Question

An aerated lagoon (a sewage treatment pond that is mixed with air) is being proposed for a small lake community in northern Wisconsin. The lagoon must be designed for the summer population but will operate year-round. The winter population is about half of the summer population. Based on these design assumptions, the volume of the proposed lagoon is 3420 m

3
. The daily volume of sewage in the winter is estimated to be 300 m
3
. In January, the temperature of the lagoon drops to
0
∘
C
​

​
but it is not yet frozen. If the temperature of the wastewater flowing into the lagoon is 15
∘
C, estimate the temperature of the lagoon at the end of a day. Assume the lagoon is completely mixed and that there are no losses to the atmosphere or the lagoon walls or floor. Also assume that the sewage has a density of 1000 kg/m
3
and a specific heat of 4.186 kJ/kg−K

Answer 1

The temperature of the lagoon at the end of the day can be estimated to be 1.362°C.

Step-by-step explanation:

To estimate the temperature of the lagoon at the end of the day, we need to consider the energy exchange that occurs. The energy gained by the lagoon can be calculated using the formula:

Energy gained = mass of sewage x specific heat x temperature change

First, we can calculate the mass of sewage using the volume and density:

Mass of sewage = daily volume of sewage x density = 300 m3 x 1000 kg/m3 = 300,000 kg

Next, we can calculate the energy gained by the lagoon:

Energy gained = 300,000 kg x 4.186 kJ/kg-K x (15°C - 0°C) = 18,837,000 kJ

To estimate the temperature of the lagoon at the end of a day, we divide the energy gained by the total mass of the lagoon:

Temperature change = Energy gained / (total mass of lagoon x specific heat)

Temperature change = 18,837,000 kJ / (3420 m3 x 1000 kg/m3 x 4.186 kJ/kg-K) = 1.362°C

The temperature of the lagoon at the end of a day can be estimated to be 1.362°C.