## Abstract

Rocket-propelled vehicles capable of travelling a kilometre or more in a ballistic ‘hop’ with propellants acquired from the Martian atmosphere offer the potential for increased mobility and planetary science return compared with conventional rovers. In concept, a radioisotope heat source heats a core or ‘thermal capacitor’, which in turn heats propellant exhausted through a rocket nozzle to provide thrust. A systematic study of the thermodynamics, heat transfer and selection of core materials for a Mars hopper was undertaken. The aim was to advance the motor design and assess technical risks and feasibility. Analytical and numerical motor models were developed; the former to generate thermodynamic performance limits, an ideal hop distance and plot a materials selection chart using simple explicit relations. The numerical model assessed the effect of core configuration and geometry. A hop coefficient *C*_{hop} is shown to characterize the effect of core geometry independently of core material and temperature. The target hop distance of 1 km is shown to be robust. A moderate advantage to pebble-bed cores over a core consisting of straight channels was suggested. High-performance engineering ceramics such as boron carbide offer the longest hop providing the core temperature can be increased significantly above 1200 K.

## 1. Introduction

Exploration of the surface of Mars has been driven by a diverse but interlinked range of scientific objectives as wide ranging as geology, climate, radiation environment, the search for indications of biological life and identification of *in situ* resources to support future manned or long-duration missions. In addition to numerous static probes and orbital observations, mobile exploration conducted or proposed has used planetary rovers of generally increasing size, scientific payload and mission duration but no step change in raw speed or mobility (Zubrin *et al.* 2000; Shafirovich *et al.* 2006; Yu *et al.* 2010*a*,*b*). The European ExoMars rover (scheduled to commence surface operations in 2019) is expected to cover only 100 m d^{−1} on rough terrain (Grant *et al.* 2010). Among the most challenging of proposed future mission concepts are those aiming to collect and return a diverse range of material samples to earth for detailed analysis (Des Marais *et al.* 2008). Enhanced mobility therefore both is an enabling technology for Mars sample return (MSR) missions (Grant *et al.* 2010) and offers exciting new opportunities for scientific investigations. The Martian atmosphere is composed mainly of CO_{2} (Hunt 1974). Given this *in situ* resource, several authors have identified the potential for a ballistic ‘hopping’ vehicle using this CO_{2} as a propellant, proposing hop distances of 0.5–20 km (Zubrin *et al.* 2000; Landis & Linne 2001; Shafirovich *et al.* 2006; Yu *et al.* 2010*a*,*b*).

Zubrin *et al*. (2000) conducted an experimental study including bench tests of cold gas and heated thrusters and both tethered and free flights of a simple demonstration vehicle. The thruster took the form of a pre-heated bed of high specific heat capacity material in pellet form. The bed was heated electrically in the tests reported. The thermal storage concept was selected on the grounds of simplicity and projected performance. A feasibility study was performed on the thermal insulation required to maintain the hot-bed at a useful temperature and a preliminary insulation design proposed. Alumina, graphite and steel hot-bed materials were tested following initial heating to 973 K. In all cases, the performance fell short of initial predictions. The authors attribute this to difficulties in achieving sufficient heat transfer between hot-bed and propellant. Fundamentally, this work was an important early experimental concept demonstration, and proposed that a mature ‘gas hopper’ concept could be expected to achieve a hopping range of over 10 km. A further proposed concept (Landis & Linne 2001) sees solar-generated electricity used to split CO_{2} into O_{2} oxidizer and CO fuel, which is then burnt in a conventional rocket motor. The design hop range was 0.5 km. The use of *in situ* acquired CO_{2} as the oxidizer for a magnesium powder-fuelled motor has also been proposed (Shafirovich *et al*. 2006; Yu *et al.* 2010*a*). Analytical studies assessed the required masses of propellant and CO_{2} acquisition systems under various constraints. Key variables considered were the number and distance of ‘hops’, total mission duration, time undertaking science and power to mass ratios for the CO_{2} acquisition systems. In the case presented, the total mission duration is limited by the mass of fuel (magnesium powder) carried. These studies show total ranges of around 10–15 km, using between five and 15 hops, i.e. with a hopping distance of the order of 1 km. Because of the finite fuel resource, the total number of hops is limited. A hopper propelled by heated CO_{2} has also been subject to further thermodynamic investigation (Yu *et al.* 2010*b*) focused on obtaining the overall thermodynamic performance (including propellant acquisition) in terms of specific thrust and thrust-specific power consumption. Design variables considered were ambient temperature and pressure, temperature and pressure at the entry to the rocket nozzle (representing the motor performance) and the flow rate and compression ratio of the coolant in the CO_{2} acquisition cycle. Detailed design of the rocket motor, in particular heat storage and transfer, was not within the scope of this study. The basic concept proposed battery-stored electrical heating of the propellant as it passes through the motor. This represents a significant challenge and technical risk for battery technology and heat transfer surface design. A common challenge for all hopper concepts is achieving adequate energy storage and heat transfer in a sufficiently small mass to allow a useful hopping distance.

