The hydraulics of the pyramids is simple but subtle, there is only one moving part = a float in the water.

Powered by human energy, in spite of its simplicity one will note later that this elevator outperformed to date all the others of its kind on earth .

Three generations were in use consecutively:

The submersible float with keel and ballast in the pyramid of Saqqara

The submersible float guided in all the following pyramids

The oscillating float in the 3 first stages of the pyramids of Cheops and Khaffre.

` Here after the description of the principles of the Saqqara's submersible float:`

Now let’s look at the operating conditions of this elevator for the 11 oriental wells.

The section of the well being 3.5 M² the float section could have been 3 M² to let a functional gap.

The length of the well is the length of the float of which 20 m are taken by the stem, so there remains 13 m for the hull and the keel containing the ballast, if we leave 3 m for the ballast, it will remain at most 10 m for the hull section 3 M², which can receive a maximum thrust of 30 t which sets the upper limit of the weight of the moving assembly.

This hull is not a traditional shipbuilding, but a special adaptation. In fact, the hull of a boat traditionally contains a load, it is thus 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 entirely immersed, which is (fortunately) never the case in an “ordinary” ship.

The hull here will be an assembly of low density wood or cork embedded in a hard wood frame but hollowed for a part at its bottom. I will admit for simplicity that it weighs 30% of its volume be 9 t.

The 3 M² tray in wood can weigh 500 Kg.

The stem will be made of lattice made with wooden beams of 10 cm on the side (the ideal would have been in bamboo stems pierced), let’s say that its base is 1.5 x 1.5 m and that there is a belt every 2.5 m with a beam diagonally, each belt is 13 m developed there are 8 in number leading to 104 m of beam for the trellis, plus 80 m of vertical beams let say 200 m length of beams in all, which represents a volume of 2 M³ and a weight of 1.5 t.

Sunk into the water of the well, once the float in low position, the rod will receive 2 t of buoyancy, but the overall buoyancy volume must be constant what ever was the float position in the pit.

It was therefore necessary to decrease the buoyancy of the hull when at the bottom. What can be done by hollowing the lower part of the hull trapping a certain volume of air. When sunk the pressure of water compresses the volume of air decreasing the buoyancy. This volume was in accordance with the buoyancy of the stem.

From the high position to the low position, the absolute pressure at the bottom of the float doubles, but between the float in the high position and the float not yet put in the water the pressure of the air volume doubles also, so for 2 M³ of variation the air pocket at the end of the race, it will be necessary to build a air volume of 8 M³, occupying 3 m at the lower end of the hull, so there is still 7 m length of sealed hull.

which represents an overall buoyancy of 21 M³ + 4 M³ of air = 25 M³, which gives the maximum weight of the mobile equipment of 25 t. If we remove the weight of the float 9 t, the rod and the plateau 2 t, there remains 14 t for the ballast and the payload.

Full loaded, the draft of the hull is almost its length, so the thrust center is very close to its half length, at 5 m under the bridge almost confused with its center of gravity. The center of gravity of the tray will therefore be 25 m from the center of thrust 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 torque of 1.5 × 15 = 22.5 t × m, or a total of 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 is still 8 t to share between the load and the ballast to balance the payload, the center of gravity of the payload may be 26 m from the center of thrust, the equilibrium 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 corresponds to the balance, load exactly centered in the axis of the float.

But of course, this condition is never reached, there will always be an offset of the load on the tray, moreover the examination of the pyramid shows that the average filling block weighs about 300 KG.

Taking 0.6 t for the payload requires 2.4 t of ballast for balancing it, leaving a surplus of 4 t, which for a tolerable lateral displacement for the tray of 0.1 m due to the offset of this load allows a reversal torque of 4 × 0.1 × (6.5 / 26) = 0.1 t × m, ie a tolerable offset of 17 cm of this load of 0.6 t without the tray rubbing against the walls.

**In conclusion:**

The calculation above, does not claim to represent what really happened in these wells, however it represent with a certain realism, the maximum performance conditions available to the builders with their technology of the moment.

