aa r X i v : . [ phy s i c s . a pp - ph ] A ug Using fuel cells to power electric propulsion systems
Lui Habl ∗ Abstract
The origin of fuel cell technology has a notable connection to the history of spaceflight, having beenused in remarkable programs such as Gemini, Apollo and the Space Shuttle. With the constant growth ofthe electric propulsion technology in the last years, one natural application for fuel cells to be consideredwould be the electrical feeding of those thrusters for different mission profiles. In this article we explorein details this possibility, showing what would the necessary characteristics of such a device be in orderto improve mission parameters, as payload ratio and thrust, among others. In the first section, a briefreview of the applications of fuel cells in the space industry is shown, and the classical analytical modelingof these devices is presented. In the second part, two case studies illustrating most of the possible waysto use fuel cells in conjunction with electric propulsion systems are shown , and the possible advantagesand limitations of the applications are demonstrated analytically. The result of the analysis shows that,in the case where the fuel cell reaction products are disposed of and the propulsion has its own feedsystem, the application of fuel cell technology would bring no advantages for most kinds of missions.On the other hand, when the case where the fuel cell exhaust is used as a propellant is considered, it isshown that it is possible to improve mission parameters, such as thrust and specific impulse, if certainconditions are met.
The birth of the modern fuel cell concept dates back to 1838, when the Welsh scientist Sir William Grove firstproposed and demonstrated what is known today as the Grove cell [1]. Despite the fact that research in thisfield continued to be developed through the years, just in the 1960s, with the work of the chemists ThomasGrubb and Leonard Niedrach of General Electric (GE) inventing the polymer electrolyte membrane (PEM)fuel cells, the first commercial systems were developed, finding their application on the Gemini missions.The space and fuel cell technologies remained connected until nowadays, and, for example, missions asthe Apollo and the Space Shuttle employed hydrogen-oxygen alkali fuel cells (AFC), which were the onlypower system type available at that time that was able to meet the necessary energy requirements [2].With the involvement of the aerospace, energy and automotive industries, fuel cell systems achieved a highdevelopment level, being used today in a large range of applications, in situations where electrochemicalconversion is deemed convenient.In general, the operation of a fuel cell follows a scheme where the reactants are injected on the outersurface of the anode and cathode, which are composed by a catalyst layer. At the anode, the fuel is oxidizedproducing electrons, while ions, through diffusion, migrate in the electrolyte to the cathode. The electrons areprovided to the load, creating the useful current, and come back to the cathode catalytic layer to recombinewith the oxidizer, completing the so-called cathodic oxidation-reduction reaction. As the voltage of a singlefuel cell is always limited by the electrochemical potential of the reaction, usually smaller than 1V, it isnecessary to make different arrangements to guarantee the necessary output voltage level. The series, orseries-parallel, combination is called a fuel cell stack. On the other hand, the current, and therefore thepower, level is solely determined by the area of the exposed electrodes and the reactants mass flow rateprovided to the cells.Through the years, many different classes of fuel cell systems were developed, with diverse unique char-acteristics, and most of them being categorized by the electrolyte material employed [2]. The most populartypes are described below [3]: ∗ MSc student, Physics institute, [email protected]. Alkaline fuel cell (AFC): is a low temperature fuel cell, usually operating between 60-250 o C. Has itselectrolyte composed usually of a solution of potassium hydroxide and water, which is highly sensibleto contamination by CO . To avoid poisoning, the AFC must run on highly pure oxygen, thus beingadequate mostly to space applications. On the other hand, this type has a high efficiency whencompared to others with minimum reaction losses. • Solid oxide fuel cell (SOFC): is a high temperature fuel cell, operating between 600-1000 o C. Itselectrolyte is composed of Yttria and stabilized Zirconia, with its major poison being sulfur. Thismaterial is very resistant to CO , thus this type of fuel cell is very tolerant to a variety of differentfuels. The main disadvantage, besides the high operational temperature, is the long startup time,which limits the application of this fuel cell type on systems with intermittent operation. • Phosphoric acid fuel cell (PAFC): is a medium temperature fuel cell, with its operational temperatureranging between 150-230 o C. The electrolyte is composed of a solution of phosphoric acid in a siliconcarbide matrix. Sulfur and CO are the greatest sources of poisoning in this fuel cell, still, in normaloperational mode the electrolyte can withstand up to 2.5% of CO contamination, allowing more flexibleapplications. These cells shown