Temperature difference across a drum wall during heat up 3

[marcopoly]

Hi all,

I'm trying to model what I think should be a simple case but I just can't get this going.

I've got a steel boiler drum wall for which I know how the internal temperature tracks over time. The boiler is in start up and the temperature is heating up inside. I can assume that the external wall is well insulated and adiabatic.

I'm trying to calculate the peak temperature difference from inside to outside.

I can simplify the ramp rate to 8°C per minute instead of using empirical data if that helps (but it doesn't help me!).

The solution would start at steady state (no temp difference) and eventually reach a steady state but as I'm interested only in differential temp (for stress calculation) I'm not interested in steady state solutions.

To me the simple 1D heat equation would suffice (drum diameter is sufficiently big to consider the wall as flat so cartesian coordinates work too) but I can't get it to work! I'm sure that in university this was considered a simple problem, can anyone help?

Marc

 

[athomas236]

So what is your question.

Best regards,

athomas236

 

[IRstuff]

Have you calculated the Biot number? I think that will determine whether you can get away with a lumped model or need a more discretized model.

Note that assuming the liquid temperature is constant up to the wall is not necessarily accurate, IF the wall Biot number is low.

TTFN
faq731-376

 

[athomas236]

IRstuff is right, I remember making that mistake back in 1995.

Best regards,

athomas236

 

[davefitz]

The problem was solved a while ago. The basis of the TRD 301 annex 1, and EN 12952-3 is that a steady , monotonic increase in fluid temperature eventually leads to a steady monotonic increase in metal temperaure throughout the heated cylinder , with a steady ( quasi-static) temperature differntial from ID to OD. This then allows the designer to calculate the thermal stress, the fatigue damage, and the max allowable ramp rate for given amount of acceptable fatigue damage.

The lowest cost copy of that EN is the "prEN 12952-3:2008 (E)", at $45.00. I suggest you find a copy of that for the simplifed calculation procedure.

 

[corus]

If a lumped mass model can be used then the assumption is that there is no temperature difference across the thickness and hence no stress, unless the thermal stresses arise from a difference in mean temperature within the structure or due to restraint from expansion. Similarly if there is a steady increase in temperature and the outer surface is adiabatic then there must be no temperature gradient across the wall, and hence no stress.
If you want a complete solution then you'd need to use separation of variables technique. Wikipedia has a solution for 1D here

http://en.wikipedia.org/wiki/Heat_equation.

The constants you'd derive from initial conditions and surface boundary conditions.

 

[ione]

Good points from corus . No restraint and wall performing adiabatically imply no stress.

A couple of interesting links below

http://www.ewp.rpi.edu/hartford/~poworp/Project/4. Supporting_Material/References/Kandil.pdf

http://ethesis.nitrkl.ac.in/1486/1/Analysis_of_transient_heat_conduction_in_different_geometries.pdf

 

[athomas236]

Well done ione, we used a similar approach to Kandil for monitoring main steam pipe stresses back in 1982.

It is worth noting that with steam drums there are also temperature differences between the top and bottom sections because of different internal heat transfer co-efficients in the steam and water sections.

Best regards,

athomas236

 

[corus]

ione,
I looked at the first reference and the explicit finite difference method they use for solving the transient is known to be unstable, requiring limits on the time step used, if my memory is correct. A better way would have been to use an implicit scheme where the temperature has to be solved from a set of equations at each time step. These methods are unconditionally stable but requre more computational effort.

In addition, the assessment of the stresses in the cylinder wall would require that the equivalent linear bending stress across the wall would have to be calculated and separated from the peak stress that would occur from a thermal shock. These stresses are subject to different limits in a prssure vessel code.

 

[marcopoly]

Thanks to everyone for such a rapid response. I guess when I wrote this post last night it was late, and I may not have been so clear.

I am looking at the stresses induced in a (HRSG boiler) drum wall (approx 1800mm od 85mm thick) during a start up (hence transient from 20°C to 280°C in an hour or so).

The lumped mass model is inappropriate since i'm trying to find the difference between internal wall temp (known from probes) and external metal wall temp (padded with insulation).

