Balance component

Each chemical species, in the system, can be described by means of a component balance around an arbitrary, well-mixed, balance region, as shown in Fig. 1.16. 

In the case of chemical reaction, the balance equation is represented by 

Expressed in terms of volume, volumetric flow rate and concentration, this is equivalent to 

In the case of an input of component i to the system by interfacial mass transfer, the balance equation now becomes  

A plant discharges an aqueous effluent at a volumetric flow rate F. Periodically, the effluent is contaminated by an unstable noxious waste, which is known to decompose at a rate proportional to its concentration. The effluent must be diverted to a holding tank, of volume V, prior to final discharge, as in Fig. 1.14 (Bird et al. 1960). 

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The mass-balance restrictions are the C balances written for the C components present in the system. (Since we will only deal with non-reactive mixtures, each chemical compound present is a phase-rule component.) An alternative is to write (C — 1) component balances and one overall mass balance. 

Vapor and liqmd streams and respectively are in equilibrium with each other by definition and therefore are at the same T and P. These two inherent identities when added to C-component balances, one energy balance, and the C phase-distribution relationships give... 

The general component- balance around a section of stages from stage n to the top of the column is... 

An overall component balance gives the average or accumulated distillate composition I o.avg... 

General Component Balance For a spherical adsorbent particle ... 

First, the old standby methods of checking the overall individual component balances and checking dew and bubble points will help verify distillate and bottoms concentrations. The total overhead (distillate plus reflux) calculated dew point is compared to the column overhead observed temperature and the bottoms calculated bubble point is compared to the column bottom observed temperature. If the analyses are not felt to be grossly in eiTor. the following method wfill also prove very helpful. 

Therefore, the sum of the component balances is the total material balance while the net rate of change of any component s mass within the control volume is the sum of the rate of mass input of that component minus the rate of mass output these can occur by any process, including chemical reaction. This last part of the dictum is important because, as we will see in Chapter 6, chemical reactions within a control volume do not create or destroy mass, they merely redistribute it among the components. In a real sense, chemical reactions can be viewed from this vantage as merely relabeling of the mass. 

OVERALL COMPONENT BALANCES (MOLS) ------- BEFORE FINAL FORCING -----... 

NORMALIZED PRODUCT STREAMS ----- AFTER COMPONENT BALANCES FORCED... 

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. 

The reactor yield is then determined by performing a component balance. The amount of C5+ in the gasoline boiling range is calculated by subtracting the C4 and lighter components from the total gas plant products. Example 5-4 shows the step-by-step calculation of the component yields. The summary of the results, normalized but unadjusted for the cut points is shown in Table 5-4. 

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

In reactor design, we are interested in chemical reactions that transform one kind of mass into another. A material balance can be written for each component however, since chemical reactions are possible, the rate of formation of the component within the control volume must now be considered. The component balance for some substance A is... 

See Figure 1.2. A component balance can be expressed in mass units, and this is done for materials such as polymers that have ill-defined molecular weights. Usually, however, component A will be a distinct molecular species, and it is more convenient to use molar units ... 

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. 

A batch reactor has no input or output of mass after the initial charging. The amounts of individual components may change due to reaction but not due to flow into or out of the system. The component balance for component A, Equation (1.6), reduces to... 

The ideal, constant-volume batch reactor satisfies the following component balance ... 

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... 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now 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... 

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

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

The component balance for a batch reactor. Equation (1.21), still holds when there are multiple reactions. However, the net rate of formation of the component may be due to several different reactions. Thus,... 

Suppose there are N components involved in a set of M reactions. Then Equation (1.21) can be written for each component using the rate expressions of Equations (2.7) or (2.8). The component balances for a batch reactor are... 

The component balance for a variable-volume but otherwise ideal batch reactor can be written using moles rather than concentrations ... 

Volume changes also can be mechanically determined, as in the combustion cycle of a piston engine. If V=V(i) is an explicit function of time. Equations like (2.32) are then variable-separable and are relatively easy to integrate, either alone or simultaneously with other component balances. Note, however, that reaction rates can become dependent on pressure under extreme conditions. See Problem 5.4. Also, the results will not really apply to car engines since mixing of air and fuel is relatively slow, flame propagation is important, and the spatial distribution of the reaction must be considered. The cylinder head is not perfectly mixed. 

The component balance will be based on the molar flow rate ... 

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... 

The reaction terms are evaluated at the outlet conditions since the entire reactor inventory is at these conditions. The set of component balances can be summarized as... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

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