Dimensionless numbers: Silent service providers

In several domains of engineering, dimensionless numbers are collections of variables that provide order-of-magnitude estimations of a system's behaviour. They are frequently a ratio between two physical parameters and are often calculated by combining coefficients from differential equations. Dimensionless numbers lower the number of variables that represent a system, minimising the amount of experimental data needed to make scalable system correlations from physical occurrences.

Let us pick up a few typical dimensionless numbers and see how they define a process.

 Significance of dimensionless numbers

The most common dimensionless group in fluid dynamics is the Reynolds number (Re). It represents the ratio of inertial forces to viscous forces, Duρ/ μ, where ρ is the fluid density, u is the average fluid velocity, D is cross-sectional length of the system, and μ is the dynamic fluid viscosity. The three regimes of Reynolds number, laminar, transition and turbulent are described based on differences in pressure drops due to inertial forces and viscous forces. Large length to hydraulic diameter ratios, greater than 70, give Re > 2300, although since none of the microsystems operated in fully developed turbulent regimes (Re ≈ 4000) the value had little significance. Typically, as a system is scaled down the influences of inertial forces decreases, while viscous forces become more dominant. The summary of this important dimensionless number is as follows:

The Reynolds number is a dimensionless number used to categorize the fluids systems in which the effect of viscosity is important in controlling the velocities or the flow pattern of a fluid. The Reynolds number is the ratio of inertial forces to viscous forces.

Re = Inertial force / Viscous force = DV ρ/ µ

The above equation can also be written as

Re = DV/v  

Here, µ is the dynamic viscosity of the fluid, and ρ is the density of the fluid. The ratio ν = µ/ρ = is termed the kinematic viscosity.

Prandtl number

The Prandtl number (Pr) is a dimensionless number defined as the ratio of momentum diffusivity to thermal diffusivity. It is a property of a fluid.

The Prandtl number is given as:

Pr = Cp x µ /k

Cp is specific heat, µ is dynamic viscosity and k is thermal conductivity.

Prandtl number is also a ratio of velocity or hydrodynamic and thermal boundary layers

Prandtl number plays an important role in heat transfer as it is connected to heat transfer coefficient ‘h’.

Nu[L] = h x L / k = 0.023 x Re^4/5 x Pr^n, Nu is Nusselt number.

How does one dimensionless number link with another?

Dittus Boelter equation

The Dittus Boelter equation combines Reynolds and Prandtl numbers to produce Nusselt number. Through empirical relationships, dimensionless numbers can be related to other dimensionless variables or quantities. The Dittus–Boelter equation, for example, is an explicit function for determining the Nusselt number from the Reynolds number and the Prandtl number in turbulent flow.

Nu =0.023 x Re^4/5 x Pr^n

where n = 0.4 for heating of the fluid and n = 0.3 for cooling of the fluid. For the Dittus–Boelter equation, knowledge of Re and Pr gives the Nusselt number.

Sieder-Tate equation

The Sieder–Tate correlation is another means for estimating the Nusselt number for turbulent flow from the Reynolds number and Prandtl number, where μ is the fluid viscosity at the bulk fluid temperature and μS is the fluid viscosity at the heat-transfer boundary surface temperature

Nu = 0.027 x Re^4/5 x Pr^1/3 x (μ/ μs)^0.14

The Dittus–Boelter equation is an example of an explicit function, while the Sieder–Tate correlation is an example of an implicit function.

An implicit function is a function with several variables, and one of the variables is a function of the other set of variables. An explicit function is defined as a function in which the dependent variable can be explicitly written in terms of the independent variable.

Credit: Google