NUMERICAL SOLUTION OF ONE-DIMENSIONAL REACTIVE FLOWS IN ROCKET ENGINES WITH REGENERATIVE COOLING

Transcription

1 Proeedings of the 11 th Brazilian Congress of Thermal Sienes and Engineering -- ENCIT 2006 Braz. So. of Mehanial Sienes and Engineering -- ABCM, Curitiba, Brazil, De. 5-8, 2006 Paper CIT NUMERICAL SOLUTION OF ONE-DIMENSIONAL REACTIVE FLOWS IN ROCKET ENGINES WITH REGENERATIVE COOLING Luiano Kiyoshi Araki Programa de Pós-graduação em Métodos Numérios em Engenharia, UFPR, Curitiba, PR, Brazil Carlos Henrique Marhi Departamento de Engenharia Meânia, Universidade Federal do Paraná, Curitiba, PR, Brazil Abstrat. The maximum temperature in the internal struture and the pressure drop on the oolant flow along the hannels are two of the most important design parameters for roket engines with regenerative ooling. A one-dimensional mathematial model for the reative flows in a LOX/LH 2 -roket engine with regenerative ooling is presented. This problem inludes the ombustion gases flow in the roket diverging-onverging nozzle, the oolant flow in hannels and the heat transfer through the walls (from ombustion gases to oolant). For the ombustion gases flow, three different physial flow models are used: a frozen, an equilibrium and a non-equilibrium one. Several hemial reation shemes are also taken into aount for eah study, for the evaluation of effets of the speies and reation equation systems on the final numerial solution. Besides the influene of physial and hemial models on the answers, the effets of the adopted grids over the final results are also disussed. Therefore numerial errors estimative are also made and the physial and hemial models results are ompared. Numerial results are obtained with a finite volume software, implemented with seond order sheme for variables, o-loated grid arrangement for all speed flows and Fortran 95 language. Keywords. Finite volume method, CFD, equilibrium flow, non-equilibrium flow, numerial error estimate. 1. Introdution The thrust hamber is the key subassembly of a roket engine, where the liquid propellants are metered, injeted, atomized, vaporized, mixed and burnt, to form hot reation gas produts (Sutton and Biblarz, 2001). By speeding up these gases, thrust fore is obtained for orbit and payload definition. The knowledge of hemial and physial phenomena onerned about suh gases flow is, therefore, important for both roket design (assoiated to material definition and ooling system design) and mission goals ahievement. Heat is transmitted to all internal hardware surfaes exposed to hot gases, namely the injetor fae, the hamber and nozzle walls. Only 0.5 to 5% of the total energy generated in the gas is transferred as heat to the hamber walls (Sutton and Biblarz, 2001). This amount of energy, however, an inrease the wall temperatures until material failure. Beause of this, most of roket engines have ooling systems, whih allow a longer lifetime for the entire devie. One of the most used ooling systems for large roket engines is the regenerative one. The predition of heat transfer harateristis in a regeneratively ooled roket ombustion engine is one of the most important and most hallenging tasks in the design work of a high performane engine (Fröhlih et al., 1993). The problem involving reation gas produts flow and heat transfer to ooling system may be divided up into three sub-problems, namely: the reative ombustion gases flow through the thrust hamber (whih inludes the ombustion hamber and the nozzle engine); the heat ondution from hot gases to the oolant, through the wall struture; and the turbulent oolant flow, in the regenerative ooling system. The solution for the whole problem an be obtained by solving, iteratively and in sequene, the three sub-problems, allowing them to interat, until its onvergene. Although two and three-dimensional models are ommonly used, one-dimensional models are still employed in roket engines projets, being orreted by empirial oeffiients (Fröhlih et al., 1993; Sutton and Biblarz, 2001). Studies involving one-dimensional softwares are, beause of this, still useful. Based on this, in this work an onedimensional model, whih inludes different physial and hemial shemes, is purposed for solving the oupled problem of reation gas produts flow and heat transfer to the thrust hamber struture and oolant. Four physial models (one-speies with variable properties, frozen, loal equilibrium and non-equilibrium flows) are studied. Both frozen and loal equilibrium flows are ideal models; they differ, however, about how the hemial reations take plae during the flow. For both frozen and loal equilibrium physial models, nine different hemial reation shemes are used. These shemes are the same presented before by Marhi et al. (2005), and inlude from three to eight hemial speies and take into aount from null to eighteen hemial reation equations. For nonequilibrium model, five different hemial shemes are used, inluding six or eight hemial speies, whih totalise five hemial shemes. At this work, however, only two hemial models are studied (one six-speies and one eight-speies for frozen and equilibrium flows and two six-speies for non-equilibrium flow). Aording to Marhi et al. (2005) and Araki and Marhi (2006), six and eight-speies models present good auray with CEA results (Glenn Researh

