1.3 Dimensional homogeneity and dimensionless quantity

WHAT IS DIMENSIONAL HOMOGENEITY ? WHAT IS DIMENSIONLESS QUANTITY ? Dimensions and Units

 A dimension is the type of physical quantity.  A unit is a means of assigning a numerical value to that quantity.  SI units are preferred in scientific work. Primary Dimensions

 In fluid mechanics the primary or fundamental dimensions, together with their SI units are:  mass M (kilogram, kg)  length L (metre, m)  time T (second, s)  temperature Θ (kelvin, K)

 In other areas of physics additional dimensions may be necessary. The complete set specified by the SI system consists of the above plus  electric current I (ampere, A)  luminous intensity C (candela, cd)  amount of substance n (mole, mol) Dimensional Homogeneity

 In any equation, each additive term must have the same dimensions.

 When adding or subtracting values, the units of each value must be similar to be valid.

 A dimensionally homogeneous equation is an equation which has balanced units on each side. For example, the very simple equation: F=ma Dimensional Homogeneity

 Has Force in Newtons on one side and on the other has mass (in kg) and acceleration (in m/s2).

 so we can examine the dimensions on each side of the equation N=kg×m/s2

 1 newton is 1 kg.m/s2 so this equation is said to be dimensionally homogeneous. Not Dimensionally Homogeneous example 1:

 now, say for we are doing a question and we are given a mass in tonnes in a question, we check for dimensional homogeneity N=t×m/s2

 one newton is not 1 t.m/s2 so we need to do some unit conversion before we can apply this formula to our given information.

http://learntoengineer.com/note/Dimensional_Homogeneity Not Dimensionally Homogeneous example 2:

 Now, lets say for example we are now incorrectly given the mass of an object in m/s (velocity). Obviously mass is not measured in m/s but for some reason that is what it has been to be determined in this example.

 So lets see if our force equation will work and be dimensionally homogeneous N=m/s×m/s2 Therefore:N=m2/s3

 Because 1 Newton is not 1 m2/s3 this equation is invalid and not dimensionally homogeneous.

http://learntoengineer.com/note/Dimensional_Homogeneity Example 1:

 A (m) =2 B(s)+5

 What should the units for constants 2 and 5 have to be for the equation to be valid?

 Answer:

 2 (m/s) and 5 (m). “Dimensionless” Quantities

 Dimensional quantities can be made “dimensionless” by “normalizing” them with respect to another dimensional quantity of the same dimensionality.

Example: speed V (m/s) can be made "dimensionless“ by dividing by the velocity of sound c (m/s) to obtain M = V/c, a dimensionless speed known as the Mach number. M>1 is faster than the speed of sound; M<1 is slower than the speed of sound.

Other examples: percent, relative humidity, efficiency

2  Equations and variables can be made dimensionless, e.g., Cd = 2D/(ρv A)

 Useful properties:  Dimensionless equations and variable are independent of units.  Relative importance of terms can be easily estimated. List of dimensionless quantities

Name Standard Symbol Field of application

Abbe number V optics (dispersion in optical materials)

climatology, astronomy (reflectivity of Albedo α surfaces or bodies) motion of fluids due to density Archimedes number Ar differences

Atomic weight M chemistry

flow of bulk solids such as grain and Bagnold number Ba sand. [5] experimental aerodynamics; the cosine Berchak number[citation needed] Be of the angle formed by a yarnometer[1]

Biot number Bi surface vs. volume conductivity of solids

Bond number Bo capillary action driven by buoyancy [6]

heat transfer by conduction from the Brinkman number Br wall to a viscous fluid Exercise / Homework

1. Answer below question:  Identify below is dimensional homogeneity or not dimensional homogeneity (Also show method calculations): 1. F = ma (Given m = m and a = m/s2) 2. F = ma (Given m = kg and a=m/s) 3. F = ma (Given m = kg and a = m/s2) 4. m = F/a (Given F = lb.m/s2 and a = m/s2) 5. m = F/a (Given F = kg.m/s2 and a = s2/m) 6. a = F/m (Given F = kg.m/s2 and m = m) 7. a = F/m (Given F = kg.m/s2 and m = kg) 2. List down 10 others dimensionless quantity that you can find.

Name Standard Symbol Field of application THANK YOU.