Radioisotope heating and thermoelectric energy systems are a well-established space technology (Bennett 1990; Lenard 2008), with radioisotope selection and novel encapsulation being investigated for the next generation of space nuclear power systems (O’Brien *et al.* 2008). Such techniques offer significant potential for both heating the core and powering the CO_{2} compression. In a radioisotope thermal CO_{2} propelled hopper, the thermal capacitor, or ‘core’, must store thermal energy as sensible heat and transfer this to the propellant. The simplest method of heating this material is to disperse the radioisotope through it using advanced manufacturing techniques such as spark plasma sintering (SPS; O’Brien *et al.* 2009). This approach has the advantage of inherent simplicity, reliability and longevity. The particular challenges are in heat transfer surface design and material selection.

This aim of the work presented herein is to assess the fundamental feasibility and performance of the radioisotope thermal CO_{2} rocket motor itself, identifying key performance targets and technical risks. The objectives are:

— To develop a basic thermodynamic and heat transfer model for the ‘heat capacitor’ core, which takes the concept of a radioisotope dispersed throughout the heat storage material as a basis but would also be applicable to cores heated electrically or by a remote radioisotope source via heat pipes.

— To compare ‘channel’ and ‘pebble-bed’ core geometries.

— To perform preliminary geometric sizing of the core for a baseline hopper vehicle configuration and assess the sensitivity to geometric parameters.

— To conduct a systematic materials selection of candidate ‘heat capacitor’ materials and assess the effect on performance.

This paper will not consider the detailed trajectory of the hopper vehicle, the thermal management requirements for the vehicle or motor or the time taken to acquire propellant or reheat the core for the next hop. The effect of the heat capacitor material on the radiation dose to personnel and equipment will be considered only qualitatively as a basis for future work.

## 2. Method

A schematic of the motor concept is shown in figure 1, indicating the station numbering convention adopted in modelling the motor. Two core configurations are assessed within the same overall motor geometry.

The channel core consists of straight channels through the block of heat capacitor material. For manufacture, the core could be constructed from a number of short subassemblies of hexagonal form that could be mechanically fixed or brazed together, a configuration previously adopted in space nuclear reactor concept studies (Lawrence 2008). Overall mechanical constraint would require thermally insulating load-bearing connections to the pressure vessel with appropriate allowance for thermal expansion. The pebble-bed core is a collection of spherical pebbles of heat storage material with the flow path formed by the interstitial voids between spheres. This configuration has been previously investigated for space nuclear reactor concepts (Lawrence 2008). Pebbles may be stacked in an ordered arrangement or allowed to arrange themselves randomly. They will be subject to overall mechanical constraint by grids permitting fluid flow at the top and bottom, and by grids or a solid channel wall at the side. As for the channel core, insulated load transfer to the pressure vessel will be required. Additional mechanical constraint could be achieved by joining the pebbles by brazing or welding. Both core concepts will present significant challenges in detailed mechanical design and manufacture, but this is not expected to significantly influence a first-order thermofluid performance model.

