Pulsed nuclear thermal rocket

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

${\displaystyle E_{\text{pulse}}=c_{f}M_{f}\Delta T}$

where:

${\displaystyle E_{\text{pulse}}}$ is the sensible heat stored after pulsation,

${\displaystyle c_{f}}$ is the fuel heat capacity,

${\displaystyle M_{\text{f}}}$ is the fuel mass,

${\displaystyle \Delta T}$ 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

${\displaystyle E_{\text{stationary}}=\chi _{l}lt}$

where:

${\displaystyle \chi _{l}}$ is the linear power of the fuel (power per length of fuel),

${\displaystyle l}$ is the length of the fuel,

${\displaystyle t}$ is the residence time of the propellant in the chamber.

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

${\displaystyle M_{f}=\pi R_{f}^{2}l\rho _{f}}$

and the linear power given by [2]

${\displaystyle \chi _{l}=4\pi \kappa _{f}(T_{f}-T_{s})}$

Where:

${\displaystyle R_{f}}$ is the radius of the cylindrical fuel,

${\displaystyle \rho _{f}}$ the fuel density,

${\displaystyle \kappa _{f}}$ the fuel thermal conductivity,

${\displaystyle T_{f}}$ is the fuel temperature at the center line,

${\displaystyle T_{s}}$ is the surface or cladding temperature.

Therefore, the energy ratio between the pulsed mode and the stationary mode, ${\displaystyle N={\frac {E_{\text{pulse}}}{E_{\text{stationary}}}}}$ yields

${\displaystyle N={\frac {c_{f}\rho _{f}R_{f}^{2}}{4\pi \kappa _{f}(T_{f}-T_{s})}}\left[{\frac {\Delta T}{t}}\right]}$

Where the term inside the bracket, ${\displaystyle \left[{\frac {\Delta T}{t}}\right]}$ is the quenching rate.

Typical average values of the parameters for common nuclear fuels as MOX fuel or uranium dioxide are:[3] heat capacities, thermal conductivity and densities around ${\displaystyle c_{f}\simeq 300J/(mol\cdot K)}$,${\displaystyle \kappa _{f}\simeq 6W/(K\cdot m^{2})}$ and ${\displaystyle \rho _{f}\simeq 10^{4}kg/(m^{3})}$, respectively., with radius close to ${\displaystyle R_{f}\simeq 10^{-2}m}$, and the temperature drop between the center line and the cladding on ${\displaystyle T_{f}-T_{s}=600K}$ or less (which result in linear power on ${\displaystyle \chi _{l}\simeq 45000W/m)}$. With these values the gain in energy is approximately given by:

${\displaystyle N\simeq 6\times 10^{-3}\left[{\frac {\Delta T}{t}}\right]}$

where ${\displaystyle \left[{\frac {\Delta T}{t}}\right]}$ is given in ${\displaystyle K/s}$. Because the residence time of the propellant in the chamber is on ${\displaystyle 10^{-3}s}$ to ${\displaystyle 10^{-2}s}$ considering subsonic velocities of the propellant of hundreds of meters per second and meter chambers, then, with temperatures differences on ${\displaystyle \Delta T\simeq 10^{3}K}$ or quenching rates on ${\displaystyle \left[{\frac {\Delta T}{t}}\right]\simeq 10^{6}K/s}$ 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., ${\displaystyle 10^{2}\leq N\leq 10^{3}}$.

Quenching rates on ${\displaystyle \left[{\frac {\Delta T}{t}}\right]\geq 10^{6}K/s}$ are typical in the technology for production of amorphous metal, where extremely rapid cooling in the order of ${\displaystyle 10^{6}K/s\leq \left[{\frac {\Delta T}{t}}\right]\leq 10^{7}K/s}$ are required.

