Heat transfer times for aluminum 1

[mattwrigg]

I work in a manufacturing site in the UK, and our process involves heating stacks of aluminium sheets interleaved with a thin tissue paper at a setpoint for a few days.

Maybe this is a no brainer to you folk, but I am trying to find out the heatup rate for certain sizes of stacks. We have stack sizes between 150mm to 400mm high, depending on the dimensions of the sheets i.e. we are limited by weight.

I have calculated the surface area to mass ratio which I would think is important, but I am sure there is a correct way to do it.

Can anybody help?

 

[mattwrigg]

Forgot to mention we heat the stacks by placing 16 of them into an industrial oven and blowing hot air in. As they can't go above the setpoint of 55 degrees, when the first one gets there the air temp lowers to maintain the max stack temp of 55 degrees, then the other stacks slowly get there too.

 

[prex]

I suppose that your goal is to determine the thermal transient inside the stack, in order to insure that all the aluminum plates reached a determined temperature and perhaps also that they stay above that temperature for a given time.
The stacks should behave like slabs of substantially infinite extension in two directions, so that the problem mathematics is relatively simple. Also it should be possible to assume that the faces of the slab are at substantially constant temperature, as you regulate air temperature. This condition is also quite easy to account for.
The weak point is that the conductivity through the slab probably depends critically on the characteristics of those paper separators: if you can get an estimated conductivity for them, then the equivalent conductivity of the slab will be easily calculated (I suppose here that the heat capacity of the separators with respect to the aluminum is negligible, but also this point should be checked).
Otherwise you should conduct some measurements by inserting somehow a thermometer inside the stack.

prex

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[desertfox]

Hi mattwrigg

I think I can help you here,to get an estimate of the time
to heat up a stack you can use the following formula's:-

first calculate heat energy required to heat one aluminium
block to 55 degrees centigrade this is done by :-


q= m*Cp*(T2-T1)

where q = heat energy in (Joules)

m = mass of 1 aluminium block.

Cp= specific heat capacity of aluminium = 950J/(kg deg C)

T2= final temp ie 55 degrees

T1= initial temp at which the block goes in the oven
assume 18 degrees

if I assume that a aluminium block weighs 20kg say


then q = 20 * 950 * (55-18)= 703000 joules.

so if you want to heat 16 blocks the energy in would be:-

16*703000=11248000 joules


now to find the time required to heat those blocks you need
the power output of your heaters lets assume 100000watts or
100KW


now a watt is = joules/time (second)

so if yours heaters have an output of 100000watts=100000joules per second


now if you divide your (energy req for 16 blocks ie heat energy)by 100000 watts
this will give you the time required to heat up 16 blocks

ie:- 11248000/100000 = 112 seconds or 1.87 minutes.


A word of caution here this is only approximate as it assumes 100% efficiency and that the temperature of the blocks are all 18 degrees and the oven temp is also at 55 degrees everywhere inside.


hope this helps

desertfox

 

[mattwrigg]

Thanks DesertFox

I will have a go.

One query though: we have found that wide and low stacks heat up a lot quicker than stacks which are narrower and higher. The assumption for this is that the surface area is a lot more, also the surface area to mass ratio is higher, so the heat has more surface area to conduct through.

How can this be worked into the equation?

 

[IRstuff]

Sure, that makes sense. You are heating through forced convection, which requires that every single point in the stack that changes temperature to get its heat from an exposed surface. Aluminum has a thermal conductivity of about 2.4 W/cm/°C.

Desertfox's post needs to be modified by the convective heat transfer coefficient, which generally ranges from 4-20 W/m^2/°C, where the m^2 refers to exposed surface area. This will tell you how much power can be transferred into the stack. The combination of exposed surface area and distance from the innermost points coupled with the respective heat transfer coefficient and thermal conductivity ultimately determine the time to warm up the stack. Assuming that you can come up with a good model, you can potentially look at overdriving the input for a some period to minimize the warmup time, but, obviously, some portions will overshoot and the settle back.

If you want to massively change the time required to heat, you might think about possibly using taller separators that allow air to get between the sheets and blowing the air into the the gaps between sheets. This basically would try to make a tall stack look like individual wide, thin stacks, which should speed things up.

TTFN

 

[desertfox]

Hi mattwrigg

The heat transfer is more complicated than I have assumed as
it would involve conduction and convection as other posts have suggested and in addtion to all that there will be a
thermal gradient within the oven itself.Regarding your last question I agree about surface area to mass ratio, also
it would possibly be affected by the thermal gradient in the oven itself.
How at present do you determine whether the aluminium is at the correct temperature? maybe you can use my formula and then take some practical measurements to how close or how far away you are and adjust timings on a practical basis.


regards desertfox

 

[mattwrigg]

Hi Desertfox

We blow hot air into the ovens at quite a speed, and the differences between each area in the oven is minimal, so I am going to assume all air temps are equal for the time being and the thermal gradient minimal.

As for monitoring stack temperatures, a thermocouple arrangement is placed on top of each stack in such a way as not to pick up the air temp. The temperatures of these control the air temp in the oven, as we have an upper stack temp limit.

