For each small length dz of the tube, permeation causes water to flow through the tube wall at the following molar flow rate (Crank, 1975): 1dn˙perm(z)=2πφlndout/dinPVT-pH2O(z)dz. Here, ϕ is the H2O permeability of the tube material, dout and din are the tube's outer and inner diameters, PV(T) is the vapor pressure of water at the temperature T of the saturator (Wagner and Pruss, 2002), and pH2O(z) is the partial pressure of water at position z inside the tube. (The notation uses p for partial pressure and P for total pressure.) At 25 ∘C the vapor pressure of water is PV=3.2 kPa. Equation (1) assumes that the temperature and total pressure are the same inside and outside the tube so that the permeation flow is driven by only the partial pressure of water. The permeation flow is zero when the partial pressures inside and outside the tube are equal, i.e., when the chemical potentials are in equilibrium.

At the tube entrance (z=0) the gas is dry, and at the tube exit (z=L) the gas is humid. As explained later, the actual mole fraction at the entrance was 0.43 ppm. Solving Eq. (1) gives the water partial pressure at the exit: 2pwet=PV1-exp-LLsat.

Here Lsat is the saturation length given by 3Lsat=lndout/din2πRTφV˙wet, where R is the universal gas constant, and V˙wet is the volume flow rate of the gas. For a long tube with L≫Lsat, the exiting gas is saturated, and the water partial pressure equals the vapor pressure. For clarity, we relate the volume flow rate V˙ to the molar flow rate n˙ by defining the volume flow rate to be that of an ideal gas, namely 4V˙≡RTPn˙. A mass flow controller (MFC) often reports the molar flow rate as a volume flow rate by using a reference temperature and pressure, say T0≡273.15 K and P0≡101325 Pa.

As shown in Fig. 1, mixing the humid gas flow with a dry gas flow produces the total flow rate, V˙total=V˙wet+V˙dry. If the total pressure after mixing is P, then the final mole fraction is 5x=pwetPV˙wetV˙total=PVPV˙wetV˙total1-exp-LLsat. To understand Eq. (5), consider its two limits. If the permeability is small, then L≪Lsat, and 6x≅2πφLPVlndout/dinn˙total. In this limit, the mole fraction xH2O depends on the tube dimensions and on temperature through the factors PV(T) and ϕ(T). Some commercial low-humidity reference standards operate in this limit (permeation tube generators; see Disclaimer).

The opposite limit of Eq. (5), L≫Lsat, occurs when the permeability is large so that 7x=PVPV˙wetV˙total=PVPn˙wetn˙total. Operating in this limit has two advantages. First, it is simple. The mole fraction depends on the two ratios, PV/P and n˙wet/n˙total, and not on the material properties or dimensions of the tube. Second, it offers the possibility of using the generator as a primary standard because the vapor pressure PV(T) is well known, and the total pressure P can be measured accurately. The smaller uncertainty in the ratio is possible if the individual uncertainties account for a possible scale error that is common to both flow meters. For example, if the standard used to calibrate both flow meters was in error by 2 %, that error will cancel out of the ratio.

Equation (5) assumes that one knows the flow V˙wet at the generator exit, but in practice one knows only the flow V˙0 at the generator entrance. The generator adds a water vapor flow V˙H2O so that, in the ideal-gas limit, the flow exiting the tube is 8V˙wet=V˙0+V˙H2O=V˙01-pwet/P. (Equation (8) comes from V˙H2O/V˙wet=pwet/P). Using Eq. (2) in Eq. (8) gives the total flow rate exiting the generator: 9n˙total=n˙dry+n˙01-PVP1-exp-LLsat-1.

Using Eq. (9) in Eq. (5) gives the mole fraction 10x=PVPn˙0n˙dry1+n˙0n˙dry-PVP1-exp-LLsat, which in the limit of saturation is 11x=PVPn˙0n˙dry1+n˙0n˙dry-PVP. This expression depends only on the ratio of the vapor pressure of water PV to the measured total pressure P and the ratio of flow rates n˙0/n˙dry.

