Dimensionless groups, relations

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

Important dimensionless groups related to the leveling agent that emerge from the... 

In this case m = 5 and k = 3, and so there are only two independent dimensionless groups relating the five variables. The tt method does not tell us what the groups are, only how many exist. It will be useful to have a characteristic velocity from among the variables. For... 

As an additional example to illustrate the use of this method, let us consider a fluid flowing in turbulent flow at velocity v inside a pipe of diameter D and undergoing heat transfer to the wall. We wish to predict the dimensionless groups relating the heat-transfer coefficient h to the variables D, p, k, and v. The total number of variables is... 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Friction Factor and Reynolds Number For a Newtonian fluid in a smooth pipe, dimensional analysis relates the frictional pressure drop per unit length AP/L to the pipe diameter D, density p, and average velocity V through two dimensionless groups, the Fanning friction factor/and the Reynolds number Re. 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

Generally, it is rarely possible to satisfy all of the various similarity eriteria in seale-up, espeeially those related to ehemieal and thermal dimensionless groups. An alternative solution is to determine the effeet of eaeh group and perform intermediate seale experiments to determine the effeet of the eritieal groups. 

Using the mathematical technique of dimensionless group analysis, the rate of mass transport (/ m) in terms of moles per unit area per unit time can be shown to be a function of these variables, which when grouped together can be related to the rate by a power term. For many systems under laminar flow conditions it has been shown that the following relationship holds ... 

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensional analysis which enables the variables in a problem to be grouped into the form of dimensionless groups. Since the dimensions of the physical quantities may be expressed in terms of a number of fundamentals, usually mass, length, and time, and sometimes temperature and thermal energy, the requirement of dimensional consistency must be satisfied in respect of each of the fundamentals. Dimensional analysis gives no information about the form of the functions, nor does it provide any means of evaluating numerical proportionality constants. 

With some algebra, the parameters used in this expression can all be related to four dimensionless groups used in Example 11.10 ... 

The dimensionless group hD/k is called the Nusselt number, /VNu, and the group Cp i/k is the Prandtl number, NPl. The group DVp/p is the familiar Reynolds number, NEe, encountered in fluid-friction problems. These three dimensionless groups are frequently used in heat-transfer-film-coefficient correlations. Functionally, their relation may be expressed as... 

It is important to realize that the process of dimensional analysis only replaces the set of original (dimensional) variables with an equivalent (smaller) set of dimensionless variables (i.e., the dimensionless groups). It does not tell how these variables are related—the relationship must be determined either theoretically by application of basic scientific principles or empirically by measurements and data analysis. However, dimensional analysis is a very powerful tool in that it can rovide a direct guide for... 

The original seven variables in this problem can now be replaced by an equivalent set of four dimensionless groups of variables. For example, if it is desired to determine the driving force required to transport a given fluid at a given rate through a given pipe, the relation could be represented as... 

Another dimensionless group, analogous to the specific speed, that relates directly to the cavitation characteristics of the pump is the suction specific speed, Nss ... 

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,... 

The most satisfactory way of representing the relation between drag force and velocity involves the use of two dimensionless groups, similar to those used for correlating information on the pressure drop for flow of fluids in pipes. 

The decrease in the exit concentration with decreases in the extraction pressure seen in Figs. 17 and 18 is a consequence of the fact that the driving force for mass transfer is directly related to the partial pressure of the volatile component in the vapor phase, which, in this case, is constant and equal to the extraction pressure. In fact, reasonably good agreement between the data in Fig. 17 and the predictions of Eq. (38) can be obtained provided it is assumed that the dimensionless group (ki ATlk y p/L) is independent of pressure. This point is illustrated in Fig. 19, which is a plot of Eq. (38) for Pe =. The value used for (ki Aj/k v(kp/L) was chosen so as to obtain the asymptotic value of wi in Fig. 17. 

Dimensional analysis rearranges these physical quantities in the form of related dimensionless groups ... 

The basis of the scale-of-agitation approach is a geometric scale-up with the power law exponent, = 1 (Table 1). This provides for equal fluid velocities in both large- and small-scale equipment. Furthermore, several dimensionless groups are used to relate the fluid properties to the physical properties of the equipment being considered. In particular, bulk-fluid velocity comparisons are made around the largest blade in the system. This method is best suited for turbulent flow agitation in which tanks are assumed to be vertical cylinders. 

The use of various dimensionless groups for scaling-up can also lead to contradictory results. If two different values for a parameter are estimated by means of two dimensionless groups, then the value that satisfies both phenomena related to these groups should be selected. 

---