitable for application in fuel element modeling. A consistent and well-v~idat~ set of correlations for the relevant properties is necessary in order to
B.V. worth-HoIland)
262
D.J. Senor et al. / Thermophysical propeq
accurately model fuel element performance. The correlations should be valid for the entire temperature range of interest, including start-up and transient overpower scenarios. The temperatures encompassed by such considerations range from 300 K to the melting point of the alloy (approximately 2700 K). To model the cladding as closely as possible, only data for Nb-1Zr were used in the construction of the correlations. Niobium data were employed only as a means of validation to ensure the reliability of the Nb-1Zr data. The thermodynamic properties of fuel element cladding materials are of primary importance during reactor transients. The behavior of the cladding specific heat is of interest under these conditions and a description of the temperature dependence of enthalpy is necessary to perform energy balances between the fuel, cladding and coolant. The cladding entropy and Gibbs free energy are of interest primarily at high temperatures in an accident scenario when chemical reactions between the various components of the reactor system are possible. To determine the strains introduced by changes in cladding temperature, knowledge of the thermal expansion behavior is required. The variation of cladding density with temperature is important for the modeling of many steady-state and transient scenarios and knowledge of the temperature dependence of the lattice parameter aids in modeling the migration of atomic species through the cladding.
2. Experimental data
A literature review revealed only one data set for the specific heat of Nb-1Zr [ll]. The measurements were made using a transient technique which involved passing a large current through a hollow cylindrical specimen in short pulses and measuring the temperature rise of a water jacket. The length and magnitude of the pulse were adjusted to yield the desired specimen temperature which was measured using optical pyrometry. Test temperatures ranged from 1500 to 2700 K. The nominal zirconium content of the specimen was 1.05% by weight. Total impu~ties amounted to less than 0.17%, with tantalum comprising the majority at 0.09%. Micrographs of the specimen indicated that considerable grain growth occurred as a result of the pulse heating. The data assume a sawtooth pattern when plotted as a function of temperature, but this does not appear to have been the result of any physical phenomenon.
~orr~luhons for Nh- I 9; %r crll~
.?.L Lineur
thermul
expi~rwron
Two data sets were available in the literature for the thermal expansion of the Nb-1Zr alloy [12,13] as a function of temperature. Ewing et al. [12] tested asfrabricated materials in three separate tests with a base temperature of 297 K. Test temperatures ranged from 57.5 to 1627 K and all specimens had less than 0.20% impurities. Fisher and Achener [13] tested two specimens in an argon atmosphere. The specimens were cut at right angles to each other and represent cross sections parallel and perpendicular to the rolling direction. The amount of cold work applied to the specimens was not reported. The base temperature was 297 K. and test temperatures ranged from 465 to 1305 K. No microanalysis was reported. Bigelow et al. [74] report a single data point at a temperature of 1366 K. 2.3. Density No information was available in the literature concerning the temperature dependence of density for the Nb-1Zr alloy. Several different values for the room temperature density were reported which range from 8400 to 8660 kg/m3, but the most commonly referenced value was 8570 kg/m3 [4,15-181. Since this was the most frequently reported value and represents a median for the entire range of reported values, it was used as the reference room temperature density in the development of the density correlation. 2.4. Latlice parameter Hobson [19] measured the lattice parameter of NbIZr as a function of oxygen content at 1873 K. He reports a lattice parameter of approximately 3.302 A at 1873 K for very low oxygen impurity levels. No additional microanalysis was provided.
3. Existing correlations 3.1. Specific heat Cezairliyan constructed a third-order polynomial fit to his own specific. heat data [ll]. Although the correlation accurately follows the trend of the experimental data, it becomes unstable below 1500 K. This behavior is inconsistent with the trend of the data and is the result of employing a third-order polynomial beyond the range of data on which it was based. The primary temperatures of interest for space reactor cladding
263
D.J. Senor et al. / Thermophysicalproperty correlationsfor Nb-I % Zr alloy
material are approximately 1000 to 1500 K and the unstable nature of the Cezairhyan correlation in this range renders it unsuitable for fuel element modeling applications. For this reason, a new correlation based on the Cezairliyan data was needed that would provide stability in the temperature regime below 1.500 K. 3.2. Linear thermal expansion A correlation for the linear thermal expansion of Nb-1Zr was developed by the Thermophysical Properties Research Center (TPRC), and reported by Toulaukian and Ho [lo]. It was a second-order poly nomial in terms of AL/L, but was converted for use in this work to the mean coefficient of linear thermal expansion. They report the correlation to be applicable for annealed material, but the database employed in its formation was not explicitly defined. Touloukian and Ho report the data of both Ewing et al. [12] and Fisher and Achener [13], but the trend of the correlation does not follow that of either of the data sets and for this reason a new fit to the data was deemed necessary.