The preceding equations assume ideal-gas behavior and do not account for four effects: (1) the nonideal behavior of the carrier gas and water vapor, (2) intermolecular forces in the mixture of water vapor and carrier gas, (3) the action of the pressure on the condensed water (the Poynting effect), and (4) the solution of the carrier gas in the water (the Raoult effect). (See, for example, Hyland, 1975; Wylie and Fisher, 1996; Koglbauer and Wendland, 2007, 2008; Lovell-Smith et al., 2016). Correcting Eq. (11) for these effects can be accomplished by the change 12PVP→fWPVP, where fW is the “pressure enhancement factor”. With this correction, Eq. (11) becomes 13x=fWPVPn˙0n˙dry1+n˙0n˙dry-fWPVP.

The carrier gas in the present study was argon, the same gas that was used in the international project to redefine the kelvin in terms of the Boltzmann constant (Pitre et al., 2017). The spherical resonators used in that project are similar to those used in the differential microwave hygrometer shown in Fig. 2. Unlike for air, the values of fW for argon have not been measured directly, but there are measurements near 25 ∘C (Koglbauer and Wendland, 2008) of a related quantity, the concentration enhancement factor gw. Those measurements agree with values of fW that we derived from calculated values of the argon–water virial coefficient Baw. In addition to being available over a wider range of temperature, the calculated values have a smaller uncertainty than the measured values. We do not discuss further the values of fW for argon that we derived, but we note that the present corrections due to the enhancement factor were less than 1.0 %, and they contributed less than 0.3 % to the relative uncertainty in the humidity.

Gas from the Ar bottle (x<0.5 ppm from specifications) entered the thermal enclosure and the humidity generator. The output of the humidity generator exited the thermal enclosure and went to the chilled-mirror hygrometer through a 1 m capillary with an inner diameter of 1.3 mm. The output of the hygrometer re-entered the thermal enclosure through a similar capillary. The MFC at the exit (maximum flow 2000 cm3 min-1) was controlled by an algorithm that stabilized the pressure P3 measured just before the MFC by a precise pressure gauge (Paroscientific 745-100A).

We minimized the number of components to avoid unwanted sources of water vapor; for example, we removed an over-pressure safety valve because atmospheric water vapor could penetrate its rubber seal. We used small-diameter capillaries to limit the surface area for adsorption and desorption. Even after reducing the surface area of the stainless-steel components, the time to achieve a humidity below 10 ppm was several hours.

The thermal enclosure, which was based on the description given in Berg et al. (2014), controlled the temperatures of the humidity generator, the gas manifold, the MFCs, and the pressure gauges near 28 ∘C. Its height and area were respectively 0.63 m and (1.27×0.66) m2, and its walls were rigid sheets of 25 mm polyisocyanurate insulation. The temperature was stabilized by a commercial controller (Arroyo Instruments model 5305; see Disclaimer) that drove a thermoelectric cooler (Laird model AA-034-12-22). The control thermistor was located in the stream of air that entered the cooler. Measurements at other locations found that the temperature was stable throughout the enclosure to approximately ±0.03 K. Without the cooler, the power dissipated by the enclosed instruments would raise the temperature to 6 K above ambient. Despite the use of four small stirring fans, the enclosed electronics created a gradient of 1 K in the enclosure, with the coldest point located at the output of the thermoelectric cooler.

The thermal enclosure was made large enough to include the temperature-sensitive microwave hygrometer and its associated tubing. A much smaller enclosure containing only the saturator would have been sufficient to keep its output constant.

The humidity generator comprised the saturator and three MFCs. It first divided the input gas into two streams, dry and saturated. The dry stream was controlled by an MFC with a maximum flow of 2000 sccm (Bronkhorst F-201CV). (1 sccm ≅0.74 µmol s-1 is the molar flow rate corresponding to 1 cm3 min-1 of an ideal gas at 0 ∘C and 1 atm.) The saturated stream was controlled by two MFCs in parallel, one with a maximum flow of 2 cm3 min-1 and the other with 200 cm3 min-1 (Alicat MC). All three MFCs were calibrated at the factory for argon flow. The saturated stream mixed with the dry stream after traveling through a capillary heated to 20 K above the enclosure temperature to avoid condensation. The range of possible mole fraction was 0.7 ppm<x<9000 ppm, where the minimum mole fraction corresponded to the minimum MFC setting of 0.015 cm3 min-1. We operated the generator with total flow rates from 50 to 300 cm3 min-1.

