Differential element of volume

Consider a reaction represented by A +. . . - products taking place in a PFR. Since conditions may change continuously in the direction of flow, we choose a differential element of volume, dV, as a control volume, as shown at the top of Figure 2.4. Then the material balance for A around dk is, from equation 1.5-la (preceding equation 2.3-3) ... 

As a basis, a differential element of volume, dVr = dxdydz, or time, dt, is identified. In cylindrical geometry, the element of volume may be a thin disk, Adz,. if changes occur only in an axial direction, or a ring 2-rrrdrdz, if changes occur both radially and axially. 

Material balances for a reaction of order n at steady state are applied over differential elements of volume in the form,... 

Where the composition within the reactor is uniform (independent of position), the accounting may be made over the whole reactor. Where the composition is not uniform, it must be made over a differential element of volume and then integrated across the whole reactor for the appropriate flow and concentration conditions. For the various reactor types this equation simplifies one way or another, and the resultant expression when integrated gives the basic performance equation for that type of unit. Thus, in the batch reactor the first two terms are zero in the steady-state flow reactor the fourth term disappears for the semibatch reactor all four terms may have to be considered. 

In a plug flow reactor the composition of the fluid varies from point to point along a flow path consequently, the material balance for a reaction component must be made for a differential element of volume dV. Thus for reactant A, Eq. 4.1 becomes... 

To develop the performance equation, we combine the rate equation with the material balance. Thus for steady-state countercurrent operations we have for a differential element of volume... 

This equation is not appropriate if all five of these conditions are not met. We can relax the third and fourth restrictions for the PFTR by considering the differential element of volume dV = At dz rather than the differential element of length dz. The mass-balance equation at a position where the fluid has moved from volume V to volume V + d V then becomes... 

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. 

It should be recalled that the differential element of volume in polar coordinates is r sin 0 dr d0d. 

If the compositions vary with position in the reactor, which is the case with a tubular reactor, a differential element of volume SV, must be used, and the equation integrated at a later stage. Otherwise, if the compositions are uniform, e.g. a well-mixed batch reactor or a continuous stirred-tank reactor, then the size of the volume element is immaterial it may conveniently be unit volume (1 m3) or it may be the whole reactor. Similarly, if the compositions are changing with time as in a batch reactor, the material balance must be made over a differential element of time. Otherwise for a tubular or a continuous stirred-tank reactor operating in a steady state, where compositions do not vary with time, the time interval used is immaterial and may conveniently be unit time (1 s). Bearing in mind these considerations the general material balance may be written ... 

The basic equation for a tubular reactor is obtained by applying the general material balance, equation 1.12, with the plug flow assumptions. In steady state operation, which is usually the aim, the Rate of accumulation term (4) is zero. The material balance is taken with respect to a reactant A over a differential element of volume 8V, (Fig. 1.14). The fractional conversion of A in the mixture entering the element is aA and leaving it is (aA + SaA). If FA is the feed rate of A into the reactor (moles per unit time) the material balance over 8V, gives ... 

In a tubular reactor, the reactants are fed in at one end and the products withdrawn from the other. If we consider the reactor operated at steady state, the composition of the fluid varies inside the reactor volume along the flow path. Therefore, the mass balance must be established for a differential element of volume dV. We assume the flow as ideal plug flow, that is, that there is no back mixing along the reactor axis. Hence, this type of reactor is often referred to as Plug Flow Reactor (PFR). 

A wedge-shaped differential element of volume in spherical polar coordinates is shown in Fig. 10.8. 

A differential element of volume is conventionally written as dx. Sometimes the notation dV is used for S to indicate the surface area enclosing the volume V. 

As the concentrations are constantly changing along the reactor, the same is true of the rate of chemical production Rj of the constituent Cj. A mass balance on a differential element of volume dV must therefore be carried out, in which the rates Rj can be considered to have well-defined values. 

Use the above parameterization (assuming that parameters per differential element of volume, surface and integrated curvature are shape-independent) in Eq. (17) to calculate the liquid-drop energy associated with neutral clusters, and then add to it the charging energy according to Eq. (26) to determine the total LDM energy E (available experimental values for a and W can also be used). 

On the other hand, for the ideal tubular reactor with a plug-flow-like profile (PER), the material balance has to be made over a differential element of volume, dV = Adz, where A is the cross-sectional area of the bioreactor and dz is a differential thickness of the bioreactor (Figure 7.4). The material balance thus becomes... 

When the composition is uniform throughout the reactor (that is, independent of position), the equation can be applied to the whole of the working volume of the reactor. Thus the element of volume equals the volume of reactants. Where the composition is not uniform, the material balance must be made on a differential element of volume and the resulting equation integrated. 

Consider a differential element of volume at some point along the length of the reactor, where dV AcdL and Ac rZ /4. The rate at which heat is generated in this element by the exothermic chemical reaction is... 

Figure 17.7 (a) Feed channel and flows through a differential element of volume dz) and (b) temp-... 

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