Hot-Cold Cans

The following experiment illustrates the flow of energy into and out of a system. We will develop a MATLAB model of this experiment and explore the strengths and limitations of the model.

• Get pairs of identical cans (I use 16 ounce soup cans) and remove labels. Spray paint one can black and leave the other silver.
• Fill with about 100 ml of room temperature water and place a thermometer in each can.
• Place the two cans each about 3 cm from a 100W light bulb.
• Record the temperature as a function of time (it will rise quickly at first and then reach an equilibrium after a few hours). If available, this is a good use of an electronic thermometer which can be monitored by the computer. If doing it manually, record the temperature every few minutes for about an hour, and then less often for later times. You should also record the final temperature of the cans (for example, after letting it sit overnight).
• Note that the black can heats up faster than silver. The energy absorbed is in the form of visible & infrared radiation which depends on the "color" of the surface.
• Now fill each can with hot tap water (about 45 degrees C), and again record temperature versus time.
• Note that cans cool at nearly the same rate. Heat loss is due to convection, conduction and radiation (to some extent). These depend primarily on the temperature difference between an object and its surroundings and are much less sensitive to the color of the cans.

This simple experiment illustrates several important points.

• In a real physical situation, there are often several mechanisms that are competing in the energy gain/loss. Each of these may depend of different physical properties of the system.
• We can develop a mathematical model which describes the experiment (in this case, the energy flow into & out of the cans and its temperature dependence)
• Using the model, we can develop a program which allows us to extract quantitative results and to make physical predictions.
• The simple model, however, may fail to fit the data perfectly and thus we would need to refine the model
• This experiment has applications in the real world - passive solar energy storage system for a home which is 55 gallon drums (painted black) filled with water exposed to the sun during the day and radiating heat at night.

Our goal is to develop a Matlab model to describe the hot/cold can experiment. This type of a model is often called an energy balance model. In an energy balance model, there can be heat flowing into or out of the system by a variety of physical processes. In the Steady State (or Equilibrium) case, these heat flows into and out of the system are equal and the temperature of the system is constant. If the heat flows are not equal (as is the case when the can is heating up or cooling down), the system will be gaining or losing energy and this will result in a temperature change.

We need to define a few terms before we can proceed with developing a mathematical model:

The Power (P) is the amount of energy flowing into/out of the system during some time period. We can write this as a mathematical equation:

P = Q/t

(1)

where Q (units of Joules) is the amount of energy flow and t is the time interval.

If the system absorbs a net amount of heat Q the temperature of the system will change by an amount T which is given by

T = Q/C

(2)

where C is called the heat capacity of the object. The heat capacity is the product of the m the mass of the object and c is its specific heat. For water, the specific heat is 4186 J/kg^oC

We will use the following algorithm to develop a computer simulation of a heat flow problem:

1. Set initial conditions (such as temperature) and parameters (mass, specific heat) for the system
2. Request user input for unknown parameters (such as the heat-loss coefficient)
3. Set the temperature to its first value
4. Use the current temperature to calculate the power gained (P_Gain) and lost (P_Loss) by system
5. The net heat flow during a time interval t is (P_Gain-P_Loss)*t, so the change in temperature is
   
   

   T = (Pgain - Ploss)t / C

   (3)

   
   
6. The new temperature is the old temperature plus T.
7. The new temperature can be stored in an array, printed or put onto a graph of temperature vs. time.
8. If the experiment is not over (not at the last time interval), continue from step 4 above using the new temperature to calculate the power, etc.
9. If the experiment is over (at the last time interval), make plots of temperature vs. time and see how well they represent the data. If they do not, run the program again using a different estimate for the parameters in step 2.

For the can filled with hot water, it will lose heat by Convection, Conduction and Radiation. For an object near room temperature, we often group these together and approximate these in an equation called Newton's Law of Cooling:

PLoss = k (T - TS)

(4)

where T is the temperature of the system, T_S is the temperature of the surroundings, and k is the heat-loss coefficient. In other words, the larger the difference between the temperature of the object and its surroundings, the faster heat is lost from the object. The heat-loss coefficient, k, can sometimes be estimated from physical equations, but it is often found empirically by doing a measurement. This is what we are going to do.

Cooled Cans If you have performed the experiment yourself, use an editor to make a file called, say, cans.dat which has three columns separated by spaces: Time, Temperature of Silver Can and Temperature of Black Can. If you do not have such a data set, you can use the data set cooled_cans.dat for the temperature vs. time for silver and black cans filled with water which start from a temperature of about 45^oC. The situation for this data set is as follows:

• m = 0.1 kg of water
• T_S = 21^oC was the temperature of the room
• t = 60 seconds for the data (but we can choose a larger or smaller t in our computer model

The MATLAB program heatloss.m will calculate the model given by the equations above. Put this M-file somplace in your MATLABPATH, or modify that path to include location of M-file. Look at the proram to see how it works. To use this program you do the following:

1. Start Matlab. If the M-file is in your path, you should see the header to the file when you type help heatloss. The header explains the inputs and the output
2. Save the files cooled_cans.dat and hot_cold_can.m someplace from where you can access them with Matlab. Each should have three columns of numbers separated by spaces, the first being the time followed by the temperature of the silver and black cans respectively.
3. Type
   load cooled_cans.dat;
   This will create a matrix cooled_cans with 3 columns containing the experimental data. Similarly for heated_cans.dat. You may extract the data by collecting the columns of each array into vectors, e.g. execute command
   time_cool = cooled_cans(:,1);
temp_silver_cool = cooled_cans(:,2);
temp_black_cool = cooled_cans(:,3);
Make sure the times are in seconds.
4. To plot the data, type, e.g.,
   figure
plot(time_cool,temp_silver_cool)
xlabel('Time (Seconds)');
ylabel('Temperature (Centigrade)');
title('Hot Can Cooling Down');
It should plot a smooth curve going from the initial to final temperature (for my data set this is 45^oC to 21^oC over a period of about 36,000 seconds (10 hours)).
5. Run the program as explained in its header to produce theoretical temperature time-series for various heat-loss coefficients (kappa). To compare it to the actual data, type (assuming you have stored theory in temp_th and time axis in time_out):
   hold
plot(time_out,temp_th,'r--')
Run the program several times until you can find a value for the heat-loss coefficient which fits the data for the silver can.
6. Repeat the procedure for the black can.

As you are working on the problem, some questions you can be asking yourself:

• Can the model fit the data over the whole time range? (For example, the command axis([0 3600 20 50]); will allow you to display the first 3600 seconds of the data).
• Why or why doesn't it fit?
• In particular, what is the fit like in the first 15-20 minutes (make plots). What about the last few hours? When is there the largest deviation between the model and the data?

Heated Cans: When the cold can absorbs energy from the light bulb, it will gain energy at a constant value. The terms that would physically come into play would be

P_Gain = P_bulb * f(SolidAngle) * (1-)

where

• P_bulb = 100W is the amount of power emitted by the light bulb
• f(SolidAngle) is the fraction of solid angle for visible and infrared light to arrive at the can. This will be the same for both cans if they are at the same distance from the bulb.
• is the fraction of the light which is reflected from the surface without being absorbed. This will depend on factors such as the "color" and "roughness" of the surface. In the global climate model we will study later, this factor is called the Albedo.

Make a new program heatgain.m (start by modifying the old one) to allow you to put in a constant value for P_Gain and try to use this to model the temperature vs. time data for cold cans heating up (either use your own data set, or you can use the file heated_cans.dat). You can probably use the same heat-loss coefficient as you did from the cooling can experiment.

This model is completely based on that described by Tom Huber.