Thermodynamic Properties of Pentaerythritol-Based Species Involved

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Ind. Eng. Chem. Res. 2010, 49, 12294–12301

Thermodynamic Properties of Pentaerythritol-Based Species Involved in Degradation of Aviation Gas Turbine Lubricants Spiridon Siouris* and Christopher W. Wilson Department of Mechanical Engineering, The UniVersity of Sheffield, Mappin Street, Sheffield S1 3JD, England, United Kingdom

The work in this Article deals with the estimation of thermodynamic data for the major species found during lubricant degradation. The organic group R on which the species were based was the pentaerythritol molecule. A computer code was developed that utilizes zero-order and bond additivity methods for the estimation of thermodynamic properties. The validation of the code was carried out against thermodynamic data for species found in published databases and showed that the computed values for heat capacity, enthalpy, and entropy were in close agreement with the reference values. Having gained confidence in the reliability of the developed code, the thermodynamic properties of species involved in aviation gas turbine lubricants were estimated and are presented in this Article. This work is the first part toward a study for the development of a lubricant degradation model in which the changes in the thermodynamic properties, and therefore heat transfer, due to chemical reactions are accounted for. Introduction When lubricants are subjected to high temperature operating conditions, such as those found in aviation gas turbines, their chemical constituents become more reactive, leading to chemical reactions in the bulk and additives.1 These reactions occur with and without the presence of oxygen, resulting in thermal and oxidative degradation, respectively. The degradation of a lubricant can affect its mechanical and thermal properties, and consequently the engine’s component wear, temperatures, and overall operation. For this reason, there have been a number of studies that focus on lubricant degradation so that this phenomenon, and its effects, can be better understood. Typical lubricant degradation studies that deal primarily with experimental methods have been employed with actual engines2 and bench tests.3–5 Furthermore, small scale reactors have been used where lubricant samples are heated to promote thermooxidative degradation.6–9 These studies focus on the investigation of the effects of lubricant degradation and also on the factors that affect it from an engineering perspective. In addition to this, there are chemical kinetic models that have also been developed which can reliably describe the degradation process from a molecular perspective.1,10–16 Apart from lubricants being used to reduce friction and minimize the wear of the sliding components, another important aim of these is to transfer heat from the hot to the cooler components in an engine. In the case of high temperatures, such as in gas turbines, the lubricant’s thermodynamic properties become an important consideration, even more so if there is deposit formation.17,18 It can be understood that if numerical simulations of chemical kinetic models for lubricant degradation are required to include such phenomena, then the thermodynamic properties of the relevant species become a requirement for such modeling. This is evident inchemicalreactionmodelingprogramssuchasCHEMKIN-PSR19,20 where such thermodynamic data are needed to allow the inclusion of the conservation of energy in the calculations. The work in this Article focuses on this aspect of simulations that will subsequently be used toward the development of an improved model for lubricant degradation and deposition. * To whom correspondence should be addressed. Tel.: +44(0)114 222 7815. Fax: +44(0)114 222 7890. E-mail: [email protected].

Studies most commonly deal with a generic organic base R,14–16 and unless it is defined by a formula and chemical structure, it is not possible to assign its appropriate thermodynamic properties. This is because the thermodynamic properties depend on the structure of the molecule.21 Therefore, the work in this Article aims to provide thermodynamic data for the species involved in lubricant degradation so that chemical reaction modeling can be coupled with heat transfer, and also energy calculations to account for the effects of endothermic or exothermic reactions that would enhance the accuracy of numerical simulations. To carry this out, the first task in this work is to review the relevant literature to identify the species that are found during lubricant degradation. Such information is then combined with studies that focus on generic lubricant degradation mechanisms to produce a mechanism with fully defined species in terms of their formula and structure. Once these species have been defined, then this information can be used toward the estimation of their thermodynamic properties based on group additivity methods.21,22 Lubricant Degradation Mechanism As stated previously, lubricant degradation has also been investigated from a chemical perspective with the aim of determining the associated reaction mechanisms. Several studies have focused on a multistep mechanism1,13,23,24 that consists of the following reactions: initiation: RH f R · + H ·

(1)

propagation: R · + O2 f ROO ·

(2)

termination: ROO · + H · f ROOH

(3)

R · + R · f RR

(4)

R · + ROO · f ROOR

(5)

