Experimental Investigation of Volume Estimation Methods for Hydrogen Refueling Stations

Refueling step

Yamaguchi et al. (2018) point out that for volumes larger than 100 L, 200 g of hydrogen injected might not be enough to perform a precise volume estimation during the startup time.

They suggested a method for the HRS to perform a CHSS volume estimation after the startup time to respect the 200 g injected mass criterion from SAE J2601 (SAE International, 2020). The SAE J2601-5 standard (SAE International, 2024a) adapted the work of Yamaguchi et al. (2018) to suggest a similar method.

If we consider a simplified HRS piping and instrumentation diagram (P&ID), shown in Figure 1, a control valve is located upstream of the dispenser. The piping system located between the dispenser and the CHSS (breakaway, hose, nozzle, etc.) is assimilated into an equivalent virtual valve in terms of head losses. Lastly, the CHSS volume is considered a single tank to simplify the following development.

Figure 1

At the dispenser, the pressure P_disp (MPa (g)), temperature T_disp (°C) and mass flow rate ${\stackrel{\cdot }{m}}_{\mathit{\text{disp}}}$ (kg/s) of the refueling flow are measured continuously. Likewise, the ambient temperature T_amb (°C) is measured by the HRS.

After the connection pulse, the control valve is closed, and the pressure of the gas in the dispenser P_disp (MPa) equalizes the pressure of the gas in the tank P_tank (MPa). This moment is noted t₀ (s), then,

To estimate a tank volume, the intent is to raise the pressure in the tank. The tank pressure elevation ΔP_tank (MPa) must be sufficient to be accurately measured by a pressure sensor. In the SAE J2601-5 (SAE International, 2024a), a pressure elevation of 5 MPa is targeted.

To get a pressure elevation ΔP_tank in the tank, a refueling step is required. To ensure safety, the refueling uses a conservative APRR.

Due to the head losses between the dispenser and the tank, the pressure in the tank is lower than the pressure at the dispenser. Consequently, to reach a targeted pressure elevation ΔP_tank [MPa] in the tank, a pressure overshoot ΔP_disp (MPa) is targeted at the dispenser.

In the SAE J2601-5 (SAE International, 2024a), this overshoot is expressed as a function of the initial pressure P_tank(t₀) (MPa), such as for a refueling up to 700 bar (H70),

In the study, only H70 refueling will be considered. Accordingly, equation (2) will be employed to calculate the dispenser overshoot pressure elevation.

The refueling step ends when the dispenser reaches the pressure P_disp(t₀) + ΔP_disp. Then, the pressure in the dispenser drops and equalizes the pressure in the tank. This moment is noted t₁ [s], then

and finally,

The injected mass Δm (kg) in the tank is obtained via a numerical integration of the mass flow rate ${\stackrel{\cdot }{m}}_{\mathit{\text{disp}}}$ (kg/s) measured at the dispenser,

These refueling steps must be considered independently of the volume estimation calculation. A primary purpose of this study is to verify that the refueling step suggested by the SAE J2601-5 allows for an exploitable pressure tank elevation ΔP_tank and injected mass Δm.

Hence, once a pressure tank elevation ΔP_tank and injected mass Δm are obtained, the method to calculate the tank volume V (m³) differs depending on how the temperature in the tank is calculated.

Volume estimation step

Let us consider the hydrogen in the tank as a uniform region, that is, 0-dimensional (0D) modeling. The mass in the tank can be expressed as

where ρ (kg/m³) is the density of hydrogen and T_tank (°C) is the temperature inside the tank. During the refueling step, the density and the mass vary, but the volume remains constant. This leads to

Then the evaluation of the volume depends on the evaluation of T_tank(t₀) and T_tank(t₁), which are unknown in the absence of communication between the HRS and the FCEV.

For Yamaguchi et al. (2018) and in the standard SAE J2601-5 (SAE International, 2024a), the initial tank temperature T_tank(t₀) is considered equal to the ambient temperature, that is,

This assumption is also made in this study. However, additional studies could be carried out to investigate the impact of the initial gas temperature estimation on the volume estimation, using, for instance, the hot/cold soak concept from the SAE J2601 (SAE International, 2020).

