System of Partial Differential Equations

[Bashyboy]

System of PDEs--Heat Equation For Two Objects

Hello everyone,

Before is a system of partial differential equations; to be specific, it is this system:

$\frac{\partial U_A }{\partial t} = - \frac{k_B}{k_A} \alpha_A \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)$

and

$\frac{\partial U_B }{\partial t} = \alpha_B \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)$

I am not very certain as to how to solve this--as a matter of fact, I do not even know if it is possible to solve this. So, does this system have a solution $U_A(x,y,z,t)$ and $U_B(x,y,z,t)$? And if it does, could someone help me with solving it, such as providing hints or suitable reading materials? I would certainly appreciate it.

 

[Bashyboy]

I think I might have figured it out: would I solve the second partial differential equation, and then use the function that I would find during the process, and then use that to solve the first partial differential equation?

 

[pasmith]

Hello everyone,

Before is a system of partial differential equations; to be specific, it is this system:

$\frac{\partial U_A }{\partial t} = - \frac{k_B}{k_A} \alpha_A \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)$

and

$\frac{\partial U_B }{\partial t} = \alpha_B \left( \frac{\partial^2 U_B}{\partial x^2} + \frac{\partial^2 U_B}{\partial y^2} + \frac{\partial^2 U_B}{\partial z^2} \right)$

I am not very certain as to how to solve this--as a matter of fact, I do not even know if it is possible to solve this. So, does this system have a solution $U_A(x,y,z,t)$ and $U_B(x,y,z,t)$? And if it does, could someone help me with solving it, such as providing hints or suitable reading materials? I would certainly appreciate it.

The second equation is the heat equation.

The second equation gives $\nabla^2 U_B = \frac{1}{\alpha_B} \frac{\partial U_B}{\partial t}$, which on substitution into the first equation gives $$\frac{\partial U_A}{\partial t} = -\frac{k_B \alpha_A}{k_A \alpha_B} \frac{\partial U_B}{\partial t}$$ from which it follows that (assuming $\frac{k_B \alpha_A}{k_A \alpha_B}$ to be independent of time)
$$\frac{\partial }{\partial t} \left(U_A + \frac{k_B \alpha_A}{k_A \alpha_B} U_B\right) = 0.$$
The quantity in the brackets is thus a function of space only, and is therefore determined by the initial conditions.

 

[Bashyboy]

So, the functions $U_A$ and $U_B$ are both independent of time? Really? Hmmm...That is not what I expected. Perhaps the way in which I am approaching this problem is wrong.

 

[Bashyboy]

What I am trying to do is calculate the time it takes for two objects call them A and B, which are initially at different temperatures, to reach thermal equilibrium. My starting assumption was that the rate at which heat flows from object B is equal to negative the rate at which heat leaves A. This idea could also be expressed in the heat flux equation (Fourier's heat conduction equation, I think):

$\dot{\mathbf{q}}_B = - \dot{\mathbf{q}}_A$

From this I could get

$\frac{\partial U_A}{\partial x} = - \frac{k_B}{k_A} \frac{\partial U_B}{\partial x}$

$\frac{\partial U_A}{\partial y} = - \frac{k_B}{k_A} \frac{\partial U_B}{\partial y}$

$\frac{\partial U_A}{\partial z} = - \frac{k_B}{k_A} \frac{\partial U_B}{\partial z}$

Differentiating each one with their respective spatial variable, we get

$\frac{\partial^2 U_A}{\partial x^2} = - \frac{k_B}{k_A} \frac{\partial^2 U_B}{\partial x^2}$

and so forth.

From these premises, I set up the heat equation for both objects, A and B, and made the substitutions using the above relationships.

 

[Bashyboy]

In light of this new information I have provided, should this thread be moved into, say, the classical physics thread?

 

[AlephZero]

So, the functions $U_A$ and $U_B$ are both independent of time?

No. It just says that If the temperature of $U_A$ changes, then $U_B$ changes in proportion, but in the opposite direction.

With the benefit of hindsight (a wonderful exact science!), that should be obvious from what you are modeling:

My starting assumption was that the rate at which heat flows from object B is equal to negative the rate at which heat leaves A.

