Applications of Buckingham – π Theorem

“Buckingham – π theorem is a mathematical theorem used in dimensional analysis that predicts the number of non-dimensional groups that must be functionally related from a

set of dimensional parameters that are thought to be functionally related”(Yunus A. Cengel, 2008)

In real life engineering problem, when equations for a system are unavailable and experimentation is the only method of obtaining reliable results, tests are performed on a geometrically scaled model, rather than a full-scale prototype. This saves both time and money. For modeling up a system, dimensional analysis is a powerful tool with primary purpose (Yunus A. Cengel, 2008) of:

Generating non-dimensional parameters that help in the design of experiments (physical and/or numerical) and in the reporting of experimental results.

Obtaining scaling laws so that prototype performance can be predicted from model performance.

Predict (sometimes) the trends in the relationships between parameters.

In order to scale down a prototype to a similar model, all three similarities i.e. geometric similarity, kinematic similarity, dynamic similarity must be achieved in addition to the similarity of non-dimensional parameters (π). These parameters are of two types (Yunus A. Cengel, 2008):

Dependent (π1) Independent (π2, π3,…, πk)

Functional relationship between the two types is: π1 = f (π2, π3,…, πk)

Buckingham – π theorem provides the relation for the number of independent non-dimensional parameters. The theorem states:

“n physical quantities with r base dimensions can generally be arranged to provide only (n - r) independent dimensionless parameters also referred to as πs”

(Kumar, 2004)

In mathematical terms, if there are n physical variables q1, q2,…, qn, the functional relationship may become:

f (q1,q2,…, qn) = 0

If there are r fundamental dimensions, there will be (n-r) independent dimensionless groups (π1, π2,…, πn-r), and the functional relationship may be written as:

f (π1, π2,…, πn-r) = 0

The Group method or the Rayleigh’s Indicial Method may be used to group the recognized set of variables into a functional form of independent non-dimensional parameters.

The theorem provides a method of making dimensionless groups and dimensionless numbers, which are of considerable value in developing relationships in chemical engineering (J M Coulson, 2009).

BibliographyJ M Coulson, J. F. (2009). Chemical Engineering. Noida: Elsevier.

Kumar, K. .. (2004). Engineering Fluid Mechanics. New Delhi: Eurasia Publishing House.

Yunus A. Cengel, J. M. (2008). Fluid Mechanics; Fundamentals and Applications (In SI Units). New Dehli: Tata McGraw-Hill Publishing Company Limited.

Example:Consider a fluid of density ρ and viscosity μ, flowing in a pipe of diameter D with a velocity u. The relationship between these variable affecting the flow of fluid may be written as:

f ( ρ, μ , D ,u )=0

Writing their dimensions:

ρ M/L3μ M/LTD Lu L/T

The relation includes three variables (M, L and T), therefore according to Buckingham – π theorem:

No. of independent Dimensionless groups = 4 - 3 = 1

The dimensional matrix, with rows corresponding to the dimensions (M, L and T) columns to the physical variables ρ , μ ,D ,u ; will become:

[ 1 1 0−3 −1 10 −1 0

01

−1]For nullity, assuming the (3 x 4) matrix multiplied to the dimensional matrix :

[ 1 1 0−3 −1 10 −1 0

01

−1] [abcd ] = [0000]

Multiplying the matrix, we get linear equations:

a + b = 0 …. I-3 a – b + c + d = 0 … II

- b – d = 0 … IIISolving these equations simultaneously:

From I: a = - b

From III: b = - d → a = dFrom II: -3 a – (-a) + c + (a) = 0

-3 a + 2 a + c =0- a + c = 0 → a = c

Using iteration, let a = 1:

[abcd] = [ 1−11

1]

f=ρa . μb .D c . ud

f=ρ1 . μ−1 . D1 .u1

f= ρu Dμ

This is equals to Reynolds’s Numbers which is a ratio of inertial forces to viscous forces and explains the characteristics of the fluid flow.