Comparison Between Numerical-and Experimental Methodologies for Total Enthalpy Determination in Scirocco PWT

4. Enthalpy Measurement and Rebuilding

As previously mentioned, when discussing the enthalpy of the test flow, reference is generally made to two types of enthalpies: mass-averaged enthalpy and centerline (core) enthalpy. For the estimation of mass-averaged enthalpy, the sonic throat method and the energy balance method are used, while for the centerline total enthalpy determination the heat transfer method is employed, which represent the standard characterization method in Scirocco. Details regarding the methodologies and the uncertainty estimation will be provided below.

4.1. Mass-Averaged Enthalpy Measurement

4.1.1. Sonic throat method

The sonic throat method, is based on the relation between the mass-averaged total enthalpy and the cross-sectional area of the throat in m², the total pressure in the arc heater in Pa and the air mass flow rate injected in the column in kg/s. The assumption of isentropic expansion under thermodynamic equilibrium is explicit in the formulation of the sonic flow method, and a correction for frozen species is applied above enthalpy levels by Winovich over the enthalpy range between 1000 and 10,000 Btu/lbm (equivalently 2.3 to 23 MJ/kg). The equation is:

$$H_{st}^0={\left(0.293\cdot A\cdot {\displaystyle \frac{P^0}{\dot{m}}}\right)}^{2.519}$$

(1)

Despite these limitations, only simple state measurements are required for this method.

The instruments used in the measurements are an absolute pressure transducer and an air flow meter, listed in Table 2, along with other instruments and the uncertainties specified by the manufacturers.

As calculated, the total instrumental percentage uncertainty is $U_{H^0}=\pm 0.253\%$. However. Furthermore, a linear interpolation of the data has been calculated. The available data do not fit well with a linear regression model given the instrumental uncertainty of only 0.253%. In order to fit the available data with a linear regression, the average percentage error relative to the regression line is approximately 8.86%. It is important to note that such models are designed to provide an approximate estimate of the mean enthalpy value; however, they are presented here for the sake of completeness.

4.1.2. Energy Balance Method

The indirect measurement of the mass-averaged total enthalpy ($H_{eb}^0$) of the Scirocco plasma flow is performed through the Energy Balance in a relatively simple way: by dividing the net power input by the mass flow rate in a so-called Energy Balance Method [3]. In this case, the $H^0$is measured by dividing the total power input into the flow by the total mass flow rate of the test gas. The total power input is the electrical power provided to the electrodes minus the power losses in the cooling circuits. The relation that explains this methodology is reported below:

$$H_{eb}^0={\displaystyle \frac{V\cdot I-\Delta T^{\mathrm{arc}}\cdot {\dot{m}}_{H_{2}O}^{\mathrm{arc}}}{\left({\dot{m}}_{\mathrm{AIR}}+{\dot{m}}_{Ar}\right)}}$$

(2)

Below is a list of the quantities involved:

• $V$ is the voltage of the Power Supply System between anode and cathode bar, V;
• $I$ is the electrical current of the Power Supply System of the anode bar, A;
• ${\dot{m}}_{H_{2}O}^{\mathrm{arc}}$ is the water flow rate of the arc heater cooling system, (m³/h);
• $\Delta T^{arc}$ is the delta-temperature of the water-flow of arc heater cooling system, (K);
• ${\dot{m}}_{\mathrm{AIRC}}$ is the air mass-flow rate of the arc heater complex, (kg/s);
• ${\dot{m}}_{Ar}$ is the argon mass-flow rate of the arc heater complex, (kg/s);

Figure 1 should help to clarify the measurement scheme:

The explicit formula for calculating the uncertainty via error propagation is not reported here for brevity, and overall uncertainty is approximately ±10.2% of the value of H₀. For this level of error, the data fit a linear regression. This is a high value and the greatest weight on this result is determined by the differential thermopile for the measurement of $\mathsf{\Delta }T$(82.93% of the uncertain weight). The strong dependence on the temperature difference measured in the cooling water represents the main limitation of this methodology.

4.2. Centerline Enthalpy Measurement - Heat Transfer Method

In the Scirocco facility, the standard methodology for directly measuring enthalpy through thermal flux is the heat transfer method, which correlates the stagnation-point heat flux measured on the probe with the flow enthalpy.

The experimental setup is sketched in Figure 2. The distance from the nozzle throat to the probe position was x=2.8 m.

