Mass-balance principles

In this section the application of the total mass balance principles is presented. Consider some arbitrary balance region, as shown in Fig. 1.14 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). 

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. 

Fig. 14.9 Mass balance principle and stable burning point in a rocket motor.

A simple approximation of the thermoforming process is based on a mass balance principle. To illustrate this concept, let us consider the thermoforming process of a conical object, as schematically depicted in Fig. 6.25. 

For an individual species c in a multi-component mixture the species mass balance principle postulates that the time rate of change of the mass of a system of species c is equal to the rate at which the mass of c is produced by homogeneous chemical reactions. 

Figure 2.6. Schematic representation of the inverse mass balance model. If we know that the spring at the foothill is evolved from rain water, possible mass transfer reactions can be modeled from the mass balance principle.

Inverse mass balance modeling here only employs the mass balance principle thermodynamics and equilibrium are not considered. Inverse models are usually nonunique. A number of combinations of mass transfer reactions can produce the same observed concentration changes along the flow path. Mass transfer reactions here refer to the reactions that result in the mass transfer between two or more phases, such as the dissolution of solid and gas or precipitation of solids. Chapter 9 describes the details of the models and shows a few examples. 

Garrels and Mackenzie (1967) introduced inverse mass balance modeling into geochemistry. They showed that if the chemistry of the start and end solutions are known, possible mass transfer reactions that had produced the compositional differences and the extent to which these reactions had taken place could be deduced from the mass balance principle. 

The choice of model should be based on biological, physiological, and pharmacokinetic plausibility. For example, compartmental models may be used because of their basis in theory and plausibility. It is easy to conceptualize that a drug that distributes more slowly into poorly perfused tissues than rapidly perfused tissues will show multi-phasic concentration-time profiles. Alternatively, the Emax equation, one of the most commonly used equations in pharmacodynamic modeling, can be developed based on mass balance principles and receptor binding kinetics. 

The amount of solute deposit Md on the membranes may be described (using mass balance principles) by... 

We can proceed from both the mass balance principle and from the defini-ton of Let us divide Eq. (3.24) by... 

This chapter is organized as follows. In Section 4.1 we describe the fundamentals of mass transfer, such as the various definitions for concentrations and velocities. Pick s first law of diffusion, and the microscopic mass balance principle. 

This section includes the terminology for concentrations, velocities, and fluxes and their relationships. Although the discussion of new physical situations is limited, knowledge of the definitions is necessary for the next sections. Tick s first and second laws and the microscopic mass balance principle are introduced. Finally, simple cases based on the analogy between heat and mass transfer are analyzed. 

Mass balance Apphed to the control volume, the principle of consei vation of mass may be written as (Whitaker, Introduction to Fluid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1968, Krieger, Huntington, N.Y., 1981)... 

Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 

Yield and other mass-related metrics such as atom economy, reaction mass efficiency and mass intensity have been examined by Constable et al with regard to their significance concerning greenness and costs. The importance of using a (product) concentration term, which can be mass intensity or mass index, is additionally emphasized by Laird et al This is in compliance with Winterton, who in his twelve more green chemistry principles demands the establishment of full mass balances. 

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

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

The energy and mass balance equations for reacting systems follow the same principles, as described previously in Secs. 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 mass balance equation, provided additional data, relating liquid density to concentration are available. 

Principles of mathematical modelling 2 Probability density function 112 Process control examples 505-524 Product inhibition 643, 649 Production rate in mass balance 27 Profit function 108 Proportional... 

This simple example illustrates the basic principles of water network design for maximum reuse for a single contaminant. A number of issues need to be considered that would apply to more complex examples. Consider Figure 26.25 involving three water mains and three operations. Operation 2 above the pinch terminates at a concentration less than the concentration for the high concentration water main. The outlet of Operation 2 must not be fed directly into this final water main. The basis of the mass balance from Figure 26.17 dictates that all streams must achieve the concentration of the water mains into... 

Some early calorimeters use thermal methods based on principles of heat and mass balance (12) and temperature rise of a constant flow of air through the combustion chamber (13). These calorimeters suffer from many drawbacks associated with their design. Heat and mass balance requires numerous measurements to account for all heat and mass flows. In most cases, thermal lag and losses in the equipment occur, which are not easily calculated. 

Three different principles govern the design of bench-scale calorimetric units heat flow, heat balance, and power consumption. The RC1 [184], for example, is based on the heat-flow principle, by measuring the temperature difference between the reaction mixture and the heat transfer fluid in the reactor jacket. In order to determine the heat release rate, the heat transfer coefficient and area must be known. The Contalab [185], as originally marketed by Contraves, is based on the heat balance principle, by measuring the difference between the temperature of the heat transfer fluid at the jacket inlet and the outlet. Knowledge of the characteristics of the heat transfer fluid, such as mass flow rates and the specific heat, is required. ThermoMetric instruments, such as the CPA [188], are designed on the power compensation principle (i.e., the supply or removal of heat to or from the reactor vessel to maintain reactor contents at a prescribed temperature is measured). 

The chapters have been organized in such a way that, after basic principles have been introduced (Mass Balance), modeling methods (Linear Algebra, Numerical Analysis, Probability and Statistics) are presented before more specifically geochemical... 

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