From this mechanism, it can be seen that the initiation step (eq 1) describes the decomposition of RH into H · and (R · ) free radicals. The newly formed R · then combines with an oxygen molecule (propagation step, eq 2) to produce a peroxide free radical (ROO · ). The termination steps involve synthesis

10.1021/ie101236n  2010 American Chemical Society Published on Web 10/21/2010

Ind. Eng. Chem. Res., Vol. 49, No. 23, 2010

reactions of the previously formed free radicals to produce ROOH (eq 3), RR (eq 4), and ROOR (eq 5). The R organic base can be defined as pentaerythritol (C13H20O8) for synthetic lubricants.25 On the basis of this information, the general reaction mechanism for lubricant degradation, eqs 1-5, can be adapted to produce eqs 6-10, respectively. C13H20O8 f C13H19O8 · + H ·

(6)

C13H19O8 · + O2 f C13H19O10 ·

(7)

C13H19O10 · + H · f C13H20O10

(8)

C13H19O8 · + C13H19O8 · f C26H38O16

(9)

C13H19O8 · + C13H19O10 · f C26H38O18

(10)

The chemical structure of the reactant species in the initiation step has been previously defined.25 In addition to this, considering that the most susceptible site for attack is the β-acyl bonds,1,23 the chemical structure of the rest of the reactants is defined as illustrated in Figure 1. Thermodynamic Property Estimation Having derived the species taking part in lubricant degradation in eqs 6-10, the task is to then assign values for their thermodynamic properties. Thermodynamic property databases26–28 can provide accurate information for a large number of species. However, when dealing with complex molecules such as those found during lubricant degradation,25 there are cases where thermodynamic data do not exist. This poses a barrier for simulations to include heat transfer and energy conservation due to the effects of endothermic or exothermic reactions. To overcome this problem, scientists have developed estimation methods that enable the derivation of thermodynamic properties of many types of pure or compound gas, liquid, and solid substances.21,22 It is such methods that have been applied in this study to provide thermodynamic property data for the species that are associated with lubricant degradation as described in eqs 6-10. Thermodynamic property estimation methods can be divided into zero-order, bond, first level group, and second level group additivity methods. Each of these methods require different amounts of structural information about the molecule to estimate its properties. Zero-order methods require only information on the kind of atoms in the molecule, for example, C, H, or N.29 Bond methods rely on information for the type of atoms and also for the type of bonds that the atom is associated with in the molecule.21,30 First and second level group methods are similar to bond methods, but these use additional information for two (first level) or three (second level) adjacent bonds of atoms.31–35 Figure 2 illustrates the extent of information required for the example of pentaerythritol, and Table 1 lists popular estimation methods along with their average absolute percentage errors.29 The main body of work in this Article resulted in the development of a computer code for the estimation of thermodynamic properties for any complex molecule. This code was designed with the requirements of minimal input information regarding the molecular structure of the species, while maintaining high accuracy for the resulting thermodynamic properties. The program produces an output based on the Gordon and McBride format,20,39 which is the format used for input to the PSR code. The estimation methods currently employed are for the derivation of heat capacity29 and for enthalpy and entropy.31 These tables provide statistical data for functional groups such as (for example, -CH3, -CH2-) for temperatures at 300, 400, 500, 600, 800, 1000, and 1500 K.

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Once the thermodynamic properties have been estimated at the above temperatures, the code produces three polynomial fits from these data to describe the heat capacity, enthalpy, and entropy for each of the species. These three polynomials are expressed as functions of time, and the following section presents the derivation of these. Five coefficients (a1-a5) are used for fitting the heat capacity data, and an additional two coefficients, a6 and a7, are used for the enthalpy and entropy polynomials, respectively. Fitting of the nonlinear data was carried out by using the Levenberg-Marquardt algorithm.40 Heat Capacity. The derivation of the molar heat capacity C °pk for the kth species starts with the assumption that standardstate thermodynamic properties are only a function of temperature at constant pressure. Applying this assumption, the molar heat capacity of a species can be expressed in polynomial form: Cpko ) R

N

∑a T

n-1

n

(11)

n)1

Assuming a fourth-order polynomial, eq 11 becomes: Cpko ) a1 + a2T + a3T 2 + a4T 3 + a5T 4 R

