Mathematical material balance

Mathematically, multiplicities become evident when heat and material balances are combined. Both are functions of temperature, the latter through the rate equation which depends on temperature by way of the Arrhenius law. The curves representing these b ances may intersect in several points. For first order in a CSTR, the material balance in terms of the fraction converted can be written... 

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

References A variety of mathematical methods are proposed to cope with hnear (e.g., material balances based on flows) and nonhnear (e.g., energy balances and equilibrium relations) constraints. Methods have been developed to cope with unknown measurement uncertainties and missing measurements. The reference list provides ample insight into these methods. See, in particular, the works by Mah, Crowe, and Madron. However, the methods all require more information than is tvpicaUy known in a plant setting. Therefore, even when automated methods are available, plant-performance analysts are well advised to perform initial adjustments by hand. 

Step 3 Refer to the Process Flow Sheets. All material balances are logical. The process flow sheets are the basis for rationalizing the mathematical statement form of material balances. 

The failure to identify the necessary authigenic silicate phases in sufficient quantities in marine sediments has led oceanographers to consider different approaches. The current models for seawater composition emphasize the dominant role played by the balance between the various inputs and outputs from the ocean. Mass balance calculations have become more important than solubility relationships in explaining oceanic chemistry. The difference between the equilibrium and mass balance points of view is not just a matter of mathematical and chemical formalism. In the equilibrium case, one would expect a very constant composition of the ocean and its sediments over geological time. In the other case, historical variations in the rates of input and removal should be reflected by changes in ocean composition and may be preserved in the sedimentary record. Models that emphasize the role of kinetic and material balance considerations are called kinetic models of seawater. This reasoning was pulled together by Broecker (1971) in a paper called "A kinetic model for the chemical composition of sea water."... 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Figure 26.53 presents a superstructure for the design of an effluent treatment system involving three effluent streams and three treatment processes17. The superstructure allows for all possibilities. Any stream can go to any effluent process and potential bypassing options have been included. Also, the connections toward the bottom of the superstructure allow for the sequence of the treatment processes to be changed. To optimize such a superstructure requires a mathematical model to be developed for the various material balances for the system and costing correlations included. Such a model then allows... 

In addition to the time representation and material balances, scheduling models are based on different concepts or basic ideas that arrange the events of the schedule over time with the main purpose of guaranteeing that the maximum capacity of the shared resources is never exceeded. As can be seen in Figure 8.5 and Table 8.1, we classified these concepts into five different types of event representations, which have been broadly utilized to develop a variety of mathematical formulations for the batch scheduling problem. Although some event representations are more... 

Briggs and Haldane (1925) proposed an alternative mathematical description of enzyme kinetics which has proved to be more general. The Briggs-Haldane model is based upon the assumption that, after a short initial startup period, the concentration of the enzyme-substrate complex is in a pseudo-steady state. Derivation of the model is based upon material balances written for each of the four species S, E, ES, and P. 

Data for the process of dehydrogenation of ethylbenzene to styrene in a tubular packed reactor are given by Jenson Jeffreys (Mathematical Methods in Chemical Rngineering, 424, 1977). The energy and material balances are like... 

Empirical Models vs. Mechanistic Models. Experimental data on interactions at the oxide-electrolyte interface can be represented mathematically through two different approaches (i) empirical models and (ii) mechanistic models. An empirical model is defined simply as a mathematical description of the experimental data, without any particular theoretical basis. For example, the general Freundlich isotherm is considered an empirical model by this definition. Mechanistic models refer to models based on thermodynamic concepts such as reactions described by mass action laws and material balance equations. The various surface complexation models discussed in this paper are considered mechanistic models. 

Ka2. °r even < K 2. Such models are mathematically similar to the monoprotic model, since the diprotic model becomes similar to a monoprotic model if the acidity constants in the diprotic acid model are such that the neutral XOH group is insignificant in the material balance equations. 

Subsequently, the condition of complete separation has to be coupled with the material balances derived for the nodes of the SMB unit and implemented in the Equilibrium Theory Model for Langmuir-type systems. That leads to the set of mathematical conditions given below, which the flow rate ratios have to fulfil in order to achieve complete separation, in particular ... 

