Material balance principles

In this section, the application of the total material balance principle is presented. Consider some arbitrary balance region, as shown in Fig. 1.11 by the shaded area. Mass accumulates within the system at a rate dM/dt, owing to the competing effects of a convective flow input (mass flow rate in) and an output stream (mass flow rate out). 

For fast irreversible chemical reactions, therefore, the principles of rigorous absorber design can be applied by first estabhshing the effects of the chemical reaction on /cl and then employing the appropriate material-balance and rate equations in Eq. (14-71) to perform the integration to compute the required height of packing. 

In this chapter we will apply the conservation of mass principle to a number of different kinds of systems. While the systems are different, by the process of analysis they will each be reduced to their most common features and we will find that they are more the same than they are different. When we have completed this chapter, you will understand the concept of a control volume and the conservation of mass, and you will be able to write and solve total material balances for single-component systems. 

The Flory principle allows a simple relationship between the rate constants of macromolecular reactions (whose number is infinite) with the corresponding rate constants of elementary reactions. According to this principle all chemically identical reactive centers are kinetically indistinguishable, so that the rate constant of the reaction between any two molecules is proportional to that of the elementary reaction between their reactive centers and to the numbers of these centers in reacting molecules. Therefore, the material balance equations will comprise as kinetic parameters the rate constants of only elementary reactions whose number is normally rather small. 

Materials balance—This technique, in principle, is developed to its fullest extent, but it is extraordinarily sensitive to uncertainties in the data it uses. Better characterization of all pathways and chemical reactions would help, as would more accurate measurements of flows through these paths. 

The bread and butter tools of the practicing chemical engineer are the material balance and the energy balance. In many respects chemical reactor design can be regarded as a straightforward application of these fundamental principles. This section indicates in general terms how these principles are applied to the various types of idealized reactor models. 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

The principle of the component material balance can also be extended to the atomic level and can also be applied to particular elements. 

While the principle of the material balance is very simple, its application can often be quite difficult, ft is important therefore to have a clear understanding of the nature of the system (physical model) which is to be modelled by the material balance equations and also of the methodology of modelling. 

The energy and material balance equations for reacting systems follow the same principles, as described previously in Sections 1.2.3 to 1.2.5. 

These equations complete a preliminary model for the mixer. Note that it is also possible, in principle, to incorporate changing density effects into the total material balance equation, provided additional data, relating liquid density to concentration, are available. 

The term in the square bracket is an effective diffusion coefficient DAB. In principle, this may be used together with a material balance to predict changes in concentration within a pellet. Algebraic solutions are more easily obtained when the effective diffusivity is constant. The conservation of counter-ions diffusing into a sphere may be expressed in terms of resin phase concentration Csr, which is a function of radius and time. 

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... 

Performances of dryers with simple flow patterns can be described with the aid of laboratory drying rate data. In other cases, theoretical principles and correlations of rate data are of value largely for appraisal of the effects of changes in some operating conditions when a basic operation is known. The essential required information is the residence time in the particular kind of dryer under consideration. Along with application of possible available rules for vessel proportions and internals to assure adequate contacting of solids and air, heat and material balances then complete a process design of a dryer. 

The second equation takes note of material balance, a principle based on the fact that all the added solute must be accounted for even though it is now present as ions. In this example, material balance means that the concentration of Cl- ions is equal to the concentration of the HC1 added initially (all the HC1 molecules are deprotonated). If we denote the numerical value of that initial concentration by [HCl]initial, the material balance relation is [Cl-] = [HCl]initia. We can combine it with the preceding equation to write... 

All the methods presented above are based on the same principle i.e., the material balance using the lever rule. Therefore these are not limited to equilateral triangles, but equally valid for scalene triangles which frequently appear when dealing with subsystems. 

In this case, as shown in Figure 4, the subsystems are stoichiometry, material balance, energy balance, chemical kinetics, and interphase mass transfer. The mass transfer phenomena can be subdivided into (1) phase equilibrium which defines the driving force and (2) the transport model. In a general problem, chemical kinetics may be subdivided into (1) the rate process and (2) the chemical equilibrium. The next step is to develop models to describe the subsystems. Except for chemical kinetics, generally applicable mathematical equations based on fundamental principles of physics and chemistry are available for describing the subsystems. 

When deriving a material balance equation, the rate of transformation of each component in a reactor is normally governed by the mass action law. However, unlike for the reactions in which only low molecular weight substances are involved, the number of such components in a polymer system and, consequently, the number of the corresponding kinetic equations describing their evolution are enormous. The same can be said about the number of the rate constants of the reactions between individual components. The calculation of such a system becomes feasible because certain general principle can be invoked under the description of the kinetics of the majority of macromolecular reactions. Let us discuss this principle in detail. 

For planar or spherical electrodes, where the mass transport is a diffusion function in one dimension, it is possible to solve the diffusion equation as a function of time. In Section 3 the principles of how the cyclic voltammetric peak current could be calculated for a simple electron transfer reaction were presented. It is also possible to solve the material balance equations for the spherical electrode at steady state for a few first-order mechanisms (Alden and Compton, 1997a). In order to tackle second-order kinetics, more complex mechanisms, solve time-dependent equations or model other geometries with... 

Bringing any of the variables xu uly y2, —y3, vu or v2 into the basis will decrease (v 2 + v 3). However, xlt y2, v2, and v3 are ruled out by the complementary slackness principle, leaving only ux and y3. We know that —y3 will be in the optimal basis, since it is associated with the material balance constraint which is always binding. Hence we choose to bring in — y3, displacing v 3, since... 

Material Balance Equation for a Constant Volume Reservoir and with No Initial Gas Cap. Reservoir material balances are based on the principle of conservation of mass which asserts that the total mass of a system remains constant during a chemical or physical change. Before applying this principle to a constant volume reservoir with no gas cap the following terms need to be defined. 

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