**3.1 Establishment of diffusion model**

2 Mass Transfer in Chemical Engineering Processes

*DAB* —the diffusion coefficient of component A in component B, 2 1 *m s* . Therefore, Fick's law says diffusion rate is proportional to concentration gradient directly and the ratio coefficient is the molecular diffusion coefficient. The Fick's diffusion law is

*A A*

*A <sup>n</sup> <sup>p</sup> <sup>c</sup>*

*A <sup>D</sup> dp <sup>N</sup>*

*z p <sup>A</sup> <sup>A</sup> <sup>p</sup>*

*<sup>A</sup> A i*

*<sup>A</sup> A i*

When area A is constant, eq. 10 become a basic equation of one-dimensional unsteady state

0

Similarly, we can obtain the liquid phase diffusion, which is written as follows:

2 11 1 2 *c c Ac* <sup>1</sup> *<sup>D</sup> t Az z z* 

*<sup>A</sup>*

*RTz* ( *Gk* -mass transfer coefficient) ,then:

Fick also presented a more general conservation equation:

diffusion, which is also known as Fick's second law.

*dz* —concentration gradient of component A at z-direction, <sup>3</sup> *kmol m m* / / ;

*A A*

*A*

*i A*

*dc NJ D dz* (2)

*v RT* (3)

*RT dz* (4)

*<sup>D</sup> N dz dp RT* (5)

*<sup>D</sup> Nz p p RT* (6)

*<sup>D</sup> N pp RTz* (7)

*N k <sup>A</sup> GA i p p* (8)

*N kc c <sup>A</sup> Li A* (9)

*t xL* 0 0, (10)

*<sup>A</sup> dc*

called the first form. Gas diffusion:

For:

We can obtain:

Define *<sup>G</sup> <sup>D</sup> <sup>k</sup>*

Where *<sup>L</sup>*

*<sup>D</sup> <sup>k</sup> z*

In 2007, through the PVT experiments of molecular diffusion, Southwest Petroleum University, Dr. Wang Zhouhua established a non-equilibrium diffusion model and obtained a multi-component gas diffusion coefficient. The establishment of the model is shown in fig.1, with the initial composition of the known non-equilibrium state in gas and liquid phase. During the whole experiment process, temperature was kept being constant. The interface of gas - liquid always maintained a balance, considering the oil phase diffuses into the vapor phase. When the diffusion occurs, the system pressure, volume and composition of each phase will change with time until the system reaches balance.

Fig. 1. Physical model schematic drawing

As shown in fig.1, *<sup>i</sup> x* and *<sup>i</sup> y* are i-composition molar fraction of liquid and gas phase respectively. *Coi* and *Cgi* are i-composition mass fraction of liquid and gas phase respectively. ni is the total mole fraction of i-composition, mi is the total mass fraction of icomposition. *Lo* and *Lg* are the height of liquid and gas phase respectively. *<sup>b</sup>* , defined as *L t <sup>o</sup>* / , is the rate of movement of gas-liquid interface. *z* , *<sup>o</sup> z* and *<sup>g</sup> z* are coordinate axis as shown in fig.1.

Research on Molecular Diffusion Coefficient

Fig. 2. Flow chart of calculation procedure

oil and gas phase

oil and gas phase

NO

coefficient of each component

of Gas-Oil System Under High Temperature and High Pressure 5

1 start

2 giving the values of all phase and components' basic parameters at t0

components in oil and gas at t1

gas, and boundary parameters at t2

3 calculating Ci, ni and the distribution of all

4 calculating Ci, ni and the distribution of all components in oil and

5 calculating the P at the first and the second time step

6 judging P1-P2<δ

7 making the time and space variables dimensionless

8 calculating the diffusion coefficients Di of each component in

END

YES

9 giving a value of Rc

10 calculating Ci,ni, Cbi and nbi (at boundary)and fugacity

12 calculating Ci,ni,the distribution of each component in

YES

11 judging phase equilibrium

13 calculating the P in the PVT cell

If there is component concentration gradation, diffusion between gas and liquid phase will occur. Under the specific physical conditions of PVT cell, when gas phase diffuses into oil phase, the density of oil phase will decrease. According to the physical characteristics of diffusion, the concentration of light component in oil phase at the gas-liquid interface is higher than that of oil phase at the bottom of PVT cell, that is to say, the vector direction of concentration gradient of light component in oil phase is consistent with the coordinate direction of oil phase *<sup>o</sup> z* . From the above analysis, we can see oil density along the coordinate direction is gradually decreasing, so there is no natural convection. The established models with specific boundary condition are as follows: Oil phase:

$$\begin{cases} \frac{\partial C\_{o\boldsymbol{\ell}}}{\partial t} = \frac{\partial}{\partial \boldsymbol{z}\_o} \left[ D\_{o\boldsymbol{\ell}} \frac{\partial C\_{o\boldsymbol{\ell}}}{\partial \boldsymbol{z}\_o} \right] \\ C\_{o\boldsymbol{\ell}} \left( \boldsymbol{z}\_o, \mathbf{0} \right) = C\_{o\boldsymbol{\ell}}^1 \left( \boldsymbol{z}\_o \right) \\ \frac{\partial C\_{o\boldsymbol{\ell}} \left( \boldsymbol{0}, t \right)}{\partial \boldsymbol{z}\_o} = \mathbf{0} \\ C\_{o\boldsymbol{\ell}} \left( \boldsymbol{L}\_o, t \right) = C\_{o\boldsymbol{\ell}} \end{cases} \tag{11}$$

Gas phase:

$$\begin{cases} \frac{\partial C\_{gl}}{\partial t} = \frac{\partial}{\partial z\_s} \left[ D\_{gl} \frac{\partial C\_{gl}}{\partial \overline{z}\_s} \right] \\ C\_{gl} \left( z\_s, 0 \right) = C\_{gl}^1 \left( z\_s \right) \\ C\_{gl} \left( 0, t \right) = C\_{gbl} \\ \frac{\partial C\_{gl} \left( L\_g, t \right)}{\partial z\_s} = 0 \end{cases} \tag{12}$$

<sup>1</sup> *Coi* , <sup>1</sup> *Cgi* are i-component initial molar concentration of oil and gas phase, respectively, <sup>3</sup> *kmol m*/ .