A first principle mathematical description of a CSTR is based on balance equations expressing the general laws of conservation of mass and energy. Assuming that n components are mixed, the material balance of the i-component, taking into account all forms of supply and discharge in the volume V of the... 

An expert, given time to do so, may utilize calculations to develop inference results. For example, a material balance calculation around a process unit may indicate a measurement inconsistency. To mimic this expertise, general mathematical operations on combinations of measurements or functions of measurements are implemented in the parallel processor also. 

TKISolver has also had heavy use in the material balance course in chemical engineering, and in a mathematical methods course in materials engineering. Graduate students in chemistry are using it in research projects in spectroscopy and kinetics. 

The material balance equations for the FCC unit are easily expressed in terms of the yield equations presented earlier. If F represents the total inlet feed (barrels per day, BPD) to the FCC unit and Y, is the yield of product i as read from Tables 2.3 and 2.4, then the production of product i can be simply obtained by multiplying the feed to the unit by the yield of product i (i.e., FY ). Such material balance equations must be written for all units of a refinery in order to prepare a mathematical... 

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 performance of a given column or the equipment requirements for a given separation are established by solution of certain mathematical relations. These relations comprise, at every tray, heat and material balances, vapor-liquid equilibrium relations, and mol fraction constraints. In a later section, these equations will be stated in detail. For now, it can be said that for a separation of C components in a column of n trays, there still remain a number, C + 6, of variables besides those involved in the dted equations. These must be fixed in order to define the separation problem completely. Several different combinations of these C + 6 variables may be feasible, but the ones commonly fixed in column operation are the following ... 

When a more detailed analysis of microbial systems is undertaken, the limitations of unstructured models become increasingly apparent. The most common area of failure is that where the growth is not exponential as, for example, during the so-called lag phase of a batch culture. Mathematically, the analysis is similar to that of the interaction of predator and prey, involving a material balance for each component being considered. 

For any system or process, the law of conservation of mass enables a mathematical expression of the operation by a series of equations derived from a total or overall material balance and a material balance for individual components within the system. The energy balance provides an additional independent overall expression. This often represents the additional tool by which systems or unknown parameters can be solved for. 

The computer program for the material balance contains several parts. First, a description ofeach item of equipment in terms of the input and output flows and the stream conditions. Quite complicated mathematical models may be required in order to relate the input and output conditions (i.e. performance) of complex units. It is necessary to specify the order in which the equipment models will be solved, simple equipment such as mixers are dealt with initially. This is followed by the actual solution of the equations. The ordering may result in each equation having only one unknown and iteration becomes unnecessary. It may be necessary to solve sets of linear equations, or if the equations are non-linear a suitable algorithm applying some form of numerical iteration is required. 

In addition to the 1 4 ratio of the e—e and ee peak areas, there are a number of other quantitative relationships between resonance areas. For example, there must always be a 1 2 relationship between the areas of the e—m and em peaks. Furthermore, material-balance calculations must agree with the peak assignments. These requirements, in addition to the ones discussed in the previous paragraph, lead to an unambiguous structural assignment of the NMR spectra and thus a correct mathematical description of the system. 

The semibatch reactor is a cross between an ordinary batch reactor and a continuous-stirred tank reactor. The reactor has continuous input of reactant through the course of the batch run with no output stream. Another possibility for semibatch operation is continuous withdrawal of product with no addition of reactant. Due to the crossover between the other ideal reactor types, the semibatch uses all of the terms in the general energy and material balances. This results in more complex mathematical expressions. Since the single continuous stream may be either an input or an output, the form of the equations depends upon the particular mode of operation. 

A differential equation describing the material balance around a section of the system was first derived, and the equation was made dimensionless by appropriate substitutions. Scale-up criteria were then established by evaluating the dimensionless groups. A mathematical model was further developed based on the kinetics of the reaction, describing the effect of the process variables on the conversion, yield, and catalyst activity. Kinetic parameters were determined by means of both analogue and digital computers. 

In its general form, the mathematical model describing the overall and component material balances of this process can be written as... 

We denote by 07 = Hi/HijS the dimensionless variables corresponding to the energy flow rates Hiy i = 1,..., N (the subscript s denotes steady-state values). Appending a generic representation of the overall and component material-balance equations, with xtfc IRm being the material-balance variables, the overall mathematical model of the process in Figure 6.1 becomes... 

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