S3 Analysis for Estimating Composition of Each Phase from Sampling Measurements

VII.S3 Analysis for Estimating Composition of Each Phase from Sampling

liquid-phase sample is the density of pure polyol at atmospheric pressure𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝_{𝑎𝑡 𝑚}) scaled by the ratio of the volume of polyol𝑉

𝑠𝑎𝑚 𝑝

𝑝 𝑜𝑙 𝑦 in the sample to the total sample volume𝑉

𝑠𝑎𝑚 𝑝

𝑡 𝑜𝑡 . The volume of polyol in the sample is 𝑉^{𝑠𝑎𝑚 𝑝}

𝑝 𝑜𝑙 𝑦 =𝑉^{𝑠𝑎𝑚 𝑝}

𝑡 𝑜𝑡 −𝑉^{𝑠𝑎𝑚 𝑝}

𝐶 𝑂2 −𝑉^{𝑠𝑎𝑚 𝑝}

𝐶5 (VII.4)

The volumes of CO₂ and cyclopentane in the sample can be estimated by dividing the density of each measured by GC (𝜌^{𝐺 𝐶}

𝐶 𝑂2(𝑝) and 𝜌^{𝐺 𝐶}

𝐶5(𝑝)) by den- sity of the pure component estimated by the assumptions above. Specifically, 𝑉

𝑠𝑎𝑚 𝑝 𝐶 𝑂2 /𝑉

𝑠𝑎𝑚 𝑝

𝑡 𝑜𝑡 ≈ 𝜌^{𝐺 𝐶}

𝐶 𝑂2(𝑝)/𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝_{𝑎𝑡 𝑚}) and 𝑉

𝑠𝑎𝑚 𝑝 𝐶5 /𝑉

𝑠𝑎𝑚 𝑝

𝑡 𝑜𝑡 ≈ 𝜌^{𝐺 𝐶}

𝐶5 (𝑝)/𝜌^𝐿

𝐶5(𝑝_{𝑎𝑡 𝑚}). Therefore, the estimate for the density of polyol in the liquid sample is

𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝) ≈ 𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝_{𝑎𝑡 𝑚}) 1− 𝜌^{𝐺 𝐶}

𝐶 𝑂2(𝑝) 𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝_{𝑎𝑡 𝑚}) −

𝜌^{𝐺 𝐶}

𝐶5(𝑝) 𝜌^𝐿

𝐶5(𝑝_{𝑎𝑡 𝑚})

!

(VII.5)

where each quantity in equation VII.5 is known from measurement. The weight fractions of each component can then be computed by dividing the density of that component in the sample by the total sample density𝜌_{𝑠𝑎𝑚 𝑝}(𝑝) =𝜌_{𝑝 𝑜𝑙 𝑦}(𝑝)+𝜌^{𝐺 𝐶}

𝐶5(𝑝)+

𝜌^{𝐺 𝐶}

𝐶 𝑂2(𝑝).

Estimating the Vapor Density after Depressurization

Due to the challenges of maintaining pressure inside the sampling apparatus, the sample of vapor from the head space depressurized to atmospheric pressure inside the six-port gas-sampling valve. Consequently, the GC measured a much lower density of CO₂and cyclopentane in the sample than expected. To correct the effect of depressurization on the densities, we assume that the head space can be treated as a binary mixture of CO₂ and cyclopentane and use a PC-SAFT model fit to such data [1] to estimate the total density 𝜌^{𝑝 𝑐−𝑠𝑎 𝑓 𝑡}

𝑡 𝑜𝑡 of the vapor phase under the known pressure and temperature of the Parr reactor. We assume that the weight fractions of CO₂and cyclopentane remain the same under depressurization. Then we scale the measured densities by the ratio of the PC-SAFT prediction for the total density to the measured total density𝜌^{𝐺 𝐶(𝑉)}

