Water at 20 $$20 ^ { \circ } \mathrm { C }$$ C flows between 80-cm-wide horizontal parallel plates 12 mm apart such that a pressure drop of 100 Pa occurs over a distance of 2 m. If the top plate is moving at 1.8 m/s inThe direction of the flow, the flow rate is nearest:
A) $0.0144 \mathrm {~m} ^ { 3 } / \mathrm { s }$
B) $0.0104 \mathrm {~m} ^ { 3 } / \mathrm { s }$
C) $0.0084 \mathrm {~m} ^ { 3 } / \mathrm { s }$
D) $0.0054 \mathrm {~m} ^ { 3 } / \mathrm { s }$

Question

Water at 20 $$20 ^ { \circ } \mathrm { C }$$ C flows between 80-cm-wide horizontal parallel plates 12 mm apart such that a pressure drop of 100 Pa occurs over a distance of 2 m. If the top plate is moving at 1.8 m/s inThe direction of the flow, the flow rate is nearest: 
A) $0.0144 \mathrm {~m} ^ { 3 } / \mathrm { s }$
B) $0.0104 \mathrm {~m} ^ { 3 } / \mathrm { s }$
C) $0.0084 \mathrm {~m} ^ { 3 } / \mathrm { s }$
D) $0.0054 \mathrm {~m} ^ { 3 } / \mathrm { s }$

Quizplus · Accepted Answer

$0.0144 \mathrm {~m} ^ { 3 } / \mathrm { s }$
$\text {Equation (7.4.7) is used to find the velocity distribution to be}$
$$\begin{aligned}
u ( y ) & = \frac { 1 } { 2 \mu } \frac { \Delta p } { L } \left( y ^ { 2 } - a y \right) + \frac { U } { a } y \
& = \frac { 1 } { 2 \times 10 ^ { - 3 } } \frac { - 100 } { 2 } \left( y ^ { 2 } - 0.012 y \right) + \frac { 1.8 } { 0.012 } y = 25000 \left( 0.012 y - y ^ { 2 } \right) + 150 y
\end{aligned}$$
$\text {where $d p / d x = - 100 / 2 = - 50 \mathrm {~Pa} / \mathrm { m }$. The flow rate is found by integrating this}$ $\text {velocity distribution over the area:}$
$$\begin{aligned}
Q & = \int _ { 0 } ^ { 0.012 } \left[ 25000 \left( 0.012 y - y ^ { 2 } \right) + 150 y \right] 0.8 d y \
& = - 20000 \left( \frac { .012 ^ { 3 } } { 3 } - \frac { .012 ^ { 3 } } { 2 } \right) + 120 \frac { .012 ^ { 2 } } { 2 } = 5.76 \times 10 ^ { - 3 } + 8.64 \times 10 ^ { - 3 } = \underline { 0.0144 \mathrm {~m} ^ { 3 } / \mathrm { s } }
\end{aligned}$$

Water at 20 20∘C20 ^ { \circ } \mathrm { C }20∘C

Water at 20 $20 ^ { \circ } \mathrm { C }$