2 Center/NASA, 2005), for frozen and equilibrium flows with adiabati walls, reason for onsidering only six and eightspeies models in this work. All the models have been implemented using Compaq Visual Fortran The implemented software, alled RHG1D 3.0, was run at a PC Pentium IV, 3.4 GHz, with 4 GB RAM. Results are ompared to those ones obtained for adiabati walls simulations and numerial error estimates, related to all the results and based on GCI estimative (Roahe, 1994), are also presented. 2. Mathematial model The basi priniples of the roket propulsion are essentially those of mehanis, thermodynamis and hemistry (Sutton and Biblarz, 2001). The flow and heat transfer problem in roket thrust hambers an be divided up into three different (but oupled) problems, for whih there are independent mathematial (and numerial) models: (1) the reation gas produts (ombustion gases mixture) flow through the thrust hamber; (2) the heat ondution from hot gases to the oolant; and (3) the oolant flow through the regenerative ooling system. Figure 1 shows, shematially, both the reation gas produts and the oolant flows. Figure 1.Thrust hamber engine with regenerative ooling system (transversal setion) Reation gas produts flow The mathematial formulation for a one-dimensional, single-speies flow (or a multi-speies frozen or equilibrium flow) through the nozzle engine is based only on four equations: onservation of mass, onservation of momentum, onservation of energy and state (the one-speies gas and/or the multi-speies gas are treated by the perfet gases law), given in the sequene: d ( ρ u S) = 0, (1) dx d dx dp ρ, (2) dx ( u S u) = S + F ' d dp ( ρ u S T ) = u S + q'+ ne, (3) dx dx p S eq / P = ρ R T, (4) where: ρ, u, P and T are four dependent variables, related to density, veloity, pressure and temperature (in this order); x is the axial oordinate along the gases flow (Fig. 1); S is the internal ross-setional area of thrust hamber struture; R is the gas onstant (or gas mixture onstant, for frozen and equilibrium flows); p is the onstant-pressure speifi heat; F ', q' and S eq/ne are related to the visous fores, the heat loss to the walls and the soure term orrespondent to equilibrium or non-equilibrium onditions (this term is neessary only for equilibrium or non-equilibrium flows), respetively, and are evaluated by the following relations:

3 π F ' = f ρ u 8 u D, (5) wall ( q + q ) q ' = u F ' + S, (6) h r S eq / ne N d hi i= 1 dx = N S hi w& i, i= 1 ( ρ u S Y ), i for loal equilibrium flow; for non-equilibrium flow; (7) where: f is the Dary frition fator; D is the diameter of the ross-setion; S wall is the internal wall surfae by length unity (along x-axis, Fig. 2); N is the total number of speies existent in the flow; h i is the enthalpy for eah hemial speies i; Y i is the mass fration for eah hemial speies i; w& orresponds to mass rate generation of speies i; and i q h and q are the onvetion and radiation heat fluxes to internal walls. r Figure 2. Cooling hannels and their geometrial parameters. As an be seen previously, the onservation of energy, Eq.(3), has the temperature as unknown and not the enthalpy nor the internal energy, as normally used. The major advantage of suh variation is about the temperature determination, whih an be obtained diretly from the numerial model, not depending on the enthalpy (or internal energy) values. Besides the gas produts flow, hemial reation shemes have an important role on final results. Frozen, equilibrium and non-equilibrium models, in different levels, take into aount hemial reations. Depending on the adopted hemial reation sheme, different hemial omposition and physial properties are obtained or, at least, larger or smaller CPU time efforts are verified. For the frozen model, by hypothesis, the flow from ombustion hamber to thrust hamber exit is so fast that there is no time for hanges in hemial omposition of reation gas produts. For the loal equilibrium model, in ounterpart, hemial equilibrium omposition must be evaluated for every single ross-setion of the thrust hamber. An intermediate (and more realisti) model between loal equilibrium and frozen ones is the non-equilibrium. For this model, hemial omposition must be evaluated for every single rosssetion of the thrust hamber as done for loal equilibrium; the equilibrium omposition, however, is not reahed and, for eah single hemial speies, mass formation rates are evaluated. For frozen and equilibrium models, nine different hemial reation shemes were implemented, taking into aount from 3 to 8 hemial speies and from null to eighteen hemial reation equations, as an be seen at Table 1 where N is the number of hemial speies and L is the number of hemial reation equations. Chemial reation shemes implemented in software Mah1D 5.0, for both frozen and loal equilibrium flows, are the same presented by Marhi et al. (2005). For non-equilibrium flow, only six and eight-speies models were onsidered, totalising five. For both frozen and equilibrium flows, different hemial shemes mean at least one different hemial reation equation (even when the same number of hemial speies and the same number of hemial reation equations are onsidered). For non-equilibrium, otherwise, different hemial shemes mean, at least, different forward reation onstant: based on this, model 3 ould be split up into two ones (the other non-equilibrium models are 5, 7 and 10). At non-equilibrium flow model, forward and bakward reation onstants (kf j and kb j, respetively) have an important role for mass generation rates evaluation, being related to eah other by the progression rate of reation j (γ):

