A rectangular mine airway of 2.0 m width and 2.5 m height has a bend with deflection of π/4 radian. If the radius of curvature of the bend is 4.0 m, the shock factor of the bend is... A rectangular mine airway of 2.0 m width and 2.5 m height has a bend with deflection of π/4 radian. If the radius of curvature of the bend is 4.0 m, the shock factor of the bend is (round off to three decimals) Read more

Steps to Solve

1. Identify the dimensions and parameters
   The dimensions of the rectangular airway are:

   • Width ( W = 2.0 , \text{m} )
   • Height ( H = 2.5 , \text{m} )
     The bend has a radius of curvature ( R = 4.0 , \text{m} ) and a deflection angle ( \theta = \frac{\pi}{4} , \text{radians} ).

2. Calculate the cross-sectional area of the airway
   The cross-sectional area ( A ) can be calculated as:
   $$ A = W \times H = 2.0 , \text{m} \times 2.5 , \text{m} = 5.0 , \text{m}^2 $$

3. Find the hydraulic radius (R_h)
   The hydraulic radius for a rectangular section can be calculated as:
   $$ R_h = \frac{A}{P} $$
   The wetted perimeter ( P ) for a rectangular channel is defined as:
   $$ P = W + 2H = 2.0 , \text{m} + 2 \times 2.5 , \text{m} = 7.0 , \text{m} $$
   Now, substituting the values:
   $$ R_h = \frac{5.0 , \text{m}^2}{7.0 , \text{m}} \approx 0.7142857142857143 , \text{m} $$

4. Calculate the shock factor (S)
   The shock factor can be calculated using the formula:
   $$ S = \frac{R_h}{R} \cdot \theta $$
   Substituting the values gives:
   $$ S = \frac{0.7142857142857143 , \text{m}}{4.0 , \text{m}} \cdot \frac{\pi}{4} $$
   $$ S \approx \frac{0.7142857142857143}{4.0} \cdot 0.7853981633974483 $$
   $$ S \approx 0.014 , (\text{approx}) $$

5. Round the answer
   The final calculated shock factor should be rounded to three decimal places, which is:
   $$ S \approx 0.014 $$