Analytical mathematics for physical understanding, versus abstract numerical computation

‘Nuclear explosions provide access to a realm of high-temperature, high-pressure physics not otherwise available on a macroscopic scale on earth. The application of nuclear explosions to destruction and warfare is well known, but as in many other fields of research, out of the study of nuclear explosions other new and seemingly unrelated developments have come.’ – Harold L. Brode, ‘Review of Nuclear Weapons Effects’, Annual Review of Nuclear Science, Vol. 18, 1968, pp. 153-202.

Summary.

For high overpressures R = {[75E(g - 1)t²]/(8p r _o)}^1/5, U = (2/5)R/t.


Introduction. The sound wave is longitudinal and has pressure variations. Half a cycle is compression (overpressure) and the other half cycle of a sound wave is underpressure (below ambient pressure). When a spherical sound wave goes outward, it exerts outward pressure which pushes on you eardrum to make the noises you hear. Therefore the sound wave has outward force F = PA where P is the sound wave pressure and A is the area it acts on.

Note the outward force and equal and opposite inward force. This is Newton’s 3rd law. The same happens in explosions, except the outward force is then a short tall spike (due to air piling up against the discontinuity and going supersonic), while the inward force is a longer but lower pressure. A nuclear implosion bomb relies upon Newton’s 3rd law for TNT surrounding a plutonium core to compress the plutonium. The same effect in the Higgs field surrounding outward-going quarks in the ‘big bang’ produces an inward force which gives gravity, including the compression of the earth's radius (1/3)MG/c² = 1.5 mm (the contraction term effect in general relativity). Fundamental physical force mechanisms have been developed in consequence:

Sir G. I. Taylor, in his analysis of the Trinity nuclear test, observed in 1950: ‘Within the radius 0.6R the gas has a radial velocity which is proportional to the distance from the centre...’ (Proc. Roy. Soc., v. 201A, p. 181.) Thus, Hubble’s ‘big bang’. The writer came across this effect in the computer outputs published by H. A. Brode in ‘Review of Nuclear Weapons Effects’, Annual Review of Nuclear Science, v. 18, pp. 153-202 (1968) and decided to study the big bang with the mass-causing gauge boson or Higgs field as a perfect fluid analogy to air. The result is a prediction of gravity and other testable data.

History.

Archimedes’ book On Floating Bodies neglected fluid mechanisms and used a mathematical trick to ‘derive’ the principle of fluid displacement he has empirically observed: he states you first accept that the water pressure at equal depths in the ocean both below floating object and in open water is the same. Since the pressure is caused by the mass above it, the mass of water displaced by the floating object must therefore be identical to the mass of the floating object. Notice that this neat proof includes no dynamics, just logical reasoning.

MECHANISM OF BUOYANCY: HOW BALLOONS RISE DUE TO GRAVITY

Archimedes did not know what pressure is (force/area). All he did was to prove the results using an ingenious but mathematically abstract way, which limits understanding. There was no air pressure concept until circa 1600 and the final proof of air is credited falsely to Maxwell's treatise on the kinetic theory of gases.

In water, where you sit in a bath like Archimedes, you just observe that the water displaced is equal in volume to your volume when you sink, or is proportional to your mass if you float. Archimedes proves these observed facts using some clever arguments, but he does not say that it is the variation in pressure with depth which causes buoyancy.

For a 70 kg person, 70 litres of air (84 grams) is displaced so a person’s net weight in air is 69.916 kg, compared to 70 kg in a vacuum.

The reason a balloon floats is because the air pressure closer to the earth (bottom of balloon) is bigger than higher up, at the top, so the net force is upward. If you have a narrow balloon, the smaller cross-sectional area is offset by the greater vertical extent (greater pressure difference with height). This is the real cause of buoyancy. Buoyancy is proportional to the volume of air displaced by an object by the coincidence that upthrust is pressure difference between top and bottom of the object, times horizontal cross-sectional area, a product which is proportional to volume. If you hold a balloon near the ground, it can't get buoyant unless air pushes in underneath it.

The reason for buoyancy is the volume of the atmosphere that it has displaced from above it to under it and therefore pushes it upwards. A floating object shields water pressure, which pushes it upward to keep it floating. This is not a law of nature but is due to a physical process.

