Transcritical Behavior of Methane in the Cooling Jacket of a Liquid-Oxygen/Liquid-Methane Rocket-Engine Demonstrator

Abstract

The successful design of a liquid rocket engine is strictly linked to the development of efficient cooling systems, able to dissipate huge thermal loads coming from the combustion in the thrust chamber. Generally, cooling architectures are based on regenerative strategies, adopting fuels as coolants; and on cooling jackets, including several narrow axial channels allocated around the thrust chambers. Moreover, since cryogenic fuels are used, as in the case of oxygen/methane-based liquid rocket engines, the refrigerant is injected in liquid phase at supercritical pressure conditions and heated by the thermal load coming from the combustion chamber, which tends to experience transcritical conditions until behaving as a supercritical vapor before exiting the cooling jacket. The comprehension of fluid behavior inside the cooling jackets of liquid-oxygen/methane rocket engines as a function of different operative conditions represents not only a current topic but a critical issue for the development of future propulsion systems. Hence, the current manuscript discusses the results concerning the cooling jacket equipping the liquid-oxygen/liquid-methane demonstrator, designed and manufactured within the scope of HYPROB-NEW Italian Project. In particular, numerical results considering the nominal operating conditions and the influence of variables, such as the inlet temperature and pressure values of refrigerant as well as mass-flow rate, are shown to discuss the fluid transcritical behavior inside the cooling channels and give indications on the numerical methodologies, supporting the design of liquid-oxygen/liquid-methane rocket-engine cooling systems. Validation has been accomplished by means of experimental results obtained through a specific test article, provided with a cooling channel, characterized by dimensions representative of HYPROB DEMO-0A regenerative combustion chamber.

1. Introduction

Technologies based on the oxygen/methane couple in the field of rocket-engine applications are attracting more and more attention in the academic community as well as industrial players in the last twenty years. In fact, there are several well-known reasons, still pushing research and economic efforts into developing and consolidating methane-fuelled space propulsion systems [1,2,3,4]. Although specific impulse is generally lower than hydrogen-based systems, methane exhibits some advantages such as storability, relative ease of handling, and soft cryogenic skills, due to the higher density and critical temperature values, respectively; if compared to rocket-grade kerosene, methane results are cleaner since it does not produce residues (e.g., coking) during the combustion, and this represents also a key factor for developing reusable rockets [5,6,7,8,9,10]. Given these reasons, the world-famous company SpaceX chose methane-based technology, concentrating investments on Raptor’s reusable engines [11].

In summary, methane-based propulsion systems display high specific impulse and thrust-to-weight ratio performances, good capabilities in terms of reusability and throttle-ability, as well as fewer concerns in terms of pollution impact on ground, atmosphere, and space, and reduced costs. Moreover, another strong point has been recently underlined since methane could be “extracted” on Mars in a view of in situ resource utilization framework, and this results into a desirable feature for ascent/descent applications in future planetary-exploration missions [12,13]. In fact, the flexibility of application makes methane an interesting solution for several missions, including access to space (from first stages of launchers to upper stages) and in-space systems (such as service modules, landing or descent vehicles, and ascent stages) [14]. Moreover, Italy is investing in the development and consolidation of methane-based space propulsion systems both on the private and public sides. The Italian Ministry of University and Research is directly funding the HYPROB (HYdrocarbon PROpulsion test Bench) Program, assigned the Italian Aerospace Research Centre (CIRA). The main goals consist of enabling and enriching the national system capabilities for the development of liquid rocket engines (LRE) and subsystems with a specific attention to the liquid-oxygen/liquid-methane couple [15]. The most significant goal is the design, realization and testing of a liquid-oxygen/liquid-methane 30 kN-thrust demonstrator (DEMO-0A), after pursuing an incremental approach strategy, contemplating the test of in-house-designed breadboards conceived to deepen the comprehension of basic physical phenomena, and critical aspects regarding Lox/LCH₄ supercritical combustion [16].

The present paper aims at describing the behavior of methane flowing as refrigerant inside a DEMO-0A cooling jacket, representative of typical LRE regenerative cooling systems, by means of 3D numerical simulations. The investigation takes into account different values of inlet conditions in terms of fluid temperature, pressure and mass flow rate in order to analyze the influence on the thermal and fluid-dynamic response of such systems and describe the coolant behavior.

