NOT COMPLETED. Continued documentation

Author: Bjarne Børresen

OpenHPL/Resources/Documents/Developer_docs/OpenHPL_Pipe.tex

@@ -222,19 +222,35 @@ \subsection{Further details}
 %
 \subsection{Friction loss of variable area pipe}
 %
-In general it is difficult to find a closed form equation for the friction force for a pipe with varying area that is completely universal.
+In general it is difficult to find a closed form equation for the friction force for a pipe with varying area that is completely universal. However, if the change i area is small compared to the pipe length we can approximate the 
 %
 \[
-F_{f}= A \cdot \rho \cdot  g \cdot h_{f}= A \cdot \rho \cdot g \left[ f \cdot \left(\frac{L}{D}\right) \frac{v^{2}}{2g} \right] =\frac{1}{A} \cdot \rho \cdot g \left[ f \left(\frac{L}{D}\right) \frac{Q^{2}}{2g} \right]
+F_{f}= A \cdot \rho \cdot  g \cdot h_{f}= A \cdot \rho \cdot g \left[ f \cdot \left(\frac{L}{D}\right) \frac{v^{2}}{2g} \right] =\frac{1}{A} \cdot \rho \cdot g \left[ f \cdot \left(\frac{L}{D}\right) \frac{Q^{2}}{2g} \right]
+\]
+%
+\[
+F_{f}= A \cdot \rho \cdot  g \cdot h_{f}= \int \frac{1}{A(x)} \cdot \rho \cdot g \left[ f \cdot \left(\frac{1}{D(x)}\right) \frac{Q^{2}}{2g} \right] dx
 \]
  Assuming a linear diameter distribution from the inlet to the outlet
  \[
  D\left(x\right)=D_{1}+\frac{\left(D_{2}-D_{1} \right)}{L}\cdot x
  \]
- or
+ and
+ \[
+  A\left(x\right)=\frac{\pi}{4} D^{2}
+ \]
+ %
  \[
   D\left(x\right)=D_{1}+\frac{\left(D_{2}-D_{1} \right)}{L}\cdot x
  \]
+%
+\[
+F_{f}= A \cdot \rho \cdot  g \cdot h_{f}=  \left[\frac{\rho \cdot L \cdot Q^{2}}{2} \right] \int \frac{1}{A(x)\cdot D(x)} dx
+\]
+%
+\[
+F_{f}= A \cdot \rho \cdot  g \cdot h_{f}=  \left[\frac{2 \rho \cdot L \cdot Q^{2}}{\pi} \right] \int \frac{1}{D^{3}(x)} dx
+\]
 %
  In principle the friction factor $f$ is a function of the Reynolds number $Re$ and relative roughness $\epsilon/D$. Ignoring this, and setting both to the constant value based on the mean value