Equations material balance equation

Reaction class Reaction type Basic differential equation Material balance equations Product distribution ... 

The material balance equation relating produced volume of oil (Np stb) to the pressure drop in the reservoir (AP) is given by ... 

Solution of the material balance equations gives the pressure and... 

Knowing the solution of the material balance equations It is easy to calcu>... 

For ease of exposition, let us limit attention to. two independent reactions--the generalization to more reactions is straightforward. Then the material balance equations take the form... 

In order to Introduce thermal effects into the theory, the material balance equations developed in this chapter must be supplemented by a further equation representing the condition of enthalpy balance. This matches the extra dependent variable, namely temperature. Care must also be taken to account properly for the temperature dependence of certain parameters In... 

As In the case of the material balance equations, the enthalpy balance can be written in dimensionless form, and this introduces new dimensionless parameters in addition to those listed in Table 11.1. We shall defer consideration of these until Chapter 12, where we shall construct the unsteady state enthalpy and material balances, and reduce them to dimensionless form. 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

Formulate the constraining material-balance equations, based on conservation of the total number of atoms of each element in a system comprised of w elements. Let subscript k identify a particular atom, and define Ai as the total number of atomic masses of the /cth element in the feed. Further, let a be the number of atoms of the /cth element present in each molecule of chemical species i. The material balance for element k is then... 

If species i is an element, AG/ is zero. There are N equilibrium equations (Eqs. [4-355]), one for each chemical species, and there are w material-balance equations (Eqs. [4-353]), one for each element—a total of N + to equations. The unknowns in these equations are the (note that y, = of which there are N, and the Xi, of which... 

Operating Lines The McCabe-Thiele method is based upon representation of the material-balance equations as operating lines on the y-x diagram. The lines are made straight (and the need for the energy balance obviated) by the assumption of constant molar overflow. The liqmd-phase flow rate is assumed to be constant from tray to tray in each sec tiou of the column between addition (feed) and withdrawal (produc t) points. If the liquid rate is constant, the vapor rate must also be constant. 

The effect of a sidestream withdrawal point is illustrated by Fig. 13-29. The material-balance equation for the column section below the sidestream is... 

Replace the holdup derivatives in Eqs. (13-149) to (13-151) by total-stage material-balance equations (e.g., dMj/dt = Vj + i + Ej- — Vj — Lj) and solve the resulting equations one at a time by the predictor step of an explicit integration method for a time increment that is determined by stability and truncation considerations. If the mole fraclions for a particular stage do not sum to 1, normalize them. 

For more precise values, computer programs can be used to calculate soluble recoveiy as weh as solution compositions for conditions that are typical of a CCD circuit, with varying underflow concentrations, stage efficiencies, and solution densities in each of the stages. The calculation sequence is easily performed by utihzing material-balance equations around each thickener. 

Evaluations of the measurements are based on the transient material balance equation for CSTR ... 

Now you can reconsider the material balance equations by adding those additional factors identified in the previous step. If necessary, estimates of unaccountable losses will have to be calculated. Note that, in the case of a relatively simple manufacturing plant, preparation of a preliminary material-balance system and its refinement (Steps 14 and 15) can usefully be combined. For more-complex P2 assessments, however, two separate steps are likely to be more appropriate. An important rule to remember is that the inputs should ideally equal the outputs - but in practice this will rarely be the case. Some judgment will be required to determine what level of accuracy is acceptable, and we should have an idea as to what the unlikely sources of errors are (e.g., evaporative losses from outside holding ponds may be a materials loss we cannot accurately account for). In the case of high concentrations of hazardous wastes, accurate measurements are needed to develop cost-effective waste-reduction options. It is possible that the material balance for a number of unit operations will need to be repeated. Again, continue to review, refine, and, where necessary, expand your database. The compilation of accurate and comprehensive data is essential for a successful P2 audit and subsequent waste-reduction action plan. Remember - you can t reduce what you don t know is therel... 

Consider a first order, exothermic reaction (aA —> products) in a CFSTR having a constant supply of new reagents, and maintained at a steady state temperature T that is uniform throughout the system volume. Assuming perfect mixing and no density change, the material balance equation based on reactants is expressed as uC g = +... 

Substituting the rate expression into the material balance equation gives uC o = + kV C and the concentration of A at any time... 

Set up the material balance equations for the competitive adsorption of A and B on an adsorbentphase.Sincetherearetwophasesandtwocomponentstheremustbefourcompo-nentequations. 

For any transient process that begins at time t and is terminated at a later time tp, the general integral material balance equation has the form... 

When we substitute this expression into the material-balance equation, we obtain [HA] = LHA]initial - [A ] = LHA]initial - ([H,Q ] -... 

BATCH MATERIAL BALANCE EQUATIONS FOR FREE-RADICAL POLYMERIZATIONS AND COPOLYMERIZATIONS... 

An extended treatment of material balance equations, with substantial emphasis on component balances in reacting systems, is given in... 

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Figure 4. Heat transfer model, energy and material balance equations, boundary and initial conditions plus physical properties.

In integral analysis concentration-versus-time (or equivalently concentration-versus-distance from the inlet of the integral flow reactor) data are known. Kinetic expressions to be determined are incorporated into the differential material balance equations ... 

For any given stage, n, the component material balance equations for each phase are thus defined by... 

When deriving material balance equations the rate of each component transformation in the reactor obeys the law of mass action. However, as distinct from the reactions with participation of exclusively low molecular weight substances, the... 

The Flory principle allows a simple relationship between the rate constants of macromolecular reactions (whose number is infinite) with the corresponding rate constants of elementary reactions. According to this principle all chemically identical reactive centers are kinetically indistinguishable, so that the rate constant of the reaction between any two molecules is proportional to that of the elementary reaction between their reactive centers and to the numbers of these centers in reacting molecules. Therefore, the material balance equations will comprise as kinetic parameters the rate constants of only elementary reactions whose number is normally rather small. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Number of variables (component flow rates) = 9 Number of independent material balance equations = 3... 

The block shown in Figure 4.6 represents any unit in an information flow diagram, and shows the nomenclature that will be used in setting up the material balance equations. 

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