Heat generation within uniform side wall of micronozzle is established and tested. Two dimensional flow analysis of gas and one-dimensional thermal analysis of uniform side wall are solved numerically and simultaneously for different configurations of heat generation. Heat generation within the entire side wall shows improvement of thrust level due to increasing both of density and pressure, although there is a degradation of Mach number and velocity and increasing in the thickness of subsonic boundary layer, dense gas keep higher thrust level at the exit section. Heat generation rate get higher capacity and more effective heat exchange at wall temperature below the material melting temperature when heat supplied to the entire wall.
Introduction
Nowadays, the need of extremely precise impulse for maneuvering the new generation of micro and nano-satellites is growing. Micronozzles with throat size of few hundred microns or below were fabricated. The flow in a MEMS supersonic micronozzle can be substantially affected by viscous effects. For various microscale nozzles reported in the literature, the Reynolds numbers are relatively low; typical values are well below 1000 and some are less than 100. As such, the magnitude of viscous losses can be significant. In the diverging nozzle section, a viscous subsonic layer may extend a sufficient distance away from the wall so as to retard the bulk flow and reduce efficiency. (Louisos and Hitt, 2008) investigated a steady viscous flow through a two-dimensional supersonic linear micronozzle, they conducted the numerical solution for range of Reynolds numbers and for expander half-angles of 10-50 deg, they found that using an expander angle larger than the traditional angle of macronozzle can compensate for the presence of the viscous subsonic layer. Improvements to the specific impulse were achieved through a combination of decreasing the nozzle length and increasing the nozzle expansion angle for low Reynolds numbers by (Ketsdever et al., 2005). The thick viscous layer growing from the sidewall in the micronozzle as the low Reynolds number effect in low-pressure measurement interacts with the shockwaves to induce a series of compression waves and a ?-shock wave, as demonstrated by (Huang et al., 2007), instead of showing a pressure jump pass through a shock wave, the pressure gradually increases when passing through the ? shock structure. They investigated that experimentally by the image visualization of supersonic flow at convergent-divergent micronozzle with various total pressure and Reynolds numbers. (Xu and Zhao, 2007) stated that, the viscous boundary layer thickness relative to the whole nozzle width on the exit plane is increased but attains the maximum value around of 0.5 and oscillates against this value with the continuous increasing of the nozzle upstream pressures. (Louisos and Hitt, 2007) found that an inherent trade-off exists between viscous losses and losses resulting from non-axial exit flow at large expansion angles. They also found that, viscous effect are more pronounced in 3D owing to the flat plate side-walls in the depth dimension, they also found that, heat loss from the flow acts to reduce viscous effects and the corresponding size of the subsonic boundary layer thus increasing micro-nozzle performance. (Alexeenko et al., 2006) predicted the importance of wall temperature in the micronozzle system; they studied time dependent performance of a high-temperature MEMS-based thruster by a coupled thermal fluid analysis of gas and walls. They found that the temperature inside the solid material approximately increases uniformly with very small changes owing to Biot number smaller than unity. They also found that the predicted thrust and mass discharge coefficient of both two dimensional and three dimensional micronozzles decreases in time as the viscous losses increases for higher wall temperatures for two different chamber pressures in different outer thermal boundary conditions, their results of effect of wall temperature on performance are agreed with the results of (Louisos and Hitt, 2007); heat losses from the high temperature gas decreases with time due to raising wall temperature and so viscous effects increases. This research is not repeating studying the relation between gas heat losses and nozzle performance, heat is supplied from nozzle wall toward gas and it increases the heat content of gas, heat is one form of energy may convert to momentum forces to increase the thrust. Current work is analyzing the performance of the nozzle in addition to changes in properties of flow when heat is generated within the wall and rejected to the gas during its expansion inside nozzle.
The intent of this study is to submit an expected solution to enhance the performance of micronozzle without increasing the pressure of the propellant reservoir. Increasing the chamber pressure requires a bigger and heavier system; however this is not desired in nano- and micro-satellites. The energy planned to be supplied to enhance the efficiency of the propulsion system is taking another form, in the space another forms of energy are more likely to be available, such as the solar energy and the generated electrical power from solar energy. Design a system supplying thermal energy across side wall to gas in order to increase the thrust of micronozzle without increasing the stagnation pressure of the system is the aim of this research. In addition to that, heating fins used previously for preheating in micronozzles increase the friction losses of gas (Bayt and Breuer, 2001b), however it seems to greatly improve the system performance as much as pressure losses reduced (Bayt and Breuer, 2001a), heating is studied here to be performed through side walls of the nozzle. Many configurations of wall heating are tested, heating is supplied through the entire wall or through a portion of the wall of convergent-divergent micronozzle; the thrust and the properties of the flow are studied accordingly.