Due to the pressure elevation inside the tank, the temperature also increases, that is, T_tank (t₁) > T_tank(t₀). However, an estimation of T_tank(t₁) is dependent on several factors such as the tank geometry (aspect ratio) and materials (tank type III, IV, etc.). It also depends on the refueling time, which can vary largely according to the ambient temperature and the precooling capacity of the HRS.

A secondary purpose of this study is to introduce different volume estimation formulations constructed from different physical hypotheses. From experimental data of refueling using different initial pressures and fuel delivery temperatures, the volume calculated from these formulations will be compared and discussed.

SAE J2601-5 formulation

The volume estimation formulation defined in the standard SAE J2601-5 (SAE International, 2024a) introduces measurement accuracy for the dispenser pressure sensor P_tol (MPa) and for the mass flow rate sensor m_tol (%).

Then, the initial gas density is calculated as

The final gas density is calculated using a temperature elevation ΔT_SAE (°C) in the tank, a function of the initial pressure P_tank(t₀) (MPa). Hence, for H70 refueling,

Then,

The CHSS volume is calculated as

It must be noted that in the SAE, the dispenser pressure tolerance and mass flow rate measurement tolerance are used to overestimate the volume. This is a conservative approach, that is, it prevents over-temperature events in the tank for SAE refueling protocols. Hence, estimating a larger CHSS volume implies a slower refueling ramp and final pressure, thus a smaller temperature elevation inside the tank.

However, this approach can be non-conservative regarding overfilling when refueling control is based on the mass flow rate. For instance, if we consider calculating the SOC for a given nominal working pressure (NWP), P_NWP (MPa) and using the mass flow rate,

An overestimated CHSS volume leads to an underestimated SOC. If the refueling control is based on this SOC, a tank overfilling event could occur. For this application, the conservative approach is to underestimate the CHSS volume. Consequently, a conservative approach depends on the event to prevent, for example, overfilling, over-temperature, and the refueling control chosen, for example, pressure ramp rate or mass flow rate.

Isothermal formulation

A simple formulation can be obtained by neglecting the temperature elevation inside the tank, that is, T_tank(t₁) = T_tank(t₀).

This hypothesis implies that the heat transferred from the gas to the walls is sufficient to maintain the temperature almost constant. This can correspond to a very slow refueling.

Because the temperature elevation inside the tank is underestimated, this formulation tends to underestimate the CHSS volume. Using equation (7), the volume can be calculated directly.

Using the first-order (FO) approximation of the mass variation from equation (12), it can be shown that

with $\frac{\partial \rho }{\mathit{\text{∂P}}} {|}_T$ evaluated at the initial time t₀.

The discrepancy between equations (18) and (19) is interesting to evaluate the error made by the first-order approximation of the mass variation. It should be noted that the subsequent adiabatic and diabatic formulations will also utilize a first-order approximation, but the suffix ‘FO’ will be omitted from the method names for simplicity.

Adiabatic formulation

Another formulation can be obtained by neglecting the heat transferred from the gas to the walls. This implies k_gS_w,int(T_g,w – T) = 0. This hypothesis can correspond to a very fast refueling.

The impact of the kinetic energy $\frac{u_{\mathit{\text{inj}}}^2}{2}$ on the gas temperature is considered negligible for a short time period.

The enthalpy injected in the tank h(P, T_inj) must be evaluated. A hot dispenser hypothesis is assumed, that is, h(P, T_inj) = h(P, T_amb).

These hypotheses tend to overestimate the temperature elevation inside the tank and, consequently, the CHSS volume. From equations (10) and (12), it can be shown that

with $\frac{1}{\left(\frac{\partial \rho }{\mathit{\text{∂P}}}{|}_T+\frac{\partial \rho }{\partial T}{|}_P\left(\frac{\mathit{\text{βT}}}{\rho c_p}\right)\right)}$ evaluated at the initial time t₀.

Diabatic formulation

During the refueling step, the gas inside the tank exchanges heat with the walls. This phenomenon is influenced by the refueling time. Moreover, according to the ambient temperature, the refueling step can vary largely.

