The hydraulics of the pyramids are simple but subtle, there is only one moving part = a float in water contained in a vertical well which makes a vertical back and forth movement from top to bottom.
Working with human energy, despite this hardiness we will see further that it led to the most efficient lifts ever made on earth to date.
Three types were used consecutively:
- The submersible float with stabilizing counterweight in the pyramid of Saqqara
- The submersible float guided without counterweight in all the following pyramids
- The float oscillating in the first three floors in the pyramid of Cheops and very probably the one of Khafre.
Saqqara submersible float:
The lifting float consists of a load-carrying plate which ends it
The rod resting on
The waterproof hull supporting it, the “deck” of which is flush with the water level in the well.
A long keel still submerged in the water extends the hull downwards.
In the lower part of the keel, is placed a very heavy ballast very probably in granite which will stabilize this “ship” whose load is very high above the deck.
The water displacement of this moving float is very small compared to its dimensions, because by its buoyancy it lifts a load whose weight is only a small fraction of the weight of the float.
Under the pyramid of Saqqarah there are 11 wells aligned under the eastern face and a central well. The vertical plane of these 11 shafts “cuts” the pyramid at a height of 20 m, which makes it possible to consider that this height is the reach of these elevators.
Let us now look at the operating conditions of this lifting float for one of the 11 eastern wells.
The section of the well being of 3.5 M² let us admit that that of the mobile set = the float plus its load, be of 3 M² to leave a functional gap in the sliding of the float in the well whose depth is 33 m.
The length of the float is equal to the depth of the well, as described above it is in 4 parts: a support plate raised by the lattice rod 20 m long = reach of the lift, the submersible waterproof hull of 10 m high, the keel with its 3 m long ballast. The hull, the outer section of which is 3 M², volume 30 M³, will be able to receive a maximum thrust of 30 t, which fixes the upper limit of the weight of the moving crew.
This hull is not a traditional shipbuilding, but a special adaptation. In fact, traditionally the hull of a boat contains a load, it is therefore hollowed out, here the load is not in the hull but on a rod placed on the deck, and in addition the hull must be fully submerged, which does not (fortunately) be ever the case in an “ordinary” vessel.
The hull here will be solid except for its bell-shaped lower part which will have to contain a certain volume of air, so it will be built on a hardwood frame filled with a sparse material, either light wood such as poplar or balsa, or bark such as cork. I will admit for simplicity that it weighs 30% of its volume or 9 t.
The 3 M² sectional tray is made of wood and can weigh 500 Kg.
The rod will be in a trellis it could have been made with wooden beams 10 cm on each side (the ideal would have been in pierced bamboo rods), let’s admit that its base is 1.5 x 1.5 m and that there is a belt every 2.5 m with a diagonal beam, each section would have made 13 m of development, 8 were needed, i.e. 104 m of beam for the trellis and 80 m of vertical beams for the 4 sides, i.e. 200 m of beam length in all , which represents a volume of 2 M³ and a weight of 1.5 t.
By sinking into the water of the well, once the mobile set is in the low position, the rod will receive 2 t of Archimedean thrust, it will therefore be necessary to reduce the volume of the hull accordingly to keep the buoyancy unchanged. This can be done by arranging in the lower part of the hull a bell-shaped volume trapping air, by sinking the water pressure compresses the air which correspondingly reduces the submerged volume of the hull.
From the high position to the low position, the absolute pressure at the bottom of the hull goes from 2 atmospheres to 4 atmospheres, but between the float in the high position and the float not yet launched the pressure of the air volume goes from 2 to 1 atmosphere, therefore for 2 M³ of variation of the air bubble at the end of the stroke, it will be necessary to fit out an air volume of 8 M³, occupying 3 m in length of the hull. This leaves 7 m of full hull length, which represents 21 M³ of volume + 4 M³ of air, or 25 M³, which gives a total Archimedean thrust of 25 t. If we remove the weights from the 9 t hull, the rod and the plate = 2t, 14 t of Archimedean thrust remains available to lift the ballast and load.
In the loaded position, the draft of the hull is almost its length, so the center of thrust is very close to the center of gravity, we will admit the balanced hull, the center of thrust is located at mid-height of the hull, or 5 m below the bridge. The center of gravity of the platform will therefore be 25 m from the center of pressure, giving a capsizing torque of 0.5 × 25 = 12.5 t × m. The center of gravity of the rod will be 15 m from the center of thrust, giving a capsizing torque of 1.5 × 15 = 22.5 t × m, or in all 35 t × m.
The center of gravity of the ballast is 6.5 m from the center of thrust, to balance the rod and the plate it should weigh 35 / 6.5 = 5.4 t rounded to 6, there are still 14 -6 = 8 t to be shared between the load and an additional ballast to balance it, the center of gravity of the load may be 26 m from the center of thrust, the balance ratio between the load and the ballast will be 26 / 6.5 = 4, which would lead to a maximum “theoretical” load of 1.6 t, for 6.4 t of ballast, this corresponding to equilibrium, load exactly centered in the axis of the float.