a high capability of heat wasting and a considerable durability, thusbeing mostly used in stationary power sources. • Molten carbonate fuel cell (MCFC): is also a high temperature system, operating in the 600-800 o Crange. Its electrolyte is usually molten alkali metal carbonates in a highly porous matrix, with itsmajor poison being sulfur. This material is very tolerant to CO poisoning and also fuel flexible, thusallowing a wide range of applications. The major disadvantage is that these systems have usually veryhigh startup time, permitting only continuous power applications. • Polymer electrolyte membrane (PEM) fuel cell: is a very low temperature cell, having its operationaltemperature in the 30-100 o C range. Its electrolyte is composed of a polymeric membrane of solidperfluorosulfonic acid, with its major poison being CO , sulfur, metal ions and peroxide. Despite beingvery sensible to several substances, this cell have very high performance levels, with an elevated powerdensity and short startup time. Because of these characteristics, and also its portability, this systemis one of the most used nowadays, especially in the automotive industry.In parallel, during the last 50 years electric propulsion systems has been getting a considerable role in thespace technology community with applications ranging from interplanetary probes, as the SMART-1 [4] andDeep Space 1 [5], to orbit raising and station keeping of telecommunications satellites [6][7]. Several differenttypes of thrusters were proposed, including [6]: Pulsed Plasma Thrusters (PPTs), MagnetoplasmadynamicThrusters (MPD-T), Field Effect Emission Propulsion (FEEP), and the two most employed nowadays, HallEffect Thrusters (HET), and Gridded Ion Engines (GIE). Each of these systems operate in a different rangeof thrust-to-power ratio, thus being ideal to different mission profiles [8].After citing both systems, one natural proposition would be their combination in order to improve thepower output temporarily to the thruster, thus creating a chemical or solar-chemical electric propulsionsystem, allowing different mission profiles that need a short high power impulse using an electric thruster.Some examples would be fast orbit raising for GEO satellites, attitude control and rendezvous maneuvers.There are several possible ways to arrange both systems and it is possible classify these different casesdepending on two basic characteristics: (1) whether the fuel cell output is re-used as a propellant; and (2)whether the power used by the thruster comes just from the fuel cell or from both the fuel cell and the solararrays. With this, four different architectures must be studied in order to completely assess the feasibilityof combination of the systems.In this article we explore in details this possibility, showing what would be the necessary characteristics ofthe device the make mission parameters (payload ratio, thrust, among others) improve. In the first section,it is shown a brief review of the applications of fuel cells in the space industry, and it is presented theclassical analytical modeling of these devices. In the second part, it is shown two case studies showing mostof the possible ways to use fuel cells in conjunction with electric propulsion systems, and it is demonstratedanalytically the possible advantages and limitations of the applications.2able 1: Specific energy values of possible reactions considering no losses Reaction Specific Energy ( η v = 1 ) Technology readiness Hydrogen ( H ) – Oxygen ( O ) 13.16 MJ/kg Commercial[2]Methanol ( CH OH ) – Oxygen ( O ) 8.77 MJ/kg Commercial[9]Hydrazine ( N H ) – Hydrogen Peroxide ( H O ) 8.86 MJ/kg Tested [10]MMH ( CH ( N H ) N H ) – NTO ( N O ) 8.74 MJ/kg Not testedAmmonia ( N H ) – Oxygen ( O ) 8.27 MJ/kg Tested [11] The modeling of fuel cells is fundamentally a complex electrochemical question and a precise computationof their performance is not necessary for the development of this study. Mench [2] and Spiegel [3] present abroad review about the modeling and construction of these systems. The simplified model presented belowis based on these two works.In order to compute the approximated performance of FCs it is necessary to take into consideration theFaraday conservation laws, and it is possible to show that mass output of a fuel cell is given by,˙ m F C = InF M
F C . (1)Where I is the produced current, n is the number of equivalent electrons produced per mole of reaction, M F C is the molar mass of the product and F is the Faraday constant. Considering that the produced poweris simply given by P F C = E c I , where E c is the voltage generated in a single cell, it is possible to determinea direct relation between power and mass flow rate as, P F C ˙ m F C = E c nFM F C . (2)Using the definition of the Gibbs energy of formation as being ∆ G = − E nF , where E is the maximumachievable voltage, it is possible to define the specific energy, in J/kg, generated by the used substances, k c = − η v ∆ G M F C , (3)where η v = E c /E is the voltage efficiency of the fuel cell. In order to assess the possible application of fuel cells to power up electric thrusters, two basic architectureswill be studied next. The first case consists on using of a fuel cell as a power source alongside with thesolar cells, with all its reaction products being rejected in a zero-velocity exhaust producing no thrust. Inthis architecture it would be also possible to consider a finite specific impulse for the fuel cell output if, forexample, it was used as a working fluid for a cold gas thruster in order to improve performance. However,for the sake of simplicity and since the impact of a low specific impulse device would have on the generalperformance of an electric propulsion system is almost negligible, we only consider here the limit case wherethe exhaust velocity tends toward zero.In the second case, the fuel cell is again used as a source in conjunction with the spacecraft solar cells,however, in this scenario the reaction products are used as the propellant for the thruster
