Equation conservation of mass

Mixture conservation of mass equation Mixture conservation of momentum equation Mixture conservation of energy equation Slip equation (concerning the difference in velocity)... 

We apply the conservation laws to two control volumes enclosing these regions. Since there is no change in area, conservation of mass (Equation (3.15)) gives, for the unbumed mixture (u) and burned product (b),... 

Although the complete solution of ux at J will not be attempted at this point, it can be shown readily that the detonation velocity has a simple expression now that u2 and c2 have been shown to be equal. The conservation of mass equation is rewritten to show that... 

The box model is closely related to the more complex airshed models described below in that it is based on the conservation of mass equation and includes chemical submodels that represent the chemistry more accurately than many plume models, for example. However, it is less complex and hence requires less computation time. It has the additional advantage that it does not require the detailed emissions, meteorological, and air quality data needed for input and validation of the airshed models. However, the resulting predictions are... 

This must equal the mass that leaves the detonation zone per second, p(D-u), where p is the density of the products of detonation. Hence, the conservation of mass equation is ... 

Autoclave processing is a process in which individual prepreg plies are laid up in a prescribed orientation to form a laminate (Fig. 5.9). The process involves consolidation of the laminate, which generally results in a three-dimensional flow field. Similar to the IP process the fiber bed is not stationary in the AP process hence, its movement has to be specifically considered when the appropriate conservation equation for this process are developed. If it is assumed that the resin has a relatively constant density (i.e., the excess resin is squeezed out before the gel point is reached) then the appropriate conservation of mass equation for this consolidating system is Equation 5.12. 

For the sector-shaped centerpiece the conservation of mass equation can be expressed as (for 6 in radians)... 

Here 0 is the sector angle (2.5° for a double-sector centerpiece), and h is the cell thickness which can vary from 12 mm (most common) to 30 mm. Since and h are constants, they drop out. Note that for a sectorshaped cell dv = Ohd(r2)/2. One can also write the conservation of mass equation for the total concentration this is done by summing Equation 18a over all q solutes. Thus... 

For the Yphantis cell one notes dv = hf(r)dr, where f(r) is the cross-sectional width. This will vary with r since the holes are curved at each end. A typical plot of f(r) vs. r is shown in Figure 2. For component i the conservation of mass equation is... 

The conservation of mass equations are used with refractometric optics. The absorption optical system will give absorbance (optical density) which is directly proportional to concentration. Unless conservation of mass equations are used, one can only obtain concentration differences from refractometric optics. The Rayleigh optical system will give information proportional to cr — cTm thus Equation 20 or 23 would have to be used to obtain cTm. Note also that the initial concentration c0 is needed. This must be measured by differential refractometry, by boundary-forming experiments, or from ultraviolet light absorption. 

These values can be used with the appropriate, conservation of mass equation to obtain (crm)id. Once this is known, the ideal concentration distribution can be obtained. For sector-shaped centerpieces the conservation of mass equation is written as... 

Equations 34 and 35 are known as the conservation of mass equations. From Equations 34 and 35 it follows that ... 

We can substitute Equation 40 into the conservation of mass equations (34 and 35) to obtain new expressions for cA and cB. Then we can substitute these new relations for cA and cB back into Equation 40 to obtain... 

Here one makes an effort to describe simultaneously transport-controlled and chemical kinetics processes (Skopp, 1986). Thus, an attempt is made to describe both the chemistry and physics accurately. For example, outflow curves from miscible displacement experiments on soil columns are matched to solutions of the conservation of mass equation. The matching process introduces a potential ambiquity such that experimental uncertainties are translated into model uncertainties. Often, an error in the description of the physical process is compensated for by an error in the chemical process and vice-versa (i.e., Nkedi-Kizza etal, 1984). 

The equations for convection are the continuity or conservation of mass equation, the momentum equations and the energy equation. From the dimensionless equation of energy, useful dimensionless numbers are obtained. 

The continuity equation is the conservation of mass equation. It is derived by a mass balance of the fluid entering and exiting a volume element taken in the flow field. In Fig. 6.1, consider a differential volume element AxAyAz. For ease of understanding, we shall consider steady, two-dimensional flow with velocity components u(x,y) and v(x,y) in the x and y directions, respectively. 

The rate expression corresponding to this scheme is derived most readily by using the equilibrium analysis of Michaelis and Menten. The conservation of mass equation for the enzyme is... 

The conservation of mass equation can be expressed in a frame of reference which moves with the fluid, the Lagrangian frame. 

Our goal is to obtain an expression in terms of total M and total L concentrations, [Mt] and [Lt], respectively, because these are measurable quantities. Under conditions where [L]T [M]T, we can assume that [L]f = [L]T. Further, we can solve for [M]f from conservation of mass, equation (3.24). The mass balance expression then... 

Regardless of the software chosen, a series of simulations should be performed over and beyond the set of conditions of interest and appropriate adjustments made to ensure that the results converge properly, all conservation of mass equations are satisfied, and the results are biologically plausible. 

A conservation of mass equation relating the total asphalt mass with the volume of naphtha (x) and volume of asphalt (y) yields... 

A complex set of equations, proposed by Riley, Stommel, and Bumpus (1949) (5) first introduced the spatial variation of the phytoplankton with respect to depth into the conservation of mass equation. In addition, a conservation of mass equation for a nutrient (phosphate) was also introduced, as well as simplified equations for the herbivorous and carnivorous zooplankton concentrations. The phytoplankton and nutrient equations were applied to 20 volume elements which extended from the surface to well below the euphotic zone. In order to simplify the calculations, a temporal steady-state was assumed to exist in each volume element. Thus, the equations apply to those periods of the year during which the dependent variables are not changing significantly in time. Such conditions usually prevail during the summer months. The results of these calculations were compared with observed data, and again the results were encouraging. 

With the growth and death rates given by Equations 25 and 26, respectively, the source term for herbivorous zooplankton biomass is given by Equation 24. The conservation of mass equation which describes the behavior of Z, is given by Equation 2, with Z, as the dependent variables replacing Pj and SZj replacing SPj as the source terms. 

The source term SNj in the conservation of mass equation for the concentration of the nutrient Nj in the fh volume segment Vj is the sum of all sources and sinks of the nutrient within Vj. The primary interaction between the nutrient system and the phytoplankton system is the reduction or sink of nutrient connected with phytoplankton growth. The rate... 

To derive the equation for the conservation of eneigy, we proceed in the same way as we derived the conservation of mass equation, regarding all variables as functions of both time and distance. Hence at any time, r, the values at the reference distance x are ... 

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