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Chapter 4: Problem 27

You are making beer. The first step is filling the glass carboy with the liquid wort. The internal diameter of the carboy is 15 in., and you wish to fill it up to a depth of 2 ft. If your wort is drawn from the kettle using a siphon process that flows at 3 gpm, how long will it take to fill?

Short Answer

Expert verified
First, calculate the cross-sectional area of the carboy. Then, calculate the volume of wort needed to fill the carboy to the desired depth. After that, convert this volume into gallons. Finally, you get the time needed to fill the carboy, which is the volume in gallons divided by the flow rate in gallons per minute.

Step by step solution

01

Calculate the Cross-Sectional Area of the Carboy

Firstly, calculate the cross-sectional area of the carboy, using the formula for the area of a circle, \( \pi r^2 \), where \( r \) is the radius. Considering the diameter is 15 inches, the radius will be half of this value (7.5 inches). Don't forget to convert the inches to feet, because our depth measure is in feet (1 foot equals 12 inches). The radius in feet is \( 7.5/12 = 0.625 \) feet. So the cross-sectional area in square feet is \( \pi (0.625)^2 \). Calculate this value.
02

Calculate the Volume of Wort Needed

Now, calculate the volume of wort needed to fill the carboy to the desired depth. This volume is the cross-sectional area times the depth, according to the formula for the volume of a cylinder. Multiply the cross-sectional area (calculated in the previous step) by the depth (2 feet). This will give the volume in cubic feet.
03

Convert the Volume to Gallons

We need to have the volume in gallons, because our flow rate is given in gallons per minute. 1 cubic foot equals 7.48052 gallons, therefore, you should multiply the volume you got in the previous step by 7.48052 to convert it into gallons.
04

Calculate the Time Needed to Fill the Carboy

Finally, calculate the time needed to fill the carboy, which is the volume (in gallons) divided by the flow rate (in gallons per minute). This will give the time in minutes.

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