For instance, let us consider a refueling with no communication and a fuel delivery temperature of –30°C. The initial measured tank pressure is P_tank = 2 MPa(g). This implies, using equation (2), that ΔP_disp = 12.33 MPa.

According to the conservative table G4 from SAE J2601-5 (SAE International, 2024a), at T_amb = 0°C, the APRR = 22 MPa/min and at T_amb = 30°C, the APRR = 7.5 MPa/min. Then, at T_amb = 0°C, a refueling step takes approximately 30 s while at T_amb = 30°C, it takes approximately 100 s.

Accordingly, a volume estimation formulation should take into account the refueling time of the refueling step.

Estimating the heat transferred from the gas to the walls is challenging. This could be performed with a 0D modeling simulation but requires knowing the geometry and the materials of the tank and the piping. Due to the absence of such information, several assumptions have to be made. Notably, the CHSS volume is an input for a 0D modeling simulation.

To address this challenge, two methods are developed in this section. The first one is a direct method, comparable with the isothermal and adiabatic formulations. The CHSS volume is directly calculated from the data collected at times t₀ and t₁. Despite requiring approximations, this method is suitable to be implemented into the programmable logic controller (PLC) of HRS.

The second method is an iterative approach. It uses a dichotomy algorithm along with a 0D modeling simulation and an initial CHSS volume guess to converge toward a precise CHSS volume.

Direct method

Like the adiabatic method, the effect of the kinetic energy $\frac{u_{\mathit{\text{inj}}}^2}{2}$ on the gas temperature is neglected. The injected gas temperature T_inj is taken as the mass-averaged temperature from the dispenser temperature:

This corresponds to a cold dispenser. From the energy balance equation (10), it can be shown that

The quantity (T – T_g,w) is approximated by

The quantity $\frac{S_{\mathit{\text{w,int}}}}{V}$ depends on the aspect ratio of the tank. The quantity k_g depends on several parameters (Woodfield et al., 2007) (e.g., the tank and injector geometry, the injection velocity, the buoyancy forces, etc.). The product $\frac{S_{\mathit{\text{w,int}}}}{V}{k}_g$ is here considered constant during the refueling step and noted Λ (W/m³/K), that is, $\Lambda =k_g \frac{S_{\mathit{\text{w,int}}}}{V}$. This leads to

Another form of equation (11) is

By substituting equation (24) into equation (25), it can be shown that

Considering $\frac{\mathit{\text{dm}}}{\mathit{\text{dt}}}\approx \frac{\mathit{\text{Δm}}}{\mathit{\text{Δt}}}$, it can be shown that,

Then, considering all thermophysical properties calculated at t₀, it can be shown that

It must be noted that if Δt → 0, then V_dia → V_adia, and if Δt → +∞, then V_dia → V_iso. The parameter Λ controls the transition rate between the two formulas. Depending on whether an overestimation or an underestimation of the CHSS volume is needed, the parameter Λ shall be adjusted.

For example, a configuration that leads to an underestimate of the heat transferred from the gas to the walls tends to overestimate the CHSS volume. This configuration is selected in this study by using the smallest surface-to-volume ratio from tank geometry assumptions from SAE J2601-5 (SAE International, 2024a), $\frac{S_{\mathit{\text{w,int}}}}{V}=6.737$ m^–1. A limited heat transfer coefficient is selected, likewise the one found for the slow refueling case issued from Monde et al. (2012), k_g = 45 W/m²/K. Finally, Λ = 303 W/m³/K.

The flowchart of the diabatic formulation is detailed in Figure 2.

Figure 2

Direct method approximated

The diabatic volume estimation is interesting for practical applications because its direct evaluation requires a limited number of operations. A PLC shall have the computation resource to perform a calculation.

However, equation (28) requires hydrogen thermophysical properties to be evaluated. To allow rapid evaluation of equation (28), a polynomial approximation method is developed.

Let us consider the approximated diabatic formula V_dia–approx (m³).

