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..