(1)	$\begin{array}{l}\displaystyle \left[\rho(p) - j_m^2 \cdot c(p) \right] \frac{dp}{dl} = \rho^2(p) \, g \, \cos \theta(l) - \frac{ j_m^2 }{2 d} \cdot f(p)\end{array}$

(2)	$\begin{array}{l}\displaystyle p(l=0) = p_0\end{array}$

and assume constant pipe inclination:

(3)	$\begin{array}{l}\displaystyle \theta(l) = \theta = \rm const\end{array}$

Let's define constant number:

(4)	$\begin{array}{l}\displaystyle G = g \cdot \cos \theta = \rm const\end{array}$

and rewrite the equation (1) as:

(5)	$\begin{array}{l}\displaystyle \frac{\left[\rho(p) - j_m^2 \cdot c(p) \right] \, dp}{\rho^2(p) \, G - \frac{ j_m^2 }{2 d} \cdot f(p)} = dl\end{array}$

The integration of the left side of (5) with the boundary condition (2) leads to:

(6)	$\begin{array}{l}\displaystyle L =\int_{p_0}^{p} \frac{ \rho(p) - j_m^2 \, c(p) }{G \, \rho^2(p) - F(\rho(p))} \, dp\end{array}$

where

$\begin{array}{l}\displaystyle F(\rho) = \frac{ j_m^2 }{2 d} \cdot f(\rho)\end{array}$

This can be further re-written as:

(7)	$\begin{array}{l}\displaystyle L = \int_{p_0}^{p} \frac{ \rho \, dp}{G \, \rho^2 - F(\rho)} - j_m^2 \cdot \int_{\rho_0}^{\rho} \frac{1}{\rho} \, \frac{d \rho}{G \, \rho^2 - F(\rho)}\end{array}$

or

(8)	$\begin{array}{l}\displaystyle L =\int_{\rho_0}^{\rho} \frac{ 1/c(\rho) - j_m^2/\rho }{G \, \rho^2 - F(\rho)} \, d\rho\end{array}$

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See also