In water the water pressure in atmospheres is 0.1D, where D is depth in metres. You are pushed up by water pressure because it is bigger further down, because of gravity, causing buoyancy. Air density and pressure (ignoring small effects from temperature variations) falls by half for each 4.8 km increase in altitude.

Air pressure falls with altitude by 0.014% per metre like air density (ignoring temperature variation). Take a 1-m diameter cube-shaped balloon. If the upward pressure on the base of it is 14.7 psi (101 kPa), then the downward pressure on the top will be only 14.698 psi (100.9854 kPa). Changing the shape but keeping volume constant will of course keep the total force constant, because the force proportional to not just the horizontal area but also the difference in height between top and bottom, so the total force is proportional to volume. Archimedes’ up-thrust force therefore equals the displaced mass of air multiplied by gravitational acceleration. Mechanisms are left undiscovered in physics because of popular obfuscation by empirical ‘laws’. Someone discovers a formula and gets famous for it, blocking further advance. The discoverer has at no stage proved that the empirical mathematical formula is a God-given ‘law’. In fact, there is always a mechanism.

Sir Isaac Newton in 1687 made the crucial first step of fluid dynamics by fiddling the equation of sound speed, using dimensional analysis lacking physical mechanism, to give the ‘right’ (already assumed) empirical result. Laplace later showed that Newton had thus ignored the adiabatic effect entirely, which introduces a dimensionless factor of 1.4 into the equation. The compression in the sound wave (which is a pressure oscillation) increases its temperature so the pressure rises more than inversely as the reduction in the volume. The basic force physics of sound waves were ignored by mathematicians from S.D. Poisson [‘Mémoire sur la théorie du son’, Journal de l’école polytechnique, 14 `me Cahier, v. 7, pp. 319-93 (1808)] to J. Rayleigh [‘Aerial plane waves of finite amplitude’, Proceedings of the Royal Society, v. 84, pp. 247-84 (1910)], who viewed nature as a mathematical enterprise, rather than a physical one.

In 1848, the failure of sound waves to describe abrupt explosions was noticed by E. E. Stokes, in his paper on the subject, ‘On a difficulty in the theory of sound’, Philosophical Magazine, v. 33, pp. 349-56. It was noticed that the ‘abrupt’ or ‘finite’ waves of explosions travel faster than sound. Between 1870-89 Rankine and Hugoniot developed the ‘Rankine-Hugoniot equations’ from the mechanical work done by an explosion in a cylinder upon a piston (the internal combustion engine). [W. J. M. Rankine, ‘On the thermodynamic theory of waves of finite longitudinal disturbance’, Transactions of the Royal Society of London, v. 160, pp. 277-88 (1870); H. Hugoniot, ‘Sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits’, Journal de l’école polytechnique, v. 58, pp. 1-125 (1889).] As a result, experiments in France about 1880 by Vielle, Mallard, Le Chatelier, and Berthelot demonstrated that a tube filled with inflammable gas burns supersonically, a ‘detonation wave’, which Chapman in 1899 showed was a hot Rankine-Hugoniot shock wave which almost instantly burned the gas as it encountered it. These studies, on the effects of explosions upon pistons in cylinders, led to modern motor transport, which cleverly utilises the rapid sequence of explosions of a mixture of petrol vapour and air to mechanically power cars. Similarly, another ‘purely’ scientific instrument, the type of crude vacuum tube screen used by J.J. Thomson to ‘discover the electron’ was later used as the basis for the first television picture tubes.

Lord Rayleigh (1842-1919) is the author of the current existing textbook approach to ‘sound waves’ and no where does he worry about sound waves being really composed of air molecules, particles! Rayleigh in fact corresponded with Maxwell, who developed the kinetic theory to predict the spectrum of air molecule speeds (Maxwell distribution; Maxwell-Boltzmann statistics), but their discussion on the physics of sound was limited to musical resonators and did not include the mechanism for sound transmission in air. Maxwell’s distribution was based on flawed assumptions, which makes it only approximate. For example, collisions of air molecules are not completely elastic, because collision energy is partly converted into internal heat energy of molecules, and thermal radiation results; in addition, molecules attract when close.