4 γ j N N, ν ij = kf j Ci kbj Ci i= 1 i= 1 ν,, ij, (8) where C i is the molar onentration of speies i in gases mixture and,,, ν ij and ν ij are, in this order, the stoihiometri oeffiients of hemial speies i in reation j existent in reagents and produts. The speies generation rate (θ j ) is obtained from the produt between Eq. (8) and the third body effetive onentration and an be estimated by: N θ j = α ij Ci γ j, (9) i= 1 in whih α ij is the effiieny of hemial speies i at a dissoiation reation j. Unfortunately, there is a lak of information about the effiieny of the moleules existent on a dissoiation reation model. Beause of this, it is ommonly adopted that the effiienies of all hemial speies are equal to unit and, in this ase, the onentration of third body speies orresponds to the whole number of existent speies (Barros et al., 1990). Table 1. Chemial reation shemes implemented in software Mah1D 5.0. Model L N Speies Observations H 2 O, O 2, H 2 Ideal model H 2 O, O 2, H H 2 O, O 2, H 2, OH H 2 O, O 2, H 2, OH, O, H 4 reations with 3 rd body Barros et al. (1990) and Smith et al. (1987) H 2 O, O 2, H 2, OH, O, H 4 reations Svehla (1964) H 2 O, O 2, H 2, OH, O, H 8 reations (4 with 3 rd body) Barros et al. (1990) H 2 O, O 2, H 2, OH, O, H 8 reations (4 with 3 rd body) Smith et al. (1987) 4 reations from model 3 and 2 from Kee et al. (1990) all the reations H 2 O, O 2, H 2, OH, O, H, HO 2, H 2 O 2 inluding 3 rd body H 2 O, O 2, H 2, OH, O, H, HO 2, H 2 O 2 18 reations (5 with 3 rd body) Kee et al. (1990) The determination of mass generation rates is obtained by the produt of Eq. (9) and moleular weight (M i ) of the speies, resulting in where L ( ν ij θ j ) w& i = M i, (10) ν ij j= 1 is alulated by ν ij = ν ij ν ij and represents the differene between formed and onsumed number of moles for a reation j. For non-equilibrium model, mass generation rates determination is fundamental for the speies onservation of mass. Equations (1) to (4) are enough for one-speies, frozen and equilibrium flows mathematial formulation; for non-equilibrium flow, however, assoiated to these four equations must be taken into aount the speies ontinuity equation: d dx ( u S Yi ) = S w& i ρ, (11) whih orresponds to the onservation of mass for eah speies separately Coolant flow Only four equations are needed for the one-dimensional study of oolant flow through the entire regenerative ooling system of a thrust hamber: the mass onservation equation, the momentum onservation equation, the energy onservation equation and a polynomial onstitutive relation for density: d ( ρ u S ) = 0, (12) ds d ds ( u S u ) dp ds ρ = S + F, (13)

5 d dp p ( ρ u S T ) = β T u S + q, (14) ds ds ρ = ρ + ρ T + ρ T, (15) where: the index orresponds to oolant properties; ρ, u, P and T are four dependent variables, related to density, veloity, pressure and temperature (respetively); s is the length of the flow along the enter of a hannel (Fig. 1); S is the internal ross-setional area of oolant hannels; F, β and q are related to the visous fores, volumetri thermal expansion oeffiient and the heat loss to the oolant hannel walls, in this order; and ρ 1, ρ and 2 ρ are onstants 3 related to the hosen oolant Heat ondution through the wall As reation gas produts have high temperature values, heat is transferred from them to thrust hamber walls by onvetion and radiation mehanisms. This energy is then onduted through walls to the oolant and transmitted to it by onvetion. The whole proess an be modelled by: q = ( q + q ) S = q S = q S, (16) h r wh w wh w where: q is the heat transfer rate through the wall; S wh is the internal wall area in ontat with reation gas produts (Fig. 2); q is the heat flux through the wall; q is the heat flux to the oolant; and S w is the effetive heat transfer area w between the wall and the oolant. More details about the employed mathematial model an be found in Marhi et al. (2000; 2004). 3. Numerial model The mathematial models for reation gas produts and oolant flows (assembly to the heat ondution through the wall) in the thrust hamber struture are disretized using finite volume method. The domain is divided into Nvol ontrol volumes, in axial diretions x and s, in whih the differential equations are integrated. For reation gas produts, a o-loated grid arrangement, appropriated for all speed flows is used (Marhi and Maliska, 1994), assoiated with a seond-order disritization sheme (CDS), with deferred orretion (Ferziger and Perić, 2001); similar treatment is given to oolant flow model, exept for the appliane for all speed flows, one the oolant flow is always subsoni. The systems of algebrai equations obtained are solved by TDMA method (Ferziger and Perić, 2001). Pressure and veloity are oupled by SIMPLEC algorithm (Van Doormaal and Raithby, 1984), in order to onvert the mass equation in a pressure (or better, in a pressure-orretion) one. So, the mass onservation equation, Eq. (1) or Eq. (12), is used for determination of a pressure-orretion ( P ' ), while veloity (u) is obtained from the momentum equation, Eq. (2) or Eq. (13), and the energy equation, Eq. (3) or Eq. (14), is taken for temperature (T) determination. Density (ρ) is gotten from the state equation, Eq. (4), or from Eq. (15). Equation (11) is also needed for non-equilibrium flow. It must be noted that veloity (u) is evaluated from very redued values (near null values) until supersoni ones, and not only for supersoni values, as ommonly found in literature (Barros et al., 1990). The iterative proess for solving the reation gas produts mathematial model provided by Eqs. (1) to (4) or Eqs. (12) to (15) onsists only on nine steps: 1. Definition of data (temperature, pressure, density, veloity) in an instant of time t. 2. Estimation of all variables in an instant t+ t (time is used as a relaxation parameter). 3. Definition of thermophysial properties (suh as the onstant-pressure speifi heat, Dary frition fator and heat transfer oeffiients by onvetion). 4. Coeffiients alulation for the algebrai system (by disretization) of the momentum equation and solution of this system by TDMA for the veloity u. 5. Coeffiients alulation for the algebrai system (by disretization) of the energy equation and solution by TDMA for temperature T. 6. Calulation of density ρ. 7. Coeffiients alulation for the algebrai system (by disretization) of the mass equation and solution by TDMA for pressure orretion P '. 8. Corretion of ρ, P and u with P '. 9. Return to item 2, until the ahievement of the desired number of iterations.

