Total Mass Balance Equation

It becomes necessary to incorporate a total mass 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 mass balance equation can generally be neglected. 

The dynamic total mass balance equation is represented by 

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

This situation is one involving both a total and a component mass balance, combined with a kinetic equation for the rate of decomposition of the waste component. Neglecting density effects, the total mass 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 mass balance equation, provided additional data, relating liquid density to concentration are available. 

When the system has been defined, an equation can be written to state the law of conservation of mass. Such a total mass balance equation can be given in verbal form as... 

In addition to the total mass balance, equations can be written to describe changes in each of the individual chemical species, or components, that are present. As with the total mass, the mass of a component can be altered by exchange with the surroundings. However, it can also be affected by chemical reactions occurring within the system, converting one component to another. The total mass of the system is not affected by such interconversions, since the mass of reactants consumed is exactly equal to the mass of products formed. In verbal form, the component mass balance for a particular component A in the system is... 

These equations apply to the total mass or mass density of the system, while we use moles when describing chemical reaction. Therefore, whenever we need to solve these equations simultaneously, we must transform our species mass balances into weight fraction when including momentum and total mass-balance equations. 

Mujtaba (1999) explained the BED operational constraint (path constraint) using two time intervals in a separation Task (producing a cut). To ensure a safe operation Mujtaba (1999) proposed the following treatment to the path constraint. The reboiler holdup profile (HTi, Figure 10.3) during any distillation Task is the solution of the total mass balance equation and can be expressed as ... 

Only a few modifications of the algorithm were required to make it applicable to absorption and reboiled absorption. The changes were mainly in the handling of the enthalpy and total mass balance equations to accommodate different specification combinations involving the reflux, heat duties, and top and bottom product flow rates. The results of two example problems, one each for absorption and reboiled absorption, are shown in Table II. 

Since the total vapor and liquid flow rates are, by definition, the sum of the component flow rates of the respective phases, the summation equations and total mass balance equation are satisfied automatically. Thus, the number of equations and variables per stage has been reduced by 2 to 2c + 1. 

The total mass balance equation with respect to the elementary local length dx ... 

The total mass balance equations, describing the relationship between the internal and external flow rates, are the same for both the TMBR and SMBR models (Eqs. 8.11, 8.13, 8.15 and 8.17). Component balances, however, have to take into account the solid streams. 

Knowing this mass loss, the total mass balance equation for the solute in the reservoir is... 

Adding the above two equations, we obtain the total mass balance equation (9.2-lb), which basically states that the total hold-up in both phases is governed by the net total flux contributed by the two phases. This is true irrespective of the rate of mass exchange between the two phases whether they are finite or infinite. Now back to our finite mass exchange conditions where the governing equations are (9.4-3a) and (9.4-3b). To solve these equations, we need to impose boundary conditions as well as define an initial state for the system. One boundary is at the center of the particle where we have the usual symmetry ... 

The nsnal total mass balance equations can be written but are not given here. Applying the generalized least squares method like in equilibrium analysis, a set of unknown parameters fPp and KJ can be found searching for the minimum of... 

Note The total mass balance equations will not have any generation/de-pletion terms, these will always be zero (as long as we do not have a nuclear reaction changing mass into energy). Of course, the same does not apply to the molar balance equations when there are chemical reactions associated with a change in the number of moles, as shown earlier. 

The overall total mass balance describes the total mass in a system. Obviously there can be only one total mass balance equation, and the net generation term is zero, which means that mass is a conserved quantity. 

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