Buckingham Pi Theorem

THE BUCKINGHAM PI THEOREM 1. The Buckingham pi theorem is a rule for deciding how many dimensionless numbers (called π’s ) to expect. The theorem states that the number of independent dimensionless groups is equal to the difference between the number of variables that go to make them up and the number of individual dimensions involved. The weakness of the theorem-from a practical point view-is that it does not depend on the number of dimensions actually used, but rather on the minimum number that might have been used. 2. Firstly one must decide what variables enter the problem. Occasionally a dimensional analysis will show that one of the selected variables should not be present, since it involves a dimension not shared by any of the other variables; but if the wrong variables go in, the wrong dimensionless numbers come out, most of the time. 3. One error to avoid in choosing the variables is the inclusion of variables whose influence is already implicitly accounted for. In analyzing the dynamics of a liquid flow, for example, one might argue that the liquid temperature is a significant variable. It is important, however, only in its influence on other properties such as viscosity, and should therefore not be included along with them. 4. The Buckingham pi theorem, if applied to the actual number of dimensions being used, tells only that there must be at least a certain number of dimensionless numbers involved. Unless one resorts to one of the tedious techniques that have been devised for discovering the minimum number of dimensions needed, the theorem gives little assurance that all the dimensionless numbers have been found-an assurance that can very quickly be secured from the step-by-step approach, if assurance is needed. 5. The method of dimensional analysis is based on the obvious fact that in an equation dealing with any system, each term must have the same dimension. For example, if ψ+η+ζ=φ is a physical relation, then ψ, η, ζ and φ must have the same dimensions. The above equation can be made dimensionless by dividing by any one of the terms, say φ: (ψ/ φ) + (η/ φ) + (ζ/ φ) = 1 These ideas are embodied in the Buckingham pi theorem, stated below: (a) Let K equal the number of fundamental dimensions required to describe the physical variables (e.g., mechanics: mass, length and time; hence K=3). (b) Let P1,P2,…..PN represent N physical variables in the physical relation f1(P1,P2,…..,PN) = 0_______________________(1) (c) Then, this physical relation may be expressed as a relation of (N-K) dimensionless products (called Π products), f2(Π1,Π2, ….., ΠN-K) = 0_______________________(2)

where each Π product is a dimensionless product of a set of K physical variables plus one other physical variable. (d) Let P1,P2,…,PK be the selected set of K physical variables. Then Π1 = f3(P1,P2,…..,PK,PK+1) Π2 = f4(P1,P2,…..,PK,PK+2) ………………………………. ΠN-K = f5(P1,P2,…..,PK,PN) (e) The choice of the repeating variables P1,P2,…PK should be such that they include all the K dimensions used in the problem. Also the dependent variable should appear in only one of the Π products. SL NO 1 2 3 4 5 6 7 8 9

QUANTITY

UNIT

BASIC UNIT

DIMENSION

FORCE ENERGY POWER PRESSURE DENSITY VELOCITY LENGTH VISCOSITY

NEWTON JOULE WATT PASCAL KG/M3 M/SEC M

Kg m / s2 Kg m2 /s2 J/s N/m2 Kg/m3 M/s m

MLT-2

ML-3 LT-1 L ML-1T-1