Chapter 3.3, Problem 17E

a.

To determine

Find the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty.

Find the relative uncertainty in the friction velocity of water flowing through a pipe.

Answer to Problem 17E

The estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 0.33\%}$.

The relative uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_{\ln F}=0.33\%}$.

Explanation of Solution

Given info:

The acceleration due to gravity is exactly $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$. The diameter of the pipe is $d=0.2\text{m}$, the length of the pipe is $l=35.0\text{m}$ with negligible uncertainty and head loss of pipe is measured to be $h=4.51\pm 0.03\text{m}$.

Calculation:

The form of the measurements of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of head loss of pipe is $h=4.51\pm 0.03\text{m}$.

Here, the measured value or mean of the head loss of pipe is $h=4.51\text{m}$ and the uncertainty in the head loss of pipe is ${\sigma }_h=0.03\text{m}$.

Measured value of the friction velocity of water flowing through a pipe:

The formula for the friction velocity of water flowing through a pipe F is $F=\sqrt{\frac{gdh}{4l}}$.

Here, $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$, $d=0.2\text{m}$,$l=35.0\text{m}$ and $h=4.51\text{m}$.

The measured value of friction velocity of water flowing through a pipe is obtained as follows:

$\begin{array}{c}F=\sqrt{\frac{gdh}{4l}}\\ =\sqrt{\frac{9.80\times 0.2}{4\times 35.0}\times h}\\ =0.118332\times \sqrt{h}\\ =0.118332\times \sqrt{4.51}\end{array}$

$=0.2513$

Thus, the measured value of friction velocity of water flowing through a pipe is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}}$.

Uncertainty:

The uncertainty of a process is determined by the standard deviation of the measurements. In other words it can be said that, measure of variability of a process is known as uncertainty of the process.

Therefore, it can be said that uncertainty is simply $\left(\sigma \right)$.

Standard deviation:

The standard deviation is based on how much each observation deviates from a central point represented by the mean. In general, the greater the distances between the individual observations and the mean, the greater the variability of the data set.

The general formula for standard deviation is,

$s=\sqrt{\frac{{\displaystyle \sum x_i{}^2-\frac{{\left({\displaystyle \sum x_i}\right)}^2}{n}}}{n-1}}$.

From the properties of uncertainties for functions of one measurement it is known that,

• If X is a measurement with uncertainty ${\sigma }_X$ and if U is a function of X then the uncertainty in the variable U is ${\sigma }_U=\left|\frac{dU}{dX}\right|{\sigma }_X$.

Here, head loss of pipe is not a constant. That is, head loss of pipe is a variable and the friction velocity of water flowing through a pipe is a function of head loss of pipe.

The uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{F=\sqrt{\frac{gdh}{4l}}}=\left|\frac{d\sqrt{\frac{gdh}{4l}}}{dh}\right|\times {\sigma }_h\\ =\left|\frac{\sqrt{\frac{gd}{4l}}\times d\left(\sqrt{h}\right)}{dh}\right|\times {\sigma }_h\\ =\left|\sqrt{\frac{gd}{4l}}\times \frac{1}{2\sqrt{h}}\right|\times 0.03\\ =\sqrt{\frac{gd}{4l}}\times \frac{1}{2\sqrt{h}}\times 0.03\end{array}$

              $\begin{array}{l}=\sqrt{\frac{9.80\times 0.2}{4\times 35.0}}\times \frac{1}{2\sqrt{4.51}}\times 0.03\\ =0.027858\times 0.03\\ =0.000836\end{array}$

Hence, the uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_F=0.000836\frac{\text{m}}{\text{s}}}$.

Relative uncertainty:

Relative uncertainty in U is the uncertainty as a fraction of the true value (mean of the measurement ${\mu }_U$). Relative uncertainty is also called as coefficient of variation. Relative uncertainty is expressed in percentage without units.

The general formula to obtain relative uncertainty is,

${\sigma }_{\ln U}=\frac{{\sigma }_U}{{\mu }_U}=\frac{{\sigma }_U}{U}$.

The relative uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{\ln F}=\frac{{\sigma }_F}{{\mu }_F}\\ =\frac{{\sigma }_F}{F}\\ =\frac{0.000836}{0.2513}\\ =0.0033\end{array}$

Thus, the relative uncertainty in friction velocity of water flowing through a pipe (F) is $\underset{\_}{{\sigma }_{\ln F}=0.33\%}$.

Estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of friction velocity of water flowing through a pipe in terms of relative uncertainty is,

$\begin{array}{c}F=\text{Measured value of }F\pm {\sigma }_{\ln F}\\ =0.2513\frac{\text{m}}{\text{s}}\pm 0.33\%\end{array}$

Thus, the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 0.33\%}$.

b.

To determine

Find the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty.

Find the relative uncertainty in the friction velocity of water flowing through a pipe.

Answer to Problem 17E

The estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 2\%}$.

The relative uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_{\ln F}=2\%}$.

Explanation of Solution

Given info:

The acceleration due to gravity is exactly $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$. The head loss of pipe is $h=4.51\text{m}$, the length of the pipe is $l=35.0\text{m}$ with negligible uncertainty and the diameter of the pipe is measured to be $d=0.2\pm 0.008\text{m}$.

Calculation:

The form of the measurements of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of diameter of the pipe is $d=0.2\pm 0.008\text{m}$.

Here, the measured value or mean of the diameter of the pipe is $d=0.2\text{m}$ and the uncertainty in the diameter of the pipe is ${\sigma }_d=0.008\text{m}$.

Measured value of the friction velocity of water flowing through a pipe:

The formula for the friction velocity of water flowing through a pipe F is $F=\sqrt{\frac{gdh}{4l}}$.

Here, $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$, $h=4.51\text{m}$,$l=35.0\text{m}$ and $d=0.2\text{m}$.

The measured value of friction velocity of water flowing through a pipe is obtained as follows:

$\begin{array}{c}F=\sqrt{\frac{gdh}{4l}}\\ =\sqrt{\frac{9.80\times 4.51}{4\times 35}\times d}\\ =0.561872\times \sqrt{d}\\ =0.561872\times \sqrt{0.2}\end{array}$

$=0.2513$

Thus, the measured value of friction velocity of water flowing through a pipe is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}}$.

Uncertainty:

Here, diameter of the pipe is not a constant. That is, diameter of the pipe is a variable and the friction velocity of water flowing through a pipe is a function of diameter of the pipe.

The uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{F=\sqrt{\frac{gdh}{4l}}}=\left|\frac{d\sqrt{\frac{gdh}{4l}}}{dd}\right|\times {\sigma }_d\\ =\left|\frac{\sqrt{\frac{gh}{4l}}\times d\left(\sqrt{d}\right)}{dd}\right|\times {\sigma }_d\\ =\left|\sqrt{\frac{gh}{4l}}\times \frac{1}{2\sqrt{d}}\right|\times 0.008\\ =\sqrt{\frac{gh}{4l}}\times \frac{1}{2\sqrt{d}}\times 0.008\end{array}$

$\begin{array}{l}=\sqrt{\frac{9.80\times 4.51}{4\times 35}}\times \frac{1}{2\sqrt{0.2}}\times 0.008\\ =0.851747\times 0.008\\ =0.05026\end{array}$

Hence, the uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_F=0.005026\frac{\text{m}}{\text{s}}}$.

Relative uncertainty:

The general formula to obtain relative uncertainty is,

${\sigma }_{\ln U}=\frac{{\sigma }_U}{{\mu }_U}=\frac{{\sigma }_U}{U}$.

The relative uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{\ln F}=\frac{{\sigma }_F}{{\mu }_F}\\ =\frac{{\sigma }_F}{F}\\ =\frac{0.005026}{0.2513}\\ =0.02\end{array}$

Thus, the relative uncertainty in friction velocity of water flowing through a pipe (F) is $\underset{\_}{{\sigma }_{\ln F}=2\%}$.

Estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of friction velocity of water flowing through a pipe in terms of relative uncertainty is,

$\begin{array}{c}F=\text{Measured value of }F\pm {\sigma }_{\ln F}\\ =0.2513\frac{\text{m}}{\text{s}}\pm 2\%\end{array}$

Thus, the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 2\%}$.

c.

To determine

Find the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty.

Find the relative uncertainty in the friction velocity of water flowing through a pipe.

Answer to Problem 17E

The estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 0.57\%}$.

The relative uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_{\ln F}=0.57\%}$.

