Mass balances simultaneous solution

The couphng equation is a vapor mass balance written at the vent system entrance and provides a relationship between the vent rate W and the vent system inlet quahty Xq. The relief system flow models described in the following section provide a second relationship between W and Xo to be solved simultaneously with the coupling equation. Once W andXo are known, the simultaneous solution of the material and energy balances can be accomplished. For all the preceding vessel flow models and the coupling equations, the reader is referred to the DIERS Project Manual for a more complete and detailed review. 

If he selects the still pressure (which for a binary system will determine the vapour-liquid-equilibrium relationship) and one outlet stream flow-rate, then the outlet compositions can be calculated by simultaneous solution of the mass balance and equilibrium relationships (equations). A graphical method for the simultaneous solution is given in Volume 2, Chapter 11. 

However, if he selects an outlet stream composition (say the liquid stream) instead of a flow-rate, then the simultaneous solution of the mass balance and v-l-e relationships would not be necessary. The stream compositions could be calculated by the following step-by-step (sequential) procedure ... 

The formal, algebraic, method. The presence of recycle implies that some of the mass balance equations will have to be solved simultaneously. The equations are set up with the recycle flows as unknowns and solved using standard methods for the solution of simultaneous equations. 

Simultaneous solution of the mass balance equation 3.87 and the heat balance equation 3.90 with the appropriate boundary conditions gives z as a function of Y. 

In Ref. 139 a purely numerical approach to the solution of the considered complex RA problem was suggested. The liquid film is treated as an additional balance region, in which reaction and mass transfer occur simultaneously. Therefore, the reactions are considered both in the liquid-bulk-phase mass balances, Eq. (Al), and in the differential balances for the liquid film, Eq. (A10). 

The pattern of flow through a packed adsorbent bed can generally be described by the axial dispersed plug flow model. To predict the dynamic response of the column therefore requires the simultaneous solution, subject to the appropriate initial and boundary conditions, of the differential mass balance equations for an element of the column,... 

Industrial design problems often occur in tubular reactors that involve the simultaneous solution of AP, energy, and mass balances. 

When a fast reaction is highly exothermic or endothermic and, additionally, the effective thermal conductivity of the catalyst is poor, then significant temperature gradients across the pellet are likely to occur. In this case the mass balance (eq 32) and the enthalpy balance (eq 33) must be simultaneously solved using the corresponding boundary conditions (eqs 34-37), to obtain the concentration profile of the reactant and the temperature profile inside the catalyst pellet. The exponential dependence of the reaction rate on the temperature thereby imposes a nonlinear character on the differential equations which rules out an exact analytical treatment. Approximate analytical solutions [83, 99] as well as numerical solutions [13, 100, 110] of eqs 32-37 have been reported by various authors. 

Mass balances are to be solved for this process. Perfect separation between methanol and the reactants is assumed. Unreacted reactants are recycled to the reactor to improve their utilization. The recycle stream within the process complicates solving the mass balances, for there is a circularity in the logic of the solution. The mass balance equations must be solved simultaneously rather than singly, or solved iteratively, as is done with a spread sheet. 

Kosinski and Bostian (12) reported the extraction of lanthanum by HDEHP from aqueous nitrate solutions over a wide concentration range. Using mass balance data and IR analysis of the organic phase, they proposed three extraction reactions which occur simultaneously ... 

Ionization of the oxide/water interface and the resultant electrical double layer have been studied intensively by a variety of techniques within the last decade. Although many electrical double layer and adsorption models have been proposed, few are sufficiently general to consider surface equilibria in complex electrolyte solutions. Recently we proposed a comprehensive adsorption model for the oxide/water interface which can simultaneously estimate adsorption density, surface charge, and electro-kinetic potential in a self-consistent manner (jL, 2, 3). One advantage of the model was that it could be incorporated within the computer program, MINEQL ( ), by adding charge and mass-balance equations for the surface. 

Thermodynamic data, whether determined through calorimetry or solubility studies, are subject to refinement as more exact values for the components in the reaction scheme, or more complete description of the solution phases, become available. Many of the solubility studies on clays were done before digital-computer chemical equilibrium programs were available. One such program, SOLMNEQ, written by one of the authors ( ) solves the mass-action and mass-balance equations for over 200 species simultaneously. SOLMNEQ was employed in this investigation to convert the chemical analytical data into the activities of appropriate ions, ion pairs, and complexes. 

To solve the mass balance, it must be accompanied by the simultaneous solution of the energy balance (i.e., the solution of Equation (9.2.9)). To do this, Equation (9.2.9) can be written in more convenient forms. Consider that the enthalpy contains both sensible heat and heat of reaction effects. That is to say that Equation (9.2.10) can be written as ... 

Example 10.2.1 illustrates the simultaneous solution of the mass and energy balances for an adiabatic, fixed-bed reactor with no fluid density changes and no transport limitations of the rate, that is, rj = 1. Next, situations where these simplifications do not arise are described. 

The solution of mass-balance systems requires the simultaneous solution of a set of equations in which the number of constraints (solutes) equals the number of mineral phases. This can be represented in the form of a matrix ... 

If it is assumed that calcite, dolomite, gypsum, and carbon dioxide are the phases to be considered, mass balance can be described by four linear equations of the form given by Equation (9). Simultaneous solution of these four equations for water chemistry changes between unmineralized rainwater to the water composition of Polk City yields the mass balance ... 

First, let us count the number of independent equations. You should recognize that not aU of the four mass balances [Eqs. (2.4a)-(2.4d)], are independent equations. Do you see how the sum of the three component balances (2.4a), (2.4b), and (2.4c) equals the total mass balance In any problem you can substitute the total material balance for any one of the component material balances if you plan to solve two or more equations simultaneously or if the substitution makes the solution procedure simpler. 

Homemade models are often mass and energy balance spreadsheets, simplified kinetic models, or the simultaneous solution of the convection diffusion and heat equations together with nonlinear isotherms. All levels of models have their place. 

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