Tube flow momentum balance

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

We will apply the steady state momentum balance to a fluid in plug flow in a tube, as illustrated in Fig. 5-6. (The stream tube may be bounded by either solid or imaginary boundaries the only condition is that no fluid crosses the boundaries other that through the inlet and outlet planes.) The shape of the cross section does not have to be circular it can be any shape. The fluid element in the slice of thickness dx is our system, and the momentum balance equation on this system is... 

This result can also be derived by equating the shear stress for a Newtonian fluid, Eq. (6-9), to the expression obtained from the momentum balance for tube flow, Eq. (6-4), and integrating to obtain the velocity profile ... 

Momentum Balance (permeation zone) Usually, pressure drops in permeation zone can be neglected. However, typical expressions are available for fluids flowing in a tube, both in laminar and turbulent conditions, and could be applied. 

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

Applying the momentum balance to the simplified tapered tube geometry of Figure 1.2, where we have only flow in and out in the x direction, gives... 

The basic equation may be derived as follows If one accelerates a certain mass flow rate, mg, of particles of a certain settling rate, Wg j, in a tube of constant cross section. A, a momentum balance yields the additional pressure drop, Apj i, at the distance, Tij, from the entry point of the particles ... 

Subsequent work took into account the velocity and shear field adjustments at the end of a die (Nickell et al., 1974). White and Roman (1976) developed a modified formulation of Tanner s theory where the melt recovers from Poiseuille flow in a tube into a state of uniaxial tension. These theories have been validated by Vlachopoulos et al. (1972). However, the die swell theory was also studied using macroscopic mass and momentum balances given by Metzner and others (Metzner et al., 1961 Graessley et al., 1970), where the balance on the fluid between the capillary exit and flow downstream formed a relationship with the die swell. The first normal stress difference is shown in Equation [4.10]. 

The momentum balance for this flow, assuming the tube is very long so that end effect is negligible, results in... 

In order to formulate the flow equations for a fluid, for instance, for the gas in the cyclone or swirl tube, we must balance both mass and momentum. The mass balance leads to the equation of continuity the momentum balance to the Navier-Stokes equations for an incompressible Newtonian fluid. When balancing momentum, we have to balance the x-, y- and -momentum separately. The fluid viscosity plays the role of the diffusivity. Books on transport phenomena (e.g. Bird et ah, 2002 Slattery, 1999) will give the full flow equations both in Cartesian, cylindrical and spherical coordinates. 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

CFD may be loosely thought of as computational methods applied to the study of quantities that flow. This would include both methods that solve differential equations and finite automata methods that simulate the motion of fluid particles. We shall include both of these in our discussions of the applications of CFD to packed-tube simulation in Sections III and IV. For our purposes in the present section, we consider CFD to imply the numerical solution of the Navier-Stokes momentum equations and the energy and species balances. The differential forms of these balances are solved over a large number of control volumes. These small control volumes when properly combined form the entire flow geometry. The size and number of control volumes (mesh density) are user determined and together with the chosen discretization will influence the accuracy of the solutions. After boundary conditions have been implemented, the flow and energy balances are solved numerically an iteration process decreases the error in the solution until a satisfactory result has been reached. 

Whereas Mersmann and Einenkel [112] correlated their experimental data on the basis of energy equations and Liepe and Joschek [334] set the power needed for particle suspension proportional to the stirrer power, Zehner [603] made use of the laws of momentum transfer. This appeared more reasonable to him, because in this case the efficiency of mechanical energy transfer did not have to be introduced. In his first fundamental investigations he utilized the jet loop, which due to the directional flow in the draught tube and in the annular space enabled easier balancing. 

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