MESH Material balance, Equilibrium

These four equations are the so-called MESH equations for the stage Material balance, Equilibrium, Summation and Heat (energy) balance, equations. MESH equations can be written for each stage, and for the reboiler and condenser. The solution of this set of equations forms the basis of the rigorous methods that have been developed for the analysis for staged separation processes. 

MESH equations A set of equations used for calculating the material, energy, and thermodynamic profiles within a distillation column. They combine material balances, equilibrium conditions, summation conditions, and heat balances, which are required to... 

The method of Gallun and Holland is the broadest application of the MESH equations in a global Newton method and may solve the widest range of columns. Formulations by Gallun and Holland (40) for distillation columns included adding the total material balance to give freedom in specifications or to substitute these for the equilibrium equations for more ideal mixtures. 

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. 

Simulation of multicomponent rectification is based on the so-called MESH equations (M = material balances, E = equilibrium, S = summation of mole fractions, H = heat balance) formulated for a single equilibrium stage (Fig. 5.2-33) as follows ... 

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. 

SC (simultaneous correction) method. The MESH equations are reduced to a set of N(2C +1) nonlinear equations in the mass flow rates of liquid components ltJ and vapor components and the temperatures 2J. The enthalpies and equilibrium constants Kg are determined by the primary variables lijt vtj, and Tf. The nonlinear equations are solved by the Newton-Raphson method. A convergence criterion is made up of deviations from material, equilibrium, and enthalpy balances simultaneously, and corrections for the next iterations are made automatically. The method is applicable to distillation, absorption and stripping in single and multiple columns. The calculation flowsketch is in Figure 13.19. A brief description of the method also will be given. The availability of computer programs in the open literature was cited earlier in this section. 

In Chapter 1 it was shown that the number of independent variables for any problem is equal to the difference between the total number of variables and the number of linking equations and other relationships. Examples of the application of this formal procedure for determining the number of independent variables in separation process calculations are given by Gilliland and Reed (1942) and Kwauk (1956). For a multistage, multicomponent column, there will be a set of material and enthalpy balance equations and equilibrium relationships for each stage (the MESH equations) and for the reboiler and condenser, for each component. 

A rigorous mathematical description of a multistage cascade involves material and energy balances and some representation of the vapor-liquid equilibrium. The resulting set of algebraic equations is called the MESH Equations, for reasons which will soon become apparent. 

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