Computational model
The computational domains are based upon the typical nozzle geometries of the microthruster prototype developed at NASA GSFC and described by (Louisos and Hitt, 2007). The expander half-angle is fixed at 15 deg. The two dimensional meshes have been developed using GAMBIT 2.4.6 grid generation software. The nozzle inlet, throat, and exit dimensions are 1103.1 �m, 90 �m, and 560�m, respectively, yield an area expansion ratio of 56:9 and are a fixed parameter in this study, span from inlet to throat of the prototype is 506.5 �m. The mesh has been changed in size horizontally and vertically to find the better mesh size. All of the mesh elements within the nozzle are quadrilateral, and outside exit plane mesh elements are unstructured quadrilateral with a maximum skewness of 0.65, as shown in Figure 1. The planar symmetry is also used to reduce the computational expenditure. In developing the final meshes, a systematic grid refinement study has been undertaken to ensure that all results are insensitive to further grid refinement. The refinement study examined grid insensitivity at both the centreline and near wall of the flow regime at the exit section.
Figure 1: model under test and its mesh
Grid-dependant Solution
The mesh is selected initially with coarse mesh horizontally and vertically, and then refined in both two directions to enhance the accuracy and stability of solution. Horizontally, the mesh cells are increased from 50 to 100 through the convergent-divergent part. Vertically, cells are increased from 100 to 400. The mesh are selected to be fine at the region of expected high velocity gradient like near wall region, and also fine grid is selected at the region around nozzle throat due to critical expected values of Mach number there, as shown in Figure 1.
Figure 2: Mach number for many selected grid sizes at the exit section (Left) around the centre (Right) near the wall.
Figure 2 (Left), Mach number around the centre converges for horizontal grid intervals increasing from 100 to 400; at the centre 300 and 400 horizontal intervals are acceptable for horizontal grid distribution. In the other hand, the vertical interval distribution is important for predicting the near wall flow properties. Figure 2 (Right) shows that solution is converged for the range of vertical intervals distribution from (50 to 100) and also affected slowly by the horizontal intervals distribution. All grids with vertical grid distribution of 50 lay in one region; however it is improved slowly by increasing the horizontal grids. The grid of 80 intervals in the vertical direction is not far from the finer grid however, it has small number of intervals on x-direction. So, 80 vertical intervals are enough to predict the precision boundary layer properties.
Thrust production equation;
?=?_(A_exit)???u(u.n) dA+?_(A_exit)??(p_exit-p_? ) dA?? Eq 1
m ?=(p_o A^*)/?(T_o ) ?(?/R (2/(?+1))^((?+1)/(?-1)) ) eq 2
The specific impulse, Isp is the ratio of the amount of thrust produced to the weight flow of the propellants. It is a measure of the fuel efficiency of a rocket engine (Balachandran, 2007). It can be obtained from:
I_sp=F/(m ?g_o ) eq 3
where:
F = gross rocket engine thrust, N
m ? = mass flow rate of exhaust gas, kg/s
go = Gravitational acceleration at sea level on Earth = 9.807 m/s�
Heat generation within wall
Table 1 Cases Nomination and Description
No Case Description Case Name Illustration diagrams
1. No heat generation within the uniform wall. Case 1 2. Heat generation of 0.001 W/mm3 within entire wall (convergent +divergent). Case 2
3. Heat generation of 1 W/mm3 within entire wall (convergent +divergent). Case 3
4. Heat generation of 5 W/mm3 within entire wall (convergent +divergent). Case 4
5. Heat generation of 10 W/mm3 within convergent wall only. Case 5 6. Heat generation of 2 W/mm3 within convergent wall plus (1mm) length extended wall. Case 6 8. Cases from 1 to 5 with inlet plane at distance of (1mm) from convergent part similar to Case 6. Case #" All corresponding cases from 1 to 7.
9. All Cases with viscosity function of temperature. Case #* All corresponding cases from 1 to 7.
Results and discussion
Temperature profile within the wall is examined for each step of increasing the rate of heat generation; the maximum temperature should not go over 80% of the melting temperature of wall material to avoid deformation of the wall under the pressure. For relatively high stagnation temperature problem, raising the rate of heat generation in the extended wall previous to nozzle more than 1 W/mm3 is impossible, the temperature of wall is increased dramatically due to accumulation of heat through the long wall, and also due to the high temperature of adjacent gas. The first three cases show insignificant changes, so the results of �Case 1� are shown instead of them.