Let us define the effective volumetric thermal capacity C_cp–eff (J/m³/K):

the non-dimensional time τ (no unit):

and a transition function F:

It can be shown that

considering

The coefficient C_iso(P, T), C_adia(P, T, T_inj) and C_cp–eff(P, T) can be evaluated using polynomial functions. The polynomial regression is performed for T ∈ [–40°C, 85°C], T_inj ∈ [–40°C, 50°C] and P ∈ [0.5 MPa(g), 87.5 MPa(g)]. Table 2 presents the coefficients for fifth-order polynomial functions. The coefficient is associated with the polynomial term (e.g., 1, T_inj, T_inj², etc).

Iterative method

If the PLC has sufficient computational resources to perform a 0D modeling simulation, an iterative approach can be performed to increase the precision.

A 0D modeling simulation is solving the mass and energy equations (9) and (10) using as inputs data from the dispenser, and producing, as outputs, the pressure and temperature inside the tank. As previously stated, it requires knowing the geometry and the materials of the tank and the piping.

In this study, a dichotomy approach is selected. The isothermal method estimates a minimum CHSS volume, while the adiabatic method estimates a maximum CHSS volume. This sets the CHSS volume interval in which the dichotomy method searches for an optimal volume. The initial guess is taken as the middle of this interval.

For a given CHSS volume, the 0D modeling simulation is calculating a pressure inside the tank P_tank–calc(t) as a function of time. At time t₁, the calculated pressure P_tank–calc(t₁) is compared to the measured pressure P_tank(t₁). (i) If superior, this means the selected CHSS volume is underestimated. The search interval is then reduced using the selected CHSS volume as the new lower bound. (ii) If inferior, this means the selected CHSS volume is overestimated. The search interval is then reduced using the selected CHSS volume as the new upper bound. Figure 3 shows the flowchart of the method.

Figure 3

According to the selected geometry and materials of the tank and the piping (e.g., hot case or cold case), the CHSS volume will tend to be overestimated or underestimated, respectively.

In this study, the influence of the tank and the dispenser is investigated by selecting:

1. A hot case tank and a cold dispenser V_{0D–HotCaseCD} constructed from the tank from SAE J2601-5 (SAE International, 2024a) (geometry from Table A7, material properties from Table A11) and assuming h_inj = h(P_disp, T_disp).

2. A hot case tank and a hot dispenser V_{0D–HotCaseHD} constructed from the same previous tank and a dispenser from SAE J2601-5 (SAE International, 2024a) (piping geometry and materials from Tables A17–A21).

3. A cold case tank and a cold dispenser V_{0D–ColdCaseCD} constructed from the tank from SAE J2601 (SAE International, 2020) (geometry from Table A3, 70 MPa, Type III) and assuming h_inj = h(P_disp, T_disp).

Type IV 243.6 L tank

The NWP of the 243.6 L tank allows investigating high initial pressures, and eight experimental refueling scenarios are considered involving different initial pressures (≈3, 11, 20, 31 MPa(g)) and different delivery temperatures (≈–20°C, –30°C). Note that due to the thermal inertia of the precooling system, the initial dispenser temperature at the very beginning of the injection is typically higher (e.g., around –10°C) before rapidly reaching the target delivery temperature.

Figure 5 presents the data issued from the mass flow rate, pressure and temperatures measured at the dispenser, and the pressure and average temperature measured in the tank.

Figure 5

It can be seen that for all tests, the 5 MPa pressure elevation expected in the tank by SAE J2601-5 was never reached. The experimental pressure elevations in the tank are between 2.23 and 3.67 MPa (as detailed in Table 3), depending on the initial conditions of each case. This could be due to the head loss performance of the employed HRS.

This default of pressure elevation induces a limited temperature elevation in the tank, below the one calculated using SAE temperature correlation; see equation (14).

This highlights that SAE temperature correlation cannot be used if the pressure elevation in the tank does not reach the 5 MPa expected value.

It can be seen that the initial tank temperature is always higher than the ambient temperature. This can be explained by the effect of the initial pulse test on the gas temperature. Hence, the pulse test consists of injecting a small amount of gas inside the tank, triggering a check valve opening and immediately stopping the mass flow rate. If the HRS is not sufficiently reactive, that is, more hydrogen mass than necessary is injected inside the tank, this step raises the gas pressure and temperature. That is what happened during the experimental tests.