It should be noted that around 0.6 t of payload, this lift could operate in “comfortable” functional conditions, which compared to the average weight of the filling blocks left the builders a great latitude of organization.

This elevator float is of great apparent simplicity, a safe and quiet operation, it must however be realized that its layout must be very precise, as ALWAYS in the pyramids, to work properly, especially the adjustment of the air volume under the hull .

**Float dynamics:**

One can choose at will (within certain limits) the imbalance in weight which will make the float to sink or float with respect to its total weight full loaded in the state of static equilibrium which is around 25 t including 0.6 t of payload. The greater the imbalance, the faster the movement will be at the cost of a drop in efficiency.

Let’s take an example to fix the ideas with operators weighing 60 KG each.

10 operators would easily hold on a plateau of 30 sqm and weight as much as 600 kg of stone.

But at 600 Kg nothing moves because the static balance is perfect, it was necessary to remove for example 30 Kg of ballast so that the mobile assembly weighing 25 t – 30 Kg floats and rises slowly.

On the other hand once arrived at 20 m, stone discharged, if one replaces the stone by 10 operators nothing moves, if one adds 30 Kg the static equilibrium is reached, nothing moves, it is necessary to add 30 Kg = the equivalent of an eleventh operator in all for the float to go down.

So to raise 600 Kg of stone, it was necessary to put on the tray 660 Kg of operators who will have to make the prior ascension to the course, an energetic efficiency of 90%

**Way down:**

It is necessary to make sure that the float has an acceptable speed when arriving at the bottom, weighing 25 t full loaded, it receives a pressure of Archimedes of 25 t which the equilibrium exactly, 30 kg of additional load make it to sink, things append as if the acceleration of gravity was then 30 / 25,000 of the normal, ie 9.82 × (40/25000) = 0.012 m / s², for a free fall of 20 m the speed at the finish will be √ ( 2 × 0.012 x 20) = 0.7 m / s or 2.5 KM / H which is safe. The race time would have been √ (2 × 20 / 0.016) = 58 s

However arrived at the low point, the float 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.

**Way up:**

With the same imbalance of 30 KG the way up will also be done in 58 s, but the float will have at the moment of the arrival a kinetic energy of 6 KJ, to avoid that this energy causes the plateau to exceed the level of the seat and then the enter a long series of oscillations, it must stopped by a stop on the course made of sufficiently heavy blocks. Thus the load can be immediately transferred to the course and the operators taking place on the plateau to sink the float 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 consumed to lift the payload, which is quite honorable for the first hydraulic lift ever made on earth, operated only by men!

**Global energetic efficiency:**

Case in point the float cycle would have been 2 minutes plus the time of loading and unloading stones, let say 4 mn in all. As a consequence the average power consumed by cycle was 118 / 240 = 0.5 KW that means that a working team of 6 would had been enough to sustain the pace. But they are 11 at least in this case, worse, they must be twice as more in order to keep the speed, a team going up on the course when the other going down on the tray. 22 people when 6 would had make it, is not brillant.

The way to improve was to weight the operators with a ballast, let say 120 kg instead of 60 Kg put to 6 instead of 11 the number of operators, the weight unbalance of the float go to 120 Kg instead of 60 Kg, passing the rising time to 41 s instead of 58 s, that change not too much the total cycle so the average power consumed pass to 0.6 KW.

At the end of the day this kind of elevator theoretically can reach an efficiency barely as high as 50%, but 25% was more likely to be reached.

This is the limitation of this principle, that will make it obsolete when it comes to double the size of the average bloc and more in the following pyramids, leading to the second generation submersible float.

**Balance of this elevator:**

In terms of investment, it is very simple and inexpensive, it takes on the ground a very small footprint, its operating mode is quiet, without difficulty.

In terms of performance, its operating cycle is of the order of 4 minutes going up and down loading and unloading of stones.

This is to raise a payload of 0.6 t to 20 m in height in 4 mn with 12 workers, which look very performing compared to using ramps and sledges, but not good enough when it comes to double the size of the pyramid in the next site à Meidum.

