Ind. Eng. Chem. Res. 1988,27, 721-723 CBHSOH, 108-95-2; H3CCONH2, 60-35-5; H&C02H, 64-19-7; HO(CHz),OH, 107-21-1; HCOZH, 64-18-6; HCHO, 50-00-0.
Literature Cited Banks, V. C.; O’Laughlin, J. W. Anal. Chem. 1957,29, 1412-1417. Imamura, S.;Tonomura, Y.; Kawabata, N.; Kitao, T. Bull. Chem. SOC.Jpn. 1981,54, 1548-1553. Imamura, S.;Sakai, T.; Ikuyama, T. Sekiyu Galzkaishi 1982a, 25, 74-80. Imamura, S.; Hirano, A.: Kawabata, N. Ind. Eng. Chem. Prod. Res. Dev. 198213, 21, 570-575. Imamura. S.: Kinunaka.. H.:. Kawabata. N. Bull. Chem. SOC.J m . 1982c,’55, 3679-3680. Imamura, S.; Doi, A.; Ishida, S. Ind. Eng. Chem. Prod. Res. Deu. 1985, 24, 75-82. Imamura, S.; Nakamura, M.; Kawabata, N.; Yoshida, J.; Ishida, S. Ind. Eng. Chem. Prod. Res. Dev. 1986, 25, 34-37. Imamura, S.: Nishimura, H.: Ishida, S . Sekivu Gakkaishi 1987.30, 199-202. Katzer. 3. R.: Ficke. H. H.: Sadana. A. J. Water Pollut. Control Fed. 1976, 48, 920-933.
72 1
Kim, K. S.; Winograd, N. J . Catal. 1974, 35, 66-72. McClune, W. F. “Powder Diffraction File”. Sets 21-22, 1980, p 375; International Centre for Diffraction Data, Pittsburgh, PA. Tagashira, Y.; Takagi, H.; Inagaki, K. Japanese Patent 75 106862, 1976; Chem. Abstr. 1976,84, 79359. 1964960p33-38. Teletzke, G. H. ‘Kyoto Institute of Technology. t Chubu University. Seiichiro Imamura,*t Ikumi Fukuda,? Shingo Ishida’ Department of Chemistry Kyoto Institute of Technology Matsugasaki, Sakyo-ku, Kyoto 606, Japan and Department of Industrial Chemistry Chubu University Matsumto-cho 1200, Kasugai 487, Japan Received for review May 1, 1987 Accepted December 29, 1987
Cramer’s Rule and Partial Thermodynamic Properties: A Revisit A simple method which formalizes the computation of partial thermodynamic properties for a single substance is presented. The method utilizes the matrix representation of the total differentials of U, A , G, H, S, and V and Cramer’s rule for solving sets of linear equations to generate the required partial derivatives. This approach allows any one of the 336 partial thermodynamic properties to be calculated in a systematic and simple way, and thus this approach lends itself to implementation on a digital computer. The evaluation of determinants for the case when the partial derivative of an energy term is taken with respect to another energy term involves significant algebraic manipulation although the technique is the same for all partial derivatives. Two examples are given to illustrate the procedure outlined in the paper. Expressions relating the first partial derivatives of thermodynamic functions to measurable quantities are well established. For a pure substance, eight common thermodynamic variables exist, i.e., U, A, G , H,S , P, T, V, and there are possible 8 X 7 X 6 = 336 first partial derivatives. Bridgman (1914,1925) has devised a system in which these derivatives can be related to measurable quantities involving V, T, and P, their mutual derivatives, Le., a and p, the heat capacity, C,, and the entropy, S . The introduction of the entropy does not pose any difficulties since it can be conveniently calculated from the heat capacity and volumetric data. Thus, he introduced 45 relationships, better known as the Bridgman relationships, for calculating the more desirable thermodynamic functions and their partial derivatives. From these relationships one can then calculate any other partial derivative. Later work by Lerman (1937) presents a method similar to that devised by Bridgman but redefined in such a manner so as to permit the resolution of the 45 forms into a few, more basic forms. Tobolsky (1942) presents another systematic method of obtaining these derivatives which is applicable to any set of independent variables. A more rigorous approach is given by Shaw (1935) which introduces the method of the Jacobian transformations, and a short overview of this technique can be found in Sandler (1977). Here we present a method by Erben (1973) which formalizes the Jacobian transformations in a simpler concept which is easier to understand and to our knowledge is not available in the literature. There is a synergism between the method of the Jacobian transformations and the method we describe below which is not immediate. However, we do not intend to demonstrate this is this communication. This method is outlined below together with two examples to illustrate the technique. Let xl, x2, and x , be three variables related by the following set of linearly independent algebraic equations: 0888-5885/88/2627-0721$01.50/0