Rayleigh’s non-molecular ‘theory of sound’ was first published in two volumes in 1877, titled (inaccurately and egotistically), The Theory of Sound. It is no more a ‘theory of sound’ than Maxwell’s elastic solid aether is the theory of electromagnetism. But it won Rayleigh the fame he wanted: when Maxwell died in 1879, his position as professor of experimental physics at the Cavendish lab was given to Rayleigh. We remember the lesson of Socrates, that the recognition of ignorance is valuable because it shows what we need to find out. Aristotle in 350 BC had proposed that sound is due to a motion of the air, but like Rayleigh, Aristotle ignored the subtle or trivial technical problem of working out and testing a complete mechanism! Otto von Guericke (1602-86) claimed to have disproved Aristotle’s idea that air carries sound, experimentally. This is another lession: experiments can be wrong. Von Guericke’s experiment was a fraud because be pumped out air from a jar containing a bell, and continued to hear the bell ring through vacuum. In fact, the bell was partly connected to the jar itself, which transmitted vibrations, and the resonating jar caused sound. Another scientist, Athanasius Kircher, in 1650 repeated the bell-in-vacuum experiment and confirmed von Guericke’s finding, this time the error was a very imperfect vacuum in the jar, due to an inefficient air pump. (More recently, in 1989, a professor of physical chemistry at Southhampton University allegedly picked up a neutron counter probe that was sensitive to heat, and obtained a gradual reading due to the effect of hand heat on the probe when placing the probe near a flask, claiming to have detected neutrons from ‘cold fusion’.)

Only in 1660 did Robert Boyle obtain the first reliable evidence that air carries sound. He did this by observing the decrease in the sound of a continually ringing bell as air was efficiently pumped out of the jar. Gassendi in 1635 measured the speed of sound in air, obtaining the high value of 478 m/s. He ignored temperature and the effect of the wind, which adds or subtracts to the sound speed. He also found that the speed is independent of the frequency of the sound. In 1740, Branconi showed that the speed of sound depends on air temperature. Newton in Principia, 1687, tried to get a working sound wave theory by fiddling the theory to fit the inaccurate experimental ‘facts’. Newton says the air oscillation is like a pendulum which has a velocity of v = (gh)^1/2, so for air of pressure p =rhg, sound velocity is (p/r)^1/2. Lagrange in 1759 disproved Newton’s theory, and in 1816 Laplace produced the correct equation. Although Newton’s formula for sound speed is dimensionally correct, it omits a dimensionless constant, the ratio of specific heat capacities for air, g = 1.4. This is because, as Laplace said in 1816, the sound wave has pressure (force/area), which alters the temperature. The sound wave is therefore not at constant temperature (isothermal), and the actual temperature variation with pressure increases the speed of sound. This is the adiabatic effect; the specific heat energy capacity of a unit mass of air at constant temperature differs from that at constant pressure, which Laplace deals with in his Méchanique Céleste of 1825. Newton falsely assumes, in effect, the isothermal relationship p/p_o = r /r _o (where subscript o signifies the value for the normal, ambient air, outside the sound wave), when the correct adiabatic equation is p/p_o = (r /r _o )^ g. Hence the Newtonian speed of sound, (p/r)^1/2 , is false and the correct speed of sound is actually higher, (g p/r

)^1/2. Of course, it is still heresy to discuss Newton’s fiddles and ignorance objectively, just as it is heresy to discuss Einstein’s work objectively. Criticisms of ‘heroes’ upset cranks.

Lord Rayleigh’s biggest crime was to reject Rankine’s 1870 derivation of the conservation of mass and momentum across a shock front. Rayleigh objected that energy cannot be conserved across a discontinuity, but Hugoniot in 1889 correctly pointed out that the discontinuity in the entropy between ambient air and the shock front changes the equation of state. Sadly, Rayleigh’s false objection was accepted by textbook author Sir Horace Lamb and incorporated in editions of Lamb’s Hydrodynamics (first published in 1879) up to and including the final (sixth) edition of 1932 (chapter X). Such was the influence of Lord Rayleigh that Sir Horace Lamb merely mentioned Hugoniot’s solution in a footnote where he dismisses it as physically suspect. Lord Rayleigh repaid Lamb by writing an enthusiastic review of the 4^th edition of Hydrodynamics in 1916 which described other books on the subject as ‘arid in the extreme’, stated ‘to almost all parts of the subject he [Lamb] has made entirely original contributions’, and misleadingly concluded: ‘the reader will find expositions which could scarely be improved.’

