First Law in Terms of Enthalpy dH = dQ + Vdp
The enthalpy is defined to be the sum of the internal energy E plus the product of the pressure p and volume V. In many thermodynamic analyses the sum of the internal energy U and the product of pressure p and volume V appears, therefore it is convenient to give the combination a name, enthalpy, and a distinct symbol, H.
H = U + pV
See also: Enthalpy
The first law of thermodynamics in terms of enthalpy show us, why engineers use the enthalpy in thermodynamic cycles (e.g. Brayton cycle or Rankine cycle).
The classical form of the law is the following equation:
dU = dQ – dW
In this equation dW is equal to dW = pdV and is known as the boundary work.
dH – pdV – Vdp = dQ – pdV
We obtain the law in terms of enthalpy:
dH = dQ + Vdp
or
dH = TdS + Vdp
In this equation the term Vdp is a flow process work. This work, Vdp, is used for open flow systems like a turbine or a pump in which there is a “dp”, i.e. change in pressure. There are no changes in control volume. As can be seen, this form of the law simplifies the description of energy transfer. At constant pressure, the enthalpy change equals the energy transferred from the environment through heating:
Isobaric process (Vdp = 0):
dH = dQ → Q = H_{2} – H_{1}
At constant entropy, i.e. in isentropic process, the enthalpy change equals the flow process work done on or by the system:
Isentropic process (dQ = 0):
dH = Vdp → W = H_{2} – H_{1}
It is obvious, it will be very useful in analysis of both thermodynamic cycles used in power engineering, i.e. in Brayton cycle and Rankine cycle.
Example: First Law of Thermodynamics and Brayton Cycle
Let assume the ideal Brayton cycle that describes the workings of a constant pressure heat engine. Modern gas turbine engines and airbreathing jet engines also follow the Brayton cycle. This cycle consist of four thermodynamic processes:

isentropic compression – ambient air is drawn into the compressor, where it is pressurized (1 → 2). The work required for the compressor is given by W_{C} = H_{2} – H_{1}.
 isobaric heat addition – the compressed air then runs through a combustion chamber, where fuel is burned and air or another medium is heated (2 → 3). It is a constantpressure process, since the chamber is open to flow in and out. The net heat added is given by Q_{add} = H_{3 }– H_{2}
 isentropic expansion – the heated, pressurized air then expands on turbine, gives up its energy. The work done by turbine is given by W_{T} = H_{4} – H_{3}
 isobaric heat rejection – the residual heat must be rejected in order to close the cycle. The net heat rejected is given by Q_{re} = H_{4 }– H_{1}
As can be seen, we can describe and calculate (e.g. thermodynamic efficiency) such cycles (similarly for Rankine cycle) using enthalpies.
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