Mass balance equation, worked example

This mass balance equation shows that material that is initially at radial position rin will move to radial position r for some downstream location, >0. A worked example of radial velocities and curved streamlines is given in Chapter 13, Example 13.10. 

Pollutants emitted by various sources entered an air parcel moving with the wind in the model proposed by Eschenroeder and Martinez. Finite-difference solutions to the species-mass-balance equations described the pollutant chemical kinetics and the upward spread through a series of vertical cells. The initial chemical mechanism consisted of 7 species participating in 13 reactions based on sm< -chamber observations. Atmospheric dispersion data from the literature were introduced to provide vertical-diffusion coefficients. Initial validity tests were conducted for a static air mass over central Los Angeles on October 23, 1968, and during an episode late in 1%8 while a special mobile laboratory was set up by Scott Research Laboratories. Curves were plotted to illustrate sensitivity to rate and emission values, and the feasibility of this prediction technique was demonstrated. Some problems of the future were ultimately identified by this work, and the method developed has been applied to several environmental impact studies (see, for example, Wayne et al. ). 

Other work has been mainly concerned with the scale-up to pilot plant or full-scale installations. For example, Beltran et al. [225] studied the scale-up of the ozonation of industrial wastewaters from alcohol distilleries and tomato-processing plants. They used kinetic data obtained in small laboratory bubble columns to predict the COD reduction that could be reached during ozonation in a geometrically similar pilot bubble column. In the kinetic model, assumptions were made about the flow characteristics of the gas phase through the column. From the solution of mass balance equations of the main species in the process (ozone in gas and water and pollution characterized by COD) calculated results of COD and ozone concentrations were determined and compared to the corresponding experimental values. 

An example of the use of these simple mass balance equations is given in the work of Eldridge and Piret [Chem, Eng, Prog, 46, 290 (1950)], who found excellent agreement between the observed and calculated extents of reaction in the hydrolysis of acetic anhydride. They carried out the reaction. 

Additional sets of equations that describe additional processes can be coupled and solved together. For example, the conservation of thermal energy and groundwater flow equations are coupled to the ADR equation and chemical mass balance equations in the work of Raffensperger (1995). 

The system of equations (20.23) complemented by the mass balance equations, has been solved numerically. As a result, equilibrium values of molar concentrations of components, and the properties of phases have been obtained. As an example, consider the calculation of the column absorber equipped with ten perforated contact plates, working in the ablation regime. The following parameter values were chosen p = 7.65 MPa T = 24 °C = 10 mill, m /day q = 217... 

A similar finite-differenced equivalent for the energy balance equation (including axial dispersion effects) may be derived. The simulation example DISRET involves the axial dispersion of both mass and energy and is based on the work of Ramirez (1976). A related model without reaction is used in the simulation example FILTWASH. 

The sequence of conversions in Figure 18.20 is used to calculate the mass or volume of product produced by passing a known current through a cell for a fixed period of time. The key is to think of the electrons as a "reactant" in a balanced chemical equation and then to proceed as with any other stoichiometry problem. Worked Example 18.10 illustrates the calculations. Alternatively, we can calculate the current (or time) required to produce a given amount of product by working through the sequence in Figure 18.20 in the reverse direction, as shown in Worked Example 18.11. 

The work discussed in this section clearly delineates the role of droplet size distribution and coalescence and breakage phenomena in mass transfer with reaction. The population balance equations are shown to be applicable to these problems. However, as the models attempt to be more inclusive, meaningful solutions through these formulations become more elusive. For example, no work exists employing the population balance equations which accounts for the simultaneous affects of coalescence and breakage and size distribution on solute depletion in the dispersed phase when mass transfer accompanied by second-order reaction occurs in a continuous-flow vessel. Nevertheless, the population balance equation approach provides a rational framework to permit analysis of the importance of these individual phenomena. 

Thermodynamic properties are characteristics of a system (e.g., pressure, temperature, density, specific volume, enthalpy, entropy, etc.). Because properties depend only on the state of a system, they are said to be path independent (unlike heat and work). Extensive properties are mass dependent (e.g., total system energy and system mass), whereas intensive properties are independent of mass (e.g., temperature and pressure). Specific properties are intensive properties that represent extensive properties divided by the system mass, for example, specific enthalpy is enthalpy per unit mass, h = H/m. In order to apply thermodynamic balance equations, it is necessary to develop thermodynamic property relationships. Properties of certain idealized substances (incompressible liquids and ideal gases with constant specific heats) can be calculated with simple equations of state however, in general, properties require the use of tabulated data or computer solutions of generalized equations of state. 

A material balance equation can be written for every identifiable species present, and so component can be understood to be either element, compound or radical. For example, in the combustion of methane, balances can be made both on the element carbon and the compound methane. When reactions are involved, it is convenient to work in terms of molar rather than mass units. 

The characteristic features of parameter estimation in a molecular model of adsorption are illustrated in Table 9.9, taking the simple example of the constant-capacitance model as applied to the acid-base reactions on a hydroxylated mineral surface. (It is instructive to work out the correspondence between equation (9.2) and the two reactions in Table 9.9.) Given the assumption of an average surface hydroxyl, there are just two chemical reactions involved (the background electrolyte is not considered). The constraint equations prescribe mass and charge balance (in terms of mole fractions, x) and two complex stability constants. Parameter estimation then requires the determination of the two equilibrium constants and the capacitance density simultaneously from experimental data on the species mole fractions as functions of pH. 

The equations presented above can be used (with or without modifications) to describe mass transfer processes in cocurrent flow. See, for example, the work of Modine (1963), whose wetted wall column experiments formed the basis for Example 11.5.3 and are the subject of further discussion in Section 15.4. The coolant energy balance is not needed to model an adiabatic wetted wall column and must be replaced by an energy balance for the liquid phase. Readers are asked to develop a complete mathematical model of a wetted wall column in Exercise 15.2.1. 

There are two distinct classes of analytic approximation that comprise the second and third approaches that were just mentioned. The first is based on the use of so-called macroscopic balances. In this approach, we do not attempt to obtain detailed information about the velocity and pressure fields everywhere in the domain, but only to obtain results that are consistent with the Navier Stokes equations in an overall (or macroscopic) sense. For example, we might seek results for the volumetric flow rates in and out of a flow system that are consistent with an overall mass or momentum conservation balance but not attempt to determine the detailed form of the velocity profiles. The macroscopic balance approach is described in detail in many undergraduate textbooks.2 It is often extremely useful for derivation of quantitative relationships among the average inflows, outflows, and forces (or rates of working) within a flow system but is something of a black-box approach that provides no detailed information on the velocity, pressure, and stress distributions within the flow domain. 

Example 3 on page 84 is an unbalanced formulae equation. This equation shows the formulae of the substances involved in the reaction, but has not been balanced to make sure the amount of each element is the same on both sides. That needs to be done to work out the ratio of the reacting masses of the reactants (Chapter 1). 

For a nuclear reaction to be balanced, the sum of all the atomic numbers on the left-hand side of the reaction arrow must equal the sum of all the atomic numbers on the right-hand side of the arrow. The same is true for the sums of the mass numbers. Here s an example Suppose you re a scientist performing a nuclear reaction by bombarding a particular isotope of chlorine (Cl-35) with a neutron. (Work with me here. I m just trying to get to a point.) You observe that an isotope of hydrogen, H-1, is created along with another isotope, and you want to figure out what the other isotope is. The equation for this example is... 

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