Dimensionless groups named

If gravitational settling can be neglected and if the droplet Reynolds number Re = payout 9s is small, then the droplet deformation and possible breakup in the flow are controlled by two dimensionless groups, namely the ratio of viscous to capillary forces, or capillary number... 

Transport Phenomena excited widespread praise and enthusiasm, as well as being a focus for the critics of the Engineering Science approach both at the time and later. In a famous review of the book, T.K. Sherwood (a sufficiently eminent chemical engineer to have a dimensionless group named in his honour ) offered the following [38] ... 

Professor Blok correctly points out that the authors have considered only two dimensionless groups namely the Reynolds Numbers of the ring and shaft. The authors believe that this simple correlation was first proposed by Lemmon eC al (1) In their paper on oil ring performance, when they showed that ring and shaft speeds, ring and shaft diameters and oil viscosity could be represented by the equation ... 

This means that the power number now becomes a function of two dimensionless groups, namely, the Reynolds number (Re) and the Froude number (Fr) ... 

This allows the pressure, which has significance only when the fluid experiencing it is named, to be replaced by a dimensionless group of properties which has significance for all fluids. Thus, the first requirement in constructing a model of a system is that the inlet pressures should be selected so that the density ratio of the two phases in the model is the same as that in the system... 

We use the recommended notation of the AIChE for dimensionless groups that are named after their originator, i.e., a capital N with a subscript identifying the person the group is named for. However, a number of dimensionless quantities that are identified by other symbols see, for example, Section IV. 

There is controvosy over the naming of Ihe dimensionless group uUDL (or its reciprocal), and, in particular, over naming it a Peclet number (see discussion by Weller, 1994)... 

The dimensionless group in the log term is called the Spalding B number, named after Professor Brian D. Spalding who demonstrated its early use [5], From Equation (9.39), the surface conditions give... 

For first order reaction in a porous slab this problem is solved in P7.03.16. Three dimensionless groups are involved in the representation of behavior when both external and internal diffusion are present, namely, the Thiele number, a Damkohler nunmber and a Biot number. Problem P7.03.16 also relates r)t to the common effectiveness based on the surface concentration,... 

Maintenance of proper temperature is a major aspect of reactor operation. The illustrations of several reactors in this chapter depict a number of provisions for heat transfer. The magnitude of required heat transfer is determined by heat and material balances as described in Section 17.3. The data needed are thermal conductivities and coefficients of heat transfer. Some of the factors influencing these quantities are associated in the usual groups for heat transfer namely, the Nusselt, Stanton, Prandtl, and Reynolds dimensionless groups. Other characteristics of particular kinds of reactors also are brought into correlations. A selection of practical results from the abundant literature will be assembled here. Some modes of heat transfer to stirred and fixed bed reactors are represented in Figures 17.33 and 17.18, and temperature profiles in... 

The dimensionless number H2 is, in fact, the well-known Reynolds number, Re, even though it appears in another form here. Now, we will explain the structure that a dimensionless number must have in order to be called a Reynolds number. (This example is equally valid for all other named dimensionless groups.) The Reynolds number is defined as being any dimensionless number combining a characteristic velocity, v, and a characteristic measurement of length, 1, with the kinematic viscosity of the fluid, v = p/p. The following dimensionless numbers are equally capable of meeting these requirements ... 

The value of dimensionless groups has long been recognised. As early as 1873, Von Helmholtz derived groups now called the Reynolds and Froude "numbers", although Weber (1919) was the first to name these numerics. 

The Froude number described above is frequently used for the description of radial and axial flotvs in liquid media when the pressure difference along a mixing device is important. When cavitation problems are present, the dimensionless group (Pj — p,) /pw - called the Euler number - is commonly used. Here p is the liquid vapour saturation pressure and p is a reference pressure. This number is named after the Swiss mathematician Leonhard Euler (1707-1783) who performed the pioneering work showing the relationship between pressure and flow (basic static fluid equations and ideal fluid flow equations, which are recognized as Euler equations). 

These eight variables have only three basic dimensions thus we should be able to find five independent dimensionless groups for this general set of fluid mechanics variables. These groups are so common that there are names for each. 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

Using such dimensionless groups, one can easily deduce the importance of fluid properties, namely, resistance to flow, on many of the production steps in the manufacture of a liquid detergent. For example, the power needed to provide agitation in a mixing vessel, known as the power number, Np, can be expressed as [33] ... 

NAMED DIMENSIONLESS GROUPS. Some dimensionless groups occur with such frequency that they have been given names and special symbols. A list of the most important ones is given in Appendix 4. 

The dimensionless group of variables defined by Eq. (3.8) is called the Reynolds niimbei It is one of the named dimensionless groups listed in Appendix 4. Its magnitude is independent of the units used, provided the units are consistent. 

The dimensionless groups in Eqs. (21.46) and (21.47) have been given names and symbols. The group k DjD is called the Sherwood number and is denoted by ATg,. This number corresponds to the Nusselt number in heat transfer. The group pjpD is the Schmidt number, denoted by It corresponds to the Prandtl number. Typical values of are given in Appendix 19. The group DGjp is, of course, a Reynolds number, IVr,. 

In his original paper, Thiele used the term modulus to emphasize that this then unnamed dimensionless group was positive. Later when Thiele s name was assigned to this dimensionless group, the term modulus was retained. Thiele number would seem... 

The inverse of the efficiency ij is a dimensionless group frequently called the Thiele modulus h. This important problem was treated independendy in 1939 by Thiele and Zeldovich. By solving the differential equation (7.3.13) without the assumption of a severe penetration effect (namely (A) = 0 at Z = L), it can be shown that indeed t) = /h for values of h larger than five. Severe penetration then means A > 5. This case is also referred to as that of the internal diffusion regime. 

The hydrodynamics of the flow between the two cylinders can be characterized by a dimensionless quantity 7h, the Taylor number, named after the mathematician G. I. Taylor. The Taylor number incorporates two dimensionless groups, the Reynolds number and a geometric ratio, and is defined as ... 

The dimensionless group ud/Dsx is called the axial Peclet number for mass Pem.ax (named after Jean Claude Peclet, see box), and d is a characteristic length. For empty tubes, d is the tube diameter dt, and for packed beds the particle diameter dp is mostly used. 

Thus the buoyancy is scaled by the density of the fluid. One may think of this dimensionless group as a reduced buoyancy, but that parlance is not used by fluid mechanicians and consequently this dimensionless group has no name. Nevertheless we will call it reduced buoyancy and accept the stigma that comes with this convenience. 

From the Song of Deborah, Judges 5 5, The mountains quaked (sometimes translated as flowed ) at the presence of the Lord. The concept of different types of deformation on different time scales and the name of the dimensionless group were introduced by Marcus Reiner in 1964, although Reiner s definition of De was the inverse of the definition now in use. 

---