The stagnation heat flux and pressure are measured using a water-cooled copper probe (Figure 3) of diameter D=10 cm injected at a distance of $x=37.5$ cm from the nozzle exit section in correspondence with the jet centerline. In the case of the Scirocco system the stagnation enthalpy in centerline of the flow is measured using a Gardon Gauge from Medtherm [27] model GT-100-8-658/756A, with an operating range of 0-3000 kW/m² and expanded uncertainty of ±3%, which provides a "direct" measurement of the heat flux at the stagnation point, of course affected greatly by the possible copper surface catalytic recombination of oxygen and nitrogen atoms. The basic Gardon Gauge design is based on a differential copper-constantan type thermocouple. A thin constantan disk membrane is soldered to a body made of copper. The copper part is embedded into the copper probe and cooled through forced convection of high-pressure water.

Table 3. Heat transfer constant for gases mixture [22].

Gas	${\mathit{K}}_{\mathit{i}}$
	lbm	g
Air	0.0461	0.1235
Argon	0.0651	0.1744

Table 3. Heat transfer constant for gases mixture [22].

Gas	${\mathit{K}}_{\mathit{i}}$
	lbm	g
Air	0.0461	0.1235
Argon	0.0651	0.1744

The Gardon gauges are equipped with control thermocouples that measure their operating temperature, which typically averages around 370 K. This value will be used as a wall boundary condition in CFD calculations. The Enthalpy is calculated from the cold wall heat flux using the Fay-Riddle formulation, in the semi empirical and simplified formulation given by Zoby [26,27], for a mixture of air and argon:

$$H_c^0={\displaystyle \raisebox{1ex}{$q_s$}\!\left/ \!\raisebox{-1ex}{$\dot{m}$}\right.}\sqrt{{\displaystyle \raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$P^0$}\right.}}\left({\displaystyle \raisebox{1ex}{${\dot{m}}_{Ar}$}\!\left/ \!\raisebox{-1ex}{$K_{Ar}$}\right.}+{\displaystyle \raisebox{1ex}{${\dot{m}}_{\mathrm{AIR}}$}\!\left/ \!\raisebox{-1ex}{$K_{\mathrm{AIR}}$}\right.}\right)$$

(3)

In this formulation, the error is essentially dominated by the uncertainty in the measurement of the stagnation heat flux $q_s$, thus leading to total uncertainty of about ±3.02% [27,28].

The Fay-Riddell model, as is well known, assumes either equilibrium or fully catalytic conditions, meaning that the gas composition near the surface adjusts to the local thermodynamic state or promotes complete recombination of dissociated species, thereby increasing the heat flux due to exothermic reactions. In contrast, Goulard [21] provides an alternative relation between stagnation point heat flux and enthalpy by assuming frozen conditions, where the gas composition remains unchanged as it approaches the surface, with an assigned catalytic recombination coefficient $\gamma$, leading to a lower heat flux. As demonstrated by Park [15], in his application, the two formulations converge for a $\gamma$ value of 0.04 for pure copper and oxygen recombination, at which point both models yield the same result.

