Thermal Conductivity and Heat Transfer in a Hot Water Bottle

[CAF123]

Homework Statement

2)A certain heat source (a heat reservoir or thermal reservoir), is always at temperature $T_0$. Heat is transferred from it through a slab of thickness $L$ to an object which is initially at a temperature $T_1 < T_0$. The object, which is otherwise thermally insulated from its surroundings, has a mass m = 0.5 kg and a specific heat capacity $c = 4 \times 10^{3} J kg^{-1}K^{-1}$. Heat is conducted through the slab at a rate (in J s^{-1}) specified by the
formula $KA((T_0 - T)/L),$ where K is the thermal conductivity of the slab, A is the area of the slab through which heat is transferred and T is the instantaneous temperature of the object.

a)Show that provided certain assumptions are made, $KA((T_0 - T)/L)\Delta t = mc \Delta{T}$
b)Show that for small t, and hence small ΔT, the above equation can be rearranged and
integrated to give $$T_2 - T_1 = (T_0 - T_1)\left(1 - \exp\left(-\frac{KA(t_2 - t_1)}{Lmc}\right)\right)$$

The Attempt at a Solution

2) a)Assuming all energy gained by the slab is then gained by the object ( and there is then no losses because of insulation), $$\frac{KA(T_o - T)}{L} \Delta t = cm (T + \Delta T - T)$$ and result follows.
b)Integrate $$\frac{KA}{L} \int_{t_1}^{t_2} dt = cm \int_{T_1}^{T_2} \frac{dT}{T_o - T}$$
which gives $$\frac{KA}{Lmc}(t_2 - t_1) = \ln\left(\frac{T_o - T_1}{T_o -T_2}\right).$$ It is given that $T_o > T_1$ so the numerator of the log is definitely positive. However, I am not so sure about the denominator. Since there are no heat losses, I think there may exist a temperature $T_2$ where $T_2 >T_o$.

Thank you.

 

[voko]

I think it is quite obvious that $T_0$ is the highest temperature possible anywhere in the system.

 

[Chestermiller]

Your final solution is correct. Just work it into the same mathematical form that they ask for.

 

[CAF123]

Hi voko,

I think it is quite obvious that $T_0$ is the highest temperature possible anywhere in the system.

Did you take the system to be reservoir + slab + object? If so, yes, it seems sensible that $T_o$ is the highest temperature since there is no external source of heat.Right?

However, if you take the object as the system, then there does exist an external source of heat so why is it that the object can never get hotter than the reservoir? (given that we don't know anything about the object)?

 

[voko]

However, if you take the object as the system, then there does exist an external source of heat so why is it that the object can never get hotter than the reservoir? (given that we don't know anything about the object)?

Your equations are already based on the assumption that there is just one source of heat, with a particular temperature. You can't have that both ways.

 

[CAF123]

You can't have that both ways.

What do you mean by this?

 

[voko]

What do you mean by this?

You cannot model your system as if it had one source with a particular temperature, and at the same time model it as if it had unknown sources.

 

[CAF123]

You cannot model your system as if it had one source with a particular temperature, and at the same time model it as if it had unknown sources.

If I take the system to be reservoir + slab + object, then the maximum temperature of this system is $T_0$ since there are no external heat sources. There exists an internal heat source (reservoir) only. Is this reasoning correct for why the temperature of the object can never exceed $T_o$?

 

[voko]

Since the system is insulated, the heat to the object can only be passed from the slab or from the reservoir via the slab. If the object is already as hot as the reservoir, can any more heat be passed to it by simple conduction? Why?

 

[CAF123]

Since the system is insulated,

What are you taking as the system here? Just the object?

the heat to the object can only be passed from the slab or from the reservoir via the slab. If the object is already as hot as the reservoir, can any more heat be passed to it by simple conduction? Why?

I would say no. If we assume the slab is a diathermal wall between the object and the reservoir, then the object and reservoir will settle to the same temperature. Since the reservoir is always at a temperature of $T_0$, by the zeroth law, the object may not become hotter.

 

[voko]

What are you taking as the system here? Just the object?

The entire system, with the reservoir and the conductor. The object alone cannot be analyzed meaningfully, I think.

 

[CAF123]

The entire system, with the reservoir and the conductor. The object alone cannot be analyzed meaningfully, I think.

Okay, thank you. Is my analysis correct in my previous post? Also, given that the system is object, reservoir and slab, would it not be correct to state that given no external heat source, it is not possible for the temperature of the system to exceed $T_0$?