Perhaps the greatest challenge to common sense by a mathematician working in fluid dynamics is Daniel Bernoulli’s 1738 Hydrodynamics that mathematically related pressure to velocity in a fluid. This ‘Bernoulli law’ was later said to explain how aircraft fly. The experimental demonstration is that if you blow between two sheets of paper, they move together, instead of moving apart as you might naively expect. The reason is that a faster airflow leads to lower pressure in a perpendicular direction. The myth of how an aircraft flies thus goes like this: over the curved upper surface of an aircraft wing, the air travels a longer distance, in the same time as the air flowing around the straight lower surface of the wing. Therefore, Bernoulli’s law says there is faster flow on the top with lower pressure down against the wing (perpendicular to the air flow), so the wing is pushed up by the higher upward directed pressure on the lower side of wing due to the slower airflow:

http://quest.arc.nasa.gov/aero/background/. But to teach this sort of crackpot ‘explanation’ as being real physic is as sadistic as saying Newton created gravitation, see http://www.textbookleague.org/105wing.htm:

‘That neat refutation of ‘the common textbook explanation’ comes from an article that Norman F. Smith, an aeronautical engineer, contributed to the November 1972 issue of The Physics Teacher. The article was called ‘Bernoulli and Newton in Fluid Mechanics’. Smith examined Bernoulli’s principle, showed it was useless for analyzing an encounter between air and an airfoil, and then gave the real explanation of how an airfoil works:

Newton has given us the needed principle in his third law: if the air is to produce an upward force on the wing, the wing must produce a downward force on the air. Because under these circumstances air cannot sustain a force, it is deflected, or accelerated, downward.

‘There was nothing new about this information, and Smith demonstrated that lift was correctly explained in contemporary reference books. Here is a passage which he quoted from the contemporary edition of The Encyclopedia of Physics:

The overwhelmingly important law of low speed aerodynamics is that due to Newton. . . . Thus a helicopter gets a lifting force by giving air a downward momentum. The wing of a flying airplane is always at an angle such that it deflects air downward. Birds fly by pushing air downward. . . .

‘Nearly 30 years later, fake ‘science’ textbooks continue to dispense pseudo-Bernoullian fantasies and continue to display bogus illustrations which deny Newton's third law and which teach that wings create lift without driving air downward.’ Also:

http://www.lerc.nasa.gov/WWW/K-12/airplane/wrong1.html

Air based explosion supersonic shock phase (high overpressure blast waves)

‘Even when the start of the motion is perfectly continuous, shock discontinuities may later arise automatically … Nature confronts the observer with a wealth of nonlinear wave phenomena, not only in the flow of compressible fluids, but also in many other cases of practical interest. One example … is the catastrophic pressure in a crowd of panicky people who rush toward a narrow exit or other obstruction. If they move at a speed exceeding that at which warnings are passed backwards, a pressure wave arises … A great effort will be necessary to develop the theories presented in this book to a stage where they satisfy both the needs of applications and the basic requirements of natural philosophy.’ – R. Courant and K.O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, 1948, pp. 1 and 433.

Sir Geoffrey I. Taylor in June 1941 wrote a secret paper on blast from a nuclear bomb, The Formation of a Blast Wave by a Very Intense Explosion, for the British Civil Defence Research Committee, Ministry of Home Security. In March 1950 it was declassified and published in Proceedings of the Royal Society (v. 201A, pp. 159-86), accompanied by a study of the fireball in the Trinity test.

Taylor used the approach of solving, part analytically and part by step-by-step numerical integration, the differential equations of motion for a blast wave in air, based on the conservation of mass, momentum, energy, and the equation of state for air. The kinetic energy of the gas per unit volume is E/V = ½

r u², where r is the density and u the wind speed. The internal heat energy per unit mass of gas is given by the equation of state for air: E/M = p/[r (g - 1)], so the heat energy per unit volume is simply E/V = p/(g - 1). The total energy in the explosion at any time (neglecting transmission of thermal radiation to great distances) is therefore the volume integral (integral over radial distance for spherical surface shells) of the sum of kinetic and internal heat energies: ò ₀^R (4p r ² ).[½r u² + p/(g - 1)]dr.