4.3 Centerline Enthalpy Measurement – CFD rebuiling

The in-house developed CFD solver, NExT, has been used to numerically simulate the Scirocco tests. This CFD tool solves the Reynolds Averaged Navier–Stokes (RANS) equations on a multi-block structured grid [29], adopting a density-based finite volume approach, with a cell centered Flux Difference Splitting (FDS) upwind [30] scheme for the convective terms. Second order formulation is obtained by means of an Essentially Non-Oscillatory (ENO) reconstruction of interface values. Chemical non-equilibrium [31] is accounted for by solving the conservation equations for the mass fractions of gas species. Dissociating air is modelled with a 5-species gas mixture (including O, N, NO, O₂, and N₂) with chemical reaction rates provided by Park [20]. The translational and rotational energy of the gas mixture is governed by a single temperature, $T$; the energy exchange between vibrational and translational modes (T_V) is instead modelled with the classical Landau-Teller non-equilibrium equation, with average relaxation times taken from the Millikan-White [32], theory modified by Park [33]. Also, the Park high-temperature limit [18] is used to prevent the relaxation from becoming faster than the collision time. The viscosity of the single species is evaluated by a fit of collision integrals calculated by Yun and Mason [34], the thermal conductivity is calculated by means of Eucken’s law, the viscosity and thermal conductivity of the gas mixture are then calculated by using the semi-empirical Wilke formulas. Finally, the diffusion of the multicomponent gas is computed through a sum rule of the binary diffusivity of each couple of species (from the tabulated collision integrals of Yun and Mason [34]). A short description of the Scirocco test rebuilding chain follows. The CFD rebuilding, Figure 4, of the selected Scirocco tests begins with the numerical evaluation of the reservoir conditions with equilibrium air calculations, imposing the total pressure and enthalpy ($P^0$, $H^0$) measured for each test: those give the inlet nozzle conditions. Starting from the inlet values as boundary conditions, an axis-symmetrical Navier–Stokes simulation of convergent-divergent nozzle is conducted to obtain the test section. The simulations are iterated by refining the estimation of $H^0$ and $P^0$, until the numerical evaluation of the $p_s$ and $q_s$ on the probe matches the experimental measurements. The probe stagnation centerline enthalpy, in turn, coincides with the core enthalpy used for the comparison experimental values. The average enthalpy value is instead obtained by averaging the enthalpy across the nozzle exit section, once the stagnation conditions are matched.

The Scirocco rebuilding simulations have been performed by using the following main physical/numerical models:

• 2D-axi RANS approach (CIRA NExT solver)
• Time marching to steady state solution strategy
• 2° order Upwind Flux Difference Splitting convective scheme
• 5-species air in thermal and chemical non-equilibrium as working gas
• Fixed temperature (T=370K) nozzle wall boundary condition
• Fixed temperature (T=370K) fully catalytic wall boundary condition for the calibration probe.

A 2D-axisymmetric domain including the nozzle, the test chamber and the calibration probe has been modelled and discretized by 34 blocks and about 110.000 cells structured multi-block mesh (on the finer level), as depicted in Figure 4. The simulations were initially conducted on a coarse grid (approximately 7,000 cells) and a medium grid (around 30,000 cells) to verify grid convergence of the solution, though the results are omitted here for brevity.

Of course, the dependence on flow solution from the grid is not the only source of uncertainty. According to a previous study carried out by Cinquegrana [25] the choice of a specific transport and chemical-kinetic model is affected by an uncertainty, which can be estimated to be below the 2% on the stagnation heat flux value. However, the level of uncertainty that allows the data to fit a linear curve is 5%. Another source of uncertainty is how the catalytic properties of the calibration probe are modelled, as it depends on the actual surface finish of the calibration probe's material (copper), adding further uncertainties to the simulation. The initial assumption is a fully catalytic response, based on the premise of a mirror-polished probe surface, in this work cases with partial catalytic behavior will also be considered.

At this time, it is worth to point out how the wall heat flux is calculated. Total heat flux is given by three terms, the convective one $q_c$ after named as ‘roto-translational’, since it is due to the roto-translational temperatures, the vibrational $q_v$ contribution, the diffusive $q_d$ contribution, after named as ‘chemical’, and the radiative $q_r$ contribution as reported in equation 4:

$$q=q_c+q_v+q_d$$

(4)

The ‘roto-translational’ wall heat flux contribution is reported in equation 5:

$$q_c=-k{\displaystyle \frac{\partial T}{\partial n}}$$

(5)

where $k$ is the mixture thermal conductivity. The vibrational contribution to the total wall heat fluxes is valued as reported in equation 6:

$$q_v=\sum _{i=1}^{N_{\mathrm{vib}}}{k}_{\nu ,i}{\displaystyle \frac{\partial T_{\mathsf{\nu }\mathrm{ib},i}}{\partial n}}$$

(6)

where $N_{\mathrm{vib}}$ are the number of chemical species with the vibrational energy mode, $k_{v,i}$ the vibrational conductivity of bi-atomic species, $T_{\mathrm{vib},i}$ the vibrational temperature of the i-th vibrational species. Finally, the diffusive, or ‘chemical’ contribution is valued as explained in equation 7:

$$q_d=\sum _{i=1}^{N_s}{h}_i\rho D_i{\displaystyle \frac{\partial Y_i}{\partial n}}$$

(7)

