A permanently installed washing machine should have its own drain and water supply, but if needed, you can make use of your kitchen sink drain and faucet as a temporary measure for a portable unit. You can even make the drain connection permanent, but you'll probably always want to be able to disconnect the washing machine supply hose so you can use the faucet for other purposes.
You need to be aware of some key code issues before you connect a washing machine to a sink drain. If you make the connection incorrectly, the drain could overflow or, even worse, siphon contaminated water into the washing machine. A drain connection with the potential for contamination is called a cross-connection, and there's a simple way to avoid it.
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Two Important Code Issues to Remember
The output from a washing machine is under pressure, and a standard 1 1/2-inch sink P-trap assembly is too small to handle it. If you're going to connect the washing machine drain hose to the sink drain before the P-trap, the entire drain, including the trap, needs to be constructed of a 2-inch pipe. An alternative is to construct an auxiliary P-trap for the washing machine, vent it with an air admittance valve and connect it to the 2-inch waste pipe into which the sink trap empties using a sanitary tee fitting.
If you connect the washing machine drain hose permanently to the sink drain, the potential for backflow exists. One way to prevent that is to make a high loop in the drain hose that rises above the flood rim of the sink before connecting it to the sink drain, much as you do with a dishwasher. This approach may not be legal in your area, though, so be sure to check first.
A Safe Way to Drain the Washing Machine
A safer alternative is to make a temporary drain connection by hooking the drain hose over the rim of the sink and clamping it in place. A washing machine drain hose comes with a curved section on the end that's supposed to hook onto a standpipe, and you can usually adapt it to fit a sink. As Balkan Sewer & Drain Cleaning puts it, you need to leave a "sizable" air gap between the end of the hose and the bottom of the sink to prevent backflow. You should also put a screen in the drain opening to collect lint and prevent the pipes from clogging.
Install a Male-to-Male Faucet Adapter for the Water Supply
You can connect one of the washing machine supply hoses to any faucet that has an aerator with external threads. Simply unscrew the aerator and screw on a sink adapter for a washing machine that allows you to attach the cold supply line. This is a male-to-male faucet adapter with two female connectors, one with standard pipe threads and one with hose threads. You can buy one at any hardware store.
The faucet needs to have an aerator for this adapter to work, and since only single-handle faucets do, you can only connect one of the washing machine's two supply hoses. You'll have the best temperature control if you connect the cold supply line, run the washing machine on cold-only settings and control the temperature from the faucet.