ASKED: To determine-
(a) Velocity of flow (V)
(b) Head lost in the pipe (HL)
(c) Kinematic viscosity of the plant ($\nu $)
Given:
Head over the pipe (H) = 9 ft
Length of the pipe (L) = 6 ft
Diameter of the pipe (D) = 0.5 in
Flow rate through the pipe (Q) = 45 ft^{3}/h
NOTE: The flow is mentioned to be laminar and minor losses are to be neglected.
The flow rate through the pipe can be expressed in terms of the velocity of flow and cross-sectional area of the pipe as:
$Q=V\times \left(\frac{\mathrm{\pi}}{4}\times {D}^{2}\right)$
$\to V=\frac{4\times Q}{\mathrm{\pi}\times {\mathrm{D}}^{2}}$
$\to V=\frac{4\times 45f{t}^{3}/h}{\mathrm{\pi}\times {\left(0.5\mathrm{in}\times {\displaystyle \frac{1\mathrm{ft}}{12\mathrm{in}}}\right)}^{2}}\times \frac{1h}{60min}\times \frac{1min}{60s}$
$\Rightarrow \overline{)V=9.167ft/s}$
The head lost in the pipe can be expressed as:
$HL=Headoverthepipe-Velocityhead=H-\frac{{V}^{2}}{2g}$
here,
g: acceleration due to earth's gravity = 32.2 ft/s^{2}
$\to HL=9ft-\frac{\left(9.167ft/s\right)}{2\times 32.2ft/{s}^{2}}$
$\Rightarrow \overline{)HL=7.695ft}$
Using the Darcy-Weisbach formula, head loss in a pipe can be given as:
$HL=\frac{f\times L\times {V}^{2}}{2\times g\times D}$
here,
$f:frictionfactor,forlaminarflowf=\frac{64}{Reynoldsnumber\left({R}_{e}\right)}$
$Reynoldsnumber\left({R}_{e}\right)=\frac{V\times D}{\nu}$
First let us determine the friction factor using the Darcy-Weisbach formula and the head loss computed above.
$\to 7.695ft=\frac{f\times 6ft\times {\left(9.167ft/s\right)}^{2}}{2\times 32.2ft/{s}^{2}\times \left(0.5in\times {\displaystyle \frac{1ft}{12in}}\right)}$
$\Rightarrow f=0.0409$
Now, using the calculated value of friction factor and the formula for friction factor, we can obtain the kinematic viscosity.
$f=\frac{64}{{R}_{e}}=\frac{64}{{\displaystyle \frac{V\times D}{\nu}}}$
$\to \nu =\frac{f\times V\times D}{64}$
$\to \nu =\frac{0.0409\times 9.167ft/s\times \left(0.5in\times {\displaystyle \frac{1ft}{12in}}\right)}{64}$
$\Rightarrow \overline{)\nu =0.000244f{t}^{2}/s}$