Lumped parameter model

[Reinoud]

Hi all,

I am working on a cryogenic distribution line (helium) and am analysing the time needed to get the thing down to nonimal conditions. I am using a lumped parameter model (global guess) and a Finite differance one for the temperature profile. Because of large derivatives in the in the finite differance one my steps have to be small (for stable predictions) so I have to calculate + 10.000.000 points through a large equation. Due to my slow computer this is not the thing I am looking for. So I am trying to adapt the lumped parameter model with relation to the heat convection coefficient which is very high for my application. I only have to proof that the convection coefficient is not of significant importance to the cool down time. I built a model for it that looks like this (energy balance):

MassFlowFluid*HeatCapacityFluid*TemperatureDifferance
=MassMaterial*HeatCapacityMaterial

IN formulae:

m*CpF(Tf)*(Tm-TF)=dT/dt*M*CpM(Tm)

where

m= Mass flow Fluid (helium)
CpF(Tf)= Heat capacitance as function of temperature
T = temp of material to cool down like headers, supports, shields
TF = Temperature of the fluid (helium) this is kept stable on 80 K for the first few steps...
M = mass of the material
CpM = Heat capacitance of the material (combined for different alloys and materials)...

I would like to fit in the convection coefficient although it is so high that is will make no differance. The other parameters can be disregarded for this model (radiation, conduction, etc)... The model is made in MathCAD and solved with a first order dierivative solver.

So could anybody give me some advice on how to do it?? Or has experience with some other model?? I also would like to have a discussion going!

Thanks in advance,
Reinoud

 

[ardilesd]

Hello, Reinoud.
Your process is not completely clear for me, e.g. I cannot see what Tm means in your model, but, assuming:
-indirect contact between Helium and teh material
-lumped parameters in the thing
-sensible of heat in transfer
-temperature of Helium does not change significatively

M.CpM.dT/dt = U.A.(Tf - T)

Being CpM and U nearly constants with T, this leads to an analytical solution:

T = Tf-(Tf-T0)exp(-U.A.t/M.Cp); T0 : initial temperature

If the temperature in helium change at the outlet, you could use:

M.CpM.dT/dt = m.CpF.(Tin-Tout)

and modify the initial Eq. to consider a mean temperature of helium for the transfer.But, in this case, you need to know the outlet temperature Tout
Normally, in the heat transfer calculations, Cp can be considered as constants in a given range of temperature
Dario

 

[poetix99]

I am trying to better understand your concerns with "h" - convective h.t. coefficient.

If you would like to satisfy yourself (or a supervisor) that h is so high as to not be a factor in the calculations, I suggest that you first try to do that to everybody's satisfaction with relatively simple dimensionless parameters and one dimensional h.t. calculation.

Calculate Reynolds number. From RE and other physical parameters calculate/estimate h. Calculate Prandtl Number (conductivity/viscosity) to know relative thickness of thermal boundary layer compared to velocity b.l. The Biot Number compares solid conductivty to convection at the surface of a solid; this relates to your implicit assumption of a uniform (bulk) metal temperature.

I think that you should be able to either "prove" that the convection could be neglected, or to adjust the bulk fluid temperature so as to account for the temperature drop of the boundary layer.

Such dimensionless analysis can be very useful, but should be done on a rational, consistent basis; the values chosen for these parameters must have some relationship to your model (of course). One will obtain relative "order of magnitude" relationships, and not absolute values. Consult a good h.t. text for further guidance.

As you have described your model, you have made other simplifications which might become limitations to the accuracy of your calculation: for example, it appears that you are using a bulk metal temperature instead of a gradient.

"The chain is as strong as the weakest link." I understnd that you would like to keep the model as "lumped parameter", but you should decide what global accuracy you will obtain from that approach before you force yourself to make local refinements, such as adding a film coefficient.

 

[Reinoud]

Hi Guys,

Thanks for the help allready it has got me thinking. I yesterday adjusted the finite differance model with some assumtions and split it up in different temperature ranges. I am now plotting 22.000.000 points but split up so the waiting is not so hard.

For the lumped model, my distribution line is 3200 m long (yep the longest in the world "CERN Geneva&quot. And considering 3200 m op st.st as a lump is somewhat strange, so I build the other one with the mass transfer included. For slow cool down.

Ardilesd --> The Cp value can be indeed considered constant for the range to say 50 K, but after that it starts to change drastically, so that is why I implemented it as a function of temperature. Eventually when the first steps to 80 K are made I will have to simulate a second step which will be cooling down to 4.5 K, and then Cp will be a real issue...

Poetix --> I indeed did the local dimensionless numbers analysis for reynolds, nusselt with different empirical equations from the Heat Transfer book of J.P. Holman (P 316). And from it the convection coefficient. They all seem to agree which eachother, the Biot number is way below the 0.1 needed for the Lumped parameter method. So that is all proven, but me as a stubborn diploma-thesis writer would like to include it in my model to get proof. Even if I know the outcome. It is considered turbulent fully developed flow. And the balance given is put in MathCAD in a way that it can be solved with the rkfixed command which is a 4th order Runge-Kutta method for solving 1st order derivative equations.

I am indeed using a bulk metal temperature, no film layers or undevelop regions, just 3200 m of pipe cooled with a single steam of helium @ 80 K... and the header will be cooled down from 300 K with gradients over 150 K in 10 meters.

For a global picture, I am simulating cooling down a seperately build cryogenic distribution line which will supply superconducting magnets with supercritical helium for the new LHC (large hadron collider, particle accellerator) to be built at CERN. The tunnel where it is installed in is 27 km long and is spilt up in 8 sectors of 3.3 km. The eventual machine will have 100 tons of helium inventory at 1.9 K....

Thanks for the feedback,
Reinoud

 

[quark]

Reinoud!

:-> That's a great place to work for and I envy you. Please let me know, if quarks are further split. (no, not having the idea of changing my handle)

I wonder what capacity vacuum pumps you are using.

Cheers,



Repetition is the foundation of technology

 

[Reinoud]

Dear Quark,

Baseline design and information are available on

http://www.cern.ch,

for Vacuum info go to intranet, find EDMS (document server) and find the LHC-ACR/vac group there everything is listed, even to the public...

Cheers,
Reinoud

 

[quark]

ty! reinoud, let me know the quarks thing also.

Regards,

Repetition is the foundation of technology