The cases are tested with two different inlet boundary conditions; specified mass flow rate and specified stagnation pressure. For specified stagnation pressure comparisons of mass flow rate and thrust are made among cases in Table 2 for both of constant and variable viscosity, mass discharge coefficient is giving evidence for the effect of viscous losses, knowing that lowering mass discharge may be the main cause of thrust decreasing, Case 4 shows thrust increasing however, discharge is decreased for the same case, pressure and density increasing at the core play a role in increasing thrust then overcoming losses as shown in Figure 6 and Figure 8, respectively. Case 6 is showing a slower decreasing in discharge and in thrust also, the losses is decreased but the specific impulse in not improved like in Cases (4) and (5). Losses for solved cases with constant viscosity appear lower than variable viscosity cases, the constant viscosity is assumed corresponding to low temperature relatively compared with real temperatures of gas appear in Figure 7. Constant viscosity results show significance of changing other properties rather than viscosity. Specific impulse is improved in cases (4) and (5) due to, thrust is increased in the case of entire heating and, discharge is decreased in the case of heated convergent. It can deduced that heating in convergent part (Case 5) may decrease the density at the throat compared with (Case 1) under approximately same pressures at throat for the two cases, lower density means lower mass flow rate. Gas pressure profile at divergent part is more flexible than throat region excluding near the wall region, pressure is raised versus heating; see Figure 8, density increases in the core proportional to pressure as appear in Figure 6. Density decreasing upstream the throat is reducing mass discharge, as density rising downstream the throat is increasing thrust.
In Table 3; Case 6 is showing chamber pressure slight higher than the theoretical pressure, the raising of chamber pressure in other cases is noticeable, which is belonging to geometry issue. There is only one interpretation of this pressure raising; the effect of choosing inlet uniform velocity boundary condition at entrance of convergent part, high velocity streamlines suddenly hit convergent wall directly, hard turning of streamlines is behind raising chamber pressure in cases 1, 4, and 5, see configurations in Table 1. Fully developed flow with lower velocities near the wall reaches convergent part in Case 6, streamlines are turning under convergence much smoother than it happens in other cases. Stagnation pressure drop through nozzle is declaring the losses for specified mass flow rate in Table 3. In the first three cases viscous losses increases with increasing area of heating, the thrust overcomes these losses and specific impulse increases slightly due to density and pressure increasing as has been deduced previously. Chamber pressure is effected by losses of convergent part much than losses at divergent part, pressure drop in Case 4 is clearly larger than drop of Case 5, but chamber pressure is raised much in Case 5 due to larger losses in convergent part, which is belonging to higher rate of heat generation (10 W/mm3) compared with (5 W/mm3) in Case 4. Variable viscosity results show higher losses and lower produced thrust due to high gas temperature throughout most of the domain. In Table 4, all cases are tested with same position of inlet boundary condition at (1 mm) previous to convergence, the extra losses of sharp turning of streamlines disappear as noticed in chamber pressures results. Comparing between Case 5 and Case 6, increasing the area of heating previous to nozzle is not more effective than heating at the nozzle itself.
Table 2 Mass Flow Rate and Thrust Variations with Cases of Specified stagnation pressure.
Constant Viscosity
Case Name Mass flow rate, g/s Thrust (?), N/m (?m ?)/m ? Cd ??/? Isp
Case 1 24.361 32.2379 0% 0.93 0% 134.9
Case 4 24.128 32.4266 -0.95% 0.92 +0.6% 137
Case 5 23.897 32.1532 -1.9% 0.91 -0.26% 137.2
Case 6 24.128 32.2000 -0.95% 0.92 -0.11% 136
Variable Viscosity ?=?(T)
Case Name Mass flow rate, g/s Thrust (?), N/m (?m ?)/m ? Cd ??/? Isp
Case1* 24.305 32.101 0% 0.927 0% 134.67
Case4* 24.049 32.144 -1.0% 0.917 +0.13% 136.3
Case5* 23.824 31.916 -2.0% 0.908 -0.57% 136.6
Case6* 24.034 31.922 -1.11% 0.916 -0.55% 135.43
Table 3 Stagnation Pressure and Thrust Variations with the Cases for Specified Mass Flow rate.
