In this article, you will learn how to keep your spacecraft safe from coding errors.

- Download Units.zip - 95 KB

## Introduction

The initial inspiration for this project is the loss of NASA's Mars Climate Orbiter in 1999. This failed to enter Mars orbit due to a mix up between metric (SI) and United States Customary Units. One sub-system was supplying measurements in pound-force seconds to another sub-system expecting them in Newton Seconds. As the probe braked to enter orbit, it travelled too close to the planet's atmosphere and either burned up or ricocheted off into solar orbit.

So I have tried to build a code library in which this kind of error should be ruled out by design. It has the following features:

- It can be used to perform many standard calculations from physics and engineering.
- It is based on dimensional analysis, so all quantities have a corresponding physical dimension, such as Length or Mass.
- It is strongly typed, so quantities of different dimension can only be combined in scientifically valid ways.
- Internally, all values are stored in S.I. (metric) units.
- Values are only converted to a particular system of units at its external interfaces, for example, when converting to and from
`string`

s.

It is written using C# version 9 and utilizes the .NET 5.0 framework.

Here is an example of the library in use:

C#

// Tsiolkovsky rocket equationMass EmptyMass = 120000.Kilograms();Mass PropellantMass = 1200000.Kilograms();Mass WetMass = EmptyMass + PropellantMass;Velocity ExhaustVelocity = 3700.MetresPerSecond();Velocity DeltaV = ExhaustVelocity * Math.Log(WetMass / EmptyMass);// DeltaV = 8872.21251 m/s

Throughout this article, and in the sample code and unit tests, I have used examples from my old grammar school physics textbook - *Nelkon and Parker Advanced Level Physics*. This was the standard sixth form physics book in Britain throughout the sixties and seventies.

## Background

The library is based on the concepts of *dimensions* and *units*.

### Dimensions

The *Dimension* of a physical quantity determines how it is related to a set of fundamental quantities such as mass, length and time. These are usually abbreviated to M, L, T, etc. New dimensions can be derived by combining these fundamental ones using multiplication and division. So:

- Area = Length x Length = L²
- Volume = Length x Length x Length = L³
- Density = Mass / Volume = M/L³ = ML⁻³
- Velocity = Length / Time = L/T = LT⁻¹
- Acceleration = velocity / Time = LT⁻²
- Force = Mass * Acceleration = MLT⁻²

And so on.

The dimension of any particular quantity can be represented as a sequence of powers of the fundamental dimensions (e.g., Force = MLT⁻² above). It is invalid to try to add or subtract quantities if their dimensions do not match. So it is invalid to **add** a mass to a volume for instance.

The International System of Units (S.I.) uses the following basic dimensions:

Dimension | Symbol | Unit | Unit Symbol |

Mass | M | kilogramme | kg |

Length | L | metre | m |

Time | T | second | s |

Electric Current | I | ampere | A |

Thermodynamic Temperature | Θ | kelvin | K |

Amount of Substance | N | mole | mol |

Luminous Intensity | J | candela | cd |

The library defines these basic dimensions, and many derived ones.

### Units

A unit system can define different basic units to correspond to the various dimensions. So whereas the S.I. system has a unit of kilogrammes for mass, the American and British systems use the pound. Similarly, we have the foot in place of the metre as the unit of length. There are also differences between the American and British systems when it comes to measurement of volume. Thankfully, the units for the other basic dimensions are the same in all three systems.

Although the library has definitions for both the S.I., American and British systems, it is possible to create and use new ones. For example, you could create a system using the Japanese shakkanho system, with the shaku (尺) as the unit of length and the kan (貫) as the unit of mass.

## Using the Code

The supplied code in the attached ZIP consists of a Visual Studio solution with two projects: the library itself and a command line programme which tests and demonstrates the library features. To use the library in your own project, add the library project file in "*\KMB.Library.Units\KMB.Library.Units.csproj*", then add the following `using`

statements to your code:

C#

using KMB.Library.Units;using KMB.Library.Units.Metric;using KMB.Library.Units.TimeUnits; // for hours, minutes etc.using KMB.Library.Units.British; // For feet and pounds. Or use USA if you prefer

### Contents of the Library

The `Units`

library defines various classes and interfaces. The primary ones are discussed here:

### class *Dimensions*

This class is used to represent a physical dimension or combination of them. It has a read-only field for the power of each dimension:

C#

public readonly short M; // Masspublic readonly short L; // Lengthpublic readonly short T; // Timepublic readonly short I; // Currentpublic readonly short Θ; // Temperaturepublic readonly short N; // Amount of Substancepublic readonly short J; // Luminous Intensitypublic readonly short A; // Angle.

Note the value for angle. Strictly angles are dimensionless, but it is convenient to treat them as having a distinct dimension. This way, we can distinguish angles from dimensionless quantities, when converting to a `string`

, for example.

