Calculation of critical slope in a prismatic channel using the Manning equation, Victor Miguel Ponce, San Diego State University

Q = (k/n) AR^2/3S^1/2

Q² = (k/n)² A² R^4/3S

R = A/P

Q² = [(k/n)² A^10/3 S] / P^4/3

S = (Q² P^4/3) / [(k/n)² A^10/3]

F = V/(gD)^1/2

F² = V²/(gD)

V = Q/A

V² = Q²/A²

D = A/T

F² = (Q²T) / (gA³)

At critical flow: F² = 1

Q²T = gA³

S = (Q² P^4/3) / [(k/n)² A³ A^1/3]

S = (g T Q² P^4/3) / [g T (k/n)² A³ A^1/3]

S_c = (g P_c^4/3) / [T_c (k/n)² A_c^1/3]

S = (g P_c^4/3) / [T_c (k/n)² R_c^1/3 P_c^1/3]

S_c = [g (P_c/T_c)] / [(k/n)² R_c^1/3]

S_c = g n² (P_c/T_c) / (k² R_c^1/3)

In SI Units:

S_c = 9.81 n² (P_c/T_c) / R_c^1/3

In U.S. Customary Units:

S_c = 32.17 n² (P_c/T_c) / (1.486² R_c^1/3)