In NEXT code, it is possible to model partial catalysis of material by means of the recombination probability $\gamma$, defined as the ratio between the flux of atoms that recombine at surface [35], $\left|M_A^{\downarrow }\right|$, with the total flux of atoms impinging the surface,$\left|M_A\right|$[3636]. It can be shown that the expression for the mass flux of atoms impinging on the surface follows the kinetic theory, and after some elaborations and according to Scott [37], can be used to derive:

$$\gamma ={\displaystyle \frac{\left|M_A\right|}{\left|M_A^{\downarrow }\right|}}={\displaystyle \frac{\left|M_A\right|}{Y_{A,w}{\rho }_w\sqrt{\frac{R_A{T}_w}{2\pi }}}}$$

(8)

In this case, the involved paths are two independent reactions, $O+O\to O_2$ and $N+N\to N_2$, each with an associated efficiency ${\gamma }_O$ and ${\gamma }_N$.

Under the very conservative assumption of a fully catalytic wall, it is possible to achieve the same wall heat flux with a lower total enthalpy. Conversely, if the wall is assumed to be partially catalytic, the match between experimental results and CFD simulations is obtained at higher total enthalpy values. To align the enthalpy values estimated via CFD with those determined through the inverse enthalpy estimation using the heat transfer method, a gamma value of 0.08 was applied. This value falls within the range suggested in the literature for slightly oxidized copper, as discussed in further detail below

5. Results and Discussion

The total enthalpy was determined using several methodologies. Figure 5 illustrates the results obtained from the experiment and the simulations, providing a comprehensive comparison of the methodologies. The data demonstrate the relationship between $H^0$ and $q_s$, highlighting the linear trends. Error bars represent the uncertainties in the measurements, which were calculated as previously explained. The CFD simulations showed the better agreement with the total enthalpy estimates obtained using the Zoby formula, suggesting that this approach can be considered reliable for the reconstruction of the thermodynamic parameters in the Scirocco PWT, but only by taking into accounts for a more realistic the catalytic effect on the surface, highlighting, the importance of considering the probe oxidation and confirming the literature findings. In the case of complete catalytic activity of the copper surface, the enthalpy is, in fact, underestimated by approximately 10%.

5.1. Mass averaged Enthalpy Measurement, and profile uniformity chacaterization

Regarding mass-averaged enthalpy methods, it was observed that both the Winowich formulas and the energy balance approach, when applied to the Scirocco facility, tend to underestimate the average flow enthalpy compared to values determined by CFD under the hypothesis of finite-rate catalyticity. This discrepancy, may be attributed to simplifying assumptions that fail to fully capture the complexity of the high-enthalpy flow generated, as well as the intrinsic limitations of the formulas themselves. These formulas are based on semiempirical data tailored to the geometry of the NASA-Ames IHF, which features a more radially shaped flow [15,16,17] profile due to a differently shaped nozzle. Consequently, they could not directly be applicable to other similar facilities.

Based on spectroscopically determined centerline enthalpy, Park [15] defines the ratio of centerline-to-average enthalpies as the centerline enthalpy determined via the spectroscopic method divided by that obtained from the sonic throat method or energy balance method, yielding a ratio of 1.4 to 1.5. Due to the higher non-uniformity in the radial flow profile, based on the plasma core profile reconstruction, the author suggests that it is reasonable for the average enthalpy to be 40% to 50% lower than the centerline value in the IHF facility [15]. In contrast, the Scirocco facility exhibits a more uniform flow, as confirmed by CFD analysis (Figure 5), spectroscopic measurements (Figure 6), and spatially resolved probe measurements (Figure 7). Based on CFD simulations, the ratio between the centerline enthalpy and the mass-averaged enthalpy along the radial profile of the flow at the nozzle exit section for the SCIROCCO facility was calculated to be 1.2. This implies that average enthalpy is expected to be 20% lower than the centerline, as confirmed by experimental data.