The class has various constructors, and also defines operators for multiplication and division:

C#

public static Dimensions operator *(Dimensions d1, Dimensions d2)...public static Dimensions operator /(Dimensions d1, Dimensions d2)...

Using this class, we can define the basic dimensions:

C#

public static readonly Dimensions Dimensionless = new Dimensions(0, 0, 0);public static readonly Dimensions Mass = new Dimensions(1, 0, 0);public static readonly Dimensions Length = new Dimensions(0, 1, 0);public static readonly Dimensions Time = new Dimensions(0, 0, 1); :

And define any derived dimensions:

C#

public static readonly Dimensions Area = Length * Length;public static readonly Dimensions Volume = Area * Length;public static readonly Dimensions Density = Mass / Volume;public static readonly Dimensions Velocity = Length / Time;public static readonly Dimensions AngularVelocity = Angle / Time; :

The overloaded `ToString()`

method of `Dimensions`

outputs the powers of each dimension:

C#

Dimensions.Pressure.ToString() // returns "M1 L-1 T-2"Dimensions.Resistivity.ToString() // returns "M1 L3 T-3 I-2"

### Interface *IPhysicalQuantity*

This interface is the basis for all physical quantities in the system. It has two properties:

C#

double Value { get; }Dimensions Dimensions { get; }

For each defined value of `Dimensions`

, there will be a corresponding structure which implements the `IPhysicalQuantity`

interface. For example, `Length`

, `Area`

, `Mass`

and so on.

### Example Physical Quantity - *Length*

The `Length`

structure implements the `IPhysicalQuantity`

interface:

C#

public readonly partial struct Length: IPhysicalQuantity

It has a read-only `Value`

property:

C#

public readonly double Value { get; init; }

And a `Dimensions`

property:

C#

public readonly Dimensions Dimensions { get { return Dimensions.Length; } }

Notice how the `Dimensions`

property returns the corresponding statically defined `Dimensions`

value.

So given this structure, we can now define a variable to represent a particular length:

C#

Length l0 = new Length(3.4); // 3.4 metres

The `struct`

defines lots of operators. For example, you can add a `length`

to another one:

C#

public static Length operator+(Length v1, Length v2){ return new Length(v1.Value + v2.Value);}

Or compare two `length`

s:

C#

public static bool operator >(Length v1, Length v2) { return Compare(v1, v2) > 0; }

Or you can create an `Area`

by multiplying two `length`

s together:

C#

public static Area operator*(Length v1, Length v2){ return new Area(v1.Value * v2.Value);}

Or a `Velocity`

by dividing a `length`

by a time:

C#

public static Velocity operator/(Length v1, Time v2){ return new Velocity(v1.Value / v2.Value);}

Here's this divide operator in use:

C#

Length l = 100.Metres();Time t = 9.58.Seconds();Velocity v = l / t; // v = 10.43 m/s

There are also various `ToString()`

and `Parse()`

methods:

C#

public override string ToString();public string ToString(UnitsSystem.FormatOption option);public string ToString(UnitsSystem system, UnitsSystem.FormatOption option);public string ToString(params Unit[] units);public static Length Parse(string s);public static Length Parse(string s, UnitsSystem system);

The formatting and parsing of `string`

s is actually delegated to the current unit system. See below.

Here are some examples to demonstrate the various options for `ToString()`

and `Parse()`

:

C#

Length l = 1234.567.Metres();string s = l.ToString(); // s = "1.234567 km" (same as BestFit)// Formatting options:s = l.ToString(UnitsSystem.FormatOption.Standard); // s = "1234.567 m" // (standard unit for length is metres)s = l.ToString(UnitsSystem.FormatOption.BestFit); // s = "1.234567 km" // (kilometres is the best fit unit // for the value)s = l.ToString(UnitsSystem.FormatOption.Multiple); // s = "1 km 234 m 56 cm 7 mm" // (use multiple units in decreasing value)// Specify the units:s = l.ToString(MetricUnits.Metres, MetricUnits.Centimetres); // s = "1234 m 56.7 cm" // British units:s = l.ToString(BritishUnits.System, UnitsSystem.FormatOption.Standard); // s = "4050.41667 ft"s = l.ToString(BritishUnits.System, UnitsSystem.FormatOption.BestFit); // s = "1350.13889 yd"s = l.ToString(BritishUnits.System, UnitsSystem.FormatOption.Multiple); // s = "1350 yd 5 in"// Specified British units:s = l.ToString(BritishUnits.Miles, BritishUnits.Feet, BritishUnits.Inches); // s = "4050 ft 5 in"// Parsingl = Length.Parse("42 m"); // l = 42 ml = Length.Parse("42 m 76 cm"); // l = 42.76 ml = Length.Parse("5 ft 4 in", BritishUnits.System); // l = 1.6256 m// This will throw an exceptionl = Length.Parse("42 m 76 kg");

Because there are so many classes, operators and methods required for the quantities, these classes are generated using the T4 Template processor. See the Code Generation section.