He concluded that the shock wave ‘radius R is related to the time t since the explosion started by the equation R = S t^2/5E^1/5

r _o^-1/5, where r _o is the atmospheric density, E is the energy released and S is a [numerically approximate] calculated function of g , the ratio of the specific heats of air.’ However, using physical understanding in place of Taylor’s obfuscating mathematical procedure below, we not only obtain Taylor’s law analytically without resorting to approximate numerical integration, but we go further than Taylor by proving the function S = [75(g - 1)/(8p

)]^1/5.

To begin with, the total mass, M, of air engulfed by the blast wave up to any moment is the normal air density,

r _o, multiplied by the spherical volume of the blast wave, (4/3)p R³, so M = r _o(4/3)p R³ [Equation 1]. The total energy, E_tot, of the fireball is equal to its kinetic energy, E_k, multiplied by the ratio E_tot/ E_k, where E_k = PV = ½ MU² (P is pressure, V is volume, and U is outward shock velocity). The equation of state for air is g = 1 + PV/ E_tot, so E_tot/ E_k = E_tot/(PV) = 1/( g - 1). Introducing E_k = ½ MU² then gives: E_tot = ½ MU²(E_tot/E_k) = ½ MU²/(g- 1) [Equation 2].

The outward velocity of the shock, U, is diminishing owing to the continuously increasing mass M of air engulfed during expansion, throughout which the energy is distributed. The total mass engulfed increases in proportion to R³, but it is the surface area of the fireball, 4

p R², which is engulfing the air, so the deceleration due to mass engulfed per fixed area of the surface of the fireball is proportional to simply R³/R² = R. Hence U is directly proportional to R/t, so U = a R/t, where a is a constant [Equation 3]. Substituting Equations 1 and 3 into 2 gives: E_tot = (2/3)p r ₀ a ²(R⁵/t²)/(g - 1) [Equation 4]. This shows that the ratio R⁵/t² is a constant for any given bomb energy, therefore R⁵ is directly proportional to t², and R is proportional to t^2/5, or R = b t^2/5 [Equation 5], where b is a constant. Differentiating this, dR/dt = (2/5)b t^-3/5 [Equation 6]. Equation 3 becomes: dR/dt = U = aR/t [Equation 7].

Substituting Equation 5 into 7 gives: dR/dt = U =a bt^-3/5 [Equation 8].

Setting Equations 8 and 6 equal gives:

a = 2/5. Putting this value for a

into Equation 4 gives

E

= 8p r _oR⁵/[75(g- 1)t²],

or

R

= {[75E(g - 1)t²]/(8p r_o)}^1/5,

proving that Taylor’s numerically-computed S is really just [75(g - 1)/(8p )]^1/5. This energy formula is valid for an air burst, so the energy is only half this value when considering the hemispherical fireball of a surface or near surface burst. Putting a= 2/5 into Equation 3 we find

U =

(2/5)

R/t,

which is useful at early times (pressure high in comparison to ambient pressure), before the shock wave has degenerated into a sound wave.

g = 1.4 at low temperature, but molecular vibration reduces it to 1.2 for air at 5,000 K. But the dissociation of molecules into ions by radiation increases g , offsetting the vibration effect (g = 1.67 for a monatomic gas), so g remains 1.4. Taylor assumed an initial air density of 1.25 kg/m³ for Trinity, but it was 1.004 kg/m³ (from recorded temperature and pressure). 1-kt yield is 4.186 x 10¹² J. The Trinity film shows that the ratio R⁵/t² is a constant (7.46 x 10¹³ m⁵/s²) between 0.38 ms (when the shock wave has formed) and 1.93 ms, before significant thermal energy emission. So the initial hydrodynamic yield of Trinity was 15-kt compared to 18.6-kt total yield determined by radiochemistry. Thus 3.6 kt or 19 % of the total yield was used in (a) melting sand into fallout on the desert floor, (b) initial nuclear radiation escaping outside the fireball, and (c) delayed radioactivity.