where R = universal gas constant (8.314 J/g mol K), 8 = Einstein temperature, T = absolute temperature. The specific heat at constant pressure is related to that at constant volume by the Nemst-Lindemann equation which adds a linear term to eq. (1). The resulting expression for c,, is given by [21]:
a@’expfQ/T) f ” = T*(exp(B/T)
bT
’
where a and b are empirical fit coefficients. The first term of eq. (2) describes the low temperature behavior of the material, while the linear term dominates the temperature regime above room temperature. For pure elements, the leading coefficient (a) theoretically assumes a values of 3R as indicated in eq. (1). An additional term to account for defect formation processes at extremely high temperatures can be added to eq. (2). The complete expression for specific heat of a solid over all temperatures is then given by [22]: a@
exp(@/T)
” = T’(exp( 0/T) 4. Correlation development
- 1)2
+bT+-
cEn RT2
exp
(3)
4.1. Specific heat
where = activation energy for defect formation (J/gmol K), = universal gas contant (8.314 J/g mol K), R T = absolute temperature, a, b, c = empirical coefficients. A more rigorous model of heat capacity is that proposed by Debye ]22] which allows for interaction between the atoms in a lattice. This theory describes the vibration of the entire lattice as opposed to only the uncoupled vibrations of the individual atoms as described by the Einstein theory. However, the approximation made by Einstein only has a noticeable effect at cryogenic temperatures, so for fuel element modeling applications, the Einstein theory is acceptable. Additionally, the Debye model is much more complex and requires much greater computational effort than the Einstein model. For these reasons, the Einstein model with the additional defect formation term as given in eq. (3) was used to develop the Nb-1Zr specific heat correlation. Since experimental specific heat data for Nb-1Zr were only available over the temperature range 1500 to 2700 K, no information about the low temperature specific heat behavior of the alloy was available. For this reason, the first term in eq. (3) was replaced by a E,
A variety of fits to the Cezairliyan [ll] data were studied including second- and third-order polyno~~, power law and exponential forms. None modeled the data well or provided adequate predictive capability over all temperatures of interest. Due to the lack of experimental data below 1500 K, a physically-based fit form was required to ensure stability at the lower temperatures. The specific heat of a solid at constant volume from low to moderate temperatures is approximated by the Einstein theory of specific heat. The explicit assumption involved is that the material Iattice is composed of N atoms which are free to vibrate harmonically in three dimensions about their lattice site. Since each atom is assumed to vibrate independentiy of all others in the lattice, the effective result is that of 3N harmonic oscillators. Based on these assumptions, the interna energy of 3N oscillators may be derived from statistical mechanics. By differentiating the internal energy with respect to temperature at constant volume, an expression for the specific heat is obtained [20]: 3R@12exp( 8/T) cv = T’(exp(B/T)
- 1)”
- 1)2 ’
(1)
264
D.J. Senor et al. / Thermaphysral
0.15”
200
400
Km
ml
lcoo
proper?v correhtions Ior Nb-- i S Zr u&v
.’ 1200
-‘. 1400
1600
‘*
’
1800
2ooO
3
“I. 2200
2400
’ * 2600 2800
Temperature (K) Fig. 1. Nb-IZr specific heat experimental data and correlations.
