I think what you are describing is a "turbo jet" in which ALL
of the air which passed thru the engine entered the combustion camber, was burnt with fuel, then expelled out the exhaust. These were used on all the early airliners such as the Commet, B707-120 series and DC8s (early series). They were total fuel hoggs. They were replaced by the low by pass engines referred to in reply 6, where only some of the air entered the combustion chamber and the rest by passed it and went directly to the jet. Hence turbo jet, fan jet, low by pass ratio and high by pass ratio engines.
As a point of intrest the recently restored QF
-XBA was built with turbo jet engines, which were later replaced with early low bypass fan jet engines. When that was done it became a B707-138B.