a corrugated metal pipe of 500 mm diameter flow half-full at a slope of 0.005. What is the flow rate for this condition
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
-
PURPOSE OF THE EXERCISE:
Demonstrate the ability to analyze open-channel flow in a partially filled pipe by applying hydraulic geometry and Manning’s equation to determine the flow rate. This exercise reinforces the concept of hydraulic radius in non‐full circular conduits. -
ENGINEERING CONTEXT:
The analysis applies to culvert and drainage design, where many metal pipes convey water at partial filling. Once mastered, you can size and evaluate open-channel conduits to ensure proper conveyance of stormwater or wastewater. -
PHYSICAL CONTEXT:
The dominant effects are gravitational acceleration driving free-surface flow and viscous dissipation losses along the wetted boundary of the pipe. -
ASSUMPTIONS:
Assume steady, uniform, one-dimensional open-channel flow with water (\rho=1000\,\mathrm{kg/m^3}) flowing in a corrugated metal pipe that is half-full; the Manning formula is applicable because the flow is long and the channel slope is uniform. -
DRAWINGS:
A useful drawing would show a cross-sectional view of a circular pipe with diameter D=0.5\,\mathrm{m}, half-filled so that the water depth is 0.25\,\mathrm{m}. Mark the radius, the free surface (the chord of the semicircle, which is not wetted), and the wetted arc. Indicate the cross-sectional area of the half-circle and the wetted perimeter used to compute the hydraulic radius. -
SOLUTION TECHNIQUE:
For a half-full circular pipe the cross-sectional area and hydraulic radius are computed by noting that if the pipe has diameter
D=0.5\,\mathrm{m}\quad\text{and radius}\quad R=\frac{D}{2}=0.25\,\mathrm{m}\,,
the area occupied by the water (a half circle) is
A=\frac{\pi R^2}{2}=\frac{\pi (0.25)^2}{2}\approx0.0982\,\mathrm{m^2}\,.
The wetted perimeter is the length of the semicircular arc,
P_{\mathrm{w}}=\pi R\approx\pi(0.25)=0.7854\,\mathrm{m}\,,
so the hydraulic radius is
R_h=\frac{A}{P_{\mathrm{w}}}=\frac{0.0982}{0.7854}\approx0.125\,\mathrm{m}\,.
Manning’s equation for uniform open-channel flow is
Q=\frac{1}{n}\,A\,R_h^{2/3}\,S^{1/2}\,,
where
• n is Manning’s roughness coefficient,
• S is the channel slope,
• A and R_h are as computed above.
Assuming a typical value for a corrugated metal pipe of
n=0.015\,,
and given
S=0.005\,,
we have
R_h^{2/3}=(0.125)^{2/3}\approx0.25\quad\text{and}\quad S^{1/2}=\sqrt{0.005}\approx0.0707\,.
Thus, the estimated flow rate is
Q=\frac{1}{0.015}\times0.0982\,\mathrm{m^2}\times0.25\times0.0707\approx66.67\times0.001737\approx0.116\,\mathrm{m^3/s}\,.
- REFLECTION:
The analysis shows that, for the given slope and half‐full condition, the estimated flow rate is about 0.116\,\mathrm{m^3/s}. Although the corrugated nature of the metal may induce additional local losses, the Manning approach provides a solid first‐approximation for open-channel conveyance.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.3: Poiseuille Flow Part 3 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow Part 1 (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
i would think a manning roughness coeff for corrugated steel would be more like 0.024 – what do you think PK Bot
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is an analysis considering your suggestion of a Manning roughness coefficient of
n=0.024\,,
instead of our initial assumption of
n=0.015\,.
Recalling Manning’s equation for open‐channel flow:
Q=\frac{1}{n}\,A\,R_h^{2/3}\,S^{1/2}\,,
the flow rate is inversely proportional to the value of n. In our earlier calculation for a half‑full corrugated metal pipe we computed
Q\approx0.116~\text{m}^3/\text{s}\;,
using n=0.015. If we now adopt n=0.024, the revised flow rate becomes
Q_{\text{new}}=\frac{0.015}{0.024}\times0.116~\text{m}^3/\text{s}\approx0.0725~\text{m}^3/\text{s}\,.
This roughly 37.5% reduction in Q reflects increased viscous losses due to a rougher interior surface.
It is important to highlight that while values for n in the literature for corrugated steel often range from about 0.015 to 0.024, the exact choice may depend on installation quality, age, and the degree of corrugation. If the field data or design conditions suggest a rougher interior (and thus higher hydraulic resistance), your suggestion of n=0.024 would be more appropriate.
To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 11.3: Poiseuille Flow Part 3 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow Part 1 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
PK Bot, I don’t talk about the Manning equation in my text. Can you
- show the Manning equation
- comment on how it is a Re-independent analysis, usually for systems with rough surfaces so epsilon/D dominates the viscous dissipation coefficient. It is typically used for rivers, storm sewers, etc.
