1. School as a Steady-State System A college has a constant undergraduate enrollment of 14,000 students. No students flunk out or trans- fer in from other colleges and so the residence time of each student is four years. How many students graduate each year? The residence time of a student is four years, the stock of students is 14,000, and the outflow rate is the graduation rate. It follows from the preceding discussion of steady-state systems that the flow rate in or out of the college is the stock divided by the residence time, or Fin Fout =M/T. For our specific problem, this formula becomes: (1) graduation rate= total stock of students residence time of students 14,000 4 yr (2) 3,500/yr. In this problem, all students spend exactly the same amount of time in college-four years. In many other steady-state situations, each unit of the substance (for example, each molecule of pollutant) comprising the stock spends the residence time in the box only in a statistical sense. The average time spent by all the units is the resi- dence time, but the individual times spent by the units may differ widely. Provided the stock is in a steady state, stock = (inflow or outflow rate) x (residence time). The flow can be of various types. The movement of students through college illustrates a type of flow in which the components of the stock pass through in an orderly manner so that each component has the same residence time. The subsequent box-model problems in this chapter illustrate the far more typical case of mixed flow, in which the inflowing units of stock mix thoroughly in a medium and have differing individual residence times. EXERCISE 1: A population of cows on a farm is in steady state. The birth rate is 7 calves per year and the average residence time for a cow on the farm is 6 years. How large is the herd? EXERCISE 2: Suppose there are 100 students enrolled in a graduate program year after year, and that each year 20 get degrees and leave, 5 flunk out, and 25 new students enter the program. (a) What is the average residence time of a student in the program? (b) If STEADY-STATE BOX MODELS AND RESIDENCE TIMES 25 all the students that flunk out do so at the end of their first year, what is the average length of time to get a degree (for those students who do get a degree)? (Hint for b: First determine the number of en- rolled students, at any specific time, who will eventually get degrees.)