Testing the Heat Flow Model
Initial Conditions
The simulation begins with 1,000 cc of water and an internal
energy content of 9,000 calories. The internal energy content is defined
relative to a base temperature of 10 degrees Celsius. The water temperature
depends on the Internal Energy Concentration which is measured in calories
per gram. At the start of the simulation, there are 9,000 calories evenly
distributed over 1,000 grams, so the concentration is 9 calories per gram.
The water temperature equation assumes that zero concentration corresponds
to 10 degrees. The specific heat of water is 1 calorie per gram per degree.
In other words, 1 calorie of heat is required to increase the temperature
of a gram of water by 1 degree. Working from a base of 10 degrees, the energy
concentration of 9 calories/gram means that the initial water temperature
is 19 degrees.
The heat flow into the glass depends on TempDif, the difference between
the air temperature and the water temperature which is 1 degree at the start
of the simulation. The initial heat flow is around 1.4 calories/sec. But
the heat loss due to evaporation is much smaller (only about 0.1 calories/sec).
With more heat flowing in than flowing out, you would expect the internal
energy content and the water temperature to increase.
Now, let's run the model over thousands of seconds to learn whether the
water temperature will eventually reach 20 degrees.
Simulation Results
Figure 5 shows the simulated behavior over 60 minutes. The air temperature
is constant at 20 degrees, and the water temperature increases over time.
The heat flows are scaled from 0 to 2 calories/second. At the beginning
of the simulation, the heat flowing in is around 1.4 calories/second while
the heat loss due to evaporation is only around 0.1 calories/second. The
net inflow is over 1 calorie/second. If these flows were to persist for
1,000 seconds, we would expect over 1,000 calories to be added to the energy
content of the water--more than enough to increase the water temperature
to 20 degrees. But the simulation shows that the temperature never reaches
20 degrees. After 15 minutes, for example, the water temperature is only
up to around 19.7 degrees. There is still a temperature difference across
the glass surface of 0.3 degrees. This means the heat flowing into the glass
is much smaller.
Figure 5. Results for One Hour of Simulated Time.
By the 30th minute of the simulation, the heat flowing into
the glass has declined almost to the same value as the heat loss from evaporation.
The water temperature appears to be approaching 20 degrees after 30 minutes,
but it is still not quite the same as the air temperature. By the 45th minute
of the simulation, the heat flows in and out of the water are nearly equal.
And the water temperature appears to be leveling off slightly below 20 degrees.
By the end of the simulation, the water temperature is 19.92 degrees.
| |
We see a small temperature difference across the surface of the glass,
and it isn't going away! |
The size and persistence of this temperature difference is the focus of
the introductory exercises.