DEVELOPMENT OF A COMPRESSED CARBON DIOXIDE PROPULSION UNIT FOR NEAR-TERM MARS SURFACE APPLICATIONS Erin Blass Old Dominion University Advisor: Dr. Robert Ash Abstract This work has focused on the development of a reusable rocket propulsion system for near term Mars surface applications by utilizing Mars atmosphere as a propellant source. Mars atmosphere is more than 95% carbon dioxide, and its low ambient temperatures mean that it is relatively easy to condense dry ice out of the atmosphere. Furthermore, the low critical temperature of carbon dioxide enables the production of a supercritical fluid by heating dry ice at constant density to temperatures only slightly higher than terrestrial ambient temperatures. The goal of this research is to develop a supersonic nozzle for reusable high-thrust propulsion. Due to the complex behavior of supercritical carbon dioxide gas, it is not possible to use the standard linear method of characteristics for nozzle design, via the ideal gas-based, Prandtl-Meyer function. Instead, a Method of Characteristics (MOC) for isentropic axisymmetric flow of a real gas has been used to develop the streamline contour of a supersonic nozzle. The boundary layer that develops along the wall of the nozzle is also taken into consideration and then combined with the non-ideal gas method of characteristics to complete the supersonic nozzle design. A Mach nozzle design has been selected because it is possible to achieve relatively high (though modest) specific impulse over longer durations before predicted performance degrades due to recondensation of carbon dioxide. Introduction There are many Martian geographic features that would be very interesting to explore, but current Mars entry descent and landing systems are not capable of placing payloads at precise locations on rough terrain and current surface rovers cannot ascend or descend large boulders, mountains or canyons. Furthermore, as demonstrated by the Mars Exploration Rovers, it is likely that even advanced wheeled systems will become stuck or trapped when they attempt to access some of the more interesting surface regions. There is a near-term need to develop high-thrust systems that can be used to propel surface vehicles in and around hazardous terrain of great scientific interest and to cover large distances. Ideally, this type of surface propulsion should use local Mars resources in order to permit extended reuse. The most readily accessible Mars resource is its atmosphere, consisting mostly of carbon dioxide. The goal of this research is to develop a supersonic nozzle for reusable high-thrust propulsion utilizing compressed carbon dioxide gas extracted from Martian atmosphere and processed as a supercritical fluid for rocket propulsion. The critical temperature of carbon dioxide is 304. K (3 o C or 88 o F), meaning that solid carbon dioxide (dry ice) in a closed container can be heated with relative ease above its critical temperature to become a supercritical fluid at very high pressures and can in that way be used as a rocket propellant. Due to the complex behavior of supercritical carbon dioxide gas, it is not possible to use the standard linear method of characteristics for nozzle design, via the ideal gas-based, Prandtl-Meyer function. Instead, a method of characteristics approach for a non-ideal gas nozzle design has been used. A supersonic convergent-divergent nozzle consisting of a subsonic, sonic, and supersonic section has been the focus of this work. In the subsonic section of the nozzle, the gas speed increases as the nozzle area decreases and that portion of the nozzle can be designed using the continuity equation for steady flow. At the throat, where the cross sectional area is a minimum, the gas velocity becomes sonic. As the subsequent nozzle cross sectional area increases the gas continues to expand and the gas flow increases to supersonic speeds. For supersonic flow, the equations governing compressible fluid flow are a set of hyperbolic partial differential equations. Hyperbolic governing equations can be solved using the method of characteristics. The method of characteristics employs characteristic lines or surfaces (for axisymmetric flow) to construct ideal streamlines. Along these characteristic lines the flow variables are continuous, but their associated derivatives are indeterminate, making it possible to use Cramer s rule to delineate the characteristic lines and compatibility equations. That is, by treating the governing equations as a set of simultaneous linear Blass