As there is insulation which will heat up very slowly due to low conduction, it is safe to ignore wall losses for the first 40 mins or so when the stresses are greatest. We are not talking about steady state or lumped mass so a steady temperature ramp on the inside causes a lag effect on the external wall, due to time to transmit/thermal diffusivity of the steel.

I am interested in this temperature gradient across the wall because of the induced stresses.

I prefer not to use Kandil if possible because I struggle to believe that 1D transient heat conduction to an adiabatic wall (with an approximated-constant temperature ramp rate) would not be solvable explicitly.

Athomas, I recognise that this would be difficult to model without some FEA or discretization... So I'm tempted to ignore it, I think the stress caused by thermal gradient through the wall during startup would be much greater than the relatively gradual difference in wall temp around the circumference of a large drum (note the upper area is almost forced convection due to risers circulating steam).

Davefitz, Thanks very much. I looked at the EN standard (I only had 2001 version) and couldn't find anything that would allow me to calculate the temperature difference: it seemed to require knowledge of midwall and inner surface temps. On the other hand TRD 301 Annex 1 was very useful has a simple calculation for a fixed ramp rate in section 5.2.2 equation 23 see attached.

I'm going to separate the pressure related term and count a start as a full cycle (on and off load) for temp difference, plus one full cycle for shut down, plus one pressure cycle for start up and shut down. I might ignore the temp difference on shutdown because it is just idle cooling and is very slow (plus the drum is insulated)

So I guess the answer to my question (according to TRD 301) is that the temperature difference is proportional to the ramp rate times the square of thickness divided by diffusivity.

Marc

 

[IRstuff]

Lumped mass does not necessarily imply zero delta T, since you have to put the thermal conduction somewhere, and it would go between a virtual inner surface and the lumping of all the mass on the outerwall.

water-->convection-->inner wall with no mass-->thermal conductivity-->outerwall with all the mass

You can then force the thermall conductivity to be at the worst-case, through the entire thickness of the wall.

TTFN
faq731-376

 

[ione]

Very good point again from corus (Have you done by chance studies on geological processes?). The implicit method does not put any restriction on time step but is more computationally demanding, while an explicit 1D transient conduction may lead to unstable solution in the event the mesh Fourier number is < 0.5.

IRstuff,
I can’t completely grab your point. Lumped capacitance model implies conduction through the wall thickness dominates over convection. The smaller the Biot number the better the approximation in taking a zero gradient temperature over wall thickness (which would be really zero for a material with infinite conductivity or zero thickness wall). It’s just an approximation and it remains that. The evaluation of the Biot number should tell how good the approximation is.

 

[chicopee]

Instead of trying to figure out the stresses based on temperature, why don't you use the ASME/NB codes on fired pressure vessels which will tell you at what temperatures should the stresses be calculated and then follow their tables of max. allowable working stresses based on temperatures and type of steel. Don't try to reinvent the wheel.

 

[davefitz]

MarcoPoly:
Your interpreation is correct

-the thermal stress is linearly proportional to the temperature difference between the mid wall metal temperature and the surface metal temperature DTi or DTo- additional modifiers are used to address the differnt therma stress concentraion factors for each different weld geometries at the tube stub to drum weld.

- that DTi is directly proportional to the ramp rate V, and the square of the wall thickness e ( or s, in the TRD)

The latest EN correlates the ramp rate V and wall thickness e to the DTi using the estimate V=TD/gamma/e^2, where TD=thermal diffusivity and gamma per figure 13.4.6 of the EN , which was calcuated using a solution of the conductive heat transfer differential equation (plot of gamma vs ratio Do/Di).

 

[marcopoly]

chicopee: I'm also trying to calculate the approximate consumed fatigue life based on existing ramp rates.

davefitz: I'm gonna buy the new standard. One gripe with 12952 i've found is that it uses unnotched steel curves for the cyclic stresses, the 'unfired' equivalent discusses both.

I must admit the ASME standard is easier to use but a lot less detailed.