Energy and the Steady Flow Energy Equation

In outcome one, we defined and generated formulae for internal energy, flow energy and enthalpy. Two other energy forms, potential energy and kinetic energy are also present in a moving fluid and these are dealt with over the next few pages.

Potential Energy

This is the energy possessed by a mass of fluid, m, by virtue of its height Z above a given datum position, so:

Total Potential Energy = mgZ (kg x m s^-2 x m) = Nm = (J)

and for a unit mass of the fluid

Specific Potential Energy = gZ (J kg^-1)

Kinetic Energy

If a fluid is in motion then it possesses kinetic energy. Thus, for a mass of fluid m, flowing with velocity C.

Total Kinetic Energy = ½ mC² (kg x m s^-1 x m s^-1) = Nm = (J)

and for unit mass of the fluid

Specific Kinetic Energy =  (J kg^-1)

Various energy forms exist in thermodynamic systems. In certain systems they may all be present. In other systems only some may be present. Not infrequently, energy forms of insignificant value may be ignored in the solution of problems.

The Steady Flow Energy Equation

The figure below represents an open system in which a steady-flow process is taking place. At entry to the system, the working fluid possesses potential, kinetic and internal energy and entry flow work is done. During its passage through the system the working fluid is considered to take in a quantity of heat Q and do external work W.

At exit from the system the working fluid will again possess potential, kinetic and internal energy and will do flow work to leave the system.

© A.Henderson, UHI

The forms of energy associated with the moving fluid mass entering the system are:

Potential energy = $\text{mgZ1 (J)}$ 

Kinetic energy = $m\frac{c_1^2}{2}\text{ (J)}$ 

Internal energy = $U_1\text{ (J)}$ 

Flow energy = $p_1{V}_1\text{ (J)}$  

Hence, total energy of the moving fluid mass entering the system:

$mg{Z}_1+m\frac{c_1^2}{2}+U_1+p_1{V}_1$ 

Also, total energy of the moving fluid mass leaving the system

$mg{Z}_2+m\frac{c_2^2}{2}+U_2+p_2{V}_2$ 

In a steady-flow system it is considered that the mass flow rate and the total energy of the working fluid remains constant throughout the process.

Initial energy of the system  +  Energy entering the system  =  Final energy of the system  +  Energy leaving the system

→ $mg{Z}_1+m\frac{c_1^2}{2}+U_1+p_1{V}_1+Q=mg{Z}_2+m\frac{c_2^2}{2}+U_2+p_2{V}_2+W$  

This is known as the steady flow energy equation.

For unit mass (1 kg) of working fluid the equation becomes:

$g{Z}_1+\frac{c_1^2}{2}+h_1+Q=g{Z}_2+\frac{c_2^2}{2}+h_2+W$  

When the mass flow rate of working fluid (m) and rates of heat input (Q) and work output (W) are given then the steady-flow energy equation can be rearranged as follows:

$Q=W+m\left[g\right(Z_2-Z_1)=\left(\frac{c_2^2-c_1^2}{2}\right)+(u_2-u_1)+(p_2{V}_2-p_1{V}_1\left)\right]$ 

Or

$Q=W+m\left[g\right(Z_2-Z_1)=\left(\frac{c_2^2-c_1^2}{2}\right)+(h_2-h_1\left)\right]$ 

Frequently in thermodynamic problems, changes in potential energy are small compared with other energy changes or even non-existent when there is no difference between entry and exit datum levels. The gZ terms can therefore be neglected or dropped and the equation shortens to:

$Q=W+m\left[\left(\frac{c_2^2-c_1^2}{2}\right)+(h_2-h_1)\right]$ 

It is important to note that, in thermofluids, the symbol H represents total enthalpy and h represents specific enthalpy. For this reason we identify height in the PE formula by the symbol Z.

Similarly, for the KE formula, the symbol C is used for fluid velocity in order to distinguish velocity from total volume, V or specific volume, v.