### (a) Fundamental performance relations for a baseline Mars hopper concept

Using the station numbering convention introduced in figure 1 and assuming steady, isentropic flow of a perfect gas, the exhaust velocity of a convergent–divergent rocket nozzle is given by (Moss 1995)
2.1
where *γ* is the isentropic index of the propellant, *R*_{0} is the universal gas constant (J K^{−1} mol^{−1}), *T*_{P3.1} is the temperature of the propellant at entry to the nozzle (K) and *M* is the molecular mass (kg mol^{−1}). The ratio of nozzle to core exit pressure *p*_{3.3}/*p*_{3.1} is a function of the selected nozzle geometry. The resulting thrust *F* and specific impulse *I*_{SP} are given by (Moss 1995)
2.2
and
2.3
where is the total propellant mass flow (kg s^{−1}), *A*_{sect,3.3} is the cross-sectional area of the nozzle exit (m^{2}), *p*_{ambient} is the ambient pressure (Pa) and *g*_{0} is the gravitational acceleration at the surface of the earth (m s^{−2}). The specific impulse allows the overall performance of a rocket-propelled vehicle to be determined via the Tsiolkovsky equation (Moss 1995)
2.4where **Δ***V* is the change in vehicle velocity and *R* is the ratio of mass at the start of discharge to the mass at the end. By neglecting drag and gravity losses, assuming a ballistic trajectory and a launch angle of 45^{°} with motor discharges at the beginning and end of the flight (where half of the **Δ***V* is used for take-off and half for landing), a simple explicit indication of hopping range *x*_{hop} is obtained from (Yu *et al.* 2010*a*)
2.5where *g* is the local gravitational acceleration, taken as 3.711 m s^{−2} on the Martian surface.

A conservative baseline Mars hopper vehicle specification is detailed in table 1. It was generated as part of a broader mission and systems engineering activity by assembling a basic mass budget using conservative sizing assumptions, generous maturity margins and using proven technology and approaches. The nominal design target was 1 km hopping range—even this conservative initial target represents an enormous increase in mobility over current rovers if it can be repeated approximately every few days. Table 1 also illustrates the design approach adopted in this paper. The fixed parameters will not be altered during the analysis while the core mass, materials and configuration form the design variables. The resulting performance parameters will therefore vary from the baseline values quoted in table 1 as the effect of the design variables is explored.

### (b) Performance limits for a hopper propelled by stored thermal energy

It is desirable to develop expressions for the thermodynamic performance limits, both for validation of the numerical model and to identify opportunities to maximize the available performance. The approach adopted was to develop bounding expressions for the temperature at core exit *T*_{P3.1} assuming ideal heat transfer. When related to hopper performance via equations (2.1)–(2.5), these allow the effects of the core mass and material properties to be assessed.

For convenience, the specific impulse can be related directly to the propellant velocity at the nozzle exit by combining equations (2.2) and (2.3) and introducing the approximation of a simplifying specific impulse coefficient *C*_{ISP},
2.6Combining equations (2.1) and (2.4)–(2.6) yields the following expression for the hopping distance:
2.7Performance envelopes can be derived by assuming that *T*_{P3.1} is equal to the initial core temperature *T*_{Cinitial} throughout the burn or by assuming overall heat balance between the propellant and core. Derivation of these relations is described in the electronic supplementary material and ultimately allows an ideal core mass *m*_{core,ideal} to be calculated,
2.8where *c*_{Pcore} is the specific heat capacity of the core material (J kg^{−1} K^{−1}, weighted to reflect the different specific heats of radioisotope and encapsulant), *m*_{p} is the total propellant mass (kg), *c*_{Pgas} is the specific heat capacity of the propellant in the gas phase, *T*_{Pinitial} is the initial temperature of the gas (K) and *h*_{lat} is the specific latent heat of vaporization of the propellant (J kg^{−1}). This allows equation (2.7) to be solved for the ideal maximum hopping distance *x*_{hop,ideal} using *m*_{core,ideal} in the mass ratio *R*. The simplest solution assumes that *T*_{P3.1} is equal to the initial core temperature *T*_{Cinitial}. The ideal hopping distance is therefore an explicit function of design parameters and independent of heat transfer effects, making it easy to calculate even at a concept design stage. Non-ideal heat transfer and other practical effects will reduce the achievable hopping distance, perhaps significantly, from this ideal value. A hopping coefficient *C*_{hop} is therefore proposed to relate the results obtained from the full numerical model to the performance limit. This provides a unifying terminology and method for future hopper design studies,
2.9

### (c) Bespoke core thermal–hydraulic numerical model

The bounding expressions developed above do not include the effect of heat transfer or the variation in thrust and specific impulse through the discharge as the core cools and the propellant tank empties. A numerical model is therefore developed to provide a more robust simulation of the performance, and ultimately to estimate the hopping coefficient *C*_{hop} for various geometric variables. The calculation approach, verification and validation of the model are further described in the electronic supplementary material.

