Conservation law for mass

Consider any region of space that has a hnite volume and prescribed boundaries that unambiguously separate the region from the rest of the universe. Such a region is called a control volume, and the laws of conservation of mass and energy may be applied to it. We ignore nuclear processes so that there are separate conservation laws for mass and energy. For mass. 

This recognition that energy and mass are equivalent is the most important conclusion of special relativity. Classically there are separate conservation laws for mass and energy, which is now replaced by a conservation law for total mass-energy. Any closed system that suffers a change in mass shows an increase in kinetic energy which may be written... 

The principles and basic equations of continuous models have already been introduced in Section 6.2.2. These are based on the well known conservation laws for mass and energy. The diffusion inside the pores is usually described in these models by the Fickian laws or by the theory of multicomponent diffusion (Stefan-Maxwell). However, these approaches basically apply to the mass transport inside the macropores, where the necessary assumption of a continuous fluid phase essentially holds. In contrast, in the microporous case, where the pore size is close to the range of molecular dimensions, only a few molecules will be present within the cross-section of a pore, a fact which poses some doubt on whether the assumption of a continuous phase will be valid. 

One of the key elements of the assessment phase of a pollution prevention program involves mass balance equations. These calculations are often referred to as material balances the calculations are performed via the conservation law for mass. The details of this often-used law are described below. 

The conservation law for mass can be applied to any process or system. The general form of the law follows ... 

This equation can be applied to the total mass involved in a process or to a particular species, on either a mole or mass basis. The conservation law for mass can be applied to steady-state or unsteady-state processes and to batch or continuous systems. A steady-state system is one in which there is no change in conditions (e.g., temperature, pressure) or rates of flow with time at any given point in the system the accumulation term then becomes zero. If there is no chemical reaction, the generation term is zero. All other processes are classified as unsteady state. 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

Apply the conservation law for mass to the control device on a rate basis ... 

Finally, the conservation law for mass may be written for any compound whose quantity is not changed by chemical reaction and for any chemical element whether or not it has participated in a chemical reaction. It may be written for one piece of equipment, around several pieces of equipment, or around an entire process. It may be used to calculate an unknown quantity directly, to check the validity of experimental data, or to express one or more of the independent relationships among the unknown quantities in a particular problem situation. 

Collection efficiency is a measure of the degree of performance of a control device it specifically refers to the degree of removal of a pollutant and may be calculated through the application of the conservation law for mass. Loading refers to the concentration of pollutant, usually in grains (gr) of pollutant per cubic feet of contaminated gas stream. 

Under steady-state conditions, this same mass flow must also leave the system. Thus, in accordance with the conservation law for mass... 

The conservation law for mass may be applied to a fluid device with one input and one output. For a fluid of constant density p and a uniform velocity v at each cross section A, the mass rate m, the volumetric rate q, and the mass flux G, are... 

Using the conservation law for mass, express the process outlet flow in terms of the process inlet flow. Also express the flow bypassing the process in terms of the upstream and process inlet flows ... 

A line diagram of the system is provided in Figure 11. Select the fluid in the bend as the system and apply the conservation law for mass ... 

The reader should note that although the number of moles on both sides of the equation do not balance, the masses of reactants and products (in line with the conservation law for mass) must balance. 

The model discussed above (total or component) involves the application of the conservation law for mass on a rate basis. For the mass balance at the outfall, the principal equation is Mass rate of substance upstream (nti, ) + mass rate added by outfall (/My,, ) = mass rate of substance immediately downstream from outfall assuming complete mixing (otd.i) o "... 

To establish the integral form of the basic conservation laws for mass, momentum and energy, the fundamental approach is to start out from a system analysis and then transform the balance equations into a control volume analysis by use of the transport theorem. However, to achieve a more compact presentation of this theory it is customary to start out from a generic Eulerian form of the governing equations. That is, the material control volume analysis is disregarded. 

Every process is subject to the laws of thermodynamics and to the conservation laws for mass and momentum, and we can expect every dynamic simulation of an industrial process to need to invoke one or more of these laws. The interpretation of these laws as they apply to different types of processes leads to different forms for the describing equations. This chapter will begin by reviewing the thermodynamic relations needed for process simulation, and it will go on to derive the conservation equations necessary for modelling the major components found in industrial processes. Finally, the different equations arising from lumped-parameter and distributed-parameter systems containing fluids will be brought out. 

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