6 Over this basi algorithm, some modifiations are neessary for reation gas produts flow, depending on the adopted physial model. For equilibrium and non-equilibrium models, step 3 inludes also the hemial omposition determination for all the domain ontrol volumes and, also here, the determination of thermoproperties must be done for every single ontrol volume. Non-equilibrium hemial omposition determination inludes, also, the use of Eq. (10), for mass generation rate evaluation, and Eq. (11), in order to estimate the hemial omposition, respeting the mass onservation law for eah speies separately. The adopted boundary onditions to reation gas produts flow inlude: definition of inlet temperature (T) and pressure (P) as funtions of stagnation parameters (T 0 and P 0, in this order); the hemial mixture omposition, given by mass frations (Y i ), is obtained from loal data (temperature and pressure); and the entrane veloity (u) is obtained from a linear extrapolation from the values obtained for internal flow. About exit onditions, for supersoni flows in nozzles, no exit boundary onditions are required; for the implementation of a numerial model, however, exit boundary onditions are needed. Beause of this, temperature (T), veloity (u), pressure (P) and mass frations (Y i ) are obtained by linear extrapolation from internal ontrol volumes. For oolant flow, some other boundary onditions are employed, as follows: for hannels entrane, both temperature and veloity are fixed (T in and u in, in this order) and inlet pressure is obtained from linear extrapolation of values obtained for internal flow; for density, both inlet and outlet values are estimated by Eq. (15); exit pressure is defined as null and both, exit temperature and exit veloity, are obtained from linear extrapolation of values obtained for internal flow. The oupling between reation gas produts and oolant flows is made by the heat ondution through the thrust hamber wall, aording to the proedures presented by Marhi et al. (2004), whih onsists on seven steps: 1. First estimative of wall temperatures distribution, for the wall in ontat with the reation gas produts. 2. Solution of reation gas produts flow. 3. Solution of oolant flow. 4. Evaluation of heat transfer rate between the reation gas produts and the oolant. 5. Evaluation of both wall temperatures: the temperature of the wall in ontat with the reation gas produts and the wall in ontat with the oolant. 6. Evaluation of heat transfer rate error, whih onsists on the differene between the reation gas produts-to-wall heat transfer rate and the wall-to-oolant heat transfer. 7. Return to item 2, until the ahievement of the desired tolerane or the desired number of iterations. 4. Definition of the problem The thrust hamber geometry used in this work is the same one presented by Marhi et al. (2000; 2004), whih onsists on a ylindrial setion, alled ombustion hamber (with radius r in and length L C ) assembled to a nozzle devie, whose longitudinal setion is defined by a osine urve (with throat nozzle radius r g and length L n ). The radius r, for x > L C, is evaluated by the following equation: r = r g + ( rin rg ) ( x L) os 2π Ln, (17) where x orresponds to the position where the radius is evaluated. Despite the ylindrial setion is alled ombustion hamber, it does not orrespond to a real ombustion hamber; the effets of fuel and oxidant injetion, mixture and burning are not onsidered. Figure 3 shows all geometrial parameters of the thrust hamber studied in this work. Figure 3. Geometrial parameters of the thrust hamber engine. About the ooling system, the number of hannels (m) plaed around thrust hamber struture. The parameters, whih define the geometrially the hannels (Fig. 2) are: the thikness (e) of thrust hamber internal wall; the thikness

7 (t) of wall that separates two neighbour ooling hannels; and the hannels height (b). The average width of eah hannel ( a ) is obtained as a funtion of previous parameters by π 2 2 a = [( r + e + b) ( r + e) ] t, (18) mb when m > 1. While the parameters b, e and t remain onstant, a does not have the same behaviour, due to radius variations. Copper-made hannels over all the length of the thrust hamber engine, aompanying the variable radius of the engine. And, while the reation gas produts flows along the x-axis in a positive diretion, the oolant flows in ounter flow, namely, in s-axis negative diretion. For all the studies presented in this work, the values used for geometrial parameters are: r g = 0.1 m; r in = 0.3 m; L C = 0.1 m; L n = 0.4 m; m = 200; b = 5.0 mm; e = 2.0 mm; and t = 1.5 mm. Based on these data, the ratio between the height and the average width of eah hannel ( b / a ) varies between 0.62 and 2.8; the engine internal wall area is 9.242x10-1 m 2 ; the base hannel area in ontat with the oolant is 7.272x10-1 m 2 and the fins surfae in ontat with the oolant is m 2. Water is taken as oolant and its mass flow rate is 1 kg/s in eah hannel, totalising 200 kg/s, with an inlet temperature of 300 K. Inside the ombustion hamber, the stagnation temperature (T 0 ) is taken as K, while the stagnation pressure value (P 0 ) is 20 MPa and liquid oxygen and liquid hydrogen are injeted at the stoihiometri omposition (OF = ). For isentropi analytial