But you probably need to think a bit more about your PDEs, because if A and B are separate objects, the two equations apply to different regions in space corresponding to the two objects, and it's not obvious how they describe how heat gets from one object to the other. That could be conduction if they have a common boundary, or radiation, or convection involving something else apart from A and B.

 

[Bashyboy]

So, would you say that I am approaching this problem incorrectly?

With the benefit of hindsight (a wonderful exact science!), that should be obvious from what you are modeling:

But you probably need to think a bit more about your PDEs, because if A and B are separate objects, the two equations apply to different regions in space corresponding to the two objects, and it's not obvious how they describe how heat gets from one object to the other. That could be conduction if they have a common boundary, or radiation, or convection involving something else apart from A and B.

Yes, the two objects are distinct, each with distinct physical properties, such as thermal conductivity, heat capacity, and density. I am also assuming the two objects form an isolated system, so that any heat leaving A will eventually reach B.

 

[Bashyboy]

What if I consider the two objects as one 'composite object', and use one heat equation. So, if object A is defined by the equation $S_1 = g_1(x,y,z)$, and object B by the equation $S_2 = g_2(x,y,z)$, then we can consider the composite object $S_1 \cap S_2$. For the initial conditions, we can constrain the temperature of the $S_1$ to be $T_1$, and $S_2$ to be $T_2$, where $T_1 > T_2$; that is, $u(x,y,z,0) = T_1 ~\forall x,y,z \in S_1$ and $u(x,y,z,0) = T_2 \forall x,y,z \in S_2$. Then, at time $t = \tau$, the time at which $S_1$ and $S_2$ come to thermal equilibrium, $u(x,y,z,\tau) = T_{EQ} ~ \forall x,y,z \in S_1 \cap S_2$.

I am not sure if my mathematical statements are correct, but hopefully the idea is conveyed.

One problem I see, however, is that will have different values of $\alpha$ for S1 and S2

 

[AlephZero]

Yes, the two objects are distinct, each with distinct physical properties, such as thermal conductivity, heat capacity, and density. I am also assuming the two objects form an isolated system, so that any heat leaving A will eventually reach B.

We didn't know that when post #2 was made. For a system of equations (ODEs, PDEs or anything else) it's a reasonable assumption all the equations apply to the same region in space and time. Post #2 would be physically meaningful if for example you sprayed liquid droplets into a gas at a different temperature.

I recommend you keep the physics clear in your mind, rather than getting lost in inventing mathematical notation. The equations in your OP don't really say how heat gets from object A to object B. They seem to be trying to say "the rate of temperature change of B at one point in space is somehow related to the heat flux in A at some other point". But if you have one equation containing $U_A(x,y,z,t)$ and $U_B(x,y,z,t)$, then $x$, $y$, $z$ and $t$ represent the same point in space and the same moment in time.

 

[Bashyboy]

Hmm, I definitely see the difficulty, now. What would you suggest I do?

 

[Chestermiller]

Post #5 explains it all, and you are definitely on the right track.

Let's take a specific example. Suppose you have two identical slabs of different materials, one at temperature T_A0 and the other at T_B0. Slab A runs from x = -L to x = 0, and slab B runs from x = 0 to x = +L. The slabs are insulated at x = -L and at x = +L, respectively. At time t equal to zero, the two slabs are brought into thermal contact, and heat is allowed to flow between them. You would like to find the temperature distribution in the slabs as a function of time and spatial position. Does this pretty much capture what you are looking for? If so, you might first be interested in considering the simpler problem in which the two slabs are made of the same material.

Chet

 

[Bashyboy]

Chestermiller: Yes, that appears to be what I am trying to do. My question is, in the scenario you have just proposed, is it possible to calculate the time at which the objects equilibrate? Also, would it still be wise to set up a heat equation for each slab?

 

[Chestermiller]

Chestermiller: Yes, that appears to be what I am trying to do. My question is, in the scenario you have just proposed, is it possible to calculate the time at which the objects equilibrate?

