The "stuff" moving through the pipes in this Stella model is temperature.
The goal of this unit is to combine the calibration of a mathematical model describing heat flow with an empirical investigation of vaporization energy. The overall strategy is to perform an easy "wet lab" which can be used to determine the rate at which heat is transferred from a hot plate to a small beaker of water. By taking periodic (say, once per minute) measurements of the temperature of the stirred water in the beaker, the student (or teacher) can create a table or graphical plot relating time to temperature. If we can determine the temperature of the hot plate (presumed to be constant over the course of the experiment) then we know both the difference in temperature between the plate and the water and the rate of temperature change in the water. Since, for temperatures midway between the freezing point and boiling point of water, the temperature of a fixed quantity of water will rise at a rate proportional to the difference in temperature between the heat source and the water, we can use our tabulated information to calibrate a mathematical model describing the water temperature as a function of time. It is then possible to predict, using the model, how long it will take for the water to heat to some specified temperature.
We will calibrate the system using temperatures well below the boiling point of water to avoid vaporatization issues. However, once the model is calibrated we can use the model to make predictions such as "when will the water temperature reach 90 degrees centrigrade?" or "When will the water boil?"
Because our model was calibrated in a situation where vaporization was not a significant factor, the model begins to fail in situations where vaporization is important. The questions involving temperatures near the boiling point of water stretch the model beyond its useful range. If the model's predictions are tested against an empirical study the model's predictions will be off by a considerable factor. This leads to an opportunity to discuss vaporization energies and the difference between energy change and temperature change.
In order to calibrate the model to be useful at the boiling point of water, it is necessary to look at an energy based model rather than a temperature based model. The tabulated data we accumlated during the initial phase of the experiment can be combined with the specific heat of the system (beaker plus water) to determine the heat conductivity between the hot plate and the beaker of water. Then we can translate the heat content of the water into temperature, extending the model to hold the water at 100 degrees centigrade until enough energy has been added to vaporize all the water in the beaker. With such a model questions can be asked like "how long will it take for half the water to boil away?" The energy based model should be robust for temperatures near the boiling point.
This kind of a lab can help demonstrate that temperature and heat content are directly related over relatively small temperature ranges, but the relationship can be more complicated if the temperature range under consideration is quite large. If a phase change occurs, the temperature of a sample may remain almost constant despite the continued flow of heat from the source to the sink.
To determine the temperature of the hot plate, set the hot plate at the desired setting and place the mineral oil bath on the hot plate with the mercury thermometer inserted into the oil. Taking readings every few minutes note when the oil bath reaches a steady state temperature. Carefully remove the oil bath.
Use this steady state temperature as the temperature of the hot plate during the mathematical modeling. Be sure to use the same setting during the empirical experiment as you use for this step.
Once the temperature of the hot plate has been determined, fill the 400ml beaker with 300ml of room temperature water and place the beaker on the already warm hot plate. Insert a mercury thermometer into the water and take temperature readings at fairly short time intervals, say every 30 second to one minute. Use the magnetic stirrer to keep the water uniformly heated. If the temperature of the water is rising slowly, use longer time intervals.
Cease the measurements when the water temperature reaches something on the order of 50 to 60 degrees centrigrade.
The Stella diagram for this difference equation is relatively straight
forward.
The "stuff" moving through the pipes in this Stella model is
temperature.
There are many ways to use the empirical data to calibrate a model like this. A nonlinear regression model could be set up to approximate a but this is probably more than what is necessary in this context. The procedure I would recommend is to solve the model for the parameter a. Each time period would yield an estimate of a so the average of these estimates ought to be close to the true value of a. If your empirical data leads to a wide range of a values, trying taking the empirical readings at less frequent time intervals, experimental error in the temperature measurement may be the cause of the wide variations.
The approximations
to a can be derived from the difference equation by computing
a=(Tw(i+1)-Tw(i))/(Tp-Tw(i))
for each time interval (except the last).
A less algebraic approach would be to intelligently guess at various values of a and run the Stella model with graphical or tabular output until a value of a is found which produces Stella output in close agreement with the empirical work.
Once the model is calibrated you can use it to make predictions about how quickly the water will warm up. The model should agree with experimental results until the water temperature approaches the boiling point. At these higher temperatures vaporization will occur, retarding the temperature gain of the water.
To extend the simple mathematical model above to account for situations where energy transfer appears in ways other than temperature change (phase changes, for example) it is necessary to base our model on energy flow. This is in agreement with the actual physics which says that energy flows between two objects at a rate proportional to the difference in the objects' temperatures.
In this new Stella model the "stuff" flowing through the pipes is heat, measured in calories. Knowing the specific heat of our system (meaning the beaker plus the 300ml of water) will allow us to compute the temperature of the system as a function of its energy content. Once this temperature hits 100 degrees, no further temperature rise will be detected as the subsequent inflows of energy will be used to vaporize the water rather than heat it further.
The difference equation describing the energy model is
Ew(i+1)=Ew(i) + r*(Tp-Tw(i))
where Ew(i) is the amount of energy in the water at the
beginning of time period i and r is the heat
conductivity, measured in calories per unit time-degrees. If the time
unit being used is minutes, then r would be measured in
calories per minute-degree. The remaining symbols retain their
meaning from the previous model.
Notice that this model requires us to be able to compute the water (and beaker) temperature at each time period. This requires determining the specific heat of the water-beaker system. One approach is to assume the beaker's heat content is insignificant and assign a specific heat of one calorie per gram-degree. Another approach is to experimentally determine the specific heat of the system by mixing known amounts of hot and cold water in the beaker originally holding one or the other. Careful measurement of the resulting temperature can be used to determine the specific heat of the glass in the beaker. This, in turn, can be used to determine the specific heat of the beaker-water system.
Using the symbol C to denote the specific heat of the
beaker-water system, the diagram of the Stella model for the energy
based model is:
The equations for this model are also interesting. In particular,
note that the temperature of the water can never exceed 100 degrees.
To determine how quickly the water will boil off, use the valve
setting to determine how quickly energy is entering the beaker. The
vaporization energy of water can then be used to determine how quick
water is being boiled each second. (This vaporization can be included
in the Stella model itself if desired.)
In the equations above are based on the assumption that the specific heat of the system is one. I also used centrigrade degrees, which means that a heat content of zero corresponds to a temperature of zero degrees centrigrade. The intial heat content of 7500 calories contained in the 300 ml (or grams) of water corresponds to an initial temperature of twenty-five degrees. The weight of the beaker should also be included in this mass, but was not in the mathematical formulation of the model.
The energy based model presented here assumes that no vaporization occurs until the water reaches 100 degrees. Clearly this is not the case, suggesting that a further extension of the model to more accurately handle high temperatures could be undertaken. Feel free to do so. It is an interesting exercise.