The order is irrelevant. First of all don't let the reservoir run empty, then there won't be any air to worry about. What you CAN do is suck most of the old fluid out first, then refill it, or just let it run out, then refill it. If you keep it full right from the start you will be constantly diluting old fluid and it will never all be gone. Even that is not a big concern. As long as you get the biggest percentage of old fluid out you'll be doing more than most of us do to our own cars. Your goal is to get out the moisture that seeps in over time. That promotes corrosion and reduces the boiling point from well over 400 degrees to 212 degrees.
Open a bleeder screw and loosen the reservoir cap to prevent vacuum from building which will impede the fluid flow. Allow it to gravity-bleed until the level in the reservoir goes down about 1/4". Most wheels will flow relatively slowly. You can speed it up with a hand-pumped vacuum pump but don't worry if you see a lot of small air bubbles in the fluid. They sneak in past the threads of the bleeder screws. When you get clear fluid, close that bleeder screw and go on to the next one.
Some manufacturers specify the order to bleed the system when it has been opened and air must be expelled. Some believe you should bleed the left front first because it's the shortest line and that's the fastest way to get one line totally air-free so pressure can start building easier. Some believe you should do the right rear first because with that one line finished the greatest percentage of air will have been removed. In the 30 years I've been rebuilding calipers and wheel cylinders, and now simply replacing them, the wheel I always bleed first is the one I'm standing closest to.
If you want to pedal-bleed with a helper it is important to never push the pedal more than half way to the floor. Crud and corrosion build up in the bottom halves of the bores where the pistons don't normally travel. Running the pedal all the way down runs the lip seals over that crud and can tear them. Additionally some master cylinders have a valve that shifts to block fluid flow to a pair of ports when there's a leak. That valve will also trip when there's an imbalance in the two hydraulic systems caused by a lack of pressure to one wheel due to pedal bleeding and an open bleeder screw. That mainly applies to GM front-wheel-drive cars. Some people think that valve will not trip if you follow a certain bleeding sequence but experience has shown that valve WILL trip when any one of the four wheels doesn't build pressure. You will not prevent that shifting valve by bleeding a certain wheel first or last.
By far the biggest concern, especially if you use something to suck out old fluid from the reservoir is to be absolutely certain you do not get any hint of a petroleum product in the fluid. That will turn all the rubber parts that contact brake fluid in the system into mush. That's a very expensive repair. Engine oil, transmission fluid, and power steering fluid are the most common contaminants but experienced mechanics will even wash their hands to prevent getting fingerprint oil on the rubber bladder seal when they poke it back into the reservoir cap.
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Thursday, March 28th, 2013 AT 7:50 AM