Experimental and Numerical Method for Estimating Diffusion

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Experimental and Numerical Method for Estimating Diffusion Coefficient of the Carbon Dioxide into Light Components Nikhil Bagalkot, and Aly Anis Hamouda Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.6b04318 • Publication Date (Web): 25 Jan 2017 Downloaded from http://pubs.acs.org on February 14, 2017

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Industrial & Engineering Chemistry Research

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Experimental and Numerical Method for Estimating Diffusion Coefficient of the Carbon

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Dioxide into Light Components

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Nikhil Bagalkota and Aly A. Hamouda b*

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b*

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a

Professor, Department of Petroleum, University of Stavanger, Stavanger, Norway N 4035. Email: [email protected], Tel. +47 51 83 22 71, Mob: +47 957 026 04

Department of Petroleum, University of Stavanger, Stavanger, Norway N 4035. Email: [email protected], Tel. +47 48628290

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Abstract

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This study addresses the diffusion coefficient of CO2 into light-hydrocarbons.

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Experiments were done under non-isothermal and non-isobaric conditions, using dynamic

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pendant drop volume analysis to measure the change in hydrocarbon drop volume due to CO2

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diffusion, for 25-45oC and 25-65 bar.

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A new numerical model was developed, where a spherical drop was used rather than

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the actual pendant shaped drop, which enabled sensitivity studies. The approach showed 3–

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6% difference in the surface area of the spherical drop compared to experimental drop. This is

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translated to less than 6% difference when compared to the published data.

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Interfacial tension investigations showed a change from a negative (decreasing) to

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positive (increasing) trend with temperature for pressures 30-60 bar. A suggested explanation

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was based on the density difference between the drop (Hydrocarbon+CO2) and the

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surrounding CO2. Further, the observed higher diffusion coefficient of n-hexane compared to

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n-decane may be attributed to viscosity.

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1. Introduction

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Molecular diffusion is a crucial and fundamental process in the applications pertaining to

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the energy sector, like CO2 flooding, carbonated water injection (CWI) and chemically

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enhanced oil recovery (Chemical EOR). Besides, molecular diffusion plays an import aspect

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in the mitigation of environmental impact especially in Carbon Capture and Storage (CCS). A

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sustainable economic outcome of the CO2 based applications relies on how the dissolved CO2

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brings about changes in physical and chemical properties of hydrocarbon like density,

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viscosity, and IFT 1. The favourable changes in physical and chemical properties due to

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dissolving CO2 into the oil phase will result in oil swelling and mobilisation of isolated oil

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ganglia

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time investigate the factors contributing to the diffusion of CO2 into oil and the factors

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influenced by the diffusion process.

2, 3

. Hence, it is critical to quantify the diffusion of CO2 in the oil phase, at the same

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Diffusion of the gas into the oil is an important process in the oil recovery and CO2 storage

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mechanisms and has been studied extensively. The diffusion coefficient quantifies the rate of

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gas dissolution in bulk liquid phase by diffusion and hence, it is important to calculate the

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value to diffusion coefficient. Numerous qualitative and quantitative methods have been

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presented previously to calculate the diffusion coefficient of gases in bulk liquids. However,

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diffusion coefficient varies with each method, indicating that the diffusion coefficient is a not

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merely a function of the physical and chemical parameters, but depends on the experimental

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or the analysis method applied, which should not be the case. Conventionally, the diffusion

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coefficient is measured by analysing the composition of the dissolved CO2 in the bulk liquid

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phase (hydrocarbon)4-6. In the compositional analysis, the CO2-hydrocarbon mixture is

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extracted from the experimental setup and analysed for the composition of CO2 using gas

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chromatography at various time level, from which the diffusion coefficient is calculated. The

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compositional analysis would perform well if the experiment were carried out at atmospheric 3 ACS Paragon Plus Environment


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conditions. However, if the experiments were to be carried out at pressures greater than the

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atmospheric, then it would be a tedious and herculean task to extract the sample from the

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setup at experimental conditions and carry out the gas chromatography at the same

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experimental conditions. The failure of accurately carrying out the extraction and

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chromatography would lead to gross error in measuring the diffusion coefficient of the CO2 in

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hydrocarbons. Further, the compositional method would turn out to be expensive and tedious,

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because of the complications involved. Once the shortcomings of the compositional method

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were realised, some new methods were proposed, where it was not essential for analysing the

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composition of the CO2-hydrocarbon mixture. Thus, eliminating the errors due to the

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experiments. To reduce the experimental time and experimental errors few studies tried to

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calculate the diffusion coefficient of gases in bulk liquids by using a model that relies

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completely on mathematical or numerical technique. Zabala, et al.

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the Fick diffusion coefficients in CO2-n-alkane binary mixtures using molecular simulation,

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with no experimental test involved. However, to achieve the objective the model requires the

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phase properties like the difference between the chemical potential, and fugacity, which

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requires solving of the complex EOS (equation of state). Hence, an exclusive experimental or

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numerical or a mathematical model have drawbacks. Pressure decay method relies on both

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experimental and numerical approach in analysing the diffusion of gases in the bulk liquids. It

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involves monitoring of reduction in gas pressure due to diffusion of the gas into the bulk

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liquid to calculate the diffusion coefficient

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setup is required, the pressure decay method requires a rather long experimental time (20

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hours to 100 hours or more). Moreover, the diffusion coefficient obtained is not for a

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particular pressure, but for a range of pressures (due to decay in pressure during the

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experiment). Hence, the pressure decay method would be unsuitable when we require a

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pressure and temperature specific diffusion coefficient of CO2 in liquids. Few studies have

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focused on determining

8-13

. Although a relatively simple experimental


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relied on different experimentally measurable properties like a change in gas volume 14, gas-

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oil interface position

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coefficient of gases in the oil. Lately, some articles have come up with a more unconventional

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method to calculate the diffusion coefficient of gases in bulk liquids. Liu, et al.

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, IFT

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, and volume of pendant oil drop

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to calculate the diffusion

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used a

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microfocus X-ray CT scanning technique to calculate the diffusion coefficient of CO2 in bulk

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n-decane and n-decane saturated porous media. Zhang, et al.

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pH value to calculate diffusion of CO2 in saline water.

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developed a method by using

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From the literature survey, the methods employed to calculate the diffusion coefficient

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may be divided into four broad categories compositional analysis, pressure decay,

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unconventional, and measuring the physical properties. Table 1 provides the comparison of

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the diffusion coefficient of CO2 in heavy oils at similar experimental conditions, carried out

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by various studies using different methods. It may be seen from Table 1 (part A) that there is

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an inconsistency in measured CO2 diffusion coefficient for similar experimental fluids and

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conditions for each of the method. Therefore, there is a lack of a well-defined and easy

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method with minimum possibility of error to quantify and interpret the vital process of

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diffusion of gases into bulk liquids. This signal for a compelling need to establish a universal,

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simpler, and reliable method to determine the diffusion coefficient for gas in a liquid. Lately,

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Yang and Gu

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determine the diffusion coefficient of CO2 in water and heavy oil by using the Axisymmetric

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Drop Shape Analysis (ADSA). Their method comprised of a combination of experimental and

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numerical approach at elevated temperatures and pressures. Yang, Tontiwachwuthikul and Gu

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the diffusion coefficients and interface mass-transfer coefficients of the crude oil−CO2 system

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at high pressures at a constant temperature. In their experiment, they first measured the

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dynamic and equilibrium interfacial tensions of the crude oil−CO2 system by using the ADSA

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and Yang, Tontiwachwuthikul and Gu

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developed a novel approach to

applied a newly developed dynamic interfacial tension method to simultaneously determine



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technique for the pendant drop case. Next, a numerical model was employed to calculate the

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dynamic interfacial tension at required time intervals, using a predetermined calibration curve

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of the measured equilibrium interfacial tension (experimental) versus the calculated

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equilibrium CO2 concentration in the crude oil. Further, the diffusion coefficient of CO2 in the

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liquid is calculated by comparing the numerically calculated and the experimentally measured

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dynamic interfacial tensions at different times. While, Yang and Gu 3 used a dynamic volume

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change of the pendant drop due to the gradual diffusion of CO2 into to it, instead of dynamic

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IFT as the measuring tool and followed the similar method as in Yang and Gu

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flexibility, simple, lesser time, and devoid of any human interference makes the above two

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methods a suitable method for calculating the diffusion coefficient of gases into the bulk

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liquids. However, the IFT is a phenomenon that is influenced by activities in the vicinity of

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the interface, whereas the dynamic volume of the pendant drop is influenced by the actives

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throughout the pendant drop. Therefore, measuring volume would better represent the

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diffusion than dynamic measurement of IFT. The method reported in the present article is the

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adaptation of the Yang and Gu 3, with modifications to the numerical model to make it

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simpler, as well as considering the influence of critical parameters. Hence, the present paper

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makes an effort to establish, modify, and more importantly extend the novel method proposed

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by Yang and Gu

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interdependency among IFT, molecular weight, viscosity, molar concentration, pressure,

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temperature, and diffusion coefficient. To the best of our knowledge, apart from the original

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works of Yang and Gu 3 this approach has not been tested for the gas-liquid system, like the

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CO2 and light hydrocarbon system used in the present study.