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

${\displaystyle P={\frac {1}{2}}Fv_{\text{e}}}$

where ${\displaystyle P}$ is the power, ${\displaystyle F}$ is the thrust and ${\displaystyle v_{\text{e}}}$ the exhaust velocity. On the other hand, thrust is given by

${\displaystyle F={\dot {m}}_{\text{p}}v_{\text{e}}}$

where ${\displaystyle {\dot {m}}_{\text{p}}}$ 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 ${\displaystyle n^{2}}$-times the propellant mass flow, and decreasing ${\displaystyle {\frac {1}{n}}}$-times the exhaust velocity. However, if the nuclear core is pulsed, thrust may be amplified ${\displaystyle n}$-times by amplifying the power ${\displaystyle n}$-times and the propellant mass flow ${\displaystyle n}$-times and keeping constant the exhaust velocity.

Pulsed nuclear thermal rocket unit cell concept for I_sp amplification. In this cell, hydrogen-propellant is heated by the continuous intense neutronic pulses in the propellant channels. At the same time, the unwanted energy from the fission fragments is removed by a solidary cooling channel with lithium or other liquid metal.

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 [4]

${\displaystyle I_{\text{sp}}\simeq c{\sqrt {T}}}$

being ${\displaystyle c}$ a constant, and ${\displaystyle T}$ the temperature of the propellant before expansion. However, the temperature of the propellant is related directly with the energy as ${\displaystyle E\simeq kT}$, where ${\displaystyle k}$ is the Boltzmann constant. Thus,

${\displaystyle I_{\text{sp}}\simeq c'{\sqrt {E}}}$

being ${\displaystyle c'}$ a constant.

In a conventional stationary NTR, the energy ${\displaystyle E}$ 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 ${\displaystyle f_{\text{n}}}$ is only around 5%, and therefore, in comparison, is almost negligible. However, if the nuclear core is pulsed it is able to produce ${\displaystyle N}$ times more energy than the stationary mode, and then the fraction of prompt neutrons or ${\displaystyle f_{\text{n}}N}$ could be equal or larger than the total energy in the stationary mode. 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, it 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 to make the propellant hotter than the fuel, which is otherwise the very limit of specific impulse in classic NTRs.

In summary, if the pulse generates ${\displaystyle N}$ times more energy than the stationary mode, the I_sp amplification is given by

${\displaystyle I_{\text{sp}}\simeq I_{\text{sp,o}}{\sqrt {f_{\text{n}}N+1}}}$

Where:

${\displaystyle I_{\text{sp}}}$ is the amplified specific impulse,

${\displaystyle I_{\text{sp,o}}}$ the specific impulse in the stationary mode,

${\displaystyle f_{\text{n}}}$ the fraction of prompt neutrons,

${\displaystyle N}$ the energy amplification by pulsing the nuclear core.

With values of ${\displaystyle N}$ between ${\displaystyle 10^{2}}$ to ${\displaystyle 10^{3}}$ and prompt neutron fractions around ${\displaystyle f_{\text{n}}\simeq {\frac {1}{20}}}$,[5] ,[6] the hypothetical ${\displaystyle I_{\text{sp}}}$ amplification attainable makes the concept specially interesting for interplanetary spaceflight.

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 ${\displaystyle I_{\text{sp}}}$ is not limited to the maximum temperature permissible by the fuel, i.e., its melting temperature.

The other nuclear rocket concept which allows a propellant hotter than the fuel is the fission fragment rocket. Because it directly uses the fission fragments as propellant, it can also achieve a very high specific impulse.

For ${\displaystyle I_{\text{sp}}}$ amplification, only the energy from prompt neutrons, and some prompt gamma energy, is used for this purpose. The rest of the energy, i.e., the almost ${\displaystyle 95\%}$ from fission fragments is unwanted energy and must be continuously evacuated by a heat removal auxiliary system using a suitable coolant.[1] Liquid metals, and particularly lithium, can provide the fast quenching rates required. One aspect to be considered is the large amount of energy which must be evacuated as residual heat (almost 95% of the total energy). This implies a large dedicated heat transfer surface.[7]

As regards to the mechanism for pulsing the core, the pulsed mode can be produced using a variety of configurations depending on 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.[8] 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.[8] If even faster pulsations are desired it would be necessary to make 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,[9] or proton and neutron beams. Frequencies on the order of 1 kHz to 10 kHz are likely choices.

• Fission fragment rocket