### Temperatures

The library contains two classes for dealing with temperatures - *AbsoluteTemperature* and *TemperatureChange*. The first is used for absolute temperatures, as you would read from a thermometer:

C#

AbsoluteTemperature t3 = 600.65.Kelvin(); // melting point of leadAbsoluteTemperature c2 = 60.Celsius(); // c2 = 333.15 K

The second is used in many formulae where it is the temperature change that is important:

C#

TemperatureChange deltaT = 100.Celsius() - 20.Celsius();ThermalCapacity tcKettle = 100.CaloriesPerDegreeKelvin();SpecificHeat shWater = 4184.JoulesPerKilogramPerDegreeKelvin();Mass mWater = 1.Kilograms();ThermalCapacity tcWater = mWater * shWater;ThermalCapacity tcTotal = tcKettle + tcWater;Energy e = tcTotal * deltaT; // e = 368208 J

### struct *PhysicalQuantity*

This is the get out of jail card for cases when the strongly typed quantities won't do. It is weakly typed so has its own property to represent the dimensions:

C#

public readonly partial struct PhysicalQuantity: IPhysicalQuantity{ public double Value { get; init; } public Dimensions Dimensions { get; init; }

Like the strongly typed quantities, it has operators for addition, etc., but these are checked at run time instead of preventing compilation. So it is possible to do this:

C#

PhysicalQuantity l1 = new PhysicalQuantity(2.632, Dimensions.Length);PhysicalQuantity l2 = new PhysicalQuantity(2.632, Dimensions.Length);PhysicalQuantity sum = l1 + l2;

But this will throw an exception:

C#

PhysicalQuantity m = new PhysicalQuantity(65, Dimensions.Mass);sum = l1 + m;

But multiplication and division will correctly calculate the correct dimensions:

C#

PhysicalQuantity product = l1 * m;string s = product.ToString(); // s = "171.08 kg⋅m"

### Shapes

The library defines some utility classes for doing calculations relating to 2-D and 3-D shapes. For example to get the area of a circle with radius 3 cm:

C#

Circle circle = Circle.OfRadius(3.Centimetres());Area area = circle.Area; // = 28.27 cm²

And here is an example using a solid 3-D shape:

C#

SolidCylinder cylinder = new SolidCylinder() { Mass = 20.Pounds(), Radius = 2.Inches(), Height = 6.Inches() }; area = cylinder.SurfaceArea; // = 101.2 in² Volume volume = cylinder.Volume; // = 75.23 in³ Density density = cylinder.Density // = 0.2653 lb/in³ Length radiusOfGyration = cylinder.RadiusOfGyration;// = 1.414 in MomentOfInertia I = cylinder.MomentOfInertia; // = 0.2778 lb⋅ft²

### class *VectorOf*

The library defines a `VectorOf`

class which can be used for directed quantities such as displacement, velocity or force. I have called it `VectorOf`

to avoid a name clash with the `System.Numerics.Vector`

class.

C#

public class VectorOf<T> where T: IPhysicalQuantity, new()

It has constructors that take either 3 scalar values:

C#

`public VectorOf(T x, T y, T z)`

Or a magnitude and direction (for 2-D values):

C#

`public VectorOf(T magnitude, Angle direction)`

Or a magnitude and two angles (inclination and azimuth) for 3-D vectors:

C#

`public VectorOf(T magnitude, Angle inclination, Angle azimuth)`

For example:

C#

// Suppose a ship is travelling due east at 30 mph and a boy runs across the deck // in a north west direction at 6 mph. What is the speed and direction of the boy // relative to the sea? var v2 = new VectorOf<velocity>(30.MilesPerHour(), 90.Degrees()); var v3 = new VectorOf<velocity>(6.MilesPerHour(), 315.Degrees()); var v4 = v2 + v3; Velocity m2 = v4.Magnitude; // 26 mph Angle a3 = v4.Direction; // 81 degrees</velocity></velocity>

Currently the `VectorOf`

class uses the `PhysicalQuantity`

type internally. This is because 'generic maths' is not supported in .Net 5. When I get around to a .Net 6 or 7 version I will define static methods in the `IPhysicalQuantity`

interface that support maths operators and then the vector maths can be re-implemented.

### class *UnitsSystem*

The library defines an `abstract`

base class for unit systems:

C#

public abstract class UnitsSystem

Subclasses of `UnitsSystem`

are responsible for converting quantities to and from `string`

s. So there are various virtual methods for `string`

conversion. There is also a static reference to the current units system, which defaults to `Metric`

.

