Total Material Balance Equation

It becomes necessary to incorporate a total material balance equation into the reactor model, whenever the total quantity of material in the reactor varies, as in the cases of semi-continuous or semi-batch operation or where volume changes occur, owing to density changes in flow systems. Otherwise the total material balance equation can generally be neglected. 

The dynamic total material balance equation is represented by 

The general component balance for a well-mixed tank reactor or reaction region can be written as 

For batch reactors, there is no flow into or out of the system, and those terms in the component balance equation are therefore zero. 

For semi-batch reactors, there is inflow but no outflow from the reactor and the outflow term in the above balance equation is therefore zero. 

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For zero flow of liquid into the tank and assuming constant density conditions, the total material balance equation becomes... 

This situation is one involving both a total and a component material balance, combined with a kinetic equation for the rate of decomposition of the waste component. Neglecting density effects, the total material balance equation is... 

These equations complete a preliminary model for the mixer. Note that it is also possible, in principle, to incorporate changing density effects into the total material balance equation, provided additional data, relating liquid density to concentration, are available. 

Calculating the total flow rates. Rearrangement of the total material balance equations parallels that of the component balance equations above. To create the tridiagonal matrix, the equations are rearranged to allow for the total vapor rates to be used as the independent valuables. Ratios of the total liquid and vapor rates become the coefficients of the vapor rate terms that go into the tridiagonal matrix. 

In matrix form, the total material balance equations are expressed... 

In the second variant, the plug flow model is considered as a series of tanks with perfect mixing flow [3.22, 3.23]. In this case, the real filter will be supposedly replaced by a series of some small filters (three in this analysis) with perfect mixing flow. Figure 3.9 shows the scheme, relations and notations used. The filtrate transfer equation has been used for the mathematical characterization of each small filter for the total material balance equation and non-steady-state solid balance equation ... 

The results obtained above for vapor and liquid enthalpies together with other equations including the total-material balance equations and the component-material balance equations were substituted into Eq. (5-58) to obtain the following form of the energy balance equations which were employed to calculate the liquid phase rates for any stage j [Pg.209]

If we expand the total material balance equation, we obtain the sum of these two back and so we are unable to solve the problem. This is because we have the unknown and only two independent equations. The unknowns are the concentrations of each component and the change in volume. Each is a function of time. We need another independent equation. To obtain this we need to think about what is happening. 

A total material balance equation can be used in place of (15-3) or (15-4). It is derived by combining these two equations and S, zy = 1.0 with (15-1) summed over the C components and over stages 1 through j to give... 

As wiil be shown in warm-up example 2 if the process includes by-passes, side-streams, and/or recycle streams (see Figure 7.16) it is advisable to include an additional process-unit (mixer) in your flow-diagram showing the mixture of one stream and, for example, a recycle stream. This advice is critical because you need to consider this "additional unit" when determining the number of independent materials balance equations NMB). in addition, if a stream is divided (e.g. side-stream), all new streams will have the same composition but you need to add one total material balance equation. When the side-stream joins the process again, it is necessary to add a new process-unit (mixer). 

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... 

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. 

To isolate a system for study, the system is separated from the surroundings by a boundary or envelope that may either be real (e.g., a reactor vessel) or imaginary. Mass crossing the boundaiy and entering the system is part of the mass-in term. The equation may be used for any compound whose quantity does not change by chemical reaction or for any chemical element, regardless of whether it has participated in a chemical reaction. Furthermore, it may be written for one piece of equipment, several pieces of equipment, or around an entire process (i.e., a total material balance). 

This combined equation represents a differential total material balance of a component, whether present as HA or the reaction product A-, within the reacting phase. The reader is referred to Olander s original paper for a more complete rationale for generating these differential component material balances for systems of reacting species near equilibrium. By using Olander s technique, the system of four differential equations with reaction terms can be simplified significantly to two differential equations with no reaction terms. 

If the initial condition of the reactor contents is known and if the feedstream conditions are specified, it is possible to solve equation 8.6.1 to determine the effluent composition as a function of time. The solution may require the use of material balance relations for other species or a total material balance. This is particularly true of variable volume situations where the following overall material balance equation is often useful. 

From the material balance, equation 14.4-1, the total volume is... 

The situation is the same as in Fig. 1.18 but without material leaving the reactor. Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total material balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

For reactions involving heat effects, the total and component material balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by where each term has units of kj/s. For steady-state operation the accumulation... 

The information flow diagram for a non-isothermal, continuous-flow reactor (in Fig. 1.18, shown previously in Section 1.2.5) illustrates the close interlinking and highly interactive nature of the total material balance, component material balance, energy balance, rate equation, Arrhenius equation and flow effects F. This close interrelationship often brings about highly complex dynamic behaviour in chemical reactors. 

Fermentation systems obey the same fundamental mass and energy balance relationships as do chemical reaction systems, but special difficulties arise in biological reactor modelling, owing to uncertainties in the kinetic rate expression and the reaction stoichiometry. In what follows, material balance equations are derived for the total mass, the mass of substrate and the cell mass for the case of the stirred tank bioreactor system (Dunn et ah, 2003). 

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... 

Total mass and component material balance equations are written for all the plates of the column, for the still and for the top reflux drum. 

The actual volume of each phase in element AV is that of the total volume of the element, multiplied by the respective fractional phase holdup. Hence considering the direction of solute transfer to occur from the aqueous or feed phase into the organic or solvent phase, the material balance equations become ... 

The material balance equations for the FCC unit are easily expressed in terms of the yield equations presented earlier. If F represents the total inlet feed (barrels per day, BPD) to the FCC unit and Y, is the yield of product i as read from Tables 2.3 and 2.4, then the production of product i can be simply obtained by multiplying the feed to the unit by the yield of product i (i.e., FY ). Such material balance equations must be written for all units of a refinery in order to prepare a mathematical... 

This equation is the total material balance for the stage. Another balance can be written by equating input to output for component i giving the component i balance ... 

Combining the top total material balance with the equation for the reflux ratio, gives an expression for the internal liquid flow L above the feed stage (assuming Ln = Ln+1 = L) ... 

The rate expressions, 2 , in the material balance equations depend on the partial pressures, Pi, of reactants and products. For reactions involving ideal gases at constant total pressure, Pt° t, the partial pressure / can be related to Fsa or Nsa as follows ... 

Note that the above model of the fast dynamics involves only the large recycle and internal flow rates u1, and does not involve the small feed/product flow rates us. Examining the material balance equations in (3.1), it is intuitive that the flow rates of the internal streams do not affect the total holdup of any component in the process, and that total holdups are affected only by the flow rates us of the... 

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. 

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