Formula for calculating air temperature when heated only momentarily by an object?

So I read that despite the temperature inside a clothing dryer ranging from 120-150 F, the temperature of the dryer’s heating mechanism itself only often hits 210 F max. That surprised me, since the air presumably only flows very momentarily over the heating element, all but instantaneously. It certainly seemed to be a lot lower than the temperature of a hair dryer, which gets hot enough that you can see the inner coils glowing red.

Is there a formula for calculating how hot air will be when exposed to such-and-such a surface for a certain length of time (i.e., just an instant?)

Can’t help with a formula but is it possible the dryer air is being recirculated and passing through the heater multiple times?

It’s not a formula you’re looking for, it’s an analysis method, and it’s complicated. Some of the air heats to very nearly the temperature of the heating element, which I’ll bet is much higher than 210 F. The molecules that hit the element get some distribution of kinetic energies including energies typical of air hotter than the element. What would be a practical approach would be to use computational fluid dynamics (CFD) to analyze the air flow and temperature through the geometry of the entire system, but this takes powerful computers, expensive software, and professional training and experience. And even so, the results are only so good. Look online for CFD and you’ll get some idea.

If you want to just consider the near instantaneous question you can consider the air as stationary, and then it becomes a much more tractable heat transfer analysis. Something worked out by Fourier.
A much simpler analysis is to simply look at the power dissipated by the heating element. That energy has to go somewhere, and there not a lot of options. Some will leave via black-body radiation, but given the element isn’t glowing visibly, that won’t be much. The rest must be leaving in the air stream. One, point - heat flows faster the higher the temperature difference. The less time any parcel of air stays near the heater the less it heat up, but the more heat is transferred overall. Eventually it should reach a steady state.

Heat conduction
is complicated enough, but in this case one material is a moving, turbulent fluid. When we need to determine what sort of heat exchanger to use we model them with CFD, mentioned above.

No, because when it passes through the dryer it picks up moisture from the clothes. If you recirculate it, you’ll just end up passing hot, wet air through the clothes and stop drying the clothes at all. The dryer has to be single-pass, bringing in fresh room-temperature air and then heating it to assure that you are hitting the clothes with air that is at less than 100% relative humidity.

As for the OP’s question, this is about convective heat transfer, which is a complicated subject. For simple geometries (e.g. forced convection over a flat plate, or buoyant convection around a sphere or cylinder), generalized relationships have been worked out that give you a reasonably accurate value for bulk heat transfer rate as a function of fluid velocity, density, and the temperatures involved. For more complicated geometries, as Napier suggests, CFD/finite-element analysis is required to properly understand the details of the flow and obtain a useful result. These are the kinds of tools that can tell you how hot the surface of the heating element really is, and how hot the air gets in the immediate vicinity of the element. Unfortunately they’re not tools that are readily available/affordable for the casual armchair user. :frowning:

I think there is a problem with your premise. According to this website, the heating element may be seen to glow red during the heating cycle.

In a darkened area, a dull red glow is approximately 1200F, so, at least according to General Electric, the heating element will heat considerably above 210F.

While the physics is complicated as others have pointed out, the engineering for the specific example (air heating by electrical heater) is fairly simple.

The equation (simplified assuming constant specific heat because of low temp delta) is :

Q = M x h x A x (delta T)

Where Q = Power output of the heating element
M = Mass flow rate of air
h = Overall heat transfer coefficient (determined using past history of similar exchangers)
Delta T = temperature difference between coil and air. You can take log mean temperature difference but since it is small, you can approximate itto delta T

h for air heaters is typically in the range of 1 to 1.5 (Btu/(ft2 hr deg-F)). Also note that the above is valid for turbulent heat transfer.

Some types of dryer do re-circulate the air, notably the condenser and heat-pump types. They remove the water from the air before recycling it. This avoids the need for a vent.

See here:

Yeah it had seemed odd that the element would only hit 210 F but that was what this source said . It seemed strange that 210 could heat it up so instantaneously; glowing red hot sounds more like it.

The element is constantly cooled by the air flowing past. If you block the air it overheats and the safety device cuts the power.

Some types of dryer do re-circulate the air, notably the condenser and heat-pump types. They remove the water from the air before recycling it. This avoids the need for a vent.

See here:

Oops duplicate post. Sorry. I don’t know how that happened.