In order to compute the overall performance of the spacecraft it is necessary to define first the basic energeticrelations between the propulsion and the FC systems. The power output of the FC, P F C , is considered tobe proportional to a generic mass flow rate input, ˙ m F C , in the form,3
F C = k c ˙ m F C , (4)where k c is a conversion factor in J/kg and also represents the specific energy of the FC as shown in thefirst section.If we assume then that a thruster of constant efficiency η T is used, and that it receives a total power P T = P SC + P F C , where Psc is the power generated by the solar cells, the resulting thrust, T , can becomputed as, T = 2 η T u e ( P SC + k c ˙ m F C ) , (5)where u e is the exhaust velocity of the thruster in m/s. Furthermore, for the sake of simplicity it is thendefined that the total power provided to the thruster P T is a multiple of the spacecraft solar cell power inthe form, P T = εP SC = P SC + k c ˙ m F C . (6)Of course, simultaneously, equation 5 and 6 also defines that the resultant thrust will also be a multipleof the thrust given by the solar power in the form T = εT SC .Using the relation above, the FC mass flow rate is then simply,˙ m F C = ε − k c P SC . (7)If it is then defined that the thrust is also given by T = ˙ m p u e , where ˙ m p is the propellant mass flowrate in the thruster, and the equations 5 and 7 are taken into consideration, it is possible to relate all majorquantities as, ˙ m F C u e = (cid:18) ε − ε (cid:19) ˙ m p η T k c . (8)Noting that all the parameters in the right hand side, except ε , can be taken as constant, it is possible tosee that this equation shows a direct relation between the three major quantities in the model: power, fuelcell mass flow rate and the generated specific impulse. Another consequence of this equation is that when ε ≫ m p / η T k c (independent of ε ), as the solar powerstarts to be insignificant compared to the power generated by the FC.To account for the influence of the power system input on the mission effectiveness itself, it is possibleto define a modified thrust equation that takes into consideration the additional propellant consumption. Ifit is considered that all the mass consumed by the fuel cell is disposed as a thruster exhaust with negligiblespecific impulse, the equivalent thrust is, T q = u q ( ˙ m p + ˙ m F C ) = u e ˙ m p , (9)where u q is defined as an equivalent exhaust velocity, that represents the combined system as a singlethruster. As shown in the relation, this equivalent exhaust velocity is defined as a ratio of propellant masflow rates times the real exhaust velocity resultant from the power increment, u e . In order improve thecomprehensibility of the model, it is interesting to introduce a set of normalized variables, χ = u q /u SEP , λ = k c ˙ m p /P SC and ν = ∆ V /u
SEP , where u SEP = p η T P SC / ˙ m p . Using equations 8 and 9, it is possible toshow explicitly that, χ = λ √ ελ + ε − . (10)Using the obtained relation for exhaust velocity it is possible to redefine then the Tsiolkovsky equationfor payload ratio, m f m = exp (cid:18) − ∆ Vu q ( ε ) (cid:19) = exp (cid:18) − ν λ + ε − λ √ ε (cid:19) , (11)4 1 R U P D O L ] H G S R Z H U İ 3 D \ O R D G U D W L R N F N N F N N F N N 3 6 & ࡆ P S Figure 1: Variation of payload mass ratio with the normalized power. Three cases are shown, with a crescentprofile k c > k , the limit case with zero derivative at ε = 1 k c = k and the pure decreasing case k c < k .The profile assumes the performance of XIPS 25-cm average I sp of 3550 seconds, thrust of 166 mN, efficiencyof 68,8%, at an input power of 4300 W.where m f and m are respectively the dry and initial mass of the spacecraft.Once that the ∆ V is considered constant for the mission, the exponential function present in the lastequation varies monotonically with u q ( ε ). With this assumption it is then possible to compute an extremumof the payload ratio by finding the extremum of u q . This can be done simply finding an ε such that, dχdε = λ ( λ − ε − √ ε ( λ + ε − . (12)Since only the numerator can make the function go to zero, the extremum can be found at, ε m = λ − . (13)As it can be seen from equation 12, when ε < ε m , u q and the payload ratio will be in a region of constantincrement and the increasing of power supplied by the FC will be advantageous in a efficiency point of view.Of course, it is also straightforward to imply that if ε m > k c > P SC ˙ m p = u SEP η T . (14)Figure 1 shows the behavior of the payload mass ratio, m f /m , as function of ε for a generic mission( ν = 0 .
