How much energy is required to heat enough water for a twenty minute shower? Before we begin the exercise we must collect all of the data. Let’s say these are the typical parameters for a Texas high-school student taking a twenty-minute hot shower:

Tap water temperature = 60℉

Hot water shower temperature = 110℉

Shower head rate = 5 gallons per minute

Student’s shower time = 20 minutes

Now that we have the data, how much energy do we need for a single shower? We must solve for the required energy measured in BTUs to heat a 20 minute shower. Use the chart below for the relevant information we will need in order to calculate (1) amount of water in pounds (2) change in temperature in degrees Fahrenheit and (3) BTUs needed

T_{init} (beginning temperature) |
60℉ |

T_{final} (final temperature) |
110℉ |

Q_{w} (velocity of water) |
5 gallons/minute |

ΔT (change in time) |
20 minutes |

ρw (density of water) |
8 lbm/gallon |

Don’t forget the “physics!” It takes 1 BTU to heat 1 lb of water by 1℉.

1 BTU / lb * ℉

## Step One: How much water will we need?

First, let’s calculate the amount of water required for a 20 minute shower in pounds (lbs). Q_{w}, Δt, and ρw are given in the table above. However, we first must find volume (V_{w}) using the equation:

##### V_{w} = Q_{w} * Δt

## click to check answer

#### 100 gallons

V_{w} = 5 gallons/minute * 20 minutes = 100 gallons

Minutes cancel out and we are left with gallons as our unit

Now that we know how many gallons are required, what about the mass of the water? You may remember the standard formula for mass: volume * density. Use the equation below to determine the mass of water (M_{w}) and check your answer before moving onto step two.

##### M_{w} = V_{w} * ρw

## click to check answer

#### 800 lbm

M_{w} = 100 gal * 8 lbm/gal

gallons cancel out and the leftover unit is lbm

## Step Two: What is the change in temperature?

We know the temperature of the water prior to and after the standard twenty minute shower, but what is the change in temperature? Use the equation below and check your answer before moving onto the final step, calculating the BTUs required.

##### ⃤ T (?? ℉) = T_{final} (℉) – T_{init} (℉)

## click to check answer

#### ⃤ T (50 ℉)

T_{final} (110 ℉) - T_{init} (60 ℉)

## Step Three: Final Calculation

Here we need to calculate how many BTUs are required to heat 800 lbs of water by 50℉. Use the equation below and check your answer to see how much energy we need for a hot shower! Remember when using the equation, cancel out units that appear on the same side of the equation.

##### BTU (???) =(800 lb) * (50℉) * (1 BTU/lb℉)

## click to check answer

#### 40,000 BTU's

lb's and ℉ cancel out so we are left with units of only BTU

Is 40,000 BTU a lot of energy? Let’s find out through work! Work is equivalent to energy, and defined as moving a force through a distance. An example of producing work is lifting a weight that stores energy. We call this potential energy. Let’s pose an interesting and fun question? How much potential energy can we store by lifting a 5lb sack of potatoes by 5 ft? Theoretically, we could convert that potential energy to another form of energy and use it to heat our shower. Let’s have some fun and learn at the same time through an energy experiment.