Dimensional analysis forced flow

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

Obtain, by dimensional analysis, a functional relationship for the heat transfer coeflicien for forced convection at the inner wall of an annulus through which a cooling liquid is flowing. 

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection can be neglected in comparison with those of forced convection. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

The procedure for performing a dimensional analysis will be illustrated by means of an example concerning the flow of a liquid through a circular pipe. In this example we will determine an appropriate set of dimensionless groups that can be used to represent the relationship between the flow rate of an incompressible fluid in a pipeline, the properties of the fluid, the dimensions of the pipeline, and the driving force for moving the fluid, as illustrated in Fig. 2-1. The procedure is as follows. 

The scope of coverage includes internal flows of Newtonian and non-Newtonian incompressible fluids, adiabatic and isothermal compressible flows (up to sonic or choking conditions), two-phase (gas-liquid, solid-liquid, and gas-solid) flows, external flows (e.g., drag), and flow in porous media. Applications include dimensional analysis and scale-up, piping systems with fittings for Newtonian and non-Newtonian fluids (for unknown driving force, unknown flow rate, unknown diameter, or most economical diameter), compressible pipe flows up to choked flow, flow measurement and control, pumps, compressors, fluid-particle separation methods (e.g.,... 

Dimensional Analysis of Forced Convection in a Single-Phase Flow... 

For heat transfer for a fluid flowing through a circular pipe, the dimensional analysis is detailed in Section 9.4.2 and, for forced convection, the heat transfer coefficient at the wall is given by equations 9.64 and 9.58 which may be written as ... 

In the development of these processes and their transference into an industrial-scale, dimensional analysis and scale-up based on it play only a subordinate role. This is reasonable, because one is often forced to perform experiments in a demonstration plant which copes in its scope with a small produdion plant ( mock-up plant, ca. 1/10-th of the industrial scale). Experiments in such plants are costly and often time-consuming, but they are often indispensable for the lay-out of a technical plant. This is because the experiments performed in them deliver a valuable information about the scale-dependent hydrodynamic behavior (arculation of liquids and of dispersed solids, residence time distributions). As model substances hydrocarbons as the liquid phase and nitrogen or air as the gas phase are used. The operation conditions are ambient temperature and atmospheric pressure ( cold-flow model ). As a rule, the experiments are evaluated according to dimensional analysis. 

The continuity equation (8.9) and the energy equation (8.12) are identical to those for forced convective flow. The x- and y-momentum equations, i.e., Eqs. (8.10) and (8.11), differ, however, from those for forced convective flow due to the presence of the buoyancy terms. The way in which these terms are derived was discussed in Chapter 1 when considering the application of dimensional analysis to convective heat transfer. In these buoyancy terms, is the angle that the x-axis makes to the vertical as shown in Fig. 8.3. 

Dimensions The numerical constants in Equation (15) are dimensional, with length in feet, mass in pounds, and time in hours. This unfortunate bit of untidiness does not detract from the utility of Equation (15), however. A proper dimensional analysis must wait until we have a better physical understanding of the mechanics of trickle flow. Presumably the characteristic length needed to make "10 3 " dimensionless would be the diameter of a disturbance that spontaneously forms in the upper part of the bed. The characteristic drag force for "3 X 105 " would presumably involve liquid-solid capillary effects. 

Additional droplet size work under flow conditions was not undertaken. The empirical expressions provided by Ingebo and Foster (10) were developed under conditions sufficiently similar to those present in the ACR to justify their use as a first approximation. Their data were derived from the injection of sprays into a transverse subsonic gas flow. They obtained the following correlation in Equations 5 and 6 between drop size parameters and force ratios by using dimensional analysis. 

One of the more beautiful techniques in fluid mechanics is the dimensional analysis, that provides - via simple calculations - a reliable judgment of the relative importance of the various forces that can drive or retard the flow of fluids. In short the procedure rests on constmction of non-dimensional groups of parameters that constitute the so-called non-dimensional munbers. The most important non-dimensional munber in fluid mechanics is the Re molds number that judges the relative importance of the inertial and viscous effects. At low values of the Reynolds number - a situation that is common to microsystems - the Navier Stokes equation can be well approximated by the Stokes equation, that, in the absence of body forces, reads ... 