2. Transcritical Behavior of Methane in a LRE Cooling Jacket

The core component of liquid rocket engines (LREs) is represented by thrust chambers that may exhibit combustion-gas temperatures up to 3600 K and pressure values of tens of bar, according to their characteristics in terms of propellants, geometry, and injection strategy. Thus, huge thermal and mechanical stresses must be managed in order to fulfil life-cycle requirements as well as performance goals [17,18]. Given these impressive operative conditions, the design of efficient cooling systems, surrounding the thrust chambers, is mandatory and the development of advanced cooling jackets, made up by narrow axial channels (Figure 1a), is linked with the comprehension of the coolant behavior. In fact, with the exception of kerosene-based engines, most LREs adopt cryogenic propellants as coolant, and this requires a deepening of the analysis of coolant behavior, caused by very different thermophysical properties according to a fixed section of the cooling jacket. Nowadays, although huge efforts have been spent, this kind of investigations remains current, especially for methane-based LREs. Indeed, data under theoretical, numerical and experimental points of view are still needed, if compared to other cryogenic propellants, such as hydrogen, to validate engineering tools and numerical codes [19,20,21]. In the case of methane, evolving in a regenerative cooling jacket, the refrigerant is injected in liquid phase, but due to the huge thermal load from combustion gases, it typically shows a transcritical behavior [22]. According to the regenerative counter-flow cooling strategy given in Figure 1b, the refrigerant (LCH₄) is inserted in the cooling channels as a compressed liquid (A), such as at temperature and pressure conditions characterized by values lower and higher than the critical values (T_cr = 190.56 K and P_cr = 4.59 MPa), respectively; then, it is heated by the heat, released by combustion gases, and exhibits a sort of phase “pseudo-change” in the cooling jacket, since both critical temperature and pseudocritical conditions are generally reached and overcome. Thus, moving from the nozzle region, a sort of stratification is gradually observed and the fluid layers (near the bottom part of the channel), behaving as a gas, coexist with fractions of fluid (in the upper part) still remaining in liquid-like conditions (B). Finally, the coolant comes out from the cooling jacket at conditions typically characterized by both temperature and pressure values higher then critical ones, and according to the cooling-system characteristics, even as a supercritical gas. Then, methane is injected into the combustor chamber by means of the injectors (C) [16,23].

Hence, the fluid experiences a sort of “pseudo-change” phase, flowing from the inlet towards the outlet section and changing from a liquid-like condition to a vapor-like one [24]. In other words, thermophysical properties (density, viscosity, specific heat, thermal conductivity, etc.) show impressive variations near the critical conditions (i.e., transcritical or near-critical conditions), as pointed out by Figure 2. However, it is important to remark the differences of behavior with respect to the liquid-to-vapor phase change, observed at subcritical pressure, which is characterized by dramatic sudden abrupt variations [25]. For the comprehension of the transcritical behavior of methane inside the cooling jacket, the pseudo-critical temperature (T_pc > T_cr) should also be considered. It represents the temperature corresponding to the condition, characterized by the maximum value of the specific heat at a specific pressure [26], because can be sometimes associated with a phenomenon called heat-transfer deterioration, strictly linked to the thermal layers occurring in a cross section [26,27,28,29,30,31,32,33]. The pseudocritical temperature value raises as a function of pressure increasing, and in parallel, the specific heat peak tends to decrease. Around the pseudocritical conditions, heat-transfer efficiency may decrease dramatically because a gas-like thin film (featured by low thermal conductivity values), developing near the bottom surface of the channels, reduces the heat transferred from the heated wall to the liquid-like core [30]. Thus, designers used to pay a lot of attention in properly conceiving the cooling passages and took sufficient margins in order to overcome possible undesired problems [31,34]. High values of channel roughness seem to suppress the deterioration mode or to mitigate its effects because of heat-transfer exchange increasing. Moreover, the use of artificial roughness may prevent the heat-transfer deterioration: as a result, wall temperature values decrease but pressure losses may significantly increase [35]. Other authors have also pointed out the effects of different configurations of inlet/outlet manifolds on the behavior of the cooling jacket [36].

It is worth highlighting that the application of the classical semiempirical correlations, adopted to calculate the evaluation of heat-transfer coefficients, does not produce reliable results in the deteriorated mode, and in addition, high values of relative roughness also represent a challenging condition from the numerical-modeling point of view. Thus, CIRA has designed, realized, and tested a dedicated breadboard, the MTP-BB (Methane Thermal Properties Breadboard) in the framework of HYPROB Program [15,37]. Experimental results were useful to collect data on transcritical behavior of methane in a relevant domain and operation, since the breadboard is representative of an LRE cooling channel in terms of geometry and thermal conditions. Moreover, a validated numerical procedure has been set up in view of being adopted to run the thermal simulations on the HYPROB final Demonstrator first unit (DEMO-0A), supporting the detailed design. Further information on the program and on the realized breadboards are available in [14,16,38].

3. Geometrical Description

The final HYPROB demonstrator is a ground engine, including a regeneratively-cooled thrust chamber characterized by a typical counterflow architecture, such as the fuel/coolant entering the cooling system from the nozzle region and exiting in correspondence with the injector head. Figure 3 depicts the assembly with the main components: the most important parts such as the igniter, the injector head, the manifolds, cooling jacket, and combustion chamber. Some details on the engine performances are shown in

Table 1. Summary regarding the main performance parameters of HYPROB DEMO.

