Reynolds number , dimensionless

Dimensionless numbers (Reynolds number = udip/jj., Nusselt number = hd/K, Schmidt number = c, oA, etc.) are the measures of similarity. Many correlations between them (known also as scale-up correlations) have been established. The correlations are used for calculations of effective (mass- and heat-) transport coefficients, interfacial areas, power consumption, etc. 

A dimensional analysis of the system will result in four dimensionless numbers, Reynolds number, Froude number and geometric dimensionless parameters given by,... 

The characteristic Nerast parameter 5, the thickness of the film around the ion exchange particle, may be converted to the mass transfer coefficient and dimensionless numbers (Reynolds, Schmidt and Sherwood) that engineers normally employ. 

Reynolds dumber. One important fluid consideration in meter selection is whether the flow is laminar or turbulent in nature. This can be deterrnined by calculating the pipe Reynolds number, Ke, a dimensionless number which represents the ratio of inertial to viscous forces within the flow. Because... 

Flow Past Bodies. A fluid moving past a surface of a soHd exerts a drag force on the soHd. This force is usually manifested as a drop in pressure in the fluid. Locally, at the surface, the pressure loss stems from the stresses exerted by the fluid on the surface and the equal and opposite stresses exerted by the surface on the fluid. Both shear stresses and normal stresses can contribute their relative importance depends on the shape of the body and the relationship of fluid inertia to the viscous stresses, commonly expressed as a dimensionless number called the Reynolds number (R ), EHp/]1. The character of the flow affects the drag as well as the heat and mass transfer to the surface. Flows around bodies and their associated pressure changes are important. 

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

The dimensionless numbers in tlris equation are the Reynolds, Schmidt and the Sherwood number, A/ sh. which is defined by this equation. Dy/g is the diffusion coefficient of the metal-transporting vapour species in the flowing gas. The Reynolds and Schmidt numbers are defined by tire equations... 

The value of tire heat transfer coefficient of die gas is dependent on die rate of flow of the gas, and on whether the gas is in streamline or turbulent flow. This factor depends on the flow rate of tire gas and on physical properties of the gas, namely the density and viscosity. In the application of models of chemical reactors in which gas-solid reactions are caiTied out, it is useful to define a dimensionless number criterion which can be used to determine the state of flow of the gas no matter what the physical dimensions of the reactor and its solid content. Such a criterion which is used is the Reynolds number of the gas. For example, the characteristic length in tire definition of this number when a gas is flowing along a mbe is the diameter of the tube. The value of the Reynolds number when the gas is in streamline, or linear flow, is less than about 2000, and above this number the gas is in mrbulent flow. For the flow... 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

An impeller designed for air ean be tested using water if the dimensionless parameters, Reynolds number, and speeifie speed are held eonstant... 

Impeller Reynolds Number a dimensionless number used to characterize the flow regime of a mixing system and which is given by the relation Re = pNDV/r where p = fluid density, N = impeller rotational speed, D = impeller diameter, and /r = fluid viscosity. The flow is normally laminar for Re <10, and turbulent for Re >3000. 

And introducing the ratio of accelerations, = ag/g, where indicates the relative strength of acceleration, ag, with respect to the gravitational acceleration g. This is known as the separation number. The LHS of equation 60 contains a Reynolds number group raised to the second power and the drag coefficient. Hence, the equation may be written entirely in terms of dimensionless numbers ... 

Archimedes number A dimensionless number that relates the ratio of buoyancy forces to momentum forces, expressed in many forms depending on the nature of the Reynolds number. 

Before discussing the on.set, and nature, of fluid turbulence, it is convenient to first recast the Navier-Stokes equations into a dimensionless form, a trick first used by Reynolds in his pioneering experimental work in the 1880 s. In this form, the Navier-Stokes equations depend on a single dimensionless number called Reynolds number, and fluid behavior from smooth, or laminar, flow to chaos, or turbulence,... 

Since the process is more complex, the proposed method may not be valid for scale-up calculation. The combination of power and Reynolds number was the next step for correlating power and fluid-flow dimensionless number, which was to define power number as a function of the Reynolds number. In fact, the study by Rushton summarised various geometries of impellers, as his findings were plotted as dimensionless power input versus impeller... 

