Material balance method equations

If the sphere in Fig. 6.2-3a is evaporating, the radius r of the sphere decreases slowly with time. The equation for the time for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation (6.2-32), where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance. (See Problem 6.2-9 for this case.) The material-balance method is similar to Example 6.2-3. The final equation is... 

The amount of nonfreezable water (defined relative to a very low temperature, e.g., - 100°C) may be determined by plotting the total melting enthalpy of water per unit weight of sample versus the sample composition and extrapolating to zero enthalpy [8], i.e., the nonfreezable water content is equated with the maximum amount of water for which no enthalpic peak has been detected [147]. We used a material balance method for this determination. During the investigation... 

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. 

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. 

The material balance was calculated for EtPy, ethyl lactates (EtLa) and CD by solving the set of differential equation derived form the reaction scheme Adam s method was used for the solution of the set of differential equations. The rate constants for the hydrogenation reactions are of pseudo first order. Their value depends on the intrinsic rate constant of the catalytic reaction, the hydrogen pressure, and the adsorption equilibrium constants of all components involved in the hydrogenation. It was assumed that the hydrogen pressure is constant during... 

The procedure is based on the theory of recycle processes published by Nagiev (1964). The concept of split-fractions is used to set up the set of simultaneous equations that define the material balance for the process. This method has also been used by Rosen (1962) and is described in detail in the book by Henley and Rosen (1969). 

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. 

The method proposed by Lewis and Matheson (1932) is essentially the application of the Lewis-Sorel method (Section 11.5.1) to the solution of multicomponent problems. Constant molar overflow is assumed and the material balance and equilibrium relationship equations are solved stage by stage starting at the top or bottom of the column, in the manner illustrated in Example 11.9. To define a problem for the Lewis-Matheson method the following variables must be specified, or determined from other specified variables ... 

With the exception of this method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column other conditions will exist at start-up, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behaviour of the column. 

If the equilibrium relationships and flow-rates are known (or assumed) the set of material balance equations for each component is linear in the component compositions. Amundson and Pontinen (1958) developed a method in which these equations are solved simultaneously and the results used to provide improved estimates of the temperature and flow profiles. The set of equations can be expressed in matrix form and solved using the standard inversion routines available in modem computer systems. Convergence can usually be achieved after a few iterations. 

For semibatch or semiflow reactors all four of the terms in the basic material and energy balance relations (equations 8.0.1 and 8.0.3) can be significant. The feed and effluent streams may enter and leave at different rates so as to cause changes in both the composition and volume of the reaction mixture through their interaction with the chemical changes brought about by the reaction. Even in the case where the reactor operates isothermally, numerical methods must often be employed to solve the differential performance equations. 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

Equations 11.1.33 and 11.1.39 provide the basis for several methods of estimating dispersion parameters. Tracer experiments are used in the absence of chemical reactions to determine the dispersion parameter )L this value is then employed in a material balance for a reactive component to predict the reactor effluent composition. We will now indicate some methods that can be used to estimate the dispersion parameter from tracer measurements. 

Table 19.4 gives the calculations of E(t), t, and of based on the histogram method. Column (4) lists the values of c j calculated from (A). Column (5) gives At, required for the calculations in subsequent columns, from equations 19.3-12 to -14. Column (6) gives values for the tracer material-balance (see below). Column (7) gives values of E t) from equation 19.3-4 with C(t) = E(t). Columns (8) and (9) give values required for the calculation of Tin equation 19.3-7, and of of in 19.3-8, respectively. 

Method (b). Numerical integration of the four equations, (3) to (6) is accomplished with ODE, either Constantinides or POLYMATH. Alternately, the material balances (11 and (2) and only the two differential equations (3) and (4) can be solved together. This procedure is better carried out with POLYMATH ODE. 

Method (a). The reaction will be conducted to 90% of equilibrium conversion. Applying material balances, the rate equation becomes,... 

Effluent concentrations from a CSTR will be found for several kinds of Input by several methods, seven cases in all. The differential equations represent the material balances in the form Input = Output + Accumulation. 

Apply the method of lines to the heat and material balances of P8.01.04. The differential equations that apply except at the center and the wall are,... 

Extending the method to a multicomponent mixture, the total material balance remains the same, but separate component balance equations must now be written for each individual component i, giving... 

Method I. To illustrate the application of these methods, the experimental data for the adsorption of H+/OH on TiC (32 ) are considered (Figure 5). Use of Method I involves the approximations in the material balance equations that, on the acidic branch of the titration curve,... 

Binary systems of course can be handled by the computer programs devised for multicomponent mixtures that are mentioned later. Constant molal overflow cases are handled by binary computer programs such as the one used in Example 13.4 for the enriching section which employ repeated alternate application of material balance and equilibrium stage-by-stage. Methods also are available that employ closed form equations that can give desired results quickly for the special case of constant or suitable average relative volatility. 

The equations are solved graphically. Values of ex are assumed and corresponding values of e2 are calculated from each equation by the Newton-Raphson method. The intersection of the curves is at ei = 0.46402 and e2 = 0.35903. The corresponding mol fractions are tabulated alongside the material balances. 

Finding the time required for a particular conversion involves the solution of two simultaneous equations, i.e. 1.24 or 1.25 for the material balance and 1.27 for the heat balance. Generally, a solution in analytical form is unobtainable and numerical methods or analogue simulation must be used. Taking, for example, a first-order reaction with constant volume ... 

To obtain the temperature and concentration profiles along the reactor, equation 1.40 must be solved by numerical methods simultaneously with equation 1.35 for the material balance. 

Equations 7.4 and 7.5 form a system of differential equations for which no analytical solution is known. Thus, the description of the behavior of the semi-batch reactor with time requires the use of numerical methods for the integration of the differential equations. Usually, it is convenient to use parameters which are more process-related to describe the material balance. One is the stoichiometric ratio between the two reactants A and B ... 

The energy balances are not solved in the same manner as the component or total material balances. With some solution methods, they are simultaneously solved with other MESH equations to get the independent cc umn variables in others they are used in a more limited manner to get a new set of total flow rates or stage temperatures. 

The theta method. This method has been primarily applied to the Thiele-Geddes equations but a form of the theta method equation has also been applied to the equations of the Lewis-Matheson method. The main independent variable of the method is a convergence promoter, theta (or 6). The convergence promoter 0 is used to force an overall component and total material balance and to adjust the compositions on each stage. These new compositions are then used to calculate new stage temperatures by an approximation of the dew- or bubble-point equation called the Kb method. The power of the Kb method is that it directly calculates a new temperature without the sort of failures that occur when iteratively solving the bubble- or dew-point equations. 

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. 

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