solution, the gas onstant (R) is taken as J/kg K and the ratio between speifi heats (γ) is The aim of numerial solutions is given by the evaluation of some parameters of interest, ited in the following: 1. Nozzle disharge oeffiient (C d ): defined as the ratio between numerial mass flux ( m& ) and the theoretial one num ( ), obtained from isentropi analytial solution: m& th C d m& m & num =. (19) th 2. Non-dimensional momentum thrust (F*): defined by the ratio between numerial and theoretial thrusts (F num and F th, in this order), given by F F num F * =, (20) th where F orresponds to the thrust values and an be obtained by the following relation between mass flux ( m & ) and exit veloity (u ex ): F = m&. (21) u ex 3. Pressure (P ex ), temperature (T ex ), veloity (u ex ) and Mah s number (M ex ) for gas produts, at the nozzle exit. 4. Maximum wall temperature (T max ): obtained by wall temperature distribution. 5. Coolant pressure drop ( P): evaluated from hannels entrane to its exit. 6. Exit oolant temperature (T ool ). 5. Numerial results and disussion Four different physial models, inluding heat transfer effets, are studied in this work: a one-speies gas with variable properties; a frozen gases mixture flow; an equilibrium gases mixture flow; and non-equilibrium gases mixture flow. All the analyses results were obtained employing a PC, with Pentium IV 3.4 GHz proessor, 4 GB RAM, Windows XP. The results are then ompared with those ones obtained by the same physial models (provided by both Mah1D 5.0 and CEA, this last one a numerial software provided by NASA), but without oolant flow effets (adiabati walls). For all the studies, the simulations were made for 3 different grids, to allow the determination of apparent and effetive onvergene orders (Marhi and Silva, 2002). Numerial error estimates, also, based on GCI estimator (Roahe, 1994), were taken for all the physial and hemial models. Some piees of information about numerial errors and error estimators an be found in Tannehill et al. (1997), Marhi (2001), and Ferziger and Perić (2001). Error estimates are very important for evaluating if two different physial (or hemial) models have the same results or the differenes an be assigned to numerial errors. The GCI estimator is evaluated by

8 φ1 φ2 GCI(φ1, p) = 3, (22) p ( q -1) where: ϕ 1 and ϕ 2 are, respetively, the numerial solutions for the refined (h 1 ) and oarse (h 2 ) grids; q is the grid refinement ratio ( q = h ); h is the grid spaing or distane between two suessive grid points; and p is related to the 2 / h1 asymptoti (p L ) or apparent (p U ) order (the lowest value between the two ones). Asymptoti error order depends on the hosen disretization model, while apparent error, for onstant refinement ratio, is evaluated by p U ϕ2 ϕ3 log 1 2 ( h1 ) ϕ ϕ =, (23) log( q) where ϕ 3 orresponds to the numerial solution for a superoarse grid. Although nine hemial shemes are available at RHG1D 3.0 software for frozen and equilibrium flows and another six shemes are implemented for non-equilibrium flow, only models 3 and 10 (with six and eight speies, respetively) for frozen and equilibrium flows and models 31 and 32 (both with six speies) for non-equilibrium flow are studied in this work. Aording to results obtained by Marhi et al. (2005) and Araki and Marhi (2006), six and eight-speies models present good auray with CEA results; beause of this, only six and eight-speies models results are reported in this work. Tables 2, 3 and 4 present the results for the variables of interest, inluding numerial error estimates, for an 80-volumes grid. The reason for using suh a grid is based on other studies (Marhi et al., 2004; Araki and Marhi, 2006), for whih numerial error estimates are about the same magnitude of experimental error; besides, CPU time onsumption for an 80-volumes grid is muh lower than the one required for a ontrol volumes, or even a 2560 one (Araki and Marhi, 2006). As an be seen from Table 2 results, heat transfer effets are not important for non-dimensional momentum thrust determination: there is no modifiation on values, when the simulations inluding or not suh effets are ompared. For nozzle disharge oeffiient, there is a tiny variation on values, whih an be bigger than the error estimates (like for frozen flow), however, whih is no greater than 1%. Larger differenes are observed (Tables 2 and 3) for loal variables of interest, suh as exit pressure, temperature and veloity. The heat transfer effets assembled to the hemial reative flow provides a redution on exit pressure, exit temperature and exit veloity values. This drop is about Pa for exit pressure (whih is equal, at maximum, to a redution of 2.6%), K for exit temperature (at maximum, a 4.5% redution) and m/s for exit veloity (at maximum, a 1.0% redution). Suh variations are motivated by the heat ondution to the oolant: one this energy is transferred, the temperature of the reation gas mixture drops and, onsequently, all the other thermophysial parameters are affeted, inluding the hemial omposition (for equilibrium and non-equilibrium models), as an be seen at Table 5. Table 2. Comparison for C d, F* and P ex (80 ontrol volumes). Results without heat transfer effets Model C d [adim.] F* [adim.] P ex [Pa] Analytial (R1) x10 