Yes, effectively. The mathematics would predict that they would never quite reach the exact same temperature, but, you can readily calculate the finite amount of time it would take for them to reach say 0.1% of the original temperature difference.

Also, would it still be wise to set up a heat equation for each slab?

Yes, definitely. That's what I would do (and have done many times). Let's see your set up of this problem, including the boundary conditions (particularly the boundary condition at the interface between the slabs).

After you get the problem properly formulated, I can give you some thoughts an how to solve the equations.

Chet

 

[Bashyboy]

I can't quite figure this out. The heat equation for A would be $\frac{\partial U_A}{\partial t} = \alpha \frac{\partial^2 U_A}{\partial x^2}$, where the unknown function $U_A(x,t)$ must satisfy the initial condition $U_A(x,0) = T_0~~\forall x \in [-L,0]$. The object B would have a similar heat equation, and the initial condition for this would be $U_B(x,0) = T_1 ~~\forall x \in [0,L]$. At the boundary between both objects, $x=0$, heat flow will occur, the heat flow being described by the heat flux equation $\dot{q}_{A,x} = - \dot{q}_{B,x}$.

I am not sure how to implement these ideas, though.

 

[Chestermiller]

I can't quite figure this out. The heat equation for A would be $\frac{\partial U_A}{\partial t} = \alpha \frac{\partial^2 U_A}{\partial x^2}$, where the unknown function $U_A(x,t)$ must satisfy the initial condition $U_A(x,0) = T_0~~\forall x \in [-L,0]$. The object B would have a similar heat equation, and the initial condition for this would be $U_B(x,0) = T_1 ~~\forall x \in [0,L]$. At the boundary between both objects, $x=0$, heat flow will occur, the heat flow being described by the heat flux equation $\dot{q}_{A,x} = - \dot{q}_{B,x}$.

I am not sure how to implement these ideas, though.

Nice job so far. You don't have to use separate subscripts for the temperatures of A and B, because the temperature will be continuous at the interface. But you do have to use separate subscripts for the thermal diffusivities of A and B, α_A and α_B, because the thermal properties in the two regions are different.

In addition to the temperature being constant at the interface, the heat flux must also be constant at the interface:
$$q=-k_A\left(\frac{∂U}{∂x}\right)_{x=0^-}=-k_B\left(\frac{∂U}{∂x}\right)_{x=0^+}$$
where the k's are the thermal conductivities. Note that, although the heat flux is constant across the interface, the temperature gradient is not.

There are also two other boundary conditions that need to be included in the formulation. These are the insulation boundary conditions at x = -L and x = +L:
$$-k_A\left(\frac{∂U}{∂x}\right)_{x=-L}=0$$
$$-k_B\left(\frac{∂U}{∂x}\right)_{x=+L}=0$$

This completes the formulation of the problem. I'll give you a chance to look this over, and then respond if you have any questions. Then we can proceed to articulating what is happening, and to discussing how to go about solving the equations.

(This analysis applies to the case where the physical properties of A and B are different. I'm assuming that you want to skip the case where the properties of the two slabs are the same. If you first want to solve the case where the properties of the two slabs are the same, please indicate that desire.)

Chet

 

[Bashyboy]

I do have one question, why does the temperature at the interface have to be constant? Is that just a simplifying assumption?

 

[Chestermiller]

I do have one question, why does the temperature at the interface have to be constant? Is that just a simplifying assumption?

It doesn't have to be constant at the interface. It has to be continuous at the interface. At finite times, temperature of A must match the temperature of B at the interface.

Chet

 

[Bashyboy]

Oh, okay. I am sorry, I misread what you said. That makes more sense.

 

[Chestermiller]

Oh, okay. I am sorry, I misread what you said. That makes more sense.

OK. So let's get started. But, first, which case do you want to consider?

1. A and B have different thermal properties

2. A and B have the same thermal properties

Chet

 

[Bashyboy]

I would choose the second alternative.

 

[Chestermiller]

I would choose the second alternative.

OK. Here we go.

First, let's check your intuition. (It is often very helpful to be able to articulate what you think is going to happen physically before you actually start solving the equations. )

What do you think the temperature at the interface between the two slabs will be after they are brought into contact?

