Conservation of mass, equation for

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. 

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... 

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... 

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. 

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... 

The steady-state volume average conservation of mass equation for component i is given by Equation (16.16)... 

The same result is found by proceeding in the usual way, developing a differential conservation of mass equation for each volume element in the CPFR, as diagrammed in Fig. 3.38. This figure shows recycling in an amount Fr [m h ] for maintaining cell culture, so that the recycling ratio is r = FJF, For the stationary state and with dV = A -dz, the equations for the conservation of cell mass, x, and substrate, s, are... 

The conservation of mass equation for this situation, which is directly applied in modeling tube reactors (F. Moser, 1977) and bubble columns (Reuss, 1976), is thus identical to Equ. 3.3a. These types of one-dimensional one-phase models are not only necessary for calculating conversion They are also very useful in, for example, calculating the /cl value of a reactor with a concentration profile ... 

Using Eq. (3.6-24), the conservation of mass equation for flows in the x and the y direction is as follows for constant density ... 

For the description of mixtures of substances, FLUENT provides the species model. This model calculates the convection, diffusion, and reaction equations for each component in a mixture. This allows the volumetric reactions, surface reactions, and reactions at phase boundaries to be modeled. For the analysis of one-phase mixtures, the conservation of mass equation for a component i can be formulated by accounting for the local mass fraction Yj ... 

Deterministic air quaUty models describe in a fundamental manner the individual processes that affect the evolution of pollutant concentrations. These models are based on solving the atmospheric diffusion —reaction equation, which is in essence the conservation-of-mass principle for each pollutant species... 

For methane at 25 °C or 298 K, cp = 2.24 J/gK. Note that on substituting for the temperatures in this steady state example it makes no difference whether K or °C units are used. This follows from the conservation of mass. However, for unsteady applications of Equation (3.40), since we have used the perfect gas law in which T is in K, we should be consistent and use it through the equations. When in doubt, use K without error. Substituting ... 

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),... 

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... 

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

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... 

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... 

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. 

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... 

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. 

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 ... 

For more complete definitions, the derivation of conservation of mass equations, and application of these equations in conventional chemical engineering analysis, the reader is referred to the classical textbook on transport phenomena [5]. The notation used in the present text follows the notation by Bird et al. 

The conservation of mass equations listed in the final row of Table 3.2 are frequently used to describe the movement of solutes through tissues and cells. These equations were developed by assuming that the tissue is homogeneous throughout the region of interest. Diffusing solute molecules must have equal access to every possible position within the volume of interest and Da must be constant with respect to both space and time. This assumption is not valid for the diffusion of certain molecules in tissues for example, consider a molecule that diffuses through the extracellular space of the tissue and does not readily enter cells. The limitations of this assumption are discussed in Chapter 4. 

Heat transfer can be analyzed based on the rate form of the conservation of energy equation for the open thermodynamic system depicted in Fig. 21.4, where liquid vaporization occurs with the extraction of a heat flux from the surface. The mass... 

By rearrangii terms in the conservation of mass equation we can now solve for the time that is required to fill the tank. 

Substituting this Langmuir equation into the conservation of mass equation at steady state (eq. 15.3-7) and solving for the solution, we obtain ... 

For low Reynolds number flows, volume average of the conservation of mass equations yields Darcy s law as the relationship between superficial velocity and pressure ... 

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