Finally, it can be noted that the ambient temperature was relatively constant with values between 27.61°C and 30.7°C. The initial gas temperatures were between 33°C and 39.9°C. The effect of the initial temperature cannot be considered well covered by these tests.

Type IV 1520 L tank

The type IV 1520 L tank is designed for hydrogen ferries. This tank volume is characteristic of a heavy-duty storage system. This aspect motivates using it to investigate the volume estimation methods. However, its NWP is only 250 bar, and this limits higher-pressure investigations.

For NWP of 250 bar, the hydrogen test bench employed was able to test custom pressure ramps and injection temperature. Accordingly, two tests were performed out of the refueling step defined in the SAE J2601-5 (SAE International, 2024a): a slow one (refueling step during 935 s) and using a less precooling (–5°C) and a fast one (refueling step of 22 s) using the maximum precooling of the station (–30°C).

A total of six experimental refueling scenarios are considered involving different initial pressures (≈2, 10 MPa(g)), different temperatures of injection (≈–5°C, –20°C, –30°C) and different refueling times. Figure 6 presents the data issued from the mass flow rate, pressure and temperatures measured at the dispenser and the pressure and average temperature measured in the tank.

Figure 6

For the type IV 1520 L tank, during cases 1–4 using the APRR from the table-based protocol, the expected pressure elevation of 5 MPa was never reached. The same observation applies to case 6, using a faster APRR than table-based protocols.

Pressure elevations in the tank are very small, between 0.25 and 2.42 MPa. This is not sufficient to ensure precise pressure measurements. These results can be explained by the large pressure drop at the dispenser caused by the large mass flow rate (20 g/s) and small injector diameter (5 mm) employed.

Only test 5, using a slow APRR, outperformed 5 MPa and reached 8 MPa. Indeed, by using a slow APRR, the mass flow rate is reduced. Consequently, the pressure drop between the dispenser and the tank is reduced. Then, overshooting the dispenser pressure to ensure a pressure elevation for a given pressure drop within an HRS in the tank leads to overshooting the pressure elevation in the tank when the real pressure drop is lower. Exceeding a pressure elevation of 5 MPa induces a gas temperature elevation (ΔT_tank = 30.51°C) superior to the one expected by the SAE (ΔT_SAE = 24.87°C).

Compared to 243.6 L tank tests, initial gas temperatures are less affected by the pulse test. This can be explained by the large volume capacity. Hence, this limits the gas temperature elevation for the same amount of mass used during the pulse test.

Finally, it can be noted that the ambient temperatures are more diverse compared to 243.6 L tank tests, with values between 12.78°C and 25.58°C.

Type IV 243.6 L tank

Concerning volume estimation calculations, as expected, isothermal formulations always underestimate the CHSS volume. The underestimation is between –22.2% and –9.6%. Then, besides giving a correct lower bound, these formulations cannot be used directly to estimate a volume if 15% discrepancy is only tolerated.

The maximal difference between the isothermal formulation and the first-order isothermal formulation is –2% (test 7, $\frac{V_{\mathit{\text{iso}}}-V_{\mathit{\text{iso}}-\mathit{\text{FO}}}}{V_{\mathit{\text{iso}}}}\approx -2\%$). This shows that the first-order approach tends to underestimate the volume. The discrepancy appears acceptable.

The adiabatic formulation always overestimates the CHSS volume. The overestimation is between 10.3% and 25.5%. Besides giving a correct upper bound, likewise the isothermal formula, this formulation cannot be used directly to estimate a volume as a 15% discrepancy is only tolerated.

Diabatic formulations produce results in between isothermal formulations and the adiabatic formulation.

Direct diabatic formulation using CoolProp provides results with <5% of error with the real CHSS volume, which is a satisfying estimation precision. The chosen parameter Λ = 303 W/m³/K provides a moderate underestimation of the CHSS volume. With this formulation, the CHSS volume is only overestimated during test 1. However, this test represents the case with the lowest gas pressure elevation in the tank (ΔP_tank = 2.23 MPa), which induces uncertainty on ΔP_tank leading deviation on the volume estimation. Hence, for all formulations, test 1 yields the highest estimated CHSS volume of all tests.

The diabatic formulation using polynomial approximations provides very close results to the one evaluating thermophysical properties using CoolProp (Bell et al., 2014).

