Calculations for heat transfer time

[Shaw123]

Dear all,

I am looking for an equation to calculate the time take to heat certain sections of cylindrical steel.

For example a solid piece of steel 300mm diameter (500mm long). If this piece was placed (from RT (20deg C)) into a charged furnace at 1200 deg C how long would it take for the centre of the metal to reach 1200 Deg C?

 

[25362]

This is a typical unsteady state heat transfer case. Don't go after equalizing temperatures since this would result in infinite time. But 1,180^oC would be reasonable.

Use Gurney-Lurie diagrams for infinitely long cylinders.
I found such in figures 10.8 or 10.10 of Lydersen's Fluid Flow and Heat Transfer, Wiley.

Looking at these graphs one sees a linear relationship between the ordinate Y=(To-T)/(To-T1) and the abscissa X=kt/[ρ]*Cp*R², on a semi-log plot.

In one case when the surface temperature is constant, fig. 10.10

To, surface temperature, K assumed constant
T, the final temperature at the center of the rod, K
T1, the initial temperature of the rod center, K
k, thermal conductivity of the steel piece, W/(m K)
[ρ]*Cp, specific heat capacity per unit volume, J/(m³ K)=Ws/(m³ K)
R, the radius of the steel rod, m
t, time, s

In another case, fig. 10.8, the same kind of diagram, but calling To, the temperature of the furnace, and T1 the starting temperature of the cylinder.

In this case the coefficent of heat transfer at the surface is also a parameter in the family of curves.

By taking data from the graphs one can build a formula.

 

[EdStainless]

As I recall you need to calcualte a Fourier modulus. For some ranges you can use Heisler charts to find the values.

= = = = = = = = = = = = = = = = = = = =
Corrosion never sleeps, but it can be managed.

http://www.trenttube.com/Trent/tech_form.htm

 

[corus]

Assuming the cylinder of thickness x retains a uniform temperature and you know the heat transfer coefficient, h, then just solve dT/dt=-h(T-1200)/p.s.x

corus

 

[Tunalover]

The steady state solution is a simple separation of variables problem with the heat equation expressed in cylindrical coordinates. You can't assume that the cylinder is infinitely long, not even close!

Once you get the steady state soln, you can seek the transient soln. Any text on conduction heat transfer would have the problem in it.


Tunalover

 

[sailoday28]

tunalover (Mechanical)has made stated an important point.

"You can't assume that the cylinder is infinitely long, not even close!"

The problem is a 2 dimensional transient. Closed form solutions for this type of problem will be pretty hard to come by.

Numerical solution, which could include varying heat transfer coefficients and varying density, specific heat and thermal conductivity might be a more reasonable approach

 

[corus]

I'm not sure what tunalover is saying as the steady state solution is that the temperature everywhere is 1200C. For this kind of problem you can assume that the radius is sufficiently large that the cylindrical form of the PDE is not required. It would also be reasonable to assume that the cylinder was infintely long and that end effects would have a negligible effect at the centre of the piece. If the thickness of the shell is small then you can assume that the temperature through the thickness is approximately uniform. The PDE for transient heat transfer then simplifies to the form I described above.

corus

 

[Tunalover]

You're right corus. The steady state soln is obvious!
I do agree that you can get a rough answer, probably within 100C by assuming an infinite cylinder.



Tunalover

 

[Shaw123]

Thanks for you help everyone!!!

Corus, please could you what the symbols signify in your above equation?

Cheers

 

[corus]

T Temperature
t time
p density
s specific heat
x thickness of cylinder
h is total heat transfer coefficient from surfaces

corus

 

[sailoday28]

corus (Mechanical)
What is the basis for these statements?
"For this kind of problem you can assume that the radius is sufficiently large that the cylindrical form of the PDE is not required.
It would also be reasonable to assume that the cylinder was infintely long and that end effects would have a negligible effect at the centre of the piece."

 

[corus]

Mostly from experience. However, you don't state the thickness so I'm guessing that the thickness is relatively small compared to the diameter, and for that diameter an XY co-ordinate system is a good approximation. Similarly the length of the cylinder seems sufficient for the centre to be remote enough from the ends not to be affected. The answer to your original question is, of course, an infinite amount of time as the solution to the ODE is an exponential which tends towards 1200 as time,t, tends towards infinity.

corus

 

[sailoday28]

Couus--The original question related to a solid piece of steel.

 

[corus]

sorry, I read cylinder to mean a cylinder as in an annular region.

For a circular section an approximate solution is to solve dT/dt=-2h.(T-Ta)/p.s.r where r is the radius.

corus

 

[rjw57]

shaw123

You don't tell us if this is a vacuum or atmosphere furnace. The answer is different depending on the type of furnace. Everyone is on the right track, but they are ignoring the combination of radiation and convection found in an atmosphere furnace (if this is what you are using). There used to be some general nomographs in Perry's as I recall that would allow you to predict heating times for some particular plain geometries in stmosphere furnaces. Also, it dealt with an h_sub_c & h_sub_r coefficient which when added together allowed you to use the formula accomodating both the effects of forced/natural convection as well as radiation.

Some time ago, I wrote a Mathcad worksheet that looked at just convective cooling of finite cylinders by calulating cooling for the intersection of an infinte cylinder and plane using a Heisler analysis. If you are interested, you can find it at:

http://www.mathcad.com/resources/mathcad_files/DisplayCategory.asp?c=48&p=0&s=0

Look for "Cooling Information for a Cylinder, Slab and Finite Cylinder Using Heisler Analysis" by Bob Wilson. You may be able to apply it to what you are doing "in reverse" however, you will need to determine how to superimpose the radiation effects if you go this direction since my worksheet assumes that convection is plainly the greater influence during cooling (actually quenching) of the cylinder. You do not have the same situation since radiation will play a large role during initial heating of your part.

Bob

Bob