The Sample Problem

• Description
• Approximate Solutions
• Finite-difference method (using a hand-calculator or spreadsheet)
• Finite-difference method (using a graphical interface, e.g. Stella 5.0)
• Exact solution (from Calculus)

Description

Take a cup of boiling-hot coffee (  assume that it behaves like an ideal liquid and doesn't evaporate away at an appreciable rate) and set it on a table. One would expect that the temperature of the cup will decrease. This behavior (cooling) will be modeled. The first activity would be to decide on what factors might affect the rate of cooling and try to find the mathematical relationships between these factors.

For a certain system, the rate of change of temperature decreases as the liquid's temperature approaches room temperature. The temperature at any given time during this process is dependent on the initial temperature, room temperature, and the time. Implicit in this description is that the coffee is in an imperfectly insulated cup or mug (if it was a perfect insulator, the temperature of the coffee would not change). The insulation value of the cup also needs to be considered and can be experimentally determined.  In fact, this is a particularly interesting cup because the all parts of the cup cool at the same rate (and so our mathematical task is greatly simplified).
 

Approximate Solutions

Using equation 1, a temperature-time profile can be easily calculated with either a handheld calculator or a variety of computer programs (e.g., spreadsheets such as Microsoft Excel, or using languages such as BASIC, C++, etc.).

As a side note, while this situation is relatively simple, the number of equations quickly increases as the model gains complexity.  A more graphical approach to modeling the system can sometimes be useful.  In this case, the Stella 5.0 program is used as an example.  The cooling coffee cup experiment can be represented by figure 1 below.  Notice that no equations or numbers are needed to initially draw the system (using this part of the program is directly analogous to using the ubiquitous Paintbrush-type program) so that the relationships between the different system components can be quickly indicated.

• "Coffee temp" is a stock or level representing the temperature of the coffee.
• "Rate of loss" represents a flow away (i.e., temperature loss or cooling) from the coffee. The flow as drawn is specifically an outflow from a stock and set to be a uniflow (one directional).  In this case, 3 external factors (indicated by the red arrows and blue circles) affect the rate of loss, namely, the coffee temperature, the insulation factor, and the room temperature.
• "Graph 1" indicates that some part of the model will be able to be visualized on the screen as the simulation is run.
• The "cloud" to the right of the rate of loss flow arrow indicates that an infinitely large sink exists to accept the output of the flow.
• Parameters such as starting time, end time, and Dt (called somewhat incorrectly dt below) are embedded into the controls of the program (pull-down menus).

The Exact Solution

If the cup insulation factor remains a constant, only the choice of Dt will affect the accuracy of the temperature-time profile. In this special case, we actual do know the "right" answer, so we can compare the effect of chosing different values for Dt.  Without going into the details of arriving at the so-called exact solution (see any calculus textbook for the details), equation 2 describes the temperature of a cup of coffee as a function of time.

Introduction

Experimental Procedure