Concerning the indirect approach, the hot case tank (HotCase) V_{0D–HotCaseHD} remains an upper bound of the CHSS volume, with CHSS overestimation between 6.9% and 20%, while the cold case tank (ColdCase) and a cold dispenser (CD) V_{0D–ColdCaseCD} remain a lower bound of the CHSS volume, with CHSS underestimation between –10.8% and –1.1%. Then, the CHSS volume is more narrowly bounded.

The hot case tank (HotCase) and a cold dispenser (CD) V_{0D–HotCaseCD} appear to be a good compromise between the two previous setups with a worst-case estimation of 6.3%.

The SAE formulation tends to overestimate the CHSS volume, which is consistent for the SAE protocols, where overestimating the CHSS volume is the safe approach. However, for these experimental tests, the SAE formulation fails to estimate the CHSS volume within a 15% error range. This can be explained by the absence of flexibility of the SAE formulation when the pressure elevation in the tank does not reach the 5 MPa requirement. Hence, the temperature elevation is not a function of the pressure elevation, the injection temperature, the ambient temperature and the refueling time. Then, it can be assumed that when the refueling step derives from a standard refueling, the SAE formulation would give erroneous results.

Type IV 1520 L tank

Table 4 presents the numerical application but for the 1520 L tank, again along with exact experimental data as input for the calculations (see Figure 6).

Concerning volume estimation calculations, the same trend as for the type IV 243.6 L is found. Isothermal formulations always underestimate the CHSS volume. The underestimation is between –27.2% and –11.5%. Again, besides giving a correct lower bound, these formulations cannot be used directly to estimate a volume if a 15% discrepancy is only tolerated.

The maximal difference between the isothermal formulation and the first-order isothermal formulation is $\frac{V_{\mathit{\text{iso}}-\mathrm{diff}}-V_{\mathit{\text{iso}}-\mathrm{direct}}}{V_{\mathit{\text{iso}}-\mathrm{direct}}}\approx -5\%$ for test 5, leading to the maximum pressure elevation in the tank.

The first-order approximation is, as expected, less precise with a large pressure elevation. Accordingly, limiting the pressure elevation to a maximum of 5 MPa is key to improving the precision of first-order approximation methods.

The adiabatic formulation always overestimates the CHSS volume. The overestimation is between 2.9% and 18.8%. Again, besides giving a correct upper bound, such as the isothermal formula, this formulation cannot be used directly to estimate a volume if 15% discrepancy is only tolerated.

Diabatic formulations produce results in between isothermal formulations and the adiabatic formulation. Direct diabatic formulation with CoolProp results are <15% of maximal errors with the real CHSS volume, which remains a satisfying estimation precision but shows less accuracy than for the 243.6 L. This can be explained by the poor gas pressure elevation obtained in the tank. The chosen parameter Λ =303 W/m³/K tends to underestimate the CHSS volume.

The diabatic formulation using polynomial approximations provides again very close results to the one evaluating thermophysical properties using CoolProp (Bell et al., 2014). A maximum deviation can be noted for test 6, reaching the 15.2% precision limit. However, for this case, the pressure elevation is not sufficient to ensure a correct pressure measurement.

The hot case tank (HotCase) V_{0D–HotCaseHD} remains an upper bound of the CHSS volume, with CHSS overestimation between 1.5% and 9.1%, while the cold case tank (ColdCase) and a cold dispenser (CD) V_{0D–ColdCaseCD} remain a lower bound of the CHSS volume, with CHSS underestimation is between –7.8% and –27.1%. Then, the CHSS volume is more narrowly bounded.

The hot case tank (HotCase) and a cold dispenser (CD) V_{0D–HotCaseCD} appear as a good compromise between the two previous setups with a worst-case estimation of –12.2%.

The SAE formulation results are not relevant. Due to the poor gas pressure elevation in the tank, the temperature elevation estimated by the SAE formulation gives at best coarse estimation values, and at worst, non-physical values. Hence, in test 6, the small tank pressure elevation and the temperature elevation predicted induce an after-refueling density lower than before the refueling. Then, the calculation gives aberrant CHSS volume estimations.