O/F	3.4	PCC	5.5 MPa
Reaction efficiency	0.98	Isp	286 s
Thrust	30 kN	Acc/At	4

. Several narrow axial channels compose the cooling systems. Each channel is generated through the copper-alloy liner, the pure-copper side ribs and the upper surface manufactured by means of electrodeposition. The close-out part is made up of electrodeposited pure nickel.

CIRA has developed a one-dimensional in-house tool with two-dimensional corrections in order to conduct the preliminary design phase of the thrust chamber and related cooling jacket [22]. With reference to Figure 4 and Figure 5, the design choice is based on imposing a constant number of channels and rib width (w) and varying the rib height (h). In this way, the optimization of cooling performances was performed focusing on the three significant sections, highlighted in Figure 5, such as the nozzle (NZ), throat (CT) and cylindrical part (CP). The considered nondimensional geometric parameters, based on reference length L (the overall thrust chamber length), are given below:

• height of channels (h/L) ranges from 0.0018 to 0.0061;
• width of channels (b/L) ranges from 0.0019 to 0.0107;
• width of ribs (w/L) equal to 0.0032;
• thickness of the liner (h₁/L) equal to 0.0020;
• height of the copper layer (h_cu/L) equal to 0.0023;
• height of the nickel close-out layer (h_ni/L) equal to 0.0034.

Data reduction was obtained by means of Reynolds number, Nusselt number, Prandtl number, friction factor, and Colburn number. In addition, dimensionless variables were considered, such as temperature, density, viscosity, thermal conductivity, and specific heat. Their formulae are reported in the following:

$$\mathit{Re}=\frac{\rho u_{av}{d}_h}{\mu }$$

(1)

$$Nu=\frac{\dot{q}{d}_h}{\left(T_w-T_{b,f}\right){\lambda }_f}$$

(2)

$$\mathit{Pr}=\frac{\mu c_p}{\lambda }$$

(3)

$$T\ast =\frac{T}{T_{cr}}$$

(4)

$$f=\frac{2\mathrm{\Delta }P{d}_h}{L\rho u_{av}{}^2}$$

(5)

$$j=\frac{Nu}{ReP{r}^{1/3}}$$

(6)

$$\rho \ast =\frac{\rho }{{\rho }_{cr}}$$

(7)

$$c_p\ast =\frac{c_p}{c_{p,in}}$$

(8)

$$\lambda \ast =\frac{\lambda }{{\lambda }_{in}}$$

(9)

$$\mu \ast =\frac{\mu }{{\mu }_{in}}$$

(10)

where u_av is the average velocity, d_h is the hydraulic diameter of the channel, and $\dot{q}$ is the heat flux evaluated on the channel surface. Moreover, T_w and T_b,f represent the channel surface temperature and the fluid bulk temperature, and ∆P is the pressure difference while T_cr and ρ_cr are the temperature and density referred to the critical point of methane, respectively. Finally, c_p,_in, λ_in and μ_in are the fluid-specific heat, thermal conductivity, and viscosity at the inlet section.

4. Numerical Modeling

The description of methane transcritical behavior inside a single cooling channel belonging to DEMO was conducted through numerical analyses performed by means of ANSYS Fluent v17 [39]. The governing equations of continuity, momentum, and energy in the 3D form were solved under the hypothesis of steady state, NIST real gas model, and turbulent flow (k-ω sst turbulence model was adopted [40]). The thermo-physical properties of methane have been considered variable and extracted through REFPROP v7.0 database [41]. A steady-state solution and a segregated method were chosen to solve the governing equations. A pressure-based method was considered while a second-order upwind scheme and the SIMPLEC coupling one were chosen for energy and momentum equations and to couple pressure and velocity, respectively [39]. The convergence criteria of 10⁻⁶ and 10⁻⁸ for the residuals of the velocity components and energy were assumed, respectively. Conduction effects were considered and roughness was imposed to the channel surfaces.

The initialization was accomplished through the inlet-section conditions in terms of fluid temperature, pressure, and single-channel mass-flow rate; their nominal values are 110 K, 16.0 MPa, and 0.02 kg/s (overall mass flow rate is equal to 1.92 kg/s), respectively. In addition, some off-nominal inlet conditions, comprised in a DEMO operating box, were considered to characterize the cooling-jacket behavior at a fluid inlet temperature equal to 100 K and 120 K, inlet pressure equal to 15.7 and 16.3 MPa. Moreover, further mass-flow-rate values, equal to 0.019 and 0.021 kg/s (for each channel), were considered to take into account a possible mass unbalance, due to the not-optimal filling of some channels fed up by the inlet manifold.