C#

public static UnitsSystem Current = Metric;

By default, the `ToString()`

and `Parse()`

methods will use the current unit system.

C#

internal static string ToString(IPhysicalQuantity qty){ return Current.DoToString(qty);}

C#

internal static PhysicalQuantity Parse(string s){ return Current.DoParse(s);}

Or you can specify which system to use:

C#

internal static string ToString(IPhysicalQuantity qty, UnitsSystem system){ return system.DoToString(qty);}

C#

public static PhysicalQuantity Parse(string s, UnitsSystem system){ return system.DoParse(s);}

By default, the unit system will perform the `string`

conversion using a lookup table of unit definitions. The unit definition uses this class:

C#

public class Unit{ public string Name; public string ShortName; public Dimensions Dimensions; public double ConversionFactor; //to convert from ISO units :

So, for example, here are some of the definitions for the metric system:

C#

public static Unit Metres = new Unit("metres", "m", Dimensions.Length, 1.0, Unit.DisplayOption.Standard);public static Unit SquareMetres = new Unit("squaremetres", "m²", Dimensions.Area, 1.0, Unit.DisplayOption.Standard);public static Unit CubicMetres = new Unit("cubicmetres", "m³", Dimensions.Volume, 1.0, Unit.DisplayOption.Standard);public static Unit Kilograms = new Unit("kilograms", "kg", Dimensions.Mass, 1.0, Unit.DisplayOption.Standard);public static Unit Seconds = new Unit("seconds", "s", Dimensions.Time, 1.0, Unit.DisplayOption.Standard);

Or similar ones for the British units:

C#

public static Unit Feet = new Unit ("feet", "ft", Dimensions.Length, feetToMetres, Unit.DisplayOption.Standard);public static Unit Inches = new Unit ("inches", "in", Dimensions.Length, (feetToMetres/12.0), Unit.DisplayOption.Multi);public static Unit Fortnight = new Unit ("fortnight", "fn", Dimensions.Time, 3600.0*24.0*14.0, Unit.DisplayOption.Explicit);public static Unit Pounds = new Unit ("pounds", "lb", Dimensions.Mass, poundsToKilogrammes, Unit.DisplayOption.Standard);

The unit system also defines a set of extension methods like this:

C#

public static Length Metres(this double v){ return new Length(v);}

That allows easy creation of a quantity from a floating point or integer value:

C#

Length l1 = 4.2.Metres();Mass m1 = 12.Kilograms();

## Code Generation

As mentioned previously, because the library has a lot of repetitive code, we use the T4 macro processor available in Visual Studio. This tool allows us to automate the creation of source code by creating a template file which contains a mix of C# code and the required output text. In general, we start with an XML file of definitions which we read, then use the template to generate the required C# classes and data.

For example, here is a line from the XML file defining the metric unit system:

XML

<unit name="Volts" shortname="volt" dimension="ElectricPotential" display="Standard" />

This template snippet will then create the static unit definitions:

C#

<#+ foreach(var ui in unitInfoList) {#>public static Unit <# =ui.longName #> = new Unit("<# =ui.longName.ToLower() #>", "<# =ui.shortName #>", Dimensions.<# =ui.dimension #>, <# =ui.factor #>, Unit.DisplayOption.<# =ui.displayOption #>);<#+ }// end foreach ui#>

Resulting in a line like this in the final code:

C#

public static Unit Volts = new Unit ("volts", "volt", Dimensions.ElectricPotential, 1.0, Unit.DisplayOption.Standard);

This technique allows us to generate the large number of operator definitions we require for each quantity class. For example, given this definition in the *Dimensions.xml* file:

XML

<dimension name="Density" equals="Mass / Volume" />

We can generate the `Density`

class and all of the following operators:

C#

public static Density operator/(Mass v1, Volume v2)public static Volume operator/(Mass v1, Density v2)public static Mass operator*(Volume v1, Density v2)

The following XML definition files are supplied:

File | Description |

Dimensions.xml | This defines the dimensions and the relations between them |

MetricUnits.xml | Unit definitions for the metric system |

BritishUnits.xml | British units like foot and pound |

USAUnits.xml | American Units. These overlap with the British units somewhat. |

TimeUnits.xml | Units of time apart from the second, such as hours and days |

## Summary Table

This table summarises the classes, dimensions, formulae and units supported by the library:

Name | Formula | Dimensions | Units |
---|---|---|---|

AbsoluteTemperature | Θ | K (Kelvin) °C (Celsius) °F (Fahrenheit) | |

Acceleration | Velocity / Time VelocitySquared / Length Length / TimeSquared Length * AngularVelocitySquared | L T⁻² | m/s² (MetresPerSecondSquared) g0 (AccelerationOfGravity) |