Explanation of Solution

Given info:

The acceleration due to gravity is exactly $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$. The head loss of pipe is $h=4.51\text{m}$, the diameter of the pipe is $d=0.2\text{m}$ with negligible uncertainty and length of the pipe is measured to be $l=35.0\pm 0.4\text{m}$.

Calculation:

The form of the measurements of a process is,

$\text{Measured}\text{value}\left(\mu \right)\pm \text{Standard deviation}\left(\sigma \right)$.

Here, for a random sample the measured value will be sample mean and the population standard deviation will be sample standard deviation.

The form of the measurements of length of the pipe is $l=35.0\pm 0.4\text{m}$.

Here, the measured value or mean of the length of the pipe is $l=35.0\text{m}$ and the uncertainty in the length of the pipe is ${\sigma }_l=0.4\text{m}$.

Measured value of the friction velocity of water flowing through a pipe:

The formula for the friction velocity of water flowing through a pipe F is $F=\sqrt{\frac{gdh}{4l}}$.

Here, $g=9.80\frac{\text{m}}{{\text{s}}^{\text{2}}}$, $h=4.51\text{m}$, $d=0.2\text{m}$ and $l=35.0\text{m}$.

The measured value of friction velocity of water flowing through a pipe is obtained as follows:

$\begin{array}{c}F=\sqrt{\frac{gdh}{4l}}\\ =\sqrt{\frac{9.80\times 4.51\times 0.2}{4}\times \frac{1}{l}}\\ =1.486573\times \frac{1}{\sqrt{l}}\\ =1.486573\times \frac{1}{\sqrt{35.0}}\end{array}$

$=0.2513$

Thus, the measured value of friction velocity of water flowing through a pipe is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}}$.

Uncertainty:

Here, length of the pipe is not a constant. That is, length of the pipe is a variable and the friction velocity of water flowing through a pipe is a function of length of the pipe.

The uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{F=\sqrt{\frac{gdh}{4l}}}=\left|\frac{d\sqrt{\frac{gdh}{4l}}}{dl}\right|\times {\sigma }_l\\ =\left|\frac{\sqrt{\frac{ghd}{4}}\times d\left(\frac{1}{\sqrt{l}}\right)}{dl}\right|\times {\sigma }_l\\ =\left|\sqrt{\frac{gdh}{4}}\times \frac{-1}{2l^{\frac{3}{2}}}\right|\times 0.4\\ =\sqrt{\frac{gdh}{4}}\times \frac{1}{2l^{\frac{3}{2}}}\times 0.4\end{array}$

$\begin{array}{l}=\sqrt{\frac{9.80\times 0.2\times 4.51}{4}}\times \frac{1}{2\times {35}^{\frac{3}{2}}}\times 0.4\\ =0.0035897\times 0.4\\ =0.001436\end{array}$

Hence, the uncertainty in the friction velocity of water flowing through a pipe is $\underset{\_}{{\sigma }_F=0.001436\frac{\text{m}}{\text{s}}}$.

Relative uncertainty:

The general formula to obtain relative uncertainty is,

${\sigma }_{\ln U}=\frac{{\sigma }_U}{{\mu }_U}=\frac{{\sigma }_U}{U}$.

The relative uncertainty in friction velocity of water flowing through a pipe (F) is,

$\begin{array}{c}{\sigma }_{\ln F}=\frac{{\sigma }_F}{{\mu }_F}\\ =\frac{{\sigma }_F}{F}\\ =\frac{0.001436}{0.2513}\\ =0.0057\end{array}$

Thus, the relative uncertainty in friction velocity of water flowing through a pipe (F) is $\underset{\_}{{\sigma }_{\ln F}=0.57\%}$.

Estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty:

The estimate of the measurement of a process in terms of relative uncertainty is,

$\text{Measured}\text{value}\left(U\right)\pm \text{Standard deviation}\left({\sigma }_{\ln U}\right)$.

The estimate of friction velocity of water flowing through a pipe in terms of relative uncertainty is,

$\begin{array}{c}F=\text{Measured value of }F\pm {\sigma }_{\ln F}\\ =0.2513\frac{\text{m}}{\text{s}}\pm 0.57\%\end{array}$

Thus, the estimate of the friction velocity of water flowing through a pipe in terms of relative uncertainty is $\underset{\_}{F=0.2513\frac{\text{m}}{\text{s}}\pm 0.57\%}$.