dimensional analysis

7/17/2019 Dimensional Analysis

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METHODS OF DETERMINATION OF

DIMENSIONLESS GROUPS

1.0 Intuitive method: This method relies onbasic understanding of the phenomenon and thenidentifying competing quantities like types of

forces or lengths etc. and obtaining ratios ofsimilar quantities.

Some examples are:

Viscous force vs inertia force, viscous force vsgravity force or roughness dimension vs diameter.This is a dicult exercise and considerableexperience is required in this case.



. R!"#ei$h method: A functional power relation

is assumed between the parameters and then thevalues of indices are solved for to obtain thegrouping. or example in the problem in example! one can write

The values of a, b, c, d, and e are obtained bycomparing the dimensions on both sides thedimensions on the ".#.S. being $ero as % terms

are dimensionless. This is also tedious andconsiderable expertise is needed to form thesegroups as the number of unknowns will be morethan the number of available equations. This

method is also called &&indicial' method



%. &u'(in$h!m Pi theo)em method:

The application of this theorem provides a fairlyeasy method to identify dimensionlessparameters (numbers). #owever identi*cation ofthe in+uencing parameters is the ,ob of an

expert rather than that of a novice. This methodis that we use in this course



THE PRIN*IPLE OF DIMENSIONALHOMOGENEIT+

The principle is basic for the correctness of anyequation. -t states &&-f an equation truly expresses aproper relationship between variables in a physicalphenomenon then each of the additive terms will

have the same dimensions or these should bedimensionally homogeneous.//



or example if an equation of the following form expresses arelationship betweenvariables in a process then each of the additive term should havethe same dimensions. -n the expression A 0 1 2 345 A 1 and (345)each should have the same dimension. This principle is used indimensional analysis to form dimensionless groups. 6quations which

are dimensionally homogeneous can be used without restrictionsabout the units adopted. Another application of this principle is thechecking of the equations derived.

7ote : Some empirical equations used in +uid mechanics mayappear to be non homogeneous. -n such cases the numeric

constants are dimensional. The value of the constants in suchequations will vary with the system of units used.



&U*,INGHAM PI THEOREM The statement of the theorem is as follows : -f a relation among n

parameters exists in the form

then the n parameters can be grouped into n – m independentdimensionless ratios or π parameters expressed in the form

where m is the number of dimensions required to specify thedimensions of all the parameters, q1, q2, .... qn. t is also possibleto form ne! dimensionless π parameters as a discrete function ofthe (n – m" parameters. #or example if there are four

dimensionless parameters π1, π2, π$ and %8 it is possible to obtain%9 % etc. as

The limitation of this exercise is that the exact functionalrelationship in equationcannot be obtained from the analysis. The functional relationship

is generally arrived at through the use of experimental results.



Dete)min!tion o- G)ou/-rrespective of the method used the following steps will systemati$e theprocedure.

Ste/ 1. "ist all the parameters that in+uence the phenomenonconcerned. This has to be very carefully done. -f some parameters areleft out % terms may be formed but experiments then will indicate

these as inadequate to describe the phenomenon. -f unsure theparameter can be added. "ater experiments will show that the % termwith the doubtful parameters as useful or otherwise. #ence a carefulchoice of the parameters will help in solving the problem with leaste;ort. <sually three type of parameters may be identi*ed in +uid +ownamely +uid properties geometry and +ow parameters like velocity and

pressure.

Ste/ . Select a set of primary dimensions (mass length and time)(force length and time) (mass length time and temperature) aresome of the sets used popularly.

Ste/ %. "ist the dimensions of all parameters in terms of the chosenset of primary dimensions. 7ext table "ists the dimensions of variousparameters involved.



Ste/ . Select from the list of parameters a set of

repeating parameters equal to the number ofprimary dimensions. Some guidelines are necessaryfor the choice.%i" the chosen set should contain all the dimensions

%ii" t!o parameters !ith same dimensions should notbe chosen. say &, &2, &$,%iii"the dependent parameter to be determined

should not be chosen.



Ste/ 2. Set up a dimensional equation with therepeating set and one of the remaining

parameters in turn to obtain n – m suchequations, to determine π terms numbering n –m. The form of the equation is,

As the "#S term is dimensionless an equationfor each dimension in terms of a, b, c, d can beobtained. The solution of these set of equationswill give the values of a, b, c and d. Thus the %term will be de*ned.



Ste/ 3. 3heck whether % terms obtained are

dimensionless. This step is essential beforeproceeding with experiments to determine thefunctional relationship between the % terms.

dimensional analysis

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