A Dimensionless Number

Consider the control volume shown in Fig. 11.7. The balance between body and surface forces acting on this control volume yields 

F/ being the inertial force, Fy the viscous force, and FB the buoyant force. 

The balance of thermal energy for the same control volume gives 

Qh being the longitudinal net enthalpy flow, 2 the enthalpy, flow across the two-phase interface, and Qk the conduction. In situations involving phase change, the Jacob number denoting the ratio of sensible heat to latent heat, 

Recalling Chapter 5, a dimensionless number resulting from Eq. (11.46) is 

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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. 

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... 

The Specific Speed is a dimensionless number using the formula above. Pump design engineers consider the Ns a valuable tool in the development of impellers. It is also a key index in determining if the pump... 

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. 

Power Number a dimensionless number used to describe the power dissipation of impeller and which is given by Np = Pg,/pN D where P = impeller power dissipation and g, = gravitational conversion factor. 

Practical probability is the limit of two ratios (Section 2.2). The numerator is the number of cases of failure of the type of interest (N) the denominator, the nonnalizing term is the time duration over which the failures occurred or the total number of challenges to the system. The former has the units of per time and may be larger than 1, hence it cannot be probability which must be less than 1. The latter is a dimensionless number that must be less than 1 and can be treated as probability. 

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. 

Discharge coefficient A dimensionless number describing the energy loss that occurs when a fluid is discharged from an orifice. 

K is a dimensionless number that establishes the regime of setding class, reference to the settling laws ... 

For the agitated and aerated vessel, the ratio of power requirements for aerated versus non-aerated systems is expressed by a dimensionless number known as the aeration rate the value is obtained from Figure 6.7. 

In 1953, Rushton proposed a dimensionless number that is used for scale-up calculation. The dimensionless group is proportional to NRe as shown by the following equation 2,3... 

In fact, the power number is a dimensionless number that is the ratio of ungassed power to gassed power in a normal bioreactor.3... 

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. 

Figure 7 shows the results of measurements of adsorption density by Parsonage, etal. [77] on a series of eighteen block copolymers, with poly(2-vinylpyridine) [PVP] anchors and polystyrene [PS] buoys, adsorbed from toluene (selective for PS) of variable molecular weight in each block. The results are presented as the reciprocal square of Eq. 28, that is, as a dimensionless number density of chains ct (d/Rg A)-2. For all but the copolymers of highest asymmetry, Eq. 28 is in good agreement with the data of Fig. 7. The high asymmetry copolymers are in the regime of the data of curves (a) and (c) of Fig. 3 where the large relative size...

Choosing as the recurring set p, N and D, then these three variables themselves cannot be grouped together to give a dimensionless number. M, L, T can now be expressed in terms of combinations of p, N, D. 

A dimensionless number known as the cavitation number (Cv) has generally been used to relate the flow conditions with the cavitation intensity [8] ... 

Mathematical modeling of the three-stage bed collapsing process led to a more quantitative characterization of powders, in the form of a dimensionless number called th e dimens ionless subsidence time ... 

Our task now is to come up with a way to quantifying the amount of nonlinearity the data exhibits, independent of the scale (i.e., units) of either variable, and even independent of the data itself. Our method of addressing this task is not unique, there are other ways to reach the goal. But we will base our solution on the methodology we have already developed. We do this by noting that the first condition is met by converting the nonlinear component of the data to a dimensionless number (i.e., a statistic), akin to but different than the correlation coefficient, as we showed in our previous chapter first published as [5],... 

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 ... 

We divide the concentration by its units to yield a dimensionless number. 

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 ... 

This number is conceptually an energy ratio, but independent of the interface heat extraction rate and thus the contact area. Since the interface heat transfer is assumed to control the solidification process of an impacting droplet, the choice of a dimensionless number should involve an evaluation of the influence exerted by this key factor. Therefore, the use of this newly defined dimensionless number is limited to an initial decision on which of the Impact number and the Freezing number is most appropriate for the application to a given material system at a know impact velocity. 

Schmidt number 3 phys chem A dimensionless number used In electrochemistry, equal to the product of the dielectric susceptibility and the dynamic viscosity of a fluid divided by the product of the fluid density, electrical conductivity, and the square of a characteristic length. Symbolized SC3. shmit. nam bar thre ) Schoeikopf s acid orgchem A dye of the following types l-naphthol-4,8-dlsulfonlc acid, l-naphthylamine-4,8-disulfonicadd,and l-naphthylamine-8-sulfonicadd may be toxic. shol.kopfs, as-3d ... 

Semenov number 1 physchem A dimensionless number used in reaction kinetics, equal to a mass transfer constant divided by a reaction rate constant. Symbolized S . Formerly known as Schmidt number 2. se-m3,n6f nom-bor won ... 

Probability—The expression for the likelihood of occurrence of an event or an event sequence during an interval of time or the likelihood of the success or failure of an event on test or on demand. Probability is expressed as a dimensionless number ranging from 0 to 1. 

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