AmountOfSubstance | N | mol (Mole) nmol (NanoMoles) | |

AmountOfSubstanceByArea | AmountOfSubstance / Area | L⁻² N | m⁻²⋅mol |

AmountOfSubstanceByTime | AmountOfSubstance / Time | T⁻¹ N | mol⋅s⁻¹ |

Angle | A | rad (Radians) ° (Degrees) | |

AngularMomentum | MomentOfInertia * AngularVelocity | M L² T⁻¹ A | kg⋅m²/s (KilogramMetreSquaredPerSecond) |

AngularVelocity | Angle / Time TangentialVelocity / Length | T⁻¹ A | rad⋅s⁻¹ |

AngularVelocitySquared | AngularVelocity * AngularVelocity | T⁻² A² | rad²⋅s⁻² |

Area | Length * Length | L² | m² (SquareMetres) cm² (SquareCentimetres) ha (Hectares) |

AreaFlowRate | Area / Time Area * VelocityGradient | L² T⁻¹ | m²/s (SquareMetresPerSecond) cm²/s (SquareCentimetresPerSecond) |

CoefficientOfThermalExpansion | Dimensionless / TemperatureChange | Θ⁻¹ | K⁻¹ (PerDegreeKelvin) |

CoefficientOfViscosity | Force / AreaFlowRate Pressure / VelocityGradient Momentum / Area MassByArea * Velocity MassByAreaByTimeSquared / VelocityByDensity | M L⁻¹ T⁻¹ | Pa⋅s (PascalSeconds) P (Poises) |

Current | I | amp (Ampere) | |

Density | Mass / Volume | M L⁻³ | kg/m³ (KilogramsPerCubicMetre) gm/cc (GramsPerCC) gm/Ltr (GramsPerLitre) mg/cc (MilligramsPerCC) |

DiffusionFlux | AmountOfSubstanceByArea / Time AreaFlowRate * MolarConcentrationGradient AmountOfSubstanceByTime / Area | L⁻² T⁻¹ N | m⁻²⋅mol⋅s⁻¹ |

Dimensionless | 1 (Ones) % (Percent) | ||

ElectricCharge | Current * Time | T I | amp⋅s |

ElectricPotential | Energy / ElectricCharge | M L² T⁻³ I⁻¹ | volt (Volts) |

ElectricPotentialSquared | ElectricPotential * ElectricPotential | M² L⁴ T⁻⁶ I⁻² | kg²⋅m⁴⋅amp⁻²⋅s⁻⁶ |

Energy | Force * Length Mass * VelocitySquared AngularMomentum * AngularVelocitySquared SurfaceTension * Area | M L² T⁻² | J (Joules) cal (Colories) kWh (KilowattHours) toe (TonnesOfOilEquivalent) eV (ElectronVolts) erg (Ergs) |

EnergyFlux | Power / Area | M T⁻³ | kg⋅s⁻³ |

Force | Mass * Acceleration Momentum / Time MassFlowRate * Velocity | M L T⁻² | N (Newtons) dyn (Dynes) gm⋅wt (GramWeight) kg⋅wt (KilogramWeight) |

FourDimensionalVolume | Volume * Length Area * Area | L⁴ | m⁴ |

Frequency | Dimensionless / Time AngularVelocity / Angle | T⁻¹ | Hz (Hertz) |

Length | L | m (Metres) km (Kilometres) cm (Centimetres) mm (Millimetres) μ (Micrometres) nm (Nanometres) au (AstronomicalUnits) | |

LuminousIntensity | J | cd (Candela) | |

Mass | M | kg (Kilograms) g (Grams) μg (MicroGrams) ng (NanoGrams) t (Tonnes) Da (Daltons) | |

MassByArea | Mass / Area Length * Density | M L⁻² | kg⋅m⁻² |

MassByAreaByTimeSquared | MassByArea / TimeSquared Acceleration * Area | M L⁻² T⁻² | kg⋅m⁻²⋅s⁻² |

MassByLength | Mass / Length | M L⁻¹ | kg⋅m⁻¹ |

MassFlowRate | Mass / Time CoefficientOfViscosity * Length | M T⁻¹ | kg/s (KilogramsPerSecond) |

MolarConcentration | AmountOfSubstance / Volume Density / MolarMass | L⁻³ N | mol/m3 (MolesPerCubicMetre) mol/L (MolesPerLitre) |

MolarConcentrationGradient | MolarConcentration / Length | L⁻⁴ N | m⁻⁴⋅mol |

MolarConcentrationTimesAbsoluteTemperature | MolarConcentration * AbsoluteTemperature | L⁻³ Θ N | m⁻³⋅K⋅mol |

MolarMass | Mass / AmountOfSubstance | M N⁻¹ | kg/mol (KilogramsPerMole) gm/mol (GramsPerMole) |