(12)

and this is the form of the C °pk/R polynomial used. Enthalpy. The enthalpy of formation of species k at temperature T can be defined as o Hko ) Hk,298 +

∫

T

Cpko dT

(13)

Cpko dT 298 R

(14)

298

Dividing by R gives o Hko Hk,298 + ) R R

∫

T

Substituting eq 12 into eq 14 and expanding the integral produces o Hko Hk,298 a2 a3 a4 + a 1T + T 2 + T 3 + T 4 ) R R 2 3 4 a5 5 a2 a 3 + T - a1298 - 2982 - 2983 5 2 3 a4 a5 4 5 - 298 - 298 4 5

(15)

term and gives Dividing by T non-dimensionalizes the H °/R k the final version of the enthalpy polynomial: Hko a3 a4 a5 a2 ) a1 + T + T 2 + T 3 + T 4 RT 2 3 4 5 o Hk,298 a2 - a1298 - 2982 +( R 2 a3 a a 4 5 1 - 2983 - 2984 - 2985) 3 4 5 T

(16)

Therefore, the a6 term, that is, the coefficient of 1/T in the polynomial, can be identified as a6 ≡

o Hk,298 a2 a3 - a1298 - 2982 - 2983 R 2 3 a4 a5 4 5 - 298 - 298 4 5

(17)

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Figure 1. Illustration of the changes in the molecular structures of the species involved in lubricant thermal-oxidative degradation.


1 R

∫

T

298

dS )

1 R

o Sko Sk,298 ) R R

o T Cpk

∫

298

∫

T

298

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dT

(23)

Cpk dT RT

(24)

T

Substituting C °pk/R from eq 12 and evaluating the integral gives o Sko Sk,298 a3 a4 + a1 ln T + a2T + T 2 + T 3 ) R R 2 3 a5 4 a3 2 + T - a1 ln(298) - a2298 - 298 4 2 a4 a5 3 4 - 298 - 298 3 4

(25)

The a7 coefficient, constant in the polynomial, can then be defined as Figure 2. An illustration of the extent of information required for thermodynamic property estimation methods for the example of pentaerythritol.

This coefficient can be calculated once the polynomial coefficients from eq 12 have been found. Entropy. The starting point for the expression of entropy, as a set of polynomials, such as in the molar heat capacity and enthalpy, can be based on the definition of the Gibbs free energy equation: T dS ) dU + P dV

(18)

Using the definition of enthalpy and differentiating both sides gives H ) U + PV

(19)

dH ) dU + V dP + P dV

(20)

Subtracting eq 20 from eq 18 and dividing by temperature T results in dS )

V dP dH + T T

(21)

Because we are dealing with constant pressure conditions (from the PSR assumptions19), dP ) 0, and therefore the previous equation can be further simplified: dH T Cpk ) T

dS )

(22)

Dividing by R and integrating both sides in the temperature range of (298, T) results in Table 1. Comparison of Accuracies between Thermodynamic Property Estimation Methods29 type of method zero-order bond first-order group second-order group

method

av. absolute percent error

Harrison-Seaton29 Benson21 Joback31 Thinh et al.36 Rihani-Doraiswamy37 Yoneda38 Benson21

3.2 7.1 1.4 1.1 3.2 1.4 1.1

o Sk,298 - a1 ln(298) - a2298 R a3 a4 a5 - 2982 - 2983 - 2984 2 3 4

a7 ≡

(26)

Results and Discussion Validation. The validation of the program can be carried out for molecules of known molecular structures that are currently in thermodynamic databases.26–28 This section provides a comparison of C °pk/R, H °/RT, and S °/R values between currently k k validated values and values produced by the developed code, over a range of temperatures. The molecules chosen for comparison are C2H4 and CH3OCH3 because there are no isomers and therefore there can be no ambiguity regarding their chemical structure and over what is being compared. The temperature range in which the comparisons were made was chosen to be between 300 and 1500 K. This range was chosen so that it would include all possible operating temperatures, and also to test the code at higher than normal temperatures to investigate its accuracy at these high temperature regimes. It has to be noted that the Gordon and McBride file format for thermodynamic properties utilizes two sets of coefficients so that low and high temperature ranges can be described with different polynomials for increased accuracy. For example, for the case of C2H4, the reference curve is a result of two polynomials for two temperature ranges. The first polynomial is for the lower temperature range (300 K, 1000 K), and the second polynomial is for the range of (1000 K, 1500 K). For the case of CH3OCH3, the reference curve is constructed from only the lower polynomial because it applies for a temperature range of 300-1500 K, as compared to (300 K, 1000 K) for the case of C2H4. Figures 3-5 present plots for comparison between the and S °/R, reference and computed values for C °pk/R, H °/RT, k k respectively. In Figure 3a, it can be seen that the computed values converge to the reference values from 300 to 1000 K, where the reference curve is constructed from the lower temperature range polynomial for C °pk/R. At 1000 K, the reference curve switches to a high temperature range polynomial, and a departure of the two curves is seen. This, however, is not observed in Figure 3b where the reference and computed curves converge as temperature increases. Convergence of the reference and computed curves can also be seen for the case of molar in Figure 4a and b for the case of C2H4 and enthalpy H °/RT k

