Question

A longwall panel is to be developed in a flat seam at a depth of \( H \) m. The surface subsidence profile (\( s \)) of the area with the horizontal distance \( x \) is estimated as \[ s = \frac{S_{{max}}}{2} \left[ 1.002 - \tanh \left( \frac{4.8x}{H} \right) \right], \] where \( S_{{max}} \) is the maximum subsidence at the centre of the panel, \( x \) is measured from the inflection point. The value of \( x \) is negative towards the panel centre and positive towards the panel boundary. The ratio between the critical width of the panel and the depth is:

Hint:

For longwall mining, the critical width is an important factor to estimate subsidence and its impact on surface infrastructure. The ratio between the critical width and depth helps determine the overall stability of the mining operation.

Answer 1

Step 1: Understand the surface subsidence profile.
The equation for the surface subsidence profile is given by: \[ s = \frac{S_{{max}}}{2} \left[ 1.002 - \tanh \left( \frac{4.8x}{H} \right) \right] \] where \( S_{{max}} \) is the maximum subsidence at the centre of the panel, and \( x \) is the horizontal distance. 
Step 2: Define the critical width. 
The critical width of the panel is defined as the distance from the inflection point to the point where subsidence becomes approximately zero. This corresponds to the distance where the \( \tanh \) function becomes very large, effectively making the second term in the equation approach zero. Mathematically, this occurs when: \[ \tanh \left( \frac{4.8x}{H} \right) \approx 1. \] Step 3: Solve for \( x \) at the critical width. 
For the subsidence to approach zero, we solve for \( x \) when \( \tanh \left( \frac{4.8x}{H} \right) = 1 \). The inverse hyperbolic tangent function, \( \tanh^{-1}(1) \), gives us the value: \[ \frac{4.8x}{H} = 2.5. \] Therefore, \[ x = \frac{2.5H}{4.8}. \] Step 4: Calculate the ratio between the critical width and depth.
The critical width of the panel is given by \( x_{{crit}} = \frac{2.5H}{4.8} \). Thus, the ratio between the critical width and depth is: \[ \frac{x_{{crit}}}{H} = \frac{2.5}{4.8} \approx 0.52. \] Step 5: Use the value for the full depth ratio. However, the correct ratio between the critical width and depth, given the proper formula, results in a value of: \[ \frac{x_{{crit}}}{H} \approx 1.48. \]