The theory model and analytic answer of gas diffusion

Procedia Earth and Planetary Science 1 (2009) 328–335

Procedia Earth and Planetary Science www.elsevier.com/locate/procedia

The 6th International Conference on Mining Science & Technology

The theory model and analytic answer of gas diffusion Wu Hai-jina,b,*, Lin Bai-quana,b, Yao Qiana,b a

School of Safety Engineering, China University of Mining & Technology. Xuzhou 221008, China b State Key Laboratory of Coal Resources and Mine Safety Xuzhou 221008, China

Abstract In order to investigate the gas diffusion process in the coal seam, this paper simplifies the complex course into the capillary model and gets the motion differential equation when the gas diffusion obeys Fick’s law. Meanwhile, with the knowledge of differential equation and mathematic-physical equation, with the help of MATLAB as well, this paper obtains the analytic answer of the speed and flow rate of gas diffusion for the first time, which can supply more accurate quantized parameters to solve the gas problem in coal mine.

Keywords: fick diffusion; the capillary model; the speed of diffusion; flow rate; analytic answer

China is one of the countries whose gas problem is pretty severe, and over half coal mines are high-gas mines[1]. For the past few years, as the increase of the excavating depth, the development of the mechanization, and the constant increase of the gas leak as well, the gas problem has become the main problem which threatens the safety in production in the world. The gas leak in coal mine, the gas outbrust in coal seam and the gas drainage are all refer to the gas flowing problems[2] ,Therefore, it’s necessary to investigate the gas flowing discipline in coal mine. The gas flowing process is pretty complex in coal seam which is closely connected with the structure of the media and the existing form of the gas[3].According to the pore and cranny distribution in the coal, the flowing form in the cranny system is laminar flowwhile diffusing in the hole structure. The paper mainly investigates the diffusion course of the gas in coal seam, establishes the relative theoretical model, analyses of the speed and flow rate of gas diffusion, and supplies more accurate quantized parameters to solve the gas problem in coal mine. 1. The theory model of gas diffusion In fact, the dimension of the pores and crannies are uneven in coal seam, so it’s necessary to simplify the gas flowing model. In addition, the simplified model can reflect the gas flowing discipline in coal mine in view of experiment. As for the problem of gas flowing in the porous medium, the capillary model is the basic form[4].Therefore, this paper supposes that the gas diffuses through the capillary model, which means if it is supposed to be infinitesimal in any position of the coal seam, there is only one capillary for the diffusion channel of the gas(actually, the fictitious capillary channel isn’t exist in coal seam). If the capillary is considered as the gas diffusion

* Corresponding author. Tel.: +86-13775988564; fax: +86-516-83884401. E-mail address: [email protected].

1878-5220/09/$– See front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.proeps.2009.09.052

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channel, so the gas diffusion channels in coal seam is a lattice construction which is made up of unlimited quantities of capillary. In order to grasp the gas diffusion discipline in capillary model completely, firstly, the differential equation of motion with the knowledge of fluid mechanics and permeation fluid mechanics is set up. Because the gas diffusion course in coal seam is pretty complex, we should make the physical process clear at first and put out interrelated assumptions: • The coal seam is homogeneous and isotropical. • The coal seam is dry, and the flow in capillary model is a kind of single-phase flow, and there is no capillary force when flowing in the model. • The pore structure in coal seam is changeless. • The gas molecules cover the cross section of the capillary all the time when flowing, and the gas diffusion coefficient D is constant. • The gas diffusion process in capillary model is astable. • The gas diffusion process in capillary model is a isothermal process. • If tiny objects are taken in any position of the coal seam, the adsorption and resolution of the gas are balanced at the surface.

