Chapter 9, Dimensional Analysis, Similitude and Hydraulic Models Video Solutions, Civil Engineering Hydraulics: Essential Theory with Worked Examples

Problem 1

The head loss of water of kinematic viscosity $1 \times 10^{-6} \mathrm{~m}^2 / \mathrm{s}$ in a 50 mm diameter pipeline was 0.25 m over a length of 10.0 m at discharge of $2.0 \mathrm{l} / \mathrm{s}$. What is the corresponding discharge and hydraulic gradient when oil of kinematic viscosity $8.5 \times 10^{-6}$ flows through a 250 mm diameter pipeline of the same relative roughness?

Alexander Allen

Numerade Educator

08:12

Problem 2

Find the pressure drop at the corresponding speed in a pipe 25 mm in diameter, 30 m long conveying water at $10^{\circ} \mathrm{C}$ if the pressure head loss in a 200 mm diameter smooth pipe 300 m long in which air is flowing at a velocity of $3 \mathrm{~m} / \mathrm{s}$ is 10 mm of water. Density of air $=1.3 \mathrm{~kg} / \mathrm{m}^3$, dynamic viscosity $=1.77 \times 10^{-5} \mathrm{Ns} / \mathrm{m}^2$. Dynamic viscosity of water $=1.3 \times 10^{-3} \mathrm{Ns} / \mathrm{m}^2$.

Ronald Prasad

Numerade Educator

08:12

Problem 3

A 50 mm diameter pipe is used to convey air at $4^{\circ} \mathrm{C}$ (density $=$ $1.12 \mathrm{~kg} / \mathrm{m}^3$ and dynamic viscosity $1.815 \times 10^{-5} \mathrm{Ns} / \mathrm{m}^2$ ) at a mean velocity of $20 \mathrm{~m} / \mathrm{s}$.

Calculate the discharge of water at $20^{\circ} \mathrm{C}$ for dynamic similarity and obtain the ratio of the pressure drop per unit length in the two cases.

Ronald Prasad

Numerade Educator

02:31

Problem 4

If, in modelling a physical system, the Reynolds and Froude numbers are to be the same in the model and prototype determine the ratio of kinematic viscosity of the fluid in the model to that in the prototype.

James Kiss

Numerade Educator

08:09

Problem 5

The sequent depth, $y_x$, of a hydraulic jump in a rectangular channel is related to the initial depth $y_i$, the discharge per unit width, $q, g$ and $p$. Express the ratio $y_s / y_i$ in terms of a non-dimensional group and compare with the equation developed from momentum principles:

$$
y_s=\frac{y_i}{2}\left(\sqrt{1+F_i^2}-1\right)
$$

Vidhi Bhatt

Numerade Educator

03:57

Problem 6

A $60^{\circ}$ V-notch is to be used for measuring the discharge of oil having a kinematic viscosity 10 times that of water. The notch was calibrated using water. When the head over the notch was 0.1 m the discharge was $2.54 \mathrm{l} / \mathrm{s}$.
Determine the corresponding head and discharge when the notch is used for oil flow measurement.

Ronald Prasad

Numerade Educator

07:18

Problem 7

The airflow and wind effects on a bridge structure are to be studied on a 1:25 scale model in a pressurised wind tunnel in which the air density is 8 times that of air at atmospheric pressure and at the same temperature. If the bridge structure is subjected to wind speeds of $30 \mathrm{~m} / \mathrm{s}$ what is the corresponding wind speed in the wind tunnel? What force on the prototype corresponds with a 1400 N force on the model? (Note the dynamic viscosity of air is unaffected by pressure changes provided the temperature remains constant.)

Khoobchandra Agrawal

Numerade Educator

04:10

Problem 8

A rotodynamic pump is designed to operate at $1450 \mathrm{rev} / \mathrm{min}$ and to develop a total head of 60 m when discharging $250 \mathrm{l} / \mathrm{s}$.

The following characteristics of a 1:4 scale model were obtained from tests carried out at $1800 \mathrm{rev} / \mathrm{min}$.

Obtain the corresponding characteristics of the prototype and state whether, or not, it meets its design requirements.
$$
\begin{array}{lccccc}
\hline \mathrm{Q}_{\mathrm{m}}(1 / \mathrm{s}) & 0 & 2 & 4 & 6 & 8 \\
\mathrm{H}_{\mathrm{m}}(\mathrm{m}) & 8 & 7.6 & 6.4 & 4.2 & 1.0 \\
\hline
\end{array}
$$

Narayan Hari

Numerade Educator

Problem 9

(a) Show that the power output, P, of a hydraulic turbine expressed in terms of non-dimensional groups in the form

$$
\mathrm{P}=\rho \mathrm{N}^3 \mathrm{D}^5 \phi\left[\frac{\mathrm{Q}}{\mathrm{ND}^3}, \frac{\mathrm{D}}{\mathrm{B}}, \frac{\mathrm{N}^2 \mathrm{D}^2}{\mathrm{gH}}, \frac{\rho \mathrm{ND}^2}{\mu}\right]
$$

Derive an expression for the specific speed of a hydraulic turbine.
(b) A 1:20 scale model of a hydraulic turbine operates under a constant head of 10 m . the prototype will operate under a head of 150 m at a speed of $300 \mathrm{rev} / \mathrm{min}$. When running at the corresponding speed the model generates 1.2 kW at a discharge of $13.6 \mathrm{I} / \mathrm{s}$. Determine the corresponding speed, power output and discharge of the prototype.

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01:03

Problem 10

The wave action and forces on a proposed sea wall are to be studied on a $1: 10$ scale model. The design wave has a period of 9 seconds and a height from crest to trough of 5 m . The depth of water in front of the wall is 7 m .

Assuming that the wave is a gravity wave in shallow water and that the celerity $\mathrm{c}=\sqrt{\mathrm{gy}}$ where y is the water depth determine the wave period, wavelength and wave height to be reproduced in the model. If a force of 4 kN due to wave breaking on a 0.5 m length of the model sea wall were recorded, what would be the corresponding force per unit length on the prototype?

Narayan Hari

Numerade Educator