Temperature change through a wall over time intervals 2

[gavinr990]

I'm trying to find the temperature inside a structure if a fire starts on the outside.

The information I have is temperature of the fire and distance from the fire to the wall. I also have (or can assume) all the information about the wall of the building (heat transfer coefficients, thermal conductivity etc.)

What I'm looking for is how the temperature inside the building increase change over time as the fire size increases, peaks and then decreases.

Thermos isn't really my strong point so any help would be fantastic.

Thanks

gav

 

[tkordyban]

It looks like people are not exactly leaping to your help on this question. The reason is probably this: the answer to your relatively simple-sounding problem is way too complicated to give in a forum like this.

Your problem requires a transient solution, which is not going to be easy to do by hand. Not only is the temperature of your building and everything in it going to change with time, but you also propose that the fire itself will change in intensity over time. That means solving the math, once it is set up, many, many times in small time steps, to resolve the temperatures as they change over time.

The heat transfer from the fire to the building is also non-linear, because most of the heat transfer will be by radiation, which is related to the fourth power of the temperature of the fire. Some heat will reach the walls as heated air flows away from the fire and contacts the building. Exactly how that flow happens depends on the local geography and how the wind is blowing, the shape and size of the fire, the shape of the building, and lots of other factors. Just knowing the distance between the building and the fire probably won't be enough.

Then you need to know the construction of the building -- the material properties of all the layers in the wall, including any air pockets or insulation. You will need the conductivity, density and specific heat of all the layers.

In transient problems any small mistakes tend to get magnified as you calculate further and further from Time=0, so it is important to know all the boundary conditions and initial condition accurately.

I hate to discourage you from trying, but I am afraid that this problem is over the head of somebody who admits "thermos is not my strong point."

 

[zekeman]

OK, you need the dimensions of the fire, its proximity to the wall and the temperature of the fire all as functions of time.
Then, a solution is readil ahievable
Give us a sense of these functions, either graphically or mathematically.
For example, you could worst case the fire to be very large with respect to the wall ( so that the view factor, F is unity) and then say the temperature falls off exponentially as T0*exp^-t/tau (or some other that is more realistic)
which is a classical time decay function. It is then possible to solve it in limited closed form by linearizing the radiant transfer.
Now if the worst case is not reasonable and you have the size function of temperature and proximity to the wall, the view factor can be written as
F(t) and the radiant heat transfer from the fire to the wall would be
F(t)*(T0^4exp^-(4t/tau)-Tw^4) and you could get a numerical solution either using a spreadsheet or a canned program.

So post what you think the fire looks like and the other parameters I mentioned and we can guide you to a solution.

 

[gavinr990]

Thanks guys.

Admittedly it looks like I've catastrophically underestimated how complicated this problem is. I assumed it was a relatively simple problem and once I had a point in the right direction I'd be flying.

Nonetheless...

I discussed this with the other guys on the project and it was agreed that we can model this fire as a constant fire (what I mean by that is that the properties/dimensions of the fire will not change with time as I stated originally) without losing the applicability of the calc. Hopefully this will simplify things considerably, although we're still looking to see how the temperature in the room changes with time.

I haven't been given the numbers yet but for argument's sake we could assume a 3m diameter by 8m high hydrocarbon fire at 1500K that is 10m from the wall of the target area. The wall is an A60 rated steel firewall. Assume a thickness of say 40mm and a density of 100kg/m3.

I was looking through some of the older threads on this forum last night and the problem is actually very similar to this one:

http://www.eng-tips.com/viewthread.cfm?qid=265250

with the exception that my fire is actually lower than the target area and that I want to see how the temperature will change over time (I guess this makes it actually significantly different!). Just to clarify, although the fire will be lower than the target area, the fire cannot actually reach the floor of the room. Thus I'm assuming only radiation onto the side wall is possible (hopefully the attached diagram will show this better) and that the radiation is equal on all points of the wall.

I hope, zekeman, you didn't spend any timefinding those equations now that I have redefined the problem significantly.

 

[corus]

Treat it as a 1D problem. You'll need the viewfactor from the target area on to the fire for that radiation, together with the emissivity of the surfaces. The remainder of the view will be radiation to ambient. The opposite side of the fire wall will also radiate away. Heat loss from both sides will alos have natural convection. In the end you'll have a 1D transient heat transfer problem involving radiation and natural convection, probably best solved with finite difference or finite element methods. In other threads it was mentioned about using charts but I'm not familiar with that.

Tata

 

[gavinr990]

Thanks corus.

A couple of (probably stupid) questions:

Is the data I've already stated enough to answer this problem? I've been trying to use the heat equation, Stefan-Boltzmann law and some radiation power equation based on Plancks Law but I always seem to be missing one or two variables.

Also, the equations mentioned above and a couple of others that I've come across all describe radiation from a black body (or grey body if I include the emmissivity factor). What equation do I calculate the radiation from the fire itself?

Thanks.

 

[zekeman]

For these problems you use the worst case of 1 for the emissivity.
But your major problem is the view factor which is the view factor (fraction of radiant energy emitted by the fire that strikes the wall.
If you can put some dimensions on the sketch you provided, then we can estimate the view factor.
Once you get that, the problem is fairly straightfoward.