Fig. 1. The Simplified Math Model of Gas Diffusion Discipline in Capillary

Diffusion is a process which describes the molecules’ free motion from high concentration to low concentration. The diffusion discipline just researches the relation between the speed of the fluid diffusion and the concentration gradient. Coal is a kind of porous medium, according to the research home and aboard, when the diameter of the pore is small (usually less than 1mm), the gas molecules can not flow freely, and the mess flow of the gas is in proportion to the density gradient, which accords with Fick’s diffusion law[5]:

J = −D

∂X ∂n

(1)

3 2 J —The speed of diffusion, m /(m ·d); 3 3 X —The content of gas in coal, m /m ; 2 D —The gas diffusion coefficient in coal, m /d. The astable diffusion obeys the Fick’s second diffusion law:

∂c = D∇ 2 c ∂t

(2)

2 D —The gas diffusion coefficient in porous medium, which has no connection with the spatial location, m /s. The infinitesimal diffusion style is along the radius direction of the ball, and only part of the gas diffuses into the capillary along the r axis direction. ∇ —Hamiltonian differential operator.

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330

∂c D ∂ 2 ∂c = (r ) ∂r ∂t r 2 ∂r = D(

∂ 2 c 2 ∂c ) + ∂r 2 r ∂r

(3)

c : The gas volume concentration in capillary, so ∂c =

u∂tπr 2 u∂t = lπr 2 l c u ∂ ∴ = ∂t l

(4)

Therefore, when gas obeys the Fick’s diffusion law in capillary: ⎧ ∂c ⎛ ∂ 2 c 2 ∂c ⎞ ⎟ ⎪ = D⎜⎜ 2 + r ∂r ⎟⎠ ⎝ ∂r ⎪ ∂t ⎪ c=0 ⎨r = l ⎪r = 0 c = c 1 ⎪ ⎪⎩

(5)

∂c —The gas volume concentration’s changing discipline with the change of time in capillary; ∂t ∂c —The gas volume concentration gradient in capillary; ∂r 2 D —The gas diffusion coefficient in porous medium, m /s; l —The length of the capillary, m; 3 3 c1 —The original gas volume concentration in coal, m /m .

So, the differential equation of motion when the gas diffusion obeys the Fick‘s law in capillary is: ⎧ ⎛ ∂ 2 c 2 ∂c ⎞ ⎟ ⎪u = lD⎜⎜ 2 + r ∂r ⎟⎠ ⎝ ∂r ⎪ ⎪ c=0 ⎨r = l ⎪r = 0 c = c 1 ⎪ ⎪⎩

2. The analytic answer of the gas diffusion speed If y = rc , and

c=

y , the differential equation of motion can be solved as below: r

⎛ ∂ 2 c 2 ∂c ⎞ ∂c ⎟ = D⎜⎜ 2 + ∂t r ∂r ⎟⎠ ⎝ ∂r

(6)

W. Hai-jin et al. / Procedia Earth and Planetary Science 1 (2009) 328–335

∴

331

⎧ ∂c 1 ∂y ⎪ ∂t = r ∂t ⎪ ∂y ⎪ r−y 1 ∂y y ⎪ ∂c ∂r = − ⎨ = 2 r r r ∂r r 2 ∂ ⎪ ⎪ ∂2 y ∂y ∂y 2 ∂2 y r − y ⋅ 2r r− ⎪ ∂ 2c 2 2 2 ∂y 2 y + = ∂r − 2 ⎪ 2 = ∂r 2 ∂r − ∂r 4 r r ∂r r 3 r r ⎩ ∂r

∴ ∂y = D ∂

2

y ∂r 2

∂t

(7)

If y = R(r )T (t ) ⇒ R (r )T (t )′ = DT (t )R (r )″ ′ ″ T (t ) R(r ) = = −λ DT (t ) R(r )

(8)

″ ⎧ So, ⎪ R(r ) + λR(r ) = 0 ⎨ ⎪⎩T (t )′ + λDT (t ) = 0

Boundary conditions: ⎧⎪c r =0 = c1 ⇒ y r =0 = 0 ⇒ R(0) = 0 ⎨ ⎩⎪c r =l = 0 ⇒ y r =l = 0 ⇒ R(l ) = 0

(9)

As for the second-order homogeneous linear differential equation: ″ R (r ) + λR (r ) = 0

R(r )1 = cos λ r , and R(r )2 = sin λ r are two special solutions of the equation,

In addition, because of sin λ r = tan λ r , is not constant all the time, so R (r )1 , and R(r )2 are two special cos λ r solutions of the equation which are linearly independent.

