Momentum balance shell

However, also momentum balance (Equation (13.11), [48]) has to be solved for both the shell and tube sides in addition to the mass balances (the molar flux,N , strictly depends on the velocity field, v). 

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. 

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... 

The momentum balance equations were not addressed, since constant values of pressure along axis in the tube and shell were assumed for preliminary studies. 

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. 

SHELL MOMENTUM BALANCE AND VELOCITY PROFILE IN LAMINAR FLOW ... 

Figure 2.9-1. Control volume for shell momentum balance on a fluid flowing in a circular tube.

Figure 2.9-2. Velocity and momentum flux profiles for laminar flow in a pipe. Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow...

Figure 2.9-3. Vertical laminar flow of a liquid film (a) shell momentum balance for a control volume A jc thick (b) velocity and momentum flux profiles.

Sec. 2.9 Shell Momentum Balance and Velocity Profile in Laminar Flow... 

The average velocity over the whole cross section of the pipe is precisely 0.5 times the maximum velocity at the center as given by the shell momentum balance in Eq. (2.9-13) for laminar flow. On the other hand, for turbulent flow, the curve is somewhat flattened in the center (see Fig. 2.10-1) and the average velocity is about 0.8 times the... 

Using this relation.ship and making a shell momentum balance on the fluid over a... 

Shell Momentum Balance for Flow Between Parallel Plates. A fluid of constant... 

Shell Momentum Balance for Flow Down an Inclined Plane. Consider the case of a Newtonian fluid in steady-state laminar flow down an inclined plane surface that makes an angle 6 with the horizontal. Using a shell momentum balance, find the equation for the velocity profile within the liquid layer having a thickness L and the maximum velocity of the free surface. Hint The convective momentum terms cancel for fully developed flow and the pressure-force terms also cancel, because of the presence of a free surface. Note that there is a gravity force on the fluid.)... 

These overall balances were powerful tools in solving various flow problems because they did not require knowledge of what goes on inside the finite control volume. Also, in the simple shell momentum balances made in Section 2.9, expressions were obtained for... 

Hence, a parabolic velocity profile is obtained. This result was also obtained in Eq. (2.9-9) when using a shell momentum balance. 

Shear thinning or pseudoplastic behavior is an important property that must be taken into account in the design of polymer processes. However, it is not the only property, and in Chapter 3 models that describe the viscoelastic response of polymeric fluids will be discussed. However, first we would like to solve some basic one-dimensional isothermal flow problems using the shell momentum balance and the empiricisms for viscosity described in this section. 

A cross section of the annular die required to produce the pipe is shown in Figure 2.7. The approximate velocity and stress profiles for this flow are sketched here also. A thin cylindrical shell is now chosen of length L and thickness Ar as shown in Figure 2.8. The shell is selected so that the surface is parallel to the flow direction. A force (or momentum) balance is now performed on the shell. Because the flow is under steady-state conditions, the forces in the z direction must sum to zero as shown below ... 

The shell force or momentum balance is performed on an element of fluid as shown in Figure 2.12, which is similar to that shown in Figure 2.8. Here the only terms that have to be considered are due to the stresses exerted by one layer of fluid on another as there are no effects due to gravity or pressure. The force or momentum balance is... 

Eqs. considered momentum and material balances in the tube core and in the shell annulus mass balances in the membrane include convective, diffusive, and reaction terms. 

The film-blowing process is used industrially to manufacture plastic films that are biaxially oriented. Many attempts have been made to predict and model this complex but important process, which continues to mystify rheologists and polymer processing engineers worldwide. A constitutive equation, able to predict well the polymer melt in all forms of deformation, is required to model the process, together with the standard conservation equations of continuity, momentum, and energy. Pearson and Petrie [125,126] were the first to predict the forces within the blown film by the use of the thin-shell approximation, force balances, and the Newtonian constitutive equation. The use of the thin-shell approximation and force balances is standard in any attempt to model the film-blowing process, and it has been used in the vast majority of subsequent studies. 

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