To properly estimate the requirements and operational costs [What Price Speed. Von Karman et ….] of an Atmospherically Buoyant Spar Platform Space Elevator (ABSPSE) [ ] it is important to formulate the multiple processes that support its main functionality especially those that are temporal dependent. One such process is that of the effusion [ ] of the Lighter-Than-Air (LTA) gas being used, Helium [ ]. Effusion [ ] is a phenomenon that will inevitably happen with any LTA aeronautical mission [ ] in which non-molecular gases (i.e.; gases consisting of single atoms and no molecules, e.g.; Hydrogen and Helium,) are used. Because of the crippling effect [ ] that effusion causes to an ABSPSE, the longer aloft an ABSPSE can be kept; the better chances for the mission’s success are to be, and so, will the business profitability follow. Moreover, the sooner the mission’s payload has reached its launching altitude the lesser is structural stability is a concern for its success. Hence, knowing how soon the ABSPSE mission will take “to be accomplished” is an important tool for estimating the project’s costs, including taking any necessary actions for the operations of an ABSPSE to be as profitable as it can be while maximizing the chances for a successful completion of the mission.
The Archimedes’ principle vs. Newton’s 2nd Law!
Although the Archimedes’ principle gives the magnitude of the acting force from below of a neutrally buoyant or levitating object, this principle does not explain the object’s related acceleration a.lev in opposite direction to nominal gravity g.n. Instead, the usual way for understanding the displacement phenomena is best done by using Newton’s Second Law for motion [ ]. It is paramount to understand the interactions of the mass displacements by the atmospheric environment and the immersed mass of the LTA object. However, to many people, this is not immediately evident because; not only the less denser object is to be accelerated by g.n, but simultaneously; is how much of the mass of the displaced “denser” fluid (or gas) is accelerated by g.n.. In the following study, the relevant equations describing the relative motions making an object buoyant are derived [ ].
Buoyancy [ ], Levitation [ ] and Upthrust [ ]!
If one has a LTA object;
• with mass m.obj,
• being fully submerged in the fluid (Earth’s atmosphere), while
• the system is in hydrostatic equilibrium, and
• Earth’s gravitational field g.n affects the whole system.
Then, apart from the gravitational acceleration acting on it,
w.obj = m.obj ∙ g.n Eq. (1)
an immersed object will experience buoyancy equivalent to W.atm by,
From the Archimedes Principle, m.atm is the displaced air mass that originally occupied the atmospheric parcel (V.atm), which is equivalent to the volume (v.obj) of the buoyed object. Therefore, the total buoyancy on the object can be expressed as;
Usually, calculating the levitating acceleration a.lev associated with Eq. (3) is done by using Newton’s Second Law as [ ];
However, this formula is incorrect, because it does not take into account the fact that gravity not only accelerates the object’s mass [ ] but it also accelerates the mass of the displaced parcel of Air. As it occurs when the atmosphere “fills in” the space vacated by the displaced object, the interaction is schematically illustrated in Fig. 1.
Figure 1. The importance of this schema is in how it make us realize the existence of inertia and drag affecting the dynamics of the buoyancy generation.
Another useful analogy to understand the dynamics is to use a weighing balance and the given masses. The object’s weight is on one side and the weight of the atmospheric parcel is on the other side as is shown in Fig. 2. Here also the resulting buoyant force F.buoy is given by the addition of the weight [ ] of the object plus the weight [ ] of the displaced atmosphere. In this analogy, the important message is that of the principle of transmissibility holding true and is represented by the lever supporting both masses. Depending on which of the two weights is heavier, one side of the scale will drop and the other rise, and since both sides are rigidly connected, both masses have to be accelerated together at the same rate (etsi; in opposite directions). The speed of levitation transmitted by the lever is obviously the resulting advantage that we are seeking. See Fig 2.
Consequently, the proper acceleration of the object is given by
Obviously, this makes much more sense than Eq. 4, as the maximum acceleration the object can achieve is –g.n (assuming the mass m.obj = 0 kg). From Eq. 4, one would derive an infinite acceleration, but this is obviously not possible because the maximum attainable acceleration for the displaced fluid is that of the free-fall acceleration g.n of our planet’s.
From the above consideration, it is clear that buoyancy always leads to a net downward acceleration of the atmospheric mass, because even if the buoyant object rises, a greater mass of displaced fluid drops at the same time. This leads to the weight [ ] reduction by the levitating object.
Equation (6). Weight reduction effect!
From the above equation, it is obvious that during the levitating motion, not taking the factor of displaced fluid into account; “energy would not be conserved” (as it would gain both potential energy (from gravity) and kinetic energy (when the parcel of fluid is dropping)).