Dimensionless number

A group of physical quantities with each quantity raised to a power such that all the units associated with the physical quantities cancel, i.e dimensionless. 

This theorem provides a method to obtain the dimensionless groups which affect a process. First, it is important to obtain an understanding of the variables that can influence the process. Once you have this set of variables, you can use the Buckingham Pi Theorem. The theorem states that the number of dimensionless groups (designated as n, ) is equal to the number (n) of independent variables minus the number (m) of dimensions. Once you obtain each n, you can then write an expression  

This result only gives you the fact that Oi can be written as a function of the other fls. Normally, the exact functional form comes from data correlation or rearrangement of analytical solutions. Correlating data using the dimensionless numbers formed by this method typically allows one to obtain graphical plots which are simpler to use and/or equations which fit to the data. If the dimensionless terms are properly grouped, they represent ratios of various effects and one can ascertain the relative importance of these effects for a given set of conditions. 

Basic dimensions include length (L), time (t), and mass (M). Variables which are independent means that you cannot form one variable from a combination of the other variables 

As an example of the method, let us examine the situation of mass transfer associated with flow across a flat plate. The variables which are important are listed in Table A.l, together with their dimensions. In this case, n = n — m = 5 — 2 = 3. 

Height average body density acceleration of gravity Crushing strength of bones 

we might suspect that dimensionless ratios like the one above would be important in predicting the behavior of a large piece of equipment from tests of a small model. Experience indicates that this is certainly the case. These dimensionless numbers have also proved invaluable in correlating, interpreting, and comparing experimental data. For example, if you were asked to compare a business venture in which you invest 4000 for a return of 400 per year with one in which you invest 6000 for a return of 650 per year, 

Furthermore, there is often a real benefit in understanding if we can show our experimental, or computational results in dimensionless form. The behavior of nature does not depend on the systerri of dimensions that we humans use to describe that behavior. Thus, the results of our observations, if they are correct, can be expressed in a form which is totally independent of the system of units we use. If they cannot, then we should wonder whether they, are correct. Finding the best way to reduce observations made in any system of units to dirhensionless form is always a good test of the experimental data, and often it is a good test of our understanding of that data. The application of 

Because they reduce the number of variables needed to describe a phenomenon and minimize the number of experiments to fit the data. 

Because they simpUiy physical understanding of a phenomenon. 

Because they can guide us in understanding which processes are dominant. 

Here we introduce you to the most common DNs with a brief explanation of their relevance. Although we employ SI units, you can use other units as long as you are consistent. The calculation of a DN does not depend on the unit system used. 

The Reynolds number is probably the most common DN in process engineering. One of its characteristics is that you can determine if the flow is laminar or turbulent 

Myllykangas, J., Aineensiirtokertoimen, neste—ja kaasuosuuden seka aineensiirtopintaalan kor-relaatiot eraissa heterogeenisissa reaktoreissa. Rep. Lab. Ind. Chem., Abo Akademi, Abo, Finland, 1989. 

Gas-Liquid Reactors, Chemical Reactor Design for Process Plants,Vo. 1 Principles and Techniques, p. 627, Wiley, New York, 1977. 

Geddes, R.L., Local efficiencies of bubble plate fractionators, Trans. Am. Inst. Chem. Eng., 42, 79,1946. 

Hikita, H., Asai, S., Tanigawa, K., Segawa, K., and Kitao, M., The volumetric liquid-phase mass-transfer coefficient in bubble columns, Chem. Eng. /., 22,61,1981. 

Ozturk, S.S., Schumpe, A., and Deckwer, W.-D., Organic liquids in a bubble column Hold-ups and mass-transfer coefficients, AIChF /., 33,1473,1987. 

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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. 

Dimensionless numbers are not the exclusive property of fluid mechanics but arise out of any situation describable by a mathematical equation. Some of the other important dimensionless groups used in engineering are Hsted in Table 2. 

An estimate of the relative importance of convection and radiation can be obtained from the ratio of the radiation-to-convection transfer rates. This dimensionless number reduces to... 

Dimensionless Numbers. With impeller diameter D as length scale and mixer speed N as time scale, common dimensionless numbers encountered in mixing depend on several controlling phenomena (Table 2). These quantities are useful in characterizing hydrodynamics in mixing tanks and when scaling up mixing systems. 

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. 

To be able to decide which compressor best fits the job, the engineer must analyze the flow characteristics of the units. The following dimensionless numbers describe the flow characteristics. 

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 advantage of the use of dimensionless numbers in calculations such as these is that they may be applied independent of the physical size of the system being described. 

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... 

There are tluee dimensionless numbers used in these equations, and their dehnitions are ... 

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... 

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 ... 

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. 

As one of two possible dimensionless numbers is now known, the second one can be obtained by dividing both sides of the equation through by the remaining values ... 

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 ... 

Substituting these dimensionless numbers into Equation 7-12 yields,... 

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. 

The dimensionless number Le is called the Lewis number (m Russian literature it is called the Luikov number). The Lewis number incorporates the specific heat capacity of humid air pCp (J/m C), the diffusion factor of water vapor in... 

The above shows how the dimensionless numbers are used to provide the most accurate solution. Collecting these definitions together,... 

TT-theorem) or from the governing equations of the flow. The latter is to be preferred because this method will give a sufficient amount of dimensionless numbers. Furthermore, it will connect the numbers to the physical process via the equations and give important information in cases where it is necessary to make approximations. 

It is observed that the following dimensionless numbers appear in the equations ... 

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure ot the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity Uq. The normalized velocity can be defined by m u f po //ao where f, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (AT is the temperature difference between the hot and the cold surface) ... 

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