A new correlation for the specific heat of metals, metal oxides and metal fluorides as a function of temperature

S. I. Abu-Eishah¹, Y. Haddad, A. Solieman, and A. Bajbouj^{2 1} Department of Chemical & Petroleum Engineering, UAEU, Al-Ain, UAE, siam20022001@yahoo.com
² Chemical Engineering Department, Jordan University of Science & Technology, Irbid 22110, Jordan

Abstract ¾ The objective of this work is to find a suitable correlation that best fits the specific heat of metals, metal oxides and metal fluorides as a function of temperature. It was found that a multilinear regression model of the form C_P = aT^be^cTe^d/T has the lowest deviation from experimental data compared to other correlations including a 4^th to 6^th-order polynomial regression model. The coefficient of determination, R², was very close to unity in most cases and the average of the absolute relative errors, AARE, was less than 5% for the specific heat of most of the systems studied. The overall AARE was about 1.8% for metals and 3% for metal oxides and metal fluorides, which is within the experimental error.

Key Words ¾ Correlations. Metals. Metal Fluorides. Metal Oxides. Specific Heat.

I. Introduction

Materials are diverse in our life and have many uses. Many applications of metals, ceramics, fluxing materials and composites are based upon their unique thermophysical properties. Specific heat, thermal conductivity and thermal expansion are the properties that are often critical in the practical utilization of solids as materials of construction (Abu-Eishah, 2001a). These properties depend upon the state, chemical composition, and physical structure of that material. They also depend on temperature and to a lesser extent on pressure, to which the material is subjected. In the design of rocket-engine thrust chambers, for example, considerable attention must be given to the effect of temperature on the thermophysical properties of its structure. The primary concern of engineers is to match the material properties to service requirements of the component, knowing the conditions of load and environment under which the component must operate. Engineers must then select an appropriate material, using tabulated test data, as the primary guide (Abu-Eishah, 2001b).
Specific heat is the property that is indicative of materials ability to absorb heat from external surroundings. The specific heat of a material is largely determined by the effect of temperature on the vibration and rotational energies of the atoms within the material, the change in energy level of electrons in the structure of the material, and changes in atomic positions during formation of lattice defects (vacancies and interstitials), order-disorder transitions, magnetic orientation or polymorphic transformations (Richerson, 1992).
In a previous work, Abu-Eishah (2001a) proposed a multilinear correlation to fit the thermal conductivity of metals as a function of temperature and found it among the best. In this study it is intended to check the suitability of such a multilinear correlation to fit the specific heat of metals, metal oxides and metal fluorides as a function of temperature.

II. Proposed Fitting Equations

The theories of the specific heat of metallic and non-metallic solids are covered in detail by Touloukian and Ho (1972a,b). The theoretical equation that represents the specific heat, in general, is given in Touloukian and Ho (1972a) as

C_P = aT + bT ³ + c/T ² (1)

The terms on the right-hand side of Eq. (1) belong to the electronic, lattice combination, and nuclear combination parts of the specific heat. Up to the knowledge of the authors, Eq. (1) was not used as is to fit the specific heat experimental data. Perry and Green (1997) give C_P for pure compounds (metallic and non-metallic solids) as a linear equation of the form (C_P = a + bT) for some compounds and by a nonlinear equation of the form (C_P = a + bT + c/T ²) for others. The temperature range (starting at 273 K), the values of the coefficients a, b and c, and the uncertainty (%) of these correlations are also given in Perry and Green (1997) and summarized in Appendix. In this work, a wider and more comprehensive temperature ranges were covered compared to those used in Perry and Green (1997).
The multilinear fitting equation proposed in this study has the form

C_P = aT^b e^cT e^d/T (2)

If the exponential terms in Eq. (2) are expanded by a Taylor's series, then we get

C_P = aT ^b[A + BT + CT ² + ... + D/T + E/T ² + ...] (3)

which can be rewritten as

C_P = A^'T ^b + B^'T ^b+1 + C^'T ^b+2 + ... + D^'/T ^b-1 + E^'/T ^b-2 + ... (4)

where, A^' = aA, B^' = aB, etc. By comparing Eqs. (1) and (4), all terms in Eq. (1) are almost there in Eq. (4), but Eq. (4) has extra merits; it is more than just a polynomial, a power or an exponential function, it is a combination of all of these functions. It can have positive and negative exponents, with integer and non-integer values. Also, while Eq. (1) may predict negative specific heat at low temperatures, the parameter a in Eq. (2) is always nonnegative, which is needed for presenting always positive thermophysical data such as specific heat. Taking the logarithm of both sides of Eq. (2) gives

ln C_P = ln a + b lnT + cT + d/T (5)

T in Kelvin and a, b, c and d are fitting constants. That is, Eq. (5) can be rewritten as