𝑡 𝑜𝑡 =𝜌^{𝐺 𝐶(𝑉)}

𝐶 𝑂2 +𝜌^{𝐺 𝐶(𝑉)}

𝐶5 (where (V) indicates that the measurement is taken of the vapor-phase sample). Then the estimate for the true density of component𝑖is

𝜌^{𝑡𝑟 𝑢 𝑒}

𝑖 ≈ 𝜌

𝑝 𝑐−𝑠𝑎 𝑓 𝑡 𝑡 𝑜𝑡

𝜌

𝐺 𝐶(𝑉) 𝑡 𝑜𝑡

! 𝜌

𝐺 𝐶(𝑉)

𝑖 (VII.6)

Estimating Density of CO₂in ISCO Pump

Initially, we believed that the amount of CO₂ in the Parr reactor could be estimated simply by using the CO₂ equation of state to calculate the density and multiply that by the volume dispensed by the ISCO pump into the Parr reactor.

This method clearly overestimates the actual amount of CO₂ in the Parr reactor because a substantial amount of CO₂ leaked during the experiment. Additionally, the ISCO pump was likely partially liquid and partially vapor, so determining the overall density of the dispensed fluid was ambiguous. Therefore, we tried two other methods to estimate the amount of CO₂ in the Parr reactor. The first used the change in density of CO₂ in the vapor phase of a CO₂–C5 binary coexistence at the pressure and temperature before and after adding CO₂, assuming that CO₂ and C5 had equilibrated immediately after adding CO₂or C5 to the Parr reactor or venting and that no CO₂entered the liquid polyol-rich phase. The second used the PC-SAFT model developed by Dr. Huikuan Chao to estimate the composition of the vapor phase. This method is somewhat circular, however, because it relies on the model that the measurements attempt to validate. Nevertheless, it provided a rough estimate of the composition. In both of the latter cases, the estimates suffered from not accounting for the possible presence of a third phase, as depicted in Figure VII.11.

The first method used to estimate the mass of CO₂ in the Parr reactor was estimating how much CO₂was dispensed from the ISCO pump into the Parr reactor based on the equation of state of CO₂[2]. The volume and pressure were recorded from the sensor readouts on the ISCO pump both before and after injection of CO₂ into the Parr reactor. The temperature was assumed to remained constant at the lab temperature (about 21 ^◦C). Based on the equation of state of CO₂, the beginning and final masses of CO₂in the ISCO pump were estimated, and the difference was taken as an estimate for the amount injected into the Parr reactor.

This method assumed that:

1. The ISCO pump was liquid-full of CO₂ and therefore contained a single, homogeneous phase of CO₂at all times

2. The Parr reactor did not leak

3. The temperature of the ISCO pump was constant and homogeneous throughout the reservoir

4. The pressure transducer of the ISCO pump did not drift

The first assumption was certainly false after the ISCO pump is refilled because the liquid CO₂ from the tank must expand to fill the dead volume. Addi- tionally, the pressure of the tank is not sufficient to re-condense that vaporized CO₂. Whether the CO₂became homogeneously liquid when pressurized to 1000 psi and above before injection was not clear and should be tested with another ISCO pump.

The second assumption was definitely false given the detection of a vapor leak through a needle valve on the gas-sampling port of the Parr reactor. The amount of leaked CO₂was estimated based on differences in pressure betwee injections of CO₂and C5, but these estimates have not been validated by other means.

The third (3) and fourth (4) assumptions are fairly robust, as the steel syringe of the ISCO pump conducts heat well enough to maintain thermal equilibrium with the laboratory and periodic checks of the pressure transducer reading when emptying the ISCO or loading with the liquid CO2 tank at a known pressure did not show signs of drift beyond 10 psi, which would have a negligible effect on the estimated amount of CO₂injected in the Parr reactor.

Overall, this first estimation method is likely an overestimate of the true mass of CO₂ in the Parr reactor because of the limitations of assumptions (1) and (2) discussed.

