Hydraulic valve lifters are a wonderful invention introduced by Pierce-Arrow in 1933. Briefly, this is what they do:
Four-cycle engines have a camshaft which opens and closes the intake and exhaust valves through connecting parts that comprise the rest of the valvetrain. Because the optimum valve opening is relatively precise, and because the valvetrain parts are subject to thermal expansion and contraction, a specific amount of space known as valve lash must be provided on a cold engine.
Hydraulic valve lifters or tappets are also known as hydraulic lash adjusters, because that is what they do. When a valve is closed oil can enter the lifter pushing a cylinder within it until the rocker arm (in the case of overhead valve engines) contacts the valve stem, producing zero valve lash. When the camshaft pushes on the tappet its fill port is blocked, so it functions as a solid lifter would. This process is automatic, and does away with the need for periodic valve adjustment.
This makes for a very quiet valvetrain. Except in rare conditions where the tappet "pumps up," it also prevents serious damage to valves and even pistons that can occur when the valve lash is set too tight. Other things being equal engines equipped with hydraulic lifters tend to be durable, since the pitfalls of improper valve adjustment are avoided.
Of course, hydraulic lifters do wear out. How long they will last depends on engine design, maintainence, and conditions of use. As a general rule of thumb auto engines with 75-100 K miles are candidates for valve lifter replacement. When you begin to hear valve noises, you should inspect the entire valvetrain, replacing parts as needed. There is no reason most engines won't go 400 K miles if this is done.
This is a surprisingly easy process with inline engines, although it's a bit more complicated than changing spark plugs or oil filters. With a V-6 or V-8 engine, you may have to remove the intake manifold, in which case you should be familiar with the operation of a torque wrench. Depending upon your specific engine, you may have to remove a number of parts to gain access to the valve cover and pushrod cover. Consult your repair manual for details.
On a typical OHV engine, removing the valve cover will reveal the rocker arms, the pushrods, and the valvestems with their associated springs and keepers. These will be filthy, so it's a great benefit to clean them now using aerosol carburetor cleaner for starters. Since carb cleaner tends to be corrosive, I like to follow that with a thin coating of Marvel Mystery Oil.
Much is made of the importance of keeping the pushrods in their original positions and alignments. This is because small differences in wear patterns will only work against you if you mix them up. It's probably a good idea to just replace the lot of them. Pushrods are pretty cheap compared to the labor involved in getting to them, and they are likely to be filthy inside and out. You want your valvetrain to wind up at the tight end of the specified valve lash range -- wear will tend to make it looser over time.
Once you've removed and either catalogued or discarded your pushrods, you can access the tappets by reaching down through the cylinder head, preferably with a tappet remover, but you can't get them out. To do that, you have to remove the pushrod cover. This will also be filthy, and needs a thorough cleaning. Oil lubricates the entire valvetrain, as well as acting as the hydraulic fluid in the tappets, so cleanliness is essential to a lasting repair.
Pull the old tappets, and replace them with new ones. Coat the new tappet with engine assembly lube or regular oil. You may want to coat the bottom where the tappet contacts the cam with a bit of axle grease. You don't want any excessive friction or galling from a dry start, so apply lube generously wherever new parts contact anything. Reassemble the engine in the opposite order you dismantled it.
This is only intended as a general guide, but you can see that depending on your specific engine, there are relatively few steps to this whole procedure, and you should be able to manage it if you take your time and address any unforseen "bumps in the road" as they occur. Given that your beloved car or truck might last longer than you do it's a pretty worthy project.
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