Working with equations : Physical units

Physical units

Unit definitions

Component values in Simile, and parameters used in equations, may be assigned physical units (e.g., cubic metres) by the modeller. There is a large set of such units built into Simile, and it can be extended with extra unit definitions provided by the modeller. The supplied units will be used to provide consistency checking and unit conversions where appropriate.

First off, there are the baseline units. These are the ones in terms of which all other units are defined. Simile comes with five baseline units predefined, metre (m), gramme (g), second (s), kelvin (k) and radian (rad). These are measures of length, mass, time, temperature and angle respectively. You can use either the short or long form to specify the units of a value.

Next there are the decimal multiplier prefixes, such as micro (). All those defined in SI are available. To use a decimally multiplied unit, simply put its prefix in short form before the short form of the unit itself, for instance kg = 1000 grammes (kilogramme). Some combinations of a multiplier and a base unit have a long form defined, e.g., kilogramme (kg), centimetre (cm). Again you can use either the short or long form to specify your units. Note that where appropriate we provide both the UK and US spellings.

Finally there are derived units, which are defined in terms of numerical constants and other units. You can write a derived unit using the short forms of other units (including other pre-defined derived units), integers and the symbols * (multiply), / (divide) and ^ (power). You can only use ^ with an integer exponent, as it works as a shorthand for multiplying, e.g., m^3 (cubic metre) means the same as m*m*m. There are also a number of pre-defined derived units. These include W (watt), psi (pound per square inch) and so forth. They may have a long form, and you can use either short or long form, and precede the short form with a decimal multiplier prefix, e.g., MJ for megajoule. Short forms of derived units can also be combined using symbols as above, e.g., W/m^2 = watts per square metre.

The definitions of all units are provided in the file in the Functions directory of the Simile installation. If you want to add extra units for your own application domain, create a file with a .pl extension in the appropriate place (see User-defined functions) and add the definitions in the same format as the built-in unit definitions. Note that all definitions define a short form in terms of other short forms -- you specify the longhands separately if they are needed. You cannot do anything with longhand forms except use them on their own to specify the units of model quantities -- all definitions, combinations and decimal multiplications must be done on short forms.

Effects of setting units

There are two mechanisms within Simile for checking that physical units are compatible and for including a unit conversion where appropriate.

  • Mechanism I is a method of using units to perform an ad-hoc conversion between values in different systems, without cluttering equations with conversion constants. This mechanism performs conversions between different Simile components.

  • Mechanism II is a system of entering equations with dimension checking, enforcing consistency and performing the required conversions automatically. This mechanism performs conversions within a component's equation.

The first mechanism is always active. To activate the second mechanism, select the "Yes" option in the drop-down field labelled "Use units in math" in the Model...Properties dialogue box. The options are "Default", "Yes" or "No". The default is to inherit the setting from the parent submodel. In a top-level submodel (i.e. desktop), the default is "No". It is possible to have submodels with equation dimension checking and submodels without in the same model.

When you enter a new equation with the equation dialogue, you can enter a unit specification in the units field. If you do not do so, or if you use the equation bar, the value is given the default units for its equation. In the equation box, you can also specify units for the values associated with incoming influences. If you do not provide units for them, they get the same units as the component they come from.


Mechanism I : Ad-hoc conversions


If you enter units for a value from an incoming influence, and the value of the component from which that influence comes also has units, Simile will perform a compatibility check and add a conversion if one is required. This makes it straightforward to include unit conversions when integrating models from different sources; provided the appropriate units are specified on each side of any link between them, the conversions will be built in automatically.

If the units at the source are real (i.e., it has no physical units) then it can be given any physical units at the destination and no conversion will be done. But if it does have physical units, then any units given to it at the destination must be compatible with them. That is to say they must have the same physical dimensions. For instance, mph (miles per hour) are compatible with m/s (metres per second) because they are both measures of speed. If one component has units of mph, and another has an influence from it which is given units of m/s, then the value from that influence that is used in the equation will be the result of converting the source value from mph to m/s, i.e., dividing it by about 2.25. However mph is not compatible with, say, km, because km is a measure of distance, not speed.

