There are all kind of mathematical models: some are based on physical laws, some are derived empirically and some are in-between. There are various pros and cons in choosing a modeling approach. Physics-based models, for example, are very concise but they are notoriously hard to construct (many problems, for example, come from combining equations for different components). Statistics-based models such as the ones from machine-learning can be often automatically or semi-automatically built but they are often imprecise or incorrect. One of the goals of the refrigeration project is to determine which modeling approach is better for diagnostics.
We will start building a physics-based model by analyzing the behavior of each component in an refrigerator. Let's start with the thermostat, which as we will see is one of the most complicated components.
The thermostat is the "brains" of the refrigerator. It is responsible for switching the compressor on and off and keeping the temperature in the refrigerator close to a value set by the user. One of the criteria for evaluating the performance of a thermostat is the variance in the temperature. A good thermostat should keep the temperature almost constant. The control problem is, of course, subject to constraints as a good thermostat should not switch on and off the compress too often.
There are different types of thermostats: the main distinction is if they are mechanical or electrical. Each thermostat includes a sensor, a switch and some control mechanism (or software) that actuates the switch based on the sensor readings. Figure 1 shows the type of thermostats used in our refrigerators.
The type of the temperature sensor that is used by our refrigerators is very similar to the old mercury thermometers. As mercury is hazardous, the thermostat sensor is filled with gas. The type of gas was not identified for the purpose of this study. The gas is often nitrogen.
Modern refrigerators use semiconductor temperature sensors, micro-controllers, and relays. The problem with these thermostats is their complexity: a refrigerator controlled with such a sensor can fail due to a software bug. They also need separate power supplies for the micro-controllers and the sensors which wastes additional energy.
Our refrigerators use old-style mechanical thermostats. Part of the disassembled thermostat is shown in figure 2. The thermostat has a temperature adjusting knob, a sensing tube, and a spring-mechanism for shortening and opening the electrical terminals that start and stop the compressor. There is a single screw for calibrating the temperature. The thermostat comes factory-calibrated.
This type of thermostat is a relatively complex thermal/mechanical/electrical component. Consider, for example, the debouncing mechanism shown in figure 3. It consists of a spring-like mechanism that prevents sparkling due to small gap between the conductive plates.
Figure 4 plots data from three different refrigerators. The left plot shows at what temperature the thermostat switches the compressor on for various possible settings of the control-knob (higher knob number means colder). The right plot shows at what temperature the thermostat switches the compressor off for the full range of control-knob settings. This difference in on and off temperatures comes from the mechanical design of the thermostat.
The three thermostats shown in figure 4 behave in a very different manner. The first thermostat is the best, where the temperature of the refrigerator is linear as a function of the control-knob setting. The second thermostat has a very strange non-linear behavior where positions one and two are indistinguishable and positions four and five are also very similar. The third thermostat, when set in its coolest regime does not switch the compressor off, a behavior different from the other two refrigerators. That is the reason why the box-plots in the second row of figure 4 are missing their right-most bars (the box-plots visualize switching behavior and this thermostat does not switch). This makes the behavior of the second thermostat, set to the lowest temperature, hard or impossible to distinguish from a short-circuited thermostat.