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. The

in a CO2 – light hydrocarbon system, with added analysis on

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Table 1: Diffusion coefficient (m2/s) obtained by different studies with CO2 as environmental phase. PART A (CO2 – hydrocarbon) Gas Liquid Experimental Phase Phase Conditions

Analysis Method Schmidt, Leshchyshyn and Puttagunta 4 (Compositional) Upreti and Mehrotra 8 (Pressure Decay) Upreti and Mehrotra 13 (Pressure Decay) Yang and Gu 3 Pendant Drop Analysis Liu, Teng, Lu, Jiang, Zhao, Zhang and Song 17 CT Scan

CO2 CO2 CO2

Bitumen Hamaca Oil Athabasca Bitumen

CO2

Heavy oil

CO2

n-decane

Diffusion Coefficient (10 -9 m2/s)

50 bar, 20 200oC

0.28 – 1.75

35 bar, 21oC

4.8

40 – 80 bar, 25 – 90oC 20 – 60 bar, 23.9oC 10 – 60 bar, 29oC

0.17 – 1.08 0.199 – 0.551 0.38 – 2.29

Part B (CO2 – n-decane) Grogan, Pinczewski, Ruskauff and Orr Jr 15 (variable pressure) Renner 14 (Pore scale, IFT measurement) Liu, Teng, Lu, Jiang, Zhao, Zhang and Song 17 (Pore scale) Zabala, Nieto-Draghi, de Hemptinne and Lopez de Ramos 7 (Numerical/Simulation)

CO2

n-decane

13 – 50 bar, 25oC

3.21 – 5.71

CO2

n-decane

15 – 60 bar, 37oC

1.97 – 5.05

CO2

n-decane

10 – 60 bar, 29oC

0.38 – 2.29

CO2

n-decane

0.2 – 0.8 CO2 saturation

1.9 – 4.1

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In the literature, there is a limited data of CO2 diffusion coefficient in n-decane, n-

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heptane, and n-hexane (light hydrocarbons). Few studies that have been carried out are

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presented in Table 1 (Part B) along with the type of experimental method applied. The

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diffusion coefficient of these studies may be uncertain due to the methods applied and

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parameters considered. The methods used by these studies are entirely numerical 7, variable

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pressure

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out at a constant temperature. As observed in the case of CO2 – n-decane, there is a difference

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in the calculated diffusion coefficient for similar experimental conditions. Hence, the

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diffusion coefficient of CO2 in light hydrocarbons (n-decane, n-heptane, and n-hexane) are

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, IFT movement

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, and pore scale

14

. Further, all these studies have been carried



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scarce and inadequate, particularly at the high pressure and varied range of temperatures. The

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purpose of the present study is to measure the diffusion coefficient of CO2 in n-decane, n-

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heptane, and n-hexane at a wide range of temperatures (25 oC, 35 oC, and 45 oC) and

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pressures (25 – 65 bar). The influence of IFT, viscosity of oil, viscosity of CO2-hydrocabon

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mixture, CO2 concentration in hydrocarbon, swelling of the drop, and the molecular weight of

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hydrocarbon on the diffusion coefficient of CO2 in hydrocarbon has been discussed. Here a

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pendant hydrocarbon drop is introduced into a see through a high-pressure chamber

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containing CO2, and then the changes in IFT and volume of the pendant drop are measured at

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regular intervals. The dynamic volume data of the drop is further used in the numerical model

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to calculate the diffusion coefficient of CO2 in oil at a required pressure and temperature.

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Further, special attention is given to understanding the swelling of pendant oil drop, time is

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taken for swelling to stabilise, time take for achieving equilibrium IFT, and how factors

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change with pressure, temperature and type of oil. The aim of the article is to find the

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diffusion coefficient of CO2 in light hydrocarbons, and further, identify the factors

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influencing and influenced by the process of diffusion. IFT is one among important parameter

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when studying molecular diffusion. However, few studies have looked into the effect of

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temperature on the IFT of CO2-hydrocarbon due to its unpredictable trend with temperature.

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Some studies have reported a decrease an inverse relation of IFT with temperature, while the

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others have reported a direct relation. The results presented in the present analysis is an

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attempt to resolve the controversial impact of the IFT-temperature relation.

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2. Theory

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Axisymmetric Drop Shape Analysis (ADSA) technique is employed in this work to

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quantify the diffusion of CO2 in light hydrocarbon drop. In the ADSA method, an

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axisymmetric pendant light hydrocarbon drop is created at the end of the capillary tube in a

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high-pressure chamber filled with CO2. When the hydrocarbon drop comes in contact with the 8 ACS Paragon Plus Environment

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CO2 at system pressure and temperature, a diffusion driven mass transfer of CO2 into the light

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hydrocarbon drop occurs. The diffusion of CO2 triggers a series of physical and chemical

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changes in a light hydrocarbon drop. A well-defined and collective experimental method and

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a computational scheme are employed to measure the diffusion coefficient of CO2 in a light

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hydrocarbon drop. The subsequent sections explain in detail about the experimental and

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numerical (computational and theoretical) model applied in the present work.

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2.1 Physical system

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Figure 1A represents the schematic diagram of a cross-section of the pendant hydrocarbon

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drop, surrounded by the CO2 in a high-pressure chamber (VC) (PVT cell). In the Fig. 1A, Pd

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and Pg represent physical region occupied by drop phase (hydrocarbon) and surrounding gas

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phase (CO2) respectively. In Fig. 1, r is the radial coordinates, z is the axial coordinate, R

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represents the radius of the spherical drop, and H represents the diameter of the cell. In this

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work, the following justifications were made in the defining the process of diffusion of CO2

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from the surrounding gas phase into the drop phase. First, the diffusion is the sole means of

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mass transport across the CO2 – hydrocarbon interface, this is achieved by eliminating density

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driven convection (small volume of pendant drop is used) and thermal convention

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(temperature is kept constant) 3. Second, no chemical reactions transpire during the

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experiment. Therefore, the mechanism is entirely physical in nature. Finally, CO2-

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hydrocarbon liquid interface is at quasi-equilibrium state 10.



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Figure 1A: Pendant hydrocarbon drop surrounded by CO2 in the high-pressure chamber and equivalent; 1B: Equivalent spherical drop surrounded by CO2 for numerical analysis.

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2.2 Experimental Setup

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2.2.1 Material

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In the present analysis n-decane, n-heptane, and n-hexane have been used as pendant

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drop phase. The light hydrocarbons used are manufactured by Merck KGaA, with a purity of

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99%. The CO2 gas (PRAXAIR) forms the surrounding gas phase and is greater than 99.9%

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pure. During the analysis of the experiment, the values of viscosity and density of all the

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fluids at various temperature and pressures have been obtained from the NIST Chemistry Web

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Book 20.

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2.2.2 Apparatus

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In the experimental part of this work, a High-Pressure Pendant Drop Apparatus (PD-

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E1700 LL-H) manufactured by EUROTHECHNICA and KRUSS is utilised. The setup is

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employed to capture and evaluate the dynamic drop volume, drop surface area, and interfacial

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tension as a result of the diffusion of the CO2 in the hydrocarbon at various pressures and

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temperatures. Detailed schematics of the experimental setup along with its essential

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components used in the present study is presented in Fig. 2. The major component of the 10 ACS Paragon Plus Environment

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setup is the corrosion resistant see through cylindrical high-pressure visual chamber (VC),

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with limiting pressure and temperature of 68.9 MPa and 180oC respectively, having a

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diameter of 18 mm. The temperature inside the high-pressure chamber is controlled by a

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NiCr-Ni thermocouple, fitted with a digital indicator. The pressure in the chamber is

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maintained by an external pump (maximum pressure of 32 MPa, GILSON) connected to the

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CO2 cylinder during the process of experiment.