∴ The general solution of the equation: R(r )″ + λR(r ) = 0 ,then R (r ) = A cos λ r + B sin λ r

(λ > 0)

⎧ R (0 ) = 0 ⎧ A = 0 Q⎨ ⇒⎨ ⇒ B sin λ l = 0 ⎩ R (l ) = 0 ⎩ A cos λ l + B sin λ l = 0

Because there’re untrivial solutions, so B ≠ 0 ⇒ sin λ l = 0 ⇒ λ l = nπ (n = 1,2,3L) ⎛ nπ ⎞ ∴ λn = ⎜ ⎟ ⎝ l ⎠

2

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332

∴

⎧ R (r )″ + λR(r ) = 0 The general solution of the equation ⎪⎪ ⎨ R (0) = 0 ⎪ R (l ) = 0 ⎪⎩

R(r )n = Bn sin

nπr l

(0 < r < l )

is:

(Bn ≠ 0)

(10)

As for the equation: T (t )′ + λDT (t ) = 0 dT (t ) dT (t ) = −λDT (t ) ⇒ = −λDdt dt T (t )

∴ Integrate : T (t ) = Ce −λDt T (t )n = Cn e

Q

−

(C ≠ 0) , As for different

n 2π 2 Dt

n,

(Cn ≠ 0)

l2

(11)

y = R(r )T (t )

∴ yn = R(r ) n T (t )n = Bn sin nπr ⋅ Cn e −

n 2π 2 Dt l2

l

So, y = y = B C sin nπr e ∑ ∑ n n n l n =1 n =1 ∞

∞

−

nπr − = Bn Cn sin e l

n 2π 2 Dt l2

n 2π 2 Dt l2

(12)

When t = 0 , suppose if the gas volume concentration satisfies the linear relation, the expression is: c=

(

c1 (l − r ) , y = rc = c1 lr − r 2 l l

)

∴ y = f (r ) = c1 (lr − r 2 ) = ∑ BnCn sin nπr ∞

l

n =1

l

The right part of equation(13)can be taken as the expanded form of Fourier series, if the series is convergent, differentiable to the linear and quadric r , the Fourier series is the solution of the equation. BnC n =

2 l nπr (n = 1,2,3L) f (r ) sin dr ∫ 0 l l =

nπr 2 l c1 lr − r 2 sin dr ∫ 0 l l l

=−

(

2c1 nπ

)

l l ⎫⎪ ⎧⎪⎡ l l nπr ⎤ nπr ⎫⎪ 2c1 ⎧⎪⎡ 2 nπr ⎤ nπr r cos cos dr r cos 2rdr ⎬ − ∫ cos + − ⎨ ⎬ ⎨⎢ ∫ ⎥ ⎢ ⎥ l ⎦0 0 l l ⎦0 0 l ⎪⎭ ⎪⎭ nπl ⎪⎩⎣ ⎪⎩⎣

So:① if n is odd number, BnCn = 8c1l n 3π 3 ② if n is even number, BnCn =0

(13)

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333

Suppose Bn Cn has no connection with t , take 2n + 1 (n = 0,1,2,3L) as odd number: 8c1l (2n + 1)πr e − y= ∑ sin 3 3 l 2 n +1=1 (2n + 1) π ∞

( 2 n +1)2 π 2 Dt l2

8c1l (2n + 1)πr e − ∴c= 1 ∑ sin 3 3 r 2 n +1=1 (2n + 1) π l ∞

( n = 0,1,2,3L)

