Material balance, differential, point

Design Procedure. The packed height of the tower required to reduce the concentration of the solute in the gas stream from to acceptable residual level ofjy 2 may be calculated by combining point values of the mass transfer rate and a differential material balance for the absorbed component. Referring to a sHce dh of the absorber (Fig. 5),... 

The material balance for a PFR is developed in a manner similar to that for a CSTR, except that the control volume is a differential volume (Figure 2.4), since properties change continuously in the axial direction. The material balance for a PFR developed in Section 2.4.2 is from the point of view of interpreting rate of reaction. Here, we turn the situation around to examine it from the point of view of the volume of reactor, V. Thus equation 2.4-4, for steady-state operation involving reaction represented by A+. ..- vcC +. . ., may be written as a differential equation for reactant A as follows ... 

Distribution planning covers transportation and inventory planning within the network, as well as the material balance calculation between sales, production and procurement. Global transportation planning considers the lead times between continents resulting in transit inventories differentiation of sent and received transportation quantities as shown in the requirements in section 4.1.5. Inventories are managed at the defined transfer point locations either with static or dynamic boundaries. 

Petersen [12] points out that this criterion is invalid for more complex chemical reactions whose rate is retarded by products. In such cases, the observed kinetic rate expression should be substituted into the material balance equation for the particular geometry of particle concerned. An asymptotic solution to the material balance equation then gives the correct form of the effectiveness factor. The results indicate that the inequality (23) is applicable only at high partial pressures of product. For low partial pressures of product (often the condition in an experimental differential tubular reactor), the criterion will depend on the magnitude of the constants in the kinetic rate equation. 

In a plug flow reactor the composition of the fluid varies from point to point along a flow path consequently, the material balance for a reaction component must be made for a differential element of volume dV. Thus for reactant A, Eq. 4.1 becomes... 

It may be worth while to review the different kinds of multiplicity involved in the symbols appearing in Eqs. (3-6) and (3-15). Equation (3-6) is merely a shorthand way of writing the material balance for each of the key components, each term being a row matrix having as many elements as there are independent reactions. The equation asserts that when these matrices are combined as indicated, each element in the resulting matrix will be zero. The elements in the first two terms are obtained by vector differential operation, but the elements are scalars. Equation (3-15), on the other hand, is a scalar equation, from the point of view of both vector analysis and matrix algebra, although some of its terms involve vector operations and matrix products. No account need be taken of the interrelation of the vectors and matrices in these equations, but the order of vector differential operators and their operands as well as of all matrix products must be observed. 

The height of packing required to perform a specified amount of mass transfer can be misted to the mess transfer cnefficjent and a meterial balance for the absorbed component at any point in a tower. Referring to Fig. 6.3-2, which represents a packed absorption cower, a material balance within a differential height... 

There are many different types of reactor. One of the most important features that differentiates one kind of reactor from another is the nature of mixing in the reactor. The influence of mixing is easiest to understand through the material balance(s) on the reactor. These material balances are the starting point for the discussion of reactor performance. 

Consider an ideal PFR in the form of a tube with constant cross-sectional area Ac and a length L. A detector is located at that point. The material balance on a tracer passing through the reactor leads to the partial differential equation ... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

Huffs method11,21 (given in A5.2.2 below) is derived by noting that differential material. and energy balances give a simple expression for relief rate at the point where the temperature stops rising and begins to fall ... 

In this sense, differential equations appear more tractable since they do not require particle tracking. Indeed, the solution of the coupled equations of mass, momentum and energy balance including the material equation, properly described on a suitable finite element mesh, theoretically provides the material lines. Nevertheless, the correct description of the basic experiments often requires the use of strong nonlinear terms. Such improvements may be unsatisfying from the numerical point of view since they can lead to stiff systems of nonlinear equations and to many convergence related problems. 

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