algebraic equations and using Cramer s rule, it is possible to make the derivatives indeterminate, e.g. u / r v / x 0 / 0, along those surfaces, the characteristic and compatibility equations can be adjusted. Recently, Aldo and Argrow (995) studied plug nozzle designs for non-ideal, dense gas BZT fluids. They developed a method of characteristics approach based on earlier work by Zucrow and Hoffman (976) who developed an Euler predictorcorrector scheme to solve the real gas characteristic and compatibility equations simultaneously. That approach has been modified here for the specific case of supercritical carbon dioxide. However, the resulting streamline contour design neglects viscous interaction with the nozzle wall and hence must be corrected for boundary layer effects. The boundary layer that develops along the wall of the nozzle is also taken into consideration. Boundary layers can be either laminar or turbulent depending on the pressure gradient and the local Reynolds number. At low Reynolds numbers, the boundary layer is laminar and the streamwise velocity increases uniformly moving away from the wall of the nozzle. At higher Reynolds numbers, the boundary layer becomes turbulent and the streamwise velocity is characterized by unsteady swirling flows inside the boundary layer. Using three different throat diameters of,, and 4 mm, with specified density, viscosity, and speed of sound at the nozzle throat, this study has determined that the flow in the sonic section of each nozzle throat produces Reynolds numbers, ranging from 3,00,000 to,800,000, that are considered to be turbulent. There are many approximate methods for estimating the boundary layer growth along the wall of a supersonic nozzle, and most of these have been described and summarized more than 50 years ago by Rogers and Davis (957). In nozzle design it is often assumed that the boundary layer thickness is negligible at the throat due to the favorable pressure gradient upstream of the throat. While it is true that the favorable pressure gradient inhibits the growth of the boundary layer, experiments have shown that the boundary layer thickness is only effectively zero when the exit design Mach number is above 3, as demonstrated by Rogers and Davis (957). In this design the boundary layer thickness at the throat was calculated using a displacement thickness equation model from McCabe (967). Subsequently, the average boundary layer growth along the nozzle contour was estimated using the methods of Rogers and Davis (957), then iterated to adjust for displacement thickness pressure effects. These boundary layer calculations were then combined with the non-ideal gas method of characteristics based streamline contours to complete the supersonic nozzle design. Method of Characteristics A Method of Characteristics (MOC) for isentropic axisymmetric flow of a real gas has been used to develop the frictionless flow streamline contour for our supersonic nozzle. First, using the above mentioned assumptions, the equations for continuity, Euler s momentum equation, irrotationality for an axisymmetric flow, and a speed of sound relation were developed as below. ( ρu) ( ρv) x r ρv 0 r () V u v dp ρ VdV ρ d ρ d () u v r x (3) p dp a (4) ρ dρ s Eqs. -4 are the governing equations used along with some mathematical techniques, such as Cramer s rule, to develop the characteristic and compatibility equations: C characteristic equation: dr dx ( θ µ ) λ tan C compatibility equation: a v ( a ) du [ uv ( u a ) ] dv 0 (5) u λ dx (6) r Where C and C - are the left and right running characteristics as sketched in Fig. (taken from Aldo and Argrow, 995). The gas is assumed to enter the supersonic portion of nozzle across a flat sonic surface (straight line OA in Fig.), with uniform properties and velocity, flowing parallel to the x-axis. The gas then expands and accelerates through the nozzle before exiting from the Blass
nozzle with a uniform flow crossing the conical terminating C characteristic surface, with an exit Mach number of M f. For a two-dimensional nozzle, the centered expansion generated by the sharp throat is a Prandtl-Meyer expansion. In the axisymmetric case, the flow at the wall is locally two-dimensional, thus at the throat the expansion is locally Prandtl- Meyer. Fig. Supersonic flow field geometry, upper halfplane Shown below are the Finite Difference Equations corresponding to the characteristic and compatibility relations of Eqs. 5 and 6. and r λ x, () Q u R v S x 0, () λ θ µ Q R S ( ) tan (3) u a (4) uv ( u a )λ a v (5), (6) r where the or denotes C or C - characteristics. It should be noted that (6) applies only to axisymmetric nozzle