Dimensionless numbers turbulent

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... 

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... 

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. 

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,... 

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. 

In the discussion above, we have considered only the velocity field in a turbulent flow. What about the length and time scales for turbulent mixing of a scalar field The general answer to this question is discussed in detail in Fox (2003). Here, we will only consider the simplest case where the scalar field 4> is inert and initially nonpremixed with a scalar integral length scale that is approximately equal to Lu. If we denote the molecular diffusivity of the scalar by T, we can use the kinematic viscosity to define a dimensionless number in the following way ... 

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 ... 

Heat transfer and its counterpart diffusion mass transfer are in principle not correlated with a scale or a dimension. On a molecular level, long-range dimensional effects are not effective and will not affect the molecular carriers of heat. One could say that physical processes are dimensionless. This is essentially the background of the so-called Buckingham theorem, also known as the n-theorem. This theorem states that a product of dimensionless numbers can be used to describe a process. The dimensionless numbers can be derived from the dimensional numbers which describe the process (for example, viscosity, density, diameter, rotational speed). The amount of dimensionless numbers is equal to the number of dimensional numbers minus their basic dimensions (mass, length, time and temperature). This procedure is the background for the development of Nusselt correlations in heat transfer problems. It is important to note that in fluid dynamics especially laminar flow and turbulent flow cannot be described by the same set of dimensionless correlations because in laminar flow the density can be neglected whereas in turbulent flow the viscosity has a minor influence [144], This is the most severe problem for the scale-up of laminar micro results to turbulent macro results. 

Reynold s number dimensionless number used in scaling fluid systems and in determining the transition point from laminar to turbulent flow. 

There are two types of convection, free and forced (Holman, 2009 Incropera et al., 2007 Kreith and Bohn, 2007). Free (natural) convection occurs when the heat transferred from a leaf causes the air outside the unstirred layer to warm, expand, and thus to decrease in density this more buoyant warmer air then moves upward and thereby moves heat away from the leaf. Forced convection, caused by wind, can also remove the heated air outside the boundary layer. As the wind speed increases, more and more heat is dissipated by forced convection relative to free convection. However, even at a very low wind speed of 0.10 m s-1, forced convection dominates free convection as a means of heat loss from most leaves (0.10 m s-1 = 0.36 km hour-1 = 0.22 mile hour-1). We can therefore generally assume that heat is conducted across the boundary layer adjacent to a leaf and then is removed by forced convection in the surrounding turbulent air. In this section, we examine some general characteristics of wind, paying particular attention to the air boundary layers adjacent to plant parts, and introduce certain dimensionless numbers that can help indicate whether forced or free convection should dominate. We conclude with an estimate of the heat conduction/convection for a leaf. 

Next, we introduce a dimensionless number that describes flow characteristics, such as whether the flow will be laminar or turbulent. This quantity indicates the ratio of inertial forces (due to momentum, which tends to keep things moving) to viscous forces (due to friction, which tends to slow things down) and is known as the Reynolds number ... 

In hydrodynamics the transition from laminar to turbulent flow is characterized by a dimensionless number called the Reynolds number. Re. 

The transition from laminar flow to turbulent flow in a forced convection situation is determined by which one of the following dimensionless numbers ... 

Pioneering work by Osborne Reynolds in the late 19th century added considerably to the understanding of fluid flow in relation to surfaces, and established concepts on which subsequent theoretical, empirical, and practical work could be based. The principal finding was in connection with fluid flow in pipes, visually demonstrating the difference between laminar and turbulent flow. Reynolds discovered that the dimensionless number that now bears his name, the Reynolds number Re), defined the flow condition in a tube. 

The introduction of different parameterizations for the turbulent viscosity parameter leads to different modifications of the correlations for the dimensionless numbers. By setting the viscous Prandtl and Schmidt number equal to unity and noting that the integral term in the denominator is simply the velocity (5.276) that can also be expressed by = = we get... 

When dealing with turbulent flows all the relevant dimensionless numbers are evaluated with the available quantities. For example, in DNS, the fluid and particle instantaneous velocities will be employed, whereas in large-eddy simulation (LES) or in Reynolds-average Navier-Stokes-equations (RANS) simulations the filtered or Reynolds-average values will be used. 

The rest of the chapter follows our approach to Chapters 5 and 6. First, we study laminar two-phase flow by approximate analytical means. Next, we consider the dimensionless numbers appropriate for two-phase. Finally, we correlate experimental data on turbulent two-phase in terms of these numbers. 

So far, we have studied a number of illustrative examples for two-phase laminar heat transfer following the analytical approach we used in Chapter 5. For two-phase turbulent heat transfer we use an approach based on two-length scale dimensional analysis and the correlation of experimental data in terms of dimensionless numbers resulting from this analysis. 

In dealing with viscoelastic fluids, especially under turbulent flow conditions, it is necessary to introduce a dimensionless number to take account of the fluid elasticity [29-33], Either the Deborah or the Weissenberg number, both of which have been used in fluid mechanical studies, satisfies this requirement. These dimensionless groups are defined as follows ... 

When the convection is free, the value of the coefficient of convection h in the fluid is finite, and it depends on the dimensions of the rubber. The convection may be either turbulent or laminar. In both cases, a new dimensionless number is obtained... 

Reynolds number for flow is a dimensionless number characterizing the turbulence in the reactor. When the Reynolds number is below 2500, the flow regime is described as laminar, and suitable scale-up approaches are used. The Reynolds nuiriber range, 2500 to 10,000, is considered by many as transitional, and engineers are very careful not to operate in that regime because of its imique characteristics. Reynolds numbers above 10,000 describe fully developed turbulence. 

Dimensionless numbers are pure numbers that are expressed as ratios of physical properties and are used to classify or understand a system. Already, in Chapter 6, we defined the Reynolds number as the ratio of the fluid s inertial forces, pud, to its viscous force and used it to differentiate between the laminar, intermediate, and turbulent flow regimes. In the laminar flow regime, viscous forces dominate but, as the velocity increases, the Reynolds number increases and inertial forces predominate in the turbulent regime. 

Correlations, dimensionless numbers, and regime maps have been developed for many of these applications. The Reynolds number was introduced in Chapter 5 to differentiate flow regimes in pipes—laminar versus transition versus turbulent. In Chapter 6, the notion of particle Reynolds number was mentioned with respect to drag force and rotameters. 

Reynolds number, a ratio of momentum forces to viscous forces, Nr or Re = (p)(V)D/(x, where p is fluid density, V is fluid velocity, u is fluid viscosity (absolute), and D is some significant dimension such as the diameter of a pipe. Units used must all cancel out, that is, make Re si dimensionless number. Example Re = flb/ft ) X (ftdir) X ft/(lb/hr ft). Try canceling out the same units in numerator and denominator, and you have no units left—a dimensionless number. As an example, the change from laminar to turbulent flow inside a pipe (where D is the inside diameter of the pipe) is in the range Re = 2100 to 3000, no matter what units are used. 

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