Chapter 9, Problem 1P

To determine

The average velocity of the water.

Explanation of Solution

Given:

The pressure drop $\left(\frac{dp}{dx}\right)$ is $100\text{ Pa}$.

The temperature of the water $\left(T\right)$ is $20°\text{C}$.

Calculation:

Consider the flow is laminar.

Since the plates are fixed, $\frac{dh}{dx}=0$.

Refer Appendix A, “Physical Properties of Water vs. Temperature (SI Units)” at $20°\text{C}$,

The density of the water $\left(\rho \right)$ is $998.3{\text{ kg/m}}^3$.

The dynamic viscosity of water $\left(\mu \right)$ is $1.00\times {10}^{-3}\text{ N}\cdot {\text{s/m}}^2$.

Calculate the average velocity.

  $\begin{array}{c}V=-\frac{a^2}{12\mu }\frac{dp}{dx}\\ =\frac{{(4\text{m})}^2}{12\left[1.00\left({10}^{-3}\right)\text{N}\cdot \text{s}/{\text{m}}^2\right]}\left(\frac{-100\text{ Pa}}{0.9\text{m}}\right)\\ =\frac{{(4\text{m}\times \frac{1\text{ m}}{1000\text{ mm}})}^2}{12\left[1.00\left({10}^{-3}\right)\text{N}\cdot \text{s}/{\text{m}}^2\right]}\left(\frac{-100\text{ Pa}\times \frac{\text{ 1}\text{N}/{\text{m}}^2}{\text{Pa}}}{0.9\text{m}}\right)\\ \text{=0}\text{.148 m/s}\end{array}$

Thus, the average velocity is $\underset{\_}{\text{0}\text{.148 m/s}}$.

Checks the assumption by calculate the Reynolds number.

  $\begin{array}{c}\mathrm{Re}==\frac{\rho V_a}{\mu }\\ =\frac{\left(998.3\text{kg}/{\text{m}}^3\right)(0.1481\text{m}/\text{s})(0.004\text{m})}{1.00\left({10}^{-3}\right)\text{N}\cdot \text{s}/{\text{m}}^2}\\ =592<1400\end{array}$