The Eco-Park Water System

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Oakridge Eco-Park has a complex water system that maintains equilibrium between three interconnected reservoirs: A, B, and C. The system works as follows:

Reservoir A receives 500 liters of rainwater per hour.
Water flows from A to B at a rate of 20% of A’s current volume per hour.
Water flows from B to C at a rate of 30% of B’s current volume per hour.
Water evaporates from C at a rate of 150 liters per hour.

The park rangers want to ensure the system remains in equilibrium, where the amount of water entering each reservoir equals the amount leaving.

Initial volumes:
Reservoir A: 2000 liters
Reservoir B: 1500 liters
Reservoir C: 1000 liters

Questions:

Calculate the amount of water flowing out of each reservoir after one hour, given the initial volumes.
Is the system currently in equilibrium? Explain why or why not.
If not in equilibrium, calculate the equilibrium volumes for each reservoir. (Hint: In equilibrium, the change in volume for each reservoir should be zero.)
How long (in hours) will it take for Reservoir A to reach within 1% of its equilibrium volume, assuming the current flow rates?
The park is considering adding a small stream from Reservoir C back to A. What should the flow rate of this stream be to maintain the equilibrium you calculated in question 3?
If a drought reduces the rainwater input to Reservoir A by 20%, how would this affect the equilibrium volumes of each reservoir? Calculate the new equilibrium volumes.

Solution

1. Water flowing out after one hour:
A to B: 20% of 2000 = 400 liters
B to C: 30% of 1500 = 450 liters
C evaporation: 150 liters

2. The system is not in equilibrium because:
A: Input (500L) > Output (400L)
B: Input (400L) < Output (450L)
C: Input (450L) > Output (150L)

3. Equilibrium volumes:
Let a, b, c be the equilibrium volumes for A, B, C respectively.

A: 500 = 0.2a
a = 2500 liters

B: 0.2a = 0.3b
500 = 0.3b
b = 1666.67 liters

C: 0.3b = 150
500 = 150
c = 1666.67 liters

4. Time for A to reach equilibrium:
Using the exponential approach formula:
t = -ln(0.01) / 0.2 ≈ 23.03 hours

5. Flow rate from C to A for equilibrium:
To maintain equilibrium, the flow from C to A should equal the net inflow to C:
Flow C to A = 500 – 150 = 350 liters per hour

6. New equilibrium with 20% less rainwater:
New input to A: 400 liters/hour

A: 400 = 0.2a
a = 2000 liters

B: 0.2a = 0.3b
400 = 0.3b
b = 1333.33 liters

C: 0.3b = 150
400 = 150
c = 1333.33 liters