KS500 Maths Problem - dh skeptical about the power of Mumsnet

[indignatio]

In the 18 years I have known him, dh has never admitted to being unable to solve a maths problem. However, today he has just such a problem. He was going to phone his Dad but he is on holiday and I'm not sure MiL would count this as the emergency for which she gave me the telephone numbers.
It really pains him to admit that he doesn't know the answer and laughed at me when I suggested Mumsnet. I have explained to him that the power of Mumsnet is so great that he will be given the answer - PLEASE let me me right

OK the problem is as follows - prepared by dh as I haven't a clue what he is going on about:

Temperature profile within a block of isotropic solid material

A block of material, cross section area A, density D, thickness X, specific heat capacity C and thermal conductivity k is heated on one side to achieve a rate of temperature rise of T degrees Celsius per second.

Assume no heat loss from the block and that the whole block starts from a single temperature.

Question 1
What is the temperature differential between the heated side and the unheated side as a function of time?

Question 2
Assuming that the steady state is reached, what will be the temperature differential (as a function of time) if the temperature of the heated surface is then held constant?

(Please use SI units)

I'm looking at this like I can solve it.

I can't. Of COURSE I can't [maths eejit]

But someone will be able to, just you watch.

I can't either but is there some info missing, e.g. if it were about waves you would have a formula prompt somewhere telling you velocity = frequency x wavelength.

Sorry no help here - far too much maffs for me

Is that really English?? Sounds like one of those scary questions on University Challenge when you've forgotten what the original question was and end up staring gormlessly at Paxman.

Thanks so far

Lio - dh says that it is the algebraic formula for calculating it that he needs

I have just read this to dh over the 'phone. He says it sounds like a pretty standard heat transfer question, but as he's busy emptying my dad's loft he can't quite get to it just yet. He also muttered something about having to do ot over the phone. Excuses, excuses. However, he is back this evening and will be happy to work it through if no-one else has provided a solution by then. There must be some other engineers round surely?

Fourier's law? Here's wiki: Fourier's law

Sorry - my books are all packed away somewhere, so am relying on memory which has been wrecked by kids, alcohol, all that good stuff.

Thanks - would like to talk to him this evening if I don't get the answer before then. Please send him my commiserations about having to be in a loft.

Indignatio's dh

Thanks for the link - I understand the starting point (the basic diffusion laws), but can't wade through the algebra to get to a final equation e.g.

Temperature differential = TDC exp(X^2/t)/k

indignatio's dh

Ahh. Is this an OU question? 'Fraid I'm just off to bed, but sneaky OU maths/chem/physics questions are my speciality. If it's not sorted by my morning I'd be delighted to have a go. Mumsnet will not let you down.

Is it OU? I'm doing OU at the moment, but not this sort of stuff!

Not OU - but a practical application for DH engineering work

Dh thinks he might be there with the answer - lots of scribbles on a page which make no sense to me - please someone else get there before he does !!

Oh that makes more sense, doesn't it? Wondered why you didn't have the starting equations.

You can cat me if you'd like. (Did this sort of thing in a previous lifetime - feel a bit self conscious about working it out here and not sure I could without my old books.)

Thanks Yajorome - how do I CAT (well how does DH as I am just the conduit)

OK - sorted out the CAT issue - message on the way to you

Sorry, you lost me at 'temperature profile'!

I opened this with one eyebrow raised at the very idea that someone's dh could be so silly as to underestimate the power of mumsnet! Someone WILL know this, you watch.

DH and yajorome are CATing - what have I started? Anyone else want a go at this? Apparently the first question is answered but not the second.
I will cut and paste the current thinking in case it makes more sense to you than it does to me

DH says:
I would like to know the temperature differential as a function of time. I expect it has an exponential term which starts at zero and approaches the final steady state value as time tends to infinity. The thermal conductivity (for solid materials) is k (Wm-1K-1). I have ignored the heat transfer coefficient to the surface of the block and assume that the surface is in close contact with another solid body that rises at T (c/s)

I think that I may have got the answer to the first question (steady state temperaure differential) by assuming that the whole block continues to rise in temperature at the same rate as the heated surface (T degrees per second). It must eventually reach this state, but I don't know how long it takes to do this.

Then I calculate the thermal mass of the block (=Area x thickness x density x specific heat capacity)
The heat flow Q into the block at the heated surface must be equal to the thermal mass times the rate of temperature rise (which is constant across the block)

Hence Q = AXDCT

Q = heat flow (watts)
A = Area (square metres)
X = Thickness (metres)
D = Density (kg/m3)
C = Specific heat capacity (J/kg/C)
T = Rate of temperature rise (C/s)

This heat flow is also equal to the heat conducted through the material at the surface of the block so

Q = kA x temperature gradient = AXDCT

The temperature gradient is constant through the block in steady state and is the temperature differntial from the heated surface to the non-heated surface divided by the thickness X

Rearranging all this I get

Differential temperautre = TDCx^2/k

I would be interested to know the equation for the transient leading up to this steady state, giving the temperature differntial as a function of time.

The same diffusion law should be OK for the second part (what happens when the temperature at the heated surface stops rising after the steady state temperature increase), but I don't really know how to get from that to the equation for the differential temperautre as a function of time. Again it probably has an exponential term that tends to zero as time tends to infinity

Lovin' this thread.

is this a secret agent code or a sign from mars? my eyes r a boggling at this! wot the f*k? r u all really so brainy? bh! it could be russian for all i know and i'm not daft - doing a degree so can't be that thick! amazing.

BBS - is your dh home ?

my dh thinks he knows the answer - see following - If anyone spots a fatal flaw in the maths then do please post

Many thanks

Steady state differential temperature = TDC(X^2)/(2k)

Thanks to Yajorome