The use of the NIST real gas model allowed to manage both the liquid and the vapor phase, and thus describes the transcritical operating conditions of the working fluid. It is important to highlight that a proper use of the real gas also affects the initialization of the simulations and the convergence history. With reference to the aforementioned Figure 4 the considered solid material for the liner part is a copper alloy (CuCrZr) while the electrodeposited layers are made up of pure copper and pure nickel. All the thermophysical properties of such materials were considered temperature-dependent and calculated through a dedicated characterization activity [14]. Results were in line with literature data, as shown in [42]. The computational domain includes both the solid and fluid parts. A half channel was considered to reduce the computational effort, taking advantage of the geometrical and thermo-fluid dynamic symmetry. The following boundary conditions have been assigned, as described by Figure 6:

−

inlet section: uniform velocity (at a fixed mass flow) and uniform temperature profile;

−

outlet section: pressure outlet static;

−

channel surfaces: velocity components equal to zero;

−

liner surface: input heat flux as depicted by Figure 5;

−

upper and lateral surfaces: heat flux equal to zero.

The liner surface is heated by the input heat flux, depicted in Figure 5, representing the nominal profile. It derived from CFD reactive simulations on the DEMO combustion chamber side. A constant wall-temperature value of 300 K was imposed [23]. This conservative approach was considered useful to assess the design of the cooling-jacket configurations; then, a weak thermal coupling was also considered for further analyses, as suggested by [22], and to accomplish the comparisons given in the “Results and Discussion” section. Thus, a boundary condition in terms of convective heat-transfer coefficient of the hot-gas side was calculated by properly scaling the “design” nominal heat flux (calculated at T_hg = 300 K) through the average hot-gas temperature inside the chamber, evaluated by means of RPA code [43]. The following formula, indeed, has been applied for some significant sections (C, I, O, T), given in Figure 5.

$$h_{c,hg}\stackrel{.}{=q_{nom}}/\left(T_{aw}-T_{hg}\right)$$

(11)

With T_aw equal to 3543 K (outlet section—O), 3538 K (beginning of chamber cylindrical part—C), 3370 K (throat—T), 2456 K (Nozzle exit—I) and T_wg = 300 K. Intermediate values between two significant sections were calculated by a linear interpolation according to the values given above.

A mesh-sensitivity analysis was accomplished considering the nominal conditions (nominal input heat flux, T_in = 110 K, P_in = 16.0 MPa and m = 0.01 kg/s—half channel). Three structured mesh distributions, generated by following the mesh suggestions given by Ansys Fluent user’s guide in the case of rough channel walls [39], were considered: they have about 1.3, 2.7, and 5.6 million of nodes, respectively. Finally, the second grid case ensured a good compromise between the machine computational time and the accuracy requirements, as shown by results given in

Table 2. Results of grid-independence analysis.

Type of Mesh	ΔP [MPa]	Outlet Fluid Bulk Temperature Tb,f out [K]	Liner Maximum Temperature Tw,hg max	Channel Bottom Wall Maximum Temperature Tw,ch max
1—coarse	5.051	420.1	600.4	555.4
2—fine	5.094	420.4	610.8	562.7
3—finest	5.096	420.5	611.5	563.4

. Figure 7b allows to give an idea on the nodes’ distribution in the axial and transverse direction and on the attention paid for meshing expansion/contraction sections and ensuring mesh elements’ continuity.

5. Methodology Validation

The validation of the numerical procedure was performed by considering the modeling applied to the MTP-BB numerical rebuilding campaign [37] (Figure 8). That breadboard was specifically designed and tested to collect data on the transcritical behavior of methane and setting up the numerical approach to be used for the DEMO cooling system. The body is made of a copper alloy, and on the top, a rectangular narrow channel, characterized by dimension comparable to DEMO cooling channels, was milled. The heat flux is imposed to the basement by means of ten electrical cartridges (able to supply a maximum power of 12 kW), located in specific slots. Both geometry and operative conditions are representative of typical ones experienced in the LRE cooling system, and in particular, in the DEMO cooling system.

The test campaign was successfully performed, considering the following test parameters:

• inlet mass flow rate = 0.015, 0.020, 0.025 kg/s;
• outlet pressure = 8.0, 10.0, 12.0, 15.0 MPa;
• inlet fluid temperature = 120, 130, 140 K;
• impressed electrical power = 0 (cold flow), 12 kW.

Results from the numerical rebuilding activity in terms of fluid temperature, pressure drops, and channel wall-temperature values were in good agreement with the experimental data, as also reported in [38]. In fact, the maximum discrepancies in terms of fluid-temperature differences between inlet and outlet sections and pressure drops are equal to about 1.3% and 4.0%, respectively, while differences in terms of the channel’s bottom surface is maximum 1.0%. Further information is available in [37].

