RE: Modeling concrete examples
Bjorn Simonsen(2003.11.13;12:50 EuSt)–
[From Bill Powers(2003.11.13.0815 MST)]
<Let's talk about the room temperature control. The first thing you have to set up in the model is the physics.
A furnace, when turned on, sends heat at a constant rate into the room air. Heat is a unit of energy, measured
in calories. The rate at which the furnace puts heat energy into the room air is measured in calories per
second. If F stands for the rate of heat emission from the furnace in calories per second, then the amount of
heat energy emitted in "dt" seconds is just F*dt.
Heat energy is being lost through the walls and windows at a rate proportional to the temperature difference
between inside and outside times the wall area A in square cm. The rate of heat loss is measured in calories
per second. If T1 is the inside temperature and T2 is the outside temperature (both Celsius), and thermal
conductivity is K, and the wall area is A, the rate of heat loss is K*A*(T1 - T2), where T2 is smaller than T1.
The amount of heat lost in dt seconds is K*A*(T1 - T2)*dt.
The _net_ flow of heat energy is the furnace output minus the heat loss rates, or F - K*A*(T1 - T2). So the
total amount of heat energy H that flows into the air in a time of "dt" seconds is
H = [F - K*A*(T1 - T2)]*dt. Note that it could be negative if the furnace doesn't put out heat at a high enough rate.
The room has a certain mass of air in it, and the air has a specific heat that determines how many calories are
require to raise the temperature of each gram of mass by 1 degree C. To compute the temperature rise, you
divide the amount of heat put into the air by the total mass in grams, and multiply by the specific heat Let S be
the specific heat, and M the total mass. The temperature rise for an amount of heat x is x*S/M. So if X is the
net heat flow in time dt, the room air temperature rise in time dt would be
change in T1 = [F - K*A*(T1 - T2)]*S/M*dt.>
Specific heat is how many calories are required to raise the temperature of each gram by 1 degree C. OK.
If you in your formula multiply the amount of heat, x with S/M and x has the denomination cal/second (F=210,000 Calories/second), the outcome will be
(FS/M-KA(T1-T2)S/M)dt[= (FS/M - LA*(T1-T2)*S/M)*dt
(cal/second)(cal/(gramdegree))/gram - (cal/cm2seconddegree)cm2degree*(cal/(gram*degree))/gram]=
Calcal/secondgamdegreegram - calcal/secondgram*degree
Did you misreport the formula?
Besides
<For air, the numbers you need are:
Specific heat S: 0.24 cal/gram
density 0.0011 grams/cubic cm
Volume of room 50,000,000 cubic cm (4 x 5 x 2.5 meters)
Mass of air = density * volume.
The thermal conductivity of the walls is K: 1.2404
Here I have a problem with the denomination. Since you use calories, I thought it was 1.2404 cal/(cm2*second). The walls are 450,000cm2. The thermal loss from the walls will be 5,581,800 cal each second if the inside temperature is 4 and the outside temperature is -6 degrees. This is not reasonable if the furnace gives 21,000 cal/second.
I will use 1.2404 Wm2=J/(second*m2)
We will say arbitrarily that
Wall area A: 450,000 square cm (ignore floor and ceiling) =450000/(100100) m2 = 45 m2
Finally, the furnace output rate when turned on is
F = 210,000 Calories/second That’s like a small room heater. = 210000/0.24 J = 875000 wattsecond
You are the engineer, Bill . But this seems to me to be a powerful furnace (875 kwatt?)
I have used F = 210 calories/second
………………………………………………………
<I suggest that as a first step you simply set up a model of this part of the system. Starting with a given inside
and outside temperature, you can turn on the furnace and start calculating the room temperature at intervals
of one second (dt = 1). For each calculation, you compute the net heat gain and the temperature change,
then add that change to the room temperature T1 to get the temperature for the next time around. Calculate
1000 values in a table of temperatures,
I think the numbers above a realistic, but I could have made mistakes. You can judge by seeing how much
the room warms up in 15 minutes (1000 second). Try varying the outside temperature to see the effect.>
I have set up a model of this part of system. It is 1 MB so I have placed it in http://home.c2i.net/bjornsimonsen/termostat5.xls
When you have this much working we can talk about the temperature control system
I am thankful for comments from you.
PS. I have taken the spreadsheet away from the net, but I can’t stop working with it.
[From Rick Marken (2003.11.14.1210)]
I think it would be interesting to see if we could use this prototype control system as the basis for
>building Bill's thermostat model. I'll give it a try too. Let's see what we come up with.
I hope you have some comments to my simple use of your model. I have put the slowing factor, leakage (through the wall 
) in my physical calculations. I am not sure about the gain in a technical model. And as you see, there is no disturbances other than the furnace which I have put in the feedback loop.
Bjorn