4 One-speies, variable Properties (R2) ± 3x ± 4x x10 4 ± 4x10 1 Frozen Flow mod. 3, 4, 5 and ± 3x ± 4x x10 4 ± 1x10 2 Frozen Flow mod. 9 and ± 3x ± 4x x10 4 ± 1x10 2 CEA (frozen flow) x10 4 Equilibrium Flow mod. 3, 4, 5 and ± 1x ± 1x x10 4 ± 5x10 2 Equilibrium Flow mod. 9 and ± 1x ± 1x x10 4 ± 5x10 2 CEA (loal equilibrium flow) x10 4 Non-equilibrium Flow mod ± 3x ± 5x x10 4 ± 7x10 1 Non-equilibrium Flow mod ± 3x ± 5x x10 4 ± 6x10 1 Results taking into aount heat transfer effets Model C d [adim.] F* [adim.] P ex [Pa] One-speies, variable Properties (R2) ± 3x ± 4x x10 4 ± 8x10 1 Frozen Flow mod ± 3x ± 4x x10 4 ± 1x10 2 Frozen Flow mod ± 3x ± 4x x10 4 ± 1x10 2 Equilibrium Flow mod ± 1x ± 1x x10 4 ± 5x10 2 Equilibrium Flow mod ± 1x ± 1x x10 4 ± 5x10 2 Non-equilibrium Flow mod ± 3x ± 5x x10 4 ± 2x10 2 Non-equilibrium Flow mod ± 3x ± 3x x10 4 ± 2x10 2 (R1): Rg = J/kg K; (R2): Rg J/kg K (equivalent to ombustion gases mixture for the ideal model) Table 4 provides the numerial results for oolant properties (as well as for heat transfer parameters). As an be seen, numerial results are quite independent from the physial model hoie; even for exit oolant temperature, for

9 whih the results are different (based on numerial error estimates), the maximum differene among models is only 0.54 K. Larger differenes are found for the maximum heat flux (q max ) (from the reation gas produts to the oolant): it ahieves 8.3x10 5 W/m 2 (omparing the one-speies gas and frozen flow models); the effet of suh variation on the maximum wall temperature, however, is quite small: only 8.5 K (for the same physial models) and, if frozen and equilibrium results are ompared, this differene is even smaller: 5.5 K. Based on these results and on CPU time onsumption, whih is presented at Table 6, for oolant and wall properties definition, the frozen flow model is a good hoie (even for an initial analysis): its results are quite the same of the other physial models and its CPU time onsumption is the lowest one (even smaller than the monogas with variable properties). Table 3. Comparison for T ex, u ex and M ex (80 ontrol volumes). Results without heat transfer effets Model T ex [K] u ex [m/s] M ex [adim.] Analytial (R1) One-speies, variable Properties (R2) 1800 ± ± ± 1x10-2 Frozen Flow mod. 3, 4, 5 and ± ± ± 1x10-2 Frozen Flow mod ± ± ± 1x10-2 CEA (frozen flow) Equilibrium Flow mod. 3, 4, 5 and ± 3x ± ± 2x10-3 Equilibrium Flow mod. 9 and ± 3x ± ± 2x10-3 CEA (loal equilibrium flow) Non-equilibrium Flow mod ± 1x ± ± 1x10-2 Non-equilibrium Flow mod ± 1x ± ± 1x10-2 Results taking into aount heat transfer effets Model T ex [K] u ex [m/s] M ex [adim.] One-speies, variable Properties (R2) 1730 ± ± ± 1x10-2 Frozen Flow mod ± ± ± 2x10-2 Frozen Flow mod ± ± ± 2x10-2 Equilibrium Flow mod ± 8x ± ± 2x10-3 Equilibrium Flow mod ± 8x ± ± 2x10-3 Non-equilibrium Flow mod ± 1x ± ± 1x10-2 Non-equilibrium Flow mod ± ± 2x ± 1x10-2 (R1): R g = J/kg K; (R2): R g J/kg K (equivalent to ombustion gases mixture for the ideal model) Table 4. Comparison for P, T ool, q max and T max with heat transfer effets (80 ontrol volumes). Model P [Pa] T ool [K] q max [W/m 2 ] T max [K] One-speies, variable Properties (R2) 8.4x10 5 ± 3x ± 2x x10 7 ± 5x ± 3x10-1 Frozen Flow mod x10 5 ± 3x ± 2x x10 7 ± 1x ± 7x10-1 Frozen Flow mod x10 5 ± 3x ± 2x x10 7 ± 1x ± 7x10-1 Equilibrium Flow mod x10 5 ± 3x ± 2x x10 7 ± 4x ± 3 Equilibrium Flow mod x10 5 ± 3x ± 2x x10 7 ± 4x ± 3 Non-equilibrium Flow mod x10 5 ± 3x ± 2x x10 7 ± 9x ± 4x10-1 Non-equilibrium Flow mod x10 5 ± 3x ± 2x x10 7 ± 9x ± 4x10-1 (R2): R g J/kg K (equivalent to ombustion gases mixture for the ideal model) Table 5. Reation gas produts omposition at nozzle engine exit (80 ontrol volumes). Model H 2 O O 2 H 2 OH O H HO 2 H 2 O 2 O CEA < x effets CEA < < Frozen Flow, without heat transfer effets Equilibrium Flow, without heat transfer Non-equilibrium Flow, without heat transfer effets Frozen flow, taking into aount heat transfer effets Equilibrium flow, taking into aount heat transfer effets < Non-equilibrium flow, taking into aount heat transfer effets

10 Table 6. CPU time physial models; simulations inluding heat transfer effets (80 ontrol volumes). Physial Model Monogas, variable properties Frozen Flow Equilibrium Flow Non-Equilibrium Flow Chemial Iterations for: Reation produts flow Coolant flow Global iterations CPU time --- 6,000 1, s 3 5,000 1, s 10 5,000 1, s 3 15,000 1, h 10 15,000 1, h 31 5,000,000 1, day 32 4,000,000 1, h From Table 5, it an be noted that the reation gas produts ompositions, for both equilibrium and non-equilibrium flows, taking into aount heat transfer effets, are only a little different from those obtained for adiabati walls. It is related to the small drop at the temperature profile through the entire nozzle engine, onsidering heat transfer effets, as an be seen at Fig. 4. The lower the temperature, the smaller the speies dissoiation effets; beause of this, all the speies mass frations drop when the heat transfer effets are onsidered, but the H 2 O one. Figure 4. Temperature distribution of reation gas produts through the nozzle engine. Beside the CPU time onsumption for physial and hemial models studied in this work, Table 6 provides the number of iterations for solving eah