Since there are three variables and two equations describing the system, we can only solve two of them in terms of the third one. Suppose we want to solve for x1 and x 2 in terms of x,; we can rearrange eq 1and 2 in the following manner: 2x1 + x 2 = -x3 (3) Xi
+ 3x2 = 4 x 3
(4)
or in matrix notation
where A is the square matrix premultiplying the vector x = [xl x2IT(T = transpose). A well-known method for the solution of linear algebraic equations is Cramer’s rule (cited in Wylie (1966)): The mechanics of this method are as follows: First find the determinant of A, det A = [ (2)(3) - (1)(1)] = 5. Then move vector b to the column for which the solution is desired, e.g., xl, and form matrix Al as follows:
where det Al = [(-1)(3) - (-4)(1)] = 1. The solution for x1 is then expressed by x ~ / x ,= det A,/det A =
y5
Similarly the solution for x2 can be found by substituting vector b in lieu of the second column, giving x 2 / x 3 = det A2/det A = -y5 0 1988 American Chemical Society
722 Ind. Eng. Chem. Res., Vol. 27, No. 4, 1988
Now we would like to expand this concept to the evaluation of partial thermodynamic properties. Let us choose the eight fundamental thermodynamic variables used in describing a pure component: U = internal energy, G = Gibbs free energy, A = Helmotz"freeenergy, H = enthalpy, S = entropy, P = pressure, T = temperature, and V = volume. One can write the total differential of U, A, G, H , S, and V as follows: dU=-PdV+TdS
(5)
dA = -P d V - S d T
(6)
dG=-SdT+VdP
(7)
dH=VdP+TdS
(8)
d S = Cp/T d T - CUVdP
(9)
dV= aVdT-PVdP
(10)
By analogy to the above solution of x1/x3 or x z / x 3 , the ratio of the two variables now becomes a partial derivative. This implies that if we first calculate the determinant, det D, of the above 6 X 6 matrix and the determinant, det D,, of the matrix obtained by replacing the column as by the column aT and forming det D,/det D, the answer will be dS/aT, and since we held P constant, our solution is nothing but (aS/aT),:
[:
1 -T
det D =
0 =1
and
det D1 = 0 C,/T
[:v
3
0 = C,/T
Therefore (aS/ar), = det D,/det D = C,/T. We would like to note that with the above six equations it is possible to solve for all of the partial derivatives, i.e., (au/aT),, (aA/aT),, (aG/aT),, (aH/aT),, (aS/aT),, and (aV/aT),. The determinants of D and D, can be readily obtained from their minors expansion. Let us evaluate another partial derivative, namely (aA/dP)w This example is chosen from Sandler (1977) where it is solved by Jacobian transformations. Although the problem setup is clear and concise, the manipulation of the matrices is not easy. First, we change the sign of the elements of column 6 (vector b ) in matrix B and pass it to the right-hand side. Second, we eliminate the fourth column since H is held constant. We now have a reduced system in the form of D y = b , and it can be solved similarly to the previous example: '
where (Y = 1 / V (aV/aT), and p = -1/V (dV/dP)T. These equations are derived from the well-known Maxwell relations and can be found in any book on thermodynamics. Equations 5-10 can be rewritten as dU P dV- T dS = 0 (54
+ dA + P d V + S d T = 0 dG + S d T - V d P = 0
(sa)
(74
dH- V d P - T d S = 0 dS - Cp/T d T + a V dP = 0
(8d
d V - a V d T + PV dP = 0
(loa)
(94
The idea behind the present method is as follows: Since there are six fundamental equations (eq 5a-loa) in eight variables, namely, dU, dA, dH,dG, dS, dP,dT, and dV, one can solve any of these variables with respect to another one if a third one is held constant (suggesting its rate of change in zero). Thus, the set of eq 5a-10a is reduced to a system of six equations with six unknowns, which has a unique solution. Equations 5a-10a can be expressed in a matrix notation as follows: 0
1
0
0
0
0 0 0 0
0 0 0 0
1 0
0 1 0 0
0 -T 1 0
0 0
s - v s 0
F
-V 0 CYV -C,/T p v - a v
0 0 0 1
dA dG dS dFJ dT dV
where B is the 6 X 8 matrix and y = [dU dA dG dH dS dP d T d v T . Columns 1-8 represent dU, dA,dG, dH,ds,dP,dT, and dV, respectively. Suppose we want to find the partial derivative (aS/aT),. We first eliminate the sixth column from matrix B (since pressure is held constant and thus dP = 0). Then we carry the seventh column to the right-hand side by changing its sign (since dS is solved in terms of dT). The system of equations can be then solved by the method outlined above. We now have six equations in six variables which are described as follows: au aA aG aH as av aT 1 0
0 0
0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0
0 1 0 0
-T 0
0
P P
-T
0 0
1 0
0
1
0 -S -S 0 CPI T ffV
det D =
kT
0 -C,/T --cy V
!]