Table 2 gives the specified performances of the MFCs, which we did not verify directly. However, as discussed later, comparing the generator to the dew-point hygrometer showed that the MFC calibrations had drifted outside their specifications. Fortunately, the comparison data had sufficient redundancy that the drifts could be modeled by a linear function of flow rate so that the generator's performance could still be tested. Recognizing the possibility of MFC errors is important, and the discussion on uncertainty suggests various ways to reduce such errors.

The saturator consisted of a 1 L glass dewar that contained a commercial sealed platinum resistance thermometer (pt1000) and 5.07 m of silicone (PDMS) tubing (Saint-Gobain product Versilic), with inner and outer diameters of 4 and 6 mm. Although the tubing had an acid-acrylic odor, we assumed that any outgassing had a negligible effect. Both were immersed in chromatography-grade or “ultra-pure” water. See Fig. 3. The saturator temperature and pressure were not directly controlled. The pressure drop along the tubing was negligible, ∼ 60 Pa for a flow of 200 sccm. The water temperature was typically 0.5 K higher than the surrounding enclosure due to the heat dissipated by nearby components, especially that of the heated capillary. The standard uncertainty in the thermometer's calibration was 0.021 ∘C. We assumed that stratification of the water temperature contributed negligible additional uncertainty because the thermometer was located near the exit of the tube; this assumption was supported indirectly by the consistency obtained when using different flow rates through the saturator. A simpler way to control the saturator temperature would have been to use an ice bath in the dewar. Three bellows-sealed valves were used to isolate the saturator for various tests.

The humidity generator was compared against a commercial chilled-mirror hygrometer (MBW model 373LX; see Disclaimer) with an accuracy of 0.1 ∘C in the nominal flow range from 200 to 2000 cm3 min-1. The hygrometer determined the water mole fraction from the hygrometer's measurements of pressure and dew-point temperature. The manufacturer's calibration with air spanned dew and frost points from -90 to +20 ∘C; a calibration made 8 months later by CETIAT found deviations of less than 0.1 K. We adjusted the hygrometer's values by the small difference between the enhancement factors of air and argon: 14fW(Ar)-fW(air)=-0.0005+(1.1×10-5K-1)×(T-273.15K). As mentioned Sect. 2.3, we used literature values for fW (air), and we derived the values of fW (Ar) from values of the argon–water virial coefficient Baw.

Figure 4 shows how the saturator performance depended on the length L of the silicone tube. The tube was held in water at 28.5 ∘C, the dry flow rate V˙dry was fixed at 200 cm3 min-1, and the wet flow rate V˙wet was varied by controlling MFC 1 and MFC 2. The reference hygrometer measured the water mole fraction of the resulting mixture. It was not used to measure the mole fraction of the gas stream exiting the saturator because the temperature of the saturator (28.5 ∘C) was above that of the hygrometer. Adding the dry flow also ensured that the flow rate of the mixture fell within the reference hygrometer's nominal flow range (200 to 2000 cm3 min-1).

Three tube lengths were used: 6, 50, and 507 cm. The mole fraction calculated for these lengths agreed with the mole fraction measured by the reference hygrometer when the water permeability ϕ of the tube was set at (950±50)×10-14 mol s-1 m-1 Pa-1. This value is consistent with those found elsewhere for PDMS; see Table in the Appendix. All three tubes agree at low flow rates, where the slope derived from Eqs. (7) and (12) is simply 15dxdV˙wet=1V˙totalfWPVP.

