Component balances flow system

Equations (1.1) to (1.3) are diflerent ways of expressing the overall mass balance for a flow system with variable inventory. In steady-state flow, the derivatives vanish, the total mass in the system is constant, and the overall mass balance simply states that input equals output. In batch systems, the flow terms are zero, the time derivative is zero, and the total mass in the system remains constant. We will return to the general form of Equation (1.3) when unsteady reactors are treated in Chapter 14. Until then, the overall mass balance merely serves as a consistency check on more detailed component balances that apply to individual substances. 

A batch reactor has no input or output of mass after the initial charging. The amounts of individual components may change due to reaction but not due to flow into or out of the system. The component balance for component A, Equation (1.6), reduces to... 

Application of the general component balance, Equation (1.6), to a steady-state flow system gives... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

The resulting model would therefore consist of component balance equations for the soluble component written over each of the many solid and liquid subsystems of the packed bed, combined with the component balance equation for the coffee reservoir. The magnitude of the recirculating liquid flow will depend on the relative values of the pressure driving force generated by the boiling liquid and the fluid flow characteristics of the system. 

For batch reactors, there is no flow into or out of the system, and those terms in the component balance equation are therefore zero. 

Apparatus and Procedure. The kinetic studies of the catalysts were carried out by means of the transient response method (7) and the apparatus and the procedure were the same as had been used previously (8). A flow system was employed in all the experiments and the total flow rate of the gas stream was always kept constant at 160 ml STP/min. In applying the transient response method, the concentration of a component in the inlet gas stream was changed stepwise by using helium as a balancing gas. A Pyrex glass tube microreactor having 5 mm i.d. was used in a differential mode, i.e. in no case the conversion of N2O exceeded 7 X. The reactor was immersed in a fluidized bed of sand and the reaction temperature was controlled within + 1°C. 

The flows in and out can be both convective (due to bulk flow) and molecular (due to dithision). We can write one eomponent continuity equation for each component in the system. If there are NC components, there are NC component continuity equations for any one system. However, the am total mass balance and these NC component balances are not all independent, since the sum of all the moles times their respective molecular weights equals the total mass. Therefore a given system has only NC independent continuity equations. We usually use the total mass balance and NC — 1 component balances. For example, in a binary (two-component) system, there would be one total mass balance and one component balance. 

Subsequently, analytical expressions for the time dependence concentration of all components in the system were obtained based on mass balance principles and also considering the reactor type, the flow rates of the feed streams, and the concentrations of substrates. Using these models we found that the basic system considered is able to perform several informationprocessing functions, such as division, rectification, and switching. 

As mentioned above, stack models are useful for analyzing full system performance including perhaps auxiliary components in the system such as compressors. In terms of equations, almost all of the models use simple global balances and equations because single cells are not the focus of the models thus, they use equations similar to eqs 21 and 78. In terms of other equations, normally they use typical flow and heat balances as well as the appropriate current and voltage relations, which take into account how the cells are connected together. The stack models can be separated into two categories, those that consider the stack and those that consider... 

The energy balance and individual components are illustrated in Figure 3.1. The energy balance shown in the figure is for an open flow system. For a nonflow (or closed) system, the energy balance would appear as in Figure 3.2. 

These rates are related to the material balance by considering steady state plug flow systems. The differential material balance on component A in a heterogeneous reaction is... 

We may model complex systems by top-down or bottom-up approaches. In the top-down approach, we describe the components from the systemic behavior of the actual system. For example, from the flow balance analysis in a steady-state bacterium, we learn the input and output flows, topology of the network, and the rates of maty metabolic reactions. 

These basic rate models were Incorporated Into a differential mass balance In a tubular, plug-flow reaction. This gives a set of coupled, non-llnear differential equations which, when Integrated, will provide a simulation model. This model corresponds to the Integral reactor data provided by experimentation. A material balance Is written for each of the four components In our system ... 

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