The model uses a simple finite-element approach whereby liquid CO_{2} enters the core at vapour-line conditions from the tank. A stepwise integration of propellant enthalpy through the core length (*z*-direction) is performed using simple but appropriate heat transfer, friction factor and pressure-loss assumptions and correlations (Moody 1944; Gnielinski 1976; Swamee & Jain 1976; KTA 1981, 1983; Ozisik 1985; Incropera *et al*. 2007). Variations of core temperature are calculated in finite steps in the time domain using heat balance. The aim of the model is to provide a good system-level representation of the performance appropriate for early-stage conceptual design. The solution is iterative to allow for variations of material and propellant properties with mean temperatures and the variation in mass flow rate during the discharge. The overall specific impulse of the motor is used to determine hopper performance via equations (2.4) and (2.5). It is given by a sum of the impulse over each time step (Moss 1995)
2.10

### (d) Material properties

Materials property expressions suitable for application at the relatively high temperatures of interest were collated from a variety of sources including the literature and supplier datasheets. Polynomial expressions describing property variation with temperature were either obtained directly from the literature or generated by a least-squares fit to tabulated data. Primary data sources suitable for concept studies were obtained from the open literature for the CO_{2} propellant (Kaye & Laby 1995; Union Engineering 2007; National Institute of Standards and Technology 2008) and encapsulant materials (Neely *et al.* 1950; Roth 1982; Karditsas & Baptiste 1995; International Nuclear Safety Centre 1997; Basak *et al.* 2003; National Institute of Standards and Technology 2008; Dorn *et al.* 2009).

## 3. Initial heat capacitor material selection

### (a) Thermal energy storage

Material property charts can be plotted for a very wide range of mechanical, thermal and other material properties (Ashby 1992). Their value stems not only from selecting and comparing a database of material properties for a given duty, but also from identifying strategies for expanding the range of properties by modifying the chemistry, microstructure or architecture of a material (Fleck *et al*. 2010). The aim of the current study is to undertake preliminary materials selection based on the requirement for storage of sensible heat. There is a wealth of potential for extending this approach to encompass mechanical properties, and to guide extending the property space through a number of materials engineering techniques (including composite materials) should the hopper concept progress.

Meaningful material property selection requires an appropriate understanding of the influence of materials properties on system-level performance (Ashby 1992). The explicit thermodynamic performance boundaries generated above can provide this direct input via solutions of equations (2.7) and (2.8). The key variables that will be intimately linked with properties of the core material are *T*_{Cinitial} and *c*_{Pcore}. The ideal maximum hopping distance *x*_{hop,ideal} was calculated for a range of values of these variables and found to be a close approximation to
3.1where *A* and *B* are constants, functions of the propellant and baseline hopper properties. For a CO_{2} propellant and the hopper properties in table 1, a least-squares fit gives *A*=0.840, *B*=−1.46. Intuitively, the energy limit on hopper performance is maximized by selecting a material with both a high specific heat capacity and a high melting point. The latter allows an increase in *T*_{Cinitial} up to some limit that, in practice, must be some way below the melting point of both radioisotope fuel and encapsulating heat capacitor material. A selection criterion *S* can be derived directly from equation (2.1) by incorporating representative temperature constraints
3.2where *c*_{P} is the specific heat capacity of the material (J kg^{−1} K^{−1}), *T*_{Emelt} is the melting temperature of the encapsulating thermal capacitor material (K), *T*_{Emeltmargin} is a margin applied to this melting temperature (K), *T*_{Fmelt} is the melting temperature of the radioisotope fuel (K) and *T*_{Fmeltmargin} is the corresponding margin (K). An alternative practical constraint could readily be substituted for the fuel limit term if required. Figure 2 shows contours of increasing *S* plotted by rearranging equation (3.2).

Beryllium, its nitride, oxide and carbide compounds, boron carbide and nitride and graphite are the group of materials offering generally the highest performance in figure 2. The shape of the contours is due to the temperature constraints imposed—in the left-hand region of the plot, the performance is limited by encapsulant melting, and in the right-hand region, fuel melting provides the limitation. As a result, increases in encapsulant melting temperature do not offer improved energy storage above the limit represented by the flattening of the *S*-contours.

A notable conclusion from figure 2 is that high-performance metal alloys—with the exception of beryllium—and refractory metals offer generally very poor thermal storage performance compared with high-performance engineering ceramics.

### (b) Mechanical requirements

The core material is not required to carry primary spacecraft or motor loads or form part of the pressure vessel. The major load case for the core is expected to be thermal shock as a result of rapid temperature changes during the discharge. Thermal shock resistance is a function of material and component geometry, thermal conductivity and coefficient of linear expansion (Katz 2002). A thermal–mechanical materials selection may be pursued in a later phase of work. The approach adopted in this paper is to evaluate a wide range of candidate materials to mitigate the risk of any one factor, including thermal shock resistance, affecting the feasibility of the hopper concept.

