### A Differential Equation for Downhill Compressible Flow in Pipes and Its Solution

#### Abstract

A general differential equation for the downhill flow of any compressible fluid has been developed. The development of the equation was made by the combination of Euler equation for the steady flow of any fluid, the Darcy-Weisbach equation for the loss of pressure head during fluid flow in a pipe and the equation of continuity of a compressible fluid. The differential equation was first numerically solved by use of the classical fourth order Runge-Kutta method in the form of a FORTRAN program. A test of the accuracy of the solution was made by changing the direction of flow to downhill, of a problem taken from the book ,“Natural Gas Production Engineering”. The pressure drop when flow was uphill was given in the book. The pressure drop from the program was less than that in the book. This was to be expected because the pressure drop when gravity aided the flow should be less than when gravity opposed it. The program also showed that the length of a pipe can be up to 5700ft and the calculated pressure drop is still accurate. Next, the Runge-Kutta solution was translated a formula valid in any consistent set of units. The formula was tested with a problem from the book “Fluid Mechanics and Hydraulics”. The test showed the formula to be accurate. The formula was also converted to oil field units and results from it were in close agreement wit the output of the computer program.

**Keywords**: Compressible flow, Pipes, down hill