Computer assessment of nuclear weapons effects

The detailed simulation of the formation and propagation of a spherical blast wave in the air by computer step-by-step integration was accomplished by Dr Harold L. Brode at the RAND Corporation in 1955. Brode set up differential equations to represent the effect of very small changes in time upon the air pressure, temperature (internal energy), density, and motion of the blast wave spherically from a point-source energy release in the air. Leonard Euler derived the equations for continuity and momentum in a compressible but frictionless fluid in 1755, both for fixed and moving (‘Lagrangian’) coordinate systems. However, J. L. Lagrange developed the theory without giving credit to Euler, so they are now known as the ‘Lagrangian equations’, and they describe the history of the particles in a fluid. The net acceleration is given by: a = d² x/dt² = (1/r ) dp/dx, where r is density, p is pressure and x is displacement. The differential equations were simply statements of the old physical principles of conservation of mass, momentum, and energy in the blasted air, with the best known ‘equation of state’ of air, which describes the relationship between temperature, pressure, and density. The basic equation of state is the ideal gas equation based on the discoveries of Boyle and Charles, (pressure) = (constant).(temperature).(density), but this simple formula becomes very inaccurate at extremely high temperatures due to energy loss in breaking down the diatomic air molecules into atoms, with progressive ionisation at higher temperatures. Viscosity needs to be introduced, since air is not an ideal fluid in a shock wave. The first treatment of the viscosity of fluids were the Navier-Stokes equations, due to Claude L. M. H. Navier in 1821, S. D. Poisson in 1829, and George Gabriel Stokes in 1845. Maxwell derived the coefficient of viscosity from his work on the kinetic theory of air in 1866.

Brode developed a digital computer program which numerically simulated the evolution and history of a blast wave in the air by storing data on the pressure, temperature and density of air at a series of small intervals fom the explosion point outwards, for example, at 10 cm intervals. The differential equations for the gradient of pressure are accurate enough to be used over such small intervals of distance. The computer program simulates the flow of energy in the first time interval, which again is a small enough interval, say 1 microsecond, that the differential equations are accurate. The original air temperature, pressure, and density at each distance from origin point are the normal atmospheric values (temperature 15° C or 288 K, pressure 101 kPa, and density 1.225 kg/m³ ). The calculated changes to these normal values at 1 microsecond after energy release are calculated and stored in the computer memory for every distance to the outer radius of the blast wave. Then these data are used for the next step of the computer calculation, for 2 microseconds after energy release. The computer therefore calculates a sequence of sets of data showing the properties of the blast wave with distance, for step-by-step increases in time.

This is the only way that the blast radius, peak overpressure, temperature and density can be accurately determined from the differential equations for any time after explosion, because differential equations are not valid for long distances or long times. It is only by a computer laboriously applying them, step-by-step, to very small distance changes and very small time changes, that they become accurate. This is the type of problem where a digital computer makes possible the solution of previously useless theory. Electronic digital computers therefore permitted a revolution in the practical use of theoretical physics. Accurate knowledge of blast waves allowed the measured blast data at nuclear tests to be accurately related to the energy release, provided that the calculated blast wave was used only for distances beyond the fireball.

A later improvement consisted of replacing the original equation for the conservation of energy in the blast wave with a more accurate equation which included the emission and absorption of thermal energy at each point, since heat is radiated by the high pressure blast wave which forms the fireball until it cools: H.L. Brode, et al., ‘A Program for Calculating Radiation Flow and Hydrodynamic Motion’, RAND Corporation, research memorandum RM-5187-PR, 1967. This improvement made the computer simulate the thermal radiation emission from the fireball and its effect on the blast wave at high pressures, thereby simulating the combined thermal and blast nuclear explosion phenomena. For example, this computer program can be used to study the effect of detonating an explosion at very high altitude upon the rate of thermal energy release from the fireball, where the low air density limits the air blast effects but allows thermal radiation to propagate very easily. It also allows measured nuclear test data from surface bursts to be analysed to accurately determine the amount of energy loss due to soil melting and cratering:

‘The shocked air and the violent expansion of the heated fireball dictate the nature of the thermal radiation... the cratering action puts vast amounts of earth material into the early fireball, thus further influencing the thermal radiation and the fallout. All effects begin with the nuclear reactions and their radiations, and all those listed are affected by the blast. Many features, in turn, have some influence on the blast itself.’ – Dr Harold L. Brode, ‘Review of Nuclear Weapons Effects’, Annual Review of Nuclear Science, Vol. 18, 1968, pp. 153-202

In the next instalment we shall see that for all overpressures: P =

å {[(g- 1)E/V]^n/3P_o^{1 –(n/3)}}, t = r/[0.340 + (0.0350/r^3/2) + (0.0622/r)], P_t = P_max [(1 – t/D_p⁺)e^-at] / [1 + 1.6(P_max /P_o).(t/ D_p⁺)], etc.