Here's a pan on a stove top. The pan's body and handle are made out of the same material.
The burner has been turned on for a little while, and the pan body is hot enough to cook
this egg. But what about the handle? Is it too hot to touch? Or is it cool enough to
hold? Let's find out.
This video is part of the Equilibrium video series. It is often important to determine
whether or not a system is at equilibrium, to do this we must understand how a system's
equilibrium state is constrained by its boundary and surroundings.
Hi, my name is John Lienhard, and I am a professor of Mechanical Engineering at MIT.
Today, I am going to talk to you about Equilibrium and Steady State. It's a common misconception
that Equilibrium and Steady State are the same. In this video, we'll take a closer look
at these concepts and see how they differ. In order to understand this video, you should
be familiar with the First and Second Laws of Thermodynamics, the meaning of thermal
equilibrium, and the concept of heat flow rate, which is heat transfer per unit time.
After watching this video, you will be able to identify whether a system is at equilibrium
or at steady state, and to describe the difference between Thermal Equilibrium and Steady State.
This metal bar has been sitting in a large room for a long period of time. In this example,
we can assume that the room, which acts as the surroundings, is very large, much larger
than the bar, which acts as our system. Heat transfer from the bar to the room acts as
energy transfer across the system boundary. We'll assume that the volume of the bar and
pressure of the room are constant. In other words, no mechanical work is done.
The temperature of the bar is 25°C and the temperature of the room is also 25°C. Even
though the molecules in the air are colliding with the atoms on the surface of the bar,
there is no net transfer of energy between the room and the bar. The temperature of both
the bar and the room remain constant at 25°C. In thermodynamic terms, we say that the bar
and the room are in thermal equilibrium. Because there is no difference in temperature,
there is no heat transfer between the bar and the room. What if we wanted the temperature
of the bar to be larger than that of the room, and we wanted it to remain that way? Pause
the video and think about how we might do this.
Of course, if we simply place a bar at 45°C in the room, the small bar will lose energy
to the large room until both the bar and room are once again at thermal equilibrium at a
temperature approximately equal to but ever so slightly higher than 25°C.
You might have suggested connecting some sort of heating element to the bar. Let's try this
and see what happens. Here's the bar after it has been left on a heater for a long time.
As you might have expected, the bar is hottest where it's touching the heater, and coolest
at the opposite end. There is a temperature gradient between these two points, but this
gradient doesn't change over time. In this case, would you say the bar is at
thermal equilibrium with the room? Why or why not? Pause the video here and discuss
this with a partner. We know that energy is being continuously
transferred from the heater to the metal bar. And we also know that the temperature profile
of the bar is not changing; in other words, it's not getting any hotter or cooler. That
means the bar must be transferring the energy it's gaining from the heater to the air in
the room. And that means it cannot be in thermal equilibrium with the room.
Now, in order for the temperature profile to be constant in time, the heat flow rate
from the heater into the bar must be constant and equal to the heat flow rate from the bar
to the room. We could use a new term to describe these conditions -- we could say that the
bar is at steady state. A quantity is at Steady State when it is constant
with respect to time. In other words, the partial derivative of the quantity with respect
to time is zero. The metal bar is at steady state because the time derivative of its temperature
at any point is zero. This happens when the heat flow rate into the bar and out of the
bar are equal and constant.
Let's go back to the heated pan which was left on the stove for a long time. Do you
think the pan handle will be too hot to touch, or cool enough to hold?
Use the concepts of thermal equilibrium and steady state temperature to think of circumstances
in which the handle might be too hot, or cool enough, to hold. You might consider how the
handle is attached to the pan, or how well the handle transfers heat to the room. Pause
the video here and discuss.
How many scenarios did you come up with? There are many, but let's start with two extreme
cases.
What would happen if the handle and pan were in perfect thermal contact, and both were
completely isolated from the room? Eventually, the handle would reach the same temperature
as the pan. This is what that would look like: the temperature of the pan and handle are
identical and constant. So, the handle is in thermal equilibrium with the pan. What
can you say about the transfer of heat from the pan to the handle once the pan and handle
are in this situation? Pause the video and discuss. The rate of heat transfer must be
zero -- if it wasn't, the handle would keep getting hotter, which is impossible, assuming
that the temperature of the pan is constant. The heat flow rate from the handle to the
room is also zero, because we assumed the handle is isolated from the room.
So, the heat flow rates into and out of the handle are exactly the same - zero and constant.
And steady state occurs when the flow rate in equals the flow rate out, even if they're
both zero. So this scenario, which we've been describing as "equilibrium," is really just
a limiting case of steady state. And that's true for all systems at thermal equilibrium:
they are all at steady state with respect to temperature. But the opposite is not true:
not all systems at steady state are in thermal equilibrium. In this case, the handle will
obviously be too hot to hold, since it's the same temperature as the pan.
