# How to figure out tube length to heat Argon to 700 C from ambient

1. Oct 23, 2014

### mesa

If one were trying to heat Argon gas from ambient (~20oC) through a mild steel tube of some diameter 'd' how long would it need to be? Additional unknown includes flow rate of Ar gas so some general function would be adequate. Unfortunately my thermo-book is limited and does not go into enough detail to solve this problem.

I played around with a few static equations to get a general idea of J/s heat transfer from the steel wall to the Ar gas but this does not account for flow rate however it could likely be converted with some work. Rather than spend too much time on this I was hoping someone could chime in with a heat transfer by convection equation for a laminar flow of Ar in a tube of some radius 'r' and flow-rate 'f' where the tube itself is in a steady state at 800oC. It could be assumed that due to the nature of how the tube is heated and the thermal conductivity of steel being 3 orders of magnitude greater than Ar that this temperature will remain constant throughout the process.

I expect the length of the tube of any substantial diameter (greater than 1/4") will likely need to be very long due to the nature of Ar's thermal conductivity. To help aid in heat transfer it would seem reasonable to add steel wool to the inside of the tube in order to create turbulent flow and add a large surface area with minimum 'thickness' of the gas to the thermally conducting steel. To what extent would probably be very difficult to calculate but if a reasonable length tube can be figured for laminar flow then the addition of the steel wool should greatly increase the efficiency of a tube of the same length, hopefully it doesn't work out to something like a mile long :)

2. Oct 23, 2014

### SteamKing

Staff Emeritus
I can't comment on your idea for an argon 'heater', but you should be aware that mild carbon steel loses a lot of its strength when the temperature goes above 550 C. Operating a device at 700 C or 800 C without using special alloy steel would be impractical, if not impossible. Definitely not recommended.

3. Oct 23, 2014

### mesa

Yes, creep resistance of steel is typically about 550 C but the tube will be subjected to very low pressure (less than 1 bar), do you think that would be an issue when taking into consideration regular creep resistive loads?

Either way I should have mentioned this is a low pressure system, thanks for pointing that out!

4. Oct 24, 2014

### mesa

I know my question about building a thermal exchanger for Ar is probably a bit unusual but you are exactly the person I was hoping would respond. Perhaps if we instead talk about the same situation but putting thermal energy into steam through a pipe?

Worst case I can do this the old fashioned way and build something and test it but I would first like to know if you have any concerns about exceeding the creep resistance temperature of this alloy even though the pressure will be moderately low.

5. Oct 24, 2014

### SteamKing

Staff Emeritus
As much as I'd like to help, I'm afraid my Heat Transfer knowledge is a bit rusty and was only employed at college over 30 years ago.

What struck me about your post was the elevated temperatures at which you wanted to use mild steel to fabricate your argon heater. All I'm saying is that mild steel loses strength as its temperature rises, and a max. operating temp. of about 550 C seems to be the point at which mild steel is no longer used and other types of steel alloys are employed.

Now, your steel may not collapse physically, but if your heater goes thru many thermal cycles from cold to hot back to cold, the reduced strength of mild steel might allow cracks to form as the metal becomes fatigued. Similar problems are encountered when mild steel structures must withstand operating in cold environments: as the temperature drops, the metal becomes very brittle and careful attention must be paid to detail design to prevent the formation and growth of cracks.

6. Oct 24, 2014

### mesa

Well, I figured it out to be 40K / 1m length 1/4" ID tube so a length of 170m would be required for laminar flow :O

This is actually a great result. The reasoning for a pre-heat of the argon was due to it's use in a closed crucible for agitation of a molten magnesium alloy. Since it is so averse to the take up of thermal energy it seems likely that bubbling ambient Ar gas through less than 10cm of liquid metal would hardly be an issue as far as cooling off the melt so (presuming the calculations are correct) the need for the pre-heat of the Ar is unnecessary, hooray! Confirmation will be through a molten lead Ar agitation test sometime next week.

I did some digging on this and came across the attached pictures from a book in our campus library. The crucible is a 2.5" thick low carbon steel used for preparing Magnesium melts that can hit the same temperature range. Nice to see industry getting their monies worth out of their equipment :O

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7. Oct 24, 2014

### SteamKing

Staff Emeritus
Yes, you can use mild steel as a crucible to melt magnesium (M.P. = 650 C), but notice, the crucible is 2.5" thick, which is rather heftier than 1/4" mild steel tubing, where the wall thickness can be as little as 0.01 inch.

