Clarification about Heat Conduction equations

[WK95]

How do I relate Fourier's Law of Heat Conduction for 1-D Heat Conduction with the Heat Conduction Equation in a large plane wall and energy balance equation?

Fourier

Energy Balance
energy in - energy out = system energy change
rate of energy in - rate of energy out = rate of system energy change

Heat Conduction Equation

So is Q_dot.cond is the rate of heat transfer through a distance x by conduction due to a temperature difference. That means there is heat energy flowing from into the body from the higher temperature side and exiting through the lower temperature side. Does that mean Q_dot.cond = Q_dot.x + Q_dot.x+deltax (The first two terms of the Heat Conduction equation except they are summed) which is the sum of energy in one direction from high to low.

Asked another way, if I were considering heat conduction through some body, how is the energy balance equation applied to such a body?

Trying to approach this confusion intuitively, I'm imagining a fully enclosed metal pot of really hot water but with water that is not in motion (no convection) surrounded by a cooler body of water in an insulated metal pot. At some instant, the hot water in the inner pot has a temperature T_I and the hot cooler water in the outer pot has a temperature T_O. due to the temperature difference, I can use Fourier's equation to determine the rate of heat transfer by conduction from the hot water through the pot to the outside air. But where I'm confuse about the topic of conduction is when I consider how the energy of the pot with the energy balance equation. Heat has to flow into the pot then out of it into the air due to the temperature difference. At all times, I'd imagine that (T_O+T_I)/2 is constant and as the hot water cools and the cooler water warms until equilibrium is reached. So that means the metal pot separating the two bodies of water isn't changing in temp. In other words, for that pot, the rate of energy of the pot system separating the two bodies of water is equal to zero.

So going back to the original question, since the rate of energy change of the pot system is equal to zero and there is not heat generation, Q_dot.in - Q_dot.out=0. I also know that Q_dot.conduction=-kA(dT/dx) and heat conduction through some body has energy going in then energy out. So is it valid to say that Q_dot.conduction=Q_dot.in + Q_dot.out, e.g. with the two terms being or equal magnitude?

 

[Mapes]

Asked another way, if I were considering heat conduction through some body, how is the energy balance equation applied to such a body?

It sounds like you want to determine the temperatures of the regions of water and the pots, and possibly their internal temperature distributions, over time.

In general, for a sufficiently simple geometry and conditions (e.g., a single symmetric shape), the differential Fourier's equation can be integrated (or solved another way) to produce the temperature profile. For more complicated problems, you can solve the differential equation(s) numerically or use the finite element method. In your case, you might have conduction equations for the hot water, the hot pot, the cool water, and perhaps the cool pot if you can't assume its temperature is uniform. At interfaces, the temperatures are equal, and energy balance equations need to be satisfied.

If you assume the water is well mixed with a uniform temperature, your problem simplifies somewhat. You can treat the water as simply two temperatures on either side of the inner pot wall. The temperature in the inner pot wall would then change over time. (The profile would be linear with a plane wall but is instead somewhat nonlinear because of the curving wall of the pot. However, you might wish to ignore this subtlety.)

Incropera and deWitt's book on heat transfer (and most heat transfer textbooks) covers all of these issues pretty clearly with examples.

So is it valid to say that Q_dot.conduction=Q_dot.in + Q_dot.out, e.g. with the two terms being or equal magnitude?

No, you wouldn't add the terms. It's the same energy moving in different locations, one on one side of the wall and the other side of the wall.

I'd imagine that (T_O+T_I)/2 is constant

This is also not correct unless the amounts of water are equal, the thermal mass of the pots is insignificant, zero heat transfer occurs to the outside, and you assume the specific heat of water is independent of temperature.