Material Balances and Stoichiometry
In dealing with food and agricultural wastewater, whether formulating treatment and utilization strategy or planning the initial stage of a comprehensive management project, a basic understanding of the effects of mass flow rate or loading factors on process designs is essential.
Stoichiometry is the material accounting for a chemical reaction. Given enough information, one can use stoichiometry to calculate masses, moles, and percents within a chemical equation that is an expression of a chemical process. Consider a simple reaction where a reactant A converts into resultant B (Equation 1.4):
aA b bB
where a and b are termed as stoichiometric coefficients and thus positive proportionality constants. Equation 1.4 tells us that for every a moles of reactant A consumed there will be b moles of resultant B produced. If, initially, A has a mole concentration of NA0 and B has a starting concentration of NB0, then at any given time the reactant A and resultant B will be NA and NB. They are related to each other by the following expression (Equation 1.5):
In this expression, (NA0 — NA) represents the consumption of A in moles at the time while (NB — NB0) accounts for the gain of B in moles. Equation 1.5 may be used to calculate NA or NB when other terms in Equation 1.5 are known. For a more general chemical reaction with the following form (Equation 1.6):
there will be the following (Equation 1.7):
(Na0  NA)/a = (Nb0  NB)/b = (Nc  Nc0)/c = (Nd  ND0)/d
All terms are in moles.
Stoichiometric equations stipulate the important principle of mass conservation: mass can neither be created nor be destroyed; it can only transform from one form or state to another. However, a stoichiometric expression can provide only a snapshot of the underlying chemical reaction at a given time; it does not reveal how fast the chemical reaction occurs. For that attribute, we introduce a new term called chemical reaction rate. Consider the chemical reaction we used in Equation 1.4:
aA b bB
In this case, we denote the rate of consumption of A per unit volume (molar unit) in a reactor as rA and the rate of generation of B per unit vol ume in the reactor as rB. And we know the following by intuition and the stoichiometric equation (Equation 1.8):
It should be emphasized that all units discussed so far are molebased. In many biological wastewater treatment process designs and calculations, the units are most likely massbased. The relationship between massbased units and molebased units is shown in Equation 1.9:
[molebased units] = [massbased units]/[molecular weight]
It is, however, difficult to establish the exact molecular structures of all microorganisms involved in a wastewater treatment process; therefore, massbased units have to be used. In this scenario, stoichiometric equations cannot be used; the relationship between reaction rates needs to be obtained from experiments.
Stoichiometry is a specific form of material balance for reactions and is expressed in molebased units. In realworld situations, those reactions take place in reactors or other forms of containers. Their designs and layouts will affect the amount of materials consumed and new substances generated in the reactions. Because of this realization, we shall use mass balance equations to describe macroscopically the dynamics of materials in a treatment system. We usually start developing mass balance equations on the treatment system with a control volume, a representative portion of the real system that can be integrated over the entire domain of the system. The changes of materials in the control volume should satisfy the law of mass conservation (Equation 1.10):
[species in]  [species out] + [generation] = [species accumulation]
In mass units, Equation 1.10 can be expressed mathematically as the following (Equation 1.11):
min  mout + rA Vc = d(CVc)/dt where min is the mass flow rate of species entering the control volume, mout is the mass flow rate of species exiting the volume, Vc is the control volume, and C is the mass concentration of the species. With appropriate boundary conditions of the system, fluid flow characteristics, and the initial condition of the species, Equation 1.11 can be integrated over these conditions to yield the quantities of the variables in the equation.
Equation 1.11 depicts an unsteady state system where the amount of the species varies with the reaction time. For a steady state system, Equation 1.11 is reduced to the following (Equation 1.12):
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