constant independent of temperature in the development of the correlation. The linear and high temperature terms were retained and the defect activation energy was implemented as an empirical coefficient. The final form of the correlation is given by:
correlation for the relative specific enthalpy of Nb-1Zr is given by: h(T)
-
h(298 K)
= -75.01 i 0.2441r+
CP= 0.2441+ 5.105 x 10-Y -t 2.784 x 10’ 7-z
2.553 x 10-5T2 2.295 x lo4
2.295 x lo4 T
T
kJ,‘kg K,
where T is the temperature in Kelvin. The relative standard deviation d the fit with respect to the Cezairliyan data over the temperature range 1500 to 2700 K is 0.98%, which is the same as the Cezairliyan [II] correlation. The correlation is valid from room temperature to the melting point of the alloy (298 to 2700 K). The expe~menta~ data are shown in fig, I along with the Cezairliyan correlation and the curve represented by eq. (4). 4.2. Enthafpy The enthalpy correlation was developed by integrating the specific heat correlation given in eq. (4) from the base temperature to an arbitrary upper limit temperature. The base temperature was selected to be room temperature (298 K) since most energy balance calculations require relative enthalpy values. The resulting
kJ/b
(5)
where T is the temperature in Kelvin. Since eq. (5) was derived from the specific heat correlation, it has the same temperature range of validity (298 to 2700 K). At low temperatures, the exponential term in eq. (5) yields values several orders of magnitude less than those given by the other terms and is thus negligible at these temperatures. At high temperatures, the exponential term represents approximately 3% of the total value and cannot be neglected. In some instances. it is necessary to perform calculations using an absolute enthalpy value in which the enthalpy is defined to be zero at 0 K. Absolute enthalpy predictions may be obtained from eq. (5) by neglecting the constant term. The relative enthaipy correlation given by eq. (6) is shown as a function of temperature in fig. 2. 4.3. Entropy The entropy correlation was derived from the specific heat correlation using the definition of absolute entropy
D.J. Senor et al. / T~er~op~~s~cal property cor~ei~tions for Nb- 1 X 2r alloy
265
900
800 700
200
400
600
800
loo0
1200
1400
1600
1800
2ooO
2200
2400
2600
2800
Temperature (K) Fig. 2. Nb-1Zr relative specific enthalpy correlation.
1201: s(T)
=s,t
TCp(V jTo--?--_ dT',
(6)
where se is the entropy at the reference temperature T,. The se term may be set equal to zero if the base temperature is taken to be 0 K. However, the function for specific heat must be valid over the entire integral. Since the specific heat correlation developed in this work is not valid at low temperatures, the reference temperature was taken to be 298 K. No entropy or low temperature specific heat data were available in the literature for Nb-1Zr. It was therefore necessary to use niobium specific heat data to calculate a room temperature entropy for use in the development of the Nb-1Zr entropy correlation. Niobium specific heat data from 10 to 300 K 1231 were fit using eq. (2). The resulting correlation yielded a leading coefficient (a) of 0.2431 kJ/kg K and an Einstein temperature (0) of 175 K. As mentioned previously, for pure elements the leading coefficient should approach a value of 3R. For niobium, this corresponds to 0.2685 kJ/kg K. Values of the niobium Einstein temperature reported in the literature range from 188 to 210 K [24]. For both the leading coefficient and the Einstein temperature, the empirically determined values are in reasonably good agreement with the expected values. The specific heat at room temperature for niobium predicted by the fit to the Chou et al. 1231 data is 0.2717 kJ/kg K; the specific heat of Nb-1Zr given by eq. (4) is 0.2593 kJ/kg K, a
difference of only 4.6%. This is remarkably good agreement considering the Nb-1Zr correlation was developed with only high temperature data (1500 to 2700 K). For these reasons, the fit to the low temperature niobium data of 1231 was deemed suitable to provide a room temperature entropy for the development of a Nb-1Zr entropy correlation. The resulting low temperature niobium specific heat fit was integrated from 10 to 298 K to calculate a room temperature entropy for niobium as indicated in eq. (6). The lower limit of 10 K was selected for two reasons. First, the value of the integrand approaches infinity as temperature decreases, so to avoid any nonphysical nume~ca~ effects the integration was performed at a temperature above which the integrand becomes unstable. Second, any effects associated with superconducting niobium were avoided since the critical temperature of elemental niobium is appro~mately 9.5 K [22]. The room temperature entropy of niobium was calculated to be 0.4115 kJ/kg K. This value was used as se in eq. (6) along with the Nb-1Zr specific heat correlation given by eq. (4). The integraf was evaluated with a base temperature of 298 K to yield a correlation for Nb-1Zr entropy as a function of temperature. The resulting expression is given by: s(r)
= 0.2441 In(T)
+ 5.105 X 10P5(T)
- 0.9947 kJ,‘kg K,
(7) where T is the temperature in Kelvin. An additional exponential term results after performing the integra-
500
loo0
1500
2mo
2500
3000
Temperature (K) Fig. 3. Nb-1Zr absolute specific entropy correlation.