Figure 4 shows that the model agrees with all the measurements. At lower flow rates the mole fraction was the same for all three tube lengths L, which means that the partial pressure at the tube exits, pH2O(L), was the same for all three tube lengths. Equation (1) assumes that the values of T and P at the exit were equilibrated across the tube wall, in which case the common partial pressure was simply the vapor pressure of water. In other words, pH2O(L)=PV, and the gas at the tube exit was saturated.

Figure 4 demonstrates that the tube length of 507 cm used for the measurements described below was sufficient to saturate the gas. According to Eq. (3), a gas flow as large as 10 sccm, which produced a mole fraction of x≈2000 ppm, caused a saturation length, the 1/e length of Eq. (2), of only 45 cm.

These tests were performed with the thermal enclosure temperature-controlled at 28.0 ∘C, corresponding to water temperatures in the uncontrolled dewar in the range (28.8±0.1) ∘C. The humidity was varied by changing the ratio V˙0/V˙dry while keeping constant the sum V˙0+V˙dry. The total flow rate was varied from 50 to 300 cm3 min-1, and the saturator input was varied from 0.015 to 50 cm3 min-1 at the points given in Table 3. To ensure equilibrium, each point was an average taken during 24 h.

For each pair of wet and dry flow rates, Eq. (13) was used to calculate the mole fraction x from the temperature T and pressure P of the saturator and the ratio of flow rates n˙0/n˙dry. The calculated values were obtained by making four small corrections to the model, as discussed in the following section.

The model (Eq. 13) required corrections for the humidity of the input “dry” gas, the permeation of argon through silicone, the diffusion of water vapor through argon, and the pressure drops caused by flow through capillaries. The first correction was simply 16x→x+xdry, where xdry=0.43 ppm was the mole fraction measured for all the gas bypassing the saturator (valve V3 closed).

The second correction seemed necessary because, upon opening the dewar, we always observed gas bubbles on the external surface of the immersed tubing. We attributed that effect to permeation of the carrier gas through the walls of the tubing. The argon permeation may have caused a second effect: small temperature spikes in the saturator temperature that disappeared when the tubing was removed. Perhaps they occurred when a rising bubble changed the temperature distribution in the water. In any case, the spikes had a negligible effect on the average temperature because they had amplitudes <0.1 K, and they occurred only a few times per hour.

As discussed in the Appendix, the permeability ϕAr of argon through PDMS allowed the carrier gas to diffuse out of the tube and into the surrounding water at the rate 17n˙perm,Ar=2πLpout-pinlndout/dinφAr, where pin and pout are the partial pressures of Ar inside and outside the tube. The applied correction was simply 18n˙Arsat=n˙Ar0-n˙Arperm, where n˙Ar0 and n˙Arsat denote respectively the argon molar flow rates at the entrance and exit of the saturator (see Fig. 5). If the outside pressure had remained at 100 kPa while the tube's interior pressure was at 108 kPa, the loss of argon would have been continuous, with the value 19n˙Arperm=1.2×10-7mols-1=0.16sccm.

In this limit, no humid gas would leave the saturator if the argon input was less than 0.16 sccm.

We did not know the value of n˙Arperm because the dewar was not tightly sealed, which caused its interior pressure to be somewhere between 100 and 108 kPa. We therefore allowed n˙Arperm to be a free parameter with a value somewhere in the range of 0<n˙Arperm<0.16sccm. Consistent with that range, the fitted value was 0.024 sccm.

The third correction accounted for the diffusion of water through argon along the saturator's exit capillary. In the absence of convective flow out of the saturator, or n˙sat=0mols-1, water will diffuse from the saturator along the exit capillary at the rate 20n˙H2Ocap(0)=DALPVRT=5.1×10-11mols-1.

Here D is the diffusion coefficient of H2O vapor in Ar gas (O'Connell et al., 1969), PV is the vapor pressure of water at the saturator temperature, A=1.27×10-6 m2 is the capillary cross-section area, and L=1.0 m is the capillary length. The corresponding volume flow rate is 21V˙H2Ocap(0)=7×10-5sccm, which for a typical dry flow rate of 200 sccm corresponds to a mole fraction of 0.34 ppm.