flow; for two-dimensional flow, S is zero. Construction of the flow field begins by separating the centered expansion into equally-spaced speed increments. Each speed increment V has a related flow turning angle increment θ. The angle θ is computed from the relation θ θ V V M dv V (7) By selecting evenly spaced increments of V, we can use the trapezoidal rule to solve Eq. 7 which can be rewritten as: where, or M d θ dv f ( V ) dv (8) V M f ( V ) (9) V θ θ V V f ( V ) dv (0) Next we use the second-order-average-property Euler Predictor scheme, based on the development presented in Zucrow and Hoffman (976). This procedure shows how the characteristic and compatibility equations can be solved simultaneously. Fig. Schematic representation of intersecting characteristic surfaces Now if we know the values of x, r, u, and v for locations and in Figure, the left running characteristic from will intersect with the right running characteristic from at some point 4, as shown in Fig. (taken from Zucrow and Hoffman, 976). We can determine the location of point 4 by writing Eq. in terms of points, and 4, in finite difference form, i.e.: ( r4 r) ( x4 x) and ( r r) ( x ) Then, λ (7) λ (8) 4 4 x ( θ µ ) and ( θ µ ) where λ tan (9) λ tan (0) v tan u θ. () Blass 3
Hence, we can solve for x 4 and r 4, and the compatibility equation can be used to solve for u 4 and v 4, the finite difference form of the compatibility equations can be written in terms of points, and 4 as: Q Q ( u4 u) R ( v4 v) S ( x4 ) 0 ( u u ) R ( v v ) S ( x x ) 0 x 4 4 4 Now, the wall contour corresponds to the streamline that passes through points A and C in Fig.. The streamline is also determined by using the averageproperty Euler predictor-corrector scheme to integrate the equation dyw dx tanθ () w marching from the initial condition θ w θ * at x 0 to the exit condition θ w 0 at x x f. Subscript w refers to quantities at the wall and subscript f refers to the exit conditions. The NIST Thermophysical property tables for carbon dioxide have been used here to estimate the properties throughout the nozzle. The strongly varying thermophysical properties are what cause this linear looking method of characteristics to become challenging. That is, the departure from linear theory occurs because the speed of sound for carbon dioxide varies strongly with temperature and pressure near supercritical and phase change states. Results The supersonic nozzle design will actually be a convergent-divergent nozzle consisting of a subsonic or convergent section and a supersonic or divergent section. In order to determine the conditions at the throat we need to calculate the properties and flow variations in the subsonic part of the nozzle. We have used a nozzle entrance diameter of 9.05 mm (i.e. an entrance radius of 0.00955 m [0.375 in]), with a throat diameter of mm (radius is mm). Furthermore, we will assume that the initial (stagnation) pressure is 3.03 MPa (4500 psi) and the stagnation temperature is 450.7 K. A subsequent isentropic expansion to sonic conditions is characterized by the data in Table. Table : Subsonic Calculations P (MPa) T (K) ρ (kg/m 3 ) M 3.050.7 44.0 0 (4500 psia) 0.68 45. 356.67 0.7 (3000 psia) 5.5 (50 psia) 390. 99.83.0 Table contains representative carbon dioxide property data using the NIST software for computing thermodynamic and transport properties of pure fluids over an anticipated supersonic operating pressure range. Note that the critical pressure for CO is 7.38 MPa (070.0 psia) and the critical temperature is 304. K (3 o C or 88 o F). Therefore, Table shows that anticipated nozzle exit conditions below the critical temperature and pressure can be encountered. Table : NIST Properties P (MPa) T (K) ρ (kg/m^3) M 5.5 390. 99.83.0 (50 psia) 4.3 38.0 8.48.09 (050 psia).76 373. 63.93.9 (850 psia).38 363.4 44..9 (650 psia) 9.997 35.5.97.4 (450 psia) 8.6 340. 00.4.53 (50 psia) 7.4 36. 75.99.67 (050 psia) 5.86 309.5 49.78.83 (850 psia) 4.48 89.5.59.0 (650 psia) 3.79 (550 psia) 77.55 06.5. Then using the angle-velocity relation, Eq. 7, we get the conditions for a centered expansion at the throat. Next using the second-order-average-property Euler Predictor scheme to solve Eqs. 5 and 6 we can calculate the results shown in Figs. 3 and 4, predicting the supersonic nozzle contour for a mm throat diameter. Figures 5 and 6 are the predicted thermodynamic property variations through the nozzle. Blass 4
