Diffusion of heat from a point source being heated continuously

[Dishsoap]

Homework Statement

In essence, I'm trying to make a model of how the heat from a point source diffuses through air. It should be a function of both distance and time (I'm assuming either a 1/r^2 or an exponential dependence on distance).

Homework Equations

I found this equation:

\Phi (x,t) = (1/√4πkt) exp(-x^2/4kt)

I have two issues. First of all, at x,t=0 (initial conditions of the point source) the temperature of the point source should be a constant. I assume it's a Dirac delta function, which makes me think this isn't the correct equation. In addition, the point source is also being heated while the heat is being diffused. This isn't a homework question, just a problem I'm working on for an internship.

The Attempt at a Solution

From the above equation and Newton's law of cooling, I have reason to think that the decay is exponential, but have no idea how to quantify this. I considered adding a term to the previous equation, but that still doesn't account for it heating up.

 

[Chestermiller]

Homework Statement

In essence, I'm trying to make a model of how the heat from a point source diffuses through air. It should be a function of both distance and time (I'm assuming either a 1/r^2 or an exponential dependence on distance).

Homework Equations

I found this equation:

\Phi (x,t) = (1/√4πkt) exp(-x^2/4kt)

I have two issues. First of all, at x,t=0 (initial conditions of the point source) the temperature of the point source should be a constant. I assume it's a Dirac delta function, which makes me think this isn't the correct equation. In addition, the point source is also being heated while the heat is being diffused. This isn't a homework question, just a problem I'm working on for an internship.

The Attempt at a Solution

From the above equation and Newton's law of cooling, I have reason to think that the decay is exponential, but have no idea how to quantify this. I considered adding a term to the previous equation, but that still doesn't account for it heating up.

The equation you wrote looks to me more like the solution for a sudden impulsive point injection of heat, rather than a continuous point heat source. But, before you start looking at the transient solution to the problem, why don't you solve for the final steady state profile, which should give you some really good insight into what the transient solution should be like. So, if you have a continuous spherically symmetric point source of heat Q at the origin, what is the differential heat balance equation for the steady state temperature profile, and what is the solution to that equation?

Chet

 

[Orodruin]

As Chet is hinting at, your expression is the Green's function of heat conduction/diffusion. It describes the distribution of heat/temperature/concentration based on a single point impulse. When $t \to \infty$ it does approach a delta distribution. It should be noted that diffusion is not the only mechanism for heat transfer in a gas ...

Why would the temperature of the point source be a constant different from infinity? A point source in itself is not very physical when it comes to heat diffusion, but may be a good approximation in some cases.

 

[Dishsoap]

The actual system is a resistor hanging in free space, but for the purpose of modelling the diffusion I felt it appropriate to consider it a point source.

What do you mean by "solve for the final steady state profile"? The system is changing with time (as heat is being added), so I'm not sure that it qualifies as a steady state. There are other changes occurring (this resistor is in a distillation column, so the dynamics in there are haywire) but for now I'm just trying to get a simple model.

 

[Chestermiller]

The actual system is a resistor hanging in free space, but for the purpose of modelling the diffusion I felt it appropriate to consider it a point source.

What do you mean by "solve for the final steady state profile"? The system is changing with time (as heat is being added), so I'm not sure that it qualifies as a steady state. There are other changes occurring (this resistor is in a distillation column, so the dynamics in there are haywire) but for now I'm just trying to get a simple model.

What I'm saying is that, if you wait long enough, the temperature profile will stop changing with time. That means that the system will have reached steady state. Do you know how to solve a steady state heat conduction problem in spherical coordinates?

Also, if the resistor is hanging in air, then, of course, natural convection is going to be a factor. Do you know where to look to find the heat transfer solution for natural convection from a sphere?

Chet

 

[Dishsoap]

What I'm saying is that, if you wait long enough, the temperature profile will stop changing with time. That means that the system will have reached steady state. Do you know how to solve a steady state heat conduction problem in spherical coordinates?

Also, if the resistor is hanging in air, then, of course, natural convection is going to be a factor. Do you know where to look to find the heat transfer solution for natural convection from a sphere?

Chet

I agree with the first notion, but in a system where heat is constantly being added I don't see how it would ever stop changing with time.

And no, I'm very new to this topic. Would you kindly point me in the right direction?

 

[Chestermiller]

I agree with the first notion, but in a system where heat is constantly being added I don't see how it would ever stop changing with time.

Well, you may not see it, but, at long times, the temperature for pure radial heat conduction in spherical geometry asymptotically approaches a final steady state profile. Do you know the differential equation for steady state radial heat conduction in spherical geometry?

And no, I'm very new to this topic. Would you kindly point me in the right direction?

See Bird, Stewart, and Lightfoot, Transport Phenomena.

Chet

 

[nickelnine37]

Hey guys - I know I'm five years late to the party here, but for anyone who's still looking for a solution, I've written a blog post about this very question.

https://nickelnine37.github.io/the-diffusion-equation.html

In summary, the initial equation describes a single finite quantity of heat released at a single point in time, in one dimension. For a continuous source you need to integrate over time. So for a source that releases heat at a constant rate $q$ from $t=0$ to $t=\tau$ from a position $\mathbf{r'}$, the temperature/concentration/whatever is described by

$$A(\mathbf{r}, t) = \frac{q}{(4\pi D)^{d/2}} \int_{0}^{\min(\tau, t)} \exp\Big[-\frac{ |\mathbf{r}-\mathbf{r'}|^2}{4D(t-t')}\Big] \frac{\mathrm{d}t'}{(t-t')^{d/2}}$$

where $d$ is the number of dimensions. For each value of $d$ there is a different solution. It can be solved for all $d\neq 2$ by substituting

$$
\mathrm{d}u = \frac{\mathrm{d}t'}{(t-t')^{d/2}}
$$

and then writing it in terms of the error function.

$$
\text{erf} (x) ={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-z^{2}}\,\mathrm{d}z
$$

For $d=2$ it becomes

$$
A(\mathbf{r}, t) = \begin{cases} -\frac{q}{(4\pi D)} \text{Ei}\big(-\frac{r^2}{4Dt}\big) & \text{if} \, \, t < \tau \\[0.2cm]
\frac{q}{(4\pi D)} \Big( \text{Ei}\big(-\frac{r^2}{4D(t-\tau)}\big) - \text{Ei}\big(-\frac{r^2}{4Dt}\big) \Big) & \text{if} \,\, t > \tau \\ \end{cases} \\
$$

where $\text{Ei}(x)$ is the exponential integral. Hope this helps someone!