Accumulation of mass

General Equation of Motion. Neglecting relativistic effects, the rate of accumulation of mass within a Cartesian volume element dx-dy-dz must equal the sum of the rates of inflow minus outflow. This is expressed by the equation of continuity ... 

Rate of accumulation of mass /dMj of component i within the system/ dt /... 

A simplified version of the model in Table IX, neglecting accumulation of mass and heat as well as dispersion and conduction in the gas phase, predicts dynamic performance of a laboratory S02 converter operating under periodic reversal of flow direction quite well. This is shown by Fig. 13 taken from Wu et al. (1996). Data show the temperature profiles in a 2-m bed of the Chinese S101 catalyst once a stationary cycling state is attained. One set of curves shows the temperature distribution just after switching direction and the second shows the distribution after a further 60 min. Simulated and experimental profiles are close. The surprising result is that the experimental maximum temperatures equal or exceed the simu-... 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

Coating Mass Uniformity and Distribution. Basically, there have been two approaches to model the accumulation of mass (coating material) on the surface of bed particles (i) the use of population balances and (ii) the probabilistic modelling of the spray-particle interaction. We will look at each of these approaches and see how it may be possible to combine these methods to give a fuller picture of coating performance. 

For fed-batch fermentation, the model equations need to include the continuous feeding of sterile substrate to the fermenter, but zero outflow. The increase in volume (total accumulation of mass) that occurs in the fermenter due to the feeding is represented by a total mass balance relationship. 

Gabor, H., and Abraham, S., 1986, Effect of dietary menhaden oil on tumor ceU loss and the accumulation of mass of a transplantable mammary adenocarcinoma in BALB/c mice, J. Natl. Cancerinst. 76 1223-1229. 

We next have to consider the continuity equation, which students first encounter seriously in introductory chemistry and physics as the principle of mass conservation. For any fluid we require that the total mass flow into some element of volume minus the flow out is equal to the accumulation of mass, and we either write these as integral balances (stoichiometry) or as differential balances on a differential element of volume. 

Physically, this equation states that the rate of accumulation of mass (represented by density) within the control volume is equal to the net amount of mass that flows across the control surfaces that bound the control volume. The leading negative sign on the right-hand side accounts for the fact that a positive value of the control-surface integral indicates flow out of the control volume. 

In the natural environment, the diffusion of organic chemicals can be retarded by sorption to soil and sediment grains. To account for this process, Eq. (70) is modified to consider the accumulation of mass in the sorbed phase ... 

The TSM resonator can be operated in liquid to measure either (I) the accumulation of mass onto the surface ffom the liquid phase, or (2) properties of the contacting liquid itself. In this section we derive the equivalent-circuit model for the resonator contacted by a semi-infinite Newtonian fluid. A Newtonian fluid is one in which the shear stress and the gradient in fluid velocity are related by a constant, independent of amplitude or frequency [IS] ... 

The terms involving lim.. o volume integrals remaining in equations (55)-(58), account for the possibility of surface forces, surface reactions, or a nonzero time rate of accumulation of mass, momentum, energy, or a chemical species at the interface. For example, if the interface is taken to be a solid surface at which chemical reactions are proceeding at a finite rate, then Wj- - w i — y ), where w- is the surface reaction rate (mass of species i produced per unit area per unit time), y is the coordinate normal to the surface, y = y at the surface, and 3 is the Dirac -function. By substitution, we then find that in equation (58) lim o di = ds, which is... 

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. 

Notice that if a process has a single input stream and a single output stream and there is no accumulation of mass in the system (so that mi = /hout = rh), the expression for AH of Equation 7,4-14a simplifies to... 

To summarize, the conditions under which Equations 11.3-12 and 11.3-13 are valid are (a) negligible kinetic and potential energy changes, (b) no accumulation of mass in the system, (c) pressure independence of U and / , (d) no phase changes or chemical reactions, and (e) a spatially uniform system temperature. Any or all of the variables T, T, , Q and (or IV) may vary with time, but the system mass, M, the mass throughput rate, m, and the heat capacities, C and Cp, must be constants. 

Eq. (8) states that the accumulation of mass in both free and adsorbed form (LHS) is balanced by the diffusion rate into the particle of gas (first term in RHS) and the adsorbed species (second term in RHS). The diffusion rate of adsorbed species in particle is related to the observed adsorbed concentration (Eq. 6) so that the effect of size exclusion (Eq. 5) has... 

The chemical species set of a state is the set of chemical constituents that are associated with the system description of that state. The values of the attributes chemical-species-set, operating-conditions, and system-volume provide the (n + 2) independent variable quantities that are necessary to define a thermodynamic state. An important feature of this representation is that each state is described by a vector of intensive and extensive variables. The intensive vector defines the operational state of the process, while the extensive vector defines the maximum accumulation of mass and energy that can occur. This is bounded by flowrate, reaction rate and physical size of the process equipment. The values of these variables are accessed through the attributes interval flowrate vector, interval accumu-... 

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