- comment on inconsistencies in language. In Manning’s equation, the hydraulic radius of a circle isn’t equal to the radius of a circle. In my text, the hydraulic radius is defined to be equal a circle’s radius and the hydraulic diameter is defined to be equal to a circle’s diameter
- write a formula for f_D in terms of g, n, and R_h
- list typical Manning roughness coefficients for concrete, earth, riverbeds, dense vegetiation, smooth metal, corrugated metal, etc.
- comment on how similar you would expect an answer to be when calculated with a Haaland or Colebrook-White approach vs. a Manning approach.
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
Below is a discussion addressing your points.
- Manning’s Equation for Uniform Open‐Channel Flow is usually written as
Q=\frac{1}{n}\,A\,R_h^{2/3}\,S^{1/2}\,,
where
• Q is the volumetric flow rate,
• n is Manning’s roughness coefficient,
• A is the cross‐sectional area of flow,
• R_h is the hydraulic radius (defined as the ratio of A to the wetted perimeter), and
• S is the channel slope (assumed equal to the energy grade line slope).
-
Unlike viscous dissipation factor formulations based directly on the Navier–Stokes equations, Manning’s analysis is Reynolds‐independent. That is, for channels in which the roughness dominates (so that the nondimensional roughness \varepsilon/D controls viscous dissipation), the empirical relationship is valid regardless of the Reynolds number. It is common practice for rivers, storm sewers, and other natural or civil engineering open-channel systems where inertial effects are secondary to surface roughness.
-
In many texts—including Professor Kirby’s—you may notice a language inconsistency when comparing definitions. In the conventional Manning approach, the hydraulic radius is defined as
R_h=\frac{A}{P_{\mathrm{w}}}\,,
so for a full circular pipe one obtains R_h=\frac{\pi D^2/4}{\pi D}=\frac{D}{4}. In Professor Kirby’s text, however, the hydraulic radius is sometimes taken to be equal to the pipe’s radius (i.e. D/2) and a separate “hydraulic diameter” is defined as equal to the actual diameter. This distinction is important when comparing results; one must always check which convention is being used before applying an equation.
- An expression for the viscous dissipation (or Darcy) factor f_D may be derived by identifying the balance between wall shear work and the gravitational driving force. In uniform flow the bed shear stress is given by
\tau_w=2\,\rho\, U^2\, f_D\,,
and for open channels the shear stress can also be expressed as
\tau_w=\rho g R_h\, S\,.
Equating these and solving for f_D gives
2\,\rho U^2\, f_D=\rho g R_h\, S\,.
Using Manning’s expression for the mean velocity
U=\frac{1}{n}\, R_h^{2/3}\, S^{1/2}\,,
substitute and cancel common factors to obtain
2\,\left(\frac{1}{n^2}\,R_h^{4/3}\,S\right)\, f_D=\; g R_h S\,.
This immediately leads to
f_D=\frac{8g\,n^2}{R_h^{1/3}}\,.
- Typical values for Manning’s roughness coefficient are as follows:
• Smooth concrete (finished): approximately 0.012 to 0.015
• Earth (natural channels): roughly 0.020 to 0.035
• Riverbeds (with stones and mild vegetation): approximately 0.030 to 0.050
• Dense vegetation (weeds or undergrowth): on the order of 0.080 to 0.100
• Smooth metal surfaces: around 0.010 to 0.013
• Corrugated metal pipes: often in the range 0.015 to 0.024
- When comparing results obtained via a Haaland or Colebrook–White approach with those from Manning’s equation, one should note that both methods aim to characterize flow resistance in the presence of wall roughness. In fully rough turbulent flows—when the nondimensional roughness \varepsilon/D dominates—both approaches yield friction factors that are nearly independent of Reynolds number. Although Manning’s equation is an empirical fit intended for open-channel conditions, a well‐conducted calculation using either the Colebrook–White (or Haaland) relation or the Manning approach will produce answers of similar order of magnitude (typically within 10–20% of each other) when rough-wall effects are dominant.
To formulate this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2). Below, I have organized the sections based on their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 6.4: Bernoulli constant drop in pipes Part 3 (HIGHLY RELEVANT)
- Appendix C.2: Dimensional Analysis to Generalize Experiments (HIGHLY RELEVANT)
- Chapter 6.2: Modified Bernoulli equation Part 2 (HIGHLY RELEVANT)
- Chapter 11.3: Poiseuille Flow Part 3 (HIGHLY RELEVANT)
- Chapter 6.4: Bernoulli constant drop in pipes Part 2 (HIGHLY RELEVANT)
Disclaimer:
See What is Professor Kirby Bot for information about Professor Kirby Bot and its answers.
my text isn’t inconsistent, Manning just used a goofy definition of “hydraulic radius” in the late 19th century