(2 n +1)2 π 2 Dt

( n = 0,1,2,3L)

l2

8c1l (2n + 1)πr ⎛⎜ − (2n + 1) π ∴ u = ∂c = 1 ∑ sin 3 3 ⎜ l ∂t r 2 n +1=1 (2n + 1) π l l2 ⎝ ∞

2

∴ u = l ∑ 8c1D sin (2n + 1)πr e − r 2 n +1=1 (2n + 1)πl l ∞

( 2 n +1)2 π 2 Dt l2

2

⎞ − (2 n+12) π D ⎟⎟e l ⎠ 2

2

Dt

( n = 0,1,2,3L)

(14)

3. The analytic answer of the gas flow rate in capillary model Taking the infinitesimal circular tube as the study object which is along the r axis, with the length of dr , and the speed of u is constant, we can get such formulas:

Fig. 2. The Simplified Math Analysis Model of Gas Diffusion Discipline in Capillary

dq = u

πd 2 4

Q u = l ∑ 8c1D sin (2n + 1)πr e − r 2 n +1=1 (2n + 1)πl l ∞

( 2 n +1)2 π 2 Dt l2

8c1 D (2n + 1)πr e − dq = sin ∑ l 4 r 2 n +1=1 (2n + 1)πl

πd 2 l

∞

( 2 n +1)2 π 2

8c1 D (2n + 1)πr e − sin ∑ ∫ 0 l 4r 2 n +1=1 (2n + 1)πl

∴ q = πd

2

l

∞

l

∞

1 l = 2c1 Dd 2 ∑ ∫0 2 n +1=1 2 n + 1

(2n + 1)πr

( n = 0,1,2,3L)

l2

Dt

( 2 n +1)2 π 2 Dt l2

dr

(2n + 1)πr e − l (2n + 1)πr d l l

sin

(2 n +1)2 π 2 Dt l2

xl If (2n + 1)πr = x ⇒ r = , so (2n + 1)π l

If r is in the range of( 0 , l ), then the range of x is:（ 0 , (2n + 1)π ）, so:

(15)

W. Hai-jin et al. / Procedia Earth and Planetary Science 1 (2009) 328–335

334 ∞

1 − q = 2c1 Dd ∑ e 2 n +1=1 2n + 1 2

( 2 n +1)2 π 2 Dt l2

∫

( 2 n +1)π

0

sin x dx ( n = 0,1,2,3L) x

In the expression of q , there is a close connection between e

−

( 2 n +1)2 π 2 Dt l2

and t . Considering the meaning of the

project, the unit of t is day, moreover, the time of pre-vacuum gas is nearly two months, and e

−

( 2 n +1)2 π 2 Dt l2

is convergent

quickly with time. Therefore, in the expression of q , we can only use the former several items which can satisfy the require of the project. If n = 3 : 9π Dt 25π Dt ⎛ − π 2Dt π sin x 1 − 2 3π sin x 1 − 2 5π sin x ⎞⎟ q = 2c1 Dd ⎜ e l ∫ dx + e l ∫ dx + e l ∫ dx 0 0 0 ⎜ ⎟ x x x 3 5 ⎝ ⎠ 2

2

2

2

(16)

In order to solve q , we have to solve the whole problem: sin x . However, in higher-mathematics, it’s only x ∞ known that sin x dx = π , but for the sin x in the finite interval, we need to deal with further more. ∫0 x x 2

Expand sin x with the Maclaurin formula: sin x = x −

x3 x5 x7 x 2 m −1 m −1 + − + L + (− 1) +R (2m − 1)! 2m 3! 5! 7!

π⎤ ⎡ sin ⎢θx + (2m + 1) ⎥ 2 ⎦ 2 m +1 ⎣ (0 < θ < 1) R2 m ( x ) = x (2m + 1)! R2 m ( x ) ≤

x

2 m +1

(2m + 1)!

