HySafe Wiki | D68 / DimensionalAnalysis

The actual physical relationship between observed and relevant independent dimensional quantities parameters characterising a problem should not depend on the choice of the units of measurement. This means that this relation can always be formulated as a relation between dimensionless quantities. The reformulation with dimensionless quantities implies a reduction of the numbers of parameters as in the original set of dimensional parameters the units of measurements were additional hidden (parasitic) quantities. Some dimensional parameters take the role of the units of measurement.

The universal and simplified formulation of the basic relations is exactly the attractive capability of the dimensional analysis. The mathematical implementation of this concept is the Buckingham Pi-theorem.

Pi-Theorem

All mechanical problems can be described with quantities having units composed of the units of the basic set of quantities length L, mass M, and time T. In some cases the replacement of the mass M by the force F might be advantageous. However, this will not change the approach in principle.

The engineering SI units for these 3 basic units are $$ [L]=m $$ $$ [M]=kg $$ $$ [T]=s $$ Together with the unit for the electrical quantity current {I=A$}, this forms the so-called mk(g)sA-system of basic units. Alternative basic units could be cm, g, and s like used for the physical c(m)gs-system. In the following we will use exclusively the mksA-system, also recommended in the international standards (SI).

All units of the relevant quantities including the observed quantity may be expressed by products of these basic units. Example for these dimensional expressions are: $$ [v]=[L]^1[M]^0[T]^{-1}=m^1kg^0s^{-1} $$ $$ [F]=[L]^1[M]^1[T]^{-2}=m^1kg^1s^{-2} (= N) $$ $$ [p]=[L]^1[M]^{-1}[T]^{-2}=m^{-1}kg^1s^{-2}(= Pa) $$

Step 1

Before the actual dimensional analysis based on the Buckingham Pi-theorem, a sound selection of the relevant quantities has to be done. We separate the observed key parameter and remove all dimensionless parameters from this set for the further consideration. This will yield the set of dimensional relevant parameters A_1, A_2,...., A_n.

Exactly this initial step requires a good physical background knowledge with an insight in the relevant physical phenomena. Thereby this methodology is more knowledge or experience based than the other similarity methods described later.

Now, we assume that there is a functional relationship for the observed dimensional quantity A: $$A=f(A_1, A_2,...., A_n)$$.

Step 2

Set up a matrix consisting of 3 rows for the basic units and n+1 columns for all dimensional quantities in the above relation.

	$$A$$	$$A_1$$	$$A_2$$	...	$$A_n$$
m	$$a_1^0$$	$$a_1^1$$	$$a_1^2$$	...	$$a_1^n$$
kg	$$a_2^0$$	$$a_2^1$$	$$a_2^2$$	...	$$a_2^n$$
s	$$a_3^0$$	$$a_3^1$$	$$a_3^2$$	...	$$a_3^n$$

Where the matrix a_m^n contains the exponents in the dimensional expressions introduced above.

Step 3

Check the rank of the matrix a_m^n. In most cases it should be 3. In special cases, like static or 0-dimensional problems, there might be a reduced rank. Practically speaking this means that one (or more) line(s) of the matrix above drop out. The actual rank determines the new dimension m.

Step 4

Determine a suitable reference system consisting of m basic dimensional quantities, columns of the matrix a_m^n respectively. Exchange columns so that the reference system consisting of the m columns reside on the left hand side of the matrix. Use linear combinations of the rows or a Gaussian substitution to make the left hand side mxm sub-matrix a unity matrix or a upper triangular sub-matrix at least.

Step 5

The homogeneous system $$ a_m^n b_n = 0$$ provides definitions for (n+1)-m dimensionless quantities \Pi_i. To this end for each dimensionless quantity the remaining (n+1)-m elements of the solution vector have to be fixed. The following transposed b vectors reduce the degree of freedoms such that the linear system may be solved and from each solution a dimensionless parameters is derived: $$ (b_0,b_1,b_2;1,0,0,...,0); $$ $$ (b_0,b_1,b_2;0,1,0,...,0); $$ $$ (b_0,b_1,b_2;0,0,1,...,0); $$ ... $$ (b_0,b_1,b_2;0,0,0,...,1); $$

Step 6

Solve the system for each of the reduced unknown b vector above. The result for each system is a dimensionless quantity $$ \Pi_i = a_i * a_0^{b_{0,i}}* a_1^{b_{1,i}}* a_2^{b_{2,i}}; i = m..n+1 $$ In some cases these dimensionless parameters might look unknown. However, with some experience one can fix the right hand side vectors in Step 5, such that more conventional dimensionless parameters are directly delivered. If not this is managed Step 7

Step 7

From suitable products of the derived dimensionless parameters including those from the initial list. known dimensionless \Pi-numbers may be produces.