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Figure 2: Schematic representation of the experimental setup used in the present study.

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In the experimental setup, the VC is placed between a high-resolution camera (CF03)

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and a light source. The camera captures the digital images (at a predefined frame rate) of the

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pendant hydrocarbon drop, as the CO2 from the surrounding phase diffuses in the drop phase.

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A KRUSS DSA 100 software is used to analyse the acquired images and compute the dynamic

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volume, surface area, and IFT profile of the pendant hydrocarbon drop. The described

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experimental method of measuring the volume of the drop and IFT is completely automated,

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making it an efficient in terms of avoiding human errors.

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2.2.2 Procedure

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Prior to the start of the experiment, the entire system is tested for any leaks at high pressures.

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Following are the steps carried out during the process of experiment.

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1. The high-pressure chamber (VC) and the line connecting the chamber are cleaned by

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deionized water, then acetone and finally dry air is blown to remove any residual

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moisture.

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2. Hydrocarbon (n-decane, n-hexane, and n-heptane) is filled in a hand driven piston

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pump (PG drop) with a capacity of 35 ml. The PG drop is connected to VC to

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introduce the drop.

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3. The required temperature inside the high-pressure chamber is set on the digital thermostat. 4. Once the temperature is set, the high-pressure chamber is pressurized with CO2 at the required pressure. 5. The hydrocarbon pendant drop is introduced into VC at the experimental pressure and temperature.

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6. Great care is employed to make sure that the capillary line through which pendant

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drop is introduced is at almost the same pressure (not more than 0.5 bar greater) as that

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of the VC when the drop is being introduced in VC. This would eliminate any errors

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occurring due to a large difference between the drop and surrounding gas phases.

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7. As soon as the pendant drop is formed, the camera along with the DSA 100 software

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starts to capture the high-resolution digital images of drop for the further analysis.

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8. From the density and viscosity data of drop and surrounding phases, the software

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calculates the dynamic volume and IFT data. The evolution of the volume is used in

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the numerical model to quantify the diffusion of CO2 in hydrocarbon.


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2.3 Mathematical Model

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Natural diffusion is a concentration driven process, and it is important to know the

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concentration of CO2 in the hydrocarbon drop as a function of time and space to understand

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the diffusion process. Series of mathematical equations are adopted in this study, which

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represents the mass transfer of diffused CO2 across the CO2-hydrocarbon interface. These

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equations help in calculating the spatial and temporal-dependent concentration of CO2 in the

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drop phase. With the passage of time and the continuation of the diffusion, the concentration

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of CO2 in drop phase gradually increases. The diffusion proceeds until the drop is saturated

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with the CO2 i.e. no or minimal CO2 concentration gradient exists across the interface. Fick’s

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second law of diffusion adequately describes the mass transfer process from the interface to

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the centre of the hydrocarbon drop. As seen from Fig. 1A the pendant drop is symmetric

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about the z-axis. Hence, the cylindrical coordinate system (r, θ, and z) can be adopted rather

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than a complicated Cartesian coordinate system (x, y, and z)

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adopted from 3 and Eq. (1) represents the diffusion process.

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2 2 ∂C  1 ∂C ∂ C ∂ C = D (t )  + + 2 ∂t ∂r ∂z 2  r ∂r

16

. The mass transfer model is

 , 

(1)

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where D(t) is the diffusion coefficient (m2/s); C is the concentration of CO2 in the

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hydrocarbon drop phase (kg/m3). The pendant hydrocarbon drop and the surrounding gas

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phase constitute a symmetry about the vertical axis (z) (θ = constant) through the centre of the

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pendant drop (Figure 1). Therefore, it is a justified to consider being a 2D axisymmetric

280

instead of a complicated 3D Cartesian system. Consequently, the diffusion of CO2 into the oil

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phase is an unsteady 2D axisymmetric system in the cylindrical coordinate system.

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At the commencement of the diffusion process (t = 0 s), there is no diffusion of CO2

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in the hydrocarbon drop. Therefore, the concentration of CO2 in the pendant drop is zero at t

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C (r , z , t = 0) = 0

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(2)

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The surrounding gas phase and drop phase are in thermodynamic equilibrium at the

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interface 21. Hence, the concentration of the CO2 at the interface remains constant as long as

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the pressure and temperature of the system are maintained constant. Equation (3) represent the

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boundary condition at the interface.

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C ( r = Ri , z = Ri , t > 0) = Co,

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where Ri is the radius of the drop, and Co is the concentration of CO2 at the interface. To

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address the continuity at the centre of the drop (r=0), a zero (constant) flux boundary

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condition is assumed.

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∂C (r = 0, z , t ) =0 ∂r

(3)

(4)

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Because of the mass transfer of CO2 into the drop phase, its volume increases as a

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result of the reduction of viscosity and swelling of the oil drop. Equation (1) along with its

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boundary and initial conditions (Eq. (2) – (4)) are solved, to obtain the time and space

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dependent concentration of CO2 in the drop. A volumetric average of the CO2 concentration

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in the pendant drop (Cavg) is calculated at required time steps by using the Eq. (5).

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Cavg (t ) =

∫∫

( r , z )∈Pd

C (r , z ) rdrdz Co

(5)

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The Cavg (mm3) indicates the amount of CO2 diffused in pendant drop at a given time t.

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The presence of CO2 in the hydrocarbon reduces the viscosity and finally leads to an increase

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in the volume of the drop or the swelling of the drop. Hence, the concentration of CO2 present

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in the drop may be used to calculate the swelling factor (SF), which effectively represents the

305

ratio of the volume of the CO2 saturated hydrocarbon to the initial volume of pure

306

hydrocarbon. Equation (6) accounts for the swelling factor 3.


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Vexp (t ) − Vo  Cavg (t ) dt ∫0 Vexp (t ) 2 T

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SF = 1 +

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where Vexp(t) is the volume of hydrocarbon drop obtained experimentally at time t, and T is

309

the total time of the experiment or simulation; Vo is the volume of hydrocarbon pendant drop

310

(experimental) at time t=0.

Cavg 2 (t )  dt ∫0 Vexp (t )2

T

(6)

,

311

The diffusion of CO2 in the hydrocarbon drop causes an increase in the volume of

312

hydrocarbon. At any instant of time, the volume of the hydrocarbon drop is the summation of

313

the initial volume of drop (Vo) and the increase in volume caused by the dissolved CO2 in the

314

hydrocarbon. The increase in volume may be represented as the product of volume average of

315

concentration (Cavg), and the volume of the CO2 saturated hydrocarbon to the initial volume of

316

pure hydrocarbon (SF-1). Equation (7) calculates the volume of the drop at any given time t 3.

317

V (t ) = Vo + ( SF − 1) ⋅ C avg (t )

(7)

318

Further, an optimisation function (F) (objective function) is developed based on the

319

difference between numerically calculated (V(t)) and experimentally determined (Vexp(t)) the

320

volume of the hydrocarbon drop at time t. The minimum of the optimisation function (Fmin) is

321

further utilised to determine the diffusion coefficient 3.

322

F =

1 T

T

∫ 0

2

Vexp ( t ) − V ( t )  dt *100% Vexp ( t ) 2

(8)

323

The optimisation function is a dependent on V(t) and Vexp(t) (Eq. (8)). Once the

324

experimental volume data at different time steps has been obtained, the optimisation function

325

depends entirely on the numerical volume V(t). From the Eq. (7) and (8), it may be observed

326

that the objective function depends on the Cavg and the swelling factor (SF) 3. Further, Cavg is

327

function of assumed value of the diffusion coefficient. Therefore, F = f (D, SF) and D and SF

328

may be used as the parameter to obtain the minimum objective function (Fmin). The values of 15 ACS Paragon Plus Environment


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Page 16 of 58

329

the D and SF for which the objective function is minimum (Fmin), represent measured CO2

330

diffusion coefficient and oil-swelling factor, respectively. Figure 3 shows the schematic

331

representation of the process involved in the calculation of the diffusion coefficient of CO2 in

332

hydrocarbon phase.

333

With the commencement of the diffusion of CO2 into the pure hydrocarbon, the pure

334

hydrocarbon liquid changes to a dilute mixture containing CO2 and hydrocarbon, which leads

335

to modifications in density and viscosity of the CO2 + hydrocarbon mixture. The extent of

336

viscosity alteration due to the diffusion of CO2 will reveal a lot about the degree of diffusion,

337

changes in IFT, and the change in the volume of the hydrocarbon. In this work, the viscosity

338

of the hydrocarbon drop phase is calculated by Eq. (9) proposed by Herning and Zipperer

339

23

340

hydrocarbon mixtures and 5 % maximum deviation 22.

