# Dimensionless constants

Dimensionless constants or groups are formulas used in determination of flows.  They are called constants. They are parameters that are derived for the determination of process flow in fluid and heat transfer. They have no unit. In heat transfer application, we consider it under forced convention. Also, in fluid we use it to determine the nature of flow such as flow being laminar or turbulent. The following includes some if the dimensionless constants:

• Nusselt number (Nu): this is the ratio of heat transfer by convection to heat conduction.

Nu= εconvcond  = hϭ/k

The increase in Nusselt number the more effective convection of heat becomes. when the Nusselt number becomes one(1), then it is represented by heat transfer by conduction. Local Nusselt number is the number at any point (x) along a plate, where x measured from the edge of the plate. From experimental data local Nusselt number of laminar and turbulent or combined of both is given by the following formulas:

Laminar= 0.332Re0.5Pr1/3

Turbulent= 0.0296Re0.5Pr1/3

Combined laminar and turbulent flows: hL/k= (0.037Re0.8-87)Pr1/3

h is the average transfer coefficient.

• Prandtl number (Pr): this is defined as the ratio of thickness of velocity at boundary layer to the thickness of thermal boundary layer in fluid system.

Pr =molecular diffusivity of momentum/molecular diffusivity of heat

Pr = µcp/k

Where µ is the dynamic viscosity

K is the thermal conductivity

Cp heat capacity at constant pressure

The prandlt number of some fluids ranges from 0.01 for liquid metals to more than 100,000 for heavy metal. Thermal boundary layer is thicker in liquid metals and thinner in oil relative to boundary layer.

Reynolds number (Re): this is the ratio of shear force to viscous force. This number is essential in fluid flow. When Reynolds number is greater than 2000, the flow is turbulent but when less than 2000, the flow is laminar.

Re = ρVL/µ

Where v=velocity of fluid

µ= dynamic viscosity

L= Length of pipe.

The relationship between prandtl and Reynolds number is given by

Nu = cRemPrn   where c, m and n are constant.

Others include:

• Mach number = u/a
• Biot number = hs/k    (for slap).
• Heat transfer factor =  h/cG (cpμ/k)2/3  /μ)
• Mass transfer factor =KM/G (μ/Dvρ)2/3
• Flow number =q/nD3a
• Schmidt number =U/Dvρ
• Sherwood number =KcD/Dv
• Weber number =DρV2

Prandtl number for gases at 1 atmosphere and 273K

 GASES pr Air 0.69 methane 0.75 Water vapor 1.06 ammonia 0.86 Oxygen 0.70 Nitrogen 0.70

Schmidt number for some gases in air at 273k and I atmosphere

 Gases Sc Acetic acid 1.24 ammonia 0.61 Acetone 1.6 Toluene 1.86 Propane 1.42 phosgene 1.65

Data is obtained from heat transmission by Mc Adams W.H. third edition, Mc Graw- Hill book company. New York city, 1954.

Sherwood. T.K and Pig-ford R.L. Absorption and extraction, secod edition.