 

[voko]

Okay, thank you. Is my analysis correct in my previous post?

It is correct in the sense that at equilibrium, the temperature will have to be $T_0$ everywhere. But that does not mean, per se, that the temperature somewhere cannot be higher than $T_0$ while on the way to equilibrium.

That requires some other proof.

Also, given that the system is object, reservoir and slab, would it not be correct to state that given no external heat source, it is not possible for the temperature of the system to exceed $T_0$?

That is correct, but you have omitted some important details. Such as what law of thermodynamics is at play here.

 

[CAF123]

That requires some other proof.

What sort of proof? Can you elaborate?

That is correct, but you have omitted some important details. Such as what law of thermodynamics is at play here.

The first law of thermodynamics. The system is isolated so no heat exchange to the surroundings.

 

[voko]

What sort of proof? Can you elaborate?

Sorry, I cannot elaborate on the proof. You will have to :)

You have to prove that the temperature in this system cannot exceed $T_0$ even before it reaches equilibrium.

The first law of thermodynamics. The system is isolated so no heat exchange to the surroundings.

That is not enough to make $T_0$ the max temperature. Why can the slab not transmit more heat to the object at any given temperature, say $T_0$, and so make it even hotter?

 

[Chestermiller]

This is a standard problem in heat transfer. The system is the mass. Heat is being conducted from the reservoir to the mass through the slab. The formulation of the problem implies that the thermal inertia (ρC_p) of the slab can be neglected so that the temperature gradient within the slab is always linear. The formulation also implies that the thermal conductivity of the mass is very high, so that, throughout the process, the temperature within the mass is essentially uniform. The temperature within the mass as well as within the slab can, at no location and no time, be higher than the reservoir temperature. This is implicit in the problem formulation as well as the problem solution.

 

[CAF123]

Sorry, I cannot elaborate on the proof. You will have to :)

You have to prove that the temperature in this system cannot exceed $T_0$ even before it reaches equilibrium.

The temperature of the reservoir is always $T_0$. Assuming that the temperature of the system can exceed $T_0$, then that means there must exist more heat energy than what we had orginally, (the reservoir is the only source of heat) so contradicting the first law. (Hmm.. is that right?) Or in an isolated system, entropy never decreases. So if the temperature exceeded $T_0$ and then at a later t decreased, this would contradict the second law of thermodynamics.

That is not enough to make $T_0$ the max temperature. Why can the slab not transmit more heat to the object at any given temperature, say $T_0$, and so make it even hotter?

I think the question you pose is exactly why I thought that there may exist a $T > T_0$ in the first place. Although with a bit of further reading, is it because in an isolated system, as we have here, equilibrium is at maximum entropy and so the state of the system cannot become more disordered? (which would happen if the object was allowed to become hotter?)

Sorry for the late response, but I had to read up on the second law before I could answer.

 

[voko]

The temperature of the reservoir is always $T_0$. Assuming that the temperature of the system can exceed $T_0$, then that means there must exist more heat energy than what we had orginally, (the reservoir is the only source of heat) so contradicting the first law. (Hmm.. is that right?)

Nope. Because the reservoir is at a constant temperature no matter how much heat it discharges, it is effectively an infinite energy supply. So conservation of energy won't work here.

Or in an isolated system, entropy never decreases. So if the temperature exceeded $T_0$ and then at a later t decreased, this would contradict the second law of thermodynamics.

Imagine an anvil and a slab in some isolated space, both at an equal temperature. The anvil is flying at a constant speed towards the slab. When the anvil hits the slab, some parts of the anvil and the slab will get hotter. But then they will get colder. No contradiction.

I think the question you pose is exactly why I thought that there may exist a $T > T_0$ in the first place. Although with a bit of further reading, is it because in an isolated system, as we have here, equilibrium is at maximum entropy and so the state of the system cannot become more disordered? (which would happen if the object was allowed to become hotter?)

Let's put it this way: can heat be transferred from some object to some other object, if the the former object's temperature does not exceed the latter's, purely by conduction, i.e., without any mechanical work? Which law of thermodynamics is involved here?

 

[Chestermiller]

The temperature of the reservoir is always $T_0$. Assuming that the temperature of the system can exceed $T_0$, then that means there must exist more heat energy than what we had orginally, (the reservoir is the only source of heat) so contradicting the first law. (Hmm.. is that right?) Or in an isolated system, entropy never decreases. So if the temperature exceeded $T_0$ and then at a later t decreased, this would contradict the second law of thermodynamics.


