The distributed load a carton puts on the conveyor, in pounds per running foot.

Roller and belt ratings are given per foot, so you convert the carton length from inches to feet and divide the weight across it. The heaviest carton at the shortest length is the worst case, because that packs the most weight into the least support. Checking the average would hide the carton that actually overloads a roller.

The maximum roller center spacing that still keeps at least three rollers under the product.

A minimum of three rollers must contact the product at any point along the run, or the carton teeters and dips between rollers. Divide the leading dimension by three and you get the widest spacing that guarantees that support. The minimum package sets it. After a 90 degree transfer the carton travels hard way, so the takeaway rollers must be sized to the hard way dimension, which is smaller, forcing tighter centers than the trunk line.

The between-frame width a curve needs so the worst-case carton corner clears the outside rail.

In a curve the carton pivots and its outside corner swings toward the outer rail. Pythagoras on that corner, at the inside radius plus the carton width and half its length, gives the reach of the corner. Subtracting the inside frame position converts reach into required frame width. You round the result up to the next catalog width, never down, because rounding down puts the corner into the rail.

The theoretical incline angle at which a carton tips forward over its front edge.

A carton tips when its center of gravity passes over its front bottom edge. Model the center of gravity at one third of the length and the arctangent of length over three times height gives the tipping angle, converted from radians to degrees. The short, tall carton has the smallest tumble angle and is the worst case, so design every incline to the minimum tumble angle in the mix with a safety margin, never to the theoretical limit.

The gap between cartons after a belt speed change.

When cartons move from a slow belt to a fast belt they spread apart, and the amount they spread depends on how long each carton spent on the slow belt. The term SpeedOut times L over SpeedIn is the distance the leading carton travels on the fast belt while the trailing carton finishes clearing at the slow speed. Accelerating grows the gap, decelerating shrinks it. The min carton produces the smallest gap, so the tool feeds that gap forward as the binding induction gap.

The maximum cartons per minute a belt can carry at a given speed and gap.

Pitch is length plus gap, the distance from one carton to the next. Dividing belt speed by pitch in feet gives cartons per minute. You use the max carton length for the design rate because the longest carton produces the fewest cartons per minute, the worst case. If the theoretical rate is below the required rate, raise belt speed or cut the gap, and never design to the minimum.

The center to center distance between cartons.

Everything about spacing derives from pitch, and pitch is simply the gap plus the carton length. If a pitch or a rate looks wrong, go back and check the gap first, because a bad gap propagates into every rate and sorter number downstream.

The minimum belt speed needed to physically move enough cartons per minute to meet the rate.

Carton length times rate is the feet of product per minute the belt must carry, converted from inches by dividing by twelve. The 1.15 safety factor adds headroom for the losses the ideal formula ignores, such as slippage and imperfect presentation. CFPM is a floor, never the run speed, because the sorter has to run faster still once the speed gap ratio is applied.

How much faster the sorter must run than the induction belt to hold the gap that already exists.

The ratio of pitch to carton length is the multiplier the sorter must apply to keep the same gap while the carton is short. A small carton carrying a large gap produces a high ratio and a high required speed, which is why the min carton is the binding case. The tool wires the induction gap to the min carton's gap produced, the smallest gap in the mix, so the ratio is computed against the worst case.

The actual operating speed the sorter must run for a carton column.

Multiply the minimum conveyor speed by the speed gap ratio and you get the run speed that both moves enough product and holds the gap. The tool chains this at full precision from the raw inputs, so it reads lower than a hand example that first rounds CFPM. The max carton drives CFPM, the min carton drives SGR, and the column with the highest required speed governs. Round the governing speed up to the next practical belt speed.

The gap between cartons once the sorter is running at its operating speed.

Belt speed over rate gives the pitch the sorter is producing, and subtracting the carton length leaves the gap. This is informational in the r4.1 tool. The pass and fail checks compare the gap produced at induction, the smaller and more conservative number, not this larger gap the sorter opens up.

The minimum gap the selected sorter model needs to complete a divert before the trailing carton enters the zone.