MolarSpecificHeat | ThermalCapacity / AmountOfSubstance Pressure / MolarConcentrationTimesAbsoluteTemperature | M L² T⁻² Θ⁻¹ N⁻¹ | J⋅K⁻¹⋅mol−1 (JoulesPerDegreeKelvinPerMole) |

MomentOfInertia | Mass * Area | M L² | kg⋅m² (KilogramMetreSquared) |

Momentum | Mass * Velocity | M L T⁻¹ | kg⋅m/s (KilogramMetresPerSecond) |

Power | Energy / Time ElectricPotential * Current ElectricPotentialSquared / Resistance | M L² T⁻³ | W (Watts) kW (Kilowatts) |

PowerGradient | Power / Length | M L T⁻³ | kg⋅m⋅s⁻³ |

Pressure | Force / Area MassByArea * Acceleration | M L⁻¹ T⁻² | Pa (Pascals) mmHg (MillimetresOfMercury) dyn/cm² (DynesPerSquareCentimetre) |

Resistance | ElectricPotential / Current | M L² T⁻³ I⁻² | Ω (Ohms) |

ResistanceTimesArea | Resistance * Area | M L⁴ T⁻³ I⁻² | kg⋅m⁴⋅amp⁻²⋅s⁻³ |

ResistanceToFlow | MassFlowRate / FourDimensionalVolume | M L⁻⁴ T⁻¹ | kg⋅m⁻⁴⋅s⁻¹ |

Resistivity | Resistance * Length ResistanceTimesArea / Length | M L³ T⁻³ I⁻² | Ω⋅m (OhmMetres) |

SpecificHeat | ThermalCapacity / Mass | L² T⁻² Θ⁻¹ | J⋅kg⁻¹⋅K⁻¹ (JoulesPerKilogramPerDegreeKelvin) |

SurfaceTension | Force / Length Length * Pressure | M T⁻² | N/m (NewtonsPerMetre) dyne/cm (DynesPerCentimetre) |

TangentialVelocity | Velocity | L T⁻¹ | m/s (MetresPerSecond) cm/s (CentimetersPerSecond) kph (KilometresPerHour) |

TemperatureChange | Θ | K (Kelvin) °C (Celsius) °F (Fahrenheit) | |

TemperatureGradient | TemperatureChange / Length | L⁻¹ Θ | m⁻¹⋅K |

ThermalCapacity | Energy / TemperatureChange | M L² T⁻² Θ⁻¹ | J/K (JoulesPerDegreeKelvin) cal/K (CaloriesPerDegreeKelvin) |

ThermalCapacityByVolume | ThermalCapacity / Volume MolarConcentration * MolarSpecificHeat Pressure / AbsoluteTemperature | M L⁻¹ T⁻² Θ⁻¹ | kg⋅m⁻¹⋅K⁻¹⋅s⁻² |

ThermalConductivity | EnergyFlux / TemperatureGradient PowerGradient / TemperatureChange | M L T⁻³ Θ⁻¹ | W⋅m⁻¹⋅K⁻¹ (WattsPerMetrePerDegree) |

Time | T | s (Seconds) ms (MilliSeconds) min (Minutes) hr (Hours) day (Days) week (Weeks) yr (JulianYears) aₛ (SiderialYears) | |

TimeSquared | Time * Time | T² | s² |

Velocity | Length / Time | L T⁻¹ | m/s (MetresPerSecond) cm/s (CentimetersPerSecond) kph (KilometresPerHour) |

VelocityByDensity | Velocity / Density | M⁻¹ L⁴ T⁻¹ | kg⁻¹⋅m⁴⋅s⁻¹ |

VelocityGradient | Velocity / Length | T⁻¹ | Hz (Hertz) |

VelocitySquared | Velocity * Velocity | L² T⁻² | m²⋅s⁻² |

Volume | Area * Length | L³ | m³ (CubicMetres) cc (CubicCentimetres) L (Litres) |

VolumeFlowRate | Volume / Time Pressure / ResistanceToFlow | L³ T⁻¹ | m³/s (CubicMetresPerSecond) cc/s (CubicCentimetresPerSecond) |

## More Examples

Here are some more examples using the library, based on questions from *Nelkon and Parker*.