6. Results and Discussion

Results for numerical simulations, considering the nominal operating conditions of the cooling jacket (T_in,f = 110 K, P_in = 16.0 MPa and overall m = 1.92 kg/s, corresponding to 0.02 kg/s for each cooling channel), are pointed out. Furthermore, other working points, included in the operating box of HYPROB DEMO, have been considered since the demonstrator could be tested also under the following conditions, determined by the firing test bench characteristics: inlet fluid-temperature values equal to 110 K ± 10 K and inlet pressure equal to 16.0 MPa ± 0.3 MPa. Furthermore, overall mass-flow-rate values equal to 0.020 kg/s ± 0.001 kg/s were considered to analyze the effect of mass unbalance in the filling of the cooling channels, connected to the inlet manifold. In this way, the effects of different inlet conditions on cooling-jacket response may be evaluated. The test matrix is summarized in

Table 3. Test matrix of the numerical campaign.

Run	Inlet Temperature		Inlet Pressure		Mass Flow Rate		Imposed Heat Flux
	Tin,f [K]	Tin,f* (Tin,f/Tin,f−nom)	Pin [MPa]	Pin* (Pin/P,nom)	m [kg/s]	m* (m/mnom)
1	110	1.00	16.0	1.00	0.020	1.00	Nominal
2	110	1.00	16.0	1.00	0.020	1.00	Weak Coupling
3	100	0.91	16.0	1.00	0.020	1.00	Nominal
4	120	1.09	16.0	1.00	0.020	1.00	Nominal
5	110	1.00	15.7	0.98	0.020	1.00	Nominal
6	110	1.00	16.3	1.02	0.020	1.00	Nominal
7	110	1.00	16.0	1.00	0.019	0.95	Nominal
8	110	1.00	16.0	1.00	0.021	1.05	Nominal

. Results include axial profiles for the liner and channel wall temperature, fluid bulk temperature, local and average convective heat-transfer coefficient, local and average Nusselt number, most significant thermophysical properties, and static pressure. Moreover, temperature field plots, including some transversal slices of significant interest, are presented together with some information about thermophysical properties.

6.1. Effects of Fluid Inlet Temperature

Figure 9 depicts the axial profiles of the hot-gas wall, bottom channel, and fluid bulk temperature, considering different values of inlet fluid-temperature values ranging from 100 K to 120 K (runs 1-4). Relative maxima in terms of liner wall temperature and channel bottom surface temperature are attained at the throat zone (x/L = 0.61) and in the region where of hot gases reattach on the combustion chamber wall (x/L = 0.13); if the nominal conditions are considered (T_in,f = 110 K − T_in,f* = 1.00), the corresponding temperature maximum values are equal to about 530 K and 610 K, respectively. Because slight fluctuations in terms of inlet temperature values as well as pressure may be imposed by the firing test bench, T_in,f is reduced to 100 K (T_in,f* = 0.91) or increased to 120 K (T_in,f* = 1.09). The effect is significant, considering the domain from half of the nozzle region, and becomes more evident at the throat zone, where the differences between the nominal conditions and the other operating points may also attain 25 K because of slight changing in fluid thermophysical properties. For what concerns the fluid bulk-temperature profiles, it may be observed that temperature increases more than linearly in the nozzle region, while in the proximity of throat region, the slope tends to change because the methane begins to locally change its conditions from a “liquid-like” to a “gas-like” fluid: in fact, the liquid methane enters the jacket but reaches the critical temperature (190.56 K) approximately in the throat region. From this section, the fluid tends to behave like a highly compressible fluid near the bottom wall of the channel (the hotter one), and as a liquid moving towards the upper wall (the colder one). Moreover, the temperature seems to increase almost linearly. When the fluid moves towards the outlet section, a larger fraction tends to show a vapor-like behavior. Finally, methane traverses the last part of the cooling jacket, such as the cylindrical part, in supercritical gas conditions, since temperature and pressure are much higher than the critical values.

Regarding the static pressure profiles carried out by Figure 10, it is possible to highlight that pressure drops are very low in the nozzle region since the fluid is still in the liquid phase. However, the most significant raise is observed in the convergent and cylindrical parts, because density tends to decrease very rapidly and velocity increases towards the outlet section. This behavior is slightly more evident if the fluid inlet temperature increases.

Figure 11 describes the local and average profiles of convective heat-transfer coefficient and Nusselt number as a function of axial coordinate. The maximum values of the convective heat-transfer coefficients are attained in the throat section, in line with design-phase choices; then, they tend to increase from x/L = 0.36 towards the exit because of the fluid velocity growth, occurring in the cylindrical part. In this case, the inlet temperature has a significant impact in the nozzle region and on the peak value near the throat, which is the highest if T_in,f = 100 K (T_in,f* = 0.91), corresponding to about 10% higher than the case with T_in,f = 120 K (T_in,f* = 1.09); moreover, for T_in,f = 120 K a slight increase in heat transfer coefficients with respect to the other conditions is observed at the beginning of the cylindrical part because of the acceleration of fluid towards the exit, due to the observed low density values. In order to discuss the Nusselt profiles, depicted in Figure 11b, it is worth to highlight that the flow regime is fully turbulent, as exhibited by the Reynolds number profiles depicted by Figure 12. The maximum values are observed in a section slightly moved towards the convergent with respect to the throat because the dimension of the channel is comparable (hence, the heat-transfer coefficients and velocity) but the thermal conductivity begins to rapidly to decrease, as well as the density. In fact, a relative maximum of Reynolds is also detected at x/L = 0.56.