Each sorter model has a physical divert mechanism with its own timing, and wider cartons need more gap to clear. The r4.1 workbook carries a lookup by model and by max carton width band, so the required gap comes from the manufacturer's own spec rather than a single rule of thumb. The values below are manufacturer-specific reference data, kept verbatim from the workbook.

The gap the widest carton needs to clear the divert without hitting the next carton.

As a carton turns onto the divert, its width projects along the line of travel by the width times the sine of the divert angle. That projection is the room the next carton must stay clear of. Two inches of margin are added on top. The gap check must satisfy both this geometric requirement and the model minimum, and whichever is larger governs.

The belt speed the takeaway spur must run.

The carton leaves the sorter at an angle, so only the cosine component of its velocity points down the spur. To keep the carton moving at the sorter's pace along the spur, the spur must run faster by one over the cosine of the divert angle. Specifying the spur at sorter speed is always wrong. At 30 degrees the spur runs about 15 percent faster, at 22 degrees about 8 percent faster.

How far sideways the transfer must move the worst-case carton.

The carton can sit anywhere across the belt, and the worst case is the side opposite the divert direction. That worst-case offset is half of the difference between the overall width and the between-frame width, added to the frame width, which is the distance the mechanism must clear.

The total time the transfer is busy, from the moment it lifts until it is back in the ready position.

The cycle is the lift time, plus the time to travel the lateral distance at the transfer speed, plus the lower time. The middle term converts the lateral distance to feet and divides by the speed in feet per second. During this whole window the trunk line cannot safely deliver the next carton, so this number sets the gap the trunk must hold.

The minimum gap the trunk line needs between cartons for the transfer to finish a full cycle in time.

Cycle time times trunk speed is the distance the trunk travels while the transfer is busy, and dividing by five converts feet per minute and seconds into the inches the trunk covers. Four inches of margin are added, and that four inches is a floor. On critical transfers add 8 to 10 inches more for product variation and belt slippage. If the available gap is short, the carton collides on every cycle, so reduce trunk speed, add gap upstream, or use a faster transfer.

The minimum length of skewed roller conveyor needed to align the narrowest carton across the belt.

A skewed section walks a carton sideways a little with each roller until it rides against the guide. The narrowest carton has the farthest to travel, so the frame width minus the min width, scaled by how many rollers per length, gives the travel distance, and the max length is added so the longest carton is contained during the traverse. This is a minimum, so add length for connection to adjacent sections.

How many seconds a carton takes to travel a given distance at a given speed.

Convert distance to feet and speed to feet per second and divide, and you have the travel time. This is the response window between a scan point and a divert. If the controls need three seconds to return a sort decision and the travel time is five, there are two seconds of margin. If belt speed changes anywhere between the points, recalculate, and never estimate travel time from memory.

The yearly labor cost the system removes or redeploys.

Positions times the fully loaded hourly rate times the hours that actually run gives the annual labor value. Fully loaded means wages plus the burden on top of them, not the offer-letter number. Confirm the headcount, the rate, and the hours with the customer in writing, because a payback built on a headcount nobody confirmed is a guess wearing a spreadsheet.

The total yearly savings the system produces, modeled from three sources.

Annual savings is not one number the customer hands you. You build it from labor, from the chargebacks the system prevents by routing on confirmed scan data, and from throughput, but only where added capacity converts to revenue or avoided overtime. If the extra cartons per minute do not turn into money, they do not belong in the number.

The number of years of savings it takes to recover the install cost.

Divide what the system costs by what it saves each year and you have the simple payback. It is deliberately simple, because the hard part is not the arithmetic, it is confirming the inputs. There is no industry-standard payback. The approval threshold is a number each company sets from its own cost of capital, so you find the customer's threshold and build to it.

The simple annual return, the reciprocal of payback.

Savings over investment expresses the same relationship as payback from the other direction, as a percent return per year. It is a simple return, not a discounted cash flow or an internal rate of return, so use it alongside the payback and the customer's threshold, not instead of them.