Constant Viscosity
Case Name Inlet Po, (kpa) ?Po, (kpa) Thrust (?), N/m ??/? Isp ,s
Case 1 264.495 91.335 34.067 0% 132.6
Case 4 265.560 98.397 34.375 +0.9% 133.8
Case 5 267.007 94.386 34.308 +0.7% 133.5
Case 6 254.493 89.85 32.790 -3.75% 127.5
Variable Viscosity ?=?(T)
Case Name Inlet Po, (kpa) ?Po, (kpa) Thrust (?), N/m ??/? Isp
Case 1* 266.799 95.924 34.215 0% 133.1
Case 4* 268.571 108.466 34.497 +0.82% 134.26
Case 5* 269.266 102.757 34.363 +0.43% 133.74
Case 6* 255.646 98.896 32.663 -4.75% 127.02
Table 4 Extended Wall Geometry Results for Specified Mass Flow Rate and Constant Viscosity
Case Name Inlet Po, (kpa) ?Po, (kpa) Thrust (?), N/m ??/? Isp
Case 1" 254.318 88.464 32.772 0% 127.44
Case 4" 254.468 95.444 32.982 +0.64% 128.26
Case 5" 254.677 91.667 32.785 +0.04% 127.5
Case 6 254.493 89.85 32.790 +0.05% 127.5
Thrust profile at the exit section of selected cases of heat generation within the uniform side walls is shown in Figure 4. It is shown that, local thrust at the core region of entire heated nozzle walls (Case 4) overcomes the others, whereas it decreases at the boundary layer region. Mach number profile at exit section results are not agree with thrust profiles, as shown in Figure 5, the lower Mach numbers are observed for entire heated nozzle (Case 4) at the exit section. The interpretation of the conflict between the results of thrust profile and Mach number profile is the exserting of the significance of the other elements of thrust production; these elements appear in the equation of defining thrust (Eq. 1), they are density, static pressure and sound speed.
The ideal gas relation is employed in this problem; pressure is related proportionally to temperature for constant density or for slowly changed density, while density is related reversely to temperature for constant pressure or for slowly changed pressure. To study the variations of the three properties together, it should notice whether each change is rapid or slowly compared with other rest properties, the contours of them within the flow should be studied carefully. Increasing the temperature at the wall is normally decreasing the density at the boundary layer region for sufficient heating as appear in Figure 6, but the core density is observed to get none expected rising of density even, the higher predicted temperature at the core belongs to Case 4 as shown in Figure 6Figure 7. The interpretation of raising density of the core is also belonging to ideal gas relation, pressure contours are shown in Figure 8 for the three cases, pressure to density ratio is raised everywhere in the expander proportionally to the temperature raising by heating, taking place high pressure changes to follow temperature changes are impossible in y-direction due to rapid expansion in x-direction, so density fall down due to sharp raise of temperature at boundary layer region where the pressure cannot follow the rapid changes in temperature near the wall, while both of density and pressure are raised at the core with majority to the pressure against temperature due to smooth temperature elevation.. However the subsonic boundary layer looks to be slightly thicker for entire heated nozzle (Case 4) as shown in Figure 5, and the thrust is degraded at the boundary layer region as shown in Figure 4 for the same case, but it is thought that this amount of degradation is caused mainly by the drop of density near the wall.
The sound speed is function of specific heat and both of them are function of static temperature, sound speed contours for Case 4 and Case 5 are approximately identical at boundary layer region, as they following temperature, while Case 4 sonic speed is higher at core region as shown in Figure 9. Raising amount of heating is keeping higher sonic speed at the core, rising the sonic speed is behind the reduction of Mach number, this reduction be more significant when it is associated with velocity decreasing at the core as shown in Figure 10. However, velocity of increased wall heating at boundary layer region shows a little bit increasing, subsonic boundary layer get thicker slightly for Case 4, as shown in Figure 2.
Figure 3 Wall Temperature Profile for Selected Cases.
Figure 4 Thrust Profile at the Exit section of the Nozzle for Selected Cases of Heat Generation.
Figure 5 Mach number Contours For Different Cases of Heat Generation Within Side Walls
Figure 6 Density Contours For Different Amounts Of Heat Generation Within Side Uniform Walls.
Figure 7 Static Temperature Contours for Different Amounts Of Heat Generation Within Side Uniform Walls.
Figure 8 Static Pressure Contours For Different Cases of Heat Generation Rate within Uniform Side Walls.
Figure 9 Sound Speed Contours for Different Cases of Heat Generation within Uniform Side Walls
Figure 10 Velocity Contours For Different Cases of Heat Generation within Uniform Side Walls.
Conclusion
It was believed that only preheating the gas entering the nozzle may cause raising the thrust. According to one-dimensional analyzing, preheating raises the stagnation temperature, this causes raising Mach number which leads to improving of thrust. The current work proving that even the heating of the entire wall can improve the thrust and specific impulse. The tracking of the contours of the properties is showing that; not only significant changes of velocity and Mach number affect the thrust, but the density firstly and pressure secondly are playing significant role dominating the thrust. Through the expanding process it is found that; density can be elevated associated with increased pressure during smooth elevating of temperature at the core. The heat can be added more effectively in the entire wall of the nozzle due to high temperature drop at the expander, also heat exchange between the wall and gas is stronger due to higher temperature difference. The configurations of adding heat to gas through the wall are proposed and tested, the best configuration is concluded; generating heat throughout the entire wall is showing the higher effectiveness and larger capacity of heat generation.