But of course, this condition is never reached, there will always be an off-centering of the load on the plate, moreover the examination of the pyramid shows that the largest filling block weighs around 300 KG.
Taking 0.6 t of payload, 2.4 t of ballast is needed for balancing, leaving a surplus of 4 t, which for a tolerable lateral displacement of the plate of 0.1 m (gap between the plate and the wall of the well) due at an offset of this load, allows an overturning torque of 4 × 0.1 × (6.5 / 26) = 0.1 t × m, i.e. a tolerable offset of 17 cm for this 0.6 t load without the plate rubbing against the walls .
The above calculation does not pretend to represent what really happened in these wells, however it identifies with a certain realism the maximum performance conditions accessible to manufacturers with this elevator technology.
It will be noticed that around 0.6 t of load, this lift could operate in “comfortable” functional conditions, which in relation to the weight of each filling blocks left the manufacturers a great deal of flexibility in organization.
This lifting float is of great apparent simplicity, safe and quiet in operation, it should however be realized that its sizing must be very precise, as ALWAYS in pyramids, to function correctly, in particular the adjustment of the variable volume under the float.
You can choose at will (within certain limits) the load imbalance which will cause the float to float or sink in relation to its total weight in static equilibrium load which is 25 t therefore 0.6 t of payload. The greater the imbalance, the faster the movement will be against a drop in efficiency.
Let’s take an example to fix the ideas with operators weighing 60 KG.
10 operators would fit easily on a 3 M² tray and weigh as much as 600 Kg of stone.
But at 600 kg nothing moves because the static equilibrium is perfect, for example 30 kg of ballast had to be removed so that the mobile unit weighing 25 t – 30 kg floats and rises slowly.
On the other hand once arrived at 20 m, stone unloaded, if one replaces the stone by 10 operators nothing moves, if one adds 30 Kg to it the static equilibrium is reached, nothing moves, it is necessary to add another 30 Kg in all is an eleventh operator for the plate to descend.
So to raise 600 Kg of stone, it was necessary to lower 660 Kg of operators who had to climb to the course beforehand with their feet, i.e. an energy efficiency of 90%
It is necessary to ensure that the float had an acceptable speed when reaching the bottom, weighing 25 t in load, it receives an Archimedean thrust of 25 t which balances it exactly, 30 kg of additional load makes it sink, everything behave as if the acceleration of gravity were then 30 / 25,000 of normal, or 9.82 × (30/25000) = 0.012 m / s², for a free fall of 20 m the speed at the end will be √ ( 2 × 0.012 x 20) = 0.7 m / s or 2.5 KM / H which does not present any danger. The travel time would have been √ (2 × 20 / 0.016) = 58 s
However arrived at the low point, the assembly will have acquired a kinetic energy of 1/2 × M × V² or 6 KJ, this kinetic energy will be absorbed by the bottom of the well and lost.
With the same imbalance of 30 KG the ascent will also be done in 58 s, but the float will have at the time of arrival a kinetic energy of 6 KJ, to prevent this energy from making the plate exceed the level of the seat then the enter into a long series of oscillations, it will be stopped by a stop on the seat made of sufficiently heavy blocks. Thus the load can be immediately transferred to the seat and the operators take their place on the platform to make the float sink again.
The potential energy given to a load of 0.6 t is 0.6 × 9.82 × 20 = 118 KJ in 58 s which corresponds to a “useful” power of 118/58 = 2 KW, which is quite honorable for the first hydraulic lift ever made on earth, operated only by men!
Overall energy efficiency:
A round trip from the float would have taken 2 minutes to which is added the time of loading and unloading the stones, say 4 minutes in all.
The average power consumed would then have been 118/240 = 0.5 KW, which could have been bearable for a team of 6 operators.
But in this example there are 11 and in addition it is necessary to double the number, a team which goes down on the plate when another climbs on the seat to keep the cycle time. That is to say 22 mobilized when 6 would have been enough.
If the operators had been ballasted at 120 Kg, 6 instead of 11 would have done the trick, the load imbalance would have been 120 Kg instead of 60, reducing the rise time by a third, i.e. 80 s cycle time. instead of 2 minutes.
In the end, leading to 12 people mobilized when 7 would have theoretically been sufficient.
In the end a theoretical return of 50% seems achievable, but in my opinion it is rather 25% which practically could be achieved.
It is this limitation in energy efficiency that made this design of the float obsolete when it came to the next pyramid in Meidum to multiply by 2.3 the volume of stones to be handled.
Assessment of this elevator:
In terms of investment, it is very simple and inexpensive, it takes up very little space on the ground, its mode of operation is quiet and safe.
