At the inlet boundary of a fluid domain, the Inflow boundary condition defines a heat flux that accounts for the energy that would normally be brought by the fluid flow if the channel upstream to the inlet was modeled.

The application of the Danckwerts condition on the enthalpy allows to express the normal conductive heat flux at the inlet boundary as proportional to the flow rate ρu and the enthalpy variation ΔH between the upstream conditions and inlet conditions:

The enthalpy variation between the upstream conditions and inlet conditions, ΔH, depends in general both on the difference in temperature and in pressure, and is defined as:

where Tupstream is the upstream temperature, T is the inlet temperature, pustream is the upstream absolute pressure, pA is the inlet absolute pressure, Cp is the fluid heat capacity at constant pressure, ρ is the fluid density, and αp is its coefficient of thermal expansion. See Equation 4-5 for details about the definition of the enthalpy.

For large flow rates, advective heat transfer dominates over conductive heat transfer at the inlet. In this case, the left-hand side of Equation 4-19 is small compared to its right-hand side. As the heat capacity Cp is positive, in the absence of pressure contribution to the enthalpy, this induces the following constraint on temperature

For low flow rates or in the presence of large heat sources or sinks next to the inlet, the conductive heat flux cannot be neglected. The first integral in Equation 4-20 has for effect to adjust the inlet temperature to balance the energy brought by the flow at the inlet and the energy transferred by conduction from the interior.

Pressure losses in the virtual channel upstream to the inlet boundary are handled through the pressure contribution to the enthalpy (the second integral in Equation 4-20).