tion, but cont~butes less than one percent to the value of s(T) at all temperatures. It has therefore been neglected in eq. (7). The entropy correlation is shown as a function of temperat~e in fig. 3. The co~ela~on is valid only over the temperature range 298 to 2700 K since the specific heat correlation is valid only over this range.
4.4. Free energy Gibbs free energy (c) and the free energy function (FEF) may be derived from enthalpy and entropy. The definition of Gibbs free energy is given by [2Oj: G-h-Ts,
Fig. 4. Nb-1Zr Gibbs free energy correlation.
(8)
267
D.J. Senor et al. / Thermophysical property correlations for Nb- I % Zr alloy where G = specific Gibbs free energy, h = absolute specific enthalpy, T = absolute temperature, s = absolute specific entropy. In eq. (8), both enthalpy and entropy are defined to be zero at 0 K. Manipulation of this equation results in the following relationship defining the free energy function: FEF=
-
G-h(298 T
K)
=s-
h(T)
- h(298 K) T ’ (9)
where FEF is the free energy function. Both free energy quantities are useful in chemical reaction calculations. To derive the correlation for Gibbs free energy, eq. (5) without the constant term and eq. (7) were inserted into eq. (8). The resulting expression is given by: G(T)
= 1.2388(T) - 0.2441(T)
- 2.552 x 10-5(T)2 ln( T) 2.295 x lo4 T
kJ/kg,
(10) where T is the temperature in Kelvin. To develop the free energy function correlation for Nb-lZr, eqs. (5) and (7) were inserted into eq. (9). The
500
1000
resulting
expression
is given by:
FEF = - 1.2388 + 0.2441 ln( T) 75 .Ol + 2.552 x 10-5(T) + 7 _ 1.213 x 10s exp - 2.295 x lo4 T T i
kJ/kg
K (11)
where T is the temperature in Kelvin. Both the Gibbs free energy and free energy function correlations are valid over the temperature range 298 to 2700 K and are shown in Figs. 4 and 5, respectively. 4.5. Linear thermal expansion The two data sets available in the literature for the thermal expansion of Nb-1Zr exhibit completely different trends. The Fisher and Achener [13] data set is concave down, while the Ewing et al. (121 data exhibit a concave upward trend. Although both experiments were performed as a component of a larger project, much greater emphasis on detail was given to the Ewing et al. data. For example, no specimen characterization was reported for the Fisher and Achener tests. The Ewing et al. data are a compilation of three separate tests and the results are very consistent. Two of the tests were performed with quartz measuring rods and one with a sapphire measuring rod. Additionally, Ewing et al. report that some of their previous thermal expansion test results on Nb-1Zr were rejected because they were in
1500
2m
Temperature (K) Fig. 5. Nb-1Zr free energy function correlation.
2500
3ooo
268
D.J. Senor et al. / Thermophysical
property
correlatrons
for Nh- I B %r ~IIO,,
8.5
7.0 -
*
Ewing et al. (1965)
q
Fisher and Achener
(1965)
*-_- TPRC Correlation 0 6.5 400
600
800
IOCQ
1200
Temperature Fig. 6. Nb-1Zr
mean coefficient
of linear thermal
error; this again supports Ewing et al. since they were able to detect errors in and reject portions of their data. The data point from Bigelow et al. [14] was not considered for inclusion in the correlation since it appears to be an evaluation of the TPRC correlation at 1366 K. Based on these criteria, only the Ewing et al. data were employed to form the correlation. A variety of fit forms were examined, but a secondorder polynomial was selected as the best fit to the Ewing et al. data. Since the data only extend from 575 to 1627 K, there was some concern about the stability of the polynomial outside this range. However, the minimum of the polynomiat occurs far below the lowest temperature of interest and at high temperatures the curve is also well behaved. The correlation for the mean coefficient of linear thermal expansion for Nb-1Zr is given by: ty, = 6.62 + 3.64 x W4( + 2.75 x 10-‘(T)’
7-) 10-“/K,
(12)
where T is the temperature in Kelvin. The relative standard deviation of the correlation is 0.96% with respect to the Ewing et al. [12] data. Fig. 6 shows all the available data along with the TPRC correlation and the curve produced by eq. (12).