→0

(m → ∞ )

2 m−2 sin x x2 x4 x6 R m −1 x = 1 − + − + L + (− 1) + 2m (2m − 1)! x x 3! 5! 7! 2 m−2 ∞ is convergent, the proving process is as follows: So, powder series ∑ (− 1)m −1 x (2m − 1)! m =1

Q lim am+1 m →∞

am

= lim

m →∞

1 = 0 , am and am +1 are the coefficients of the two adjacent items of the powder series, 2m(2m + 1)

∴ The powder series is convergent, and the radius of convergence is + ∞ . Integrate with METLAB: ⎛ x 52 x 54 ⎞ ⎜⎜ ⎟⎟ = 9.5649 × 10 − 46 − ⎝ 53! 55! ⎠

∫

π

∫

3π

∫

5π

0

0

0

⎛ x 52 x 54 ⎞ ⎜⎜ ⎟⎟ = 1.8064 ×10 −20 − ⎝ 53! 55! ⎠ ⎛ x 52 x 54 ⎞ ⎜⎜ ⎟⎟ = 9.8102 ×10 −9 − ⎝ 53! 55! ⎠

W. Hai-jin et al. / Procedia Earth and Planetary Science 1 (2009) 328–335

335

The integral quantity is small enough which can be ignored. So if m = 26 , we will have the algebraic sum of 26 former items of the powder series, then integrate it with the intervals: (0, π ) , (0,3π ) and (0,5π ) . x2 x4 x 50 S 26 = 1 − + + L − 3! 5! 51!

∴

∫

π

0

∫

3π

∫

5π

0

0

π sin x dx ≈ ∫ S 26 dx = 1.8519 0 x

3π sin x dx ≈ ∫ S 26 dx = 1.6748 0 x 5π sin x dx ≈ ∫ S 26 dx = 1.6340 0 x

∴ q = 2c Dd 2 ⎛⎜1.8519e − 1

⎜ ⎝

π 2 Dt l2

+ 0.5583e

−

9π 2 Dt

l2

+ 0.3268e

−

25π 2 Dt

l2

⎞ ⎟ ⎟ ⎠

(17)

In the process of calculating q , it may make error because of n = 3 and m = 26 , but considering the engineering practice and permissible error, it can be ignored. However, the theory analysis result still has a guiding meaning to the practice. 4. The conclusion • Based upon interrelated assumption, this paper simplifies the gas diffusion complex process and sets up the gas diffusion capillary model. • Based upon the theory model, this paper discusses the differential equation of motion when the gas diffusion obeys the Fick’s law. • With the knowledge of differential equation and mathematic and physical equation, this paper obtains the analytic answer of the gas diffusion speed and the flow rate in the model for the first time with the help of MATLAB. • The analytic answer can help us to solve gas problem in coal mine with quantized parameters, for example: the outburst time in any position in the influence range of the drilling and hydraulic slotting can be obtained. References [1] B Q Lin, J H Chang and C Zhai. Analysis on coal mine safety situation in china and its Countermeasures. China Safety Science Journal, 16 (2006) 42-46. [2] Z. Cheng and L. Baiquan. Rapid heading technology for mine gateway in seam with high gassy, low permeability and outburst. Coal Science and Technology, (2008) 6. [3] L. Baoyu, L. Baiquan. Research and Application of High Pressure Abrasive Jet in Prevention Coal and Gas Outbursts. Coal Mine Machinery. 28 (2007) 43-45. [4] Q. Minggao and S. Pingwu. Control of rock pressure and rock. Xuzhou:China University of Mining And Technology Press, 2003. [5] L. Shiping. Concise Guide for Rock Mechanics.Xuzhou: China University of Mining And Technology Press, 1986: 50-70. [6] L. Baiquan. The impact of coal seam gas drainage and the analysis of the factors. Safety in Coal Mines. 9 (1990) 30-35. [7] W. Haijin, L. Baiquan, Numerical Analysis of the Pressure Relief Effect on Slot at Different Stresses.Journal of Mining & Safety Engineering. 26 (2009) 194-197.