All considerations made in Sections 7.1 and 7.2 should convince that chemical propulsion is inadequate to explore interplanetary space and perform planetary missions within reasonable times and budgets. The main reason is low Isp, at most of order 450 s. The Tsiolkowski equation predicts most of the mass of propellants will be spent accelerating the propellants themselves, and that the payload will be a small fraction of the initial mass, of order 1-3% for AV of order 7-8km/s. The Tsiolkowski relationship is:
where M is the initial mass of the spacecraft and Isp is in seconds. The exponential dependence means dramatic reduction in mass ratio, even simply doubling the /sp. Therefore, any advance in propulsion concepts to explore the Solar System must satisfy two separate conditions:
(1) Propellant consumption must be ''low'', that is, /sp must be as high as possible.
(2) Thrust must be "high" to ensure acceleration and AV needed by the mission.
Meeting these two conditions poses a severe power requirement, since power P - (Ve)3 - (/sp)3. In fact, if /sp (Ve) could be doubled for the same thrust F, the propellant consumption m (in kg/s) could be halved, because m Ve = F = Thrust = constant (7.5)
but the power demand would increase eight times. So, increasing propulsion efficiency (that is, /sp) means reducing the mass flowrate of propellants, not the power required to accelerate them. The power will inexorably increase.
Remember the second limitation of chemical propulsion is "slow" interplanetary travel. In the present context, what can be defined "fast" is: 1-3 years at most for unmanned probes, and several months to a year for manned vehicles. This means that any advanced propulsion system must economically enable A V much larger than 10 or 12km/s, in fact many tens of km/s. In Section 7.1 we have seen that to achieve these speeds a propulsion system must be capable of sustained acceleration for days or even weeks, with a commensurate power requirement.
Now, nuclear power converts fuel mass into energy according to E = mc2; the J available in fission is of order 8.2 x 1013J/kg using U, almost 10 million times larger than in combustion. This factor alone does justify propulsion based on nuclear reactions. However, how to exploit such J is one of the key questions. For instance, Section 7.2 pointed out that typical nuclear reactors cannot operate at temperatures much higher than, say, 2,500 K. So, at a first glance, a clear advantage of replacing H2/O2 combustion, characterized by similar temperatures, with nuclear heating, as done in so-called nuclear thermal rockets (NTR), is not evident. However, in NTR the propellant can be pure hydrogen, and its molecular weight, 2, is much lower than the average 9 or 10 of the burnt gas produced by an H2/O2 rocket. At similar temperature, an NTR ejecting pure hydrogen will have Isp higher by the square root of the ratio (9 to 10)/2, i.e., by a factor of about 2.2. In fact, the best Isp of LRE is about 450 s; the Isp of NTR tested in the past was of order 900 s. Furthermore, above 2,500 K a certain fraction of hydrogen begins to dissociate into H atoms (MW = 1), so that Isp grows a little more, perhaps near 950 s.
Isp in this range is very appealing for interplanetary travel, since the mass ratio following acceleration to a specified velocity is inversely proportional to Isp according to the Tsiolkowski relationship (7.4) already seen:
MinitiaA Minitial ( DV
From Isp = 450 s of a chemical rocket to 1,000 s of a nuclear thermal rocket means the total mass of propellants needed to inject into LEO a given payload may be reduced by a factor of 2.5. This is as if the gross weight of the US Shuttle at lift-off (about 2,800 tons) was reduced to 800 tons. Thus, both physics and engineering point to nuclear propulsion as the key to practical space exploration [Powell et al., 2004a].
The fundamental limitation of chemical propulsion is "low" Isp. One might ask, what is the explanation for this limitation. Aside from its units, for an ideal (complete, isentropic one-dimensional) expansion in a nozzle, Isp coincides with the exhaust velocity, Ve. This velocity is limited because it determines the kinetic energy of the flow, and this energy cannot be higher than that gases reach inside the thrust chamber because of chemical heat release. That is, in the chamber the heat released forms molecules of average mass m, possessing high translational, rotational and vibrational energy (call all of them internal energy E), and very little organized flow velocity. When the hot gas expands in the nozzle, molecular collisions gradually force all molecules to acquire the same orderly flow velocity at the expense of internal (disordered) energy. At the nozzle exit, in the ideal case this velocity is Ve = (2 E/m)1/2 if we neglect relativistic effects. The ratio E/m is the energy density, J, and, try as we might, even with H2/O2, J cannot reach above 107 J/kg. Ultimately, the limitation on V and Isp is due to the potential of the electro-weak force, because it is this force that shapes chemical bonds.
The next question is then what can be expected from choosing as energy source the only alternative, that based on the nuclear ''strong'' force.
In all three nuclear processes of Section 7.2 energy is released by converting fuel mass into energy. When the 235U nucleus fissions after colliding with neutrons, the total mass of its fission fragments is slightly less than its initial mass. A certain percentage, a, of the mass disappears, converted into kinetic energy and other forms of energy (call them all KE) of the fission fragments, according to KE = mc2. Since c, the speed of light, is 3 x 108m/s, the energy released is "large" on a human scale. Relativistically speaking, the mass lost corresponds to a decrease of the potential energy of the nuclear force binding neutrons and protons. In fact, while in Newtonian physics mass and energy are separate quantities, each separately conserved in any transformation, in relativistic physics it is the sum mc2 KE
that is conserved. Note that m is the relativistic mass, i.e., the rest mass, mo, divided by v/1 - (V/c)2.
Splitting the atom (fissioning) transforms potential energy of the nuclear force in KE of the fragments, their J of order 1013 J/kg already mentioned. The potential energy in a mass m of fuel is the fraction a mc2:
Fuel potential energy = a mfuel c2
The effect of fission is to convert the potential energy of the nuclear force (binding nucleons together) into kinetic energy of fragments (e.g., nuclides, neutrons, photons, ...). The KE of fragments, through collisions, converts into internal energy of a fluid or propellant, present as a mass Mp, and finally becomes orderly motion of particles ejected at speed Ve, or V for short. To calculate the ideal velocity V reached by a mass Mp of propellant after am mass of fuel fissions, a relativistic energy balance must be written. Approximating (for simplicity) KE with only 0.5mV2, that is, neglecting neutrino and photon energies, the energy balance becomes [Bruno, 2005, 2008]
where mo and Mpo are the fuel and the propellant mass at rest. Rewriting this equation, a preliminary result is that
4a2 1 c
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