# The Relationship of Power and Heat to Length of a Folded Resistor

This question is a follow-up to a previous question of mine (The Relationship of Power and Heat to Length of a Resistor).

We make heaters in our lab. They are 1.5 inches long and are made using 0.005-inch-diameter (36 AWG) nichrome wire that is folded back-and-forth onto itself five times so that if you were to look at a cross-sectional area of the heater, you would see six wires.

We want to start making some heaters that are twice as long (3 inches) but still have the same heat output per cross-sectional area of heater. In other words, we want to keep the power of these heaters constant for a given point along the heater; since the heaters are twice as long, they should put out a total of twice the heat, but at a given point along the heater, they will be outputting the same heat. The heat output per cross-sectional area of heater should be the same for both heaters.

We currently use a 12-volt battery to power the 1.5-inch heaters (they are in parallel). The resistance of these 1.5-inch heaters is about 23 ohms, and thus the current through each one is about 0.52 amps. These heaters are powered for 3 seconds every 15 minutes so that they provide heat pulses for our sensors.

If I make the 3-inch heaters the same way (i.e., if I fold the same exact nichrome wire 5 times so that a cross section of the heater has 6 wires, but, for these 3-inch heaters, each fold is twice as long), I believe I would have to double the voltage to get the same heat output per cross-sectional area of heater because of the reasons mentioned in my previous post (The Relationship of Power and Heat to Length of a Resistor). The fact that the nichrome wire is folded is throwing me off, though - am I correct in my thinking, or is there something else I'm missing? If I'm missing something, what am I missing, and what is the reasoning behind it? Thank you!

Doubling your voltage is the simplest strategy on paper, and electrically should work just fine. I see two caveats:

### Thermal effects

There are a variety of ways the geometry of the system might cause it to scale non-linearly. Two wires physically parallel to each other could dissipate heat differently than a single, longer, straight wire. From your description of the change you're proposing, and the duty-cycle of the device, this probably wont matter, but it doesn't sound hard to physically check.

### Engineering constraints

There are lots of reasons you might not want to switch to a 24V power supply. In that case you'll need a different strategy. Elliot's answer is perfectly valid; note that he's proposing you put two of the old units electrically in parallel, where your solution is electrically equivalent to putting two of the old units in series.
You could also change the wire you're using, or how you're using it. You want the same power density across twice the area, so you want twice your old power.
$$P=IV=\frac{V^2}R$$ Of course nichrome will have varying R for a given temperature, so it won't be quite that easy.

Yes, you will need to double the voltage to get the same power output if you double the length of the wire.

On the other hand, you could connect two 3-inch heaters in parallel and operate them from your 12 V source. The battery would be required to provide twice the current in this case...if the voltage drops at the higher current level then you won't get quite twice as much power.

• Thanks. I like this idea, but all of the heaters (including the 1.5-inch ones) are in parallel.... I believe this would only work if our current design was to power all of the heaters in series. We don't put the heaters in series because if one breaks, all of the other heaters wouldn't work either. Dec 13 '18 at 14:41