Q = (k/n) AR

^{2/3}S

^{1/2}

Q

^{2} = (k/n)

^{2}
A

^{2} R

^{4/3}S

R = A/P

Q

^{2} = [(k/n)

^{2}
A

^{10/3} S] / P

^{4/3}

S = (Q

^{2} P

^{4/3}) / [(k/n)

^{2}
A

^{10/3}]

F = V/(gD)

^{1/2}

F

^{2} = V

^{2}/(gD)

V = Q/A

V

^{2} = Q

^{2}/A

^{2}

D = A/T

F

^{2} = (Q

^{2}T) / (gA

^{3})

At critical flow:
F

^{2} = 1

Q

^{2}T = gA

^{3}

S = (Q

^{2} P

^{4/3}) / [(k/n)

^{2}
A

^{3} A

^{1/3}]

S = (g T Q

^{2} P

^{4/3}) / [g T (k/n)

^{2}
A

^{3} 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 n

^{2} (P

_{c}/T

_{c}) / (k

^{2}
R

_{c}^{1/3})

In SI Units:

S_{c} = 9.81 n^{2} (P_{c}/T_{c}) /
R_{c}^{1/3}

In U.S. Customary Units:

S_{c} = 32.17 n^{2} (P_{c}/T_{c}) / (1.486^{2}
R_{c}^{1/3})