That is not the most well-written article I have read. Not wanting to nit-pick it, but I suspect the writer has misunderstood some of the information that was used as a basis, but without any cites, it’s hard to tell.

Maybe this? This in entirely speculation on my part, but it is quite possible the temperature shut-off device is designed to cut-out at 210F as a safety feature (it would be hard for the clothes to catch fire if the heat chamber wouldn’t go over 210F). Maybe the article’s author misunderstood the over-temperature cut-out as representing the maximum temperature of the heating element.

You described that 210F number as the temperature of the “heating mechanism.” That number almost certainly indicates the maximum temperature of air immediately after flowing over the heating element…the element itself is much hotter than that. Other posters are right that it’s probably closer to 1200 degrees F.

Heating elements are reasonable approximations of black bodies. The Draper pointis the temperature at which objects start to glow visibly. It’s about 800 Kelvin or 977 F. When you see a hair dryer’s heating element glowing, it’s definitely hotter than that.

Heating elements in air have to be very hot partly because air is a poor conductor of heat and has a low heat capacity. You need a really hot element to make the air usefully hot.

Heating elements in water (which has high thermal conductivity and a very high heat capacity) don’t get nearly as hot, even though they’re likely transferring at least as much heat. An electric kettle and a hair dryer may both consume 1200 watts, but dumping that heat to water doesn’t require the large temperature differential required to dump the same amount of heat to air.

If you restricted your heating element to 210 F, it wouldn’t make the air nearly hot enough for a clothes dryer unless you added a truly giant heat exchanger. It’s much easier and cheaper to use a hot element than to go crazy with a heat exchanger.

I’d say it’s easy to tell that the author doesn’t understand the details here. To wit:

He’s quite obviously guessing at things he doesn’t understand. He seems to be under the impression that the heating element can’t or shouldn’t get hotter than boiling water. Plus, the repeated use of “might” indicates to me that the author understands that he’s speculating. He also doesn’t understand conjugate heat transfer, but it would be unreasonable to expect otherwise in this context.

I am frequently on review teams for Oil & Gas Plants and have reviewed the design of electrical heaters, both for air heating and hydrocarbons.

The goal for the electrical heaters is to minimize radiant heat transfer (and hence the temperature) because with radiant heat you are heating up everything around the element and not just the flowing gas.

Here is a worked out example for designing a heating element to raise the air temperature to 200 deg C (392 F) and the calculated wire temperature is 348 deg C (658 F). That’s a delta T of 148 deg C (270F).

The air in the home dryers is not allowed to go above 220 F (See for example) for fire protection reasons.

Although I have not done the detailed calculations, I would say that you will see approximately the same delta T. So the wire temperature for the home dryer will be about 220+270 ~ 500 degrees F.

I don’t doubt your expertise or experience in the petrochemical industry. Yet I’m standing by my assertion that the dryer heating element is well above 500 degrees F.

The poster who goes by “Excavating” supplied a link to a GE dryer FAQ along with some choice quotes. Not to put too fine a point on it:

I mean, the glow is red, so I doubt it’s Cherenkov radiation…

Here is chart of steel color versus temperature.

As you can see from the chart, steel “glows” at temperatures below 500 F.

I have no problem with you standing by your assertion. However I am not seeing any scientific evidence here.

Look at the footnotes at the bottom of your reference:

So, below 800F, you are talking about reflected light; that is, the color of the steel once it has cooled back to room temperature. You have to get above 800F for there to be any emitted light (glowing). And, if there is just about any ambient light, the reflected light from ambient will be greater than the incandescent light, so the object will appear black or sliver, depending on the emissivity of the object. This is true until the amount of light emitted by the glowing object significantly exceeds the ambient light.

No, it does not, at least not in the visible range of the EM spectrum. As Excavating pointed out, your chart—sourced from a site for model railroad enthusiasts—conflates color due to oxidation with incandescent colors. (Edward Tufte would be horrified!)

The oxidation colors are responsible for the iridescent-looking color patterns one often sees on motorcycle exhaust pipes.

It’s almost like you’re unfamiliar with the both Planck’s law and Draper point. I linked to the Draper point Wikipedia article upthread, FWIW.

Besides, your claim (via the chart) that steel glows at 390 degrees F doesn’t match observation. If it did, the steel racks in your oven would glow when baking a pizza at 450 F. Even at night with the lights out, the only thing glowing in my oven is the heating element.

That’s generous of you. And I agree: you’re not seeing any scientific evidence.