2) employing an ion thruster with similar characteristics of the 25-cm XIPS thruster, developed byL3 Communications, that has an average I sp of 3550 seconds, thrust of 166 mN, efficiency of 68,8%, at aninput power of 4300 W [12]. The figure demonstrates the three main behaviors of the equation 11 dependingof the value of k c compared to k = 2 P SC / ˙ m p , as predicted by the requirement in the expression 14.If the model is expressed in terms of absolute power, using again ε = P T /P SC , it is possible to considerthe case where the FC is the only power source of the propulsion system, without any input from thespacecraft’s solar cells ( P SC = 0). With this equation 6 becomes P T = k c ˙ m F C . Since equation 13 maximizes χ it is possible to say that, when P SC = 0, u q maximum will happen with P T = k c ˙ m p , which is the sameof ˙ m F C = ˙ m p . Using the equation 10 one can find that the theoretical maximum for the equivalent exhaustvelocity in the system without solar cell power input is,5 í N F 0 - N J 0 D [ L P X P , V S T V Ș 7 Ș 7 Ș 7 Ș 7 Figure 2: Maximum specific impulse for the case where the FC is the only power supply. u q,m = r η T k c . (15)Figure 2 plots the equivalent specific impulse, I sp,q = u q /g , for a range of specific energy values. It ispossible to see that for all the real values of specific energy shown in table 1, the attained equivalent specificimpulse is considerably small when considering that this is an electric propulsion mission.In fact, if it is taken into consideration that the theoretical maximum exhaust velocity obtained in achemical thruster can be expressed as u ch,m = p ( − H /M ) [8], where ∆ H is standard enthalpy offormation and M is the exhaust molar mass of the propellant, it is possible to see that, u ch,m u q,m = 2 r ∆ H ∆ G . (16)Showing that, if P SC = 0, the maximum specific impulse obtained in the chemical thruster using thesame fuel and oxidizer will always be higher, since | ∆ G | ≤ | ∆ H | for spontaneous reactions[2]. This issimply justified by the fact that in the chemical thruster the energy is converted directly and its exhaust isnot disposed as in this study of case.With the general definitions, where the FC is the secondary supply, it is possible to compute the ap-proximate time needed for the execution of a given maneuver. Considering that both the ˙ m p and ˙ m F C areconstant throughout all the maneuver, the total mass of equivalent propellant used in the maneuver can bedefined as m q = ( ˙ m p + ˙ m F C )∆ t , where ∆ t is the time in seconds, and considering τ = ∆ t ˙ m p /m f . Usingagain the simple Tsiolkovsky equation for propellant mass the approximate time for the maneuver is then, τ = λλ + ε − (cid:18) exp (cid:18) ν λ + ε − λ √ ε (cid:19) − (cid:19) . (17)The analysis of the equation obtained for time is slightly more complex than the one for exhaust velocityand an analytical expression for the exact location of its extrema might be overcomplicated for the contextof an initial study. Nevertheless, in order to derive a second general requirement for the performance of FC,now bound to the transfer time, it is possible to analyze the derivative of the function 17 only at the point ε = 1. Thus, at this point, the derivative of the function 17 is given by, dτdε (cid:12)(cid:12)(cid:12)(cid:12) ε =1 = 1 − e ν λ + νe ν (cid:18) − λ λ (cid:19) . (18)6f d ∆ t/dε > λ > (cid:18) − − e − ν ν (cid:19) . (19)Comparing the obtained condition with the expression 14, it is possible to observe that the values aresimilar, but corrected by a factor influenced mainly by the ∆ V of the maneuver and performance characteris-tics of the used thruster. It is important to note that the obtained requirement does not provide a guaranteethat the application of FCs would be advantageous for the mission, but rather provides a general lowerboundary for the performance of the system. On the contrary of the previous requirement, in expression 14,the present one depends on the mission profile, by the ∆ V .In order to increase the fidelity of the estimation it is possible to include the growth of the power systemand FC mass with the increase of total power. Considering a linear variation of the mass of both systemand that α