Chapter 9, Problem 9.1P

Interpretation Introduction

Interpretation:

The given motor power is to be verified to be adequate to drive the agitator at the given speed.

Concept Introduction:

The formula to calculate the Reynolds number for an agitator is:

  $\mathrm{Re}=\frac{n{d}_a^2\rho }{\mu }$   ........ (1)

Here, $n$ is the speed of the agitator in r/s, $d_a$ is the diameter of the agitator, $\rho$ is the density of the fluid, and

  $\mu$ is the viscosity of the fluid.

Now, power consumption by the agitator is calculated as:

  $P=N_P{n}^3{d}_a^5\rho$   ........ (2)

Here, $N_P$ is the power number that depends on the Reynolds number of the fluid. Its value is taken from the standard graph specific for a type of an agitator.

Answer to Problem 9.1P

The given motor with 8 kW power is not adequate to drive the agitator at the required speed as the power required is more than the available power of the motor.

Explanation of Solution

Given information:

Tank diameter, $d=1.2\text{ m}$

Height of the tank, $h=2\text{ m}$

Depth of latex filled in the tank, $h_f=1.2\text{ m}$

Viscosity of latex, $\mu =10\text{ P}$

Density of latex, $\rho =800\frac{\text{kg}}{{\text{m}}^3}$

The tank is not baffled.

Diameter of the propeller, $d_p=360\text{ mm}$

Height of the propeller in the tank, $h_p=360\text{ mm}$

Pitch is 1:1

Agitator speed, $n=800\text{ r/min}$

Three-blade propeller is used.

Available power from the motor, $P_{available}=8\text{ kW}$

Convert the speed of the propeller from r/min to r/s as:

  $\begin{array}{c}n=800\frac{r}{\text{min}}\times \frac{\text{min}}{60\text{ s}}\\ =13.33\frac{\text{r}}{\text{s}}\end{array}$

Use equation (1) to calculate the Reynolds number for the propeller as:

  $\begin{array}{c}\mathrm{Re}=\frac{n{d}_p^2\rho }{\mu }\\ =\frac{\left(13.33\frac{\text{r}}{\text{s}}\right){\left(360\text{ mm}\times \frac{\text{m}}{1000\text{ mm}}\right)}^2\left(800\frac{\text{kg}}{{\text{m}}^3}\right)}{10\text{ P}\times \frac{\text{Pa}\cdot \text{s}}{10\text{ P}}}\\ =1382\end{array}$

From the plot of $N_P$ versus $\mathrm{Re}$ for unbaffled propeller with three blades, the value of $N_P$ corresponding to the calculated value of Re is taken as:

  $N_P=0.73$

Use equation (2) to calculate the required power for the given agitator as:

  $\begin{array}{c}{P}_{req}=N_P{n}^3{d}_p^5\rho \\ P_{req}=\left(0.73\right){\left(13.33\frac{\text{r}}{\text{s}}\right)}^3{\left(360\text{ mm}\times \frac{\text{m}}{1000\text{ mm}}\right)}^5\left(800\frac{\text{kg}}{{\text{m}}^3}\right)\\ P_{req.}=8364\text{ W}\\ P_{req.}=8.364\text{ kW}\end{array}$

Since the required power is more than the available power, the motor available to drive the agitator is not adequate.