Chapter 5. Equilibrium Diagrams -- 5 Exercises

1. Urban Dynamics

The equilibrium diagram is adapted from Gerald Barney's (1974, p. 37) explanation of the dynamic equilibrium in the flows of workers into and out of an urban area. This diagram portrays a city of just over 1 million workers in dynamic equilibrium.These particular flows appeared in a computer simulation of "revival" policies in Forrester's Urban Dynamics. The stocks represent thousands of workers in three categories of employment. The flows are measured in thousands of workers per year. Recall from chapter 2 that a flow with two arrowheads is a biflow -- a flow that may be positive or negative. Also, recall that that the white arrowhead portrays the positive direction. Since the urban system is in dynamic equilibrium, you should be able to answer the ?? in the diagram:

What is the value of the "net departures of managers"? What is the value of the "net departures of laborers"?

2. The Dutch Health Care System

The equilibrium diagram is adapted from Luc Verburgh's (19794 p. 139) explanation of the dynamic equilibrium in the flows of patients in the hospital sector of the Dutch health care system. The stock at the top of the diagram shows 6,428 people waiting for admission to a hospital. This stock is increased by referrals from medical specialists which is set at 803 people/week in this example. The stock is drained when the patients are admitted to the hospital. The average time waiting for admission is influenced by the occupany of the hospital.

What is the flow of "patients admitted"? What is the "average waiting time for admission"?

The second stock shows 1,304 people in the hospital. Since the system is in dynamic equilibrium, you can replace two more ?? in the diagram:

What is the value of the "patients leaving hospital"? What is the value of the "average length of stay in hospital"?

Verburgh explains that the around 2.5% of the patients leaving the hospital are referred to GPs (General Practitioners). Depending on the occupancy, around 90% could be referred back to medical specialists and 7.5% could be discharged to return to home. With these assumptions, you can fill in the rest of the equilibrium diagram:

What is the value of the flow "referred to general practitioners"? What is the value of the flow "referred back to medical specialists"? What is the value of the flow "discharged to return to home"?

3. Physiological Example

Turn to the "first model" of body temperature control explained in the physiology exercises on this website.

Do the heat flows in and out of the body core produce a dynamic equilibrium?

If not, suppose you were to add a flow of heat into the core from shivering. (Shivering is an involuntary contraction of the muscles to release chemical energy into the core.) What is the value of the shivering heat flow to achieve dynamic equilibrium in the body's core?

4. The Ragwort Plant and Its Seeds

Turn to the exercises on the Ragwort Plant. You will see a complicated model with four stocks and and ten flows. When the model reaches equilibrium, there will be 4 leaved plants and 1 flowering adult plant in the test area. Now, suppose we were to double the number of "invading seeds" from 5,000 seeds/yr to 10,000 seeds/yr. What do you think will happen to the number of flowering adults in the test area when the system reaches equilibrium?

This is a complicated question, but you might work through the diagram based on a gut instinct that there will be twice as many flowering adults. Change the number of invading seeds to 10,000 seeds/yr and the number of flowering adults to 2 plants. Then work your way through the rest of the equilibrium diagram to check whether your gut instinct is reliable.

5. Equilibrium in a Three Bottle System with Evaporation

Recall the equilibrium conditions for the two bottles system shown on page 53 of the book. You simulated this system for exercise 6, Appendix C, page 328 in the book. Now consider a system with three bottles and evaporation. The inflow to the first bottle is the same as before, constant at 5 cc/second. The overflows from bottles 1 and 2 are the same as before. Now imagine that the fluid is highly volatile, so we should account for evaporation from the fluid surfaces in all three containers. The evporation rate is 0.1 cm/second. The third container collects the overflow from the 2nd bottle. It is a cone chaped container with a very large capacity,so we don't need to worry about overflow. Questions: 1) What is the overflow from the 1st bottle when the system reaches equilibrium? 2) What is the overflow from the 2nd bottle when the system reaches equilibrium? 3) What is the evaporation from the surface of fluid in the 3rd container? 4) What is the surface area of the fluid in the 3rd container?

6. Build the Three Bottles Model and Verify Your Description of Equilibrium

Expand the "Two Bottles Model" from Exercise 6 in Appendix C to account for evaporation from all three surfaces. Introduce a third stock to accumulate the fluid in the third container. Let's assume that the cone is shaped at a 45 degree angle, so the following geometric formulas apply:

let V = Volume of fluid in the cone
let H = Height of fluid in the cone
let A = surface Area of fluid exposed at the top of the cone
Then:
V = A*H/3 which means that the volume of the cone is one-third of the volume of a cylinder with similar A and H,
and
H = Cube Root (3*V/3.1346) where 3.1346 is pi, the Greek letter used to characterize circles. By the way, Stella has a "square root" function in the the list of special "built-in" functions, but not a cube root. To take the cube root, you may use the ^ symbol which stands for "raise to the power." For example, X^3 stands for Xcubed and X^(1/3) stands for the cube root of X.
Turn in a copy of your stock and flow diagram and your equations. Simulate the model for a sufficiently long period to allow the system to reach equilibrium. Turn in a time of the overflows from bottles 1 and 2, the evaporation from the fluid in the cone and the surface area of the fluid in the cone.