341

µ drop = 

342

where µCO2 and µHC (cP) are the viscosity of CO2 and the hydrocarbon in at the required

343

pressure and temperature respectively. xCO2 and xHC are mole fraction of CO2 and the

344

hydrocarbon at the required pressure and temperature respectively. MCO2 and MHC are

345

molecular weight of CO2 and the hydrocarbon at the required pressure and temperature

346

respectively.

22

. Viscosities obtained from Eq. (9) have an accuracy of 1.5 % average deviation for

 ( µ co ⋅ xco ⋅ M co ) + ( µ HC ⋅ xHC ⋅ M )  2 2 2 HC  ,   ( xco2 ⋅ M co2 ) + ( xHC ⋅ M HC )   P ,T

(9)

347

Apart from the alterations in the viscosity of the CO2 + hydrocarbon mixture (drop),

348

there will be a change in density of the drop as the diffusion of CO2 progress into the

349

hydrocarbon drop. Zolghadr, et al. 24, related the change in density of the hydrocarbon + CO2

350

mixture to the changes in IFT, and the IFT in turn, could be linked to the diffusion process.

351

Hence, it is critical to analyse the influence of changing density of the drop on the diffusion of


Page 17 of 58

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352

CO2 in the hydrocarbon. Equation (10) gives the analytical equation for the diffusion of the

353

CO2 + hydrocarbon drop using the volume fractions derived from experiments 25 .

354

ρ drop = ( (VCO ⋅ ρCO ) + (VHC ⋅ ρ HC ) )

355

where VCO2 and VHC are the volume fractions of CO2 and hydrocarbon in the drop respectively

356

at given pressure and temperature. ρCO2 and ρHC are the individual densities of CO2 and

357

hydrocarbon in the drop respectively at given pressure and temperature.

2

2

P ,T

,

(10)

358 359 360

Figure 3: Systematic representation of the process involved in the calculation of the diffusion coefficient of CO2 in oil phase



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361

Page 18 of 58

2.4 Numerical Model

362

Two significant changes have been made in the numerical model in the present work

363

compared to Yang and Gu 3. First, a spherical hydrocarbon drop is considered for the

364

numerical analysis instead of the actual shape of the pendant drop. Figure 1B shows the

365

equivalent spherical drop surrounded by the CO2 used for the numerical analysis. In the Fig.

366

1B, R is the radius of the spherical drop. Second, a simple finite difference method (FDM) is

367

used to discretize the partial differential equations instead of the finite difference (FEM)

368

The most attractive characteristic of FDM is that it will be easy to implement compared to

369

FEM. A well thought process is involved in transforming the experimental pendant drop to an

370

equivalent drop for the numerical analysis. First, the initial volume, surface area, and the

371

radius of the pendant drop at the required pressure and temperature are obtained from the

372

experimental data. The experimental radius of the drop is used to get the surface area and

373

volume of the equivalent drop for numerical analysis. The surface area of the drop defines the

374

rate of diffusion. Therefore, it is used as the comparison parameter between pendant drop

375

(experimental) and spherical drop (numerical). From the comparison made between the

376

surface area of initial pendant drop and equivalent spherical drop an error of 3 – 6% was

377

found depending on the pressure of the system. Therefore, it is justified to use a spherical

378

drop for the numerical analysis instead of the actual pendant shape. Spherical drop

379

idealisation reduces the effort and complexity, at the same time maintaining minimal error in

380

the calculation of diffusion coefficient. Table 2 shows the diffusion coefficient obtained by

381

Yang and Gu

382

values) used in Yang and Gu 3. It may be observed that the diffusion coefficient obtained by

383

simpler spherical drop with FDM model matches well (6 % error) with the value obtained by

384

Yang and Gu

385

credibility of the present model. To avoid numerical complexity the effect of shape and

3

3

3, 19

.

and that obtained by the present work for the same system (experimental

using the actual shape of the pendant drop with complex FEM, establish the


Page 19 of 58

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386

interface movement of the drop are neglected. The pendant hydrocarbon drop is assumed to

387

be unchanged in the numerical model 3, 16.

388

Yang et al. (2005) Present Model

Table 2 Verification of the present numerical model Diffusion Coefficient Objective Function % (m2/s), D F -9 0.29 *10 0.073 0.275 *10 -9 0.066

Swelling Factor SF 1.084 1.093

389 390

With the establishment of the mathematical model and obtaining of the experimental

391

data, the following steps are carried out in the numerical model to derive the diffusion

392

coefficient.

393

1.

394

hydrocarbon (Eq. 1) is discretized using unconditionally stable finite difference method

395

(FDM, Crank - Nicolson Method).

396

2.

397

different time steps (Vexp (t)) are applied as an input to the numerical model.

398

3.

399

divided into an equally spaced subinterval.

400

4.

401

(5). Cavg (t) is further used to estimate the SF and V(t) using Eq. (6) and (7) respectively.

402

5.

403

the drop are compared to obtain the best fit using Eq. (8).

404

6.

405

value of Fi is plotted against Di to get Fmin.

406

7.

407

coefficient of the CO2 in hydrocarbon at the required pressure and temperature.

408

8.

The partial differential equation representing the diffusion of CO2 into the

The initial experimental volume of the pendant drop (Vo) and subsequent volumes at

A predefined range of diffusion coefficient is set Di ( n > i > m), and the range is

For each value of Di, the numerical model is solved, and Cavg (t) is calculated from Eq.

The values of the numerically (V(t)) and experimentally (Vexp (t)) calculated volume of

For every value of Di, a corresponding value of Fi and SFi is obtained. Then, each

The diffusion coefficient for which Fmin is obtained is the optimum diffusion

For better accuracy subintervals between adjacent values of Di may be made small.



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409

The governing Eq. (1) is solved for the given set of initial and boundary conditions (Eqs. (2) –

410

(4)) to investigate the diffusion of CO2 in the hydrocarbon drop. A semi-implicit finite

411

difference formulation is adopted to discretize the governing partial differential equation.

412

Crank-Nicolson discretization scheme is used for the second order terms representing

413

diffusion in drop phase (first term on the RHS of Eq. (1)). A two point backward differencing

414

is used discretize the temporal term.

415

3. Results and Discussions

416

3.1 Analysis of the volume of pendant drop

417

Reproducibility is a major factor governing the precision of experimental setup and

418

the credibility of the obtained values. The increase in the volume of the hydrocarbon drop due

419

to the diffusion of CO2 is the principle parameter involved in the estimation of the diffusion

420

coefficient. Hence, at a pressure and temperature, if the volume profile does not vary for

421

different experimental runs, the diffusion coefficient will remain consistent. Figure 4, shows

422

the evolution of pendant drop volume for three different pendant drops Drop 1, Drop 2, and

423

Drop 3 of n-decane at P = 45 bar and T = 45 oC. The three pendant drops (Drop 1, Drop 2,

424

and Drop 3) represent the volume change profile of three different n-decane pendant oil drops

425

under the same conditions to establish the reproducibility. It may be noted that the three drops

426

have a similar increase in volume profiles, with a negligible difference, showing a high degree

427

of reproducibility of the experimental setup and hence, the computation of the diffusion

428

coefficient. Further, for the equilibrium volume of the three drops, i.e. value of volume above

429

which there is no or little change in pendant drop volume. This is in the range of 1.1 to 1.12

430

indicating a difference of 1.7%. A similar reproducibility was attained for all pressures,

431

temperatures, and drop phase fluids (n-hexane and n-heptane).

432


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433 434 435

Figure 4 Comparison between the relative change in volume to establish the reproducibility of the obtained data for n-decane, at P = 45 bar and T= 45 oC.

