Dimensional Analysis

Dimensional Analysis relies on the fundamental principle that any equation or relation between variables must be dimensionally consistent; that is to say that each term in the relationship must have the same dimensions. The corollary of this principle is that if the whole equation is divided through by any one of the terms, each remaining term in the equation must be dimensionless. Such dimensionless groups, or dimensionless numbers as they are called, are of considerable value in developing relations in chemical engineering.

The requirement of dimensional consistency places a number of constraints on the form of the functional relation between variables in a problem and forms the basis of the technique of dimensionless analysis which enables the variables in a problem to be expressed in dimensional groups. Since the dimensions of the physical quantities can be expressed in terms of the fundamental units, length, mass, time and temperature, the requirement of dimensional consistency must be satisfied in respect of each of these fundamentals. Dimensional analysis, however, will give no information about the form of the functions, nor will it provide a means of evaluating numerical proportionality constants.

The study of problems in fluid dynamics and in heat and mass transfer is made difficult by the many parameters which appear to affect them. By grouping the variables into dimensionless groups it is possible to reduce the effective number of variables. Thus the experimeter can greatly reduce the number of observations to be made by varying the dimensionless groups, rather than observing the effect of varying each dimensional property.

e.g. For a smooth pipe flow $\Delta P = fn \left(d, l, u, \mu, \rho \right)$

The form of the function is unknown, but since any function can be expanded as a power series, the function can be regarded as the sum of a number of terms each consisting of products of powers of the variables. The simplest form of relation will be where the function consists simply of a single term.

\begin{displaymath}{\rm ie } \hspace*{2cm} \Delta P = constant \times d^\alpha l^\beta u^\gamma \rho^\delta \mu^\epsilon \end{displaymath}

The requirement of dimensional consistency is that the term on the right must have the same dimensions as that on the left i.e. the dimensions of pressure.

Dimensionally

$\Delta P = M L^{-1} T^{-2}$ u = L T-1
d = L $\mu = M L^{-1} T^{-1}$
l = L $\rho = M L^{-3}$


\begin{displaymath}{\rm ie } \hspace*{2cm} M L^{-1} T^{-2} = L^\alpha L^\beta \l... ...M L^{-3} \right)^\delta \left(M L^{-1} T^{-1} \right)^\epsilon \end{displaymath}

The consistency must apply to each of the fundamental units

\begin{displaymath}{\rm for \ M} \hspace*{2cm} 1 = \delta + \epsilon \end{displaymath} (1)


\begin{displaymath}{\rm for \ L} \hspace*{2cm} -1 = \alpha + \beta + \gamma - 3 \delta - \epsilon \end{displaymath} (2)


\begin{displaymath}{\rm for \ T} \hspace*{2cm} -2 = -\gamma - \epsilon \end{displaymath} (3)

Thus we have three equations in five unknowns which may be solved in terms of any two unknowns. Solving in terms of $\beta$ and $\epsilon$.


\begin{displaymath}{\rm from \left(1 \right)} \hspace*{2cm} \delta = 1 - \epsilon \end{displaymath}


\begin{displaymath}{\rm from \left(3 \right)} \hspace*{2cm} \gamma = 2 - \epsilon \end{displaymath}


\begin{displaymath}{\rm from \left(2 \right)} \hspace*{2cm} \alpha = -1 - \beta ... ...+ 3 \left(1 - \epsilon \right) + \epsilon = - \beta - \epsilon \end{displaymath}

Therefore

\begin{displaymath}\Delta P = constant \times d^{- \beta - \epsilon} l^\beta u^{2 - \epsilon} \rho^{1 - \epsilon} \mu^\epsilon \end{displaymath}

or

\begin{displaymath}\frac{\Delta P}{\rho u^2} = constant \times \left(\frac{l}{d} \right)^\beta \left(\frac{\mu}{\rho u d} \right)^\epsilon \end{displaymath}

Since $\beta$ and $\epsilon$ are arbitary constants the groups $\frac{\Delta P}{\rho u^2}$, $\frac{l}{d}$ and $\frac{\mu}{\rho u d}$ are all dimensionless. The latter term in its inverted form is easily recognisable as the Reynolds Number.

Thus

\begin{displaymath}\frac{\Delta P}{\rho u^2} = constant \times \left(\frac{l}{d} \right)^\beta \left(\frac{\rho u d}{\mu} \right)^{-\epsilon} \end{displaymath}

More generally, however

\begin{displaymath}\frac{\Delta P}{\rho u^2} = fn \left(\frac{l}{d}, \frac{\rho u d}{\mu} \right) \end{displaymath}

The number of dimensionless groups is normally the number of variables less the number of fundamentals.