The principal safety requirement for deployment of a radioisotope power source in space is containment of the radioisotope under heating and impact loads associated with launch vehicle failure, inadvertent spacecraft re-entry and consequent ground impact (Bennett 1990; Lenard 2008). To date, radioisotope thermal generators have used multi-layered configurations consisting of carbon/graphite ceramic matrix composite ablative ‘aeroshells’ and refractory metal containment shells. Iridium-alloy-welded vessels have been successfully applied in this role (Lynch 1998), while encapsulating the isotope in a tungsten matrix was evaluated at laboratory scales as an alternative approach (O’Brien *et al.* 2009). The encapsulating thermal capacitor material could form part of the containment system. Equally, the multi-layered configuration adopted to date provides sufficient design flexibility that this is not an essential feature of the initial material selection. The approach adopted was to operate all thermal capacitor materials at a nominal margin (*T*_{Emeltmargin}=350 K) below their melting temperatures and assess the sensitivity of performance to larger margins. Refractory metals and metal alloys may also have a significant role as surface coatings to manage corrosion and reactions with the CO_{2} propellant. In these contexts, their contribution to the overall thermal storage capability will be negligible.

### (c) Nuclear property requirements

A further desirable feature of the encapsulating material is that it should not increase the overall radiation dose to vehicle instrumentation or assembly and integration staff by releasing neutrons in response to alpha-particle flux (*α*,*n* reaction) from the radioisotope (O’Brien *et al.* 2008). Furthermore, an encapsulant that is a strong neutron absorber may provide more flexibility in selection of radioisotopes previously discounted on account of higher neutron emission fluxes owing to (*α*,*n*) reactions or spontaneous fission, e.g. curium-244.

### (d) Material design cases

In order to assess Mars hopper vehicle performance, assess sensitivity to enforced selection of a non-optimum material and validate the materials selection approach, four material design cases were selected based on figure 2 and the general requirements of §3*b*,*c* above.

#### (i) Beryllium

Beryllium, or its alloys, offers the highest specific heat capacity at the expense of relatively low melting temperature and significant health, safety and environmental concerns. Exposure limits, focused on inhalation of dust or machining debris, are in place for workers exposed during manufacturing (Fulton & Goldberg 2009). The European Space Agency explicitly lists beryllium as a safety hazard and prohibits its use without prior approval on low-gravity platforms (Ceglia *et al.* 2005). To perform a role as a thermal capacitor, significant quantities of material will probably be required, so selection of this material could be considered a significant technical risk. A further disadvantage of beryllium is the characteristic (*α*,*n*) reaction in response to flux from the radioisotope and small neutron absorbance cross section: the very effect from which the neutron itself was discovered (Chadwick 1932; Martin 2006; Goldberg 2009). Additional shielding or an interface material between the dispersed radioisotope particles and the encapsulant would, therefore, be required to reduce radiation dose to instruments or assembly staff. Beryllium has nevertheless been proposed in previous hopper studies (Zubrin *et al.* 2000) and is included as a reference case for motor sizing.

#### (ii) Boron carbide

Boron carbide is representative of a number of high-performance ceramics that achieve the highest value of *S* based on the assumptions used in plotting figure 2, i.e. providing the thermal capacitor can be operated up to around 1800 K. Boron carbide is an established high-performance ceramic in relatively widespread use. Of notable interest is that boron carbide sees widespread application in nuclear reactor control rods owing to its large neutron absorption cross section (Thevenot 1990). Boron carbide is also compatible with advanced manufacturing techniques such as SPS (Hayun *et al.* 2010), which has been proposed as a particularly promising manufacturing process for encapsulated radioisotope fuel (O’Brien *et al.* 2009).

Boron nitride, graphite and beryllium compounds have broadly comparable thermal storage performance, with advantages and disadvantages in each case. The overall hopper performance for this group of core materials will be well represented by boron carbide.

#### (iii) Silicon carbide

Silicon carbide is broadly representative of a number of engineering ceramics that do not achieve quite the thermal performance as the options above, but could nevertheless provide a viable solution. This group includes alumina and silicon nitride, and more exotic options such as beryllide and diboride compounds. The diborides are likely to have the advantage of neutron absorption common to compounds containing boron. Owing, in part, to good thermal shock resistance, silicon nitride has been applied for complex, load-bearing components in automotive turbocharger and aerospace auxiliary power units with temperatures exceeding 1100 K (Katz 2002): up to 1580 K has been reported in a small-scale rocket nozzle test (Eckel 2000). Silicon carbide is selected to represent this group of materials as a design case to provide a quantification of the impact of the risk that boron carbide or an equivalent is unsuitable and to validate the materials selection technique.

