#### Simple units conversion

#### A paradox with temperature units

#### Is there a problem?

#### Resolving the ambiguity

#### Algebra of temperatures and temperature differences

#### A new unit for temperature difference

Unit conversion are simple (in most cases): Converting a quantity, say length, from one unit (e.g. inches) to another (e.g. meters) involves multiplying the quantity by a factor (namely 0.0254). This is very convenient because the following calculations can be done without any ambiguity:

Why am I even writing about this? This stuff is trivial.

Well, when it comes to temperature, things get a bit tricky. Consider a set of similar equations involving temperatures in degrees Kelvin and Celsius. Define

The result of subtracting the two will depend on interpretation of what these quantities mean.

For example, if T1 represents the air temperature of 0°C, T:=T1-T2, would indicate the result of temperature change by 10°C, yielding air temperature of -10°C. Expressing this temperature in different units will yield 263.15K, 14°F, etc.

If we look at the equation ΔT:=T1-T2, and interpret it to mean the temperature difference of air (at the temperature of 0°C) and a heating element (at temperature of 10°C).

The temperature differential would then be -10°C. Expressing this quantity in other units will yield -10K, -18°F, etc.

Something is not adding up here! We assigned the same values to T1 and T2 here, and got different results for the difference T1-T2 when expressed in K, °F, etc.

The apparent paradox lies in the fact that there are two different conversions between degrees Kelvin and degrees Celsius or Fahrenheit depending whether we are referring to a temperature of an object or a substance or a temperature difference.

When converting from °C to K, if we interpret a quantity as a temperature of an object or a substance we use conversion K = °C + 273.15. However, if the quantity represents a temperature difference, the conversion factor is 1. Thus in K, the above equations would look like:

In the first example, and the second example would look like:

Before I go into how to resolve the ambiguity in temperature conversions, is this a problem? Usually not if doing calculations by hand or using only degrees Kelvin. However, if one wants to use a calculation tool to do such calculations and automatic conversions, the tool needs to know what the author had in mind. This is useful to automate the process of conversion enabling the author to display results in desired units, or to catch the possible errors. When communicating with engineers in different countries, it important to enable conversions to unit system that is most commonly used. This makes documents easier to read. The units should always be explicit, i.e. never say temperature is 10, as this is guaranteed to cause errors.

If we look at closely at the examples above, the hints to interpret T1, T2 and the result were subtle. In one case, I used T to name the result, implying it is a temperature of an object or matter. And in the other I used ΔT to hint that the result was a temperature difference. That’s too subtle and it would not work in situations like T+(T1-T2).

One approach comes from an observation that it (almost – see below) never makes sense to add two temperatures together. Using T for temperature and ΔT for temperature difference, we can derive the following simple algebra:

**T1 + T2** yields an error

**T1 – T2** yields **ΔT**

**ΔT1 + T2** yields **T**

**ΔT1 – T2** yields an error

**T1 +/- ΔT2** yields **T**

**ΔT1 +/- ΔT2** yields **ΔT**

Such system would still require declaring some of the quantities as either a temperature or temperature difference, but would help catch mistakes. It also has a drawback in not allowing temperatures to be added in computing an intermediate result such as average temperature.

Alternative approach takes directly from the difference in converting from degrees Celsius to Kelvin when talking about a temperature and a temperature difference. Specifically, one can define two units:

°C – for the temperature in degrees Celsius, and

Δ°C – for the temperature difference in degrees Celsius

Thus we can define T2 as either:

Note that, for degrees Kelvin, such distinction is not necessary. With this system we can disambiguate the example above:

Assuming that T2 is a temperature difference,

And assuming T2 is a temperature,

This solves the problem, but has a downside: an engineer using a calculation tool, must now be careful and specific in specifying different units for temperatures and temperature differences – something that we don’t normally worry about. Maybe there is a better way…

Visit our dedicated page focused on engineering unit conversion.