 

[gavinr990]

I've revised my sketch with the dimensions. We can assume a wall thickness of 40mm. Thanks again.

 

[zekeman]

Oh, you need some reasonable estimate of the fire's duration and a rough estimate of the structure that encloses the target area including the depth dimension.

Even though the view factor I derived is low,for very large fire durations, you would get unreasonably high values of temperature.

 

[gavinr990]

Yes, of course, sorry. The room is 7.5m wide (it will be at ambient conditions and, if its relevant, it will be largely free of equipment and so on i.e there is not a huge separator or other vessel sitting in the middle of the room). Unfortunately this fire is likely to be of significant duration. The calculations that I've done suggest 3.5 - 4 hours.

 

[zekeman]

Sorry for not getting back sooner.Also, I am using units I am more comfortable with so if you have conversion problems feel free to ask for help.[The Brits stuck us in the US with these ridiculous units]

I have developed a model for this problem which starts with getting a view factor. Assuming the fire dimensions you gave and doing some scaling of your sketch, I got 0.148 view factor and since the fire area facing the firewall is about 1/2 the area of that wall, I get a flux (assuming unity emisivity) of
6300BTU/hr-ft2.
The final equations I got were
7.5dTw/dt=6300-.173*10^-8*(Tw+460)^4_-5.5*(Tw-T0)
assuming only the wall heat capacity is the mass thermal storage.
T1=TW(.44/.75)+70*31/75=386
T2=T0+0.1*(T1-T0)=119 F
where
Tw=firewall temperature
T1 air gap temperature
T2= target temperature
T0= ambient temperture assumed at 70 F and initial room temperature.


The time constant is roughly 3/4 hr so a 4 hour fire translates to using the steady state solution where
Tw=610F
T1=558F
T2=.1*558+.9*70=119 F
Hope this satidfies your needs.
However,if you need transient data we can linearize the basic equation for an easy solution or solve the nonlinear equation for t as a function of Tw.

 

[gavinr990]

Believe me zekeman, when I'm getting free advive I'm not going to complain about waiting a couple of days to get it! I'm thouroughly grateful for any help I cn get for this. Also, being British, I suppose its my fault you're using the old fashioned units as well!

The problem is really about trying to establish how long this building will remain usable as a temporary shelter during a fire. This is why I'm looking for a transient solution so I can see what the temperature is after say 10mins, 20mins, 1hour, 2hours etc. The plan is to get some formulae to put into a spreadsheet that will calculate temperature with respect to time.

Also, is it possible for you to provide some more details as to how you got from the numbers that I gave you to your post above? If its going to take a lot of time don't worry about it but I'd like to try and learn something here rather than just getting a free answer.

Thanks.

 

[zekeman]

"Also, being British, I suppose its my fault you're using the old fashioned units as well!"

Gavin,

I'll forgive you this one time and I will answer your question but I must tell you that without any information on the firewall and the other walls I guesstimated the thermal properties, so the steady state solution I posted is not accurate, but I believe a good starting point.

If you could give me a good description of especially the firewall and its opposite wall that contains the air gap plus an estimate of the walls and roof of the target area, I will refine the answer and walk you through the process.

 

[gavinr990]

We believe that all walls (including the one between the air gap and target area) are A60 firewalls. I'm treating them all as 40mm (1.57") steel with a density of 100kg/m3 and a thermal conductivity of 0.4 W/m/K (0.231 Btu/ft/F) and a heat capacity of 460 J/kg/K (0.110 Btu/lb/F). Unfortunately I don't have any details as to how either of these vary with temperature. Will this have asignificant impact on the results? Is this what you meant by thermal properties or is there more that I've missed?

 

[zekeman]

I have a thermal network which I put together a while ago and am trying to upload it. I drew it in "PAINT" and will give it a try here. If it works, I will explain it shortly.

 

[zekeman]

Gavin,
I had a few errors in the thermal network above and am uploading a revised copy.

I used a network analyzer ( SPICE) to get the transient and the numbers are not pretty.

The target room temperature rises in less than 1/2 hour to exceed 140 deg F with a steady state approaching 170 F. The gap

If you are still interested I will try to explain the network.

Just to make sure,all the walls are all madeup of steel clad
+40 mm of insulation.

 

[zekeman]

I also meant to say the temperature in the gap goes to over 500 F in even less time.

 

[gavinr990]

zekeman,

Yes, I had a feeling the numbers wouldn't look good. If you could try and explain how you got to the numbers that would be fantastic. I can also confirm that it is just 40mm insulation in the walls.

Thanks for all of this.

gav

 

[Yonathan]

you can do something simple,
put a small water tank near the wall (outside and inside) and measure the temperature raise over time, this will give you an estimation for the heat (per unit area assuming that the temperature along the wall is uniform),
from there on you can use a simple heat balance from any textbook. you also need estimate the amount of heat losses from the room (open windows, wall thickness, ...)

 

[gavinr990]

Thanks for the suggestion Yonathon. Unfortunately this is an offshore module and I'm working from the office and I don't think the client would be too keen to carry out my experiments for me!

Out of interest though, I don't quite understand the theory behind this. How would the water tanks allow me to estimate the rate of heat transfer from a fire?