Accumulation rate process mass balance

Time-Averaged Properties. The unsteady-state macroscopic mass balance for mobile component A is applied to the quiescent liquid, where the rate of interphase mass transfer via equation (11-205) is interpreted as an input term due to diffusion across the gas-liquid interface. There are no output terms, sources, sinks, or contributions from convective mass transfer in the macroscopic mass balance. Hence, the accumulation rate process is balanced by the rate of interphase mass transfer across time-varying surface S t), where both terms have dimensions of moles per time ... 

This is the integral form of the mass transfer equation within an arbitrary control volume V (f). Notice that there is a term of the form / pi(n Vsurface) dS in the accumulation rate process and in the net rate of input due to mass flux acting across the time-varying surface S(t). These terms are present because the surface that bounds the control volume is in motion. The fact that they cancel provides quantitative support for the claim that the final form of the mass transfer equation is independent of the characteristics of the control volume. All surviving terms in the mass balance. 

While we laud the virtue of dynamic modeling, we will not duphcate the introduction of basic conservation equations. It is important to recognize that all of the processes that we want to control, e.g. bioieactor, distillation column, flow rate in a pipe, a drag delivery system, etc., are what we have learned in other engineering classes. The so-called model equations are conservation equations in heat, mass, and momentum. We need force balance in mechanical devices, and in electrical engineering, we consider circuits analysis. The difference between what we now use in control and what we are more accustomed to is that control problems are transient in nature. Accordingly, we include the time derivative (also called accumulation) term in our balance (model) equations. 

Students frequently have difficulty when they first encounter the concept of recycle because they find it hard to understand that material can circulate in a system without an accumulation of mass. If you have this difficulty, you might find it helpful to refer back to the flowchart of Figure 4.5-1. Observe that even though there is material circulating within the system, there is no net accumulation 110 kg of material enters the system each minute, and the same mass leaves each minute. Within the system there is a net circulation rate of 120 kg/min, but the circulation has no effect on the overall process material balance. 

As the rate constant kads increases rapidly with overpotential, the adsorption process must eventually become constrained at higher overpotentials by diffusion of SPS in the electrolyte. Mass balance between the catalyst accumulating on the interface and diffusing through the electrolyte is given by ... 

In this article a simplified mass balance has been used to describe the net transport of sand over an accreting mud bottom. The combination of these two sedimentary processes controls the transition from sand to mud on the floor of the Sound. The distribution of sand may be described with three parameters an advection velocity of sand grains, an eddy-diffusion coefficient for mobile sand, and a rate of accumulation of marine mud. (Only the ratios of these quantities are needed if the distribution is in a steady state.) The motion of sand is thereby represented with both a deterministic part and a statistical part. The net, one-way advection of sand is the result of the superposition of an estuarine circulation on the tidal stream, and unpredictable variations in the rate of sand transport are represented as an eddy-diffusion process. Sand is immobilized when it is incorporated into the permanent deposit of marine mud. 

In practical terms, an indoor air quality model should provide a reasonable description of the mass balance of the test chamber experiments, trying to address factors such as material emissions, airflows into and out of the chamber and chemical/physical decay or other removal and/or transformation processes of the VOCs. VOC concentrations are increased by emissions within the defined volume of the chamber and by infiltration from external air to the chamber. Similarly, concentrations are decreased by transport via exiting chamber air, by removal to chemical and physical sinks within the chamber air, or by transformation of a VOC to other chemical forms. A general mass balance equation concerning the concentration of a VOC in a test chamber can be written in the form of one or more differential equations representing the rate of accumulation and the VOC gain and loss. This concept for a VOC concentration C (mass units/ m ) in a chamber of volume V (m ) is translated into the following differential equation ... 

The overall mass balance for the process dynamics is the following rate of solute in by flow - rate of solute out by flow = rate of accumulation of solute in the fluid phase and in the solid phase ... 

This analysis begins with the unsteady-state mass balance for component i in the well-mixed reactor. At high-mass-transfer Peclet numbers, which are primarily a function of volumetric flow rate q, the rate processes of interest are accumulation, convective mass transfer, and multiple chemical reactions. Generic subscripts are... 

