Equilibrium stage model MESH equations

The equilibrium stage model of Figs. 15.1 and 15.3 is again employed. However, rather than solving the N(2C + 3) MESH equations simultaneously, we combine (15-3) and (15-4) with the other MESH equations to eliminate 2N variables and thus reduce the problem to the simultaneous solution of N(2C +1) equations. This is done by first multiplying (15-3) and (15-4) by Vj and Lj, respectively, to give... 

Tne MESH Equations (the 2c + 3 Formulation) The equations that model equilibrium stages often are referred to as the MESH equations. The M equations are the material balance equations, E stands for equilibrium equations, S stands for mole fraction summation equations, and H refers to the heat or enthalpy balance equations. 

The class of simultaneous solution methods in which all of the model equations are solved simultaneously using Newton s method (or a modification thereof) is one class of methods for solving the MESH equations that allow the user to incorporate efficiencies that differ from unity. Simultaneous solution methods have long been used for solving equilibrium stage simulation problems (see, e.g., Whitehouse, 1964 Stainthorp and Whitehouse, 1967 Naphtali, 1965 Goldstein and Stanfield, 1970 Naphtali and Sandholm, 1971). Simultaneous solution methods are discussed at length in the textbook by Henley and Seader (1981) and by Seader (1986). 

Condensers and reboilers may be modeled as equilibrium stages. The independent variables for such a stage are the mole fractions (x,y, y y 2c), the temperature (7 1), the flow rates (Vj, Lj 2), and the stage pressure Pj 1). The corresponding 2c + 4 independent equations are the MESH equations (of Section 13.4.1) plus a pressure drop equation. 

The MESH equations constitute a system of n (2-k +3) equations. A mixture with 5 components fractionated in a column with 50 equilibrium stages ( = 50) is modeled by 650 algebraic equations. Furthermore, there is a large number of additional equations for modeling the phase equilibrium and the enthalpies. 

Rigorous models of staged distillation processes are formulated by setting up material balance equations, equilibrium relations, summation equations, and enthalpy balance equations (MESH equations). In these models, the extent of nonlinearity may be very severe, particularly for azeotropic and reactive distillation systems. MESH system based mathematical models can thus yield multiple solutions (multiple steady states), a fact which has been observed by many researchers. 

While there are only small thermal fluctuations in equilibrium, shear can enlarge them on small spatial scales (> the mesh size I in the following simulations) in an early stage t < yxo). In later times the fluctuations on various spatial scales appear and the system tends to a strongly fluctuating, dynamical steady state. Hence, random-source terms in the dynamic equations are indispensable at the onset. In our simulations we have therefore added the Gaussian random source terms,/y(r, t) in (3), V ffR(r, t) in (8) or (12), and 0(r, t) in (13) or (15), which are related to l/ j, rjQ, and C, respectively, to satisfy the fluctuation-dissipation relations. In our model, however, even after the dynamic equations are made dimensionless, these random source terms are still proportional to a common parameter e = which has not yet been specified, d be-... 

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