The reckless jumper:

C#

// A person of mass 50 kg who is jumping from a height of 5 metres// will land on the ground// with a velocity = √2gh = √ 2 x 9.8 x 5 = 9.9 m/s , assuming g = 9.8 m/s.Mass m = 50.Kilograms();Length h = 5.Metres();Acceleration g = 9.80665.MetresPerSecondSquared();Velocity v = Functions.Sqrt(2 * g * h); // v = 9.90285312 m/s// If he does not flex his knees on landing,// he will be brought to rest very quickly, say in// 1/10th second. The force F acting is then given// by momentum change/time = 50 * 9.9 / 0.1 = 4951 NMomentum mm = m * v;Time t = 0.1.Seconds();Force f = mm / t; // f = 4951.42656 N

And the flying cricket ball:

C#

// Suppose a cricket ball was thrown straight up with an initial velocity,// u, of 30 m/s.// The time taken to reach the top of its motion can be obtained from the equation// v = u + at.// The velocity, v, at the top is zero; and since u = 30 m and// a = —9.8 or 10 m/s²(approx),// we have 30 - 10t = 0.// Therefore t = 30 / 10 = 3s// The highest distance reached is thus given by// d = ut + 1 / 2 at ^ 2 = 30x3 - 5x3 ^ 2 = 45 m.var u = 30.MetresPerSecond();var g = 9.80665.MetresPerSecondSquared();var t = u / g; // t = 3.05914864 svar d = u * t + -g * t * t / 2.0; // d = 45.8872296 m

Surface Tension:

C#

// Calculate the work done against suface tension in blowing a bubble of 1 cm in diamter// if surface tension of a soap solution = 25 dynes/cm.Length r = 1.Centimetres() / 2;SurfaceTension surfaceTensionOfSoapSolution = 25.DynesPerCentimetre();// The initial surface area is zero// The final surface area = 2 x 4π x 0.5² = 2π sq cm.Area a = new Sphere(r).Area * 2;// Therefore work done = T x increase in surface area = 25 x 2π = 157 ergs.Energy e = surfaceTensionOfSoapSolution * a; // 157.1 erg

Young's Modulus:

C#

// If a 2kg weight is attached to the end of a wire of length 200cm and diameter 0.64mm// and the extension is 0.6mm then what is the Young's Modulus E of the wire?Force f = 2.KilogramWeight();Area a = Circle.OfDiameter(0.64.Millimetres()).Area;var stress = f / a;Length l = 200.Centimetres();Length e = 0.6.Millimetres();var strain = e / l;// E = (2000 x 980 / π x 0.032²) / (0.06/200) = 2e12 dynes/cm²var E = stress / strain; // 2.032E+12 dyn/cm²

Viscosity:

C#

// What is the frictional force over an area of 10 sq cm in water at 10 C between two// layers of water 0.1 cm apart which move with a relative velocityh of 2 cm per sec?// Assume viscosity of water is 0.013 poises.Area a = 10.SquareCentimetres();Velocity dv = 2.CentimetersPerSecond();Length dx = 0.1.Centimetres();VelocityGradient vg = dv / dx;CoefficientOfViscosity c = 0.013.Poises();Force f = c * (a * vg); // 2.6 dyn

Fick's First Law:

C#

// In a region of an unsaturated solution of sucrose the molar concentration gradient is -0.1 mol/L/cm.// What quantity of sucrose molecules pass through an area of 1cm² in 10 minutes?MolarConcentration c = 0.1.Mole() / 1.Litres();MolarConcentrationGradient cg = -c / 1.Centimetres();Area a = 1.SquareCentimetres();Time t = 10.Minutes();AreaFlowRate d = 0.522e-9.SquareMetresPerSecond(); // diffusion coefficientDiffusionFlux j = d * cg;AmountOfSubstance n = j * a * t; // -313.2 nmol

## Points of Interest

### Unit Tests

The sample program also tests the library, but does not use a unit testing framework. Instead, it uses a simple static class `Check`

which allows us to write code like this:

C#

Check.Equal(42.0, d5, "wrong value for d5");

This will throw an exception if the first two arguments are not equal.

### Performance

I had hoped that by creating immutable data types and making copious use of the aggressive inlining and aggressive optimization hints that the performance of the quantity classes would be comparable to the performance of 'raw' doubles. But this has turned out not to be the case. To test this, I implemented the same rocket simulation twice, once using plain doubles and again using the quantity classes. In a release build, the version using `double`

s is around 6 times faster. The reason can be seen by examining the code generated for some typical arithmetic. For example, this code:

C#

double d1 = 4.2;double d2 = 5.3;double d3 = 6.4;double d4 = d1 + d2 + d3;

Generates code for the addition like this:

00007FFCCC4B6A46 vmovsd xmm3,qword ptr [rbp-8] 00007FFCCC4B6A4B vaddsd xmm3,xmm3,mmword ptr [UnitTests.Program.TestDouble()+0B0h (07FFCCC4B6AC0h)] 00007FFCCC4B6A53 vaddsd xmm3,xmm3,mmword ptr [rbp-10h] 00007FFCCC4B6A58 vmovsd qword ptr [rbp-18h],xmm3

Whereas the same formula using the class library:

