Shell balance

These detergent range (C C ) odd and even linear internal olefins are fed to oxo-alcohol plants to produce C22 C2 semilinear alcohols. Most of the alcohols are ethoxylated and sold into detergent markets (8). Shell balances carbon numbers by a combination of the ethylene oligomerization extent. 

A steady-state shell balance on species j in the element of length dz between z and... 

For the PFTR we wrote the mass balance on species j by writing a shell balance between position z and z + dz. After letting ds —> 0 we obtained the expressions... 

Figure 5-2 Energy balance in a PFTR. A shell balance is made on an element of volume dV between Z and + dz. Species flow Fj and enthalpy flow in and out of this dement of volume are balanced hy species and energy generated by reaction.

For a spherical pellet we must find Ca R), where R is a radius within the pellet, which has total radius R,. Then we proceed as before to find rj. For a sphere the shell balance is set up in spherical coordinates to yield... 

A shell balance on species j between z and z dz yields the expression... 

This is the hoop stress for thin-waUed vessels, i.e., t < r/4. For thick-walled vessels, see Ashby, p. 396 [1]) The longitudinal stress can be obtained from a similar shell balance (see Figure 8.7)... 

Fig. 3.7 Shell balance for the diffusion limited immobilized enzyme...

Fig. 8.5 Shell balances around a differential element of a filter.

A diagram of the shell balance (material balance) for simultaneous diffusion and first order reaction of component A in a. sphere is shown in Figure 6.3.6. The material balance in the spherical shell is given by ... 

The shell balance method will be used to examine steady laminar flow of a fluid in a pipe. For the geometrical system illustrated in Figure 3B-1 and for steady laminar fully developed flow of a fluid, a shell momentum balance can be conducted (Bird et al., 1960 Geankoplis, 1983) using the cylindrical coordinates, r, 6, andz. The momentum balance is conducted on a control volume shell at a radius r with dimensions Ar and Az. 

We now proceed to perform our shell balance on A. The area that appears in the balance equation is the total area (voids and solids) normal to the direction of the molar flux ... 

To describe this problem in mathematical terms, either the differential species mass balance (1.39) can be reduced appropriately or alternatively a species mass shell balance over a thin layer, Az, can be put up and combined with Pick s law. The resulting equation for steady diffusion in the thin layer is of course the same in both cases. The simple ordinary differential equation is integrated twice with the appropriate boundary conditions in order to get a relation for the concentration profile that is needed to determine the diffusive flux. 

Determine the concentration profile and the flux of A to the surface using ta) shell balances and (b) the general balance equations. 

FIGURE 2.2-2 One-dimensional shell balances (a) rectangular and (6) cylindrical coordinates. 

Although general and rather elegant derivations of the differentia] balances are available,6 7 it is more instructive to use the simple shell" balance approach. Figaro 2.2-2 depicts situations where changes in density, velocity, composition, and temperature occur continuously and in only one spatial direction, either x or r, The change whh lima of ihe total mass within the volume element of Fig. 2J2-2a. owing to the difference in mass flow into and out of Ihe volume, is... 

This sama shell balance can be applied in curvilinear cootdinele systems as illustrated in Fig. 2.2-26. In this case the change of area with radial position must he tafcco into account, that is,... 

A shell balance applied to the mass of eny particular species i would yield, in rectangular conrdjnaies. 

The differential linear momentum balnece for a multicomponent mixture can he derived from a shell balance approach or from a more sophisticated analysis using continuum mechanics. Such formulations are tabulated in many texts, for example, that by Bird el al.,1 and will not be presented here in any detail. By way of illustration the x component of the diffeiemie] momentum balance in rectangular coordinates would be... 

In equations 1.2a-1.3b, we have used c i to represent the bulk concentration of species A (moles per unit volume), and c to represent the surface concentration of species A (moles per unit area). The nomenclature for the homogeneous reaction rate, and heterogeneous reaction rate, Rj, follows the same pattern. The surface concentration is sometimes referred to as the adsorbed concentration or the surface excess concentration, and the derivation (Whitaker, 1992) of the jump condition essentially consists of a shell balance around the interfacial region. The jump condition can also be thought of as a surface transport equation (Slattery, 1990) and it forms the basis for various mass transfer boundary conditions that apply at a phase interface. 

Before exploring the general problem in some detail, we consider the typical intuitive approach commonly used in textbooks on reactor design (Carberry, 1976 Fogler, 1992 Froment and Bischoff, 1979 Levenspiel, 1999 Schmidt, 1998). In this approach, the analysis consists of the application of a shell balance based on the word statement given by... 

If the plastic bead covering the tip of the thermocouple described in Problem 1.4 is quite large, and since plastic usually sustains a very low value of thermal conductivity, then the simple lumped model solution becomes quite inaccurate. To improve the model, we need to account for thermal conductivity in the (assumed) spherical shape of the plastic bead, (a) Assuming the bead is a perfect sphere, contacted everywhere by external fluid of temperature 7), perform a shell balance on an element of volume 4irr Ar and show that... 

Setting the shell balance at the position r, we obtain the following mass balance equation... 

The transient mass balance equation inside the capillary is obtained by setting a shell balance in the capillary ... 

C3. Derive Eq. (17-31) from shell balances. You should obtain a second-order equation. Do the first integration and apply the boundary condition R = 1.0 at z = 0. 

In this discussion overall or macroscopic mass balances were made because we wish to describe these balances from outside the enclosure. In this section on overall mass balances, some of the equations presented may have seemed quite obvious. However, the purpose was to develop the methods which should be helpful in the next sections. Overall balances will also be made on energy and momentum in the next sections. These overall balances do not tell us the details of what happens inside. However, in Section 2.9 a shell momentum balance will be made to obtain these details, which will give us the velocity distribution and pressure drop. To further study these details of the processes occurring inside the enclosure, differential balances rather than shell balances can be written and these are discussed in other later Sections 3.6 to 3.9 on differential equations of continuity and momentum transfer. Sections 5.6 and 5.7 on differential equations of energy change and boundary-layer flow, and Section 7.5B on differential equations of continuity for a binary mixture. 

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