SC and α F C are the specific power of each of them respectively, it is possible to final to definethe dry mass of the spacecraft as, m f = m u + α SC P SC ε + α F C P SC ( ε − , (20)where m u is the useful mass. Using this definition with equation 11, it is possible to define a correctedpayload mass ratio equation that expresses the amount of dry mass that can be used for other purposes thanpower and fuel cell systems, m u m = exp (cid:18) − ∆ Vu q ( ε ) (cid:19) − α SC P SC ε + α F C P SC ( ε − m . (21)Using a similar approach of the derivation of the condition 19, it is possible to define a requirement forthe specific power values that ensures an increasing profile of the payload ratio for a given 1 ≤ ε < ε m , ddε (cid:18) m u m (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ε = ε m = (cid:18) ν ε / m (cid:18) − ε m + 1 λ (cid:19) − φ (cid:19) m f m = 0 , (22)where φ = ( α SC + α F C ) P SC /m f . Giving the following condition to ensure that the extremum of thefunction is at ε m , λ = ν ( ε m + 1) ν − ε / m φ . (23)From this relation it is straightforward to observe that, to ensure that it is possible to select specificpowers with positive values, one must have a positive value in the right hand side. One way to ensure this ishaving k c > ( ε m + 1) P SC / ˙ m p , which shows close concordance with the condition expressed in 14. Since thedefinition of useful mass ratio does not modify the function of payload ratio in equation 11, the expressionof the maneuver time continues to hold true. In this case, it is considered that the propellant fed to the thruster is composed by the exhaust of the fuelcell, ˙ m F C , and a separate feeding system, ˙ m p . In the first part we study a combination of both flows inorder to determine the impact of adding a fuel cell to a given propulsion system. Next, in the second part,we consider the propellant to be exclusively the exhaust of the fuel cell.It is important to note that output product substance may not be directly compatible to an electricthruster, nevertheless several works in literature demonstrate the usage of common reaction products, suchas water and carbon dioxide, as a propellant for different types of thruster. We leave for a future work theconsideration of the performance loss due to the usage of alternative propellants.Taking into consideration again the definition of thrust efficiency, the exhaust of the system can be simplycalculated as, 7 e = r η T P SC ˙ m T + 2 η T k c ˙ m F C ˙ m T , (24)where ˙ m T is the total propellant mass flow rate, given by ˙ m T = ˙ m p + ˙ m F C . It is interesting to not thatthe first term in the square root represents a “solar electric propulsion” part and the second term is relativeto a “chemical propulsion” part. For P SC = 0 or very high ˙ m F C , the expression becomes very similar to anideal chemical thruster, and, if it is considered that ˙ m T = ˙ m F C we obtain u e = p η T k c , (25)which holds some resemblance with the expression for the thermal acceleration u e ≈ p c p T demonstratedby Jahn[13]. Comparing this expression with equation 15 it is clear that, as expected, using the fuel cellexhaust as propellant instead of discarding it, a specific impulse 2 times higher is obtained.When k c = 0 the relations are reduced to the equations of a common electric thruster. The specificimpulse and the thrust of a propulsion system in this configuration will always be higher than a systemusing only solar power, since the energy from the chemical reaction is added to the thrust.Considering a similar approach to case 1, it is possible to define again P T = εP SC and ˙ m F C = ( ε − P SC /k c , and rewrite equation 24 in the form, u e = s η T εP SC ˙ m p + ( ε − P SC k c . (26)Defining once more the normalized variables λ = k c ˙ m p /P SC and χ = u e /u SEP it is possible to write theexhaust velocity expression as, χ = r λελ + ε − . (27)Noting the resemblance between equations 27 and 10 it is possible to observe that both architectureshave a similar behavior. However, in contrast to the first case, equation 27 does not present any extremumpoints making χ increase or decrease monotonically with ε . To guarantee that χ always increases we cananalyze its derivative in the form, dχdε = ( λ − λ χ ( λ + ε − . (28)Which then imposes a restriction for the fuel cell given by λ >