To combine the effects of convection and diffusion, we used the steady-state one-dimensional convection–diffusion equation, 22Dd2Cdz2-vdCdz=dCdt=0, where C is the H2O concentration in mol m-3, v is the flow velocity, and z is the distance along the capillary. After setting the capillary entrance concentration at C(0)=PV/(RT) (saturation) and the exit concentration at C(L)=0 (merging with the dry gas stream), one obtains the H2O flow when the convective flow is not zero, 23n˙H2Ocapn˙sat=PVPn˙sat1-exp-PVPn˙satn˙H2Ocap0, where the total flow rate out of the saturator is 24n˙sat=n˙Arsat+n˙H2Osat=n˙Arsat1-PVP. This correction was significant only for mole fractions x<2 ppm.

The above corrections lead to the following generalization of Eq. (11): 25x=xdry+n˙ArsatPVPn˙dry1-PVP+n˙Arsat1-exp-PVPn˙Arsatn˙H2Ocap0. (The enhancement factor fW, not shown here for simplicity, was included in our analysis.)

The fourth correction accounted for the pressure drops caused by flow through capillary 2 and capillary 3. These corrections were applied to the pressure P in the saturator and the pressure Pcm in the chilled-mirror hygrometer. Due to capillary 3, Pcm was higher than the controlled pressure P3 by about 1 %, or specifically 26Pcm-P3=kcap3n˙, where kcap3 is the capillary flow coefficient of capillary 3. Similarly, the pressure P in the saturator was higher than P3 by 27P-P3=kcap2+kcap3n˙. As shown by Fig. 6, the capillary flow coefficients had similar values: 28kcap2=8.9Pasccm-1  and  kcap3=8.6Pasccm-1.

These values are consistent with that calculated from the length and inner radius of the capillary. We characterized the uncertainty in the capillary pressure drops by their difference. At the typical total flow rate of 200 sccm, the capillary pressure drops increased the uncertainty in the pressure P in the saturator by approximately 29u(P-P3)=(0.3Pasccm-1)(200sccm)=60Pa.

We compared the water mole fraction calculated for the generator to that measured by the chilled-mirror hygrometer. Figure 7 (upper) shows the initial comparison, which used the nominal factory calibrations for the MFCs. The points obtained with MFC 1 and MFC 2 deviate from zero in different directions, and the points obtained with MFC 1 at different total flow rates have a scatter of ∼ 10 %.

We attributed these inconsistencies to miscalibration of the MFCs, and we attempted to correct the nominal flow rate V˙nom for each MFC by a linear function of flow, 30V˙=a+bV˙nom, where a and b are the coefficients in Table 4. The values of b for MFC 1 and MFC 2, though inconsistent with the manufacturer's specification given in Table 2, were needed to obtain agreement between similar values of humidity created with different total flow rates. We note that the MFCs were used after the manufacturer's warranty period, and similar shifts in the MFC calibrations were seen in the year preceding these measurements.

Figure 7 (lower) shows the corrected differences, which used the MFC correction coefficients a0, b1, and b2 shown in Table 4. This led to much better agreement, and with two exceptions (at 70 and 910 ppm), the differences are within ±2 ppm for x<1000 ppm. The only other adjustable parameter was the Ar permeation parameter n˙Arperm. As discussed in the previous section, its fitted value fell within the range expected from the permeability of Ar through PDMS.

Figure 8 shows the calculated-to-measured ratio of mole fractions for the same data. With one exception (at 70 ppm), the corrected ratios fall within 1.00±0.01 for x>50 ppm. At smaller mole fractions, the deviations are larger but still correspond to a mole fraction error of approximately ±1 ppm, or a flow error of only about 1 % of the full-scale flow of MFC 1. We speculate that those deviations were caused by irreproducibility or by nonlinearity not described by the linear correction of Eq. (30).

A concern is that the MFC corrections could hide an error in the model of the generator. However, there are several reasons that the corrections can be attributed to errors in the MFCs and not to an error in the model.

The nonzero value of the offset a0 (MFC 0) was necessary to obtain agreement between different total flow rates at the same mole fraction.