Chapter 9, Problem 9.1PP

Compute points on the velocity profile from the pipe wall to the centerline of a $2$ -in Schedule $40$ steel pipe if the volume flow rate of castor oil at $77°F$ is $0.25f{t}^3/s$. Use increments of $0.20$ sssssssin and include the velocity at the centerline.

To determine

To compute: the velocity profile from the pipe wall to the centerline of a 2 in schedule 40 steel pipe.

Answer to Problem 9.1PP

The velocity profile from the pipe wall to the centerline of a 2 in schedule 40 steel pipe $v=10.72ft/s$, $N_R=253$ $\to$Laminar

Explanation of Solution

Given:

To consider the value 2-in schedule 40 steel pipe chart $D=2.067in=0.1723ft$, Kinematic viscosity of fluid $\nu =7.31\times {10}^{-3},$ volume flow rate of castor oil $Q=0.25f{t}^3/s$, Full radius $r_o=D/2=1.034$ in;

Concept Used:

Annuity problem requires the use of the local velocity for Laminar flow equation as follows:

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Here,

Velocity $=$

   $U$, Average velocity= $v$, Radius= $r$, $r_o=$ full radius

Calculation:

As per the given problem

   $Q=0.25f{t}^3/s$, $D=2.067in=0.1723ft$

Annuity problem requires the use of this formula

   $A=\frac{\pi D^2}{4}$

Substitute these values in the formula

   $A=\frac{\pi {\left(0.1723ft\right)}^2}{4}=0.02333f{t}^2$

   $A=0.02333f{t}^2$

Annuity problem requires the use of this formula

   $Q=0.25f{t}^3/s$, $A=0.02333f{t}^2$

   $v=Q/A$

Substitute these values in the formula

   $v=\frac{0.25f{t}^3/s}{0.02333f{t}^2}=10.72ft/s$

   $v=10.72ft/s$

Annuity problem requires the use of the local velocity for Laminar flow equation as follows:

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

To find out velocity at different radius

   $r=0.0$

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{0.0in}{1.0338in}\right)}^2\right]=21.44ft/s$

   $U=21.44ft/s$

Annuity problem requires the use of this formula

   $r=0.20in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{0.20in}{1.0338in}\right)}^2\right]=20.64ft/s$

   $U=20.64ft/s$

Annuity problem requires the use of this formula

   $r=0.40in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{0.40in}{1.0338in}\right)}^2\right]=18.23ft/s$

   $U=18.23ft/s$

Annuity problem requires the use of this formula

   $r=0.60in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{0.60in}{1.0338in}\right)}^2\right]=14.22ft/s$

   $U=14.22ft/s$

Annuity problem requires the use of this formula

   $r=0.80in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{0.80in}{1.0338in}\right)}^2\right]=8.60ft/s$

   $U=8.60ft/s$

Annuity problem requires the use of this formula

   $r=1.00in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{1.00in}{1.0338in}\right)}^2\right]=1.38ft/s$

   $U=1.38ft/s$

Annuity problem requires the use of this formula

   $r=1.0338in$, Average velocity $v=10.72ft/s$, $r_o=D/2=1.034in$.

   $U=2v\left[1-{\left(\frac{r}{r_o}\right)}^2\right]$

Substitute these values in the formula

   $U=2\left(10.72\right)\left[1-{\left(\frac{1.0338in}{1.0338in}\right)}^2\right]=0.00ft/s$

   $U=0.0ft/s$

Annuity problem requires the use of this formula

   $v=10.72ft/s$, $D=2.067in=0.1723ft$, $\nu =7.31\times {10}^{-3}$

   $N_R=vD\rho /\eta$

   $\Rightarrow$ $N_R=vD/\nu$

   $N_R$ $\to$ Reynolds Number, $\nu$ $\to$ Kinematic viscosity of fluid

Substitute these values in the formula

   $\Rightarrow$ $N_R=vD/\nu$

   $N_R=vD/\nu =\left(10.72\right)\left(0.1723\right)/7.31\times {10}^{-3}=253$

   $N_R=253$ $\to$Laminar

Conclusion:

The velocity profile from the pipe wall to the centerline of a 2 in schedule 40 steel pipe,, $N_R=253$ $\to$Laminar