Airflow at takeoff
The CFM56-5C, the largest CFM56, powers A340. Its thrust is about 145 kN at takeoff. It has an air mass flow of 485 kg/s (source). A A340 has four engines.
Fortunately this thrust is not needed after takeoff and the engine airflow is considerably lower in cruise. I don't have the actual figure but we can do a quick estimate using the fuel consumption. It can be assumed relatively safely that air is in proportion of fuel.
According to this document, for the CFM56-5C:
- T/O thrust is 150 kN, fuel consumption 2.3 kg/s
- Cruise thrust is 40 kN, fuel 0.5 kg/s.
(meaning cruise at 27% T/O thrust requires 22% of the T/O fuel, the engine is more efficient in altitude.)
Airflow in cruise
So we can roughly estimate air mass flow in cruise to be about 485 * 0.22 = 107 kg/s.
This is supported by figures found online in non authoritative sources, like page 245 of this book:
Bypass flow vs core flow
After the fan, air is separated in two flows: core (hot/primary) and bypass (cold/secondary) according to the bypass ratio.
The CFM56-5C bypass ratio is 6.4:1. So:
The part used in the core for combustion, bleed air and core cooling is 485/7.4 = 66 kg/s
The part bypassing the core, and providing only thrust by fan acceleration is 419 kg/s.
For comparison the Trent XWB (A350 XWB), a larger engine developing 430 kN at takeoff, has an air mass flow of 1,440 kg/s (source).