1400
1600
1800
(K)
expansion
experimental
data and correlations.
expansion of the alloy and its density at room temperature. The density correlation was derived from the definition of the mean coefficient of linear thermal expansion. The explicit assumptions involved were an initial perfectly cubic unit cell and isotropic expansion with temperature. The base temperature was taken to be the same as that of the data used in the formation of the thermal expansion correlation (297 K). The density at this temperature was taken to be 8570 kg/m3 as discussed previously. After simplifying the expression for the density at any temperature, the following equation results:
P(7’) = (1 +
a;;!297))’ wm3+
(13)
where T is the temperature in Kelvin and a is the thermal expansion correlation given by eq. (12). Although eq. (13) is not extremely complicated, it does require computation of the thermal expansion correlation To facilitate use in fuel modeling applications a simpler expression was developed by fitting a series of values generated by eq. (13). Of the fit forms examined. a linear correlation provided the best agreement with the trend of eq. (13), provided a simple fit form, and guaranteed stability over the valid temperature range. The resulting Nb-1Zr density correlation is given by:
4.6. Density
p = 8637 - 0.200( 7’) kg/m3,
A relationship for the temperature dependence of density for Nb-1Zr was derived based on the thermal
where T is the temperature in Kelvin. Since the density correlation employs an experimental room temperature density, the lower limit of validity is 297 K. However.
(14)
269
D. J. Senor et al. / Thermophysical property correlations for Nb- I 5%Zr alloy
800
loo0
1200
Temperature (K) Fig. 7. Nb-1Zr
density
the thermal expansion correlation is valid only to 1627 K, this upper limit applies to the density correlation as well. Fig. 7 shows the density correlation as a function of temperature over this range.
since
4.7. Lattice parameter The available data for the lattice parameter of Nb1Zr do not offer any insight into the temperature dependence of the property. For this reason, a correlation
correlation.
for the lattice parameter of Nb-1Zr was derived from the density correlation. The correlation was derived following the same assumptions made for the derivation of the density correlation. In a manner similar to that used for the density correlation, a simple fit was applied to the resulting expression to yield a more manageable form. The Nb-1Zr lattice parameter correlation is given by: a = 3.293 + 2.622 X 10M5(T)
A,
(15)
3.3x
3.325
3.275
3.250 400
600
800
Ical
1200
1400
1600
Temperature (K) Fig. 8. Nb-1Zr lattice parameter experimental data and correlation.
1800
2ooo
270
D.J. Senor et al.
property correlations for N&l
/ Thermophysical
where T is the temperature in Kelvin. Eq. (12) has the same temperature range of validity as the density correlation (297 to 1627 K). Fig. 8 shows the lattice parameter correlation as a function of temperature along with the low oxygen content Hobson [19] data point at 1873 K for comparison.
