Editing Enthalpy (section)

===Open systems===
In [[thermodynamic]] [[Open system (systems theory)|open systems]], mass (of substances) may flow in and out of the system boundaries. The first law of thermodynamics for open systems states: The increase in the internal energy of a system is equal to the amount of energy added to the system by mass flowing in and by heating, minus the amount lost by mass flowing out and in the form of work done by the system:
<math display="block">
 \mathrm{d}U = \delta Q + \mathrm{d}U_\text{in} - \mathrm{d}U_\text{out} - \delta W,
</math>
where {{mvar|U}}{{sub|in}} is the average internal energy entering the system, and {{mvar|U}}{{sub|out}} is the average internal energy leaving the system.

[[Image:First law open system.svg|250px|thumb|right|During [[Steady-state (chemical engineering)|steady, continuous]] operation, an energy balance applied to an open system equates shaft work performed by the system to heat added plus net enthalpy added]]

The region of space enclosed by the boundaries of the open system is usually called a [[control volume]], and it may or may not correspond to physical walls. If we choose the shape of the control volume such that all flow in or out occurs perpendicular to its surface, then the flow of mass into the system performs work as if it were a piston of fluid pushing mass into the system, and the system performs work on the flow of mass out as if it were driving a piston of fluid. There are then two types of work performed: ''flow work'' described above, which is performed on the fluid (this is also often called ''{{mvar|pV}} work''), and ''mechanical work'' (''shaft work''), which may be performed on some mechanical device such as a turbine or pump.

These two types of work are expressed in the equation
<math display="block">
 \delta W = \mathrm{d}(p_\text{out} V_\text{out}) - \mathrm{d}(p_\text{in} V_\text{in}) + \delta W_\text{shaft}.
</math>
Substitution into the equation above for the control volume (cv) yields
<math display="block">
 \mathrm{d}U_\text{cv} = \delta Q + \mathrm{d}U_\text{in} + \mathrm{d}(p_\text{in} V_\text{in}) - \mathrm{d}U_\text{out} - \mathrm{d}(p_\text{out} V_\text{out}) - \delta W_\text{shaft}.</math>

The definition of enthalpy {{mvar|H}} permits us to use this [[thermodynamic potential]] to account for both internal energy and {{mvar|pV}} work in fluids for open systems:
<math display="block">
 \mathrm{d}U_\text{cv} = \delta Q + \mathrm{d}H_\text{in} - \mathrm{d}H_\text{out} - \delta W_\text{shaft}.
</math>

If we allow also the system boundary to move (e.g. due to moving pistons), we get a rather general form of the first law for open systems.<ref>
{{cite book
 |first1=M. J. |last1=Moran
 |first2=H. N. |last2=Shapiro
 |year=2006
 |title=Fundamentals of Engineering Thermodynamics |edition=5th
 |url=https://archive.org/details/fundamentalsengi00mora_077
 |url-access=limited
 |publisher=John Wiley & Sons
 |isbn=9780470030370
 |page=[https://archive.org/details/fundamentalsengi00mora_077/page/n141 129]
}}
</ref>
In terms of time derivatives, using [[Notation for differentiation#Newton's notation|Newton's dot notation]] for time derivatives, it reads:
<math display="block">
 \frac{\mathrm{d}U}{\mathrm{d}t} = \sum_k \dot Q_k + \sum_k \dot H_k - \sum_k p_k\frac{\mathrm{d}V_k}{\mathrm{d}t} - P,
</math>
with sums over the various places {{mvar|k}} where heat is supplied, mass flows into the system, and boundaries are moving. The {{mvar| {{overset|'''.'''|H}}{{sub|k}}}} terms represent enthalpy flows, which can be written as
<math display="block">
 \dot H_k = h_k \dot m_k = H_\text{m} \dot n_k,
</math>
with <math>\dot m_k</math> the mass flow and <math>\dot n_k</math> the molar flow at position {{mvar|k}} respectively. The term {{math|d''V''{{sub|''k''}}/d''t''}} represents the rate of change of the system volume at position {{mvar|k}} that results in {{mvar|pV}} power done by the system. The parameter {{mvar|P}} represents all other forms of power done by the system such as shaft power, but it can also be, say, electric power produced by an electrical power plant.

Note that the previous expression holds true only if the kinetic energy flow rate is conserved between system inlet and outlet.{{clarify|reason=This new type of energy, kinetic energy, was not mentioned before. Is it part of {{mvar|U}}? Does it need to be conserved, or just the net flow across boundary be zero?|date=March 2015}} Otherwise, it has to be included in the enthalpy balance. During [[Steady-state (chemical engineering)|steady-state]] operation of a device (see [[Turbine]], [[Pump]], and [[Engine]]), the average {{math|d''U''/d''t''}} may be set equal to zero. This yields a useful expression for the average [[Power (physics)|power]] generation for these devices in the absence of chemical reactions:
<math display="block">
 P = \sum_k \big\langle \dot Q_k \big\rangle
  + \sum_k \big\langle \dot H_k \big\rangle
  - \sum_k \left\langle p_k \frac{\mathrm{d}V_k}{\mathrm{d}t} \right\rangle,
</math>
where the [[angle bracket]]s denote time averages. The technical importance of the enthalpy is directly related to its presence in the first law for open systems, as formulated above.