Y = a₁ X₁ + a₂ X₂+ a₃ X₃ + a₄ X₄ (6)

where a₁ = ln a, a₂ = b, a₃ = c, a₄ = d, and Y = ln C_P, X₁ = 1.0, X₂ = ln T, X₃ = T, and X₄ = 1/T. That is, all terms in Eq. (6) are linear in X_i, i = 1, 2, 3, and 4; from which the multilinear name of the proposed model is derived. In addition to the multilinear regression, and for comparison purposes, an nth-order polynomial model of the form

C_P = B_{0 + B1}T + B₂T ²+ B₃T ³ + ... + B_nT ⁿ (7)

is used in this study. Here B₀, B₁, B₂ ... B_n are the polynomial fitting parameters.
Although the polynomial regression method is well known and easy to implement on digital computers, its main disadvantages are more fitting parameters are needed to get higher accuracy, and it might give impractical (or unrealistic negative) values for the predicted property. Polynomials are based on power laws and diverge greatly at or near the end points of the data region. They are thus poor candidates for fitting "smooth" curves. In addition, polynomials force a certain number of inflection points that may not be in the "real" behavior of the physical property.
No body can claim any physical significance of the parameters of the proposed model, at least for the time being, but this model, which is a combination of power and exponential series, is characterized by (a) smaller number of fitting parameters and (b) a more realistic representation of the experimental data (no negative values, for example). The main disadvantage of this method is that it does not properly fit sharp changes in the physical properties.

III. Results and Discussion

The experimental data for the specific heat of metals, metal oxides and metal fluorides were taken from Touloukian and Ho (1972a,b). Information about the purity, composition, and specifications concerning the samples used originally for experimental analysis as well as the reported error (last column of Appendix) are available in Touloukian and Ho (1972a). Throughout the analysis of results, the following basic definitions have been used:

Absolute Error, AE = |C_Pexp - C_Pcal|

Absolute Relative Error,
ARE = |C_Pexp - C_Pcal|/C_Pexp

Average of Absolute Errors,
AAE =(Absolute Errors)/M

Average of Absolute Relative Errors,
AARE =(Absolute Relative Errors)/M

M is the number of data points in a given set of data. The coefficient of determination, R², is defined in terms of the symbols used in Eq. (6) as

(8)

and the standard error of estimate, SSE, is defined as

(9)

where , and i = 1, 2, ..., M and j = 1, 2, ... 4.

The experimental data used for metals, metal oxides and metal fluorides were taken from Touloukian and Ho (1972a,b) on the basis of very similar purity and composition of the chosen samples (curves). It should be mentioned first that, up to the knowledge of the authors, there is no single equation that fits all the temperature range of the specific heat of the studied systems. Equation (1) is just a proposed theoretical formula, but not used to fit the experimental specific heat data of the studied systems.
For all the metal samples used, any impurity in the sample is less than 0.2% each, and the total impurity in any sample is less than 0.5% (Touloukian and Ho, 1972a,b). Table 1 shows the calculated fitting parameters for the specific heat of metals, the coefficient of determination, R², and the average of the absolute relative error, AARE. R² values are very close to unity, the values of AARE are less than 5%, and the overall AARE is about 1.8%, which is within the experimental error. The standard error of estimate, SEE, defined by Eq. (9) is shown on the last column of Table 1. The values of SEE are generally low, and vary in accordance with the AARE values. A comparison between the calculated and experimental data for the specific heat of metals is shown in Figs. 1 to 3, where the match is thought to be sufficient.

Fig. 1: 1^st set of calculated heat capacity of metals as a function of temperature using proposed multilinear regression model vs. experimental data (Touloukian and Ho, 1972a).

Fig. 2: 2^nd set of calculated heat capacity of metals as a function of temperature using proposed multilinear regression model vs. experimental data (Touloukian and Ho, 1972a).

Fig. 3: 3^rd set of calculated heat capacity of metals as a function of temperature using proposed multilinear regression model vs. experimental data (Touloukian and Ho, 1972a).

In order to get the best fit, the data points for some metals (see Table 1) were divided here into two sets; low temperature range and high temperature range. On the other hand, when the full range of temperature for those metals were considered, the AARE jumps to above 10% and reaches 31%, see Table 2.
In order to compare the proposed model in Eq. (3) or (5) with that given in Perry's Handbook (1997), one needs to consider the temperature range used for both equations. The temperature range used for Perry's Handbook equation starts from 273 K and above, while that for Eq. (5) may start at as low as a few degrees K. Thus the comparison might not be valid for many of the studied systems. Anyway, the proposed model in Eq. (5) fits the specific heat of aluminum and molybdenum, for example, better than Perry's Handbook equation for the temperature range shown in Fig. 4.
It should be mentioned, as shown in the last column of Appendix, that the reported error for the studied metals is ranging from 0.1 to £5%, and some samples have no reported error (Touloukian and Ho, 1972a). To make things shorter, no summary tables are included here for the polynomial regression results of the specific heat of the studied metals.

Fig. 4: Calculated specific heat vs. temperature for Al and Mo using (C_P = a + bT + c/T ²) proposed in Perry and Green (1997) and Eq. (5) proposed in this work.