The same sort of check is also performed when a flow joins a compartment. Because the compartment's value is a function of the flow's value over time, the flow's units must be compatible with the units for the rate of change of the compartment over time. For instance it would be OK for the compartment to have units of litre and a connecting flow to have units of ft^3/day, because the first is volume and the second a rate of change for volume. In this case a conversion will be done on the amount of change due to the flow before adding it to or subtracting it from the compartment. But they could not both have units of litre, because the flow must be compatible with the compartment's rate of change rather than its actual value. In a future version of Simile we may allow a flow to have two different sorts of units at once, one for each end, to capture the situation in which it transforms one substance into another, e.g., a chemical reaction. In the mean time, if a flow has no physical units then no conversion will be performed at either end.


Mechanism II : Fully consistent physical dimensions


When units checking within math is selected, setting an equation causes it to be checked for consistency of units. This works by generating a physical sort of units for each subexpression in the equation starting with the units of the incoming parameters, and ending with a default sort of units for the result.

When adding or subtracting two quantities, both quantities must have the same units. There is one exception: if one quantity is a numerical constant, it automatically takes the units of the other. If the two quantities have the same dimensions but do not also have the same units, it is necessary also to specify a conversion for one or both to the same units. This can be done by changing the units associated with a parameter as described in the last section. When that is done, a conversion will be applied to the parameter value before it is used in the equation, so it becomes compatible with the quantity added to or subtracted from it. For example, adding one distance in feet to another in metres, it is necessary first to either convert the former to metres, the latter to feet, or both to a third unit, such as km. The result will be in the units of both the arguments.

When multiplying or dividing two quantities, the units of the result are determined by multiplying or dividing the units of the quantities themselves. e.g., the result of dividing a parameter with units metre by one with units metre/second will have units second.

Explicit conversion of dimensions

Although it should always be possible to create a model in which strict conservation of physical units applies, it is often the case that a model will include some ad-hoc representation of a complex relationship for which the exact equations are not available. In this case, the units of an equation's result may bear no relationship to those of its arguments. For instance, you might have a flow into a model compartment with dimensions mass/time, and the rate of this flow is set by a sketch graph against time. A sketch graph, like trig and exponential functions, must have a unitless argument, and produces a unitless result. However, the result of the time() function has dimensions time, and so cannot be used directly as the argument. For this kind of situation, you can include a unit name in the equation, with double-quotes around it to distinguish it from a parameter name, to represent unity with those units. So the argument for the graph function would be time()/"day" if the x-axis values on the graph represented numbers of days. Similarly, the result of the graph function is a unitless value, so it must be given units to allow it to make sense to the rest of the unit-checking-enabled model. This can be done by multiplying it by unit specifications -- for instance if the y-axis values on the graph represented mass transfer rates in kg/hour, the equation would be graph(...)*"kg"/"hour".  Of course the flow might go into a compartment whose units were some measure of mass other than kg, bit Simile would perform the appropriate conversions in this case.

If all the compatibility requirements are met, the end result of this process is a sort of units for the result of the equation. This is used as the default sort of units for the value being calculated. If you supply another sort of units for this value, it must be compatible with the default in terms of dimensions, and if the two are not the same an extra conversion may be applied after the equation has been calculated and before its value is assigned to the component. Note that the default units generated for the equation may not be in the most readable form, for instance if a result in watts is generated the units from the components might actually be something like N*m/s. In such a case you can set the units manually to the normal form, and no conversion will be done because they are mathematically equivalent.

In the physical sciences, many models can be completely expressed in terms of physically meaningful quantities. With this sort of model, it is a good idea to enable units checking within math, as it provides an extra safeguard against mistakes. In ecology and the social sciences, it is more common to use ad-hoc parameterisation that defies physical interpretation. In these cases, it is probably not useful to use this system, which is why it is switched off by default.

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