Conservation of energy in a pipe

To derive the equation for the conservation of eneigy, we proceed in the same way as we derived the conservation of mass equation, regarding all variables as functions of both time and distance. Hence at any time, r, the values at the reference distance x are  

Rgure 3.4 Element of pipe at an angle e to the horizontal, subject to a heat flux and doing useful work variations in pressure, specific volume, velocity and specific internal energy. 

The principle of conservation of energy is the same as stated at the beginning of Section 3.4, namely  

Noting first that pAc = (Ac/v)pv, and then using the identity h= u+ pv allows us to re-express (3.60) as  

Dividing by St and letting St — 0 gives the partial differential equation  

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