Early 240D (1974 and 1975) and the preceeding car, the 220D, had a mechanically driven vacuum pump that exhausted into the intake manifold. The pump has a diaphragm that has engine oil on one side, and if it fails, at idle it will suck engine oil into the intake manifold. Enough to make a thick billowing white cloud. On the road at speed there is no manifold vacuum in a Diesel so the smoking is much less dramatic. Come to a stop though, and you can be lost in a cloud of thick smoke.

I am not sure if the 1981 240D has such a device or not, however, if it does you will be able to see a ~3/4 inch thick nylon tube that runs from the vacuum pump on the front of the block to the intake manifold. The diaphragm is cheap and relatively easy to replace, so, if this is your problem it might be a blessing.

Other than that, smoking from a car that starts well and has power is an enigma. There must be some other aspect of performance that is affected by the smoking, which is either too much fuel or engine oil consumption. Neither is a likely event in an otherwise healthy engine.

Good luck and let us know how the car behaves on the road. Jim

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Own:
1986 Euro 190E 2.3-16 (291,000 miles),
1998 E300D TurboDiesel, 231,000 miles -purchased with 45,000,
1988 300E 5-speed 252,000 miles,
1983 240D 4-speed, purchased w/136,000, now with 222,000 miles.
2009 ML320CDI Bluetec, 89,000 miles

Owned:
1971 220D (250,000 miles plus, sold to father-in-law),
1975 240D (245,000 miles - died of body rot),
1991 350SD (176,560 miles, weakest Benz I have owned),
1999 C230 Sport (45,400 miles),
1982 240D (321,000 miles, put to sleep)