TI-84 Evo BASIC: Hydraulic Radius and Chezy-Manning (Circular Pipes)
Introduction
The program, HYDRAD makes three calculations concerning the flow of water through a sloped pipe:
* Wetted perimeter of the pipe. This is the arc length of the pipe that is touched by the flowing water. (U.S. units: feet)
* Hydraulic radius, which is mathematically defined as the cross area of the water flow (not necessarily the entire cross area of a pipe) divided by the wetter perimeter. (U.S. units: feet)
* Flow rate of the water through the pipe. (U.S. units: cubic feet per second)
The flow rate depends on the type of material the water flows through.
The general equation is:
Q = α ÷ n * A * Y^(2/3) * √S
where:
α = coefficient which varies depending on what type of units are used. If US units, specifically feet, are used, then α = 1.49 (ft^(1/3)/s). If SI units, specifically meters, are used, then α = 1.00 (m^(1/3)/s). (Unusual powers of units.)
n = Manning’s Roughness Coefficient which is dependent on the material the water flows through. Specific values for certain materials include:
Concrete: n = 0.013
Cast Iron: n = 0.014
Steel: n = 0.010
(Lindeburg)
The coefficient is unit-less.
A = the flow area, which is the total cross-section are which water can flow through. In the case of the pipe, A = π * r^2.
Y = the hydraulic radius, or the “radius” of the wet area that the water touches. In general, the hydraulic radius is the wetted area divided by the wetted perimeter.
In the case of the pipe, the area of the wetted segment is calculated as R = r^2 * (Θ – sin Θ) ÷ 2, which Θ is the angle of the wet pipe in radians. The wetted perimeter is Wp = r * Θ. (see diagram below)
S = slope of the pipe, which should not be 0. In this case, downward slope is considered to be positive. (feet/feet or meters/meters)
TI-84 Evo BASIC Program: HYDRAD (US Units)
Code:
ClrHome
Disp “CHEZY-MANNING”, “PART-FILLED PIPE”,
“RADIANS SET”
Radian
Disp “LENGTHS IN FEET”
Input “PIPE RADIUS? “, R
Input “HEIGHT OF H20? “, H
Input “SLOPE? “, S
Menu(“SELECT PIPE MATERIAL”, “CONCRETE”, 1,
“CAST IRON”, 2, “STEEL”, 3, “OTHER N VALUE”, 4)
Lbl 1: .013 → N : Goto 0
Lbl 2: .014 → N : Goto 0
Lbl 3: .01 → N : Goto 0
Lbl 4: Input “N? “, N : Goto 0
Lbl 0
2 * cos^-1(1 – H / R) → Θ
R * Θ → W
R² * (Θ – sin(Θ)) / 2 → A
A / W → Y
(1.49 / N) * (π * R²) * Y^(2/3) * √(S) → Q
Disp “WETTED PERIMETER:”, W, “HYDRAULIC RADIUS”, Y,
“FLOW RATE:”, Q
Examples
Example 1:
pipe radius: r = 2.4 feet
water height: h = 0.9 feet
slope: 0.13
concrete pipe
wetted perimeter: 4.299191011 feet
hydraulic radius: 0.5463307184 feet
flow rate: 499.7552462 ft³/s
Example 2:
pipe radius: r = 1.5 feet
water height: h = 0.02 feet
slope: 0.4
custom n: n = 0.2
wetted perimeter: 0.4904439191 feet
hydraulic radius: 0.0132918217 feet
flow rate: 5.90983011 ft³/s
Example 3:
pipe radius: r = 7.5 feet
water height: h = 3 feet
slope: 1.01
steel pipe
wetted perimeter: 13.90942827 feet
hydraulic radius: 1.808870611 feet
flow rate: 39284.78283 ft³/s
Sources
EngineerExcel. “Chezy Equation: Application In Open Channel Flow Calculations” 2026. Retrieved July 7, 2026. https://engineerexcel.com/chezy-equation/
Lindeburg, Michael R, PE. Quick Reference for the Civil Engineering PE Exam. Professional Publications, Inc. Belmont, CA. 1999. ISBN 1-888577-47-9. pg. 19.
National Weather Service. “Normal Depth Calculator” Weather.gov > Alaska-Pacific RFC > Normal Depth Calculator. Anchorage, AK. Retrieved 5, 2026. https://www.weather.gov/aprfc/normaldepthcalc
Wikipedia. “Chézy formula” Edited March 12, 2026. Retrieved July 5, 2026. https://en.wikipedia.org/wiki/Ch%C3%A9zy_formula
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Eddie
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