Heat transfer: transient state

[Biker]

Hello,

I have studied about heat transfer through conduction only in steady state but I wondered about this problem that I created.

Suppose you have a box that is insulated from all sides but Suppose a constant heat flow from that wall. Inside that box you have M kg of water at the same temperature initially as the wall.

lets say the temp at the inner wall is T2 and from outside T1. T1>T2. Calculate the time it takes for the water to reach a certain temp.

Or you can change the problem to having T1 to be constant at all times.

In both cases, I needed the distribution of temperatures along the thickness of the wall. In basic terms, How do you solve this kind of problems?

Excuse me if I am saying non-sense stuff. We are only required to solve steady state questions but I just wonder.

 

[Nidum]

Look at a simpler problem first :

A constant temperature source + an intermediate layer of heat conductive material + a metal block . All lined up in series and fully insulated from any external heat sources or sinks .

With the temperature source being at a high temperature and both intermediate layer and metal block being initially at a uniform low temperature can you describe in words what you think will happen as heat from the source flows into the layer and block ?

 

[Biker]

Look at a simpler problem first :

A constant temperature source + an intermediate layer of heat conductive material + a metal block . All lined up in series and fully insulated from any external heat sources or sinks .

With the temperature source being at a high temperature and both intermediate layer and metal block being initially at a uniform low temperature can you describe in words what you think will happen as heat from the source flows into the layer and block ?

Yes, The heat starts to flow from the high to low and the metal blocs and the layer starts to heat up. So some of the flow is heating up the metal to a certain temp and the rest is heating the layer to the same temp.

The flow of the heat gets reduced because now the temp difference is lower which continues to decreases until the temp on the two sides is equal.

One question though, Doesn't it take some fairly small time for the layer and the metal block to have the same temp? How do you take that into account?

 

[Chestermiller]

Suppose in your original problem description, the insulating wall is fairly thin and its heat capacity is very low, so the temperature profile within the wall responds virtually instantaneously to the temperatures on its two sides (so that its temperature profile is always linear). Suppose also that there is a stirrer within the water so that its temperature is always uniform at T, even all the way up to the wall. What is the instantaneous rate of heat flow to the water if its temperature at the inner wall is T, and its temperature at the outer wall is T2?

 

[Biker]

Suppose in your original problem description, the insulating wall is fairly thin and its heat capacity is very low, so the temperature profile within the wall responds virtually instantaneously to the temperatures on its two sides (so that its temperature profile is always linear). Suppose also that there is a stirrer within the water so that its temperature is always uniform at T, even all the way up to the wall. What is the instantaneous rate of heat flow to the water if its temperature at the inner wall is T, and its temperature at the outer wall is T2?

Sorry for being late.

$\frac{KA}{L} (T_2- T) = m_{\text{water}} c \frac{dT}{dt} + \frac{m_{\text{wall}} c \frac{dT}{dt}}{2}$

So I assumed that at any time t both the water and the wall have the same temperature.

If I solve this derivative of temperature, I get a exponential function which seems to be true. Is that correct?

Again, Sorry for being late.