GPMs of water with elevation changes.

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Using a 4" hose instead of a 2" hose can increase the flow rate when pumping water from a fire hydrant, but the extent of this increase depends on the restriction imposed by the 2" meter compared to the 500 feet of 4" pipe. The elevation rise of 500 feet equates to approximately 217 psi, which could significantly impact flow rates. It's essential to consider whether the hydrant's pressure is boosted and to have pressure drop or Cv data for the flow meter to make an informed decision. Ultimately, the choice of hose size should be based on these pressure considerations and flow restrictions. Proper authorization is required to operate a fire hydrant.
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If I have a 2" water meter restrickting flow out of a fire hydrant and want to pump water 1 mile away with an elevation rise of 500' will my GPM's at the outlet increase if I use a 4" hose coming out of the 2" meter or should I stay with 2" hose?
 
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NOTE: Only operate a fire hydrant if you are authorized to do so!

The flow rate will increase some if you run 4 inch pipe. It depends on the magnitude of the restriction of your 2" flow meter compared to 500 feet of 4 inch pipe.

Your 500 ft rise is worth about 217 psi whcih may be far larger than the meter or pipe. Is the pressure from the hydrant boosted?

Do you have pressure drop or Cv data for your flow meter?
 
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