In fact, considering the Reynolds number profiles carried out by Figure 12, they tend to rapidly increase from the inlet section (where fluid is injected as a compressed liquid) towards the throat region, where thermophysical properties sharply change, the channel cross-section area attains the minimum value, and the heat load reaches its peak. Moreover, near-critical conditions are detected from half-convergent, and this means that viscosity decreases more significantly than density, as depicted in the following figures. Then, Reynolds number keeps very high, moving towards the cooling-jacket exit, where supercritical vapor conditions are observed. However, the maximum values of Reynolds numbers are detected for the condition, characterized by the highest fluid inlet temperature. If the Prandtl number profiles are observed, they tend to decrease from typical values, comparable to liquid in the inlet region, to a plateau, exhibited near the throat where critical temperature is on average reached, as shown by the aforementioned figures. The effect of decreasing the inlet temperature on Pr can be observed in the nozzle region, where values keep higher than the other considered cases as the fluid maintains “liquid-like” conditions for larger parts of the cooling jacket. At the beginning of the convergent part, the Prandtl number tends to slightly decrease, exhibiting values typical of gases.

Figure 13 summarizes the aforementioned results in terms of friction factor and Colburn number. Results for both nondimensional numbers remark the strong influences for the effect of entrance region (x/L = 1), the variation of hydraulic diameter along the cooling jacket, the channel roughness and the properties changing along the channel, which does not allow for the establishment of a real fully developed flow, as underlined in [37] since the fluid near the bottom surface of the channel tends to behave like a gas moving towards the exit.

The aforementioned behavior can be deepened by analyzing the axial profiles of the most significant thermophysical properties, such as density, viscosity, specific heat, and thermal conductivity, carried out by Figure 14. Density, viscosity, and thermal conductivity decrease monotonically at about one order of magnitude towards the outlet; and the lower the inlet fluid temperature, the greater the property value at a fixed section. In near-critical conditions, attained in the divergent-throat region, variations are very noticeable, especially from x/L equal to about 0.60. In particular, viscosity and thermal conductivity values reduce very fast in near-critical conditions; then, because the fluid is composed of a supercritical hot gas in the last part of the cooling-jacket channel, they keep almost constant, as reported in [30,31]. The highest values of specific heat, indicating the pseudocritical conditions, are detected at about x/L = 0.50–0.55. If the inlet temperature increases, peaks of specific capacity tend to move towards the throat section and reach for the highest values. Thus, pseudocritical conditions, which are to be taken into consideration to evaluate possible thermal deterioration modes, are attained at about x/L = 0.52 for the nominal point and x/L = 0.49 and 0.54 for T_in,f = 100 K (T_in,f* = 0.91) and 120 K (T_in,f* = 1.09), respectively.

Figure 15 depicts the dimensionless temperature fields, considering the nominal working conditions, and some transversal slices of interest are highlighted. In particular, seven significant sections are chosen, and their position corresponds to the outlet (x/L = 0.00), reattachment point region (x/L = 0.13), a section where the pseudocritical conditions inside the channel are observed on average (x/L = 0.52), the throat one (x/L = 0.61), a section representative of the divergent region (x/L = 0.75), and the inlet (x/L = 1.00). In this way, together with Figure 16, it is possible to appreciate the most stressed zones, which are concentrated in the throat section and in the cylindrical part of the thrust chamber, and more specifically, near the outlet section in correspondence of the reattachment point (x/L = 0.13). Temperature fields show lower temperature values for the close-out walls, made of copper and nickel, respectively, and located on the top of the jacket; and the gradual heating of the working fluid, flowing towards the cooling-jacket exit. Methane is injected as a compressed liquid (x/L = 1.00) and reaches the critical values in the throat region (x/L = 0.61) on average. However, it is possible to observe the transcritical conditions of methane at x/L = 0.75, where some parts of fluid near the bottom wall of the channel have already reached the critical temperature. Thus, a sort of thermal stratification is evident [20,21,30], as pointed out by Figure 16d: very high gradients are observed near the bottom surface of the channel since some parts of the fluid near the bottom walls as well as side walls tend to behave like a “gas”, while in the upper zones of the channel, where temperature values are lower, methane tends to remain in “liquid-like” conditions. From the throat region, larger parts of fluid begin to behave like a “vapor” and after x/L = 0.52, the fluid is fully composed by a supercritical gas because the temperature values overcome the pseudocritical one all over the fluid domain, as also confirmed by density axial profiles, plotted in Figure 14, and the dimensionless fields reported hereinafter in Figure 17. Fluid exhibits very low values of density and tends to accelerate towards the exit (in the cylindrical part of the thrust chamber, ρ* is almost one order of magnitude lower than the inlet section), partially compensating the effects of less favourable conditions for the heat exchange, given by thermophysical properties characterizing a gas in comparison with a liquid.

