The relationship between flow rate (Q), pressure (P), and a sprinkler's orifice constant (K-factor) is defined by the hydraulic formula Q = K√P. To find the required pressure, this formula must be rearranged to solve for P, resulting in P = (Q/K)².

Given:

Flow (Q) = 17 gpm

K-factor (K) = 4.2

Calculation:

P = (17 gpm / 4.2)²

P = (4.0476)²

P ≈ 16.38 psi

Therefore, a pressure of 16.38 psi is required to deliver 17 gpm from a 4.2 K-factor sprinkler.

A. 4.52 psi (0.31 bar): This value is the result of an incorrect mathematical calculation and does not follow the standard hydraulic formula P = (Q/K)².

B. 9.22 psi (0.63 bar): This result is mathematically incorrect. It may stem from a misapplication of the formula, such as incorrectly manipulating the square root or the variables.

D. 36.86 psi (2.54 bar): This value is significantly higher than the correct pressure and represents a substantial error in applying the hydraulic calculation formula.

1. NFPA 13, Standard for the Installation of Sprinkler Systems (2022 ed.). Section 28.2.2.1, "Sprinkler Discharge," explicitly provides the formula for determining the pressure (p) required for a specific flow (q) from a sprinkler with a known K-factor (k): p = (q/k)².

2. SFPE, Handbook of Fire Protection Engineering (5th ed.). Chapter 21, "Automatic Sprinkler System Calculations," Section 21.3.1, "Hydraulic Calculation Principles." This section details the fundamental orifice flow equation used for sprinkler calculations as Q = K√P, which is the basis for the calculation performed. (DOI: 10.1007/978-1-4939-2562-421).