Reynold s number It is a dimensionless number that is significant in the design of any system in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid. It is equal to the density of a fluid, times its velocity, times a characteristic length, divided by the fluid viscosity. This value or ratio is used to determine whether the flow of a fluid through a channel or passage, such as in a mold, is laminar (streamlined) or turbulent. 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

The dynamical regimes that may be explored using this method have been described by considering the range of dimensionless numbers, such as the Reynolds number, Schmidt number, Peclet number, and the dimensionless mean free path, which are accessible in simulations. With such knowledge one may map MPC dynamics onto the dynamics of real systems or explore systems with similar characteristics. The applications of MPC dynamics to studies of fluid flow and polymeric, colloidal, and reacting systems have confirmed its utility. 

L/pj-A)(S/psA), liquid-solids velocity ratio, dimensionless Number of heat-transfer stages, dimensionless = hdp/kg, Nusselt number, dimensionless Pressure drop, gm-wt/cm2 = Cpu kg, Prandtl number, dimensionless = dpiipj U, Reynolds number, dimensionless S Mass velocity of solids, gm/cirf sec... 

For steady injection of a liquid through a single nozzle with circular orifice into a quiescent gas (air), the mechanisms of jet breakup are typically classified into four primary regimes (Fig. 3 2)[4°][41][22°][227] according to the relative importance of inertial, surface tension, viscous, and aerodynamic forces. The most commonly quoted criteria for the classification are perhaps those proposed by Ohnesorge)40] Each regime is characterized by the magnitudes of the Reynolds number ReL and a dimensionless number Z ... 

F), in addition to the Reynolds and Weber numbers, to fully describe a droplet spreading and solidification process upon impact on a substrate. They introduced two new dimensionless numbers, defined as ... 

In this correlation, the material properties are evaluated at the melting temperature. The left hand side of the correlation is the dimensionless minimum melt superheat. The right hand side of the correlation is also dimensionless, and represents a combination of the Prandtl number, Euler number, Reynolds number and Nusselt number, as well as temperature and length ratios TJTG and l0/d0. The correlation is accurate within 10%. In addition, considering the effects of the surface roughness of nozzle wall, the pre-basal coefficient in the regression expression has been increased by 25% in order to predict a safe estimate of the minimum melt superheat. 

The second method uses dimensionless numbers to predict scale-up parameters. The use of dimensionless numbers simplifies design calculations by reducing the number of variables to consider. The dimensionless number approach has been used with good success in heat transfer calculations and to some extent in gas dispersion (mass transfer) for mixer scale-up. Usually, the primary independent variable in a dimensionless number correlation is Reynolds number ... 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

Dimensionless numbers, such as Reynolds and Froude numbers, are frequently used to describe mixing processes. Chemical engineers are routinely concerned with problems of water-air or fluid mixing in vessels equipped with turbine stirrers where scale-up factors can be up to 1 70 (3). This approach has been applied to pharmaceutical granulation since the early work of Hans Leuenberger in 1982 (4). 

Hydrodynamic dimensionless numbers Examples are the Reynolds number, Froude, Archimedes, and Euler number. These dimensionless numbers have to be functions of identical determining dimensionless numbers of the same powers and with the same value of the other constant coefficients, so that the model and the object are similar. 

Dimensionless equations - some empirical and some with theoretical bases - are often used in chemical engineering calculations. Most dimensionless numbers are usually called by the names of person(s) who first proposed or used such numbers. They are also often expressed by the first two letters of a name, beginning with a capital letter for example, the well-known Reynolds number, the values of which determine conditions of flow (laminar or turbulent) is usually designated as Re, or sometimes as The Reynolds number for flow inside a round straight tube is defined as dvp p, in which d is the inside tube diameter (L), V is the fluid velocity averaged over the tube cross section (LT ), p is the fluid density (ML" ), and p is the fluid viscosity (ML T" ) (this is defined... 

As mentioned above, two distinct patterns of fluid flow can be identified, namely laminar flow and turbulent flow. Whether a fluid flow becomes laminar or turbulent depends on the value of a dimensionless number called the Reynolds number, (Re). For a flow through a conduit with a circular cross section (i.e., a round tube), (Re) is defined as ... 

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