Incremental energy problem

NASSAfellow · Feb 23, 2007

1. The problem statement, all variables and given/known data

A pond of water at 0°C is covered with a layer of ice 4.50 cm thick. If the air temperature stays constant at -11.0°C, how much time does it take for the thickness of the ice to increase to 9.00 cm?

____________hours

Hint: To solve this problem, use the heat conduction equation,

dQ/dt = kA (change in)T/x

and note that the incremental energy dQ extracted from the water through the thickness x is the amount required to freeze a thickness dx of ice. That is, dQ = LpA dx, where p is the density of the ice, A is the area, and L is the latent heat of fusion. (The specific gravity and thermal conductivity for ice are, respectively, 0.917 is 2.0 W/m/°C.)

2. Relevant equations

Heat conduction equation

3. The attempt at a solution

I am not sure how to do this, I have done extensive research but am still confused... please help. Also I don't study maths, so please help me with a word equation if possible.

Many Thanks,
Nasser

Saketh · Feb 23, 2007

This is a fun problem.

Put your expressions together (I'm not doing anything but math), and you get the following.
[tex]
LA \,dx = \frac{kA \Delta T}{x} dt
[/tex]

Separate variables, and integrate. You've already done all the physics.

NASSAfellow · Feb 24, 2007

I am not sure what this all means- I am doing AS level physics and I don't do maths, would it be possible to give me a word equation?

NASSAfellow · Feb 24, 2007

Can someone please help, or is this too challenging?

cristo · Feb 24, 2007

Is A assumed to be constant? If so, you should be able to plug dQ=LA dx into dQ/dt = kA T/x and integrate.

NASSAfellow · Feb 24, 2007

I don't do maths... i.e. Integration means nothing to me... sorry

cristo · Feb 24, 2007

What do the "d"s mean in your equation then, if you are not using calculus? Are they meant to be [itex]\Delta[/itex]; denoting "change in"?

edit: you didn't answer my question re A being constant.

NASSAfellow · Feb 24, 2007

Yes, I think so, Webassign just gave me the question to do, and the formula to use.

NASSAfellow · Feb 24, 2007

[tex]LA \,dx = \frac{kA \Delta T}{x} dt[/tex]
Someone gave me this^^, but still confused

cristo · Feb 24, 2007

Ok, I've seen your other thread-- you're studying AS Physics, so I presume you did moderately well in GCSE maths. Can you see how the equation you give above comes from your two equations in your original post? (simply substitute dQ from the second equation into the first).

Now, dx means "change in x", so rewrite it as dx=x-x₀ where x₀ is the value of x when t=t₀=0 (since we will take the original time as t=0). Thus we write dt=t (since t₀ is zero).

This should make your equation easier to handle. Try to rearrange it to make t the subject of the equation.

NASSAfellow · Feb 24, 2007

okay... I have got this far...

dt=Ldx^2/kT

Is this correct?

cristo · Feb 24, 2007

No, dx is not the same as x. Let's replace the d's in the equations in post #6, as I suggested in #7. This will give [tex]LA(x-x_0) = \frac{kA T}{x} t[/tex]. Can you rearrange this?

NASSAfellow · Feb 24, 2007

I have confused myself with density and difference in etc...

Is it this? Where is density?

Change in time = Latent heat of fusion*change in x^2 / thermal conductivity of ice * 11

NASSAfellow · Feb 24, 2007

Wait hang on, just recieved your msg.. let me do it

hage567 · Feb 24, 2007

There should be a term for density in your equation for dQ = LAdx in your original post. You sort of mentioned it in the sentence, but left it out of your equation.

cristo · Feb 24, 2007

NASSAfellow said: ↑

I have confused myself with density and difference in etc...

Is it this? Where is density?

You said this:

dQ = LA dx, where is the density of the ice, A is the area, and L is the latent heat of fusion.

Now, have you missed out the density term from the equation? My guess is that you copied the question incorrectly, and there should be a ρ somewhere.

NASSAfellow · Feb 24, 2007

Of course! I have seen it now! Sorry, when I pressed copy on the pc it did not copy:

dQ= LpA , where p is density of ice!

NASSAfellow · Feb 24, 2007

Also the delta sign did not copy for the heat conduction equation

dQ/dt = kA (Change in) T/x

NASSAfellow · Feb 24, 2007

So does this mean I am lost?

Cristo...?

NASSAfellow · Feb 24, 2007

I am lost, please help Cristo!