Now let's consider the opposite case. What would happen if the handle were completely
insulated from the pan? The handle would be at thermal equilibrium with the room, and
would stay that way no matter how hot the pan got. The temperature of the handle would
be exactly the same as that of the air. Just as in our first case, both the heat flow rate
from the pan to the handle and from the handle to the air are zero and constant. Thus, this
case of "thermal equilibrium" is really just another limiting case of steady-state temperature,
except, of course, the handle will be cool enough to hold. Both of these cases are great
thought experiments. But they could never happen in real life, because it's impossible
to completely isolate one component of a system from another. In case 1, it would be impossible
to isolate the pan and handle from the room; and in case 2, it would be impossible to isolate
the handle from the pan. So are we any closer to figuring out whether we can pick up this
pan or not? Let's keep going...
Let's go back to our first case and refine our thinking. We originally assumed that the
pan and handle were in perfect thermal contact, and that the pan and handle were totally isolated
from the room. Let's keep the first assumption, but change the second assumption. In other
words, we will allow for some flow of heat from the handle to the room. Now what happens
as we heat up the pan? We can assume that the heat flow rate from the pan to the handle
is initially larger than the heat flow rate from the handle to the room. This is reasonable
because the pan is initially much hotter than the handle.
So the handle will definitely heat up, just like our metal bar from before. As long as
the flow of heat from the pan to the handle is greater than the flow of heat from the
handle to the room, the handle will get hotter. And also just like our metal bar, it will
be hottest closest to the pan, and coolest far away from it.
As the temperature of the handle rises, the heat flow rate from the handle to the room
will also rise. And eventually, it will equal the heat flow rate from the pan to the handle.
In other words, after some length of time the handle will be losing energy to the room
at exactly the same rate as it's gaining energy from the pan. When this happens, the temperature
gradient along the handle will stop changing, and the system will be at steady state with
respect to temperature. But, remember: it will not be in thermal equilibrium. So, can
we say whether the handle is too hot to hold? No! That depends on the exact temperature
distribution when the system reaches steady state. Take a moment to sketch a temperature
profile that describes a steady-state situation wherein all or part of the handle is too hot
to touch. What values of the heat flow rate from the pan to the handle could result in
a handle that is too hot to touch. What characteristics of the pan, handle and/or room could produce
this profile? Pause the video and discuss this with a partner.
Okay, we're at our final scenario. Let's review the previous cases.
In case 1, we assumed the pan and handle were in perfect thermal contact and that the system
was completely isolated from the room. In case 2, the handle was completely isolated
from the pan.
In both case 1 and case 2, the handle was in thermal equilibrium -- there was no transfer
of heat into or out of the system. The handle was also at steady state with respect to temperature.
Make sure that you can explain why this is true.
In case 3, we went back to our assumption of perfect thermal contact between the pan
and the handle, but we allowed for some heat transfer from the handle to the room. For
this final case, let's pick and choose the most reasonable assumptions from our previous
cases and build from there. We'll assume that the handle can exchange heat with the room.
We'll also assume that the pan and handle are insulated from each other. This is reasonable,
because if you're a pan designer, you don't want your customers burning their hands! But
of course, there is no such thing as perfect insulation, so we have to allow for at least
some heat transfer between the pan and handle. If the rate of heat flow from the pan to the
handle is extremely small, what can you say about the temperature profile across the handle
once the system reaches steady state? Pause the video and sketch a possible steady state
temperature profile in the handle.
Here's one. If the rate of heat flow from the pan to the handle is very low, the system
will reach a steady state temperature well before the majority of the handle gets hot.
There will be a very small region near the pan where the handle is hotter than the room,
but most of the handle will be close to the temperature of the room. So let's answer our
original question. Will the pan handle be too hot to hold or will it be comfortable
to touch? Let's look at a thermal image of the pan after it has been left on the stove
for 30 minutes. It looks like we predicted in case 4! A very small portion of the handle
is warmer than the room, but most of the handle is at room temperature. Remember, this does
not mean that the handle is in thermal equilibrium with the room. There is a constant but small
rate of heat flow from the pan to the handle, and an equal rate of heat flow from the handle
to the room.
The handle was designed to reach a steady-state temperature that is comfortable to hold without
a glove. This is a well-designed pan!
You now have the tools to understand that Thermal Equilibrium and Steady State temperature
are different. We saw that if the temperatures of all parts of a system do not change with
time, the system can either be at steady state or at thermal equilibrium. If the net heat
flow is constant and nonzero as for our metal bar on the heater, the system will have a
steady temperature profile. If it is constant and zero, as for our metal bar at room temperature,
the system is at thermal equilibrium. So, equilibrium is just a special case of steady
state. We were able to apply our understanding of thermal equilibrium and steady state temperature
to describe the temperature profile of a pan left on the stove for a long time and to understand
the design elements that led to that profile. I hope you enjoyed this video.
Good luck in your studies.