Overall continuity

The usual density of the gas p, that is, the mass of gas per unit volume of space available to the gas, is not precisely equal to Pj because of the volume occupied by the particles. This difference already has been encountered and discussed in Section 4.2.6. Since the fraction of the total volume of physical space occupied by the particles is f=i tzr fjdrd, the fluid and gas densities are related by the expression 

According to the definition of a dilute spray, the fractional volume occupied 

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Wood has been used by mankind for millennia because of its excellent material properties. Although the use of timber in some markets has decreased, the consumption of timber overall continues to rise. Projections have been made until the middle of the 21st century that in most cases show a rise in demand for timber (in all but low economic growth models) and an increase in production (Figure 1.7) (Brooks etal., 1996). There is, however, concern that the supply of timber for industrial purposes may not be able to match demand. For example, Bowyer etal. (2003), note that there will be a shortfall in the amount of forest area providing industrial timber by the year 2100, due to the rise in human population during this time (Table 1.4). 

In the simplest incompressible flow problems under constant property conditions, the velocity and pressure fields (u and p) are the unknowns. In principle, Eq. (1-1) and the overall continuity equation, Eq. (1-9) below, are sufficient for... 

In the thin layer adjacent to the particle surface the overall continuity equation may be written (SI)... 

The continuity equation is a statement of mass conservation. As presented in Section 3.1, however, no distinction is made as to the chemical identity of individual species in the flow. Mass of any sort flowing into or out of a differential element contributes to the net rate of change of mass in the element. Thus the overall continuity equation does not need to explicitly demonstrate the fact that the flow may be composed of different chemical constituents. Of course, the equation of state that relates the mass density to other state variables does indirectly bring the chemical composition of the flow into the continuity equation. Also, as presented, the continuity-equation derivation does not include diffusive flux of mass across the differential element s surfaces. Moreover there is no provision for mass to be created or destroyed within the differential element s volume. 

The derivative can be expanded to facilitate incorporation of the overall continuity equation as... 

The usual equipment for gas adsorption is a number of vessels containing fixed beds of the adsorbent, at least two vessels for achieving overall continuous operation. Figure 15.17 shows suitable vertical and horizontal vessels. The vertical ones are less likely to form channels and usually are favored. Bed depths as high as 45 ft are in use. Horizontal vessels are preferred when pressure drops... 

Overall Continuing Medical Education Objectives 14 Individual Educational Units in Total... 

The conservation equations, to be used repeatedly, are derived in Appendixes C and D and may be summarized as follows. Overall continuity, equation (D-33), is... 

For steady flow overall continuity, equation (1), reduces to... 

In a one-dimensional, spherically symmetrical system, the overall continuity equation, equation (1-23), can be written as... 

Usual interpolation rules and definitions of velocity and pressure corrections, similar to single-phase flows (Eq. (6.29)), can be used to derive a pressure correction equation from the discretized form of the overall continuity (normalized) equation. The momentum equation for multiphase flows (Eq. (7.16)) can also be written in the form of Eq. (6.28) for single-phase flows. Again, following the approximation of SIMPLE, one can write an equation for velocity correction in terms of pressure correction,/ ... 

The pressure correction equation is solved based on the normalized overall continuity equation. 

To obtain useful expressions from the general species-based design equation, we should know the formation rate of species j, (rj), at any point in the reactor. To express (vj), the local concentrations of aU species as well as the local temperatures should be provided. To obtain these quantities, we should solve the overall continuity equation, the individual species continuity equations, and the energy balance equation. This is a formidable task, and, in most situations, we cannot reduce those... 

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