Component balance equation

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. 

For steady-state operation of a continuous stirred-tank reactor or continuous stirred-tank reactor cascade, there is no change in conditions with respect to time, and therefore the accumulation term is zero. Under transient conditions, the full form of the equation, involving all four terms, must be employed. 

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The third term on the left side of the equation has significance in reactive systems only. It is used with a positive sign when material is produced as a net result of all chemical reactions a negative sign must precede this term if material is consumed by chemical reactions. The former situation corresponds to a source and the latter to a sink for the material under consideration. Since the total mass of reactants always equals the total mass of products in a chemical reaction, it is clear that the reaction (source/sink) term (R should appear explicitly in the equation for component material balances only. The overall material balance, which is equivalent to the algebraic sum of all of the component balance equations, will not contain any (R term. 

This reaction cannot be elementary. We can hardly expect three nitric acid molecules to react at all three toluene sites (these are the ortho and para sites meta substitution is not favored) in a glorious, four-body collision. Thus, the fourth-order rate expression 01 = kab is implausible. Instead, the mechanism of the TNT reaction involves at least seven steps (two reactions leading to ortho- or /mra-nitrotoluene, three reactions leading to 2,4- or 2,6-dinitrotoluene, and two reactions leading to 2,4,6-trinitrotoluene). Each step would require only a two-body collision, could be elementary, and could be governed by a second-order rate equation. Chapter 2 shows how the component balance equations can be solved for multiple reactions so that an assumed mechanism can be tested experimentally. For the toluene nitration, even the set of seven series and parallel reactions may not constitute an adequate mechanism since an experimental study found the reaction to be 1.3 order in toluene and 1.2 order in nitric acid for an overall order of 2.5 rather than the expected value of 2. 

Reactor Performance Measures. There are four common measures of reactor performance fraction unreacted, conversion, yield, and selectivity. The fraction unreacted is the simplest and is usually found directly when solving the component balance equations. It is a t)/oo for a batch reaction and aout/ciin for a flow reactor. The conversion is just 1 minus the fraction unreacted. The terms conversion and fraction unreacted refer to a specific reactant. It is usually the stoichiometrically limiting reactant. See Equation (1.26) for the first-order case. 

Application of the general component balance, Equation (1.6), to a steady-state flow system gives... 

There are only two possible values for concentration in a CSTR. The inlet stream has concentration and everywhere else has concentration The reaction rate will be the same throughout the vessel and is evaluated at the outlet concentration, SIa = A(ctout,bout, ) For the single reactions considered in this chapter, continues to be related to by the stoichiometric coefficient and Equation (1.13). With SS a known, the integral component balance, Equation (1.6), now gives useful information. For component A,... 

In a batch vessel, the question of good mixing will arise at the start of the batch and whenever an ingredient is added to the batch. The component balance, Equation (1.21), assumes that uniform mixing is achieved before any appreciable reaction occurs. This will be true if Equation (1.55) is satisfied. Consider the same vessel being used as a flow reactor. Now, the mixing time must be short compared with the mean residence time, else newly charged... 

Assume that the entering material is rapidly mixed so that the composition is always uniform in the radial direction. The transpiration rate per unit length of tube is = q(z) with units of m /s. Component A has concentration Utrans = o-transi/) in the transpired stream. The component balance, Equation (3.4), now becomes... 

Where a chemical reaction occurs, the change, due to reaction, can be taken into account by the addition of a reaction rate term in the component balance equation. Thus in the case of material produced by the reaction... 

The resulting model would therefore consist of component balance equations for the soluble component written over each of the many solid and liquid subsystems of the packed bed, combined with the component balance equation for the coffee reservoir. The magnitude of the recirculating liquid flow will depend on the relative values of the pressure driving force generated by the boiling liquid and the fluid flow characteristics of the system. 

When there is transfer from one phase to another, the component balance equations must take this into account. Thus taking a balance for component i around the well-mixed phase G, with transfer of i from phase G to phase L, gives... 

The production rate term allows for the production or consumption of material by chemical reaction and can be incorporated into the component balance equation. Thus,... 

Assuming well-mixed conditions, the component balance equation is given by... 

Assuming a chemical reaction in the tank, in which the rate of reaction is proportional to concentration, the component balance equation now becomes... 

Consider the case of three, constant-volume tanks in series, as represented in Fig. 2.11, in which the tanks have differing volumes V], V2, V3, respectively.. Assuming well-mixed tanks, the component balance equations are... 

For a first-order process the time constant can be found from the defining differential equation as shown in Sec. 2.1.1.1. For the case of the aeration of a liquid, using a stirred tank, the following component balance equation applies... 

For unsteady-state operation the component balance equations, for each phase are now of the form... 

Adding the above two component balance equations gives the dynamic equation for the complete stage as... 

Note that the total component balance equation is linked here only to the equilibrium relationship. 

The individual component balance equations for each phase are... 

Allowing for the additional backmixing flow contributions, the component balance equation for the two phases in stage n of the cascade are now... 

Owing to the intensive agitation conditions and intimate phase dispersion, obtained within the mixing compartment, the mixer can usually be modelled as a single, perfectly mixed stage in which the rate of mass transfer is sufficient to attain equilibrium. As derived previously in Sec. 3.3.1.3, the component balance equations for the mixer, based on the two combined liquid phases, is thus given by... 

Allowing for the additional flow contributions due to the entrainment backmixing, the component balance equation.s, for any mixer, n, along the cascade, are now expressed by... 

Taking the phase flow rate, G , to represent the dispersed phase, the component balance equations now become, for any stage n... 

Considering the end regions of the column as well-mixed stages, with small but finite rates of mass transfer, component balance equations can be derived for end stage 0... 

In the component balance equations, dY]/dt will therefore be zero, whereas dXj/dt may still be quite large. This can obviously cause considerable difficulties in the integration procedure, owing to equation stiffness. 

For a simple binary distillation, the component balance equation becomes... 

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

Thus for component i of a system of j components, the component balance equation, on the nth plate, becomes... 

As explained by Franks (1972), this again shows that the component balance equations, for the different components of the mixture, will thus have different time constants, which depend on the relative magnitudes of the equilibrium constants K and which again can lead to possible problems of numerical stiffness. 

The concentration gradient terms, dC/dZ, both in and out of segment n, can be approximated by means of their finite-differenced equivalents. Substituting these into the component balance equation, gives... 

At the air-solid surface, Z=0, the drying rate is determined by the convective heat and mass transfer drying conditions and the surrounding atmosphere of the drier. Assuming that the drying rate is known, the component balance equation for segment 1 becomes... 

A component balance equation can be derived for the element AV, based on the generalised component balance expression, where for any reactant, A... 

Substituting these quantities, gives the resulting component balance equation... 

The component balance equation can also be written in terms of fractional conversion, Xa, where for constant volumetric flow conditions... 

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