Bessler's (4th) Kassel wheel Archimedes screw pump calculations

[jim_mich]

The facts, as can best be determined by scaling Bessler's drawing:

Screw length = 12 foot (144 inches)
Screw lead = 1/2 foot (6 inches)
Screw ID = 1.5 foot (18 inches)
Screw axle = 1/4 foot (3 inches)
Screw tilt angle = 20º
Wheel square pulley = 1.4 foot across corners (16.8 inches)
Screw square pulley = 2.8 foot across corners (33.6 inches)
Pulley ratio = 2:1 reduction.
Wheel speed = 20 RPM (Ref. not used)
Screw speed = 10 RPM (Ref. not used)
Screw working length = 8 foot (96 inches)
Screw reserve length = 2 foot (24 inches)
Screw maximum working length when H2O level is low = 10 foot
Volume per 'bucket' filled with water = 25% to 35%
Notes:
As used here a 'bucket' is the space available for one 'chunk' of the water to ride.
These bucket volume percentages are low and hi estimates based on research reading.
I wanted to actually calculate the volume but such is very complex requiring a computer program.
The upper percentage volume limit of any screw pump is just shy of 60%.
Most screw pumps have a slope of 30° and an ideal percentage volume around 26%.
A shallower slope allows for a larger volume percentage.
Bessler's pump is drawn as a 20° slope and thus could have had a 35% or greater H2O volume.

==============================

Notes:
I've tried to make my calculation as clear as possible.
I've tried to show all my calculations.
I've done a little bit of rounding of numbers to keep it readable.
My goal was an end result number comparable to lifting a weight from the axle.
If anyone does not understand anything then just ask.


Empty bucket volume = ( Pi × (1.5_ID ÷ 2)^2 - Pi × (1/4_Axle ÷ 2)^2 ) × 1/2_Ft_Lead_Length
Empty bucket volume = 0.859 Cubic_Foot

Bucket volume of H2O = 0.859 × 25% = 0.215 Cubic Foot (Conservative estimate)
Bucket volume of H2O = 0.859 × 35% = 0.301 Cubic Foot (Aggressive estimate)

Ref: Weight of H2O = 62.3 Lbs per cubic foot at 69ºF per machinery's handbook

Bucket H2O weight = 0.215 × 62.3 = 13.378 Lbs (Conservative estimate)
Bucket H2O weight = 0.301 × 62.3 = 18.731 Lbs (Aggressive estimate)

Ref: 1 Gallon water weighs 8.328 Lbs.
Gallons H2O per Bucket = 13.378 ÷ 8.328 = 1.61 Gallons (Conservative estimate)
Gallons H2O per Bucket = 18.731 ÷ 8.328 = 2.25 Gallons (Aggressive estimate)

Ref: 1 Liter of water weighs 2.2 Lbs
Liters H2O per Bucket = 13.378 ÷ 2.2 = 6.08 Liters (Conservative estimate)
Liters H2O per Bucket = 18.731 ÷ 2.2 = 8.51 Liters (Aggressive estimate)

Number of buckets being lifted = 8_Ft ÷ 2_per_foot = 16 Buckets (When tank is full)
Number of buckets being lifted = 10_Ft ÷ 2_per_foot = 20 Buckets (When tank is 1/2 full)

Weight H2O lifted = 16 Buckets × 13.378 Lbs_per_Bucket = 214 Lbs (Conservative with full tank)
Weight H2O lifted = 20 Buckets × 18.731 Lbs_per_Bucket = 374 Lbs (Aggressive with 1/2 empty tank)

Load with pulley reduction = 214 Lbs ÷ 2 = 107 Lbs load on wheel (Conservative with full tank)
Load with pulley Reduction = 374 Lbs ÷ 2 = 187 Lbs load on wheel (Aggressive with 1/2 empty tank)

If the bucket volume was 25% and the tank was 1/2 empty then...
Load = 107 Lbs ÷ 8 × 10 = 134 Lbs load on wheel (Conservative with 1/2 empty tank)

If the bucket volume was 35% and the tank was full then...
Load = 187 Lbs ÷ 10 × 8 = 150 Lbs load on wheel (Aggressive with full tank)

----------------------

So... the bottom line...

Assuming a typical screw pump with each bucket filled 25% and with the water tank filled as shown in the drawing, then the load was 107 Lbs, which is slightly less than lifting the 112 Lb weight.

If we assume that the water level in the tank dropped to half full then the load would have increased by about 25% to the equivalent of lifting a load of 134 Lbs, which is about 20% more than lifting the 112 Lbs weight.

Assuming a less typical screw pump with each bucket filled 35% due to the shallow slope angle and with the water tank filled as shown in the drawing, then the load was 150 Lbs, which is about 34% more than when lifting the 112 Lb weight.