FLUIDS FLOWING NORMALLY TO A SINGLE TUBE. The variables affecting the coefficient of heat transfer to a fluid in forced convection outside a tube are D , the outside diameter of the tube Cp, /r, and k, the specific heat at constant pressure, the viscosity, and the thermal conductivity, respectively, of the fluid and G, the mass velocity of the fluid approaching the tube, Dimensional analysis gives, then, an equation of the type of Eq, (12,27) ... 

Dynamic similarity ensures that the ratios of all forces, on the fluid flow and boundaries, in the prototype and scale model are the same and can be expressed as constants. Ratios of forces in fluid flows are often expressed in terms of dimensionless numbers. These dimensionless numbers are derived using what is called dimensional analysis using the Buckingham II theorem. 

Sometimes in fluid mechanics we may start with these four ideas and the measured physical properties of the materials under consideration and proceed directly to solve mathematically for the desired forces, velocities, and so on. This is generally possible only in the case of very simple flows. The observed behavior of a great many fluid flows is too complex to be solved directly from these four principles, so we must resort to experimental tests. Through the use of techniques called dimensional analysis (Chap. 13) often we can use the results of one experiment to predict the results of a much different experiment. Thus, careful experimental work is very important in fluid mechanics. With the development of supercomputers, we are now able to solve many complex problems mathematically by using the methods outlined in Chaps. 10 and 11, which previously would have required experimental tests. As computers become faster and cheaper, we will probably see additional complex fluid mechanics problems solved on supercomputers. Ultimately, the computer solutions must be tested experimentally. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

Assuming that the drag force is due to the inertia of the fluid, Cp would be constant and dimensional analysis shows that is generally a function of the particle Reynolds number Rep. The form of the function depends on the regime of the flow. This relationship for rigid spherical particles is shown in... 

Kinematic similarity is concerned with the motion of phases within a system and the forces inducing that motion. For example, in the formation of boundary layers during flow past flat plates and during forced convection in regularly shaped channels, there are usually three dominant forces pressure, inertia, and viscous forces. If corresponding points in two different-sized cells show at corresponding times identical ratios of fluid velocity, the two units are said to be kinematically similar and heat and mass transfer coefficients will bear a simple relation in the two cells. It can be shown by means of dimensional analysis that for a closed system under forced convection the equation of motion for a fluid reduces to a function of Re, the Reynolds number, which we have met in Chapter 2. To preserve kinematic similarity under those circumstances, Reynolds numbers in the two cells must be identical. 

The pressure drop per unit length of circular pipe (A/ / ) corresponding to the mean flow velocity (F) is a quantity of considerable practical importance, and was the first engineering application of dimensional analysis (Reynolds, 1883). A section of pipe sufficiently far from the inlet so that equilibrium has been established in the flow pattern is shown in Fig. 5.8. If inertia and viscous forces are included in the analysis but compressibility effects are ignored, then the important variables are those listed in Table 5.2. 

The force D) that a flowing fluid exerts upon a totally immersed body of fixed geometry, such as a sphere, will be considered next in terms of dimensional analysis. If compressibility of the fluid is ignored, the quantities of Table 6.1 suffice to define the system. 

A fluid for which p may be considered zero is called an ideal fluid. Air flowing at high speed approximates an ideal fluid and Eq. (6.2) may be used to find the drag on a body in such a case. The constant is approximately 0.1 for a sphere, but will have other values for bodies having other shapes. At very low values of R, inertia forces will be small compared with viscous forces and p need not be considered in the dimensional analysis. 

A pitot static tube (Fig. P6.10) is used to measure the volume rate of flow in a pipe Q). For the fully turbulent condition viscous forces may be ignored. Perform a dimensional analysis for the volume rate of flow (0 as a function of A/> (between A and B which is equal to yji where = specific weight of manometer fluid), D, and r for the fluid in the pipe. IfAp increases by a factor of 2, what is the corresponding increase in the rate of flow ... 

The flow pattern of the vortex motion of the gas in reverse-flow cyclone is quite complex. First, it is three-dimensional second, the flow is turbulent An exact analysis is therefore difficult Soo (1989) has summarized a fundamental analysis of velocity profiles and pressure drops in such a cyclone. He has also analyzed the governing particle diffusion equation in the presence of electrostatic, gravitational and centrifugal forces. He has then provided an analytical expression for partide collection efficiency under a number of limiting conditions. We wiU, however, opt here for a much simpler model of particle separation in a cyclone developed by Clift et id. (1991). This approach is based on a modification of the original model by Leith and Licht (1972). The model will be... 

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