#### (iv) Titanium alloy (6Al/4V)

Aerospace-grade titanium alloy is relatively inexpensive and representative of the highest value of *S* achieved by the conventional high-performance metal alloys. It is included as a design reference case and to validate the materials selection technique.

## 4. Results and discussion

### (a) Channel core geometry

The key overall geometric design variables for any core configuration are the overall diameter *D* and length *L*_{core} of the core. This study is confined to cores of cylindrical form. For the channel configuration core, the channel diameter *d* and surface roughness *δ* are the key design variables. A series of solutions of the bespoke model were undertaken to explore the effect of these variables on the baseline hopper vehicle detailed in table 1. The channel spacing was nominally set at 2*d*, which is a physically realistic limit (illustrated in finite-element models presented in the electronic supplementary material). Material properties were determined for a core material consisting of a fixed mass of the dispersed radioisotope giving a power output of 1 kW encapsulated in a varying mass of beryllium. The initial mass flow rate was selected to achieve a constant thrust-to-weight ratio at the start of discharge (i.e. for take-off) as a representative system-imposed constraint. The varying core mass was used in determining **Δ***V* and, therefore, the estimated hop distance, *x*_{hop}. These assumptions are intended to ensure that the optimization is, as far as possible, representative of the overall vehicle system. An initial core temperature of 1200 K was assumed, some 350 K below the melting point of beryllium in accordance with the assumptions used in plotting figure 2.

Figure 3*a* shows solutions of the numerical model for various combinations of core diameter and length. The agreement with the trends represented by the performance envelope is good at higher core mass where the core is sufficiently massive to ensure that the specific impulse through the burn is relatively constant. At low masses, however, the performance drops below the lower performance boundary. Small-diameter cores offer a longer hop for a given core mass. A reasonable hypothesis is that the greater length of small-diameter cores gives greater length for heat transfer to the propellant, resulting in higher nozzle entry temperatures and, therefore, overall specific impulse. It is important to note that a long, thin configuration for the core would result in disadvantages in thermal insulation design and practicality of integration.

The effect of a range of channel diameters and surface roughness is shown in figure 3*b*. The roughened tubes are assumed to incorporate a relative roughness *δ*/*d*=0.01. This is the limit of applicability of the correlation applied (Swamee & Jain 1976). The effect of channel diameter and roughness on heat transfer is clearly indicated as the curves approach the upper thermodynamic performance limits. The improved heat transfer also shifts the optimum geometry to a shorter core, which would be expected to ease integration. Rough surface results for a larger 10 mm channel are included, and show comparable, or better, performance to a smooth-walled 6 mm channel. The significance of these data is that manufacturing or vehicle integration constraints imposed on channel diameter or overall core length can be offset with the addition of surface texture. The roughness added in these calculations is a conservative first estimate limited by the surface friction relationship selected for the model (Swamee & Jain 1976). In the rough-wall data presented in figure 3*b*, the surface friction factor *f* is between 1.38 and 1.83 times the smooth-wall value. Incropera *et al*. (2007, p. 485) quote that, in practice, and given an appropriate correlation, *f* can be increased to four times the smooth-wall value in any given condition to further improve the heat transfer coefficient. This is significant because it suggests that heat transfer enhancement by surface texture changes offers further potential for improving performance in light of practical constraints likely to be imposed on the motor design. Heat transfer enhancement options for a straight channel include machined grooves, rifling, fins and coil inserts. Many of these approaches have been characterized in detail and further analysis approaches and experimental data are available (Incropera *et al*. 2007, pp. 491–494).

### (b) Pebble-bed core geometry

For the pebble-bed core, the key geometric design variables are the overall diameter *D*, length of the core *L*_{core} and pebble diameter *d*. The sensitivity to these variables was explored in a very similar manner to the investigations undertaken on the channel core, using a beryllium heat capacitor material initially at 1200 K. The pebble diameter *d* is limited by the validity of the heat transfer and pressure-loss correlations (KTA 1981, 1983) to be less than approximately *D*/20. The total number of pebbles was selected to achieve a bed porosity (void fraction) of 0.39. This is in the middle of a correlation validity of 0.36–0.42 (KTA 1983), which is itself in good agreement with studies on porosity limits for beds of randomly stacked spheres (Song *et al.* 2008). The sensitivity of the hopping distance to porosities of 0.36 and 0.42 was assessed using the model. There was negligible difference in performance to assuming a fixed porosity of 0.39.