If this was the case and the water dropped to half full, then the load would increase to the equivalent of lifting 187 Lbs, which is 67% more than lifting 112 Lbs.

So it is not unreasonable to assume that the wheel might have been able to maintain its 26 RPM speed when lifting the 112 Lb weight (witnesses said that it maintained its speed) but then also slow down when pumping water due to a much greater load that it might have been asked to carry.

At the beginning of this posting are the assumptions. At the end are the possible results. If anyone wants to, they can change the assumption and recalculate the results. Hopefully I've made this clear enough that anyone could do the math.

Also, hopefully I've not made any mistakes.

[ovyyus]

Jim, what is your estimate of water flow rate per screw revolution?

...when lifting the 112 Lbs weight (witnesses said that it maintained its speed)...

Please cite reference, thanks.

[jim_mich]

Jim, what is your estimate of water flow rate per screw revolution?

Ref: 1 Gallon water weighs 8.328 Lbs.
Gallons H2O per Bucket = 13.378 ÷ 8.328 = 1.61 Gallons (Conservative estimate)
Gallons H2O per Bucket = 18.731 ÷ 8.328 = 2.25 Gallons (Aggressive estimate)

Ref: 1 Liter of water weighs 2.2 Lbs
Liters H2O per Bucket = 13.378 ÷ 2.2 = 6.08 Liters (Conservative estimate)
Liters H2O per Bucket = 18.731 ÷ 2.2 = 8.51 Liters (Aggressive estimate)

[ovyyus]

OK, your average flow rate estimate is about 7 litres per screw revolution. That's 70 litres per minute at 10 RPM, or about 1.2 litres per second. Therefore, the wheel lifted 1.2 kg to a height of about 1 metre every second.

Work to lift 1.2 kg to a height of 1 metre per second is 12 Watts.

[daxwc]

The screw is an incline plane, it is not a correct draw of 112lbs, it takes 12ft to raise the water 6 ft.

[jim_mich]

Daxwc, you are correct that it is an inclined plane. Last night i realized my error in not including the inclined plane, but it was after 1:30 am in the morning and I was too tired to post anymore.

The screw is inclined at 20° so the incline ratio = Sin(20°) = 0.342

So... the bottom line adding the slope to the four calculations...

107 Lb × 0.342 = 36.6 Lbs (Conservative with full tank)
134 Lb × 0.342 = 45.8 Lbs (Conservative with half empty tank)
150 Lb × 0.342 = 51.3 Lbs (Aggressive with full tank)
187 Lb × 0.342 = 64.0 Lbs (Aggressive with 1/2 empty tank)

The conservative estimate uses a 25% fill of each bucket.
The aggressive estimate uses a 35% fill of each bucket.
Its sometimes possible to fill the bucket up to about 60% when using a very shallow slope and multiple leads. To determine the fill percentage estimate of Bessler's screw requires a computer program be used. My gut feeling is Bessler's 'buckets' were filled to about 45% or 50%, but only a computer program can tell me how full they were based upon the known data at the start of this thread.

Also the water moving within the screw and the rope driving over square pulleys would have produced much more friction than the simple lifting of the 112 lb weight.

Bill's method of estimating the Watts seem to not take into account all the back-torque produced by the water screw, but I'll need to think about it some more. My old brain does not work as fast and clear as it did a few years ago.

It's after 9 AM so time to start working my day job. The stock market opens in about 20 minutes.

[ovyyus]

it takes 12ft to raise the water 6 ft

The water isn't being raised 6 feet.

[daxwc]

What is the best approximation of height it is being pumped?

[jim_mich]

The water was pumped about 2-3/4 feet according the the drawing. If the tank were to become half empty then the water would be pumped almost 3-1/2 feet. The slope of the screw was 20°.

PS. Today my day job was good to me.

[ovyyus]

I averaged it at 1 metre lift.

[eccentrically1]

the water looks like it is being funneled back to the tank in the drawing. The tank would never be less than completely full.

[daxwc]

Should have been good with gold up 53 dollars on the Gold Spot (FOREX:XAUUSDO)

[jim_mich]

the water looks like it is being funneled back to the tank in the drawing. The tank would never be less than completely full.

Except for the spillage and leakage that is clearly shown being drained through the floor, along a trough, and then out the window in the lower floor.

[ovyyus]

I doubt there was much spillage or leakage that escaped the closed loop tank/return trough. The drain is for emptying the tank.

[jim_mich]

Bill, do you really thing that little return trough was catching all the water that was sploshing and splashing out of the water screw?

Did anyone stop to think about how much slippage and friction would have been present with the square pulley and rope drive system?