# Need help in cable temperature calculations

I am trying to calculate how hot will a 0.6m long 28AWG wire carrying a load of 12VDC, 0.2A get.

I understand that are other factors such as environment cooling rate, thermal resistance between air and the cable etc. Ampacity values are not really relevant to my scenario as the cable is in contact with the human body. Thus, I am more concerned about whether if the user can detect the change in the wire's temperature.

I also do not have the resources nor the proper apparatus to conduct an accurate measurement test.

An aluminium core 28AWG wire has a resistance of 0.32716 Ω/m. Power dissipation:

$$P=I^2R$$ $$P=0.2A^2\times0.32716Ω \times 0.6m$$ $$P=7.851mW$$

Found this equation here, altough it is only meant for radiative heat loss $$\dot{Q}_{12} = \epsilon A\left ( \sigma T_1^4 - \sigma T_2^4\right )$$

Based on the above, I got a value of $$\309K\$$ which means the temperature increase is about $$\4°C\$$ from an ambient temp of $$\305K\$$.

Is this an accurate reference?

Basically, I want to know if a 28AWG wire will stay cool during operation or do I need to select a lower gauge wire.

• It'll be fine as far as heat goes. Where is the 12V coming from? Make sure you consider if there are any fault conditions that could cause the 12V wire to be energized at a much higher voltage (for example if it is a power supply). If it is a battery, so much the better. But if it is a power supply plugged in to the wall then you need to think about safety very carefully. – mkeith Aug 5 '19 at 2:15
• @mkeith The source is a 12V boosted lithium battery – Wb16 Aug 5 '19 at 9:04
• Just put a fuse inline with the wire somewhere to make sure the current stays within bounds. If you think 200mA is the max normal current, maybe a 1A fuse would be appropriate, but you have to work it out for yourself. – mkeith Aug 5 '19 at 17:15

Radiation is a negligible component of heat loss for a wire close to room temperature in air. It becomes a factor for long wires in a vacuum, and high temperature differences, but that's not the case here. Convection is the primary source of heat loss for a long wire in still (or moving) air.

If it's in contact with skin, conduction will be a big factor. In air, convection is the primary factor. Convection calculations involve fluid dynamics so they're not as straightforward as you might hope for. But to get a rough estimate, consider this graph from this website:

Very roughly the temperature rise in still air is about 3.3$$\\times \text I^2\$$, so for a current of 200mA in an AWG 28 wire you could expect about a 0.15K rise. If it's in contact with a 'bag of mostly water' the conduction to the skin (and therefore the skin temperature) will have a large effect.

• Are you able to clarify on what you meant by large effect? Isn't a temperature rise of 0.15K considered insignificant? – Wb16 Aug 5 '19 at 1:35
• If it’s 0.07K that’s a large effect (half) and to a different temperature base, at least assuming the flesh is alive. In any case, even fat is much more thermally conductive than still air. – Spehro Pefhany Aug 5 '19 at 1:38
• @Wb16 "fat" may be anything with 7% to 47% water in it, and the adipocytes may be small or big. Even NASA experiments regarding astronaut safety done in this decade do not go into this level of detail. What exactly is it that you are trying to do? – Indraneel Aug 5 '19 at 2:11

7 milliWatts spread over 60 centimeters will not be detected.

The heat will be dumped into the chest, and cooled by the blood.

Sunlight is 1,000 watts per square meter, or 1,000 watts per 10,000 square cm. We easily sense sunlight, which is 0.1 watts per square cm.

Your heat density is 0.1 milliWatts per cm length or about 0.1mW per 1cm*0.2cm (assuming the insulation spreads out the heat) or about 0.5mw per cm squared.

Thus your heat flux into the skin, thru the wire's insulation, is 200X smaller than the sun's flux.

What do you mean by a wire? Is it an electrical wire, with an insulation?

As you have already said, the temperature will depend on the environment (including air flow), surface area of wire and insulation. Additionally, it will also depend on biological factors like blood flow, electrolyte status, perspiration etc.

For biological experiments, keep in mind that human pain perception for aluminum conductors is just above 42C at the chest (most sensitive) for healthy person. Pain perception is measured at epidermis/dermis interface. Make sure your subject is not hyper-sensitive to pain, e.g. due to impaired hormone status, or you might end up with an emergency situation.

As you can see, there is no simple answer, especially at the extreme limits of accuracy where you are calculating. (Additionally, your reference is not room temperature at 305K, it will be skin temperature prior to measurement).

• I am referring to a standard insulated 28AWG cable, apologies for the confusion. – Wb16 Aug 5 '19 at 1:07