problem (reation gas produts and oolant flows) separately and, also, the number of global iterations. These amounts of iterations were hosen in order to allow the onvergene of eah subproblem separately (reation gas produts and oolant flows), at eah global iteration. The number of total global iterations was hosen in order to ahieve the round-off error (exept for non-equilibrium flow, for whih a tolerane of 10-5 was speified). This proeeding was neessary to minimize other types of numerial errors than disretization ones and, in this way, to guarantee that numerial errors are made, essentially, of disretization ones. While Fig. 4 presents the temperature distribution of reation gas produts through the nozzle engine, Fig. 5 provides the wall temperature (in ontat with reation gas produts) along the whole engine. Despite the differenes on temperature values at thrust hamber entrane and exit, all the physial models have similar solutions for maximum wall temperature (atually, the differene at numerial results is smaller than 10K, as previously ommented). Although frozen flow presents lower temperatures for reation gas produts than equilibrium flow ones, wall temperatures are, at least from thrust hamber entrane to nozzle throat, higher than those obtained for equilibrium flow studies. This phenomenon is explained by the higher values for onvetion heat transfer oeffiient verified for frozen flow, whih ahieves, at maximum, 11,691 W/m 2 K at nozzle throat region (against 10,861 W/m 2 K for equilibrium flow model). Thus, even though the reombination reations, whih take plae along all the thrust hamber length and forms H 2 O speies, are exothermi and ontribute to the higher reation gas produts temperatures, the higher values for onvetion heat transfer oeffiient, verified for frozen flow model, overtake these reombination effets and its assoiated higher temperature, at least until nozzle throat region, where the highest wall temperatures are found. Otherwise, the differene between frozen flow and equilibrium flow reation gas temperatures after the nozzle throat region beomes so expressive that the effets of the higher values for onvetion heat transfer oeffiient (verified for frozen flow model) are overtaken by the higher equilibrium flow temperature values for reation gas produts. Beause of this, the wall temperatures, from the nozzle throat to the thrust hamber exit, are higher for the equilibrium flow model.

11 6. Conlusion Figure 5. Wall temperature distribution (in ontat with reation gas produts) through the nozzle engine. A software for one-dimensional hemial reative flow with regenerative ooling was implemented, using FORTRAN 95 language and Visual Compaq ompiler. Four physial models were studied: a one-speies flow (with variable properties); and the other three for reation gas produts (frozen, loal equilibrium and non-equilibrium flows). For frozen and equilibrium models, nine different hemial reation shemes were implemented, the same hemial reation shemes disussed by Marhi et al. (2005), but only two were studied (a six and a eight-speies models) and the results were ompared to those ones obtained for adiabati wall studies, inluding those ones from the NASA s software CEA. For non-equilibrium, only six and eight-speies reation shemes were employed, totalising six models, omparing the results to frozen and loal equilibrium ones. Unfortunately, beause of time restritions, only two from these six models were studied. Differently as usually found in literature, the energy equation has the temperature as unknown (and not the enthalpy nor the internal energy, as ommonly used). The major advantage of suh variation is about the temperature determination, whih is done diretly from the numerial model. Also, the veloity determination is done from the ombustion hamber (where the veloity is almost null) to the nozzle exit (supersoni flow), overing all the veloity regimes in a real engine (subsoni, transoni and supersoni ones), and not only the supersoni flow, as ommonly done. Assembling refrigeration effets to reation gas produts flow has only small influene on global parameters of interest: for nozzle disharge oeffiient, there is only a tiny variation on values, whih an be bigger than the error estimates (like for frozen flow), however, whih is no greater than 1%; for non-dimensional momentum thrust, no appreiable variation on numerial results was verified. For loal parameters of interest, otherwise, larger differenes were observed, espeially for exit temperature values, for whih a drop between 35 and 70 K at exit temperature (at maximum, a 4.5% redution, for frozen flow model) was verified. At exit veloity and exit pressure, suh variations are smaller: only a 1.0% redution for veloity (15-35 m/s) and a 2.6% redution for pressure (about Pa). The reation gas produts ompositions, for both equilibrium and non-equilibrium flows, taking into aount heat transfer effets, are only a little different from those obtained for adiabati walls. Comparing the physial models results for oolant parameters of interest, it was observed only a small variation on oolant exit temperature: 0.54 K, while the pressure drop values are the same for all the models studied. For maximum heat flux (from the reation gas produts