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Figure 3. Comparison of reference and estimated molar heat capacity C °pk/R of (a) C2H4 and (b) CH3OCH3 in the range of (300 K, 1500 K).

Figure 5. Comparison of reference and estimated molar entropy S °/R of k (a) C2H4 and (b) CH3OCH3 in the range of (300 K, 1500 K). Table 2. Comparison of Molar Heat Capacity C °pk/R between Reference and Computed Values for the Molecules of C2H4 and CH3OCH3 molecule

T (K)

C °pk/R reference

C °pk/R computed

percent deviation

C2H4

300 900 1500 300 900 1500

5.15 10.77 13.20 7.65 16.33 20.36

5.49 10.90 13.47 8.23 16.73 20.26

6.56 1.18 2.11 7.51 2.46 0.53

CH3OCH3

Table 3. Comparison of Molar Enthalpy H °/RT between Reference k and Computed Values for the Molecules of C2H4 and CH3OCH3 molecule

T (K)

H °/RT k reference

H °/RT k computed

percent deviation

C2H4

300 900 1500 300 900 1500

21.07 12.56 12.39 -73.30 -16.15 -2.28

19.69 12.25 12.30 -86.87 -20.27 -4.64

6.53 2.48 0.74 18.52 25.54 109.42

CH3OCH3

Figure 4. Comparison of reference and estimated molar enthalpy H °/RT k of (a) C2H4 and (b) CH3OCH3 in the range of (300 K, 1500 K).

CH3OCH3, respectively. Comparison plots of molar entropy S °/ k R in Figure 5a and b show an almost linear trend for both reference and computed curves. There is a notable shift of the computed values in Figure 5a for C2H4, but at closer inspection, the maximum error is approximately 6% at 300 K. Tables 2-4 present a comparison between reference and and S °/R, respectively, computed values for C °pk/R, H °/RT, k k for selected temperatures of 300, 900, and 1500 K. Table 2 shows that the maximum deviation for both molecules occurs at 300 K because this is where the absolute values are their lowest, and therefore deviations in this region can have a significant effect on the relative error. It can be seen that for C°pk/R at 900 and 1500 K, the percent errors are approximately 1-2% for both C2H4 and CH3OCH3. For the case of H °/RT k (Table 3) for CH3OCH3, although the reference and computed curves converge together as the temperature increases, the

Table 4. Comparison of Molar Entropy S °/R between Reference k and Computed Values for the Molecules of C2H4 and CH3OCH3 molecule

T (K)

S °/R k reference

S °/R k computed

percent deviation

C2H4

300 900 1500 300 900 1500

26.39 34.96 41.11 32.44 45.22 54.67

24.70 33.54 39.80 31.16 44.62 54.15

6.39 4.08 3.19 3.92 1.31 0.95

CH3OCH3

negative absolute values increase toward zero, and this influences the percent errors significantly. Although the curves converge as the temperature increases, the calculated percent error shows an increasing trend of up to 109%. However, considering the absolute values at 1500 K, which are -2.28 for the reference, and -4.64 for the computed data, it can be seen that a 2.36 difference in H °/RT is k negligible when the reference values range from -73.30 to -2.28 for CH3OCH3. In addition to this, Table 4 shows that ranges from 3.19% to 6.39%, the percent deviation of S °/R k


