4.3. Inside World of Showers

To examine the model behind the gaming simulator open World of Showers B and press the 'Model' tab on the left of the screen. There are two sectors – 'your shower' on the left and 'shower 2 – hidden' on the right. It is best to study shower 2 because this sector contains the complete balancing feedback loop for comfort seeking. (In 'your shower' the equivalent feedback loop is completed by you, the game player, and so the loop is not visible in the diagram.)

4.3.1. A Tour of Formulations in the Comfort-Seeking Loop of the Hidden Shower

Let's get into shower 2 and talk through the operating structure as shown in Figure 4.9 and the corresponding equations in Figure 4.10. The temperature gap is the difference sensed by the shower-taker between desired temperature and the actual temperature at the shower head. In the equations, desired temperature is set at 25 °C. The symbols on the left-hand side of the map show how the recognition of a temperature gap leads to a change in tap setting. This part of the map represents the decision making of a rational shower-taker who wants to stay comfortable.

Figure 4.9. Operating structure of shower 2 in World of Showers B

Source: The Dynamics of Resource Sharing: A Metaphorical Model, System Dynamics Review, 11(4) page 293. Morecroft, J.D.W., Larsen, E.R., Lomi, A. and Ginsberg. A., 1995 © John Wiley & Sons Limited. Reproduced with permission.

Figure 4.10. Equations for shower 2 in World of Showers B

The temperature gap leads to a fractional adjustment of the tap setting. Imagine what goes through the mind of the shower-taker. The water temperature is too cold. She decides to turn the tap part way round the scale in a direction that she believes will warm her up. In the equations this judgemental fractional adjustment is proportional to the temperature gap – the more degrees away from comfort, the greater the fractional adjustment envisaged, according to the judgemental calibration of the scale for the tap setting.

The shower-taker's thinking moves one step closer to action as she gauges the required adjustment to the tap setting. She glances at the tap and notes its current setting and the maximum and minimum settings to which it can be turned. If she feels too cold (temperature gap greater than zero) then she turns the tap toward the hot end of the scale. How far should she turn the tap between the current and maximum setting? Here the shower-taker's judgement is key to her later comfort! In the model, the required adjustment is equal to the difference between the maximum and current tap setting (her room for manoeuvre, or angular distance remaining, on the scale) multiplied by the fractional adjustment (her estimate of the appropriate turn of the tap as a fraction of the remaining angular distance on the scale). The equation formulation can be seen in Figure 4.10. If she feels too hot then she follows a similar line of reasoning in gauging how far to turn the tap between the current and minimum setting.

All these thoughts, comprising her judgement on tap setting, flash through her mind in an instant. In fact, she is scarcely aware of the steps in the judgement. What she knows and feels is a sensation of being too cold or too hot, and so she turns the tap. In the model, the change in tap setting is set equal to the required adjustment divided by the time to adjust the tap setting. This portion of the map represents the part of the shower system where judgement and decision making convert into action. Her hand turns the tap and the tap setting changes. The angular movement of the tap accumulates in a level that represents the position of the tap on the hot-cold scale. Then the plumbing takes over, and she awaits the consequences!

On the right-hand side of the Figure 4.9 the water temperature at the tap is shown to depend on the tap setting. The equivalent algebraic formulation is shown in Figure 4.10. The temperature at the tap depends on the flow of cold water, the temperature of the cold water, the tap setting, the maximum flow of hot water to shower 2, and the temperature of the hot water. The equation looks quite complex, but really it is just blending two flows of water, cold and hot, and calculating the resultant temperature. The flow of cold water is fixed at 15 litres per minute, at a temperature of 10 °C. The maximum flow of hot water available to both showers is set at 60 litres per minute, at a temperature of 70 °C. The flow of hot water in shower 2 is a fraction of this maximum flow, determined by the tap settings in shower 2 and in your shower (as explained below). As the tap setting changes from its minimum value of 0 (the cold end of the scale) to its maximum value of 1 (the hot end of the scale) then the water temperature moves from its minimum value of 10 °C to its maximum blended value (a weighted average of 10°C and 70 °C water).

The temperature at the shower head changes with the temperature at the tap, but only after a time delay, which is literally the pipeline delay in the pipe connecting the tap to the shower head. In the model, the pipeline delay is set at 4 seconds. As the water emerges from the shower head, we come full circle around the diagram, back to the temperature gap.

To summarise, shower 2 in World of Showers B is represented in three parts. On the left of Figure 4.9 is the behavioural decision-making process that translates a temperature gap into a required adjustment of the tap setting. At the top of the figure is the action of adjusting the tap that leads to a new tap setting. On the right of the map is the piping and water flow that converts the tap setting into hot water at the shower head.

4.3.2. Interdependence of Showers – Coupling Formulations

The crescent of links in the top right of Figure 4.9 shows the coupling connections between the two showers and Figure 4.10 shows the corresponding algebra (Morecroft, Larsen, Lomi & Ginsberg, 1995). The showers are supplied with a maximum flow of 60 litres of hot water per minute. The fraction of hot water available to shower 2 depends on the tap setting in both shower 2 and in your shower – hence the coupling. As the tap setting in shower 2 moves further toward the hot extreme of the scale, then (assuming no change in the tap setting in your shower) shower 2 gains access to a larger share of the available hot water supply. However, if the tap setting in your shower increases, then shower 2 loses a proportion of the hot water flow. Algebraically, the fraction of hot water available to shower 2 is represented as the ratio of tap setting 2 to the sum of tap setting 2 and the tap setting in your shower. The maximum flow of hot water available to shower 2 is the product of the fraction of hot water available to shower 2, and the maximum flow of hot water in the system as a whole (the shared resource constraint).

The coupling equations for your shower are similar to shower 2, but expressed in terms of the tap settings as they affect the flow of hot water in your shower. Hence, the fraction of hot water available to your shower is represented algebraically as the ratio of the tap setting in your shower to the sum of tap settings in your shower and in shower 2.