.4 x 0-3 500 (a) Pressure 400 (b) Temperature. Pressure (psi) 000 500 000 Temperature (K) 350 300 0.8 0.6 0.4 0. 0 0 0.5.5.5 3 3.5 4.5 x 0-3 Fig. 3 Supersonic Contour, upper-half Density (kg/m 3 ) 500 0 (c) Density 300 50 00 50 00 0 Enthalpy (kj/kg) 50 0 (d) Enthalpy 500 wall 480 axis 40 0 Fig. 6 (a) Pressure distribution along the wall and axis; (b) Temperature distribution along the wall and axis; (c) Density distribution along the wall and axis ; (d) Enthalpy distribution along the wall and axis 460 440 0.5 0-0.5 - -.5 0 0.5.5.5 3 3.5 4 Fig. 4 Supersonic Contour The thrust and specific impulse can be calculated using the equations below: m& ρva (3) u eq u Thrust e pe p m& & a A e (4) (5) mu eq Velocity (m/s) Mach Number 450 400 350 300 (a) Velocity 50 0 (c) Mach Number.5.5 0 Speed of Sound (m/s) 300 50 (b) Speed of Sound 00 0 wall axis Fig. 5 (a) Velocity distribution along the wall and axis; (b) Speed of Sound distribution along the wall and axis; (c) Mach Number distribution along the wall and axis I u eq sp (6) g Using eqs. 3-6 we get the following results: Using the conditions at the throat of the nozzle: m& 0.75 kg/s Now using the exit conditions: u 55.3607 m/s eq Thrust 39.9 N ( 3.46 lb) I 5.53 s sp The exit Mach number is M f.047 Blass 5
Boundary Layer Corrections In nozzle design it is often assumed that the wall boundary layer thickness is negligible at the throat, due to the favorable upstream pressure gradient. While it is true that the pressure gradient inhibits the growth of the boundary layer, some experimental evidence shows that its thickness is only effectively zero when the design Mach number is above 3. This is shown in the Fig. 7 where the measured throat boundary layer thickness in inches has been compared with Tucker s (95) turbulent theory, taken from Rogers and Davis (957). Although the nozzle considered in the figure is not consistent with the present flow, it still shows that we should not neglect the boundary layer effects at the throat initially. δ x 0.9 Re / 5 x (4) Fig. 8 Variation of Thickness Parameter g with Mach Number and N Fig. 7 Comparison of Theoretical and Experimental Boundary Layer Thickness at Throat McCabe (967) correlated the boundary layer displacement thickness measured in the throats of supersonic nozzles as a function of Reynolds number and a characteristic geometry parameter. His correlation. * δ throat h 0 R.06 Re / 5, (3) where h is the throat half height and R is the ratio of curvature at the throat to the throat half height, has been used here to estimate the displacement thickness correction at the nozzle throat. Then from Rogers and Davis (957), we can estimate the average boundary layer growth along the nozzle contour and employ an estimate of the ratio of the displacement thickness to the boundary layer * thickness, given by the shape factor g δ / δ, which is displayed as a function of Mach number and turbulent power law (for representing the turbulent boundary layer profile) in Figure 8. A nominal turbulent velocity profile power law of N 7, has been assumed here, which is the accepted value in most references for moderate Mach numbers. This is based on the /7 power law for a compressible turbulent boundary layer where the velocity profile is represented: u U y δ / 7 (5) In order to justify a turbulent nozzle boundary layer assumption, we need to estimate the throat Reynolds numbers for the expected carbon dioxide sonic conditions. Using d to represent one of our nozzle diameters (d, and 4 mm), and recognizing that the subsonic flow up to the nozzle throat is nominally the same for all three nozzles, we have: ρ a * d Re 3,00,000d µ For the throat of the nozzle, where the following carbon dioxide properties have been assumed: density, ρ 99.83 kg/m 3 Blass 6
speed of sound, a* 85.797 m/s diameter, d, or 4 (mm) viscosity, µ.69e-05 Pa/s Hence, the estimated nozzle throat Reynolds numbers are: Re 3,00,000; Re 6,400,000; and Re 4,800,000.6 x 0-3.5.4.3.. BL Correction Nozzle Wall Boundary Layer Now, shown in Figures 9-, the displacement thickness of the boundary layer found from Eqs. 3 and 4 are used to correct the potential flow outline of the nozzle profile, for throat diameters of,, and 4 mm..9 0 5 6 7 8 Fig. Comparison of Wall Contours using the Boundary Layer Analysis (4mm throat diameter) 6.4 x 0-4 6. 6 5.8 5.6 5.4 5. 5 BL Correction Nozzle Wall Boundary Layer 4.8 0 0. 