The fact that the builders placed eleven of these wells in parallel means that on the base 11 filling teams were working in parallel, and as a result, from the quarries there were eleven stone transport lines, reducing the average cycle to about 20 s, at the cost of a workforce of about 150 people

This number of eleven was worth for the beginning of the construction site because as the elevation, the end wells were no longer on the course and therefore were abandoned. When the course was 20 m high, the 4 end wells were already out of order.

In the end to raise the stones, these operators have only one thing to do = climb on the course with their legs and let down on a plateau, which looks more like a walk of health than a traumatic work like pulling a rope with a heavy load.

**Central well:**

Section 49 M², depth 33 m, the ballast was left in the pyramid with the aim of luring looters and archaeologists to believe that this volume was the vault of the king.

I claim that this volume was the ballast of a large float designed to raise much heavier loads, probably the stones of the funerary complex, up to 20 m altitude in the heart of the pyramid.

A quick ballast analysis will tell us about the maximum weight of the high stones.

The granite volume measures outside 3 × 5 × 3.8 m = 57 M³ if one took, which is common in the 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 “vault of Djoser”?

The ballast goes down to the bottom of the well with 29 m of water above, so the pressure is 2.9 Kg / cm², that is on the inner wall the largest distributed force applied 1600 KN enough to implode the volume.

To avoid this catastrophe, the builders therefore played the caution by filling the water ballast balancing the internal and external pressures, then in the end have added the cap to complete the staging for future archaeologists and looters of the present .

That’s 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 in the air 116 t and 127 full of water, but immersed it receives a thrust of 57 t so it will weight 70 t, its center of gravity being 5.5 m from the center float pressure placed above, if it had been a block of 40 M² section and 4 m height, weighing 48 t and carrying a rod 20 m high weighing 16 t, lifting a tray of 8 t.

It could therefore have on the plateau a load that should not exceed 15 t for the mobile team remains stable. In doing so the bar was set very high for the following pyramids!

To sink the float it was necessary not less than 150 operators weighted to 100 kg or a density of 3 men per square meter.

Why are stones so heavy?

The volume of a “suitable” mortuary requires a vaulted roof or double rafters, which requires substantial blocks, but there is also the massive sarcophagus of course! container and lid, we do not take a single ticket for eternity without a proper “time capsule”.

This weight compared to what we find in the pyramid of Cheops is modest because some blocks weigh three times more, but the bar is placed for the following pyramids, shame to Pharaoh who can not raise in his pyramid a stone not exceeding 15 t!

The day we go to visit the remains of Djoser, we discover in the center of this pyramid a mortuary complex worthy of buildings that are found outside in terms of size and quality of finish, but built with much larger blocks and most probably in granite and 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 likely that the actual values used by the manufacturers, apart from the section and the depth of the well have been different.

**Conclusion:**

The submersible float, the first use of the Archimedes’ thrust, to lift the stones proves to possess the qualities of simplicity of operation, reduced investment cost, as well as the footprint.

However, in view of the increasing demands of the pharaohs, it is limited in performance of flow and absolute load, while the old methods using ramps have been largely made obsolete.

When we study finely the dynamic functioning of this float we see that in its ascent, when the bridge of the float burst the surface of the water, carried by its momentum, it continues its course upward for a while then begins to oscillate at a low frequency .

For this lift the oscillation is a parasitic phenomenon that prevents the landing of the load and must be smothered.

The priests and engineers who have concentrated for years on these hydraulic problems, have finally taken advantage of this inconvenience = parasitic oscillation of the float, to design the most powerful elevator in the world that has never existed even in in our modern times, the oscillating float that will be implemented in the pyramids of Cheops and Khaffre.

However Saqqarah’s submersible float will not be abandoned as long as a second generation obtained by abandoning the ballast, will be put into service in ALL the following pyramids.

Its low cost, simplicity of operation and especially its small footprint on a reduced base will be its strengths.

Subsequently when it comes to fulfill the last purpose: lead the procession of burial of the king in his apartments for eternity, it is still this quiet float that will make the job.

To date MISSION ACCOMPLISHED!