436

Examining the growth in the volume of the hydrocarbon pendant drop due to the

437

diffusion of the CO2 is an effective approach for analysing and determining the diffusion

438

process. Experimentally and numerically obtained the volume of the pendant drop are plotted

439

against time for n-decane at a constant temperature of 25oC and pressures ranging from 30 -

440

50 bar is shown in the Fig. 5. There is an acceptable agreement between experimental and

441

numerical data. The volume of the n-decane drop increases sharply for a period (20 - 40 s)

442

depending on the pressure, and eventually approaches an equilibrium volume (time at which

443

there would be minimal or no increase in drop volume as the time progresses). As the

444

diffusion progresses, the hydrocarbon drop is gradually saturated with CO2, and there is a

445

reduction in the rate of increase in the volume of the drop. Eventually, the volume reaches a

446

state of steadiness (equilibrium volume). The process of evolution of volume from initial to

447

the equilibrium volume follows a logarithmic in nature. This phenomenon becomes 21 ACS Paragon Plus Environment


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448

predominant as the pressure increases (45 – 50 bar), where the stability in volume growth is

449

achieved at a faster rate compared to a low-pressure range (30 – 35 bar). Approximately 50 %

450

decrease in the time required to reach equilibrium (stability) was observed for 50 bar (high-

451

pressure) compared to 30 bar (low-pressure). The rapid evolution of pendant drop volume at

452

high pressure may be credited mainly to two parameters, the viscosity of the drop, and the

453

solubility of CO2 in the hydrocarbon drop. As the pressure increase, the solubility of CO2 in

454

oil increases and the viscosity of the drop decreases, suggesting an increased diffusion rate at

455

high pressures. Thus, contributing to the observed rapid increases in the volume and shorter

456

time to reach equilibrium at high pressures. Additionally, Yang and Gu 26 attributed the rapid

457

growth in volume and shorter saturation time at high pressures to a higher specific surface of

458

the pendant drop due to a smaller initial volume at high pressures.

459 460 461

Figure 5 Experimental and numerical pendant drop volume of n-decane for pressure 30 – 50 bar at T = 25oC


Page 22 of 58

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462

The results in the Fig. 6 (Fig. 6A, 6B, and 6C) represent the influence of temperature

463

on the evaluation of the volume of the pendant drop (n-decane + CO2) at a constant pressure.

464

Figure 6A, 6B, and 6C show the evolution of the pendant drop volume for three pressures 35

465

bar, 45 bar, and 50 bar respectively at three different temperatures (25oC, 35oC, and 45oC).

466

The vertical dashed line in Fig. 6 indicates the time required to achieve a state of stable

467

volume (equilibrium volume). A common trend is observed in all the Fig. 6A - 6C, for a

468

constant pressure, the time required for attaining the equilibrium volume decreases as the

469

temperature increases from 25oC to 45oC. Thus, indicating a rapid diffusive mass transfer of

470

CO2 in n-decane as the temperature increases. The observed phenomena may be credited

471

mainly to two factors, pendant drop viscosity change, and the kinetic energy. The viscosity of

472

the liquid decreases as the temperature increases, this favours a greater mass transfer of the

473

CO2 into the n-decane drop phase. Additionally, higher the temperature, the greater the kinetic

474

energy and hence, the motions of the molecules. In other words, the rate of diffusive mass

475

transfer of CO2 in hydrocarbon increases as temperature increases. An interesting observation

476

in Fig. 6 is that the magnitude of the equilibrium volume decreases with the reduction in the

477

temperature. There is a 3.2%, 5.83%, and 7.81% reduction in the magnitude of the

478

equilibrium volume (represented by dashed lines) as temperature rises from 25oC to 45oC for

479

pressures 35 bar, 45 bar, and 50 bar respectively. The observed phenomena may be attributed

480

to the decrease in the solubility of the CO2 in the n-decane with the increase in temperature

481

(25oC to 45oC). A greater solubility indicates the higher mass transfer of CO2 from the gas

482

phase (CO2) to the drop phase (n-decane). Hence, the pendant drop of n-decane at 25oC has a

483

greater magnitude of equilibrium volume (stable volume) compared to that at 45oC for

484

pressure. However, this increased the magnitude of equilibrium volume is achieved in a

485

longer time (slower rate) for 25oC compared to 45oC.



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486 487 488

Figure 6 Evolution of volume of n-decane drop at P = 35 bar, 45 bar, and 50 bar and at T = 25oC, 35oC, and 45oC.

489

Figure 7 and 8 show the link between the growth in the volume (experimentally

490

obtained) of the pendant drop and the distribution of the CO2 concentration in the n-decane

491

and n-hexane drop respectively at 35oC and 40 bar. Figure 7A, 7B, and 7C show the

492

numerically calculated distribution of the CO2 concentration in the spherical n-decane drop (a

493

quarter of a drop) at time 5 s, 15 s, and 40 s respectively. Figure 7D represents the

494

experimental observation of the increases in the n-decane drop volume as time progresses.

495

Similarly, Fig. 8A, 8B, and 8C show the numerically calculated distribution of the CO2

496

concentration in the n-hexane drop at time 5 s, 15 s, and 40 s respectively. Figure 8D

497

represents the experimental observation of the increases in the n-hexane drop volume with

498

time. Fick’s law relates the diffusive flux and the concentration gradient across the two


Page 24 of 58

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499

regions or phases. Thus, establishing the concentration gradient as the driving force for the

500

diffusive mass transfer of CO2 into the hydrocarbon drop. At the commencement of the

501

diffusion process, when the diffusive mass transfer of CO2 has begun, there is a larger

502

concentration gradient of CO2 across the CO2-hydrocarbon interface; this may be verified

503

from Fig. 7A which displays the concentration distribution of CO2 in the n-decane drop at t =

504

5 s. From the Fig. 7A it can be seen that at t = 5s (initial period) the CO2 has not penetrated

505

deep into the drop, and most of the CO2 (relative value of 1 - 0.5) is present in merely 30% of

506

the drop (within 0.45 – 0.3 mm radius), in the region near to the interface (0.45mm). Hence,

507

there exists a larger concentration gradient across the interface CO2– n-decane. Consequently,

508

a greater driving force for the CO2 mass transfer into the drop, which is the reason for the

509

rapid increase in the volume during the initial period of the diffusion process. As the time

510

progresses (diffusion progresses) there is greater penetration of the CO2 into the n-decane

511

drop. From Fig. 7B and 7C it can be noted that the CO2 concentration has spread to

512

approximately 55% of the drop at t = 15 s and more than 70% of the region at t = 40 s

513

respectively. The CO2 penetrated around 55% of the drop in 15 s and took 25 s to move from

514

55 % coverage to above 70% of the hydrocarbon drop. Indicating, that as time or diffusion

515

progresses, there is a decrease in the concentration gradient of CO2 across the interface,

516

leading to a reduction in the mass transfer rate. The numerical results in Fig. 7A - 7C coincide

517

with the experimental data of the growth of the drop volume with time (Fig. 7D). In Fig. 7D

518

there is a rapid increase during the initial phase (5 -15 s), and there is a decrease, and finally,

519

stability in volume is achieved (t > 40s) as time progresses. A similar phenomenon is

520

observed for n-hexane (Fig. 8), but there is a faster diffusive mass transfer of CO2 compared

521

to n-decane, indicating a higher rate of reduction in the concentration gradient.

522 523 524 25 ACS Paragon Plus Environment


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525 526 527

Figure 7 Growth in the volume of the drop and the CO2 concentration gradient in the drop in n-decane at t = 5 s, t = 15 s, and t = 40 s, at T = 35 oC and P = 40 bar.

528 529 530

Figure 8 Growth in the volume of the drop and the CO2 concentration gradient in the drop in n-hexane at t = 5 s, t = 15 s, and t = 40 s, at T = 35 oC and P = 40 bar. 26 ACS Paragon Plus Environment

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531

Figure 9 shows the increase in the volume of n-decane, n-heptane, and n-hexane

532

pendant drop with time at 35oC and 45 bar. It is observed that the equilibrium volume

533

(constant volume) is achieved at a faster rate as the hydrocarbon become lighter (n-decane to

534

n-hexane). Hexane takes the least time (t = 20 s), and n-decane (t = 33 s) takes the longest

535

time to attain the equilibrium. Thus, there will be a faster CO2 mass transfer for the lighter

536

hydrocarbon (n-hexane), compared to heavier (n-decane) at a pressure and temperature.

537

Viscosity is an additional factor contributing to the shorter equilibrium time for lighter

538

hydrocarbon (n-hexane) compared to heavier light hydrocarbon (n-decane). The viscosity of

539

n-hexane is 22 % lower than n-heptane and 63% lower than n-decane at 35oC and 45 bar. The

540

significant reduction in viscosity of n-hexane drives the diffusive mass transfer of CO2 into it

541

compared to heavier light hydrocarbon (n-decane). Further, Wilke and Chang

542

experimentally that diffusion coefficient of solute increases with a decrease in solvents

543

molecular weight. Hence, there is a rapid progression of the drop volume surrounded by CO2

544

for lighter hydrocarbon (n-hexane) compared to heavier light hydrocarbon (n-decane) and

545

evidently, lead to greater diffusion.

