Calculating Stripping Ratios for Irregular Geometries ©Dr. B. C. Paul Spring 2003

The problem of complex geometry Many mines are not strip coal mines Many mines are not strip coal mines Some hard to describe with cones and cylinders Some hard to describe with cones and cylinders –Had conveniently level surface Consider the Mountain Top Removal Operation Consider the Mountain Top Removal Operation

Numerical Calculation of Stripping Ratios

Average End Area from Topo Use a digitizer or Autocad Digitizing Function Use a digitizer or Autocad Digitizing Function 1700 ft Contour Area is ,860 ft ft Contour Area is ,860 ft 2 sits directly above coal seam - half way down to next slice is 0 sits directly above coal seam - half way down to next slice is 0 sits 50 ft below 1750 ft Contour Area - halfway up to next slice is 25 ft. sits 50 ft below 1750 ft Contour Area - halfway up to next slice is 25 ft. Volume ftom slice #1 = 53,452,860ft 2 * (Oft + 25ft) = 1,336,321,500 ft 3 Volume ftom slice #1 = 53,452,860ft 2 * (Oft + 25ft) = 1,336,321,500 ft 3

Next Slice 1750 ft Contour Area is 37,540,860 ft ft Contour Area is 37,540,860 ft 2 half way down to 1700 slice is 25 ft halfway up to 1800 slice is 25 ft. half way down to 1700 slice is 25 ft halfway up to 1800 slice is 25 ft. Volume from slice #2 = 37,540,860ft 2 * (25ft + 25ft) = 1,877,043,000 ft 3 Volume from slice #2 = 37,540,860ft 2 * (25ft + 25ft) = 1,877,043,000 ft 3

Remaining Slices 1800 ft Contour Area is 18,390,654ft 2 * (50ft) = 919,532,700 ft ft Contour Area is 18,390,654ft 2 * (50ft) = 919,532,700 ft ft Contour Area is 6,176,782ft 2 * (50ft) = 308,839,100 ft ft Contour Area is 6,176,782ft 2 * (50ft) = 308,839,100 ft ft Contour Area is 360,049ft ft Contour Area is 360,049ft 2 halfway down to 1850 is 25ft halfway down to 1850 is 25ft half way up to 1950 is Zoicks! There is no 1950!!! Now What do I do? half way up to 1950 is Zoicks! There is no 1950!!! Now What do I do?

Alternatives for handling the missing top slice Approximation #1 - Do nothing - ignore the next slice up Approximation #1 - Do nothing - ignore the next slice up 360,049 ft 2 * 25ft = 9,001,225 ft 3 360,049 ft 2 * 25ft = 9,001,225 ft 3 Approximation #2 - Treat the next slice up as a slice of 0 area and go half way to it. Approximation #2 - Treat the next slice up as a slice of 0 area and go half way to it. 360,049ft 2 * 50ft = 18,002,450 ft 3 360,049ft 2 * 50ft = 18,002,450 ft 3 Approximation #3 - Imagine a sort of pyramid above the top slice going halfway up to slice of 0 area - pyramid volume is 1/3 * height* base Approximation #3 - Imagine a sort of pyramid above the top slice going halfway up to slice of 0 area - pyramid volume is 1/3 * height* base 9,001,225 ft ,049ft 2 * 25 *0.333 = 11,998,632 ft 3 9,001,225 ft ,049ft 2 * 25 *0.333 = 11,998,632 ft 3

More Approximations Approximation #4 - The height of the pyramid could have been some other number including 50ft or the distance to the peak. Approximation #4 - The height of the pyramid could have been some other number including 50ft or the distance to the peak. Great - You just gave me 5 different answers - which one is right? Great - You just gave me 5 different answers - which one is right? Sum up the volumes Sum up the volumes – 1,336,321,500 –+ 1,877,043,000 –+ 919,532,700 –+ 308,839,100 –+ 9,001,225 –Total - 4,450,737,525

Concept on Handling Ends This is an engineering approximation. You may use your knowledge of the true geometry to sharpen the approximation. You may choose an approximation to keep the math simple and fast. You may make a reasonable but arbitrary choice in the presence of insufficient data. Often other estimates and judgments that must be made will have far more influence on the investors bottom line $$$$ than how you approximated a peak. This is an engineering approximation. You may use your knowledge of the true geometry to sharpen the approximation. You may choose an approximation to keep the math simple and fast. You may make a reasonable but arbitrary choice in the presence of insufficient data. Often other estimates and judgments that must be made will have far more influence on the investors bottom line $$$$ than how you approximated a peak.

Finishing the Stripping Ratio Convert OB volume to yards Convert OB volume to yards –4,450,737,525/ 27 = 164,842,131 yd 3 Get Weight of Coal Get Weight of Coal –1700 ft slice 53,452,860 ft 2 – X 5 ft – 267,264,300 ft 3 – X 80 lbs/ft 3 – / 2000 lbs/ton – 10,690,572 tons

Finalize Stripping Ratio 164,842,131 yd3/ 10,690,572 tons = 15.42:1 164,842,131 yd3/ 10,690,572 tons = 15.42:1 If Dragline Operations Cost 35 cents per cubic yard If Dragline Operations Cost 35 cents per cubic yard –15.42 * $0.35 = $5.40/ton –Minimum margin on coal to remove O.B.

The Concept When geometry gets to complex for standard formulas – use a numeric approximation to integrating what ever shape you have When geometry gets to complex for standard formulas – use a numeric approximation to integrating what ever shape you have –That’s the way all the basic geometry formulas were derived (they were simple enough to work out analytic answers) With computers today who does analytic answers With computers today who does analytic answers We’re engineers (how close do we have to be before you don’t care anymore) We’re engineers (how close do we have to be before you don’t care anymore)

What if there is no Off-the shelf contour map? Probably remember from calculus that you can integrate horizontal or vertical slices to get the same formula Probably remember from calculus that you can integrate horizontal or vertical slices to get the same formula Take Multiple Cross-Sections of your pit and do an average end area Take Multiple Cross-Sections of your pit and do an average end area

Phosphate Mine Example Do a series of cross-sections This geometry is simple Enough. Then do average end area With each cross section Weighted by thickness Half way to its neighbor

Four Basic Ways to Get Stripping Ratio 1 Dimensional Formulas 1 Dimensional Formulas –Works on coal strip pits Simple Geometric Calculations Simple Geometric Calculations Average End Area off of Topo Average End Area off of Topo Average End Area off of Cross-Section Average End Area off of Cross-Section Which One Do I Use? Which One Do I Use? Simplest One that gives you the desired accuracy Simplest One that gives you the desired accuracy