Differential volume element

A differential volume element dV in the flow field contains a mass of gas dm and a mass of solids dni. The corresponding volumes taken by gas and solids are denoted by tiV and dY. The sum of these partial volumes is the total volume of the mixture. 

When we have to deal with charge distributions rather than point charges, the definitions have to be generalized. What we do is to divide continuous charge distributions into differential charge elements /o(r)dr, and then apply the basic formula for the electrostatic field, and so on. Flere, dr is a differential volume element. Finally, we would have to integrate over the coordinates of the charge... 

Distribution Function.—Let us denote a point in space, having rectangular coordinates (x,y,z), by r the differential volume element dxdydz will be represented by dr. Similarly, the velocity (or point in velocity space) v will have rectangular components (vz,vy,vz) the volume element in velocity space, dvjdvudvz, will be represented by dv. If dN is the number of particles which are in the differential volume dr, at r, and have their velocities in the range dv, at v, then the distribution function is defined by ... 

The simplified concept of a gas-liquid-particle operation used in the following analysis is illustrated in Fig. 1. By considering a differential volume element of height dz, the following material balances may be formulated for one component in the three phases ... 

A differential balance written for a vanishingly small control volume, within which t A is approximately constant, is needed to analyze a piston flow reactor. See Figure 1.4. The differential volume element has volume AV, cross-sectional area A and length Az. The general component balance now gives... 

FIGURE 3.1 Differential volume elements in piston flow reactors (a) variable cross section (b) constant cross section. 

This section derives a simple version of the convective diffusion equation, applicable to tubular reactors with a one-dimensional velocity profile V (r). The starting point is Equation (1.4) applied to the differential volume element shown in Figure 8.9. The volume element is located at point (r, z) and is in the shape of a ring. Note that 0-dependence is ignored so that the results will not be applicable to systems with significant natural convection. Also, convection due to is neglected. Component A is transported by radial and axial diffusion and by axial convection. The diffusive flux is governed by Pick s law. 

FIGURE 8.9 Differential volume element in cylindrical coordinates. 

Unlike stirred tanks, piston flow reactors are distributed systems with one-dimensional gradients in composition and physical properties. Steady-state performance is governed by ordinary differential equations, and dynamic performance is governed by partial differential equations, albeit simple, first-order PDEs. Figure 14.6 illustrates a component balance for a differential volume element. 

FIGURE 14.6 Differential volume element in an unsteady piston flow reactor. 

Schematic representation of differential volume element of plug flow reactor.

FIGURE 10.1 Differential volume element in a chromatographic column, where... 

The material balance relationship (i.e., Equation 1.5) holds for any reactant. If the liquid in a reactor is completely stirred and its concentration is uniform, we can apply this equation to the whole reactor. In general, it is applicable to a differential volume element and must be integrated over the whole reactor. 

Ihe material balance for a reactant A for a differential volume element d V of the PFR perpendicular to the flow direction is given by... 

In the case of the continuous mixer shown in Fig. 7.31(a), we can assume that the differential volume element is, in effect, a small batch mixer similar to the one described... 

The change in size of the solid bed over a small down-channel increment will depend on the rate of melting at the solid bed-melt film interface. Consider a small differential volume element, perpendicular to the solid-melt interface (Fig. 9.32). The solid bed has a local down-channel velocity Vsz and a local velocity component into the melt film of Vsy. The barrel surface velocity Vb is resolved into down-channel and cross-channel components Vb . and Vbx. 

Fig. 9.32 A differential volume element perpendicular to the melt film-solid bed interface. Schematic view of temperature profile in the film and solid bed shown at right. Schematic views of velocity profiles (isothermal model) in the x and z directions are also shown.

Consider a differential volume element dV, all mass and energy balances can be represented as shown in Fig. 16.1. 

Given any complex system of heterogeneous catalytic first order reactions the mass balance on a differential volume element of the reactor at the height h yields the following system of differential equations for the j-th reaction component i) for the bubble phase... 

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 energy equation may be derived using the first law of thermodynamics for a differential volume element in a flow field. In the absence of radiation and heat sources or sinks in the fluid, the energy balance on a differential volume element AxAyAz about a point (x,y,z) may be expressed as... 

Consider a differential volume element AxAyAz. The conservation of mass may be stated as... 

The steady-state balance equation for a differential volume element of the fin is... 

The mathematical description of simultaneous heat and mass transfer and chemical reaction is based on the general conservation laws valid for the mass of each species involved in the reacting system and the enthalpy effects related to the chemical transformation. The basic equations may be derived by balancing the amount of mass or heat transported per unit of time into and out of a given differential volume element (the control volume) together with the generation or consumption of the respective quantity within the control volume over the same period of time. The sum of these terms is equivalent to the rate of accumulation within the control volume ... 

For a single ideal reactor, a component material balance on a differential volume element dtjf (see Fig. 5.3) becomes, for species A... 

FIGURE 5.3 Differential volume element for plug-flow... 

The heal conduction equation can pe derived by performing an energy balance on a differential volume element. The onc-dimcnsional beat conduction equation in rectangular, cylindjdcal, and spherical coordinate systems for the case of constant thermal conductivities are expressed as... 

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