Question

Locust beans having an average particle diameter of 7 mm are ground at a rate of 10 ton/h to produce an average particle diameter of 0.62 mm. The mill consumes 6.7 kW power at the given rate. For the same rate of grinding, using Rittinger’s law, the power required to grind the beans to an average particle diameter of 0.25 mm is _________kW. (Rounded off to 2 decimal places)

Hint:

When calculating the power using Rittinger’s law, remember that the power is inversely proportional to the particle size, and the law relates the surface area to the power.

Answer 1

Rittinger’s law states that the power required for grinding is proportional to the surface area of the particles, which is inversely proportional to the particle size. The relationship is given by: \[ P = k \times \left( \frac{1}{d_1} - \frac{1}{d_2} \right) \] Where:
\(P\) is the power consumed (in kW),
\(k\) is a constant,
\(d_1\) and \(d_2\) are the initial and final particle diameters, respectively.
Step 1: Calculate the constant \(k\) using the given data From the given information:
\(P = 6.7 \, {kW}\),
\(d_1 = 7 \, {mm}\),
\(d_2 = 0.62 \, {mm}\).
Substitute into the equation: \[ 6.7 = k \times \left( \frac{1}{7} - \frac{1}{0.62} \right) \] First, calculate the values: \[ \frac{1}{7} \approx 0.142857, \quad \frac{1}{0.62} \approx 1.612903 \] \[ \left( \frac{1}{7} - \frac{1}{0.62} \right) = 0.142857 - 1.612903 = -1.470046 \] Now, solve for \(k\): \[ 6.7 = k \times (-1.470046) \] \[ k = \frac{6.7}{-1.470046} \approx -4.56 \, {kW.mm} \] Step 2: Calculate the power required to grind to a particle diameter of 0.25 mm
Now, for the new particle diameter \(d_2 = 0.25 \, {mm}\), use Rittinger’s law again: \[ P = k \times \left( \frac{1}{7} - \frac{1}{0.25} \right) \] Substitute the values: \[ P = -4.56 \times \left( \frac{1}{7} - \frac{1}{0.25} \right) \] Calculate the values: \[ \frac{1}{7} \approx 0.142857, \quad \frac{1}{0.25} = 4 \] \[ \left( \frac{1}{7} - \frac{1}{0.25} \right) = 0.142857 - 4 = -3.857143 \] Now, solve for \(P\): \[ P = -4.56 \times (-3.857143) = 17.6 \, {kW} \] Thus, the power required to grind the beans to an average particle diameter of 0.25 mm is approximately 17.00 kW.