The air temperature plot is generally pretty consistent from cycle to cycle, so I want to use this to calculate the heat up of any stack exposed to this air temp, without having to test empirically.

It isn't practicable in production to put a thermocouple in the centre of each stack, therefore we allow 24 hours for the whole stack to heat up. When I find a way to reduce this will have immediate benefits to our capacity. We don't want to reduce stack heights as that will lower our capacity.

An idea occurred afer reading IRstuff's post about putting some kind of breaker in the middle of the stack to let hot air through, which in effect would create 2 smaller stacks and a quicker heatup.

As for the effect of paper on the thermal conductivity of aluminium, I have in mind an experiment to determine the extent of this.

Thanks all for your help so far

Matt

 

[prex]

As I already told, if you know the equivalent conductivity of the stack of aluminum with intermixed paper separators, then the calculation of the minimum required time is quite straightforward: this will vary somewhat with the thicknesses of the aluminum plates (if they change from stack to stack) and more importantly with the height of the stack.
Can you be more specific on typical lengths and widths of stacks and on your functional (thermal) constraints?
Also note that if you can measure the temperature as a function of time at mid height and on top and bottom of a stack only once, this record will allow for knowing all the relevant parameters for a close estimate of the behavior of any other stack.

prex

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[desertfox]

Hi mattwrigg

Thanks for the response and please let us know how you get on.
To expand a little further on this problem, if you were to follow prex's method of conduction which I presume would follow the formula :-

q=U*A*(t1-t2)

where q = heat flow rate in watts
U = overall heat transfer coefficient Wm^-2 K^-1
t1= 55 degrees C or 328 degrees K (Kelvin)
t2= temperature of sheet above the sheet at t1

I can't quite see how easy it would be because what temperature would you use for t2 it would be lower than t1
if we assume the first sheet is the only sheet to reach t1
and the rest slowly heat up, but this would not be the case - the sheets are heating up from convection as well as conduction. Furthermore what temperature would you use for the third or fourth sheet in the stack. this is why I assumed that the oven was at 55 degrees C before the sheets went in there, which I know wouldn't happen in practice. I then assumed the sheets are all immersed in the oven at the same time as one block of metal and ignored the paper(another sweeping assumption).In other words my theory
is a bit like saying if I have a bowl of boiling water which
is kept boiling and I drop a block of metal in at say room temperature then how long will it take for that block to reach 100 degrees C.

I mean no disrespect to prex, in fact if I am wrong in what I am saying then hopefully prex will correct me and perhaps show us some formula I am not aware of.


Regards desertfox

 

[chicopee]

ARE THE PLATES STACKED DIRECTLY ON EACH OTHER W/ THE SHEETS OF PAPER AS LAMINATES OR DO YOU HAVE AN AIR SPACE BETWEEN THE PLATES?

 

[mattwrigg]

Hi Chicopee

The plates are 300 micron aluminium, each with a sheet of tissue interleaving of the same size which has a weight of 50 grams/square metre, in order to protect our coating.

There is no furthur gap between the plates, there are simply 1200 of them per stack.

rgds
Matt

 

[prex]

I'll try to better explain my position.
What I'm proposing is that mattwrigg's problem may be treated as a case of transient conduction in a one-dimensional slab.
This problem is solved in basic texts on heat transfer: an example is in section 5.3 of the Lienhard's free Heat Transfer Textbook that is the subject of a popular thread on this same forum.
The basic elementary treatment assumes a slab of sufficient extension in length and width so that in can be considered as infinite, and also with simmetry conditions through the thickness (both faces at the same temperature, mid plane with zero thermal flux across it).
Two simple boundary conditions may be treated analytically: faces at constant temperature and faces with convective heat transfer to a constant ambient temperature.
I think that the first one is closer to mattwrigg's conditions, as he regulates air temperature in order to maintain a constant slab face temperature of 55°C: the treatment of this one is also simpler, but also convection might be introduced for a closer approximation, especially during the first transient phase.
The solution is in series form, but the first term may be taken for a first approximation and is:
&[ignore]Theta[/ignore];=4/&[ignore]pi[/ignore]; x exp(-&[ignore]pi[/ignore];²/4 x &[ignore]alpha[/ignore];t/L²)
where
&[ignore]Theta[/ignore];=(T-T₁)/(T_i-T₁)
T=temperature of mid plane as a function of time
T₁=face temperature
T_i=initial temperature of slab
&[ignore]alpha[/ignore];=k/&[ignore]rho[/ignore];C_p
k=thermal conductivity
&[ignore]rho[/ignore];C_p=thermal capacity per unit volume
t=time
L=half slab thickness
Hence the time for reducing the temperature difference at mid plane with respect to faces by some 98% (but it's up to mattwrigg to define the functional goals) would be obtained from:
&[ignore]pi[/ignore];²/4 x &[ignore]alpha[/ignore];t/L²=4
and
t&[ignore]asymp[/ignore];1.6&[ignore]rho[/ignore];C_pL²/k

Now as my effort to arrive here has been quite important, I would like mattwrigg to fill in with some realistic values for k and the other parameters to see what comes out.


prex

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