Dear Colleagues,

Below are some notes from my experiences from working on physical quantities and units of measure.

## Introduction

When we developed ISO 15926-2 some 10+ years ago, quantities and units were one of the most challenging areas we addressed, and we were not entirely happy with the results of our work. With the benefit now of some 10 years hindsight, here are a few of the issues I think need to be addressed in such an ontology that are not immediately evident.

## Missing scales

What is really going on with the use of quantities and units is that a quantity (say a particular degree of hotness) is mapped to a number (say 20) on a scale (say the Celsius scale). It is a desirable property of a scale that the mapping from quantities to numbers is isomorphic (1:1 relationship) so you can go back and forth reliably from quantities to numbers.

Interestingly you rarely see scales mentioned, and BIPM talk very much about physical quantities and units of measure. However, scales for different types of physical quantity can have the same unit of measure, so you do not know you have two comparable values unless you have both the type of physical quantity and the unit of measure. For example, both stress and pressure can be expressed in the unit Pascal, but a stress value is not the directly comparable to a pressure value. Even more so with dimensionless numbers that do not have units, a Reynolds Number is not comparable to a mass ratio because they have the same unit.

It is not clear to me what it means for two different scales to have the same unit of measure, when values on the two scales are not comparable.

## What are units really about?

It is interesting to notice the practical use of units, or more particularly the labels we give to units. When we do scientific or engineering calculations based on units, we quite often end up with composite units, such as DegC/s. However, when we find the same unit as a divisor and multiplier, we conventionally cancel them out. A dimensionless number is an extreme example where all the units cancel out.

The utility of this approach is that if you have a quantity expressed in a unit, say DegC/s, and you want to change the units to say DegF/hr, then you know that it is only the conversion factors for DegC to DegF and s to hr that you need to concern yourself with, the fact that there were also mass flowrates involved in the calculation is unimportant if the units cancelled out. Of course with dimensionless numbers, once calculated, they never have to be converted, but really they are just a special case.

I would argue that this conventional dimensionality, though useful for conversions, is potentially misleading, since there is a natural inclination to assume that two values expressed in the same units are comparable. On the other hand, if you consider what call full dimensionality, the difference and the reason for them becomes clear. So considering a mass ratio as M/M instead of just dimensionless, makes the distinction clear with a length ration which is L/L.

These of course are simple dimensionless numbers. Reynolds Number really is worth looking at as a more complex example. Reynolds Number for a pipe may be calculated as:

Q*D/v*A

Where:

Q = volumetric flowrate (m3/s) Dimensions (L3*T-1)

D = the pipe diameter (m) dimension (L)

V = the kinematic viscosity (m2/s) dimensions (L2*T-1)

A = pipe cross-sectional area (m2) dimensions (L2)

So although the standard dimensionality is 0, the full dimensionality is:

(L4*T-1)/(L4*T-1)

## Is a Maximum Allowable Working Temperature a Temperature?

Practical engineering quantities also present challenges. One example we used a lot was Maximum Allowable Working Temperature for a furnace tube.

Let’s consider an ordinary temperature first. A temperature is an intrinsic property of some physical object that can be determined solely by observing the physical object itself.

On the other, a maximum allowable working temperature (MAWT) depends on a number of different factors. As an example I will consider the maximum allowable working temperature of the furnace tubes for a Crude Distillation Unit.

The factors involved in determining the maximum allowable working temperature in this case are:

- The creep rate/temperature function for the furnace tube material,

- The actual wall thickness for the furnace tube,

- The minimum safe thickness for the furnace tube, determined from maximum operating temperature and pressure,

- The cost of replacing a furnace tube

- The life required for the furnace tube, determined by the time between shutdowns, and the economics of replacing furnace tubes.

By the way, nothing dramatic happens if the MAWT is exceeded by a modest amount, the creep rate will increase and the life of the furnace tube is reduced.

What is clear is that Maximum Allowable Working Temperature is not an intrinsic property/quantity, since it depends on external factors beyond the furnace tube itself, so it is not an ordinary temperature, or a subtype of ordinary temperature, despite having the same unit of measure. However, it is also clear that a Maximum Allowable Working Temperature is comparable to an ordinary temperature, in that it makes sense to compare the actual temperature of a furnace tube to its Maximum Allowable Working Temperature and ask questions like whether the actual temperature is greater or smaller than the maximum.

The key lesson is that quantities are weird, especially when you start looking at practical engineering quantities, and that in particular you should be cautious about drawing conclusions from quantities being expressed in the same Unit of Measure.

Regards

Matthew West

**Information Junction**

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