8. Oct 25, 2014

### mesa

These crucibles go up to 800 C for an occasional 'super heat' process (no different than mine). Now, yes we do have some thin wall 1/4 tube but I think it is a safe bet we would try something a good bit thicker. Let's see if we can come up with a 'general' minimum thickness based on the information we have so far.

We know in our picture that crucible measures about 4 ft in diameter and holds a 4000lb melt at temperatures no different than what our heat exchanger tube will be subjected to. Doing some quick calculations we see that crucible will be subjected to a constant pressure (at the base) of:

Next we need a general idea of our ultimate tensile strength at 800 C:
Tensile Strength apprx. = Ts/(1+36(T/1000)^6.2) or about 10% of Ts

By somewhat inappropriately borrowing Barrows formula we can make a general comparison of 'max' psi's :
P = (2*Ts*wt)/(I.D.) where wt = wall thickness, P = pressure, and ID = inside diameter
Setting 'P' equal for the crucible and pipe we get a minimum 'wt' for the pipe of a little over .01"

This is likely 'off' but it is probably safe to say 1/8" wall or thicker (an order of magnitude+ more) would work fine for our purposes.

On a completely different note, my calculations for length are completely wrong :O

9. Oct 26, 2014

### Staff: Mentor

The equations you are looking for are these:
$$WC_p\frac{dT}{dx}=\pi hD(T_w-T)$$
where W is the mass flow rate of argon, Cp is the heat capacity at constant pressure, x is axial position along the tube, h is the local heat transfer coefficient and D is the tube diameter.

You don't need to solve the problem exactly for your application. All you need is an upper bound to the length. You can obtain that by neglecting radiative heat transfer in the gas and natural convection heat transfer. You can also use the asymptotic value of the heat transfer coefficient, which is the lowest value that the heat transfer coefficient for laminar flow in a tube can have (at long distances from the tube inlet). For constant wall temperature,

hD = 3.647 k

where k is the thermal conductivity of argon. Note that your upper bound to the length will be independent of the tube diameter.

Chet

10. Oct 27, 2014

### mesa

That is exactly what I was looking for!
I came up with 1.134m to get the Ar to 700o C from 20o C with a flow rate of 2.101x10^-4 kg/s, although this is for Cp at 20o C (I am still looking for the asymptotic value at elevated temperatures... hint's are always appreciated!).

11. Oct 27, 2014

### Staff: Mentor

Argon is a mono-atomic gas, so its heat capacity should be very insensitive to temperature.
What about the thermal conductivity? How does that vary with temperature? Use the lowest value over the desired temperature range.

Chet

12. Nov 4, 2014

### mesa

I would imagine thermal conductivity would drop with increased temperature due to the KE (both rotational and along some vector) of individual atoms being higher than at lower temperatures making it increasingly more difficult to 'absorb' additional energy. Does the relation to heat capacity of a mono-atomic gas have something to do with the rotation (from some fixed outside frame of reference) of individual particles (in this case argon atoms) as compared to say a diatomic molecule?

Do you perhaps have links (or better yet good book suggestions!) on where to get heat capacity and thermal conductivity tables for different materials at different temperatures?

13. Nov 4, 2014

### Staff: Mentor

The thermal conductivity of argon increases with temperature, so to stay consistent with our concept of getting a worst-case upper bound to the length of the tube, use the lowest value at the lowest temperature, which would be your starting temperature of 20° c

14. Nov 9, 2014

### mesa

Okay, that was a bit of a surprise but after some consideration it makes sense. If a material (like our gas in this example) is heated it will have a higher kinetic energy equating to more collisions with our steel wall hence a higher likely hood of energy transfer. The thing is that even though the number of collisions increases the temperature difference is lower (for this particular case) so there would be a lower energy transfer per collision at elevated temperatures.

15. Nov 9, 2014

### Staff: Mentor

If you're interested in understanding the temperature dependence of thermal conductivity of gases, see Transport Phenomena by Bird, Stewart, and Lightfoot.

Chet

16. Nov 9, 2014

### mesa

Very much so, just picked up a hardcover copy on abe for thirteen bucks. Thanks for all the help Chet!