2
$\begingroup$

I am studying the non dimensional heat equation for a finite rod, and have the quantity $$Q=-\frac{\partial \theta}{\partial x}$$ where $\theta$ and $x$ are non dimensional temperature and distance respectively. I was wondering what this quantity means.


Question 1: Is it heat flow in the positive x direction? Does it depend on direction at all?

Question 2: Since it is non-dimensional, does that make it heat flow relative to heat flow everywhere else, and so if say one end of a rod has a lot of heat loss and the other end doesn't lose as much, then $Q$ is quite small (like $0.1$) at the end where a lot of heat is not lost, and high at the other end ($2$)? And if heat loss everywhere is very low, but the same, then $Q=1$ everywhere?

Question 3: I also believe there to be some significance to the cases $Q=0$, $0<Q<1$, $Q=1$ and $Q>1$, so if someone could explain what these cases mean too, that would be very much appreciated. :)


I asked this question on maths stack exchange too, but I think it is more suited here since I am looking to understand what it physically means.


EDIT

What do I think about these questions?

Q1) I think it represents heat flow in the positive x-direction, but I am not sure what this means at the end of a finite rod.

Q2) I think what I wrote is true (about what different values of $Q$ mean), and just wanted to check if I am right in thinking this.

Q3) I think $Q=0$ means heat stays constant at that point and doesn't flow, $0<Q<1$ means less heat flowing in the x-direction that other parts of the system, $Q=1$ means heat flow which is equal to the average heat flow over the whole system, $Q>1$ means more heat flowing the the x-direction that other parts of the system.

Are my beliefs correct?

$\endgroup$
  • $\begingroup$ Maybe you can provide what you think about those questions for us to see how you're thinking about them? $\endgroup$ – Damian Sowinski Apr 24 '17 at 18:57
  • $\begingroup$ @DamianSowinski I have edited the question to include this. $\endgroup$ – John Doe Apr 24 '17 at 19:04
  • $\begingroup$ What is the form of the dimenionless heat equation you are studying? Write the equation out and see how it relates to your $Q$. $\endgroup$ – Damian Sowinski Apr 24 '17 at 20:24
  • $\begingroup$ The equation is $$\frac{\partial\theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}$$ I don't understand what you mean by your second sentence? $$Q=-\int\frac{\partial\theta}{\partial t} dx$$ What does this mean physically? $\endgroup$ – John Doe Apr 24 '17 at 20:59
1
$\begingroup$

Well, from your definition of $Q$ you can see that the diffusion equation can be written: $$ \begin{align} \frac{\partial \theta}{\partial t} +\frac{\partial Q}{\partial x}=0\\ \end{align} $$ This is what is known as a continuity equation. It tells us a lot about $Q$. Consider a finite interval $[a,b]$. If the total amount of $\theta$ in that interval is $$ \Theta(a,b) = \int_a^b \! dx ~\theta $$ then the continuity equation demands that: $$ \frac{\partial\Theta}{\partial t} = Q(a) - Q(b) $$ Let's see how to interpret this. If $Q(a)>Q(b)>0$ then $\Theta$ will be increasing in the interval. If $Q$ represents heat flow to the right (when it is positive, left if its negative), then this equation means that there is more heat flowing into the left boundary of the interval than their is heat flowing out of the right boundary, and the total $\theta$ is increasing. Does this seem physical?

If we take $a=0$ and $b=L$, where $L$ is the length of the rod, then this equation is describing a rod that is connected to some heat source(sink) at its boundary. The total flow through these endpoints will determine how much total $\theta$ there is in the entire rod.

Notice that $Q$ is defined as the negative gradient of $\theta$. Thus when the the temperature is increasing to the right, the heat flow is negative. This is also very physical. If the right side of the rod is hotter than the left side, then heat should be flowing from hot to cold: to the left. If $\theta$ is constant throughout the rod, then the heatflow vanishes.

Concerning your statements about special values of $Q$, you need to first define what the average heat flow is: $$ \bar Q = \frac{1}{L}\int_0^Ldx ~ Q $$ The statements you made are about relationships to this quantity: points at which $Q>\bar Q$, $Q=\bar Q$, and $Q<\bar Q$. You were correct to interpret $Q=0$ as points in which heat does not flow.

Hope that helps!

$\endgroup$
  • $\begingroup$ I think this answered all of my questions, thank you! :) $\endgroup$ – John Doe Apr 25 '17 at 1:26
  • $\begingroup$ Just a follow up question regarding the average heat flow $\bar Q$. Firstly, since $Q=-\frac{\partial\theta}{\partial x}$, does this mean that $\bar Q=\frac{\theta(0)-\theta(L)}{L}$? Secondly, for an infinite rod, how could this be determined? $L\to\infty$, and $\theta(L)=0<\infty$, so would $\bar Q=0$ in this case? $\endgroup$ – John Doe Apr 26 '17 at 0:49
  • $\begingroup$ Your formula for the average heat flux is just the negative slope of the temperature at both ends of the rod. If the two ends are kept at constant temperature, and then you stretch the ends infinitely far apart, then the slope will approach $0$ and there will be no average heat flow. $\endgroup$ – Damian Sowinski Apr 26 '17 at 1:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.