Figure 17 integrates the description regarding the behavior of methane inside the cooling jacket through the fields of dimensionless density, specific heat, thermal conductivity, and viscosity. From x/L = 0.61, the fluid core is not only characterized by decreasing values of density but specific heat tends to increase. The highest values in the throat section are attained near the channel surfaces, where relative values of thermal conductivity are also locally detected, while at x/L = 0.52, where pseudocritical conditions are detected, a sort of large bubble can be observed. In the middle of the channel the highest values of specific heat are detected and the whole fluid core is characterized by low thermal conductivity since methane can be considered behaving like a supercritical gas. The simultaneous presence of zones, characterized by low thermal conductivity and high specific heat, could determine a deterioration of the fluid thermal performances and consequent local overheating phenomena, which may be difficultly managed.

In fact, as reported in literature [30], a kind of “thermal barrier” between the fluid in proximity of the hot wall and the upper part of the channel may be observed. However, for the present cooling jacket, local wall-temperature profiles do not show relevant overheating, due to the wall roughness values defined in the design phase and fluid pressure conditions adopted, which are sufficiently far from the critical values [31,37]. As a result, temperature peaks, detected between the throat region and the convergent zone, are typical of LRE cooling jackets and definitely far from the maximum allowable values for CuCrZr materials, as pointed out by Figure 9. At the outlet section of the cooling jacket, methane exits as a supercritical vapor, and in fact, very low thermal conductivity and viscosity values, equal to about 0.3 and 0.2 times the inlet values, are observed.

6.2. Effects of Inlet Pressure and Mass-Flow-Rate Variations

In this subsection, the effects of inlet pressure and mass-flow variations with respect to nominal conditions are discussed. Figure 18a shows that the influence of inlet pressure is slight on wall-temperature results and thermal performances. It is possible to point out that the aforementioned variations in terms of inlet temperature of fluid provide more noticeable effects on thermal performances than variables like the inlet pressure, because the temperature margin on critical values is lower in percentage (for example T_in,f/T_cr equal to about 0.57 on average) while inlet pressure is about three times the critical value. The effects of mass-flow rate changes are more evident as expected and the liner temperature difference between the off-nominal conditions is at most 45 K, as observed in the reattachment region of the cylindrical part, while the fluid bulk temperature exhibits ±10 K at most in the outlet section. However, if the minimum allowed mass-flow rate is imposed the maximum temperature reaches about 645 K, a tolerable value for the thrust chamber under material thermomechanical resistance point of view, while in the throat section, a peak of 560 K is detected.

For what concerns the static pressure axial profiles, given in Figure 19a, the consequent pressure drops slightly increases if the inlet pressure is reduced, since there is a slight decrease in terms of critical-condition gain. This kind of behavior is also observed in Figure 19b if mass-flow rate is reduced. In this case, the fluid tends to be hotter than the other cases at a fixed section and the density decrease balances the lower mass-flow rate, and even pressure drops slightly increase, especially in the last part of the cylindrical section where methane is in supercritical vapor conditions.

Figure 20a and Figure 21a remark that the inlet-pressure changes have a slight impact on both convective heat-transfer coefficients and Nusselt number profiles. However, the maximum values are attained for P_in = 16.3 MPa (P_in* = 1.02) because of advantageous property conditions, since the critical conditions are slightly far away with respect to the nominal and low-pressure off-nominal points.

The effects of mass-flow rate variation are more relevant, as expected, and an increase of about 11% in terms of h_c,x is observed in the throat section (x/L = 0.61) if an increase of 5% in terms of methane mass, flowing in the cooling jacket, with respect to the nominal operating point. Differences in terms of heat-transfer coefficients are very significant, not only in the throat region but also from the half part of the convergent zone until the half part of the divergent one. If the cylindrical part is considered, differences become less evident moving towards the channel exit since the effects linked to mass are compensated by the increase in fluid temperature and density decrease. Thus, a difference of only 3% is detected if the case of m = 0.019 kg/s (m* = 0.95) is compared to the one at m = 0.021 kg/s (m* = 1.05).

If the Nusselt number axial profiles are considered, it is observed that peaks are detected at the same section of the nominal point and they correspond to an increase of 5.8% and a decrease of 6.6% if mass flow rate is equal to +5% and −5% with respect the nominal conditions. In the cylindrical part, the higher the mass flow rate, the higher Nu_x results, and variations of ±5% are detected in correspondence with the reattachment-point abscissa.