= C,
and det D1 = (S + P V a ) ( V - TVa) + PCp/3V (from minors expansion) Therefore (aA/aP)H = det D,/det D = (S + PVa)(V- TVa)/C,
+ PPV
The reader is encouraged to work through this example and compare the result to the Jacobian transformation solution of Sandler (1977). It is interesting to note that while solving this example an error was found in Bridgman's original work. In conclusion, we feel that this method is more systematic and simpler to implement than the previous methods. Difficulties, however, do arise when one has to take the partial derivatives of an energy term with respect to another. In this case, the minor expansion for the determinants det D and det D, does not reduce easily, although the problem is still easy to formulate. Nomenclature A = Helmotz free energy
det = determinant C, = heat capacity G = Gibbs free energy H = enthalpy P = pressure S = entropy T = temperature U = internal energy V = volume Greek Symbols = volume expansivity /3 = isothermal compressibility cy
Ind. Eng. Chem. Res. 1988, 27, 723-126
Literature Cited
723
Union Carbide Corporation.
Bridgman, P. W. Phys. Rev. 1914, 3(4), 273. Bridgman, P. W. A Condensed Collection of Thermodynamic Formulas; Harvard University Press: Boston, 1925. Erben, T. M. Istanbul Technical University, personal communication, 1973. Lerman, F. J. Chem. Phys. 1937,5, 792. Shaw, A. N. Phil. Trans. R. SOC.1935,2344, 299. Sandler, S . I. Chemical and Engineering Thermodynamics; Wiley: New York, 1977. Tobolsky, A. J . Chem. Phys. 1942, 10, 644. Wylie, C. R. Advanced Engineering Mathematics; McGraw-Hill: New York, 1966.
* West Virginia University.
Manuk Colakyan,*' Richard Turtonj Union Carbide Corporation Technical Center S o u t h Charleston, W e s t Virginia 25314 and West Virginia University Department of Chemical Engineering Morgantown, West Virginia 26506 Received f o r review March 30, 1987 Accepted November 30, 1987
Modified Back-Calculation Method To Predict Particle Size Distributions for Batch Grinding in a Ball Mill This paper describes a back-calculation procedure based on the quasi-Newton method of optimization. The general solution of the integro-differential equation was undertaken, and with a careful study of the literature, various forms of breakage distribution function and selection function were assumed. An error function was defined as the root mean square error between the calculated and the experimental product size distribution$. The values of the parameters a t the lowest error were calculated, using the quasi-Newton optimization technique. This method was experimentally verified for different materials and for grinding processes following different forms of selection and breakage functions. The calculated size distribution compared well with the experimental size distribution. I t is concluded that the present method is more useful than other back-calculation methods and can be used for both normalized and nonnormalized breakage distribution functions. It has a potential in industrial application, as it does not demand grinding data with narrow size feed materials. Grinding of solid materials is the most energy-consuming unit operation in mineral processing and cement and allied industries. As expenditure on energy is a major portion of the total processing cost of the products, it is absolutely necessary to run size reduction processes at optimum operating conditions so that the maximum size reduction can be achieved (Devaswithin et al., 1985) at the lowest operating cost. To evaluate the optimum operating conditions, it is essential (Devaswithin et al., 1987) to predict the complete product size distribution. There are many models (Reid, 1965; Luckie and Austin, 1971; Herbst and Mika, 1970; Kapur, 1970) for the prediction of the particle size distribution for batch ball mill grinding. These models relate the product size distribution with breakage and selection functions. There are many methods available in the literature (Reid and Stewart, 1970; Austin and Luckie, 1971/1972; Herbst and Fuerstenau, 1968; Kapur, 1982; Gupta et al., 1981) for the experimental determination of the breakage distribution and selection functions. Most of these methods demand elaborate experimentation. In this paper, an optimization technique based on the quasi-Newton algorithm is presented.
Equations for the Prediction of Particle Size Distribution For a finite size interval, the batch grinding equation in continuous time form is conveniently expressed as dWi(t)/dt = -SiWi(t) +
i-1
SjbijWj(t)
j=l,i>l
(1)
where Wi(t)is the weight fraction of the total material present in the size interval i at any time t;Si is the selection for breakage or specific rate of breakage of the material in the ith size interval with units of inverse time; and bij is the breakage distribution of particles in the size interval 0888-5885/88/2627-0723$01.5Q JQ
j broken into smaller fragments falling in the size interval i, Le., b, = B, - Bi+ljwhere B, is the cumulative breakage
distribution function. For the first size interval, the solution of the batch grinding eq 1 is given (Reid, 1965) as Wl(t) = Wl(0)e-slt
(2)
The second size interval is given as
and the ith size interval is given as
where
It can be seen that the solution for all the n size intervals will contain n selection function constants and n(n - 1)/2 breakage function constants. These constants are evaluated by a back-calculation method.
Experimental Section Batch grinding experiments were conducted with a known feed size distribution in a laboratory ball mill that was 0.2 m i.d. and 0.3 m long with a provision for changing the speed of revolution of the mill. The product size distributions are measured at different times of grinding 0 1988 American Chemical Society