27

showed

546 547 548

Figure 9 Progression of volume of n-decane, n-heptane, and n-hexane at T = 35oC and P = 45 bar. 27 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

549

Page 28 of 58

3.2 Analysis of the IFT

550

IFT is critical parameter when studying molecular diffusion. However, few studies

551

have looked into the influence of temperature on the IFT of CO2-hydrocarbon due to its

552

unpredictable trend with temperature. Some studies have reported a decrease an inverse

553

relation of IFT with temperature

554

results presented in Fig. 10 is an attempt to resolve the controversial impact of the IFT-

555

temperature relation. Figure 10 represent the experimental equilibrium IFT of n-decane at

556

25oC, 35oC, and 45oC and at pressures 25 bar to 60 bar. It is observed from Fig. 10 that the

557

equilibrium IFT decreases as the temperature increases (IFT = f (1/T)) at 30 bar. However, as

558

the pressure is raised beyond 35 bar, there is an increase in the IFT with the increment of

559

temperature (IFT = f (T)), which is unlike the observations made below 35 bar (P = 30bar).

560

An interesting observation is made at 35 bar, where the IFT is almost constant as the

561

temperature increases from 25oC to 45oC. Thus, indicating a shift from negative (decrease in

562

IFT) to positive (increase in IFT) slope with temperature as the pressure increases from 30 bar

563

to 60 bar. Zolghadr, Escrochi and Ayatollahi

564

heptane, CO2+hexadecane, and CO2+diesel fuel systems. Yang and Gu

565

phenomena and credited decreasing the solubility of CO2 in hydrocarbon with an increase in

566

temperature. Yang, Tontiwachwuthikul and Gu

567

of CO2 in hydrocarbon due to increasing temperature will be more prominent than the

568

increase in the solubility of CO2 with the growth in pressure. However, the solubility is not

569

the only parameter responsible for the observed phenomena. Additionally, the observation

570

may be credited to an increase in the density difference between gas (CO2) and the drop

571

phases (CO2+hydrocarbon). Figure 11 shows the density difference between gas (CO2) and

572

drop (n-decane) phases at the same experimental conditions as that of results in Fig. 10. The

573

densities of the gas phase (CO2) at the experimental conditions are acquired from NIST

28

, while the others have reported a direct relation

24

29

29

. The

reported similar observations for CO2+n3

observed the same

emphasised that the reduction in solubility


Page 29 of 58

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574

Chemistry WebBook 20, while the densities of the drop (CO2+n-decane) phase are calculated

575

from the Eq. (11). It can be observed that the density gradient and temperature plot follows a

576

similar trend to that in Fig. 10. At P < 35 bar density difference decreases as temperature

577

increases, and above 35 bar the density difference increases as temperature increases.

578

Therefore, exhibiting a similar trend of slope shift (from negative to positive slope) as that of

579

IFT as a function of temperature (Fig. 10). Thus, confirming the density difference along with

580

the reduction CO2 solubility with temperature plays a major part in the change of IFT trend of

581

n-decane at 25oC, 35oC, and 45oC, at various pressures (25 bar – 60 bar).

582 583 584

Figure 10 Equilibrium IFT of n-decane at three different temperatures (25oC, 35oC, and 45oC) from 25 bar – 60 bar

585 586 587



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

588 589 590

Figure 11 Density difference between n-decane and CO2 at three different temperatures (25 o C, 35 oC, and 45 oC) from 25 bar – 60 bar

591

The result depicted in Fig. 12 (12A – 12D) illustrate the dependency of dynamic IFT

592

on the degree CO2 concentration in the hydrocarbon drop phase for the n-decane drop at 25 oC

593

and at P = 25 – 50 bar. Figure 12A shows the experimentally observed variation of dynamic

594

IFT for the n-decane drop at pressures ranging from 30 – 50 bar and at temperature 25 oC. It is

595

seen from Fig. 12A that the IFT decreases as the pressure increases and there is approximately

596

60 % decrease in the IFT (14.32 to 5.9 m N/m) as pressure increases from 30 bar – 50 bar.

597

The decline of the IFT with the increment in pressure may be attributed to an increase in the

598

CO2 solubility in hydrocarbon (n-decane) as the pressure increases 29. Further, from Fig. 12A

599

it is observed that there is a gradual reduction in the dynamic IFT until it reaches a stable

600

value (equilibrium IFT) over a period of 30 – 50 s depending on the pressure. At low

601

pressures (30 bar), the dynamic IFT reduction as a function of time curve is rather linear in


Page 30 of 58

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602

nature, however, at high pressure (50 bar) the profile shifts to a more of elliptical in nature.

603

Therefore, there is a gradual transformation from linear to an elliptical reduction of dynamic

604

IFT as pressure increases from 30 – 50 bar. Consequently, indicating that CO2 rapidly

605

saturates in the vicinity of the CO2 – n-decane interface at high pressures compared to low. To

606

support this theory, numerical results depicting CO2 concentration at 30 bar, 40 bar, and 50

607

bar at 25 oC are presented in Fig 12B – 12D, respectively at time t = 15 s. At the same time (t

608

= 15 s) there is a significant buildup of CO2 concentration near the interface (0.4 – 0.45 mm)

609

for 50 bar when compared to 40 bar or 30 bar (least). Hence, there is greater saturation of

610

CO2 in the drop near to the interface at high pressures (50 bar) compared to that at lower (30

611

bar) at same simulation time. Although there is a significant difference in the CO2

612

concentration distribution in the drop near the interface, there is not much of difference

613

regarding buildup of CO2 concentration near the core of the drop (0 – 0.35 mm) among 50bar,

614

40 bar or 30 bar. This further confirms the relation of CO2 concentration buildup (saturation)

615

near the interface, with the rate at which IFT attains equilibrium. Additionally, the rapid

616

stabilisation of dynamic IFT at high pressures is in tandem with that of the rapid evolution of

617

volume at high pressures (Fig. 5), which further cements the observed behaviour. Yang and

618

Gu

619

increase in pressure is the reason for a decrease in IFT with the increase in pressure.

19

suggested that the increased equilibrium CO2 concentration in the drop phase with the



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

620 621 622

Figure 12 The dynamic interfacial tension of n-decane pendant drop and the concentration of CO2 in the pendant drop at t = 15 s, at T = 25 oC and P = 30 – 50 bar

623

Figure 9 demonstrated the influence of molecular weight of light hydrocarbon used in

624

the present work on the evolution of volume of the pendant drop; a similar analysis has been

625

carried out for IFT, represented by Fig. 13. Figure 13 shows the dynamic IFT values for n-

626

decane, n-heptane, and n-hexane at 40 bar and 35oC. The equilibrium IFT is highest for n-

627

decane, lowest for n-hexane, and intermediate for n-heptane (as represented by the data points

628

in the Fig. 13). Thus, indicating a greater resistance to the mass transfer of CO2 across the

629

interface for n-decane compared to n-hexane or n-heptane at a pressure and temperature. An

630

analogy may be made from the results obtained in Fig. 9 and Fig. 13. In Fig. 9 it was 32 ACS Paragon Plus Environment

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631

discussed that time taken to attain equilibrium or stable volume has a direct relation with the

632

molecular weight of the light hydrocarbon drop (n-decane, n-hexane, and n-heptane).

633

Similarly, dynamic IFT has a direct relation with the molecular weight of light hydrocarbon

634

drop used in the present work as observed from Fig. 13. Hence, the molecular weight of the

635

hydrocarbon influences the value of the equilibrium IFT. Thus, at a pressure and temperature,

636

a molecular weight light hydrocarbon is associated with smaller IFT and a significant volume

637

increase when compared to high molecular weight hydrocarbon, indicating a high rate of

638

diffusion-driven CO2 mass transfer across the interface into the hydrocarbon drop phase.

639 640 641

Figure 13 Experimentally measured dynamic interfacial tension (IFT) values of n-decane, nheptane and n-hexane at P = 40 bar and T = 35 oC.

