Material balance with chemical reaction

Modeling in drinking water applications is largely confined to describing chemical processes. The mathematical models used in this area are based on the reaction rate equation to describe the oxidation of the pollutants, combined with material balances on the reaction system to calculate the concentrations of the oxidants as a function of the water matrix. As noted above, the reaction rate equation is usually simplified to pseudo-first order. This is based on the assumption of steady-state concentrations for ozone and the radicals involved in the indirect reaction. 

The basis for both of these observations is the law of conservation of mass, which states that mass can neither be created nor destroyed. (We will not be concerned in this book with the almost infinitesimal conversions between mass and energy associated with chemical reactions.) Statements based on the law of conservation of mass such as total mass of input = total mass of output or (Ibm sulfui/day), = (Ibm sulfur/day)oui" are examples of mass balances or material balances. The design of a new process or analysis of an existing one is not complete until it is established that the inputs and outputs of the entire process and of each individual unit satisfy balance equations. 

Translate word problems and the associated diagrams into material balances with properly defined symbols for the unknown variables and consistent units for steady-state processes with and without chemical reaction. 

Perform energy and material balances in unit operations with chemical reactions, separations, and fluid transformations (heating/cooling, compression/ expansion),... 

In this section we do simple material (weight or mass) balances in various processes at steady state with no chemical reaction occurring. We can iise units of kg, lb , lb mol, g, kg mol, etc., in our balances. The reader is cautioned to be consistent and not to mix several units in a balance. When chemical reactions occur in the balances (as discussed in Section 1.5D), one should use kg mol units, since chemical equations relate moles reacting. In Section 2.6, overall mass balances will be covered in more detail and in Section 3.6, differential mass balances. 

Here, p is mass density and yk th mass fraction, t is time and div the divergence operator v is local mass flow velocity (vector) and jk the it-th molecular diffusion flux vector, added to the term pykV representing the convection of particles Ck by the motion of a material element as a whole. So the instantaneous local change (increase) of the Ck-concentration (mass per unit volume) equals minus the amount that escapes from a volume element (the divergence term) plus the amount produced by chemical reactions. Physically, the balance makes sense if we know how the flux jk depends on the gradients (most simply by Pick s law), and how the rates of possible reactions depend on the local state of the element. If also the latter information is available then the balance takes the form of convective diffusion equation, possibly with chemical reactions. [If we have no information on the reaction rates, the w -terms can be eliminated from Eqs. (C.2) by an algebraic transformation in the same manner as in Chapter 4 indeed, it is sufficient to substitute for W, in (4.3.2) and to define the components of column vector n as follows from (C.2).] Observe finally that we have... 

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. 

Sections 2- and 3- describe how to use the relationships among atoms, moles, and masses to answer how much questions about individual substances. Combining these ideas with the concept of a balanced chemical equation lets us answer how much questions about chemical reactions. The study of the amounts of materials consumed and produced in chemical reactions is called stoichiometry. 

The Flory principle allows a simple relationship between the rate constants of macromolecular reactions (whose number is infinite) with the corresponding rate constants of elementary reactions. According to this principle all chemically identical reactive centers are kinetically indistinguishable, so that the rate constant of the reaction between any two molecules is proportional to that of the elementary reaction between their reactive centers and to the numbers of these centers in reacting molecules. Therefore, the material balance equations will comprise as kinetic parameters the rate constants of only elementary reactions whose number is normally rather small. 

Since the rate of a chemical reaction is normally strongly temperature dependent, it is essential to know the temperature at each point in the reactor in order to be able to utilize equation 8.0.1 properly. When there are temperature gradients within the reactor, it is necessary to utilize an energy balance in conjunction with the material balance in order to determine the temperature and composition prevailing at each point in the reactor at a particular time. 

The analysis of simultaneous diffusion and chemical reaction in porous catalysts in terms of effective diffusivities is readily extended to geometries other than a sphere. Consider a flat plate of porous catalyst in contact with a reactant on one side, but sealed with an impermeable material along the edges and on the side opposite the reactant. If we assume simple power law kinetics, a reaction in which there is no change in the number of moles on reaction, and an isothermal flat plate, a simple material balance on a differential thickness of the plate leads to the following differential equation... 

A technique is described [228] for solving a set of dynamic material/energy balances every few seconds in real time through the use of a minicomputer. This dynamic thermal analysis technique is particular useful in batch and semi-batch operations. The extent of the chemical reaction is monitored along with the measurement of heat transfer data versus time, which can be particularly useful in reactions such as polymerizations, where there is a significant change in viscosity of the reaction mixture with time. 

Ridelhoover and Seagrave [57] studied the behaviour of these same reactions in a semi-batch reactor. Here, feed is pumped into the reactor while chemical reaction is occurring. After the reactor is filled, the reaction mixture is assumed to remain at constant volume for a period of time the reactor is then emptied to a specified level and the cycle of operation is repeated. In some respects, this can be regarded as providing mixing effects similcir to those obtained with a recycle reactor. Circumstances could be chosen so that the operational procedure could be characterised by two independent parameters the rate coefficients were specified separately. It was found that, with certain combinations of operational variables, it was possible to obtain yields of B higher than those expected from the ideal reactor types. It was necessary to use numerical procedures to solve the equations derived from material balances. 

Equation 235 is the basic expression of material balance for a closed system in which r chemical reactions occur. It asserts that in such a system there are at most r mole-number-related quantities, S , capable of independent variation. Note the absence of implied restrictions with respect to chemical reaction equilibria the reaction coordinate formalism is merely an accounting scheme, valid for tracking the progress of each reaction to any arbitrary level of conversion. 

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