# Summary

Dimensionless numbers play an important role in analysing fluid dynamics and heat and mass transfer problems. They provide a method by which complex phenomena can be characterised, often by way of a simple, single number comparison. This article provides a brief overview of the derivation and use of dimensionless numbers.

# Definitions

: | Diameter | |

: | Friction Factor (Darcy) | |

: | Valve opening | |

: | Length | |

: | Characteristic length | |

: | Mass | |

: | Pressure | |

: | Temperature | |

: | Volumetric flow | |

: | Fluid velocity |

Greek Characters:

: | Density | |

: | Dynamic viscosity | |

: | Dimensionless Group |

Subscripts:

: | Model system | |

: | Pilot system |

# Introduction

Dimensionless numbers are scalar quantities commonly used in fluid mechanics and heat transfer analysis to study the relative strengths of inertial, viscous, thermal and mass transport forces in a system.

Dimensionless numbers are equal for dynamically similar systems; systems with the same geometry, and boundary conditions. This makes them a powerful tool for scaling operations from model to pilot and beyond.

For a list of dimensionless numbers and the formulae to calculate them, refer to our Index of Dimensionless Numbers.

# Usage

In the subsequent sections a simple system is utilised to explore the dimensionless numbers through:

- Derivation using
**dimensional analysis** - Usefulness in system
**scaling** - Application in
**analytical and empirical relationships**

The example system is a small 1/5 scale pilot scale system consisting of a pipe and a gate value with water as the fluid. Dimensionless numbers will then be used to predict key parameters on a full sized system using a different fluid from results obtained from the pilot system.

## Dimensional Analysis

We begin with dimensional analysis to understand the system and derive dimensionless groups. As it is a fluid dynamic problem, it can be described by Bernoulli’s equation the key variables are as follows:

Variable | Description | Dimensions |
---|---|---|

Pressure drop | ||

Valve opening | ||

Pipe diameter | ||

Fluid velocity | ||

Fluid density | ||

Fluid viscosity |

The table above gives us N = 6 variables ( ) and M = 3 independent dimensions (M, L, T). Buckingham theory states that if there are N variables and these variables contain M distinct dimensions, the equation describing the system will be a function of N-M = 3 dimensionless groups. Therefore, from the six variables we must form 3 dimensionless scaling groups to describe the system.

In this particular scaling problem we want the predict the system at a different size and using a different fluid. Therefore we are primarily interested in variables representing the system size ( ) and the kinematic ( ) and mass dependence ( ) on the fluid. To form our three scaling groups we multiply each of the remaining variables with our primary variables to the power of a constant and solve for their exponents such that the result is dimensionless, for example starting with we have our first dimensionless group as follows:

To ensure is dimensionless we solve a, b and c so that the exponent of M, L and T are 0 as follows:

This gives us the equations of exponents as follows:

Based on the above we determine that a = 0, b = -2 and c = -1 which substituted into the above equation results in our first scaling group:

Completing the same process for and we get the remaining two scaling groups:

It should be noted that is the inverse of Reynolds number and as is dimensionless there is no impact in substituting with its more convenient inverse .

From Buckingham theory we can describe the system as and can therefore develop a description of the system using our dimensionless groups , and :

## Scaling

Let’s assume the model introduced in the previous section is a 1/5th scale of our full scale pilot system and experimental trials with water reveal that for a given valve opening the velocity of the water was measured at 10 m/s with an associated pressure drop of 80 kPa.

What would the velocity and pressure drop be in the full scale pilot system if we substituted water for diesel? To determine this, first we would consider the relevant properties of each of the fluids:

Fluid | Density ( ) | Dynamic Viscosity ( ) |
---|---|---|

Water | ||

Diesel |

As the model and pilot system are dynamically similar (geometrically similar systems with the same boundary condition) the dimensionless numbers describing the system will be equal for both systems. Therefore, using Reynolds number ( as determined in the previous section) we have:

The velocity ratio for the pilot and model systems and scaling ratio can then be used to calculate the volumetric flow rate in the pilot system.

Finally we can calculate the pressure drop in the pilot system using .

## Analytical and Empirical Relationships

One of the most common uses of dimensionless numbers for the practising engineer is in the calculation of system properties using analytical and empirical relationships. Returning to the example from the preceding sections, consider the calculation of the pressure drop in 10 metres of pipe in the pilot system if that pipe is new DN100 Sch 40 carbon steel pipe with an absolute roughness of 0.02 mm.

The internal diameter of DN100 Sch 40 pipe is 102.26 mm, and for fluid flow in a pipe is our characteristic length in the calculate the Reynolds number.

From empirical studies we know that a Reynolds number greater than 4000 indicates we are in the turbulent flow regime. Armed with this knowledge we can then calculate the friction factor using an appropriate equation for the turbulent regime such as the Serghides equation to obtain a dimensionless friction factor of 0.017. The pressure drop in the pipe is then calculated using the following equation:

In conclusion, dimensionless numbers reduce the number of variables that are required to describe a system and therefore reduce the amount of experimental data that must be collected in order to determine analytical and empirical relationships to describe complex physical phenomena and scale systems.

# Further Reading

- Perry's Chemical Engineers' Handbook, Eighth Edition
- Chemical Engineering Volume 1, Sixth Edition: Fluid Flow, Heat Transfer and Mass Transfer
- Fundamentals of Heat and Mass Transfer

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