Chili made with ground beef, beef chunks, chicken, turkey or pork requires a liquid to tenderize the meat and finish cooking the beans. The flavors all meld together in that liquid. Chili is served over rice or pasta or mopped up with bread so liquid is an important part of the dish. Beer is often used because it adds its own taste component. While the alcohol in the beer evaporates during the cooking process, some people would rather not use it.
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Cowboys don't normally carry along a bottle of beer for their chili. Even if they did have the beer, it's doubtful they'd prefer adding it to the chili pot rather than drinking it themselves. What they do have is coffee. Brew a pot of strong coffee to replace the beer in equal measurements. The taste of the coffee won't be recognizable. The dark color adds a richness to the chili.
Chicken or beef broths substitute for beer and add flavor. If it's a vegetarian chili, use vegetable broth cubes. If you don't have any, make your own vegetable broth. Chop two carrots, two stalks of celery, an onion and a bunch of spinach, parsley or leafy greens. Cover with 2 cups of water. Bring to a boil and let simmer for an hour. Drain and strain.
Tomato Juice or Sauce
Purists would argue that tomatoes have no place in an authentic chili. However, many chili recipes do call for chopped canned tomatoes. If that's the case for you, substitute tomato juice for the beer. If you're out of juice, dilute tomato sauce in a 1 to 1 ratio. In other words for every cup of sauce, add 1 cup of water. Tomato based vegetable juices may be used as well.
Bean Cooking Liquid
Save the liquid that the beans have cooked in when you drain the beans to add to the chili. If you're using canned beans, save the liquid in the can. Use that as replacement for the beer. The cooking or canned liquid has some flavor.
Water and Salt
If you don't have anything else, 1 cup of water flavored with 1/4 tsp. of salt will replace the liquid the beer provides, but other than the salt won't add any flavor. You might want to spice it up a little, literally, by adding a 1/4 tsp. of cumin, oregano or red pepper flakes for every cup of water you add.