The pressure drop in a long, uniform length of pipe is a function of fluid density, mean velocity in the pipe, the diameter of the pipe, viscosity, the length of the pipe, and the equivalent sand grain roughness (units of length) of the pipe. The equation is shown below. Develop the dimensional matrix using M (mass), L (length), and T (time) for this phenomenon, and determine its rank. Ap = f(p.V.D.µ.L.ɛ)
The pressure drop in a long, uniform length of pipe is a function of fluid density, mean velocity in the pipe, the diameter of the pipe, viscosity, the length of the pipe, and the equivalent sand grain roughness (units of length) of the pipe. The equation is shown below. Develop the dimensional matrix using M (mass), L (length), and T (time) for this phenomenon, and determine its rank. Ap = f(p.V.D.µ.L.ɛ)
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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![The pressure drop in a long, uniform length of pipe is a function of fluid density, mean velocity in the pipe, the diameter of the pipe, viscosity, the length of the pipe, and the equivalent sand grain roughness (units of length) of the pipe. The equation is shown below. Develop the dimensional matrix using M (mass), L (length), and T (time) for this phenomenon, and determine its rank.
\[ \Delta p = f \left( \rho, \overrightarrow{V}, D, \mu, L, \varepsilon \right) \]
Here, \(\Delta p\) represents the pressure drop, \(\rho\) is the fluid density, \(\overrightarrow{V}\) is the mean velocity in the pipe, \(D\) is the diameter of the pipe, \(\mu\) is the viscosity, \(L\) is the length of the pipe, and \(\varepsilon\) is the equivalent sand grain roughness.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ebcf202-e71c-4e7a-b7be-45ee67012454%2Fee4abe35-7a31-44db-a2d1-576b70b27dd1%2F5fzis2_processed.png&w=3840&q=75)
Transcribed Image Text:The pressure drop in a long, uniform length of pipe is a function of fluid density, mean velocity in the pipe, the diameter of the pipe, viscosity, the length of the pipe, and the equivalent sand grain roughness (units of length) of the pipe. The equation is shown below. Develop the dimensional matrix using M (mass), L (length), and T (time) for this phenomenon, and determine its rank.
\[ \Delta p = f \left( \rho, \overrightarrow{V}, D, \mu, L, \varepsilon \right) \]
Here, \(\Delta p\) represents the pressure drop, \(\rho\) is the fluid density, \(\overrightarrow{V}\) is the mean velocity in the pipe, \(D\) is the diameter of the pipe, \(\mu\) is the viscosity, \(L\) is the length of the pipe, and \(\varepsilon\) is the equivalent sand grain roughness.
Expert Solution
Step 1: Introduction:
To develop the dimensional matrix for the given phenomenon, we can use the Buckingham π theorem, which states that if you have
- 'n' variables and
- 'k' fundamental dimensions
- 'n - k' dimensionless π (Pi) groups.
In this case,
- 6 variables (Δp, ρ, V, D, μ, L, ε)
- 3 fundamental dimensions (M, L, T).
So, we have:
- n = 6 variables (Δp, ρ, V, D, μ, L, ε)
- k = 3 fundamental dimensions (M, L, T)
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