The second method of estimating the amount of CO2 dispensed assumes that, because the amount of polyol in the vapor phase is negligible, the vapor phase can be approximated as the vapor phase of a CO₂–C5 binary coexistence. Under this assumption, a PC-SAFT model with parameters fitted to Eckert and Sandler’s data [1] was used to compute the vapor–liquid equilibrium of CO₂ and C5 at the pressure and temperature in the Parr reactor both before injecting with the ISCO and immediately after. Next, the difference in density of CO₂ in the vapor phase was multiplied by the estimated volume of the vapor phase, which was estimated by subtracting the estimated liquid volume from the approximate interior volume of the Parr reactor (1200 mL). The liquid volume was estimated as𝑉_{𝑙𝑖 𝑞} = 𝑚_{𝑝 𝑜𝑙 𝑦}/𝜌^{𝐻 𝑃 𝐿 𝐼 𝑆}

𝑝 𝑜𝑙 𝑦 , where𝜌^{𝐻 𝑃 𝐿 𝐼 𝑆}

𝑝 𝑜𝑙 𝑦 = 𝜌^{𝑎𝑡 𝑚}

𝑝 𝑜𝑙 𝑦(𝑇)𝑣^{𝐻 𝑃 𝐿 𝐼 𝑆}

𝑝 𝑜𝑙 𝑦 , where𝜌^{𝑎𝑡 𝑚}

𝑝 𝑜𝑙 𝑦(𝑇)is the estimated density of polyol under atmospheric pressure at the given temperature and𝑣^{𝐻 𝑃 𝐿 𝐼 𝑆}

𝑝 𝑜𝑙 𝑦 is the volumetric

fraction of polyol in the HPLIS, inferred by estimating the volumes of CO₂and C5 based on their masses measured by the gas chromatograph and their approximate densities at the given temperature.

This method makes the following assumptions:

1. There is no polyol in the vapor phase

2. The vapor-liquid equilibrium between CO₂ and C5 is achieved very rapidly (minutes)

3. The vapor-liquid equilibrium between CO₂ and C5 is not affected by polyol in the liquid phase (e.g., the polyol does not enhance adsorption of CO₂into the liquid phase)

4. No third phase forms

5. Fitting to Eckert and Sandler’s data [1] yields accurate PC-SAFT parameters for the binary coexistence

The first assumption is likely valid because of the low vapor pressure of polyol given its molecular weight of 1000 g/mol and surface tension of almost 30 mN/m. A quick sniff assures the scientist that this is indeed the case.

The second, third, and fourth assumptions have limited validity. The vapor–

liquid equilibrium will definitely be affected by the presence of polyol in the liquid as this will lower the diffusivity (hindering equilibrium between vapor-phase and liquid-phase CO₂and C5) and will affect solubility in the liquid phase. This is made clear when the PC-SAFT estimates of C5 weight fraction in the vapor phase do not match the GC estimates. The fourth assumption is likely false by the sixth injection based on preliminary evidence of the formation of a third phase then, and it is likely that the third phase is present in later measurements as well.

The fifth assumption is probably trustworthy since the data are plentiful and precise.

The advantage of this method is that it only considers changes in CO₂mass on the order of a few minutes, so we can neglect the leaking of CO2 and actually use this method as an estimate for how much CO₂leaked between injections from the ISCO.

Estimating Mass of Gas Lost through a Leak

We estimated the leak of CO₂using the CO₂–C5 binary coexistence method’s estimate of the mass of CO₂. Assuming that changes in the estimated mass are only due to the leaking of CO₂ or injections of CO₂from the ISCO pump, we took the difference in mass (in general a decrease) between injections from the ISCO and divided by the elapsed time to estimate the rate of leak of CO₂. We noticed an increase in the rate later in the experiment, around the time that we began to notice leaking through the needle valve on the gas-sampling port of the Parr reactor.