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Table 5. Coefficients for the Thermodynamic Polynomials for All Associated Pentaerythritol Species during Lubricant Degradation polynomial coefficients molecule

a1

a2

a3

a4

a5

a6

a7

C13H20O8 C13H19O8 C13H19O10 C13H20O10 C26H38O16 C26H38O18

-9.423 -9.523 -9.311 -9.210 -21.53 -21.32

0.2035 0.2018 0.2123 0.2141 0.4113 0.4219

-1.525 × 10-4 -1.533 × 10-4 -1.609 × 10-4 -1.600 × 10-4 -3.111 × 10-4 -3.187 × 10-4

5.486 × 10-8 5.606 × 10-8 5.762 × 10-8 5.639 × 10-8 11.19 × 10-8 11.34 × 10-8

-7.623 × 10-12 -7.943 × 10-12 -7.819 × 10-12 -7.499 × 10-12 -15.33 × 10-12 -15.21 × 10-12

-1.613 × 105 -1.545 × 105 -1.867 × 105 -1.960 × 105 -3.167 × 105 -3.490 × 105

35.19 35.09 32.47 31.47 57.25 54.07

and 0.95% to 3.92% for C2H4 and CH3OCH3, respectively, which is considered as a good agreement between the reference and computed data. Tables 2-4 suggest that the magnitude of errors in the developed code is similar to the errors of the additivity methods that the code uses, as seen in Table 1. The developed code can be therefore considered as reliable and suitable for thermodynamic data generation of complex molecules such as those found during lubricant degradation (Table 5). Pentaerythritol-Based Molecules. Having gained confidence in the reliability of the developed code, the computation of thermodynamic properties for the pentaerythritol-based molecules found in the reaction scheme of eqs 6-10 was carried out. The polynomial data were then plotted for the

temperature range of (300 K, 1500 K), and the results are presented in Figures 6-11. It can be seen that the curves for and S °/R do not change significantly between C °pk/R, H °/RT, k k the C13 species, but for the C26 molecules, the C °pk/R and H °/ k RT have almost doubled. The reason for this is that the more carbon atoms there are in a molecule, the more energy can be absorbed and stored as vibrational energy because there are more atoms that can vibrate. A similar phenomenon where the occurs for the enthalpy of formation H °/RT k molecules with twice the amount of carbon atoms require a much higher energy input to be formed by creating the required bonds. In addition to this, comparing Figure 6 with Figure 7, and then Figure 7 with Figure 9, it can be seen that an addition of

Figure 6. Plots of thermodynamic properties in the range of (300 K, 1500 K) for the molecule of C13H20O8.



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a hydrogen atom (Figure 6 to Figure 7) does not modify the C °pk/R, H °/RT, and S °/R as much as the addition of an oxygen k k molecule (Figure 7 to Figure 9). This can be explained by the fact that an oxygen atom is more massive than a hydrogen atom and has therefore greater inertia that can store higher amounts of vibrational energy. This, in turn, can be seen as increased heat capacity. Furthermore, the enthalpy of formation is affected due to the varying amount of internal energy the atom needs to have to collide and bond with the other atoms to form a molecule.31,41 Conclusions Aviation gas turbine lubricant base-stocks are usually comprised of molecules with a high number of carbon atoms, such as pentaerythritol. These molecules can have complex structures, and because these are used within mixtures, it is difficult to identify their thermodynamic properties for use in thermochemical simulations. When such data are not available from published thermochemical databases, these need to be generated, and for this reason property estimation methods have been developed

to allow the generation of such data. These data can then be used in chemical modeling simulations that also solve the energy equation. The work in this Article deals with providing thermodynamic data for major components during lubricant degradation, which are identified through the application of previously developed lubricant degradation kinetic schemes. This work forms the first step that is required for the development of an improved model for lubricant degradation and deposition. To carry out such a task, a computer code was developed that calculates the derived polynomial coefficients for the functions of heat capacity, enthalpy, and entropy versus temperature. The validation of the computer code was performed by comparing the thermodynamic data of the C2H4 and CH3OCH3 molecules. After assessing its reliability, the code was then used to derive the thermodynamic properties of lubricant degradation species within the range of (300 K, 1500 K). These results can then be used as inputs for modeling heat transfer phenomena that arise from lubricant degradation and


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ReceiVed for reView June 7, 2010 ReVised manuscript receiVed September 10, 2010 Accepted September 13, 2010 IE101236N

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