0.4 0.6 0.8..4.6.8 Fig. 9 Comparison of Wall Contours using the Boundary Layer Analysis (mm throat diameter) Table 3: Values for Different Throat Diameters Nozzle Throat 4 Diameter (mm) Reynolds 3,00,000 6,400,000,800,000 Number Exit Area with.5e-6 4.98e-6.99e-5 BL Correction (m ) Mass Flow Rate 0.068 0.7.086 (kg/s) Thrust (N) (lb) 35.3 7.9 40.66 3.6 56.94 6.33 Specific Impulse (s) 5.67 5.65 5.64.35 x 0-3.3.5..5..05 BL Correction Nozzle Wall Boundary Layer 0.95 0 0.5.5.5 3 3.5 4 Fig. 0 Comparison of Wall Contours using the Boundary Layer Analysis (mm throat diameter) Conclusion This research has focused on the design a supersonic nozzle for a propulsion system utilizing compressed carbon dioxide gas extracted from Martian atmosphere and processed as a supercritical fluid for rocket propulsion. Undergraduate mechanical engineering design teams at Old Dominion University have worked hard to design a system that will be used to test the supersonic nozzles once they are built, but have been using a sonic nozzle in their testing this far and have been getting good results. There design consisted of placing solid carbon dioxide (dry ice) in a tank and heating it above its critical temperature to become a supercritical fluid at very high pressures to be used as a rocket propellant. While testing the sonic nozzle there was visual and audible observations which suggested that the expanding Blass 7
carbon dioxide continued to behave like a single phase gas expansion at pressures and temperatures that should have produced two-phase nozzle flows, thus extending observed thrust duration. Therefore, one element of my completed research will be to determine whether subcooled carbon dioxide can support propulsion. For the design of the supersonic nozzle a method of characteristics for isentropic axisymmetric flow of a real gas has been used to develop the streamline contour of our supersonic nozzle. The equations for continuity, Euler s momentum equation, irrotationality for an axisymmetric flow, and a speed of sound relation were developed and used along with some mathematical techniques, such as Cramer s rule, to develop the characteristic and compatibility equations shown in eqs. 5 and 6. The second-order-average-property Euler Predictor scheme was then used to solve the characteristic and compatibility equations simultaneously. To get the conditions of the centered expansion at the throat we used the angle velocity relation in eq. 7. The average-property Euler predictor-corrector scheme is used again to integrate eq. to develop the wall contour and the NIST Thermophysical property tables for carbon dioxide have been used to estimate the properties throughout the nozzle. The results of this design technique are shown in figs. 3-6 which show the nozzle contour and the properties throughout the nozzle. However, the resulting streamline contour design neglects viscous interaction with the nozzle wall and hence must be corrected for boundary layer effects. The boundary layer that develops along the wall of the nozzle is also taken into consideration. Boundary layers can be either laminar or turbulent depending on the pressure gradient and the local Reynolds number. Using three different throat diameters of,, and 4 mm, with specified density, viscosity, and speed of sound at the nozzle throat, this study has determined that the flow in the sonic section of each nozzle throat produces Reynolds numbers that are considered to be turbulent. In this design the boundary layer thickness at the throat was calculated using a displacement thickness equation model shown in eq. 3 and the average boundary layer growth along the nozzle contour was estimated using eq. 4 and then iterated to adjust for displacement thickness pressure effects. These boundary layer calculations were then combined with the non-ideal gas method of characteristics based streamline contours to complete the supersonic nozzle design. The results of the boundary layer analysis are shown in figs. 9- which compares the size of the boundary layer along the wall, the nozzle wall contour, and the boundary layer correction due to the displacement thickness calculations. These supersonic nozzle designs will soon be built and tested. The experimental results will then be compared to the computational results and will be evaluated for future study. References Aldo, A.C. and Argrow, B. M.. Dense Gas Flow in Minimum Length Nozzles. Journal of Fluids Engineering, Vol. 7. ASME, June 995 McCabe, A.. Design of a Supersonic Nozzle. Ministry of Aviation, Aeronautical Research Council Reports and Memoranda. 967 Rogers, E.W.E., Davis, B.M.. A note on Turbulent Boundary Layer Allowances in Supersonic Nozzle Design. Ministry of Supply, Aeronautical Research Council Current Papers. 957 Tucker, M. Approximate Calculation of Turbulent Boundary Layer Development in Compressible Flow. N.A.C.A. TN. 337, 95 Zucrow, Maurice J. and Hoffman, Joe D.. Gas Dynamics, Volume. New York: John Wiley and Sons, Inc., 976. Blass 8