Batch Sparging

Batch sparging is a very easy way to go about sparging and allows you to use the cooler you converted into a Mash/Lauter tun using this tutorial. Batch sparging allows you to sparge in the same container that you mash in as long as your filter of sorts (like the Mash/Lauter tun cooler) or a false bottom. This filter or false bottom helps to keep grain out of your wort.

What You'll Need

Mash/Lauter Tun (≈ $42). Which can be built using this tutorial.
A Mash - Shown over here
Large Stainless Steel Spoon or Wooden Paddle (≈ $5 for SS Spoon)
Pitcher (≈ $5) - The bigger the better and preferrably graduated
Kitchen Thermometer (≈ $3)
Large Pot for Heating Water (≈ $30) or a Turkey Fryer with Stand (≈ $60)

The Calculations

Determine Where To Start

The first step is partially determined by your recipe. You need to know what the target amount of beer your trying to make is going to be. In most cases you will probably want to end up with around 5 gallons of beer. Since some beer will be lost when siphoning from one container to the other, add another half gallon of beer to your target volume. In other words, shoot for 5.5 gallons when making your recipe. Now add another gallon for to account for the water that will be lost during the boil. So for the purpose of the following calculations, the volume that will be used is 6.5 gallons.

Calculate The Mash Volume and Run-offs

For these calculations, two infusions will be performed but you are more than welcome to use infusionss if you perfer. The infusions serve to "wash" the sugar from the grain. The infusions occur after the mash. The whole purpose of the mash and sparging is to extract sugar from the grains in order to give the yeast something to ferment. The varying types of grains used will add to the malty flavors of the beer.
Let's calculate the Mash volume and Infusion/Run-off volumes now. To do this you need to know the combined weight of the grain you will use in your recipe. You also need to know the Grist Ratio and Absorbtion Rate you intend to use. A standard grist ratios are 1.5 quarts of water per pound of grain or 1.25 quarts/pound. For this tutorial, we'll use 1.25 qts/lb. The absorbtion rate is the amount of water that the grain will absorb during the mash. Pretty straight forward, right? This number can be extracted after brewing a couple of beers with your mash setup, but Brewology101 will try to start you in the right direction. For this tutorial .15 gallons of water per pound of grain will be used.
Pre-boil Volume = 6.5 gallons
Pounds of Grain = 13 lbs
Grist Ratio = 1.25 qts/lb
Absorbtion Rate = .15 gal/lb
Number of Run-offs = 2
Mash Volume = Pounds of Grain * Grist Ratio = 13 * 1.25 = 16.25 qts = 4.0625 ≈ 4 gallons
Absorbtion = Pounds of Grain * Absorbtion Rate = 13 * .15 = 1.95 gallons ≈ 2 gallons
Run-off Volumes = Pre-boil Volume / Number of Run-offs = 6.5 / 2 = 3.25 gallons
First Run-off = 3.25 Gallons = Mash Volume - Absorbtion + First Infusion
First Infusion = 3.25 Gallons - 4 Gallons + 2 Gallons = 1.25 Gallon
Second Infusion = 3.25 Gallons

The Process

Now that you have calculated how much water to use for each step, you're probably wondering what you do with all this water. Very simple actually. Just follow this timeline:
Heat your mash volume to your mash temperature. Typically around 165°F to reach a mash temperature of around 155°F
Pour the heated water into your mash. Stir. Let rest for 60 minutes, remembering to stir around every 15 minutes.
Heat First Infusion to ≈ 185°F to sparge at around 170 °F
Add First Infusion and rest 10 minutes.
Start heating the water for the Second Infusion. Targeting ≈ 185°F to sparge at around 170 °F
Time to Vorlauf. Slowly drain a quart of the mash from the Mash/Lauter Tun into a pitcher. Pour the wort that was just run-off back into the mash. Repeat until the wort coming out of the mash is no longer cloudy. This could take 4 or more repetitions.
Drain the entire mash/lauter tun and set aside for the boil.
Add the Second Infusion and rest 10 minutes.
Vorlauf and drain again.
Proceed to the boil as described as you would in any other brewing process. If you're not familiar with the boil process, you can see it over in this tutorial.