(has been resolved):Integral to find elbow volume

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In summary, the conversation discusses finding the elbow volume using the integral method. The correct area of the circular cross section is debated, with one side arguing for π(r/2)^2 and the other for π(r)^2/4. It is eventually concluded that the correct area is π(r/2)^2 and the volume can be calculated as (3/16)π^2r^3. The conversation also addresses a possible mistake in the calculations and the use of LaTeX.
  • #1
Homework Statement
to find elbow volume
Relevant Equations
volume integral
has been resolved
1671522065456.png

Please help me to understand which answer is correct.
I use the integral method to find the elbow volume as follows:
1671522271843.png


But my book say:
1671522299478.png
 
Last edited:
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  • #2
The area of the circular cross-section of diameter r shown in the integral is not correct, it should be ##\pi(r)^2/4##
 
  • #3
Lnewqban said:
The area of the circular cross-section of diameter r shown in the integral is not correct, it should be ##\pi(r)^2/4##
i think it should be ##\pi(2r)^2/4##
 
  • #4
tracker890 Source h said:
i think it should be ##\pi(2r)^2/4##
Think again…
The r that has been given to you is the diameter of the cross-section of the elbow.
Therefore, the area of that section should be π times the radius of that section, which in your case is r/2.
Then,
##\pi(r/2)^2=\pi(r)^2/4##
 
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  • #5
Another way to do it, leading to the same result of ##2.3562*r^3## that is shown in the book:
Straightening the elbow until it becomes a cylinder, its length should be equal to the perimeter of the center line of the elbow.
As the perimeter of that line is ##2\pi(radius~to~center)/4##, then
##Length=2\pi(1.5r)/4##
##Volume=Area~*~Length##
##Volume=(\pi/4)[(r^2)(2)(1.5)(r)]=(3/4)\pi(r^3)##
 
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  • #6
Lnewqban said:
Another way to do it, leading to the same result of ##2.3562*r^3## that is shown in the book:
Straightening the elbow until it becomes a cylinder, its length should be equal to the perimeter of the center line of the elbow.
As the perimeter of that line is ##2\pi(radius~to~center)/4##, then
##Length=2\pi(1.5r)/4##
##Volume=Area~*~Length##
##Volume=(\pi/4)[(r^2)(2)(1.5)(r)]=(3/4)\pi(r^3)##
Shouldn’t there be another π/4?
 
  • #7
Frabjous said:
Shouldn’t there be another π/4?
Perhaps.
I am very clumsy working with LaTeX.

9A0C60D6-3A6A-4D0B-BAC6-52EE0504AD48.jpeg
 
  • #8
Lnewqban said:
Perhaps.
I am very clumsy working with LaTeX.

View attachment 319111
Your answer is correct.
The cross section area = ##\pi \text{(}\frac{r}{2}\text{)}^2##
$$
\forall _{elbow}=\int_0^{\frac{\pi}{2}}{A\left( 1.5r \right) d\theta =}\int_0^{\frac{\pi}{2}}{\left( \pi \text{(}\frac{r}{2}\text{)}^2 \right) \left( \frac{3}{2}r \right) d\theta =\text{(}\frac{\pi r^2}{4}\text{)}\left( \frac{3}{2}r \right) \left( \frac{\pi}{2} \right)}=\frac{3}{16}\pi ^2r^3
$$
 
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  • #9
Lnewqban said:
Think again…
The r that has been given to you is the diameter of the cross-section of the elbow.
Therefore, the area of that section should be π times the radius of that section, which in your case is r/2.
Then,
##\pi(r/2)^2=\pi(r)^2/4##
Yes!
Your answer is correct.
 
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1. What is the purpose of finding the elbow volume in scientific research?

The elbow volume is an important measurement in various scientific fields such as biomechanics, engineering, and medicine. It is used to determine the volume of fluid or gas that can pass through a particular point in a system, which can provide insights into the efficiency and performance of the system.

2. How is the elbow volume calculated?

The elbow volume is typically calculated using the integral method, where the area under the curve of the flow rate versus time graph is integrated to find the volume. This method takes into account the changing flow rate over time and provides a more accurate measurement than other methods.

3. What factors can affect the accuracy of the elbow volume calculation?

The accuracy of the elbow volume calculation can be affected by various factors such as the quality of data collected, the assumptions made in the calculation, and the complexity of the system. It is important to carefully consider these factors and use appropriate techniques to minimize errors in the calculation.

4. How is the elbow volume measurement used in practical applications?

The elbow volume measurement has practical applications in various industries, such as in designing efficient piping systems for fluid transport, optimizing ventilation systems in buildings, and evaluating the performance of respiratory devices in medicine. It can also be used to study the flow dynamics of fluids in natural systems such as rivers and oceans.

5. Are there any limitations to using the integral method to find the elbow volume?

While the integral method is a widely used and accurate technique for finding the elbow volume, it does have some limitations. For example, it assumes a steady flow rate over time and may not be suitable for systems with fluctuating flow rates. Additionally, it requires precise data collection and may be affected by human error. It is important to carefully consider these limitations when using this method for elbow volume calculations.

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