1. The starting temperature of slab A U_0A

2. The starting temperature of slab B U_0B

3. Something in-between U_0A and U_0B

At short times, where do you think the largest temperature change will occur in each of the slabs?

1. Near the interface between the slabs

2. Far from the interface between the slabs

3. The temperatures will be uniform within each of the two slabs at all times

Chet

 

[Bashyboy]

I would expect the interface to be at some temperature between that A and B's. As for the second one, hmm...Again, I would expect the temperature change to be greatest near the interface.

 

[Chestermiller]

I would expect the interface to be at some temperature between that A and B's. As for the second one, hmm...Again, I would expect the temperature change to be greatest near the interface.

Excellent. Exactly correct.

Now, would you be willing to hazard a guess as to what the interface temperature would be for all time, once the two identical slabs were brought into thermal contact?

Considering your answer to the second question, does the following make sense to you: Shortly after the slabs are brought into thermal contact, a narrow temperature region of nominal thickness δ(t) develops on each of the two sides of the interface, respectively. Within this region, the temperature is varying very rapidly with spatial position, while beyond this region, the temperature has not yet changed significantly relative to its initial value at time zero. As time progresses, the thicknesses of these two regions increase with time until they encompass the full thickness of each slab.

Chet

 

[Bashyboy]

Would the temperature at the interference be the average of the two temperatures? Was that guess hazardous enough? :-)

Excellent. Exactly correct.
Considering your answer to the second question, does the following make sense to you: Shortly after the slabs are brought into thermal contact, a narrow temperature region of nominal thickness δ(t) develops on each of the two sides of the interface, respectively. Within this region, the temperature is varying very rapidly with spatial position, while beyond this region, the temperature has not yet changed significantly relative to its initial value at time zero. As time progresses, the thicknesses of these two regions increase with time until they encompass the full thickness of each slab.

Chet

Yes, this makes sense.

 

[Chestermiller]

Would the temperature at the interference be the average of the two temperatures? Was that guess hazardous enough? :-)

Yes. That is correct. So, for the case of two identical slabs at initially different temperatures, the temperature at the interface will be the average of the initial slab temperatures (U_A0+U_B0)/2 for all times. What this means is that we can consider each of the two slabs separately. For example, in the case of slab B, its initial temperature is U_B0, and, at time t = 0, the temperature at it surface at x = 0 is suddenly changed to (U_A0+U_B0)/2 (and held at this value for all times). The solution to this problem is also the same as for the case of a slab twice as thick 2L, and where the temperatures at its two surfaces x = 0 and x = 2L are suddenly changed to (U_A0+U_B0)/2 at time t = 0. Do you know how to solve this kind of problem using the product of a function of x times a function of t? From the solution to this problem, you can find out how long it takes for the temperature anywhere in the slab to approach as closely as you desire to (U_A0+U_B0)/2. This will be the amount of time it takes for the temperatures of the two slabs to equilibrate.

Before moving on to the problem of slabs of different properties and different thicknesses, it is worthwhile and instructive to stay with the problem of two identical slabs a little longer, and consider what happens at short times. By short times, I mean before the effective thermal boundary layer δ(t) has had a chance to grow to such an extent that its thickness becomes comparable to that of the slab L. Thus, if δ(t) << L, then far from the boundary, the temperature will still be effectively at the initial temperature of the slab. Thus, the behavior will be essentially the same as if the slab thickness L were infinite. For slab B, the solution to the differential equation for short times in this case is found to be :
$$U=U_{B0}+\frac{(U_{A0}-U_{B0})}{2}erfc\left(\frac{x}{2\sqrt{αt}}\right)$$
I hope you are familiar with the complementary error function erfc. At small values of its argument, its value approaches 1, and at values of its argument greater than about 2, its value approaches zero. So, the thermal boundary layer thickness δ is on the order of $δ(t)≈4\sqrt{αt}$.

I'm going to stop here and let you digest what I have said. But, I'd like to continue a little further after you say are comfortable with it.

Chet