A number of points emerge. 1) If the index of a particular variable is found to be zero, this indicates that this variable is not of significance in the problem (or that another variable has been overlooked).

2) If two of the fundamental variables always appear in the same combination (e.g. if L & T always appear as power of LT-1) then the number of effective fundamentals is thus reduced by one.

3) The form of the final solution will depend on the method of the solution of the simultaneous equations i.e. the dimensionless groups obtained are not unique.

Clearly the maximum degree of simplification is achieved by using the greatest possible number of fundamentals. In some instances force, or thermal energy can be used as additional fundamentals. However these should only be used with caution.

The choice of physical variables to be included must be based on an understanding of the nature of the phenomenon being studied. If an additional variable of no significance is added, the exponent relating to that variable will tend to zero. If an important variable is omitted it will be found that there is no unique relationship between the dimensionless groups.

Since linear size is included among the variables, dimensional analysis provides a convenient means of scale up. For example, for flow past an object a change in linear dimension without any change of shape or other variables will lead to a change in flow pattern. However, if the change in scale is accompanied by a change in other variables so as to keep the Reynolds Number constant, no change in flow pattern will occur.

Buckingham's $\Pi$ Theorem

"The number of dimensionless groups is equal to the number of variables minus the number of fundamental dimensions". In mathematical terms this can be experienced as follows.

If there are n variables, Q1, Q2 ..... Qn, the functional relationship between them can be written.

\begin{displaymath}f_1 \left(Q_1, Q_2, ...... Q_n \right) = 0 \end{displaymath}

If there are m fundamental dimensions, there will be n - m dimensionless groups ($\Pi_1$, $\Pi_2$ .... $\Pi_{n-m}$) and the functional relationship between the can be written.

\begin{displaymath}f_2 \left(\Pi_1, \Pi_2 .... \Pi_{n-m} \right) = 0 \end{displaymath}

The groups $\Pi_1$, $\Pi_2$, etc must be independent of one another, and no one group should be capable of being formed by multiplying together powers of the other groups.

The method involves choosing m of the original variables to form what is called a recurring set. Any set m of the variables may be chosen with the following two provisions:

a) Each of the fundamental dimensions must appear in at least one of the m variables.

b) It must not be possible to form a dimensionless group from some or all of the variables within the recurring set.

The remaining n - m variables are taken singly and formed into a dimensionless group by combination with one or more members of the recurring set. In this way the n - m $\Pi$ groups are formed, the only variables appearing in more than one group being those that constitute the recurring set. Thus, if it is desired to obtain an explicit functional relation for one particular variable, that variable should not be included in the recurring set.

In some cases the number of dimensionless groups will be greater than predicted by the $\Pi$-theorem (e.g. if 2 fundamentals appear always in the same combination).

Referring back to our previous example, $f \left(\Delta P, d, l, \rho, \mu, u \right) = 0$. ie 6 variables in 3 fundamental dimensions. This implies that we will obtain 3 $\Pi$ groups.

The recurring set must contain three variables that cannot themselves form a $\Pi$ group, eg d, u, $\rho$.

Dimensionally

u = L T-1
d = L
$\rho = M L^{-3}$

The fundamental dimensions can be expressed in terms of these variables

L = d
$M = \rho d^3$
T = d u-1

The 3 $\Pi$ groups are obtained by taking each remaining variable in turn:

$\Pi \left(\Delta P \right)$
$\Delta P$ has units M L-1 T-2
Therefore $\Delta P \rho^{-1} d^{-3} d d^2 u^{-2}$ is dimensionless $ = \frac{\Delta P}{\rho u^2}$

$\Pi \left(l \right)$
$ = \frac{l}{d}$

$\Pi \left(\mu \right)$
$\mu$ has units M L-1 T-1
Therefore $\mu \rho^{-1} d^{-3} d d u^{-1}$ is dimensionless $ = \frac{\mu}{\rho u d}$
Thus

\begin{displaymath}f \left(\frac{\Delta P}{\rho u^2}, \frac{l}{d}, \frac{\rho u d}{\mu} \right) = 0 \end{displaymath}

The $\Pi$ theorem has the advantage that the form of relationship is not constrained to be a power series (may include $\log \left(x \right), e^x$, etc).

David Balmer
Last modified: Wed Dec 2 17:01:06 GMT