To furtherly confirm jet uniformity, a high-speed CMOS camera (Phantom v4.0, B&W, 8-bit, 512x512 pixels) was employed, capturing 2000 frames at a sampling rate of 1071 Hz with an exposure time of 870μs. The maximum error was ± 5% of the full scale. It collects the radiation in the VIS and near IR spectral range corresponding to the spectral range for which the plasma emission is significant since it is transparent in the infrared spectral range as indicated in [38]. Notice that the plasma jet presents a diameter of about one meters while the depth of field of the camera corresponds to few centimeters. For this reason, no Abel inversion was needed as reported in [39,40]. In Figure 8, transversal profiles extracted from the mean radiation distribution of the free-stream plasma flow are shown. The excellent uniformity of the flow at the exit section of the nozzle in the Scirocco Wind Tunnel is influenced by several factors, including: the low half-angle and long length of the nozzle, which enable a very gradual expansion; a smooth, well-shaped nozzle throat, with profiles designed to maintain continuity with the linear section up to the second derivative; and a highly uniform and stable exit pressure, achieved through a unique and powerful vacuum system. The correlation between the data suggests that it would be possible to correct the systematic error to use Winovich's formulation in a way that aligns more closely with CFD calculations, even though the CFD results remain within the uncertainty of the formulation. The determination of average enthalpy using the energy balance method presents several limitations, particularly due to its strong dependence on the accuracy of cooling water temperature, as discussed in Section 4.1.2.

Moreover, the energy balance method presents a significant limitation in the difficulty of achieving thermal steady-state in the facility. Terms associated with electrical power dissipation due to the Joule effect have much faster time constants, akin to the rise time of a first-order system, compared to those related to water cooling. This discrepancy can lead to inaccurate enthalpy estimates and prevent the application of this methodology in short-duration tests, where the thermal steady-state of the entire facility is not achieved.

The energy balance method showed a discrepancy of up to 30% in underestimating the enthalpy value, while the sonic throat method provided a closer estimate to the CFD values, with a difference of 13% in underestimating.

5.2. Centerline Enthalpy Measurement, and surface–catalytic recombination coefficient estimation

In the characterization of arc-jet tunnels, the standard procedure used in Scirocco involves determining the centerline enthalpy using a copper calorimeter, which is recognized for its high catalytic efficiency, as documented in the literature. The local flow enthalpy is calculated through Zoby heat transfer formula given by Park [15], utilizing the heat transfer rates measured from the calorimeter alongside the assumption of full catalytic efficiency conditions on the wall.

The catalytic phenomena on surfaces significantly affect heat transfer, particularly in the context of total enthalpy determination, as shown in Figure 9. It was found that the enthalpy exhibits the same trend observed by Park [15], showing high sensitivity to the chosen γ value within the relevant range

A substantial amount of experimental data has been collected regarding the catalytic effects of oxygen and nitrogen for copper, primarily using arc-jet wind tunnels, along with side-arm reactors [41], diffusion tubes andarc-jets [42], and Plasmatron [43]. This last device enables high chemical purity in plasma flows, providing high valuable data for studying catalytic processes under such conditions. Various studies have measured the catalytic efficiency of copper oxide in different test facilities, as summarized by Chug [24].

Cheung et al. [24]summarizes the existing data of catalytic efficiency for copper (Cu) and copper oxide. The copper can be oxidized in two different types of copper oxide: cupric oxide (CuO) and cuprous oxide (Cu₂O). The author reports these values: ${\gamma }_{\mathrm{Cu}}$ ranging from 0.01 to 0.17, ${\gamma }_{\mathrm{CuO}}$ ranging from 0.01 to 0.045 and ${\gamma }_{{\mathrm{Cu}}_{2}O}$ ranging from 0.025 to 0.11, for different gas mixture, enthalpy level, wall temperature and facilities. The efficiency of copper oxide has not been measured in a well-defined environment, where flow enthalpy is independently known, the material is unequivocally copper oxide, and the surface is smooth.

Viladegut and Chazot [43] present an experimental approach to develop a catalytic model for surfaces exposed to high-temperature, low-pressure plasma flow conditions. This technique, known as the Mini-Max method, is applied at the Plasmatron facility at VKI to determine the catalytic recombination coefficient on a copper-based calorimeter. Their findings reveal a notable decrease in the recombination coefficient at elevated pressures, while it remains relatively stable across the range of power levels tested. They indicate a range for γ gamma between 0.088 and 0.026 for a copper probe, with a stagnation pressure range between 15 and 50 mbar, in good agreement with the analyzed cases for Scirocco.

In the case analyzed in this study, matching the CFD estimations with calorimetric measurements led to an estimation of γ = 0.08 for both nitrogen and oxygen recombination, as discussed in Section 4.2. This value agrees with Pope’s [44] estimation for copper and nitrogen (N₂) recombination, measured at a cold wall temperature of 350 K in an arc-jet facility.