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Example of Solving a Set of Coupled, Non-linear Equations with Units

By Philip J. Pritchard, Ph.D.

This worksheet will:

• Solve a set on nonlinear couple equations for several unknowns with different units
• Apply to fluid mechanics
• Perform using the Colebrook equation, friction factor, diameter, speed, flow rate, applied pressure, energy, Reynolds number, etc.

This worksheet illustrates PTC Mathcad's ability to solve a set of nonlinear coupled equations for several unknowns, where the unknowns each have different units.  In this example, from fluid mechanics, the unknowns are the required pipe diameter D and water speed V in the pipe to generate a specified flow rate due to an applied pressure.  In addition to the diameter D and speed V, the Reynolds number Re and friction factor f are unknowns.  The four unknowns require four equations: energy, flow rate, definition of the Reynolds number, and the Colebrook equation for the friction factor.

To begin, this worksheet lists out all the data surrounding the water flow through a pipe, including the required flow rate, applied pressure, pipe data, and estimates for the unknowns. Next the worksheet provides you with all of the equations needed for your computations including the energy equation, volume flow rate equation, definition of the Reynolds number, and the Colebrook equation for the friction factor. Next the solution is found and checked for accuracy.

All notation, formulas, data, computations, and solutions are included in this worksheet to aid you in your own calculations.