5. Correlation validation 5. I. Specific heut Comparison of the Nb-1Zr correlation to elemental niobium data for specific heat provides evidence that the trend of the correlation developed in this work below 1500 K is reasonable. The niobium data [25-281 are shown with the Nb-1Zr correlation in fig. 9. The niobium data exhibit a linear trend below approximately 1800 K. Since little difference would be expected in the specific heat behavior of niobium and Nb-lZr, the similarity between the correlation and the niobium specific heat data is encouraging. The Nb-1Zr specific heat correlation may therefore be used with confidence from room temperature to 2700 K. 5.2. Enthalpy Since no experimental enthalpy data were available, it was necessary to determine the error propagated by deriving the enthalpy correlation from the specific heat
%, Zr crll~v
correlation. To determine the propagation of error involved in deriving a correlation, the following expression may be used [29]:
(16)
cl,. =
where u,, = absolute standard deviation of derived correlation, .r = dependent variable, /, = independent variables (total of n). 0, = absolute standard deviation of independent variables. For the enthalpy correlation, the only independent variable involved was the specific heat correlation itself. By inserting the appropriate values and assuming that the specific heat correlation varies slowly over its valid temperature range, it may be shown that the relative standard deviation of the enthalpy correlation is approximately the same as that for the specific heat correlation (0.98%). 5.3. Entropy The uncertainty associated with the derivation of the entropy correlation from the specific heat correlation was determined in a fashion similar to that described for enthalpy. For entropy, however, there were two independent variables (sa and cp( T)) which had the potential to propagate error into the derived correlation,
L
0.4 s g 8 0.3 -
0
400
800
1200
IMX)
2cQo
24co
2800
Temperature (K) Fig. 9. Niobium
specific heat experimental
data and Nb-1Zr
correlation
developed
in this work.
211
D.J. Senor et al. / Thermophysical property correlations for Nb-I % Zr alroy
[32,34,35] clearly exhibit a concave upward trend and one [30] displays the opposite behavior. The other data sets either exhibit no clear trend or are fairly linear. The general trend and magnitude of the data agree reasonably well with the Nb-1Zr correlation developed in this work. This behavior provides support for the selection of the Ewing et al. [12] data as the sole basis for construction of the correlation. The Nb-1Zr thermal expansion correlation may be used with confidence over the temperature range 575 to 1627 K. However, the comparison to the niobium data indicates that the correlation should provide reasonable predictions over all temperatures of interest in space reactor fuel element modeling.
After evaluating eq. (16) with the appropriate values and converting to relative standard deviation, the error associated with the entropy correlation was calculated to be approximately 1.70%. The bulk of the increase in uncertainty over the specific heat correlation was introduced by the error associated with the calculation of the room temperature entropy from the fit to the low temperature niobium data. 5.4. Free energy The errors associated with the Gibbs free energy and free energy function correlations were determined using eq. (16). In both cases, there were two independent variables associated with the derivation of the correlations. Gibbs free energy is dependent on absolute enthalpy and entropy and the free energy function is dependent on relative enthalpy and absolute entropy. For Gibbs free energy, the relative standard deviation was calculated to be 1.70% essentially the same as the uncertainty for the entropy correlation. The corresponding relative standard deviation of the free energy function correlation was calculated to be 2.20%.
5.6. Density The error associated with the density correlation was calculated using eq. (16). The independent variables were taken to be the room temperature density and the thermal expansion correlation. However, since no reasonable estimate of the error in the room temperature density was available, any propagation of error due to this term was neglected. After evaluating the remaining terms and converting to relative standard deviation, the error associated with the thermal expansion correlation was shown to be on the order of lo-*%, indicating that virtually all the error associated with the density correlation is due to the accuracy of the room temperature density incorporated in the fit. This is a consequence of
5.5. Linear thermal expansion A comparison was made between the predictions offered by the Nb-1Zr thermal expansion correlation and thermal expansion data for niobium [30-371 as shown in fig. 10. Of the niobium data sets, three
10
9
8 Edwards et al. (1951) Tottle (1957) Heal (1958) Amonenko et al. (1964) Conway et al. (1965) Conway and Losekamp (1966) Vasyutinskiy et al. (1966) Johnson et al. (1968) 4’ 0
I
I
500
1000
I 1500
I . 2000
This Work 2500
3@30
Temperature (K) Fig. 10. Niobium thermal expansion experimental data and Nb-1Zr correlation developed in this work
272
D.J. Senar ef al. /
Therm~physicaf propertycorrelations for N/w/ %i.Zr alloy
the fact that a describes the expansion of the unit cell. The expansion (&AT) of a unit cell is on the order of 0.1% for AT on the order of 103. This represents a very small change in the cell dimensions compared to its initial size, and the error in this change is negligible compared to the error in the base dimensions. The reliability of the room temperature density employed in the density correlation was judged to be high since it was the most often quoted of the several values. Thus, the Nb-1Zr density correlation may be used with a reasonable degree of confidence within its valid temperature range. 5.7. Lattice parameler The lattice parameter measured by Hobson 1191 at 1873 K for Nb-1Zr with very low, levels of oxygen impurities is approximately 3.302 A, a difference of only 1.21% from the value predicted by the Nb-1Zr lattice parameter correlation. It should be emphasized, however, that the thermal expansion, and thus the density and lattice parameter correlations are strictly valid to only 1627 K. All three correlations have stable forms, however, and should predict reasonable values outside the validated temperature range. Thus, the agreement between the Hobson data point and the correlation prediction is en~o~a~ng. By inserting the appropriate values into eq. (16), it can be shown that the relative standard deviation of the lattice parameter correlation is approximately the same as the density correlation from which it was derived. There is little propagation of error in the derivation, and the lattice parameter correlation may be employed with the same level of confidence as the density correlation.
