Conservation of Energy

Conservation of Energy
The conservation of energy law means that energy cannot be vanished
but it converts to other form according to:
Type of flow.
Type of fluid.
Type of dimensions or shape of container.
Energy Equation:
To get simple energy equation , some assumptions should be made:
Incompressible fluid.
Ideal fluid( frictionless fluid).
That means:
No eddies appear.
No energy dissipation.
There are Two Main Equations of Energy:
Euler s Equation
Euler Eq. could be derived for one dimensional flow by using small derivative
elements as shown in the Fig.

The forces that accelerated the element:
Pressure force at each end:
F_P=PdA-(P+dP)dA=-dPdA (1)
Where:
P= Pressure
dA= area of element at each end.
Body force:
dW=?gdsdA
The force at direction of flow:
F_ws or dws=-?gdsdA(dz/ds)
F_ws=-?gdAdz (2)
Where:
?= Fluid density
ds= length of flow pipe.
dz= difference in elevation between first and last ends of flow pipe.
g= ground acceleration.
Accelerated density:
dM=?dsdA (3)
By using Newton s second law of motion, the differential force:

dF=dM.a (4)
A= acceleration, as=v dv/ds
dF=F_P+F_ws
dF=-dPdA-?gdAdz (5)
dM.a=(?dsdA)(v dv/ds) (6)
Substitute eq.6 &5 in eq. 4:
[-dPdA-?gdAdz=(?dsdA)(v dv/ds)]÷?gdA
[dP/?g+vdv/g+dz=0] (7)
Eq.7 is Euler s Equation in fluids.