However, brewing water gets much more complicated, and it's really as complicated as you want to make it. Water contributes quite a lot to the taste or perception of a beer's taste based on the style, as well as helps control the mash pH. Personally I've taken a very gradual approach to optimizing (or even understanding) the effects of the water profile on my beers. I started simply using spring water from the grocery store. That worked well enough, especially for extract brews, but as I learned more about the mineral content needed for controlling mash pH, maintaining yeast health, balancing bitterness and maltiness to name a few, I began to understand how far I was undershooting some important stuff.
The first thing to do when you start to work on your water profile is to find out what's in your water to begin with. If you're using tap water, check out your local water report. It will usually have all the pertinent mineral data. If you're on well water, you can get a cheap report from Ward Labs. You can find the water report for most commercial spring waters online, but those are usually very soft, or low in mineral content. Distilled water is good for bringing the hardness and minerals down in hard tap water since it has no dissolved solids.
If your city water uses chlorine, you'll have to pre-boil it to remove it. If they use chloramine, however, adding one Campden tab (potassium metabisulfite) per 20 gallons of water will remove it. For the most part, simply doing this will suffice for extract brewing, as a lot of the minerals needed are in the malt extract. If you're doing all-grain, it will greatly improve your control on the taste of the brew if you focus in on the specifics of your water's minerals.
Below is a list of the ions in water that are important for brewing. I've listed some recommended levels (in mg/L or ppm), as well as adjusted levels for certain types of beer styles:
Calcium (Ca++)
Very generally, calcium is important for lowerin mash pH to desired levels, as well as increasing yeast yield and growth, enhancing yeast flocculation and assisting in protein coagulation in the hot and cold breaks, among others. A rule of thumb is that the calcium ion concentration should be somewhere between 50-100ppm.
Can be supplemented using Gypsum (CaSO4) if the water is low in sulfate, or calcium chloride (CaCl).
Magnesium (Mg++)
Magnesium is goof for yeast health and flocculation, and in a lesser amount decreasing the mash pH. The nice thing is that most of the Mg should be extracted from the extract or the grain during the mash, so typically an addition is not needed. Starting levels should be between 5-30ppm.
Sodium (Na+)
Levels should be less than 100ppm or so. If your water report has extremely high levels, it could likely be due to water softening agents. If this is the case, either dilute the water with distilled water to bring it down, or try to source the water before the softening. High levels above 150ppm may give a slightly salty flavor to the beer.
Chloride (Cl-)
Not to be confused with chlorine, chloride accentuates the sweetness or maltiness of a beer, and should be present in levels between 10-100ppm. Historically, people who added a touch of table salt to domestic lagers attempted to bring out some maltiness in the flavor.
Sulfate (SO4--)
Sulfates provide a sharp hops bitterness, and levels should not exceed 150ppm for most beers. There is a debate about choride/sulfate ratios and its effect on the bitter/malty flavor. I generally balance the ion concentration of both within their normal limits, and occasionally shift the balance one way or the other depending on the recipe. An easy way to do this is using gypsum and calcium chloride, which both boost the calcium level, which is a good thing anyway if it's lacking.
Carbonate (CO3) and Bicarbonate (HCO3)
These ions are mainly important in mash alkalinity (pH buffering capacity), stabilizing and preventing a decrease in mash pH. While this is usually never a problem for most all-grain brewers, too much carbonates will not allow the pH to reach the desired 5.2-5.4 range, especially in light beers where unadjusted water does not have the acids present in highly kilned malts to neutralize the bicarbonates. Generally, light beers should have a level less than 50ppm (converted from CaCO3 if reported as such), and can be balanced with Ca/Mg to reduce the residual alkalinity (below). Very dark beers may need elevated bicarbonates to counteract the acids present in dark grain grists.
Residual Alkalinity (RA)
Residual alkalinity is a measure of both permanent water hardness (Ca/Mg) and Alkalinity (carbonates). Basically, Ca and Mg neutralize the water alkalinity by interacting with the phosphates present in the malt, while bicarbonates contribute to it. Residual alkalinity (expressed in ppm) is a rough indicator of where the mash pH will end up. The chart below shows the general relationship between hardness and alkalinity, and also demonstrates the use of darker malts (of Dublin, for instance) that counteract high alkalinity, and lighter beers (from Pilsen) that have low RA to start. If you can't combat the high alkalinity, you might also consider adding 1-2% acidulated malt to lower the mash pH.
Below is a chart from AJ Delange, an expert in brewing chemistry, showing this RA relationship among common regional water profiles. Note: Take these city water profiles with a grain of salt, as not many people know how the specific breweries treat their water and the levels may be inaccurate. In short, don't try to imitate a water profile exactly to copy a certain style.
Summary
The main takeaway here is that water chemistry can seem daunting, but is not overly complicated for homebrewers. As a extract brewer, I would suggest getting a water report if using tap water to ensure the levels are not way off. If they are, dilute with distilled water, but do not use all distilled or reverse osmosis water, as you need some ions in there. If you want to play with the concentrations, start with a small amount of gypsum and calcium chloride, no more than about 0.5 tsp of each to 5 gallons before the boil. These are your main flavor components. Adjust with recipes, higher on the sulfates for bitter/hoppy beers, higher on the chlorides for maltier beers.
For all-grain brewers, it would be useful to understand the water chemistry a bit more, and estimate your mash pH and ion concentrations. A good source for this is Bru'n Water, which is a great overview a free Excel calculator for estimating your mash and ion components. It takes a little while to get used to, but it's a great way to start. A pH meter might also be helpful on brewing day to ensure you're getting the right levels.
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