Figure 4*a* shows the influence of core size on the hopping performance for the baseline design. The performance of a core formed of 10 mm diameter pebbles is similar to one with 2 mm diameter channels. The latter would be expected to represent a more significant manufacturing challenge. The drive for a thin, long core remains, but is a weaker design driver than in the channel core configuration. Importantly, a compact pebble-bed core 0.35 m in diameter and length still offers *C*_{hop}=0.85, a significant improvement on the trends shown in figure 3. The effect of pebble diameter is examined in figure 4*b*, with smaller pebbles offering longer hop distances. Higher values of heat transfer coefficient were noted and account for this improvement. The practical limits to the utility of this conclusion include the manufacturing of such pebbles, but perhaps more significantly the difficulties in accurately accounting for the total inventory of the radioisotope during manufacture and assembly. The effect of surface roughness was not considered as this effect is not included in the correlation selected, but may provide further increases in heat transfer coefficient for a given pebble diameter.

### (c) Core and propellant temperature distributions

The variation in core and propellant temperatures for the pebble-bed motor with overall diameter *D*=0.2 m and pebble diameter *d*=10 mm is discussed in detail in the electronic supplementary material. Achieving maximum values of hop distance requires a core with sufficient mass and length to achieve sufficient energy storage and heat transfer to maintain a consistently high propellant temperature at core exit. Figures 3 and 4 show some configurations where *x*_{hop} drops below the lower performance boundary; this is on account of inadequate heat transfer.

### (d) Effect of material and initial core temperature

To assess the effect of different materials, two geometric design cases representative of channel and pebbled bed geometries with relatively low manufacturing risk were used. Both cases featured a core of overall diameter *D*=0.2 m. Rough-wall channels of diameter *d*=6 mm and pebble diameter *d*=10 mm were assumed for the channel and pebble configurations, respectively. The four material design cases identified in §3*d* were assessed at a number of temperatures up to the limiting temperatures determined in equation (3.2). The pebble-bed configuration was evaluated only with the beryllium and boron carbide encapsulants.

Figure 5*a* shows a number of performance curves and the corresponding upper performance envelopes for the channel core configuration. The materials are operated at their limiting temperatures as defined by the first term of equation (3.2). The pebble-bed core results show a very similar trend. The numerical results show excellent agreement with the materials selection chart of figure 2, with the relative hop performance closely matching the relative values of selection metric *S*. Figure 5*a* also shows that titanium alloy and silicon carbide require a notably larger core mass (approaching twice that of beryllium and boron carbide) for maximum hop distance. It is important to reiterate that this mass is already incorporated within the maximization of hop distance. The analysis has not, however, considered the second-order effect of a large increase in core mass on other items in the hopper vehicle mass budget—for example, the structural mass could increase to support higher inertia loading from a heavier core. Additional mass constraints are also likely to be imposed by the Earth launch, Mars transfer and atmospheric entry systems.

Figure 5*b* shows the predicted performance for all material and temperature data points generated using the two geometries. It is shown that *C*_{hop} can, for practical engineering design purposes, be considered independent of material and initial core temperature, and a function of the core configuration and geometry. This is an important simplification that has significant utility for conceptual design, material selection and in the preparation of a structured programme to experimentally validate designs and methods in the future.

Figure 5*c* effectively draws together all the results presented to explore the effect of initial core temperature in more detail. The materials selection approach described in §3 and figure 2 incorporated assumed temperature constraints; these are extremely difficult, if not impossible, to define with confidence at an early stage of concept development. It is reasonable to consider that the core temperature may be limited by any one of a number of additional factors. For example, assume a pebble-bed core of *C*_{hop}=0.90, the baseline hopper configuration in table 1 and an initial core temperature limited to 1000 K, similar to that already experimentally demonstrated at the laboratory scale (Zubrin *et al.* 2000). In this case, beryllium offers the longest hop at 1.04 km while an enforced change to boron carbide only reduces the hopping distance to 0.96 km—a fairly modest impact. If the limitation is related to the encapsulant material itself, some increase in design temperature is likely to be possible with boron carbide, which has a much higher melting point than beryllium. It is interesting to note that a boron carbide core requires an increase in temperature to only 1075 K to match the performance of the beryllium core at 1000 K. Refined temperature constraints could readily be used to replot the *S*-contours to repeat the materials selection.

