A cylindrical pipe has inner diameter of 14 cm. Water flows through it at a rate of 154 litres per minute. What is the speed of water in km/hr? \(\left( {{\rm{Take}}\,\,{\rm{\pi }}\,{\rm{ = }}\frac{{22}}{7}} \right)\)
0.6
Calculating Water Speed in a Cylindrical Pipe
This problem asks us to find the speed of water flowing through a cylindrical pipe, given its inner diameter and the flow rate. We are provided the diameter in centimeters, the flow rate in litres per minute, and we need to find the speed in kilometers per hour. We also need to use the value of \(\pi\) as \(\frac{22}{7}\).
Understanding the Concepts
The volume of water flowing through the pipe per unit time (flow rate) is equal to the cross-sectional area of the pipe multiplied by the speed of the water. Mathematically, this can be expressed as:
Flow Rate = Area of Cross-section \(\times\) Speed
Since the pipe is cylindrical, the cross-section is a circle. The area of a circle is given by \(\pi r^2\), where \(r\) is the radius of the circle.
Step-by-Step Calculation
First, let's list the given information and convert units where necessary.
- Inner diameter of the pipe = 14 cm
- Inner radius of the pipe, \(r = \frac{\text{Diameter}}{2} = \frac{14}{2} = 7\) cm
- Flow rate = 154 litres per minute
Unit Conversion for Flow Rate
The flow rate is given in litres per minute. To work with the radius in centimeters, it's helpful to convert the flow rate to cubic centimeters per minute (\(\text{cm}^3\text{/minute}\)). We know that 1 litre = 1000 cubic centimeters.
Flow rate in \(\text{cm}^3\text{/minute}\) = 154 litres/minute \(\times\) 1000 \(\text{cm}^3\text{/litre}\)
Flow rate = \(154 \times 1000 = 154000\) \(\text{cm}^3\text{/minute}\)
Calculate the Cross-sectional Area
The cross-sectional area of the pipe is the area of a circle with radius \(r = 7\) cm. Using \(\pi = \frac{22}{7}\):
Area, \(A = \pi r^2 = \frac{22}{7} \times (7 \text{ cm})^2\)
\(A = \frac{22}{7} \times 49 \text{ cm}^2\)
\(A = 22 \times 7 \text{ cm}^2\)
\(A = 154 \text{ cm}^2\)
Calculate the Speed of Water
Now we can use the relationship: Flow Rate = Area \(\times\) Speed. Let \(v\) be the speed of water in cm/minute.
\(154000 \text{ cm}^3\text{/minute} = 154 \text{ cm}^2 \times v \text{ cm/minute}\)
To find \(v\), we rearrange the equation:
\(v = \frac{154000 \text{ cm}^3\text{/minute}}{154 \text{ cm}^2}\)
\(v = 1000 \text{ cm/minute}\)
Convert Speed to km/hr
The question asks for the speed in kilometers per hour (km/hr). We have the speed in centimeters per minute (cm/minute). We need to convert both units.
- 1 kilometer (km) = 100,000 centimeters (cm). So, 1 cm = \(\frac{1}{100000}\) km.
- 1 hour (hr) = 60 minutes. So, 1 minute = \(\frac{1}{60}\) hour.
Speed in km/hr = Speed in cm/minute \(\times\) Conversion factor from cm to km \(\times\) Conversion factor from minute to hour
\(v (\text{km/hr}) = 1000 \left(\frac{\text{cm}}{\text{minute}}\right) \times \frac{1 \text{ km}}{100000 \text{ cm}} \times \frac{60 \text{ minutes}}{1 \text{ hour}}\)
\(v = 1000 \times \frac{1}{100000} \times 60 \text{ km/hr}\)
\(v = \frac{1000 \times 60}{100000} \text{ km/hr}\)
\(v = \frac{60000}{100000} \text{ km/hr}\)
\(v = \frac{6}{10} \text{ km/hr}\)
\(v = 0.6 \text{ km/hr}\)
Thus, the speed of water in the pipe is 0.6 km/hr.
Summary of Calculations
| Parameter | Value | Unit |
|---|---|---|
| Inner Diameter | 14 | cm |
| Inner Radius (r) | 7 | cm |
| Flow Rate | 154 | litres/minute |
| Flow Rate (converted) | 154000 | cm<sup>3</sup>/minute |
| Cross-sectional Area (A) | 154 | cm<sup>2</sup> |
| Speed (calculated) | 1000 | cm/minute |
| Speed (converted) | 0.6 | km/hr |
The final answer is 0.6 km/hr.
Revision Table: Water Flow Calculations
| Concept | Formula/Relationship | Units |
|---|---|---|
| Area of Circle | \(\pi r^2\) | Length<sup>2</sup> (e.g., cm<sup>2</sup>) |
| Flow Rate | Volume / Time | Volume/Time (e.g., litres/min, m<sup>3</sup>/s) |
| Flow Rate (in terms of speed) | Area of Cross-section \(\times\) Speed | Length<sup>2</sup> \(\times\) Length/Time = Volume/Time |
| Speed | Distance / Time | Length/Time (e.g., cm/min, km/hr) |
| Unit Conversion (Volume) | 1 Litre = 1000 cm<sup>3</sup> | - |
| Unit Conversion (Length) | 1 km = 100,000 cm | - |
| Unit Conversion (Time) | 1 hr = 60 minutes | - |
Additional Information: Fluid Flow Rate and Speed
Fluid flow rate, also known as discharge, is a measure of the volume of fluid that passes a point per unit time. It's a fundamental concept in fluid dynamics and is important in various applications like pipe sizing, irrigation, and water supply systems.
- Volume Flow Rate (Q): This is the volume of fluid passing per unit time (e.g., m³/s, litres/minute). This is what was given in the problem.
- Mass Flow Rate (\(\dot{m}\)): This is the mass of fluid passing per unit time (e.g., kg/s). It is related to volume flow rate by the fluid's density: \(\dot{m} = \rho Q\), where \(\rho\) is density.
- Flow Speed (v): This is the average velocity of the fluid particles in the direction of flow (e.g., m/s, km/hr).
For flow in a pipe, assuming the flow is uniform across the cross-section, the volume flow rate \(Q\) is given by:
\(Q = A \times v\)
Where \(A\) is the cross-sectional area of the pipe and \(v\) is the average flow speed.
In real-world scenarios, fluid flow in pipes is often not uniform across the cross-section due to effects like viscosity (creating a velocity profile where speed is highest at the center and zero at the walls). However, for problems like this one, we usually consider the average speed or assume uniform flow for simplification.
Understanding unit conversions is crucial in these types of problems, as mixing units (like using cm for radius and litres for volume) will lead to incorrect results. Always ensure consistent units before applying formulas.
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