Heat Transfer-How long will it take for a material to heat up

[Matthias85]

Homework Statement

A cylinder made out of AL 6061-T5, dimensions d 10mm, L 28mm. Is put into an oven pre-heated to 100 deg C.
How long will it take for the cylinder to reach 100 deg C ?

Homework Equations

None given, this is not a homework question or even material I have been taught.
I have found a Nusselt equation in form of:
Nu = C.(Gr.Pr)ⁿ.K

The Attempt at a Solution

Nusselt equation gave me a number of 1500 hours.

 

[Bystander]

What's your question?

 

[Matthias85]

As stated in the first post:

A solid cylinder made out of AL 6061-T5, dimensions d 10mm, L 28mm. Is put into an oven pre-heated to 100 deg C.
How long will it take for the cylinder to reach 100 deg C ?

 

[Chestermiller]

Mathematically, your analysis should predict that it will take an infinite amount of time to reach 100 C, but it will reach 99.99 C in a much shorter time than 1500 hours. Please show us the details of your calculations.

chet

 

[Matthias85]

I can't find my precious calculations. I attach my most recent calculations, which still seem to be wrong.
I am pretty sure it should not take longer than half an hour to heat up a small AL cylinder to ~100 deg C.

 

[Chestermiller]

The heat transfer coefficient looks like it's about the right order of magnitude. Don't forget that, as the temperature difference decreases, the heat transfer coefficient is going to be decreasing because of the decrease in the Grashoff number. That can be included in the analysis. According to the correlation you are using, the cylinder is horizontal. How is the cylinder levitated? Are there fans in the oven, or is it really just stagnant?

Let's see the rest of your analysis to calculate the transient temperature variation.

Chet

 

[Matthias85]

The AL cylinder is placed on a steel plate in furnace with fan turned on to ensure uniform heat distribution.

I followed your advice and taken into account decrease in Br and Gr numbers.
Time has now reduced to 7 and a half hours.

 

[Chestermiller]

The placement of the cylinder on a steel plate is not consistent with the natural convection assumption that the cylinder is levitated. The steel plate is going to interfere with the natural convective flow and slow down the rate of heat transfer. On the other hand, the operation of a fan in the enclosure is very likely to significantly enhance the rate of heat transfer (i.e., increase the heat transfer coefficient). I'm going to try a calculation with a constant heat transfer coefficient of 10 W/m^2K to see if I can confirm your results. Get back to you later.

Chet

 

[Chestermiller]

OK. With a constant heat transfer coefficient of 10 W/K(m^2), I get about 40 minutes for the temperature to reach 98 C. If the heat transfer coefficient were higher, I would get a lower amount of time.

Chet

 

[Matthias85]

Thank you for your help.
May you please share your calculations with me?

 

[Chestermiller]

Thank you for your help.
May you please share your calculations with me?

The differential equation I used was:
$$mC\frac{dT}{dt}=πDLh(100-T)$$
where m is the mass of the cylinder (6 gm), C is the heat capacity of aluminum (0.9 J/gC), D is the diameter of the cylinder (1 cm), L is the length of the cylinder (2.8 cm), and h is the heat transfer coefficient (10 W/m^2-sec-C). Based on this data, what do you calculate for the quantity $\frac{mC}{πDLh}$ (in units of seconds)?

Chet