1, or, k c > P SC ˙ m p = u SEP η T . (29)It is possible to note again the similarity between the restrictions 29 and 14, but in the present scenario k c has a more relaxed constraint, which is two times higher than in the first case. Of course, this can bejustified by the fact that the fuel cell flow is actually being used to give kinetic energy for the spacecraft,and 29 is only requiring that the energy in the fuel cell flow is higher than the energy contained in the “solarelectric propulsion” flow, given by P SC / ˙ m p .We assume now that the thruster uses exclusively the products of the fuel cell as propellant, with noexternal feeding system. To this end, it is assumed that the total propellant flow and the fuel cell exhaustis given by ˙ m p , i.e. ˙ m T = ˙ m p , ˙ m F C = ˙ m p . Substituting these definitions and dividing equation 24 by u SEP it is straightforward to define χ = r P SC + k c ˙ m p P SC = √ λ. (30)Noting that a normalized thrust is simply given by ξ = χ , a relation between specific impulse and thrustcan be written as 8 N o r m a li z ed I s p , χ λ = 0λ = 0.5λ = 1.0λ = 1.5 Figure 3: Normalized specific impulse in function of the normalized thrust for different values of λχ = 1 + λξ , (31)resembling the common relation of power, thrust and specific impulse. It is interesting to observe herethat, as expected, if the propellant is only composed by the reaction products, the trend of the specificimpulse is always crescent, once we are adding energy to the flow without any drawback.Figure 3 plots the equation for several values of λ . It is possible to observe that the energy addition fromthe chemical reaction increases the overall power when λ increases.Taking into consideration the mass of power supplies that increases with power, as in the first case, it ispossible to derive a corrected payload ratio, m u m = exp (cid:18) − ν √ λ (cid:19) − α F C P SC m λ − α SC P SC m , (32)where α F C and α SC are the specific masses of the fuel cell and the other power supplies in kg/W,respectively. Simply differentiating the obtained equation about λ it is possible to find an extremum pointwhich characterizes the optimum specific energy for a given mission. This expression is found as ddλ (cid:18) m u m (cid:19) = (cid:18) ν λ m ) / − φ F C (cid:19) m f m = 0 ; (33) λ m = (cid:18) ν φ F C (cid:19) / − , (34)where φ F C = ( α F C P SC ) /m f . Giving the condition, k c < (cid:18) m f ˙ m p ∆ Vα F C √ η T (cid:19) / − P SC ˙ m p . (35)Meaning that if k c is superior to this specified value the payload ratio will start to decrease since thefuel cell mass will get proportionally higher. Thus if the performance of the fuel cell is increased, with thedecrement of α F C , a wider range of values for k c are permitted to increase the performance of the mission.For the applicability of this scheme it is important then to guarantee that λ m is positive,9 m = (cid:18) ν φ F C (cid:19) / − > φ F C < ν . (37)Yielding the condition for the specific mass of the fuel cell in the form α F C < m f ∆ V u SEP P SC . (38)It is important to note that even after the calculated limit, if λ m is positive, there is a region where theresultant payload ratio is still higher than the payload ratio without the fuel cell action. It is possible tocalculate an approximate limit where it is still advantageous the usage of the fuel cell. We can then dividethe equation 32 by itself with λ = 0 and obtain (cid:20) − α F C P SC m f (cid:21) exp (cid:18) ν (cid:18) − √ λ (cid:19)(cid:19) = 1 . (39)Where it is imposed that this ratio should be equal the unity to represent the limits to the region wherethe performance of the system is higher than the case without fuel cell. Using a Taylor expansion to representthe exponential part, and truncating it in the second order, we can achieve the approximate value λ max ≈ ν − φ F C ν ( ν + 3) . (40)Then, if λ m >
