Generalized Material Balance

Experiments were carried out between 720° and 985°C with 1-butene and between 760°and 925°C with the mixed isomers of 2-butene, using steam dilution corresponding to a steam hydrocarbon weight ratio of between 0.17 and 1.2 g/g. All runs were isobaric at total pressure of 1.0 psig. Material balances generally fell within dz 2% for all of the experiments. Tables I and II summarize the individual run conditions, the observed yields and conversions, and the calculated rate constants for pyrolysis of 1-butene and the 2-butenes, respectively. 

For ease of exposition, let us limit attention to. two independent reactions--the generalization to more reactions is straightforward. Then the material balance equations take the form... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Material balances, often an energy balance, and occasionally a momentum balance are needed to describe an adsorption process. These are written in various forms depending on the specific application and desire for simplicity or rigor. Reasonably general material balances for various processes are given below. An energy balance is developed for a fixea bed for gas-phase application and simphfied for liquid-phase application. Momentum balances for pressure drop in packed beds are given in Sec. 6. 

For a steady-state ciystallizer receiving sohds-free feed and containing a well-mixed suspension of ciystals experiencing neghgible breakage, a material-balance statement degenerates to a particle balance (the Randolph-Larson general-population balance) in turn, it simplifies to... 

Process calculations, where material balances are performed, normally produce flow values in terms of a weight flow. The flow is generally stated as pounds per hour. Equation 2.10 can be used either with a singlecomponent gas or with a mixture. 

Generally, these concentrations are expressed in terms of moles of solute per mole of pure solvent (liquid phase) and moles of solute per mole of inert gas (gas phase), thus making the material balance calculations easier. 

In general, if a reaction leads to two or more products, and the products are not formed at equal rates, there must be an intermediate to account for the material balance. (The converse, of course, is not necessarily true, for an intermediate may be present at vanishingly low concentrations and yet be kinetically important.)... 

General Material Balances. According to the law of conservation of mass, the total mass of an isolated system is invariant, even in the presence of chemical reactions. Thus, an overall material balance refers to a mass balance performed on the entire material (or contents) of the system. Instead, if a mass balance is made on any component (chemical compound or atomic species) involved in the process, it is termed a component (or species) material balance. The general mass balance equation has the following form, and it can be applied on any material in any process. 

Differential and Integral Balances. Two types of material balances, differential and integral, are applied in analyzing chemical processes. The differential mass balance is valid at any instant in time, with each term representing a rate (i.e., mass per unit time). A general differential material balance may be written on any material involved in any transient process, including semibatch and unsteady-state continuous flow processes ... 

For any transient process that begins at time t and is terminated at a later time tp, the general integral material balance equation has the form... 

We turn now to the issue of material balance closure. Material balances can be perfect when one of the flow rates and one of the components is unmeasured. The keen experimenter for Examples 7.1 and 7.2 measured the outlet concentration of both reactive components and consequently obtained a less-than-perfect balance. Should the measured concentrations be adjusted to achieve closure and, if so, how should the adjustment be done The general rule is that a material balance should be closed if it is reasonably possible to do so. It is necessary to know the number of inlet and outlet flow streams and the various components in these streams. The present example has one inlet stream, one outlet stream, and three components. The components are A, B, and I, where I represents all inerts. 

Phase Balances for Components. Material balances can be written for each phase. For the general case of unsteady operation and variable physical properties, the liquid-phase balance is... 

The general material balance of Section 1.1 contains an accumulation term that enables its use for unsteady-state reactors. This term is used to solve steady-state design problems by the method of false transients. We turn now to solving real transients. The great majority of chemical reactors are designed for steady-state operation. However, even steady-state reactors must occasionally start up and shut down. Also, an understanding of process dynamics is necessary to design the control systems needed to handle upsets and to enable operation at steady states that would otherwise be unstable. 

The design of a reactor starts with the expression of material balance for any reactant (or product). The basis for all material balances is the law of conservation of matter, which states that matter cannot be created or destroyed in a given system (nuclear reactions are, of course, out of this dictum). Material balance is generally given as ... 

When the process occurs nonisothermally in the reactor, energy balances must be used in conjunction with material balances. The general expression for energy balance is ... 

No hard and fast rules can be given on the selection of suitable boundaries for all types of material balance problems. Selection of the best sub-division for any particular process is a matter of judgement, and depends on insight into the structure of the problem, which can only be gained by practice. The following general rules will serve as a guide ... 

Material-balance problems are particular examples of the general design problem discussed in Chapter 1. The unknowns are compositions or flows, and the relating equations arise from the conservation law and the stoichiometry of the reactions. For any problem to have a unique solution it must be possible to write the same number of independent equations as there are unknowns. 

Consider the general material balance problem where there are Ns streams each containing Nc independent components. Then the number of variables, N, is given by ... 

The general approach to the solution of unsteady-state problems is illustrated in Example 2.15. Batch distillation is a further example of an unsteady-state material balance (see Volume 2, Chapter 11). 

Then, as there is no generation in the system, the general material balance (Section 2.3)... 

This section is a general discussion of the techniques used for the preparation of flowsheets from manual calculations. The stream flows and compositions are calculated from material balances combined with the design equations that arise from the process and equipment design constraints. 

This procedure for deriving the set of material balance equations is quite general. For a process with n units there will be a set of n equations for each component. 

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