6. Summary Comparisons with niobium data indicate that the Nb-1Zr correlation developed in this work offers reasonable predictions for the specific heat of Nb-1Zr. The correlation is based on a physical fit form suggested by the Einstein theory of specific heat and is stable over the entire temperature range of interest (298 to 2700 IQ. The relative standard deviation of the correlation with respect to the experimental data is 0.98%. The Nb-1Zr enthalpy correlation was derived based on the Nb-1Zr specific heat correlation. Since the specific heat correlation has a physical fit form with ernpi~~ly determined coefficients, the enthalpy corre-
lation also has a physical basis, lending to its credibility despite the lack of experimental data to verify its predictive capability. The enthalpy correlation is valid over the temperature range 298 to 2700 K and has an approximate relative standard deviation of 1.0% based on the propagation of errors introduced in its derivation. A correlation for the entropy of Nb-1Zr was derived from the Nb-1Zr specific heat correlation. The entropy correlation is stable and yield predictions consistent with physical expectations over all temperatures of interest (298 to 2700 K). A room temperature entropy was calculated based on a fit to low temperature niobium data using the Einstein formulation. The relative standard deviation of the Nb-1Zr entropy correlation was calculated to be appro~mately 1.70%. based on the propagation of errors from the Nb-1Zr specific heat correlation and the fit to the low temperature niobium data used in its formation. Correlations for Gibbs free energy and the free energy function of Nb-1Zr were developed based on the Nb-1Zr enthalpy and entropy correlations. The correlations are stable and yield predictions which agree with physical expectations over all temperatures of interest. The free energy correlations are both valid over the temperature range 298 to 2700 K. The estimated relative standard deviations of the fits are 1.70% for Gibbs free energy and 2.20% for the free energy function. The behavior of the Nb-IZr thermal expansion correlation agrees with physical expectations and is a good empirical fit to the most reliable data with a relative standard deviation of 0.96%. The correlation is valid over the temperature range 575 to 1627 K. However, the correlation is numerically stable and yields reasonable predictions over the entire range of temperatures for space nuclear reactor applications as indicated by a comparison with niobium data at high temperatures. The accuracy of the Nb-1Zr density correlation is dependent on two factors, the standard deviation of the Nb-IZr thermal expansion correlation and the error associated with the romm temperature density value. The dominant factor in the error of the density correlation is that associated with the room temperature density, for which no estimate could be made from the availabie data. However, the level of reliability associated with the room temperature density was high, so the density correlation may be used with a reasonable degree of confidence over the temperature range 297 to 1627 K. A correlation for the lattice parameter of Nb-1Zr was derived from the Nb-1Zr density correlation. The primary component of the error associated with the lattice parameter correlation is that due to the room
D.J. Senor et al. / Therma~hys~cal property eorreiat~~~s for Nb- I % Zr alloy
temperature density value used in the formation of the density correfation. The correiation is strictly valid over 297-1627 K, but is stable and should offer reasonable predictions outside this range of temperatures based on its good agreement with a data point at 1873 K.
Acknowledgements Research sponsored by the US Department of Energy under contract DE-ACO2-86NE3796~.A~ and by the University Programs Division of the Oak Ridge Associated Universities under the Nuclear Engineering and Health Physics Fellowship Program.
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