For You Who Wants to Understand Dimensional Analysis

My research makes me should perform a dimensional analysis for variables that influence a certain variables. So that I want to share my understanding of the basic theory for solving my problems. I hope it will be useful for you who may meet the same problem.

Dimensional analysis is the practice of checking relations among physical quantities by identifying their dimensions. Let me explain an example how to perform dimensional anlysis. Here I will give you an example for pressure drop in round pipe flow.

Pressure difference dP between 2 ends of pipe in which fluid is flowing is a function of the following variables:

D = pipe diameter
l = pipe length
u = fluid velocity
rho = fluid density
miu = fluid viscosity

It will be shown by dimensional analysis how these variables are related. From the expressin above, we can write the equation like the following Eq. 1.

dP = f(D, l, u, rho, miu) ...... (1)

The first step to perform dimensional anlysis is to identify the unit of each variables.

M = mass
L = length
T = time

unit of each variables:
(note: ** = rank)

dP = M x L**-1 x T**-2
d = L
l = L
u = L x T**-1
rho = M x L**-3
miu = M x L**-1 x T**-1

so that if the dP unit is dimensionless if the unit of D, l, u, rho, miu are multiplied, we can write the equation like the following Eq. 2.

dP = constant d**a x l**b x u**c x rho**d x miu**e .......(2)

M x L**-1 x T**-2 = (L)**a x (L)**b x (L x T**-1)**c x (M x L**-3)**d x (M x L**-1 x T**-1)**e ....(3)

from the Eq. 3, we get:
M : 1 = d + e
L : -1 = a + b + c - 3d - e
T : -2 = -c - e

There are 5 variables (a, b, c, d, e) with only 3 equations. So that we solve it in term of b and e:
d = 1 - e (from equation M)
c = 2 - e (from equation T)

Substitute those two equation into equation L:
-1 = a + b + (2 - e) - 3(1 - e) - e
we get:
a = -b - e

Substitute the new a, c, and d to Eq. 2:
dP = constant d**(-b-e) x l**b x u**(2-e) x rho**(1-e) x miu**e

from this, we get:
dP = constant (l/d)**b x (miu/d.u.rho)**e x u**2 x rho

so that we can arrange like the following equation:

dP / (u**2 x rho) = constant (l/d)**b x (miu/d.u.rho)**e ..........(4)

l/d is dimensionless.
miu/d.u.rho is equal to 1/Re. Re is dimensionless.
so that the left side of Eq.4 is dimesionless.

Now we already have 3 groups dimensionless variable. For more general form, we can write Eq. 4 like the following equation.

dP / (u**2 x rho) = f( (l/d, d.u.rho/miu)

or

dP / (u**2 x rho) = f( (l/d, Re)


The more easier method to perform dimensional analysis is using Buckingham's Phi Theorem. You should know the rule first.