Fluid mechanics, equations

Analytical approach applying the three fluid mechanics equations... 

Commonly used simulation programs are based on a numerical solution of the onedimensional classical fluid mechanics equation by the method of characteristics [1-2]. For numerical simulation the piping system is devided in sections with constant cross-sectional areas which are connected by knot-elements. 

In the previous section, the importance of the uniformity of the radial flow profile was established. In the present section, the fluid mechanical equations for all four flow configurations in Figure 1 are derived and solved for comparison. The development of equations closely follows the approach of Genkin et al. (1,2). Here we extend their work to include both radial and axial flow in the catalyst bed. Following our derivation in reference (16), the dimensionless equations for the axial velocity in the center-pipe for all four configurations are (the primes denote derivative with respect to the dimensionless axial coordinate). 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

To apply the above equations to electrochemical mass transfer, the solution velocity [v in Equation (26.54)] must be known, which requires the solution of a separate set of fluid mechanics equations. Also, the transport parameters, D and M , must be specified, as will be discussed in some detail below. 

The flow sensors most commonly employed in the industrial practice are those that measure the pressure gradient developed across a flow constriction. Then, using the well-known (from fluid mechanics) equation of Bernoulli, we can compute the flow rate. Such devices can be used for both gases and liquids. The orifice plate (Figure 13.5a), venturi tube (Figure 13.5b), and Dali flow tube are typical examples of sensors based on the foregoing principle. The first is more popular due to its simplicity and low cost. The last two are more expensive but also more accurate. 

The tliree conservation laws of mass, momentum and energy play a central role in the hydrodynamic description. For a one-component system, these are the only hydrodynamic variables. The mass density has an interesting feature in the associated continuity equation the mass current (flux) is the momentum density and thus itself is conserved, in the absence of external forces. The mass density p(r,0 satisfies a continuity equation which can be expressed in the fonn (see, for example, the book on fluid mechanics by Landau and Lifshitz, cited in the Furtlier Reading)... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

Aris, R., 1989. Vectors, Tensors and the Basic Equations of Fluid Mechanics, Dover Publications, New York. 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

Dynamic meteorological models, much like air pollution models, strive to describe the physics and thermodynamics of atmospheric motions as accurately as is feasible. Besides being used in conjunction with air quaHty models, they ate also used for weather forecasting. Like air quaHty models, dynamic meteorological models solve a set of partial differential equations (also called primitive equations). This set of equations, which ate fundamental to the fluid mechanics of the atmosphere, ate referred to as the Navier-Stokes equations, and describe the conservation of mass and momentum. They ate combined with equations describing energy conservation and thermodynamics in a moving fluid (72) ... 

StoKes-Einstein and Free-Volume Theories The starting point for many correlations is the Stokes-Einstein equation. This equation is derived from continuum fluid mechanics and classical thermodynamics for the motion of large spherical particles in a liqmd. 

One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction velocity components perpendicmar to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental consei vation equations of fluid mechanics are greatly simphfied for one-dimensional flows. A broader categoiy of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. 

Macroscopic and Microscopic Balances Three postulates, regarded as laws of physics, are fundamental in fluid mechanics. These are conservation of mass, conservation of momentum, and con-servation of energy. In addition, two other postulates, conservation of moment of momentum (angular momentum) and the entropy inequality (second law of thermodynamics) have occasional use. The conservation principles may be applied either to material systems or to control volumes in space. Most often, control volumes are used. The control volumes may be either of finite or differential size, resulting in either algebraic or differential consei vation equations, respectively. These are often called macroscopic and microscopic balance equations. 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

For modeling, the similitude laws governing modeling must be followed. The topics of dynamic similitude and theory of models are discussed in most textbooks on fluid mechanics,and only the resulting equations are discussed here. 

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