642 643 33 ACS Paragon Plus Environment


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644

3.2 Analysis of the Diffusion Coefficient

645

The diffusion coefficient of CO2 in n-decane, n-heptane, and n-hexane at different

646

equilibrium pressures (25 bar to 65 bar) and temperatures (25 oC, 35 oC, and 45 oC) are

647

plotted in the Fig. 14A, 14B, and 14C, respectively. It is seen from the Fig. 14A that at a

648

constant temperature the calculated diffusion coefficient increases as the pressure increase and

649

this is true for all the hydrocarbon samples used in the present work (n-decane, n-heptane, and

650

n-hexane). This behaviour is due to increase in the equilibrium CO2 concentration (CO2

651

solubility) with pressure

652

lower at higher pressures. The viscosity of the drop is calculated using the Eq. (10) and

653

correlated with the pressure in the Fig. 15. In the Eq. (10), the drop phase viscosity is a

654

function of the mole fraction of CO2 and hydrocarbon in the drop phase; the mole fraction is

655

calculated from the volume fraction obtained from the experiments. It is seen from Fig. 15

656

that the viscosity of the drop (CO2 + Hydrocarbon) decreases with the increases in the

657

pressure (as claimed before) indicating, an inverse relation with pressure. Therefore, the

658

combined effect of an increase in equilibrium CO2 concentration (solubility) and reduction in

659

drop phase viscosity, with the increase in pressure contribute to the increase in the diffusion

660

coefficient of CO2. An interesting observation is made in the Fig. 14; unlike at low-pressure

661

(< 50 bar), the rate of increase in the diffusion coefficient does not remain same at higher

662

pressures. At higher pressures (50 – 60 bar), there is a decline in the rate of increment, and the

663

diffusion coefficient profile tapers. This decrease may be related to an increment in the

664

viscosity of pure hydrocarbons as pressure increases. The increment in viscosity of pure

665

hydrocarbon acts in a way as to oppose the effect of the reduction of viscosity of pendant drop

666

(hydrocarbon +CO2), thus lessening the rate of increase of diffusion at higher pressure.

16

. Additionally, the viscosity of the drop (CO2 + hydrocarbon) is

667


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668 669 670

Figure 14 Calculated diffusion coefficient (m2/s) of CO2 in n-decane (A), n-heptane (B), and n-hexane (C) at three different temperatures (25 oC, 35 oC, and 45 oC) from 25 bar – 60 bar

671

When we compare the values of the diffusion coefficient of n-decane, n-heptane, and

672

n-hexane (Fig 14A, 14B, and 14C), it is be observed that the diffusion coefficient of CO2 in n-

673

hexane is greater than that in n-heptane (intermediate) and n-decane (lowest) at a pressure and

674

temperature. Change in viscosity, IFT, and molecular weight may be credited for this higher

675

diffusion coefficient of CO2 in n-hexane. It is observed in Fig. 12 that hexane had the lowest

676

IFT compared to heptane (intermediate) or decane (highest) for a fixed pressure and

677

temperature. Thus, a minor resistance in shown by CO2-hexane interface compared to CO2-

678

heptane followed by CO2-decane, indicating a higher diffusion coefficient for hexane

679

compared to heptane and decane. Next, from the analysis of the Fig. 9, it is seen that the

680

diffusion coefficient of solute (CO2) increase as the molecular weight of the solvent (n-

681

decane, n-heptane, and n-hexane) decreases. Therefore, the diffusion coefficient of CO2 in 35 ACS Paragon Plus Environment


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682

hexane should be highest (among the samples considered in this work) followed by heptane

683

and finally decane, which is what has been observed in Fig. 14A, 14B, and 14C. Finally, from

684

Fig. 15 it may be seen that the magnitude of the viscosity of CO2+n-hexane drop is low

685

compared to CO2+n-heptane followed by CO2+n-decane at T = 35oC and pressure 30 – 60

686

bar. The viscosity of CO2+n-hexane is approximately 63% less than CO2+n-decane and about

687

22% less than CO2+n-heptane. Hence, the combined influence of low molecular weight

688

(among n-decane, n-heptane, and n-hexane), low viscosity, and low IFT may explain the

689

observed higher diffusion coefficient of CO2 in n-hexane than in n-heptane and n-decane.

690

From the previous discussion of IFT (Fig. 10), it was observed that above a certain

691

pressure (35 bar); IFT is directly proportional to temperature. Hence, the diffusion coefficient

692

should decrease as the temperature increases. However, this is not the case; there is an

693

increase of diffusion coefficient with the increase in temperature from 25oC to 45oC as seen

694

from the Fig 14 (14A, 14B, and 14C). Apart from the lower molecular weight the increase in

695

diffusion coefficient may be attributed to the drop in viscosity of the pendant drop (CO2+n-

696

decane) due to diffusion of CO2 at high temperatures (Fig. 16). Figure 16 shows the change in

697

the viscosity of the pendant drop (CO2+n-decane) with pressure at different temperature

698

(25oC, 35oC, and 45oC) at various pressures. It is observed that there is a gradual decrease in

699

the viscosity of the drop phase as the temperature increases. Depending on the pressure,

700

approximately 11 – 13 % reduction in viscosity of drop phase is seen when the temperature

701

increases from 25oC to 35oC and a 20 – 23 % reduction is seen when the temperature is

702

increased from 25oC – 45oC. Thus, the increase in diffusion coefficient due to a decrease in

703

the viscosity is dominant than the combined influence of, a possible decrease in diffusion

704

coefficient due to an increase in IFT and a drop in the solubility with an increase in the

705

temperature. The same holds true for heptane and n-hexane.


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707 708 709

Figure 15 Comparison of viscosity of the drop of n-decane, n-heptane, and n-hexane at T=35 o C and P= 30 bar - 60 bar

710 711

Figure 16 Viscosity of pendant drop (n-decane +CO2) at T= 25oC, 35oC, and 45oC. 37 ACS Paragon Plus Environment


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712

4. Conclusions

713

The present work uses an experimental method together with a numerical scheme for

714

measuring the diffusion coefficient of CO2 in light hydrocarbons (n-decane, n-heptane, and n-

715

hexane). The experimental work is carried out using the dynamic pendant drop volume

716

analysis (DPVA), which provides the increase in the volume of light hydrocarbon as CO2

717

dissolves in it. The experimental work has been carried out at temperatures between 25oC -

718

45oC and pressure of 25 – 65 bar.

719

A model is developed based on Fick’s law of diffusion to calculate the CO2

720

concentration in the pendant drop. In the numerical analysis, a spherical drop is assumed

721

rather than an actual pendant shaped drop, with merely a 3 – 6% error between the surface

722

area of spherical drop and the actual pendant drop. This significant change in the numerical

723

model forms a major advantage of the current work as it makes the model simple and fast, at

724

the cost of negligible error. The numerical model with the help of experimental volume data

725

calculates the concentration of CO2 in the drop phase, which is used to get the calculated

726

volume of the drop phase. An optimisation function is used to determine the difference

727

between the experimental and calculated volume of the light hydrocarbon drop phase. The

728

diffusion coefficient is used as the adjustable parameter to determine the minimum value of

729

the optimisation function. The minimum value of the optimisation function is used to obtain

730

the diffusion coefficient of CO2 in the light hydrocarbon.

731

From the obtained results, few unique and important aspects can be concluded. First,

732

the relation of the IFT with temperature is a function of the difference in density of the drop

733

phase (Hydrocarbon + CO2) and the surrounding gas phase (CO2). Similar to density

734

difference the IFT from negative (decrease in IFT) to positive (increase in IFT) slope with

735

temperature as the pressure increases from 30 bar to 60 bar. Second, the diffusion coefficient

736

of CO2 is higher for lighter hydrocarbon (n-hexane) compared to heavier (n-decane). 38 ACS Paragon Plus Environment

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737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783