I think the question you pose is exactly why I thought that there may exist a $T > T_0$ in the first place. Although with a bit of further reading, is it because in an isolated system, as we have here, equilibrium is at maximum entropy and so the state of the system cannot become more disordered? (which would happen if the object was allowed to become hotter?)

Sorry for the late response, but I had to read up on the second law before I could answer.

I'm having trouble understanding what you are saying. T (=T₂) > T₀ does not satisfy the equation for the solution (in either form). As long as t₂>t₁ (which it must, since t₁ is the initial time), T < T₀.

 

[CAF123]

Let's put it this way: can heat be transferred from some object to some other object, if the the former object's temperature does not exceed the latter's, purely by conduction, i.e., without any mechanical work? Which law of thermodynamics is involved here?

The zeroth law.

 

[voko]

The zeroth law.

There were two questions, but you only answered the second one. Regardless of the answer to the first one, I cannot see the significance of the zeroth law here.

 

[CAF123]

Hi Chestermiller,

I'm having trouble understanding what you are saying. T (=T₂) > T₀ does not satisfy the equation for the solution (in either form). As long as t₂>t₁ (which it must, since t₁ is the initial time), T < T₀.

What I should have said is that I incorrectly thought that $T_2$ could be greater than $T_0$. As $t_2 - t_1 \rightarrow \infty, T_2 \rightarrow T_0$.

What I am trying to figure out with voko is why for $t > t_1$, $T < T_0$.

 

[CAF123]

Let's put it this way: can heat be transferred from some object to some other object, if the the former object's temperature does not exceed the latter's, purely by conduction, i.e., without any mechanical work?

No, heat follows from a hotter body to a colder body.

Which law of thermodynamics is involved here?

Two laws are perhaps relevant: The zeroth law: If the bodies are in thermal contact, heat flows from hot body to cold body until thermal equilibrium.
Heat from hot to cold is essentially the statement of the second law.

 

[voko]

The zeroth law: If the bodies are in thermal contact, heat flows from hot body to cold body until thermal equilibrium.
Heat from hot to cold is essentially the statement of the second law.

Heat from hot to cold is the statement of the second law, as formulated by Clausius. The zeroth law says nothing about the direction of the heat flow.

Do you understand now why $T_2$ is never greater than $T_0$?

 

[CAF123]

Heat from hot to cold is the statement of the second law, as formulated by Clausius.

I have yet to reach Clausius' statement in my book.

Do you understand now why $T_2$ is never greater than $T_0$?

Maybe, I'll tell you my thoughts: If $T_2$ was greater than $T_0$, then that would mean heat would flow from the object to the reservoir, thus increasing the temperature of the reservoir. We know the temperature of the reservoir is fixed at $T_0$, so $T_2$ cannot be greater than $T_0$. Yes?

 

[voko]

Maybe, I'll tell you my thoughts: If $T_2$ was greater than $T_0$, then that would mean heat would flow from the object to the reservoir, thus increasing the temperature of the reservoir. We know the temperature of the reservoir is fixed at $T_0$, so $T_2$ cannot be greater than $T_0$. Yes?

If the object somehow gets hotter than the reservoir, then the reservoir by the virtue of its constant temperature will eagerly absorb any heat that is given to it.

You need to analyze how the object, in the given system, could possibly get hotter than the reservoir.

 

[CAF123]

You need to analyze how the object, in the given system, could possibly get hotter than the reservoir.

Since the system is isolated, any heat or matter external influences would not affect the object. In the given system, therefore, the only way for the object to get hotter is if there was an external agent doing work on the object.

 

[voko]

Since the system is isolated, any heat or matter external influences would not affect the object. In the given system, therefore, the only way for the object to get hotter is if there was an external agent doing work on the object.

If your system is the just the object itself, it cannot be meaningfully analyzed, as I said earlier.

If your system includes the slab and the reservoir, then your conclusion is manifestly wrong.

 

[CAF123]

If your system is the just the object itself, it cannot be meaningfully analyzed, as I said earlier.

Just out of interest, why would this be the case?

If your system includes the slab and the reservoir, then your conclusion is manifestly wrong.

My aim is to show that $T_2$ cannot be greater than $T_0$. Since I know (but have yet to understand) why, the object cannot get hotter than $T_0$ by internal influences. So the only way for it to get hotter is if there was some work done on it. I am taking the system as all three components. Why is it incorrect?

 

[Chestermiller]

Heat flows from (a) hot to cold or (b) cold to hot?