*Cobi* ,*Cgbi* are i-component molar concentration of oil and gas phase at oil-gas interface respectively, <sup>3</sup> *kmol m*/ .

In order to study the law of mutual diffusion between components, eq. 11 and 12 need to be solved. Because the velocity of gas-oil interface movement during the diffusion process is rather slow, we introduce a time step *t* . Then, we assume that gas-oil interface doesn't move, the height of oil and gas phase keeps the same, molar concentration at boundary and*Cobi* ,*Cgbi* are constant during the whole time step,. And in the next time step, refresh the *Lo* , *Lg* and their values are the calculated result of the former time step, so each component concentration of oil and gas phase can be calculated. Continue the circular calculation like this way till gas and liquid phase reach balance. The detailed calculation procedure is as follows in fig. 2.

If there is component concentration gradation, diffusion between gas and liquid phase will occur. Under the specific physical conditions of PVT cell, when gas phase diffuses into oil phase, the density of oil phase will decrease. According to the physical characteristics of diffusion, the concentration of light component in oil phase at the gas-liquid interface is higher than that of oil phase at the bottom of PVT cell, that is to say, the vector direction of concentration gradient of light component in oil phase is consistent with the coordinate direction of oil phase *<sup>o</sup> z* . From the above analysis, we can see oil density along the coordinate direction is gradually decreasing, so there is no natural convection. The

*gi gi gi g g*

*C C D tz z*

, <sup>0</sup>

<sup>1</sup> *Coi* , <sup>1</sup> *Cgi* are i-component initial molar concentration of oil and gas phase, respectively,

*Cobi* ,*Cgbi* are i-component molar concentration of oil and gas phase at oil-gas interface

In order to study the law of mutual diffusion between components, eq. 11 and 12 need to be solved. Because the velocity of gas-oil interface movement during the diffusion process is rather slow, we introduce a time step *t* . Then, we assume that gas-oil interface doesn't move, the height of oil and gas phase keeps the same, molar concentration at boundary and*Cobi* ,*Cgbi* are constant during the whole time step,. And in the next time step, refresh the *Lo* , *Lg* and their values are the calculated result of the former time step, so each component concentration of oil and gas phase can be calculated. Continue the circular calculation like this way till gas and liquid phase reach balance. The detailed calculation

<sup>1</sup> ,0

*gi g gi g*

*Cz Cz*

(11)

(12)

*oi oi oi o o*

*C C <sup>D</sup> tz z*

<sup>1</sup> ,0

*oi o oi o*

*Cz Cz*

0, <sup>0</sup>

*oi o*

*C t z C Lt C*

*C tC C Lt z*

*gi g g*

0,

*gi gbi*

,

*oi o obi*

established models with specific boundary condition are as follows:

Oil phase:

Gas phase:

<sup>3</sup> *kmol m*/ .

respectively, <sup>3</sup> *kmol m*/ .

procedure is as follows in fig. 2.

Fig. 2. Flow chart of calculation procedure

Research on Molecular Diffusion Coefficient

diffusion test of N2-Oil (20 *MPa*, 60Ԩ).

**4.3.1 Experimental apparatus** 

**4.2 Experimental temperature and pressure** 

**4.3 Experimental apparatus and experimental procedures** 

gas booster pump. The flow chart is shown in fig.3.

Fig. 3. The flow chart of diffusion experiment

**4.3.2 Experimental procedures** 

of Gas-Oil System Under High Temperature and High Pressure 7

Three groups of gas diffusion tests are conducted. The first one is the diffusion test of CO2- Oil (20*MPa*, 60Ԩ); the second is the diffusion test of CH4-oil (20 *MPa*, 60Ԩ); the third is the

Diffusion experiments are conducted mainly in DBR phase behavior analyzer. The other equipments include injection pump system, PVT cell, flash separator, density meter, temperature control system, gas chromatograph, oil chromatograph, electronic balance and

Before testing, firstly, oil and gas sample under normal temperature are transferred into the intermediate container and put the middle container in a thermostatic oven. Then the oven is being heated up to 60Ԩ for 24 hours in general. The pressure of oil and gas sample under high-temperature is increased to the testing pressure—20*MPa*. Meanwhile, the temperature and pressure of PVT cell is increased to the experimental temperature and pressure, and then, the height of plunger is recorded. Secondly, transfer the oil sample into PVT cell and record the height of plunger again when the oil sample becomes steady. The difference of the two recorded heights is the oil volume. Thirdly, transfer the gas sample into PVT cell from the top of PVT cell. During the transferring process, it is necessary to keep a low sample transfer rate so that it would not lead to convection. Record the height of plunger and liquid level once completing sample transfer. Fourthly, start the diffusion test and make a record of time, pressure and liquid level. If variation of pressure is less than 1 psi during an interval of 30 minutes, it means gas-oil have reached the diffusive equilibrium and the
