General Energy Equations
Bernoulli's equation is one of the most important relations in fluid mechanics but it only works under certain conditions, such as no shaft work and negligible heat transfer. However many situations involve addition of energy to a system (such as with pumps) or taking energy out of a system (such as in a turbine). The following is an analysis of the First Law of Thermodynamics to yield a general energy equation for fluids.
Contents 
Textbook Readings
Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 212  221.
Notation
Since a lot of the notation used in the analysis comes from Thermodynamics, a brief introduction is needed. When we talk about energy transfer, we usually talk about energy transfer by heat, denoted by Q, and work, denoted by W, both given in Joules (J). In addition, all fluids have their own internal energy, denoted by U, also given in Joules. However in many applications, it is easier to report these values as specific values, that is as units of energy per unit mass. Specific values are denoted by lower case letters so that specific heat transfer is denoted by q, specific work transfer by w and specific internal energy by u. In other situations, it is more convenient to report the energy transferred as units of energy per second (or power). These are denoted by the usual symbol with a dot on top, so that heat transfer is denoted by and work is denoted by Ẇ.
In addition, all fluids have a property known as enthalpy, denoted H (J/K), or in specific terms as h (J/kg.K). Enthalpy is a combinational property of a fluid's pressure, volume and internal energy, given by H = U + PV, or in specific terms, as h = u + Pv.
In general, all specific values can be converted to rel values by multiplying through by the mass, m. In the same way, all specific values, multiplied by the mass flow rate, ṁ, will give a result in Joules per second, or Power.
To summarise:
Energy transfers are denoted by:  Internal Energies and Enthalpies are denoted by: 








 

First Law of Thermodynamics
^{[1]}Under steady flow and a control volume, the First Law of Thermodynamics can be expressed as:
That is, in steady flow, the net rate of energy transfer to a control volume by heat and work transfer is equal to the difference between the rates of outgoing and ingoing energy flows by mass flow.
In most cases, only one inlet and one outlet exist, so that the mass flow rate in is the same as the mass flow rate out. The First Law then reduces to:
By dividing through by the mass flow rate, the equation can be rewritten in terms of specific values as:
Now, by noting that enthalpy, h, is defined as h = u + Pv = u+P/ρ and substituting and rearranging we get:
Now, the terms on the right hand side, u_{2}  u_{1}  q, represents the mechanical energy loss in the system. We can therefore define:
 e_{loss} = u_{2}  u_{1}  q
By noticing that work is transferred in to a system through pumps, and out through turbines, we can rewrite the equation as:
By multiplying through my the mass flow rate, we arrive at the general energy formula for fluids:
General Energy Equation in Terms of Heads
^{[2]}In a similar way to Bernoulli's equation, we can divide the general energy equation by the acceleration due to gravity to give all terms in terms of meters, or heads. The equation can be rewritten as:
Special Case: Incompressible Flows and No Work
^{[3]}Under incompressible flow, a fluid's density remains constant (ρ_{1} = ρ_{2}) and if there are no pumps or turbines adding energy into the system, then the general energy equation collapses into the familiar Bernoulli's equation:
Kinetic Energy Correction Factor
^{[4]}Mathematical analysis shows that the kinetic energy of a fluid given by (1/2)(V^{2} is not exact. Hence a correction factor is added to the general energy equation to correct the difference. This correction factor is denoted by alpha (α).
It is often the case that the correction factor can be estimated to be 1 since:
 Most cases deal with turbulent flow, for which α is very close to 1
 Kinetic energy term is usually small in comparison to flow and elevation energy terms
However it is advised to always add the correction factor as sometimes it is essential for correct analysis. The adapted general energy equation becomes:
and
References
"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).