5. References (1).

Riazi, M.; Sohrabi, M.; Jamiolahmady, M.; Ireland, S. In Oil recovery improvement using CO2-enriched water injection. EUROPEC/EAGE Conference and Exhibition, 2009; Society of Petroleum Engineers: 2009. (2). Sohrabi, M.; Riazi, M.; Jamiolahmady, M.; Kechut, N. I.; Ireland, S.; Robertson, G. Carbonated water injection (CWI)–a productive way of using CO 2 for oil recovery and CO 2 storage. Energy Procedia. 2011, 4, 2192-2199. (3). Yang, C.; Gu, Y. New experimental method for measuring gas diffusivity in heavy oil by the dynamic pendant drop volume analysis (DPDVA). Ind. Eng. Chem. Res. 2005, 44, 4474-4483. (4). Schmidt, T.; Leshchyshyn, T.; Puttagunta, V. Diffusivity of CO2 into reservoir fluids. Paper 1982, 82, 33-100. (5). Nguyen, T.; Ali, S. Effect of nitrogen on the solubility and diffusivity of carbon dioxide into oil and oil recovery by the immiscible WAG process. J. Can. Pet. Technol. 1998, 37. (6). Sigmund, P. M. Prediction of molecular diffusion at reservoir conditions. Part 1Measurement and prediction of binary dense gas diffusion coefficients. J. Can. Pet. Technol. 1976, 15. (7). Zabala, D.; Nieto-Draghi, C.; de Hemptinne, J. C.; Lopez de Ramos, A. L. Diffusion coefficients in CO2/n-alkane binary liquid mixtures by molecular simulation. J. Phys. B: At., Mol. Opt. Phys. 2008, 112, 16610-16618. (8). Upreti, S. R.; Mehrotra, A. K. Experimental measurement of gas diffusivity in bitumen: results for carbon dioxide. Ind. Eng. Chem. Res. 2000, 39, 1080-1087. (9). Sheikha, H.; Pooladi-Darvish, M.; Mehrotra, A. K. Development of graphical methods for estimating the diffusivity coefficient of gases in bitumen from pressure-decay data. Energy Fuels. 2005, 19, 2041-2049. (10). Zheng, S.; Li, H. A.; Sun, H.; Yang, D. Determination of Diffusion Coefficient for Alkane Solvent–CO2 Mixtures in Heavy Oil with Consideration of Swelling Effect. Ind. Eng. Chem. Res. 2016, 55, 1533-1549. (11). Zhang, Y.; Hyndman, C.; Maini, B. Measurement of gas diffusivity in heavy oils. J. Pet. Sci. Eng. 2000, 25, 37-47. (12). Riazi, M. R. A new method for experimental measurement of diffusion coefficients in reservoir fluids. J. Pet. Sci. Eng. 1996, 14, 235-250. (13). Upreti, S. R.; Mehrotra, A. K. Diffusivity of CO2, CH4, C2H6 and N2 in Athabasca bitumen. Can. J. Chem. Eng. 2002, 80, 116-125. (14). Renner, T. Measurement and correlation of diffusion coefficients for CO2 and rich-gas applications. SPE Reservoir Eng. 1988, 3, 517-523. (15). Grogan, A.; Pinczewski, V.; Ruskauff, G. J.; Orr Jr, F. Diffusion of CO2 at reservoir conditions: models and measurements. SPE Reservoir Eng. 1988, 3, 93-102. (16). Yang, D.; Tontiwachwuthikul, P.; Gu, Y. Dynamic interfacial tension method for measuring gas diffusion coefficient and interface mass transfer coefficient in a liquid. Ind. Eng. Chem. Res. 2006, 45, 4999-5008. (17). Liu, Y.; Teng, Y.; Lu, G.; Jiang, L.; Zhao, J.; Zhang, Y.; Song, Y. Experimental study on CO 2 diffusion in bulk n-decane and n-decane saturated porous media using micro-CT. Fluid Phase Equilib. 2016, 417, 212-219. (18). Zhang, W.; Wu, S.; Ren, S.; Zhang, L.; Li, J. The modeling and experimental studies on the diffusion coefficient of CO 2 in saline water. J. CO2 Util. 2015, 11, 49-53.



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784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813

(19). Yang, D.; Gu, Y. Determination of Diffusion Coefficients and Interface Mass-Transfer Coefficients of the Crude Oil− CO2 System by Analysis of the Dynamic and Equilibrium Interfacial Tensions. Ind. Eng. Chem. Res. 2008, 47, 5447-5455. (20). Lemmon, E.; McLinden, M.; Friend, D.; Linstrom, P.; Mallard, W. NIST chemistry WebBook, NIST standard reference database number 69. National Institute of Standards and Technology, Gaithersburg. 2011. (21). Jamialahmadi, M.; Emadi, M.; Müller-Steinhagen, H. Diffusion coefficients of methane in liquid hydrocarbons at high pressure and temperature. J. Pet. Sci. Eng. 2006, 53, 47-60. (22). Herning, F.; Zipperer, L. Calculation of the viscosity of technical gas mixtures from the viscosity of individual gases. Gas u. Wasserfach. 1936, 79, 69. (23). Davidson, T. A. A simple and accurate method for calculating viscosity of gaseous mixtures. 1993. (24). Zolghadr, A.; Escrochi, M.; Ayatollahi, S. Temperature and composition effect on CO2 miscibility by interfacial tension measurement. J. Chem. Eng. Data. 2013, 58, 11681175. (25). Wikibooks, T. F. T. P. Introduction to Chemical Engineering Processes/The most important point. https://en.wikibooks.org/w/index.php?title=Introduction_to_Chemical_Engineering_Processe s/The_most_important_point&oldid=2833056 (September 27, 2016), (26). Yang, C.; Gu, Y. Diffusion coefficients and oil swelling factors of carbon dioxide, methane, ethane, propane, and their mixtures in heavy oil. Fluid Phase Equilib. 2006, 243, 64-73. (27). Wilke, C.; Chang, P. Correlation of diffusion coefficients in dilute solutions. AIChE Journal. 1955, 1, 264-270. (28). Mackay, D.; Hossain, K. Interfacial tensions of oil, water, chemical dispersant systems. Can. J. Chem. Eng. 1982, 60, 546-550. (29). Yang, D.; Tontiwachwuthikul, P.; Gu, Y. Interfacial tensions of the crude oil+ reservoir brine+ CO2 systems at pressures up to 31 MPa and temperatures of 27 C and 58 C. J. Chem. Eng. Data. 2005, 50, 1242-1249.

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826 827

Table of Content Image

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Table of Contents 82x45mm (150 x 150 DPI)

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Figure 1: A: Pendant hydrocarbon drop surrounded by CO2 in the high-pressure chamber and equivalent; B: Equivalent spherical drop surrounded by CO2 for numerical analysis. 554x291mm (96 x 96 DPI)



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Figure 2: Schematic representation of the experimental setup used in the present study. 540x289mm (96 x 96 DPI)


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Figure 3: Systematic representation of the process involved in the calculation of the diffusion coefficient of CO2 in oil phase. 48x52mm (300 x 300 DPI)



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Figure 4: Comparison between the relative change in volume to establish the reproducibility of the obtained data for n-decane, at P = 45 bar and T= 45 oC. 172x142mm (300 x 300 DPI)


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Figure 5: Experimental and numerical pendant drop volume of n-decane for pressure 30 – 50 bar at T = 25oC. 174x148mm (300 x 300 DPI)



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Figure 6: Evolution of volume of n-decane drop at P = 35 bar, 45 bar, and 50 bar and at T = 25oC, 35oC, and 45oC. 397x274mm (300 x 300 DPI)


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Figure 7 Growth in the volume of the drop and the CO2 concentration gradient in the drop in n-decane at t = 5 s, t = 15 s, and t = 40 s, at T = 35 oC and P = 40 bar. 246x186mm (150 x 150 DPI)



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Figure 8 Growth in the volume of the drop and the CO2 concentration gradient in the drop in n-hexane at t = 5 s, t = 15 s, and t = 40 s, at T = 35 oC and P = 40 bar. 34x26mm (300 x 300 DPI)


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Figure 9 Progression of volume of n-decane, n-heptane, and n-hexane at T = 35oC and P = 45 bar. 172x129mm (300 x 300 DPI)



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Figure 10 Equilibrium IFT of n-decane at three different temperatures (25oC, 35oC, and 45oC) from 25 bar – 60 bar 167x118mm (300 x 300 DPI)


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Figure 11 Density difference between n-decane and CO2 at three different temperatures (25 oC, 35 oC, and 45 oC) from 25 bar – 60 bar 172x130mm (300 x 300 DPI)



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Figure 12 The dynamic interfacial tension of n-decane pendant drop and the concentration of CO2 in the pendant drop at t = 15 s, at T = 25 oC and P = 30 – 50 bar 190x174mm (150 x 150 DPI)


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Figure 13 Experimentally measured dynamic interfacial tension (IFT) values of n-decane, n-heptane and nhexane at P = 40 bar and T = 35 oC. 175x142mm (300 x 300 DPI)



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Figure 14 Calculated diffusion coefficient (m2/s) of CO2 in n-decane (A), n-heptane (B), and n-hexane (C) at three different temperatures (25 oC, 35 oC, and 45 oC) from 25 bar – 60 bar 397x274mm (300 x 300 DPI)


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Figure 15 Comparison of viscosity of the drop of n-decane, n-heptane, and n-hexane at T=35 oC and P= 30 bar - 60 bar 246x168mm (96 x 96 DPI)



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Figure 16 Viscosity of pendant drop (n-decane +CO2) at T= 25oC, 35oC, and 45oC 175x142mm (300 x 300 DPI)


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Experimental and Numerical Method for Estimating Diffusion

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