The discussion on thermophysical-property profiles along the axial coordinates allows us to justify the aforementioned results. If the pressure inlet changes, slight differences are observed in terms of density, viscosity, and thermal conductivity, as pointed out by Figure 22; while specific heat is the most affected property, and its values are higher if pressure decreases, as expected, moving towards the critical conditions.

If different values of inlet mass-flow rate are considered, density, viscosity, and thermal conductivity also show more remarkable variations, as depicted by Figure 23. The higher the mass-flow rate, the lower the fluid temperature, and consequently those properties exhibit the highest values if m = 0.021 kg/s (m* = 1.05). However, when the whole fluid domain behaves like a supercritical vapor (from about x/L = 0.30), the case with m = 0.019 kg/s (m* = 0.95) tends to show the highest viscosity and thermal conductivity average values with respect to the other cases, as expected for those physical conditions. Another effect of changing the inlet mass-flow rate is the shifting of the pseudocritical conditions inside the channel, which moves to about to x/L = 0.48 and 0.54 if m = 0.021 kg/s (m* = 1.05) and 0.019 kg/s (m* = 0.95) are considered, respectively.

7. Conclusions

The present paper describes the thermal and fluid-dynamics investigations, performed to support the development of a typical liquid-oxygen/liquid-methane rocket engine, belonging to the 30-kN thrust class and developed in the context of the HYPROB Program. The presented analyses allow an investigation on refrigerant behavior and the thermal response of the cooling jacket to combustion processes at high pressure, as well supporting verifications under the thermo-structural point of view, which are necessary to evaluate the lifecycle verifications before performing the firing tests.

Investigations were performed considering the nominal operating conditions, defined by the nominal set of inlet parameters, such as fluid temperature, pressure, and mass-flow rate (T_in,f = 110 K, P_in = 16.0 MPa, and m = 0.02 kg/s). Then, the effects of changes of such inlet variables with T_in,f, P_in, and m ranging from 0.91 to 1.09, 0.98 to 1.02, and 0.95 to 1.05 times the nominal inlet parameters, respectively, were also considered. In this way, an extended description of fluid and cooling-jacket behavior before firing tests can be accomplished. Results point out that temperature maxima are attained in the throat section, where the peak of heat flux is located (x/L = 0.61), and in correspondence with the reattachment point region (x/L = 0.13), where the coolant behaves like a supercritical vapor, due to poor thermal properties. Moreover, the coolant, injected as a compressed liquid, tends to reach the critical temperature values on average in the throat region, but some fractions of fluid, near the bottom parts of the cooling channel, are observed to be in near-critical conditions from the half nozzle (x/L = 0.75). Indeed, an evident thermal stratification inside the channel is observed near the bottom surface of the channel, and some parts of the fluid near the bottom and side walls tend to behave like a “gas”, while in the upper zones of the channel, where temperature values are lower, methane tends to remain in “liquid-like” conditions. Moreover, from that region, a dramatic decrease in terms of density, thermal conductivity, and viscosity is observed, as well as just downstream of the throat section, the Nusselt number reaches its peak in accordance with the Reynolds number relative maximum. The pseudocritical conditions are detected at about half a cooling jacket, and then, the fluid is expected to be completely in vapor conditions, as confirmed by the axial profiles and fields, given for the thermophysical properties. At x/L = 0.52, large fractions of the fluid exhibit very high values of specific heat and simultaneously also low values of thermal conductivity. However, no significant overheat phenomena, partly due to the choice of working at pressure levels three times greater than critical values, are observed, neither in that zone nor in other parts of the jacket, where largely tolerable liner temperature values are detected. Finally, methane exits as a supercritical vapor since density values are about one order of magnitude lower than inlet values (with a consequent increase in terms of pressure losses in the last part of the cooling system) as well as for thermal conductivity and viscosity.

If inlet conditions detach from nominal ones—as an effect of test-bench-scattered parameters, for example—the effects connected to changes in terms of fluid inlet temperature are more evident on thermal performances than variations of inlet pressure. The temperature margin on critical values is lower in percentage if compared with pressure ones, and this further influences the thermophysical properties. The effect on the liner-wall temperature results is significant, considering the domain from half of the nozzle region, and becomes more evident at the throat zone, where the differences between the nominal conditions and the other operating points may also attain 25 K because of slight changes in fluid thermophysical properties. For what concerns mass-flow-rate changes, if some imbalance among the channels could happen as a result of a not-optimal filling, their influence is evident and the liner-temperature difference between the off-nominal conditions is at most 45 K, as observed in the reattachment region of the cylindrical part, while the fluid bulk temperature exhibits ±10 K at most in the outlet section. An increase of about 11% in terms of convective heat-transfer coefficient in the throat section (and about 5% in terms of Nusselt number) can be observed if an increase of 5% in terms of methane mass with respect to nominal flow rate is considered. Slight variations are observed in the cylindrical part instead. Finally, the location, where pseudocritical conditions are detected, tends to move upstream if mass-flow rate decreases.