Consumption term

In 1997 it was estimated that global production of PET was about 16.7 X 10 t.p.a., of which 12 million tonnes was used in textiles, 2 million tonnes for audio and video film (with a small quantity for technical mouldings) and 3 million tonnes for packaging, particularly bottles. The tremendous growth in the bottles market from zero in the late 1970s to 1.5 million tonnes in the USA alone in 1998 is, in consumption terms, one of the most spectacular examples of growth in plastics materials in recent times and will be considered later in this section. 

If the global oxygen production caused by the small fraction of carbon fixed by photosynthesis that is buried were not balanced by an oxygen consumption term, and other processes remained the same, the O2 content of the atmosphere would roughly double in about 2 million years. This is a short time geologically, particularly because it is believed there has been... 

An equation similar to (1-3) can be written for the total concentration of soap (emulsifier) in the reactor, Sr(t), noting of course that the consumption term in the case of soap is zero, since emulsifier is conserved in the reactor. 

If a detailed potential is not required in the model, then the catalyst layer can be treated as simply the location where oxygen and hydrogen are consumed and water is produced. Hence, Faraday s law (eq 24) is used as a generation/consumption term in the... 

The generation or consumption terms are defined as follows rx = growth rate of cells rp = production rate of some product rs = consumption rate of substrate... 

For processes involving reactions, the mass balance equation, Equation (3), used in the first example is not sufficient. Equations (1) or (2) must be used. The presence of reactions means that the generation and consumption terms in these equations are non-zero. The key difference between a simple separation process and a process involving reactions is the need to define these extra terms in terms of the amounts... 

Now that all the transports of exergy across the system boundary have been evaluated, the exergy consumption term can be determined ... 

The concentration of the adsorbed species can be obtained by compiling the formation term and the consumption term of the rate for the adsorbed species, as displayed in Equations (9) and (10) ... 

The formulation of the consumption term, based on experimental evidence, goes back at least to Monod [Mol Mo2], The term takes the form... 

The object is to partition the time of an individual predator. Let N denote the number of prey caught during a food-search time period of length T. Let x denote the prey density (units are cells/unit volume), s the search rate (units are volume/time), and b the handling time (units are time/prey). One then has N = sx(T-bN). The functional response (the consumption term) for one predator is N/T or, extending over a unit volume. 

If the balance is on a nonreactive species or on total mass, the generation and consumption terms equal zero and the equation reduces to input — output. 

Since the process is at steady stale there can be no buildup of anything in the system, so the accumulation term equals zero in all material balances. In addition, since no chemical reactions occur, there can be no nonzero generation or consumption terms. For all balances, Equation 4.2-2 therefore takes the simple form input = output. 

Observe that the input and output streams shown on the chart denote the initial and final stales for this batch process. Since no reactions are involved, the generation and consumption terms of Equation 4.2-3 may be omitted so that all balances have the simple form input output ... 

The occurrence of a chemical reaction in a process brings several complications into the material balance procedures described in the previous sections. TTie stoichiometric equation of the reaction imposes constraints on the relative amounts of reactants and products in the input and output streams (if A B, for example, you cannot start with 1 mol of pure A and end with 2 mol of B). In addition, a material balance on a reactive substance does not have the simple form input = output, but must include a generation and/or consumption term. 

A number of different balances could be written on this process, including balances on total mass. C.Hs, C2H4, and H2. Of these, only the first has the simple form input = output since the three given species all participate in the reaction, the balance equation for each of them must include a generation terra (for C2H4 and H2) or a consumption term (for C2Hj). 

If molecular species balances are used to determine unknown stream variables for a reactive process, the balances on reactive species must contain generation and/or consumption terms. The degree-of-freedom analysis is as follows ... 

Once a generation or consumption term has been calculated for a species in a given reaction, the generation and consumption terms for all other species in that reaction may be determined directly from the stoichiometric equation. (We will shortly illustrate this determination.) One generation or consumption term must therefore either be specified or calculated for each independent reaction, which is why each reaction adds a degree of freedom to the system. 

