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See the documentation for the AirflowNetwork:Distribution:Linkage object. The following output variables will give you mass/volume flow rates between nodes in an airflow network:

HVAC,Average,AFN Linkage Node 1 to Node 2 Mass Flow Rate [kg/s]
HVAC,Average,AFN Linkage Node 2 to Node 1 Mass Flow Rate [kg/s]
HVAC,Average,AFN Linkage Node 1 to Node 2 Volume Flow Rate [m3/s]
HVAC,Average,AFN Linkage Node 2 to Node 1 Volume Flow Rate [m3/s]