In dealing with fire ground math for a smooth bore tip and friction loss, there are two formulas. One that calculates the flow (gallons per minute) and one that determines the friction loss based on the hose. The sum of those answers is the pressure that the fire pumper will pump at. I want to be able to determine the flow and friction loss given a pump pressure.
The formula for flow is: $$GPM=29.71\times D^2\times\sqrt{np}\;\;where\,D\,is\,nozzle\, diameter\;and\,np\,is \;nozzle \;pressure$$ and for friction loss is: $$FL=C\times\frac{Q}{100}\times2\times\frac{L}{100}\;\;where\,C\,is\,hose\,friction\,loss\,coefficient,\;Q\,is\, GPM,\,and\,L\,is\,length $$ So, if D is 1.125, np is 50, then GPM = 266
With a C of 8 and L of 100 Then FL= 43
Resulting pump pressure is 50 + 43 = 93
If NP is 45 instead of 50 with the same D, C, and L I get 252gpm, 40 FL and pump pressure of 85
So, what I want to be able to do is calculate gallons per minute with a pump pressure of 93? (Given in this case the same parameters for D, C, and L).
I guess I could try writing computer code iterating through with different nozzle pressures, adding the result to a sorted list / map and then searching the list. But, is there an easier way?
is there an easier way? Sure. Denoting $A=\frac 1{29.71 D^2}$ and $B=\frac{2CL}{100^2}$ (and assuming that $Q$ and $GPM$ are exactly the same thing), you get the equation $A^2\times GPM^2+ B\times GPM=pump\ pressure=PP$ so, by the quadratic formula, $$ GPM=\frac{-B+\sqrt{B^2+4\times PP\times A^2}}{2A^2} $$