Table 3 shows a comparison between the fittings of a 4^th-order polynomial model and the multilinear regression model for the specific heat of metal oxides and metal fluorides. The reported error for these metal oxides and metal fluorides ranges from 0.1 to 5%, except for uranium oxides (U₃O₈) where it reaches 15% (Touloukian and Ho, 1972b), and some samples have no reported error. Again as shown in Table 3, although the polynomial model has R² > 0.98, the corresponding AARE for the specific heat of some metal oxides is so high (190% for Li₂O, 102% for MgO, 120% for SiO₂ quartz glass, and 77% for SiO₂ quartz crystal). This is because of the prediction of negative values for the specific heat of those metal oxides. For other metal oxides, R² may be as low as 0.23 (for U₃O₈) while AARE = 4.6%, R² = 0.86 (for Cr₂O₃) and AARE = 7.7%, or R² = 0.9985 (for SiO₂ cristobalite) and AARE = 12.4%.
The corresponding values of R² and AARE for the multilinear method are much better (see Table 3) with no non-realistic prediction of the specific heat. The maximum AARE is less than 5% and reaches only 14.5% for Li₂O, 16.7% for MgO, and 8.8% for SiO₂ quartz glass. Again for U₃O₈, although R² is low (0.146) due to the uncertainty of the original data (15%) in Touloukian and Ho (1972b), the value of AARE is very similar to the polynomial model prediction and equals only 4.5%.
For the studied metal fluorides, Table 3 shows that the polynomial model predictions are very close to those of the multilinear model except for KF where AARE reaches 25% (because of the negative values prediction) while the corresponding value for KF using the multilinear model is only about 4.1%. Otherwise, the maximum error in the 4^th-order polynomial model predictions reaches 7.2% for CaF₂. In brief, the multilinear model has an overall AARE of only 3% for all the systems shown in Table 3, which is again within the reported experimental error.
Lastly, the calculated fitting parameters for the specific heat as a function of temperature are shown in Tables 4 and 5 for metal oxides and metal fluorides using the polynomial and multilinear regression methods, respectively.

IV. Conclusions

In this work, a multilinear regression model of the form C_P = aT^b e^cT e^d/T was used to fit the specific heat of several metals, metal oxides, and metal fluorides as a function of temperature. The coefficient of determination, R², was very close to unity for most of the systems studied. The average of the absolute relative errors, AARE, did not exceed 5% for the systems studied, except for Li₂O and MgO where it reaches 14.5% and 16.7%, respectively. The overall AARE was about 1.8% for metals, and 3.0% for metal oxides and metal fluorides, which is within the experimental error.
On the other hand, the polynomial fitting correlation, gave very close, and sometimes better, predictions for the specific heat of metals, metal oxides, and metal fluorides where the coefficient of determination, R², was very close to unity in most cases and the average of the absolute relative errors, AARE, was less than 7.7% except for thorium (8.5%), and Li₂O (190%), MgO (102%), SiO₂ quartz glass (120%), SiO₂ quartz crystal (77%), and potassium fluoride (25%). The polynomial method failed here because of the unrealistic negative values predicted for the specific heat of those systems.

Nomenclature

AARE Average of absolute relative errors
a₁, a₂, a₃, a₄ Constants in Eq. (6)
a, b, c, d Multilinear equation coefficients, Eqs. (1), (2)
A, B ... E Constants in Eq. (3)
B₀, B₁ ... B_n Polynomial coefficients, Eq. (7)
C_P Specific heat at constant pressure, kJ.kg^-1.K^-1
M Number of data points in a given set of data
R² Coefficient of determination, Eq. (8)
SEE Standard error of estimate, Eq. (9)
T Temperature, K

Subscripts

cal Calculated
exp Experimental

Table 1: Multilinear Regression Parameters and R² and AARE for the Specific Heat of Metals (cal.g^-1.K^-1 = 4.186 kJ.kg^-1.K^-1). Experimental Data from Touloukian and Ho (1972a)

Table 2: Multilinear Regression Parameters and R², SEE, and AARE for the Specific Heat of Some Metals with Full- Range Data (cal.g^-1.K^-1 = 4.186 kJ.kg^-1.K^-1).

Table 3: R² and AARE for the Specific Heat of Metal Oxides and Metal Fluorides for Polynomial and Multi-linear Fittings. Experimental Data from Touloukian and Ho (1972b)

Table 4: Multilinear Regression Parameters for the Specific Heat of Metal Oxides and Metal Fluorides (cal.mol^-1.K^-1 = 4.186 kJ.kg^-1.K^-1). Experimental Data from Touloukian and Ho (1972b)

Table 5: 4^th-order Polynomial Parameters for the Specific Heat of Metal Oxides (cal.mol^-1.K^-1). Experimental Data from Touloukian and Ho (1972b)

Appendix: Heat Capacity Coefficients for Metals, Metal Oxides and Metal Fluorides for C_P = a + bT +c/T² (cal.mol^-1.K^-1) and Uncertainty (Perry and Green, 1997), and Reported Error as given by Touloukian and Ho (1972a,b).

References
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