Method of Jacobians

Partial derivatives, as introduced in Section 2.12 are of particular importance in thermodynamics. The various state functions, whose differentials are exact (see Section 3.5), are related via approximately 1010 expressions involving 720 first partial derivatives Although some of these relations are not of practical interest, many are. It is therefore useful to develop a systematic method of deriving them. Hie method of Jacobians is certainly the most widely applied to the solution of this problem. It will be only briefly described here. For a more advanced treatment of the subject and its application to thermodynamics, die reader is referred to specialized texts. 

Stability criteria are discussed within the framework of equilibrium thermodynamics. Preliminary information about state functions, Legendre transformations, natural variables for the appropriate thermodynamic potentials, Euler s integral theorem for homogeneous functions, the Gibbs-Duhem equation, and the method of Jacobians is required to make this chapter self-contained. Thermal, mechanical, and chemical stability constitute complete thermodynamic stability. Each type of stability is discussed empirically in terms of a unique thermodynamic state function. The rigorous approach to stability, which invokes energy minimization, confirms the empirical results and reveals that r - -1 conditions must be satisfied if an r-component mixture is homogeneous and does not separate into more than one phase. 

Prove that the rate of change of pressure with respect to volume is greater in magnitude along an adiabatic path relative to an isothermal path. Use the method of Jacobians and do not restrict your analysis to ideal gases. 

In contrast to Eq. (3.22), the negative sign cancels, because the pressure p introduced the negative sign. Further, observe that the partial differential has to be evaluated at constant energy, thus we have the set of independent variable of (U, V). This is different from the definition equation of the energy, where we have the set (S, V) as independent variables. We want to express Eq. (3.27) in terms of the independent variables (T, V). Using the method of Jacobian determinants, as explained in Sect. 1.14.6, we get... 

Another way to calculate the partial derivatives is possible. Figure 15.12 represents a typical module. If a module is simulated individually rather than in sequence after each unknown input variable is perturbed by a small amount, to calculate the Jacobian matrix, (C + 2)nci + ndi + 1 simulations will be required for the ith module, where nci = number of interconnecting streams to module i and ndi = number of unspecified equipment parameters for module /. This method of calculation of the Jacobian matrix is usually referred to as full-block perturbation. 

Assuming that the original problem was solved using Newton s method, the Jacobian J, evaluated at the solution, is already known and factored into its LU components. The 3F/9a matrix is determined one column at a time by finite-difference perturbation of the parameter, analogous to the procedure described by Eq. 15.36. For each column of dF/da, a column of the sensitivity matrix 3y/3a is determined via a back-substitution using the Jacobian s LU factors. 

The first term on the right-hand side is the product of the physical problem s current Jacobian matrix and the sensitivity-coefficient matrix (i.e., the dependent variable). Assuming that the underlying physical problem (i.e., Eq. 15.58) is solved by implicit methods, the Jacobian evaluation is already part of the solution algorithm. The second term, which is the matrix that describes the explicit dependence of f on the parameters, must be evaluated to form the sensitivity equation. Note that all terms on the right-hand side are time dependent, as are the sensitivity coefficients S(t). 

Our solution method of choice is Newton s iterative method, see Section 1.2. Newton s method requires us to evaluate the Jacobian matrix Df of... 

Eqn (29) may be included into the Newton-Raphson iteration as the n-th equation to determine all the intermediate as well as the final pressure. This however, requires subsequent derivation of the extra row in the Jacobian by the differentiation of eqn (29) with respect to the vector x of all pressures. This leads to fairly involved algebraic expression, so the quickest and safest method of calculating the choked flow conditions in the line segment is by a simple single variable optimization of y from eqn (27) with respect to the final pressure. The vector x is conputed frcm eqn (24) by the straight forward Newton-Raphson iteration for each step in the single variable hillclimbing. 

The matrices indicated in equation (15) are known as Jacobians. The properties of Jacobians have been of interest to thermodynamicists for many years and may be used to find an interesting interpretation of the meanings of the shadow prices. Using the methods described in (2,8), define ... 

At this stage, we need to discuss the actual task of calculating the Jacobian matrix J. It is always possible to approximate J numerically by the method of finite differences. In the limit as Akz approaches zero, the derivative of R with respect to k, is given by Equation 7.16. For sufficiently small Ak the approximation can be very good. 

The quasi-Newton methods. In the Newton-Raphson method, the Jacobian is filled and then solved to get a new set of independent variables in eveiy trial. The computer time consumed in doing this can be very high and increases dramatically with the number of stages and components. In quasi-Newton methods, recalculation of the Jacobian and its inverse or LU factors is avoided. Instead, these are updated using a formula based on the current values of the independent functions and variables. Broyden s (119) method for updating the Jacobian and its inverse is most commonly used. For LU factorization, Bennett s (120) method can be used to update the LU factors. The Bennett formula is... 

Numerical calculation has been carried out using a software interface which is based on the so-called "Method of lines" (14). Gear s backward difference formulas (15) are used for the time integration. A modified Newton s method with the internally generated Jacobian matrix is utilized to solve the nonlinear equations. ... 

The two points retained for the next step are x and either Xq or x, the choice being made so that the pair of values/(Jc), and either/(jc ) or/(jc ), have opposite signs to maintain the bracket on x. (This variation is called regula falsi or the method of felse position.) In Figure L.7, for the k + l)st stage, x and Xq would be selected as the end points of the secant line. Secant methods may seem crude, but they work well in practice. The details of the computational aspects of a sound algorithm to solve multiple equations by the secant method are too lengthy to outline here (particularly the calculation of a new Jacobian matrix from the former one instead refer to Dennis and Schnabel"). 

In the classical Newton-Raphson technique, the Jacobian matrix is inverted every iteration in order to compute the corrections AT] and Al]. The method of Tomich, however, uses the Broyden procedure (Broyden, 1965) in subsequent iterations for updating the inverted Jacobian matrix. 

Determinants, Jacobians and Hessians are continually appearing in different branches of applied mathematics. The following results will serve as a simple exercise on the mathematical methods of some of the earlier sections of this work. The reader should find no difficulty in assigning a meaning to most of the coefficients considered. See J. E. Trevor, Journ. Phys. Chem.,-3, 523, 573, 1899 10, 99, 1906 also R. E. Baynes Thermodynamics, Oxford, 95, 1878. 

The above techniques are used quite commonly to interrelate and manipulate thermodynamic variables. Later in this chapter we introduce two additional and more general methods of manipulating thermodynamic derivatives called Legendre and Jacobian transformations. 

---