Pulsed nuclear thermal rocket

Statement of the concept

A rough calculation for the energy gain by using a pulsed thermal nuclear rocket in comparison with the conventional stationary mode, is as follows. The energy stored into the fuel after a pulsation, is the sensible heat stored because the fuel temperature increase. This energy may be written as

E pulse = c f M f Δ T

where:

E pulse is the sensible heat stored after pulsation, c f is the fuel heat capacity, M f is the fuel mass, Δ T is the temperature increase between pulsations.

On the other hand, the energy generated in the stationary mode, i.e., when the nuclear core operates at nominal constant power is given by

E stationary = χ l l t

where:

χ l is the linear power of the fuel (power per length of fuel), l is the length of the fuel, t is the residence time of the propellant in the chamber.

Also, for the case of cylindrical geometries for the nuclear fuel we have

M f = π R f 2 l ρ l

and the linear power given by

χ l = 4 π κ f ( T f − T s )

Where:

R f is the radius of the cylindrical fuel, ρ f the fuel density, κ f the fuel thermal conductivity, T f is the fuel temperature at the center line, T s is the surface or cladding temperature.

Therefore, the energy ratio between the pulsed mode and the stationary mode, N = E pulse E stationary yields

N = c f ρ f R f 2 4 π κ f ( T f − T s ) [ Δ T t ]

Where the term inside the bracket, [ Δ T t ] is the quenching rate.

Typical average values of the parameters for common nuclear fuels as MOX fuel or uranium dioxide are: heat capacities, thermal conductivity and densities around c f ≃ 300 J / ( m o l ⋅ K ) , κ f ≃ 6 W / ( K ⋅ m 2 ) and ρ f ≃ 10 4 k g / ( m 3 ) , respectively., with radius close to R f ≃ 10 − 2 m , and the temperature drop between the center line and the cladding on T f − T s = 600 K or less (which result in linear power on χ l ≃ 45000 W / m ) . With theses values the gain in energy is approximately given by:

N ≃ 6 × 10 − 3 [ Δ T t ]

where [ Δ T t ] is given in K / s . Because the residence time of the propellant in the chamber is on 10 − 3 s e c to 10 − 2 s e c considering subsonic velocities of the propellant of hundreds of meters per second and meter chambers, then, with temperatures differences on Δ T ≃ 10 3 K or quenching rates on [ Δ T t ] ≃ 10 6 K / s energy amplification by pulsing the core could be thousands times larger than the stationary mode. More rigorous calculations considering the transient heat transfer theory shows energy gains around hundreds or thousands times, i.e., 10 2 ≥ N ≤ 10 3 .

Quenching rates on [ Δ T t ] ≥ 10 6 K / s e c are typical in the technology for production of amorphous metal, where extremely rapid cooling in the order of 10 6 K / s ≤ [ Δ T t ] ≥ 10 7 K / s are required.

Direct thrust amplification

The most direct way to harness the amplified energy by pulsing the nuclear core is by increasing the thrust via increasing the propellant mass flow.

Increasing the thrust in the stationary mode -where power is fixed by thermodynamic constraints, is only possible by sacrificing exhaust velocity. In fact, the power is given by

P = 1 2 F v e

where P is the power, F is the thrust and v e the exhaust velocity. On the other hand, thrust is given by

F = m ˙ p v e

where m ˙ p is the propellant mass flow. Thus, if it is desired to increase the thrust, say, n-times in the stationary mode, it will be necessary to increase n 2 -times the propellant mass flow, and decreasing 1 n -times the exhaust velocity. However, if the nuclear core is pulsed, thrust may be amplified n -times by amplifying the power n -times and the propellant mass flow n -times and keeping constant the exhaust velocity.

Isp amplification

The attainment of high exhaust velocity or specific impulse (I_sp) is the first concern. The most general expression for the I_sp is given by

I sp ≃ c T

being c a constant, and T the temperature of the propellant before expansion. However, the temperature of the propellant is related directly with the energy as E ≃ k T , where k is the Boltzmann constant. Thus,

I sp ≃ c ′ E

being c ′ a constant.

In a conventional stationary NTR, the energy E for heating the propellant is almost from the fission fragments which encompass almost the 95% of the total energy, and the faction of energy from prompt neutrons f n is only around 5%, and therefore, in comparison, is almost negligible. However, if the nuclear core is pulsed -as previously discussed, it is able to produce N times more energy than the stationary mode, and then the fraction of prompt neutrons or f n N could be equal or larger than the total energy in the stationary mode; and because this neutron energy is directly transported from the fuel into the propellant as kinetic energy -unlike the energy from fission fragments which is transported as heat from the fuel into the propellant, then is not constrained by the second law of thermodynamics, meaning that there is no impediment to transport this energy from the fuel to the propellant even if the fuel is colder than the propellant, in other words, it is possible a " propellant hooter than the fuel" which is the very limit for specific impulse enhancement in classics NTRs.

In summary, if the pulse generates N times more energy than the stationary mode, the I_sp amplification is given by

I sp ≃ I sp,o f n N + 1

Where:

I sp is the amplified specific impulse, I sp,o the specific impulse in the stationary mode, f n the fraction of prompt neutrons, N the energy amplification by pulsing the nuclear core.

With values of N between 10 2 to 10 3 and prompt neutron fractions around f n ≃ 1 20 , , the hypothetical I sp amplification attainable makes the concept specially interesting for interplanetary spaceflight.

Advantages of the design

There are several advantages relative to conventional stationary NTR designs. Because the neutron energy is transported as kinetic energy from the fuel into the propellant, then a propellant hotter than the fuel is possible and therefore the I sp is not limited to the maximum temperature permissible by the fuel, i.e., its melting temperature.

The other rocket concept which allows a propellant hotter than the fuel is the fission fragment rocket, however, in the pulsed rocket concept, the propellant is heated instantaneously as a burst rather than by continuous heating as in the fission fragment concept, then very high propellant temperatures are theoretically attainable immediately after the pulse, which is followed by a rapid radiative cooling. Therefore, by reducing the residence time of the propellant after the pulse will reduce the losses in propellant temperature.

Other considerations

For I sp amplification, only the energy from prompt neutrons -and some prompt gamma energy, is used with this purpose. The rest of the energy, i.e., the almost 95 % from fission fragments is an unwanted energy and must be continuously evacuated by a heat removal auxiliary system using a suitable coolant,. Liquid metals, and particularly lithium, can provide the fast quenching rates required.

As regard to the mechanism for pulsing the core, the pulsed mode can be produced using a different variety of configurations depending of the desired frequency of the pulsations. For instance, the use of standard control rods in a single or banked configuration with motor driving mechanism or the use of standard pneumatically operated pulsing mechanisms are suitable for generating up to 10 pulses per minute, for the production of pulses at rates up to 50 pulsations per second, the use of rotating wheels introducing alternately neutron poison and fuel or neutron poison and non-neutron poison can be considered. However, for pulsations ranking the thousands of pulses per second (kHz), optical choppers or modern wheels employing magnetic bearings allow to revolve at 10 kHz. If even fast pulsations are desired, then, it would be necessary the use of a new type of pulsing mechanism that does not involve mechanical motion, for example, lasers (based in the 3He polarization) as early proposed by Bowman, or proton and neutron beams. For the nuclear thermal rocket frequencies in the order of 1 kHz to 10 kHz are the choice.