In terms of performance, its operating cycle is of the order of 4 minutes between raising, lowering, loading and unloading stones. This is to raise a load of 0.6 t at 20 m high with 12 to 22 workers depending on the option chosen, which seems very efficient, but not enough for the next pyramid in Meidum.
The fact that the builders placed eleven of these shafts in parallel means that on the base 11 filling teams were working in parallel, and that consequently, from the quarries there were eleven lines of stone transport. These 11 lines made it possible to reduce the average cycle time to 20 s at the cost of a workforce of 150 workers to lift the stones.
This number of eleven was valid for the start of the site because as the elevation progressed, the end shafts no longer gave onto the foundation and therefore were abandoned. When the seat was 20 m high, the 4 end shafts were already out of service.
In the end to raise the stones, these operators have only one thing to do = get on the course with their legs and let themselves go down on a platform, which looks more like a walk in the park than a traumatic job, like pulling a very heavy load with a rope on an uphill ramp.
Section 49 M², depth 33 m, the ballast was left in the pyramid with the aim of luring looters and archaeologists by suggesting that this volume was the death chamber of the king.
I claim that this volume was the ballast of a large float intended to lift much heavier loads, probably the stones of the funeral complex, up to 20 m altitude at the heart of the pyramid.
A quick analysis of the ballast will tell us about the maximum weight of high stones.
The granite volume outside measures 3 × 5 × 3.8 m = 57 M³ if one took, which is common in pyramids, a wall thickness of 1 m, the interior volume would be 11 M³.
The question arises, why this hole in the ceiling to access the “Djoser’s vault”?
The ballast went down to the bottom of the well with 29 m of water above, so the pressure was 2.9 Kg / CM², i.e. on the largest interior wall a distributed force applied of 1600 KN enough to implode the volume, it was the experience of the “barrel break” upside down.
To avoid this catastrophe, the builders therefore played it safe by filling the ballast with water balancing the internal and external pressures, then at the end added the “cork” to complete the setting for the archaeologists of the future and the looters of the times.
This is why the entrance to the “king’s vault” is on the ceiling and not on the ground floor!
With walls 1 m thick, this volume would weigh 116 t in the air and 127 full of water, but when submerged it receives a thrust of 57 t it will therefore weigh 70 t, its center of gravity being 5.5 m from the center of thrust of the float placed above, if it had been a parallelepiped 40 M² in section and 4 m high, weighing 48 t and carrying a rod 20 m high weighing 16 t, lifting a platform of 8 t.
It was therefore possible to have a load on the platform which did not have to exceed 15 t so that the mobile equipment remained stable. In doing so, the stake was set very high for the following pyramids!
To sink the float, no less than 150 operators ballasted at 100 Kg were needed, ie a density of 3 men per M².
Why so heavy stones to lift?
The volume of a “proper” mortuary chamber requires a vaulted or double-rafter roof, which requires substantial blocks, but there is also the massive sarcophagus of course! container and cover, you can’t take a one-way trip for eternity without an adequate “time capsule”.
This weight compared to what we find in the pyramid of Cheops is modest since some blocks weigh three times more, but the stake is placed for the following pyramids, shame on the pharaoh who will not be able to climb a stone in his pyramid. not exceeding 15 t!
The day we go to visit Djoser’s remains, we will discover in the center of this pyramid a mortuary complex worthy of the buildings that we find outside in terms of size and quality of finish, but built with much larger blocks. and most likely of granite and of perfectly waterproof construction, although they were raised to 20 m altitude.
NB: the values given above are limit values to illustrate the problem, it is quite probable that the real values used by the manufacturers, apart from the section and the depth of the well, were different.
The sunken float, the first use of Archimedes’ thrusts, to lift the stones, turns out to have the qualities of simplicity of operation, reduced investment cost, as well as the size on the ground.
However, in view of the pharaohs’ increasingly demanding requirements, it is limited in terms of flow rate and absolute load performance, while the old methods using ramps have been largely made obsolete.
When we carefully study the dynamic functioning of this float we see that in its ascent, when the float bridge bursts the surface of the water, carried away by its momentum, it continues its upward course for a certain time then begins to oscillate .
For this lift, oscillation is a parasitic phenomenon which prevents the load from unloading and which must be suffocated.
priestsscientists and engineers who have focused for years on these hydraulic problems, ended up taking advantage of this drawback = parasitic oscillations of the float, to design the most powerful lift in the world that has never existed even in our times modern, the oscillating float which will be implemented in the pyramids of Cheops and probably Khafre.
However, the submersible float of Saqqara will not be abandoned for all that, a second generation obtained by freeing itself from the counterweight, will be put into service in ALL the following pyramids.
Its low cost, its simplicity of operation and especially its small footprint on an increasingly reduced course will be its strengths.
Subsequently when it comes to the finality of leading the burial procession of the king in his apartments of eternity, it is again this quiet float that will take care of the task.
To date MISSION ACCOMPLISHED!