### (e) Impact of simplifying assumptions

The analysis in this paper assumes no gravity losses, and optimizes hop distance without considering external mass constraints such as those imposed by initial launch. Based on the current hopper vehicle parameters, it was estimated that gravity losses could reduce the hop distance by up to 0.2–0.5 km. However, the potential for minimizing this effect via increasing motor thrust and optimizing vehicle trajectory was not within the scope of the studies conducted to date. The main objectives of this work relate to comparative studies of motor design details and confirm, in broad terms, that the target hop distance of 1 km is feasible. Future work will refine the estimates of vehicle performance by incorporating these effects.

## 5. Conclusions

This study has systematically considered geometry design variables and materials selection for the stored thermal rocket motor propelling a ballistic Mars hopping vehicle. The study was based on a conservative baseline vehicle specification with a target hopping distance of 1 km, but the models developed are sufficiently general to have wide applicability.

Analytical expressions have been developed to describe an ideal thermodynamic performance envelope for the combined system of a Mars hopper and a stored thermal rocket motor. An ideal hopping distance was determined from simple vehicle and material properties using explicit relations (equations (2.7) and (2.8)). These should prove useful engineering design tools for future hopper concept development.

A bespoke numerical model based on proved and/or conservative correlations and assumptions was developed to investigate the combined system performance of motor and hopper in more detail and demonstrated adequate heat transfer between the core and propellant in practically realizable configurations.

A hopping coefficient *C*_{hop} was proposed to link the maximum hopping distance determined from the numerical model with the analytical ideal (equation (2.9)). This coefficient was shown to be effectively independent of core material and initial core temperature in the range of interest. It is therefore primarily a function of heat transfer efficiency and can be used to compare the effect of core configurations and geometry.

The hopping coefficient of vehicles with channel configuration cores is increased by smaller channel diameters (where the spacing between channels scales with channel diameter) and surface roughening. There is further scope for enhancing performance using greater relative surface roughness or surface features such as rifling. Manufacturing complexity will impose a practical limit.

The hopping coefficient of vehicles with pebble-bed configuration cores is increased by smaller pebbles (where the overall porosity of the bed is constant). Manufacturing complexity and nuclear material accounting requirements for radioisotope material when encapsulated into numerous small pebbles may impose a practical limit.

Within the assumptions of this work, practical pebble-bed configurations yield values of *C*_{hop} approximately 10 per cent higher than practical channel configurations, owing mainly to better heat transfer. Both configurations are viable from the thermofluid viewpoint. Optimum cores in both configurations are significantly longer than their diameter, i.e. with *L*/*D*≈4. The pebble-bed configuration is more tolerant to reductions in this value, which may give more flexibility for vehicle integration.

A materials property chart was plotted from the explicit performance expressions and used for preliminary core material selection. There was excellent agreement with the results of the full numerical modelling. The selection metric incorporated practical temperature constraints, and, while the assumptions are relatively arbitrary at this stage, entering new constraints and assessing the effect on materials selection will be relatively trivial as the concept matures.

Boron carbide and a number of alternative engineering ceramics with comparable thermal properties provide a range of viable alternatives to beryllium for the heat capacitor material. If the initial design temperature of the motor core can be increased significantly above 1200 K, the predicted hopping distance is significantly increased with the use of boron carbide compared with beryllium. Several other advanced engineering ceramics would be expected to offer similar performance to boron carbide, which should be seen as a significant reduction of technical risk.

Overall, achieving the hopping range target of 1 km is not critically dependent on a single motor configuration, core material or geometric parameter. A radioisotope thermal thruster represents one of the most novel aspects of the Mars hopper concept, and will rightly be regarded as a technical risk during development. That the thermofluid design appears relatively robust is an important conclusion, albeit one that will require much further work as the concept develops.

## Acknowledgements

The authors gratefully acknowledge the funding provided by EPSRC via grant EP/D030277/1, the input of overall system parameters by M.-C. Perkinson, J. Reed & L. Waugh (EADS Astrium) as part of the Mars hopper system studies and S. Howe and R. O’Brien (Centre for Space Nuclear Research, Idaho National Laboratory) for useful discussions. Thanks also to P. Samara-Ratna for assistance with finite-element modelling. The authors would also like to express their thanks to the reviewers for their suggested improvements.

- Received August 20, 2010.
- Accepted October 11, 2010.

- This journal is © 2010 The Royal Society