Force Flow Equation

The one-dimensional force-flow equations (13.21)-(13.26) can be readily generalized for three-dimensional gradients and flows. In this case, Ei z = dE jdz is replaced by the three-dimensional gradient V,... 

In the absence of body force the equations of continuity and motion representing Stokes flow in a two-dimensional Cartesian system are written, on the basis of Equations (1.1) and (1.4), as... 

Similarly in the absence of body forces the Stokes flow equations for a generalized Newtonian fluid in a two-dimensional (r, 8) coordinate system are written as... 

By comparing the gravity force acting on the particle (equation 24) with the resistance to liquid flow (equation 22), we obtain the average liquid velocity relative to the particles ... 

Consider a cake of moulding resin between the compression platens as shown in Fig. 4.63. When a constant force, F, is applied to the upper platen the resin flows as a result of a pressure gradient. If the flow is assumed Newtonian then the pressure flow equation derived in Section 4.2.3 may be used... 

Due to the very low volumetric concentration of the dispersed particles involved in the fluid flow for most cyclones, the presence of the particles does not have a significant effect on the fluid flow itself. In these circumstances, the fluid and the particle flows may be considered separately in the numerical simulation. A common approach is to first solve the fluid flow equations without considering the presence of particles, and then simulate the particle flow based on the solution of the fluid flow to compute the drag and other interactive forces that act on the particles. 

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

A common approximation in many flow field computations at high fluid velocities is to consider that inertial forces dominate the flow and to neglect viscous forces (inviscid approximation). Since solvent viscosity is a variable in some of the experiments discussed here, the above approximation may be not be valid throughout and viscous forces are explicitly considered in the flow equations. Results of computations showed, nevertheless, that even with viscous solvents such as bis-(2-ethyl-hexyl)-phtalate with qi = 65 mPa s, viscous forces do not affect the flow field unless tbe fluid velocity drops below a few m s"1 at the orifice. This limit is generally more than one order of magnitude lower than the actual range used in the present investigations. 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

The mass flows include fuel (F), oxygen (O2), liquid water (1), evaporated water vapor (g) and forced flows (fan). The chemical energy or firepower is designated as Q and all of the heat loss rates by q. While Figure 12.4 does not necessarily represent a fire in a room, the heat loss formulations of Chapter 11 apply. From Equation (3.48), the functional form of the energy equation is... 

The interaction of forced and natural convective flow between cathodes and anodes may produce unusual circulation patterns whose description via deterministic flow equations may prove to be rather unwieldy, if possible at all. The Markovian approach would approximate the true flow pattern by subdividing the flow volume into several zones, and characterize flow in terms of transition probabilities from one zone to others. Under steady operating conditions, they are independent of stage n, and the evolution pattern is determined by the initial probability distribution. In a similar fashion, the travel of solid pieces of impurity in the cell can be monitored, provided that the size, shape and density of the solids allow the pieces to be swept freely by electrolyte flow. 

Gibbs adsorption equation is an expression of the natural phenomenon that surface forces can give rise to concentration gradients at Interfaces. Such concentration gradient at a membrane-solution Interface constitutes preferential sorption of one of the constituents of the solution at the interface. By letting the preferentially sorbed Interfacial fluid under the Influence of surface forces, flow out under pressure through suitably created pores in an appropriate membrane material, a new and versatile physicochemical separation process unfolds itself. That was how "reverse osmosis" was conceived in 1956. 

Fig. 19-17), the electrochemi- 192°-1992 cal energy inherent in the difference in proton concentration and separation of charge across the inner mitochondrial membrane—the proton-motive force—drives the synthesis of ATP as protons flow passively back into the matrix through a proton pore associated with ATP synthase. To emphasize this crucial role of the proton-motive force, the equation for ATP synthesis is sometimes written... 

Biot and Daughaday (B6) have improved an earlier application by Citron (C5) of the variational formulation given originally by Biot for the heat conduction problem which is exactly analogous to the classical dynamical scheme. In particular, a thermal potential V, a dissipation function D, and generalized thermal force Qi are defined which satisfy the Lagrangian heat flow equation... 

Coverage has been limited to horizontal three-phase separators up to this point. Considering Fig. 4.9, oil and water must flow vertically downward and gas vertically upward. The same laws of buoyancy and drag force apply. Equation (4.3) may therefore be used in the oil phase for water separation. Equations (4.12), (4.13), and (4.7) (see Fig. 4.8) are applied to the gas phase and oil phase for oil-gas particle separations, as was equally done for horizontal separators. The equations for the horizontal separator from Fig. 4.8 may also be used for the water drop terminal velocity in the vertical separator. 

By ruling out gravitational and acceleration forces, we are left with a simple balance pressure acting against viscous forces. By considering a symmetric cylindrical tube, we have a geometry so simple that this balance can be easily formulated and the flow equations readily solved. 

In these equations, p is the pressure relative to the local ambient pressure and. as before, is the angle between the direction of the forced velocity and the direction of the buoyancy forces as defined in Fig. 9.3. The x-axis is in the direction of the undisturbed forced flow. 

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

Only the (+) sign has been accounted for on the buoyancy term, i.e., only the case where the forced flow is in the same direction as the buoyancy forces has been considered. This is because in opposing flow the outer forced flow would be in the opposite direction to the buoyancy-driven flow near the surface. In this case the boundary layer equations would not apply. 

With mixed convective flow in a horizontal pipe the buoyancy forces act at right angles to the direction of forced flow leading to the generation of a secondary motion as discussed earlier. The equations governing this type of flow will be briefly discussed in this section. 

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