Each balance on a reactive species will contmn a generation or consumption term. We will use the notation Cch4.i( oICH4) to denote the consumption of meth ine in Reaction 1, Gh o.2 (mol H2O) to denote the generation of water in Reaction 2, and so on. Note that any G and C term for a specified reaction may be expressed in terms of any other G or C term for the same reaction directly from the stoichiometric equation. For example, the generation of water in Reaction 1 may be expressed in terms of the consumption of oxygen in that reaction as... 

Since we know both the feed and output amounts of methane, a methane balance should involve only the two methane consumption terms (one for each reaction) as unknowns. Since CCH4.1 can be expressed in terms of Gco.i nd Cch4,2 can be expressed in terms of Gco,.2. the CO, CO2, and CH4 balances will yield three equations in three unknowns—nco.Gco.it nd Gco2.2-... 

You may analyze reactive processes using (a) molecular species balances (the only method used for nonreactive processes), (b) atomic species balances, or (c) extents of reaction. Molecular species balances on reactive processes are often cumbersome they must include generation and consumption terms for each species, and one degree of freedom must be added for each independent reaction. Atomic species balances have the simple form input = output and are usually more straightforward than either of the other two methods. Extents of reaction are particularly convenient for reaction equilibrium calculations. 

An integral energy balance may be derived for a closed system between two instants of time. Since energy can neither be created nor destroyed, the generation and consumption terms of the general balance (4.2-1) drop out, leaving... 

In this equation, 1.50 mol/s would be the sum of the input and generation terms in the balance on species A and 0.200Ca would be the sum of the output and consumption terms. The goal would be to solve the differential balance to determine the concentration CA(moI/L) as a function of time f(s), either as an analytical function or in the form of a table or plot. 

The functions on the right-hand sides of these equations are derived from the input, output, generation, and consumption terms in the balance equations. The solutions of the equations may be expressed as a table of yi, >2, -. .. Vn for increasing values of t or as plots of yj versus r, 2 versus t,..., y versus t. The next example provides an illustration. 

Substitute for the input, output, generation, and consumption terms in the balance equation, and convert the resulting equation into one with the form... 

To proceed further it is necessary to define the nature of the chemical production or consumption terms in (3[S]/9r)chem- that, let us consider the general term Kj in Eq. (171). 

With respect to a total mass balance, in this book the generation and consumption terms are zero whether a chemical reaction occurs in the system or not (we neglect the transfer between mass and energy in ordinary chemical processing) hence... 

With respect to a baranc iTthe to lhble ifXchernical reaction does occurryou-most likely will have to take into account the generation or consumption terms. In the absence of chemical reaction, the generation and consumption terms do not apply to a single chemical compound such as water or acetone with a chemical reaction present in the system, the terms do apply. 

From the viewpoint of both a mass balance or a mole balance for elements themselves, such as C, H, or O, the generation and consumption terms are not involved in a material balance. Finally, Eq. (2.1) should not be applied to a balance on a volume of material unless ideal mixing occurs (see Sec. 3.1) and the densities of the streams are the same. In this chapter, information about the generation and consumption terms for a chemical compound will be given a priori or can be inferred from the stoichiometric equations involved in the problem. Texts treating chemical reaction engineering describe how to calculate from basic principles gains and losses of chemical compounds. 

Most, but not all, of the problems discussed in this chapter are steady-state problems treated as integral balances for fixed time periods. If no accumulation occurs in a problem, and the generation and consumption terms can be omitted from consideration, the material balances reduce to the very simple relation... 

Note that instead of making mole balances on CO2 and O2 as in Example 2.3, we could have made mole, or mass, balances on the elements C and 0 (or oxygen expressed as O2) that would lead to the same final information. You can avoid using the generation and consumption terms in Eq. (2.1) if you make element balances. 

Steps 7, 8, and 9 A direct solution is possible by subtraction. The accumulation term in Eq. (2.1) is negative (depletion) and the generation and consumption terms are zero. Also, only transport out occurs transport in is zero. The component and total balances in kilograms are (only two are independent)... 

---