C#

Dimensionless d1 = 4.2;Dimensionless d2 = 5.3;Dimensionless d3 = 6.4;Dimensionless d4 = d1 + d2 + d3;

Generates much longer code:

00007FFCD5726B59 mov rcx,qword ptr [rsp+70h] 00007FFCD5726B5E mov qword ptr [rsp+58h],rcx 00007FFCD5726B63 mov rcx,qword ptr [rsp+68h] 00007FFCD5726B68 mov qword ptr [rsp+50h],rcx 00007FFCD5726B6D vmovsd xmm0,qword ptr [rsp+58h] 00007FFCD5726B73 vaddsd xmm0,xmm0,mmword ptr [rsp+50h] 00007FFCD5726B79 vmovsd qword ptr [rsp+48h],xmm0 00007FFCD5726B7F mov rcx,qword ptr [rsp+48h] 00007FFCD5726B84 mov qword ptr [rsp+40h],rcx 00007FFCD5726B89 mov rcx,qword ptr [rsp+60h] 00007FFCD5726B8E mov qword ptr [rsp+38h],rcx 00007FFCD5726B93 vmovsd xmm0,qword ptr [rsp+40h] 00007FFCD5726B99 vaddsd xmm0,xmm0,mmword ptr [rsp+38h] 00007FFCD5726B9F vmovsd qword ptr [rsp+30h],xmm0 00007FFCD5726BA5 mov rcx,qword ptr [rsp+30h] 00007FFCD5726BAA mov qword ptr [rsp+78h],rcx

There are lots of superfluous move instructions. Perhaps someone with a deeper understanding of the JIT compiler can shed some light on this.

### Comparison with F#

The F# language has built in support for units of measure, which also has the aim of preventing programming errors. So it is possible to write statements like this:

F#

let l1 = 12.0<m> // define a length in metreslet l2 = 7.0<m> // define another lengthlet l3 = l1 + l2 // add lengths togetherlet a = l1 * l2 // define an area (a has type float<m^2>)let v = l1 * l2 * l3 // define a volume (v has type float<m^3>)let m1 = 5.0<kg> // define a mass in kilogrammeslet d = m1 / v; // define a density (d has type float<kg/m^3>)

And given the above, this statement will not compile:

F#

let x = m1 + l1; // !! The unit of measure 'm' does not match the unit of measure 'kg'

The standard library of units defines the basic S.I. unit like metre, but does not define derived units like centimetres. You can define your own units like this:

F#

[<Measure>] type cm // centimetres

And you can use it in the same way:

F#

let l4 = 42.0<cm>

But there is no way to indicate that centimetres and metres are the same dimension. So whereas `l1`

above has type `float<m>`

, `l4`

has type `float<cm>`

, and attempting to add them will not compile:

F#

let l5 = l1 + l4; // !! The unit of measure 'cm' does not match the unit of measure 'm'

You can only get around this by defining a conversion function:

F#

let convertcm2m (x : float<cm>) = x / 1000.0<cm/m>

Then using it in the expression:

F#

`let l5 = l1 + convertcm2m(l4);`

You also have to be careful to always use the same numeric type when using units of measure. This is because in this definition:

F#

`let l6 = 5<m>`

The type of `l6`

is `int<m>`

, and this cannot be added to a value of type `float<m>`

. So this line will not compile either:

F#

let l7 = l1 + l6; // !! The type float<m> does not match the type int<m>

Finally, although the units of measure are checked at compile time, the types do not carry through to the compiled code. The values are just defined as floating point numbers. Consequently, you cannot discover at run time what the unit of measure of a value actually is. So you can only print these types of values as floating point, like this:

F#

printfn "l5 = %e" l5 // outputs "l5 = 1.204200e+001"

Even if you use the format specifier `%O`

:

F#

printfn "l5 = %O" l5 // outputs "l5 = 12.042"

So although the F# system has the same goal of preventing invalid mathematical operations, it is more restrictive due to its basis on units rather than dimensions.

## History

- 6
^{th}July, 2021: Initial version - 6
^{th}August, 2021: Added`AbsoluteTemperature`

to the library. Added a table to the article summarising the contents of the library. - 8
^{th}August, 2021: Corrected format of summary table. - 7
^{th}December 2021: Added examples of equations from calorimetry, thermal expansion, thermal conductivity and ideal gases. - 9
^{th}May 2022: Added shapes, vectors, statics, hydrostatics, surface tension, elasticity, and friction. - 1
^{st}Sep 2022: Added viscosity, osmosis and diffusion. Also updated the summary table.

I've been working on this for two years in my spare time. Currently, the library has the basics in place, and can be used for equations in dynamics, statics, heat and some electrics. I am continuing to add more derived dimensions and quantity classes to support more equations as I gradually work my way through *Nelkon and Parker*.