Heat transfer: transient state

In summary: 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?Yes, the exponential function is correct.
  • #1
Biker
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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.
 
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  • #2
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 ?
 
  • #3
Nidum said:
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?
 
  • #4
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?
 
  • #5
Chestermiller said:
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.
 
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Likes Chestermiller

Related to Heat transfer: transient state

1. What is heat transfer in the transient state?

Heat transfer in the transient state refers to the process of transferring thermal energy between two or more objects that are not in thermal equilibrium. This means that the temperature of the objects is changing over time, and heat is being transferred between them until they reach a state of equilibrium.

2. What factors affect heat transfer in the transient state?

The rate of heat transfer in the transient state is affected by several factors, including the temperature difference between the objects, the thermal conductivity of the materials, the surface area of contact between the objects, and the time duration of the heat transfer.

3. How is heat transfer in the transient state different from steady-state heat transfer?

Steady-state heat transfer refers to the transfer of heat between objects that have reached a state of thermal equilibrium, where the temperature of each object remains constant over time. In contrast, heat transfer in the transient state involves a changing temperature difference between objects and the transfer of heat until they reach equilibrium.

4. What are some common examples of heat transfer in the transient state?

Some common examples of heat transfer in the transient state include heating or cooling a pot of water on a stove, melting ice cubes in a drink, and the heating and cooling of buildings throughout the day as the temperature outside changes.

5. How is heat transfer in the transient state calculated?

Heat transfer in the transient state can be calculated using Fourier's law, which states that the rate of heat transfer is proportional to the temperature difference, thermal conductivity, and surface area of contact between the objects. Additionally, the heat transfer coefficient, which takes into account the properties of the materials and the nature of the heat transfer, is also used in calculations.

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