# Re: Heat transfer - urgent help requested

"Robert Higgins" <robert_higgins_61@xxxxxxxxxxx> wrote in message news:bb4cacf6-c4c3-44a0-a124-4514f7cf814a@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Mar 13, 5:48 am, "Androcles" <Headmas...@xxxxxxxxxxxxxxxxxx> wrote:
"flint" <fl...@xxxxxxx> wrote in message

news:60bec7e0-13ed-43f3-8d96-d7dfeda07225@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

> HI,

> Can anyone help with this? It's way beyond my capability. It's not a
> homework problem but I'd really appreciate some assistance with this.

> I need to know how long it would take for the temperature in a length
> of copper strip to distribute uniformly. By way of example - I have a
> strip of copper 20 mm wide and 0.5 mm thick and 120 mm long. The strip
> is exposed at each end for about 10mm of it's length. The middle 100
> mm are well insulated. For reason of clarity assume perfect
> insulation.

> The whole strip is stabilised at a temperature of 20 deg C. If the
> strip is now exposed to an ambient temperature of 30 deg C, i.e. by
> moving the strip to another room, how long would it take for the
> centre of the strip to warm to 30 deg?

> I know I'm asking a lot but how do you calculate this??

the rod is x long and the cross-sectional area can be any shape.

<------------------------|------------------------>

Now make the rod a uniform temperature, say d degrees.

d-----------------------d-----------------------d

If we raise the temperature at one end by 2d degrees

2d-----------------------d-----------------------d

then at some later time t the middle will become 1.5d

2d---------------------1.5d---------------------d

If we divide by d then we have

2---------------------1.5---------------------1

If we then subtract 1, we have

1---------------------0.5---------------------0

Eventually this becomes
1---------------------1---------------------1
but we are interested in finding the time for the middle temp
to rise, so let's go back to

1---------------------0.5---------------------0

Now we can say the velocity of the temperature is x/t,
or t = x/v, but we don't know v, which would depend on
the material. If it were (say) glass it would take much longer
than copper, but eventually we'd arrive at the same situation.
That should give you a starting point in solving your problem.

It says a lot about your reputation as an "engineer" that you did not
recognize the heat equation (with Dirichlet boundary conditions) on
sight. The heat equation is the simplest PDE in all of mathematical
physics.
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