Enthalpy balance equations

The equilibrium ratios are not fixed in a separation calculation and, even for an isothermal system, they are functions of the phase compositions. Further, the enthalpy balance. Equation (7-3), must be simultaneously satisfied and, unless specified, the flash temperature simultaneously determined. 

DGA Partial derivative of the enthalpy balance equation (7-14) with respect to the vapor-feed ratio. 

The principles outlined so far may be used to calculate the tower height as long as it is possible to estimate the temperature as a function of Hquid concentration. The classical basis for such an estimate is the assumption that the heat of solution manifests itself entirely in the Hquid stream. It is possible to relate the temperature increase experienced by the Hquid flowing down through the tower to the concentration increase through a simple enthalpy balance, equation 68, and thus correct the equiHbrium line in ajy—a diagram for the heat of solution as shown in Figure 9. 

Using a similar treatment as described previously in Section 4.3.5, the resulting finite difference form of the enthalpy balance equations for any element n become... 

TaUe 1. Dimensionless representation of the stationary mass and enthalpy balance equations for combined interphase and intraparticle transport and reaction (single, nth order, irreversible reactions). 

In the case of adiabatic rather than isothermal flashing, when the total enthalpy of the system rather than its temperature is specified, the equations associated with isothermal flash are solved jointly with an enthalpy-balance equation, treating the temperature as another variable. The general enthalpy balance is... 

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. 

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. 

The model of the distillation column used throughout the paper is developed by [10], composed by the mass, component mass and enthalpy balance equations used as basis to implement a SIMULINK model (figure 1) which describes the nonlinear column dynamics as a 2 inputs Q, Lvo/) and 2 output (Ad, Ab ). Implementations details for the overall column dynamics are given in [11]. 

Multi-stage separation calculations require heat balances involving transition between the phases. The enthalpy balance equations must, therefore, include heats of vaporization and condensation. Usually these heats are not calculated separately but are implied by using appropriate methods for calculating liquid enthalpies and vapor enthalpies. Consider, for instance, a system where a vapor stream and a liquid stream enter with enthalpies and // respectively, and leave as a mixed phase stream with vapor enthalpy and liquid enthalpy h2. These are total stream enthalpy rates with units such as kJ/h. If no heat is added to or removed from the system, an energy balance is written as... 

These two equations are subtracted from the enthalpy balance equation. The... 

The enthalpy balance (Equation 12.35) is satisfied, giving a residual of 8 kJ/h. Compared with the rigorous perturbed model results, the reduced model predicts the products enthalpies and temperatures fairly well. The products compositions can use some improvement. It is noted that Equation 12.38, used for calculating the stream enthalpies in the reduced model, does not take into account the effect of composition. This could be improved by using the ideal solution method as applied in Example 12.5. This method is still computationally simple enough to satisfy the reduced model conditions of reduced computing time. 

If desired, (12-79) through (12-90) can be applied with mass units rather than mole units. No enthalpy balance equations are required because ordinarily temperature changes in an adiabatic extractor are not great unless the feed and solvent enter at appreciably different temperatures or the heat of mixing is large. Unfortunately, the group method is not always reliable for liquid-liquid extrac-... 

If the heat losses from the connecting pipes are taken into consideration [45], further component models must be developed for determination of the losses. In this case the coupling equations are interpreted for the relation of loss part models and the models of the units. If the loss heat flow rate of the collector-storage pipe is c-t> th t of the storage-collector duct is x-c thos of th coUector-heat exchanger-collector pipes accordingly are c-h h-c> respectively, and the coupling equations between the units can be written for the different modes of operation on the basis of the enthalpy balance equations... 

In the combined mode I h- II of operation, the inlet temperatures can be determined from the mixing enthalpy balance equation. 

We could, for example, consider the mathematical modeling of a flash drum separator or a distillation column. The stoichiometric equations (order of magnitude 1) stay with the enthalpy balance equations (order of magnitude 10 -10 ) and significant differences in orders of magnitude are present in the resulting nonlinear system. 

Unsteady-state mass, momentum, and enthalpy balance equations in the bulk gas phase of the reaction side are written 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. 

The column section may be solved by simultaneous solution of the component mass balance and enthalpy balance equations and the vapor-liquid equilibrium relations. Additional equations include the temperature, pressure, and composition dependence of the equilibrium coefficients and enthalpies. The equations for stage j are as follows ... 

The physical property parameters as derived here can be used with the other equations of the reduced model to calculate the column section. The reduced model should be able to reproduce the base case results exactly. The calculations can be carried out iteratively using a scheme similar to the one described in Section 12.4.1. The calculations start with the base case stripping factor to solve for the product stream component rates. Equations 12.3a and 12.37b are then solved with the reduced model /f-value equations (Equations 12.39, 12.40, and 12.41) to calculate the top and bottom tray temperatures. The enthalpy balance (Equation 12.35), coupled with the enthalpy model (Equation 12.38) and the overall material balance (Equation 12.36b), is solved for the product vapor and liquid flow rates. The new product component rates are Anally used in Equation set 12.33 to calculate then is calculated from Equation 12.42. The computations are repeated until converges within an acceptable tolerance. 

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