One of the best ways to monitor a Mechanical Sampling System (MSS) is to measure and track the sample extraction ratio of the system. The extraction ratio is the ratio of the weight of the sample collected (in kg or lbs) to the weight of the lot of coal sampled. For any sampling system running properly the extraction ratio should be nearly a constant.  If a sampling system is collecting less or more coal than the sampling ratio predicts there is a problem with the system that should be corrected.

Fortunately it is not difficult to calculate what the correct extraction ratio should be for any system. Here is how you calculate the extraction ratio for a MSS. You first calculate the division ratio (k) for each stage of the sampling in your system.

The division ratio for a falling stream sampler:

• w = tip-to-tip width of the cutter (in inches, feet, mm, cm, or meters)
• t = in the operating interval from beginning of one cut to the beginning of the next (s, m, or hrs)
• vc = Velocity of the cutter in units that match the units used for t and w

All time, distance, and speed measurements must be in the same units.

Falling Stream Sampler example:

                                          Metric                                       Imperial

	Measured	for formula	Measured	for formula
w	150 mm	0.15m	6 inches	same
t	60 s	same	60 s	same
vc	0.6 m/s	same	18 ips	same

Now calculate k = 0.15/ (60*0.6) = 0.0041667k = 6/ (60*18) = 0.0055556
Unitskg/1,000 * lot size in tonneslbs/2,000 * lot size in short tons

The division ratio for a Cross-Belt Sampler:

The formula is very similar to the falling stream division formula; the only difference is the velocity value is the velocity of the belt being sampled and not the cutter velocity.

• w = tip-to-tip width of the cutter (in inches, feet, mm, cm, or meters)
• t = in the operating interval from beginning of one cut to the beginning of the next (s, m, or hrs)
• Vb = Velocity of the conveyor belt in units that match the units used for t and w.

                               Metric                                                       Imperial

	Measured	for formula	Measured	for formula
w	150 mm	0.15m	6 inches	0.5 feet
t	60 s	same	60 s	1 minute
vb	4 m/s	same	650fpm	same

Now calculate k = 0.15/ (60*4) = 0.000625k = 0.5/ (1*650) = 0.000769
Unitskg/1,000 * lot size in tonneslbs/2,000 * lot size in short tons

The division ratio for a Rotary Divider:

There are a number of types of rotary dividers, all work on similar operating principles the extraction ratio is dependent on the fixed angle of the sampler cutter(s) θ and the number of cutters n. Note the angle θ must be in radians. To convert, multiply the angle in degrees by 2π/360 to get the angle in radians.

Rotary divider – example:

	θ =		20 degrees
Converted to radians	θ =		20*(2π/360) = 0.349066 radians
	n =		2 cutters
Now you can calculate	k =		(2*0.349066)/ (2π) = 0.111111

Units - for Metric units kg/1,000 * lot size in tonnes or lbs/2,000 * lot size in short tons for Imperial unit calculations

Calculating the division ration for a mechanical sampling system

The first step is to calculate the division ratios for each of the sampling stages in the system as described above for each type of sampler. Then the division ratio for the whole system is ksystem = kprimary * ksecondary * ktertiary for a three stage sampling system and for a two stage system ksystem = kprimary * ksecondary.

The following shows an example of an extraction data calculation for a two stage sampling system and compared to the results to the final sample weight calculated by the division ratio. The division ratios are calculated at the end of each sampler section and combined at the end of the extraction data calculation. As this example shows the correct sampling ratio can be calculated simple from the data on each sampler in the system without having to know all the design data for the system.

2 stage MSS Design Extraction Data		XYZ Mining
Metric Units and ISO		ABC Plant Product Belt
Input Data
Belt Capacity	200tph
Belt Width	1500mm
Belt Speed	3.3m/s
Lot Size	2000 tonnes	Belt Loading	168.35kg/m
Type of Coal	Raw	N1	32 ISO
Number of increments	50	Minimum ISO Increments	46
Material Top Size	50mm	Minimum time to sample lot	1 hour
Primary Sampler
Cutter opening	150mm	Primary timer setting	72.00 seconds
		Cross Belt
		Primary Increment weight	25.25 kg
		Weight of Total Primary Sample	1,262.63 kg/lot
		Primary Sample flow rate	1,262.63 kg/hr
		K	0.0006313 kg/kton
Primary or Crusher Feeder
Belt Width	400mm	Sample Volume	1.58 m3/hr
Material Surcharge Angle	0 degrees	Cross Sectional Area	0.0095m2
Material Density	800kg/m	Minimum Belt Speed	0.0461 m/s
		Selected Belt Speed	0.0462 m/s
		Crusher
		Crusher Feed Rate	1,264.03 kg/h
		% of Time Crushing coal	99.89%
		Minutes Crushing coal	59.93 minutes
Secondary Feeder
Belt Width	400mm	Sample Volume	1.58 m3/hr
Material Surcharge Angle	0 degrees	Cross sectional Area	0.0095m2
Material Density	800kg/m	Minimum belt Speed	0.0462 m/s
		Selected Belt Speed	0.0462 m/s
Secondary Sampler
Cutter Opening	75mm	Secondary timer setting	23.97 seconds
Cutter Speed Falling Stream only	0.2000m/s
Number of secondary Increments	150
		Falling Stream
		Secondary Increment Weight	0.1317 kg
		Weight of Final Secondary Sample	19.75 kg
		K	0.0156424 kg/kton
		Extraction Ratio	19.75 kg/lot