Dimensions and Units

Dimensions are the basic concepts of measurement such as length, time, mass, temperature, and so on.

The units are the means of expressing the dimensions, such as feet or meters for length, or hours or seconds for time.

Dimensions are used to describe physical quantities.

A physical quantity such as length can be represented by the dimension L, for which there are a large number of possibilities available when selecting a unit, e.g, cm, m, km, ft, yard, mile. . . etc.

Dimensions and Units Dimensional homogeneity:

Every valid equation must be dimensionally homogeneous: that is, all additive terms on both sides of the equation must have the same dimensions.

Consistency:

An equation is said to be consistence if each additive term has the units.

V (m/s) = Vo (m/s) + g (m/s2) t (s)

The previous equation is both dimensionally homogenous and consistent in its units, in that each additive term has the units m/s.

Dimensions and Units

The dimensions have to be the same for each term in an equation

Dimensions of fluid mechanics are:

length

time

mass

force

temperature

L

T

M F = ma MLT-2

Dimensions and Units

Quantity Symbol Dimensions Velocity V LT-1

Acceleration a LT-2

Area A L2

Volume V L3

Discharge Q L3T-1

Pressure p ML-1T-2

Gravity g LT-2

Temperature T

Mass concentration C ML-3

Dimensions and Units

Quantity Symbol Dimensions

Density r ML-3

Specific Weight (rg) g ML-2T-2

Dynamic viscosity m ML-1T-1

Kinematic viscosity L2T-1

Surface tension MT-2

Bulk modulus of elasticity E ML-1T-2

These are _______ properties! fluid

UNITS CONVERSION

Example 1: Convert 6.7 in to millimeters

Conversion factor : 1 in = 25.4mm

Solution:

Dimensions and Units Example 2: Convert 85.0 1bm / ft3 to kilograms per

cubic meter

Conversion factor: 1ft3 = 0.304 8m3

Solution:

Dimensions and Units Example 3: Convert 23 lbm . ft/min2 to its equivalent

in kg.cm/s2

Solution:

Dimensions and Units Example 4: Convert an acceleration of 1 cm/s2 to its

equivalent in km/yr2

Solution:

Dimensions and Units

Example 5: A useful theoretical equation for computing the relation between pressure, velocity, and the altitude in a steady flow of nearly inviscid , nearly incompressible fluid is the Bernoulli relation:

𝑃𝑜 = 𝑃 + 1

2 𝜌𝑉2 + 𝜌𝑔𝑧

Where; P0 = Stagnation pressure, P = pressure in moving fluid, V= velocity, ρ = density, g = gravitational acceleration and Z = altitude

(a) Show that this equation satisfies the principle of dimensional homogeneity, which states that all additive terms in a physical equation must have the same dimensions,

(b) Show that consistent units result without additional conversion factors in SI units.

Dimensions and Units

Dynamic and Kinematic Viscosity

Kinematic viscosity (ν) is a fluid property obtained by dividing the dynamic viscosity (μ) by the fluid density

r

m

3m

kg

sm

kg

2m

sNm

2s

mkgN

[m2/s]