0, the region between λ = 0 and λ = λ max always increases the payload ratio of themission. This work presented critical study on the feasibility of using fuel cell systems to modify the performanceof electric thrusters using just basic relations of propulsion performance. In the introduction section wepresented a background discussion showing the main points about history of fuel cells and their usage onspaceflight. In the first section we showed a basic modeling procedure for the fuel cell performance based onthe chemical energy contained in the used reactants.Using this basic fuel cell model, in the second section, we begin a study of cases, the first consisting of aspacecraft using separate propellant feeding systems for the propulsion and fuel cell systems, and the secondconsisting of a thruster using the reaction product of the fuel cell as its propellant. In the first scenarioit is showed that if the fuel cell product is simply discarded, as expected, the overall performance of thepropulsion system is bound to decrease unless we possess reactants that yield energy levels superior to aquantity denoted by k = 2 P SC / ˙ m p = u SEP /η T . Considering then the performance of the hydrogen-oxygenfuel cell showed in table 1 as the maximum attainable, in order to achieve an increment in performance inthis case, the thruster should have less than 360 s of specific impulse, proving the non-applicability of thisscheme for the vast majority of the electrical thrusters available today. It is also showed in this section thatif no power is available from external sources, like solar panels, the performance of an electrical thrusterwill be always poorer than an chemical thruster with the same propellants, simply because of the relationbetween enthalpy and free energy of formation.In the next case, where it was considered that the reaction products were used as propellant, it is shownthat both the specific impulse and the thrust increase for any given k c , since we are simply adding energyto the system without any apparent drawback. Next we considered the impact of the fuel cell mass in thespacecraft performance and this imposed a restriction for the applicability of this scheme considering themission ∆ V and the specific mass of the fuel cell. In order to have an increment in performance it is shownthat the fuel cell must at least meet the requirement calculated as α F C < m f ∆ V / u SEP P SC . Lastly it wasshown if lambda is inside a certain range between 0 and λ max while λ m is positive, the payload ratio of themission increases. 10 cknowlegments ISF thanks CNPq, project number PDE(234529/2014-08), and also FAPDF project number 0193.000868/2015,call 03/2015.
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Physics of Electric Propulsion . Dover, 1964.11 ist of symbols E Maximum expected voltage, V E c Voltage in a single cell, V F Faraday constant, ≈ .
33 sA/mol I Current produced by a fuel cell, A I sp Specific impulse, s k c Specific energy, J/kg k Specific energy limit given by 2 P SC / ˙ m p , J/kg M Molar mass, kg/mol M F C
Molar mass of the fuel cell product, kg/mol m Initial mass of the spacecraft, kg m f Dry mass of the spacecraft, kg˙ m F C
Output mass flow rate of a fuel cell, kg/s m q Total mass of equivalent propellant ˙ m p + ˙ m F C , kg˙ m p Propellant mass flow rate, kg/s m u Useful mass (dry mass excluding power systems masses), kg n Number of equivalent electrons produced per mole P Power, W P F C
Power generated by the fuel cell, W P SC Power generated by the solar cells, W P T Total power, W T Thrust, N T q Equivalent thrust, N u e Real exhaust velocity, m/s u ch,m Theoretical maximum of the exhaust velocity in a chemical thruster, m/s u q Equivalent exhaust velocity, m/s u SEP
Real exhaust velocity when P T = P SC , W α SC Specific power of the solar power system, kg/W α F C
Specific power of the fuel cell power system, kg/W β Non-dimensional mass flow, β = ˙ m p / ˙ m SEP ∆ G Standard Gibbs free energy of formation, J/mol∆ H Standard enthalpy of formation, J/mol∆ V Delta-V of the mission, m/s∆ t Mission time, s ε Ratio of total power to solar cell power, ε = P T /P SC ε m Power ratio that guarantees maximum payload ratio η T Thrust efficiency η V Voltage efficiency, η V = E c /E λ Ratio of power deposited with FC and SC, λ = k c ˙ m p /P SC ν Non-dimensional V, ν = ∆ V /u
SEP ξ Non-dimensional thrust, ξ = T / ˙ m SEP u SEP φ Non-dimensional power source mass, φ = ( α SC + α F C ) P SC /m f φ F C