to the oolant), the differene more expressive: it ahieves 8.3x10 5 W/m 2 (omparing the one-speies gas and frozen flow models); the effet of suh variation on the maximum wall temperature, however, is quite small: only 8.5 K (for the same physial models). Frozen flow model presents higher values for wall temperatures from thrust hamber entrane until its throat region, although equilibrium flow model shows greater reation gas produts temperatures. This phenomenon is onsequene of the higher values for onvetion heat transfer oeffiient verified for frozen flow model and, beause of this, the highest value for maximum wall temperature was K, for frozen flow model. From the nozzle throat region to the exit thrust hamber, the effets of the higher values for onvetion heat transfer oeffiient are overtaken by those ones aused by the greater reation gas produts temperature, what explains the higher wall temperature values for equilibrium flow after the nozzle throat region. For all the studies, 80-volumes grids were employed, one they provide numerial error estimates at the same magnitude of experimental ones (Marhi et al., 2004; Araki and Marhi, 2006). Comparing numerial results (for oolant and wall properties definition) and also CPU time requirements, the frozen flow model is a good hoie (even for an initial analysis): its results are quite the same of the other physial models and its CPU time onsumption is the

12 lowest one (even smaller than the monogas with variable properties, whih demands at least 90% more time): it is at least more than 500 times faster than the equilibrium flow model and 5500 times faster the than non-equilibrium one. 7. Aknowledgement The authors would aknowledge Federal University of Paraná (UFPR), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and The UNIESPAÇO Program of The Brazilian Spae Ageny (AEB) by physial and finanial support given for this work. The first author would, also, aknowledge his professors and friends, by disussions and other forms of support. 8. Referenes Araki, L. K., Marhi, C. H., 2006, Effets of Chemial Reation Shemes and Physial Models on a One-Dimensional Flow in a Roket Engine Nozzle, Proeedings of the 27th Iberian Latin-Amerian Congress on Computational Methods in Engineering, Belém, Brazil. Paper CIL (Paper submitted.) Barros, J. E. M., Alvim Filho, G. F., Paglione, P., 1990, Estudo de Esoamento Reativo em Desequilíbrio Químio através de Boais Convergente Divergente, Proeedings of the III Enontro Naional de Ciênias Térmias, Itapema, Brazil. Ferziger, J. H. and Perić, M., 2001, Computational Methods for Fluid Dynamis, 3ed., Berlin: Springer-Verlag. Fröhlih, A., Popp, M., Shmidt, G., Thelemann, D., 1993, Heat Transfer Charateristis of H 2 /O 2 Combustion Chambers, Proeedings of the 29 th Joint Propulsion Conferene, Monterrey, USA, AIAA Glenn Researh Center, 2005, CEA Chemial Equilibrium with Appliations, available at: < Web/eaHome.htm>, aess in: Feb 16, Kee, R. J., GrCar, J. F., Smooke, M. D, Miller, J. A., 1990, A Fortran Program for Modeling Steady Laminar One- Dimensional Premixed Flames, Albuquerque: Sandia National Laboratories, SAND UC-401. Marhi, C. H., 2001, Verifiação de Soluções Numérias Unidimensionais em Dinâmia dos Fluidos, Thesis (Mehanial Engineering PhD). Universidade Federal de Santa Catarina, Florianópolis, SC. Marhi, C. H., Araki, L. K., Laroa, F., 2005, Evaluation of Thermohemial Properties and Combustion Temperatures for LOX/LH 2 Reation Shemes, Proeedings of the 26 th Iberian Latin-Amerian Congress on Computational Methods in Engineering, Guarapari, Brazil. Paper CIL Marhi, C. H., Laroa, F., Silva, A. F. C., Hinkel, J. N., 2000, Solução Numéria de Esoamentos em Motor-Foguete om Refrigeração Regenerativa, Proeedings of the 21 st Iberian Latin-Amerian Congress on Computational Methods in Engineering, Rio de Janeiro, Brazil. Marhi, C. H., Laroa, F., Silva, A. F. C., Hinkel, J. N., 2004, Numerial Solutions of Flows in Roket Engines with Regenerative Cooling, Numerial Heat Transfer, Part A, Vol.45, pp Marhi, C. H., Maliska, C. R., 1994, A nonorthogonal finite volume method for the solution of all speed flows using o-loated variables, Numerial Heat Transfer, Part B, v. 26, pp Marhi, C. H., Silva, A. F. C., 2002, Unidimensional numerial solution error estimation for onvergent apparent order, Numerial Heat Transfer, Part B, Vol. 42, pp Roahe, P. J., 1994, Perspertive: A method for uniform reporting of grid refinement studies, Journal of Fluids Engineering, Vol. 116, pp Smith, T. A., Pavli, A. J., Kaynski, K. J., 1987, Comparison of Theoretial and Experimental Thrust Performane of a 1030:1 Area Ratio Roket Nozzle at a Chamber Pressure of 2413 kn/m 2 (350 psia), Cleveland, NASA Lewis Researh Center, NASA Tehnial Paper Sutton, G. P., Biblarz, O, 2001, Roket Propulsion Elements, 7 ed., New York, John Willey & Sons. Svehla, R. A., 1964, Thermodynami and Transport Properties for the Hydrogen-Oxygen System, Cleveland, NASA Lewis Center, NASA SP Tannehill, J. C., Anderson, D., Plether, R. H., 1997, Computational Fluid Mehanis and Heat Transfer, 2ed., Philadelphia: Taylor & Franis. Van Doormal, J. P., Raithby, G. D., 1984, Enhanements of the SIMPLE method for prediting inompressible fluid flow, Numerial Heat Transfer, v. 7, pp