Chapter 2. The Big Bang

**Homework: **

1) Ch142 Final 2000: **due 2/9/2016**

2) Predicting the water level in a bathtub – an example of Steady State systems: **due 2/11/2016.**

Consider a simple bathtub with a faucet that adds water to the tub at a constant rate and a drain where water flows out at a rate proportional to the height of the water level. In this exercise you will calculate the water level in the tub as a function of time, determine the steady state water level, and calculate the residence time of water in the tub. This bathtub model is a good analogy for many dynamic environmental systems typical of the atmosphere, ocean, lakes, and living organisms. Assume that water is added to the tub from a faucet with a constant flow rate of 10 liters/minute. The inflow rate expression is a simple zero-order process where the inflow is constant and independent of water level.

(1)

The outflow is proportional to the height of the water in the tub. This is a first-order process where the higher the water level the greater the outflow rate.

(2)

Notice that the sign of the change in volume with time (dV/dt) is negative since water is leaving the tub. The rate constant k will have units of (liters/minute cm) so that the product of the rate constant and water height is outflow rate (liters/minute). From geometry we also know that the height of the water in the tub is the volume of the tub divided by the area of the tub.

When you start filling a tub the initial volume is usually zero. Adding water increases the volume and outflow decreases the volume.

Volume = initial volume + water added – water lost down the drain (3)

Equation 3 describes the conservation of mass (water) for the tub. We will write similar conservation equations for may environmental systems.

If the amount of water added is greater than the water lost then the volume will increase. As the volume increases the depth of the water in the tub will increase which will increase the outflow rate. Eventually the outflow rate will equal the inflow rate and the bathtub fill level will reach **steady state**.

Steady State is reached when INFLOW = OUTFLOW. Notice that you can set equation 1 equal to equation 2 to calculate the water height at steady state. It is interesting to think about what steady state means. For the bathtub, steady state water level is only ** “steady”** when the water is running constantly into the tub. Adding water is like adding energy to the system. Steady state systems need energy inputs to maintain their positions. This can be compared to the equilibrium water level of the tub which will be zero after the faucet is turned off.

The residence time of the water is the average time that a water molecule will spend in the tub. This can be defined as the volume of water in the tub divided by the inflow or outflow rate when the system is at steady state (volume/(volume/time) = residence time).

**Model Details:**

Water inflow rate = 10 l/minute

Water outflow rate constant (k) = 1 (l/min cm)

Area of the tub = 10,000 cm^2

Starting water volume in tub = 0 liters

time step for the model = 0.5 minutes

**Assignment:**

1) Build and Excel model of bathtub levels as a function of time. Use the attached YouTube video as a model for your program.

2) Try running the model with an initial volume of 0 liters, 100, and 200 liters. How do the simulations compare? What does this tell you about **Steady State** conditions?

3) Manipulate equations one and two to calculate the expected Steady State level of the bathtub and compare to the results obtained in question two.

4) Calculate the residence time of water in the bathtub at steady state.

Turn in the answers to the questions 1-4 on one piece of paper showing your model on the front of the page and your results on the back side of the page. Don’t waste paper!

**Resources:**

How to Build a habitable Planet – Text Resources

Lecture Slides: CH217-L1-big-bang-2016

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