The following discussion represents a detailed description of the mass balance for any species in a reactive mixture. In general, there are four mass transfer rate processes that must be considered accumulation, convection, diffusion, and sources or sinks due to chemical reactions. The units of each term in the integral form of the mass transfer equation are moles of component i per time. In differential form, the units of each term are moles of component i per volnme per time. This is achieved when the mass balance is divided by the finite control volume, which shrinks to a point within the region of interest in the limit when aU dimensions of the control volume become infinitesimally small. In this development, the size of the control volume V (t) is time dependent because, at each point on the surface of this volume element, the control volnme moves with velocity surface, which could be different from the local fluid velocity of component i, V,. Since there are several choices for this control volume within the region of interest, it is appropriate to consider an arbitrary volume element with the characteristics described above. For specific problems, it is advantageous to use a control volume that matches the symmetry of the macroscopic boundaries. This is illustrated in subsequent chapters for catalysts with rectangular, cylindrical, and spherical symmetry. 

It is evident that to avoid future environmental contamination situations as have occurred with DDT, Mirex, or polychlorinated biphenyls (PCBs), it is essential to develop a better understanding of the processes of transport, transformation, and accumulation of toxic substances in the environment. For compounds which are already present in the environment, it is desirable to assemble mass balances which describe where and at what concentrations the substances are accumulating and the rates at which they are degrading or moving between environmental compartments. For chemical substances which may be introduced in the future, it is desirable to have the capability of generating advance information on their behavior and, of course, their toxic effects. 

Most high-tonnage commodity polymers are produced in continuous processes. The feed is metered continuously into the reactor and the effluent is removed continuously from the reactor. When polymerization reaches a steady state in operation, the rate of heat generated at any point in the system is usually constant. Continuous processes have advantages of easy operation and low costs, particularly suitable for large-volume production. The mass balances of reactants and products are in a general form of accumulation = flow in - flow out + production - consumption. For example, in the continuous free-radical polymerization, the mass balances for initiator, monomer and polymer, are... 

The rate of change of volume in the tank can be written as a lumped parameter model, where all the resistance to flow is assumed to be associated with the valve, and all the capacitance of the process is assumed to be associated with the tank. This model is shown in Equations 3.1 and 3.2. The basis of Equation 3.1 is the principle of conservation, mass balance in this case (i.e. what goes in must come out or get accumulated in the system). 

The condition for the practical implementation of such a feed control is the availability of a computer controlled feed system and of an on-line measurement of the accumulation. The later condition can be achieved either by an on-line measurement of the reactant concentration, using analytical methods or indirectly, by using a heat balance of the reactor. The amount of reactant fed to the reactor corresponds to a certain energy of reaction and can be compared to the heat removed from the reaction mass by the heat exchange system. For such a measurement, the required data are the mass flow rate of the cooling medium, its inlet temperature, and its outlet temperature. The feed profile can also be simplified into three constant feed rates, which approximate the ideal profile. This kind of semi-batch process shortens the time-cycle of the process and maintains safe conditions during the whole process time. This procedure was shown to work with different reaction schemes [16, 19, 20], as long as the fed compound B does not enter parallel reactions. 

The DAE system (14.2 through 14.7) represents the system model. Differential equations (14.2) are derived from the application of conservation principles to fundamental quantities the rate of accumulation of a quantity within the boundaries of a system is the difference between the rate at which this quantity enters the system and the rate at which it comes out, plus the rate of its net internal production. In chemical process systems, the fundamental quantities that are being conserved are mass, energy, and momentum, and conservation laws are expressed as balances on these quantities. 

We have shown previously that the existence of a temperature threshold Tc results from the balance between particle influx induced by evaporation and particle depletion due to diffusion, recirculation, and deposition of particles. The velocity Vs of the contact line is directly influencing the deposition rate and recirculation and thus the Tc value. Figure 15.12 shows the evolution of Tc in the case of 500 nm PS particles assembled on PDMS templates. The influence of the particle solid content was also added in the figure in order to illustrate the impact of mass transfer on the accumulation process. 

The bubble size distribution is among the important factors controlling the interfacial mass transfer rate in gas liquid stirred tank reactors. This distribution is determined by a balance of coalescence and breakage rates. For this reason the trailing vortices play an important role in the gas dispersion processes in gas-liquid stirred tanks. This role stems from the vortex s ability to capture gas bubbles in the vicinity of the impeller, accumulate them inside the vortex and disperse them as small bubbles in the vortex tail. This ability is related to the high vorticity associated with the rotation of the vortex. Sudiyo [86] investigated bubble coalescence in a 2.6 L stirred tank. 

In an unsteady-state energy balance the same principle applies. The accumulation of energy within a process where all the energy forms are considered including kinetic, potential, heat flow rates, enthalpies, and stirrer works may result in an increase in the thermal energy and a rise in temperature. Unsteady-state heat transfer involves the transfer of heat under conditions where the temperature changes with time. For the simple case of one-dimensional conduction in a solid slab, the accumulation of heat is a product of the mass and specific heat of the material and the increase in temperature where ... 

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