2-1

Chapter 2

Unsteady State Molecular Diffusion 2.1 Differential Mass Balance When the internal concentration gradient is not negligible or Bi ≠ << 1, the microscopic or differential mass balance will yield a partial differential equation that describes the concentration as a function of time and position. For a binary system with no chemical reaction, the unsteady state molecular diffusion is given by

Ac

t

∂∂

= ∇∇∇∇⋅(DAB∇∇∇∇cA) (2.1-1)

For one-dimensional mass transfer in a slab with constant DAB and convective conditions of hm and cA,∞, equation (2.1-1) is simplified to

Ac

t

∂∂

= DAB 2

2Ac

x

∂∂

(2.1-2)

x=0 L-L

h , cm A,infh , cm A,inf

Figure 2.1-1 One-dimensional unsteady mass transfer in a slab.

Equation (2.1-2) can be solved with the following initial and boundary conditions I. C. t = 0, cA(x, 0) = cAi

B. C. x = 0, 0

A

x

c

x =

∂∂

= 0; x = L, − DABA

x L

c

x =

∂∂

= hm(cAf − cA,∞)

In general, the concentration within the slab depends on many parameters besides time t and position x. cA = cA(x, t, cA,i, cA,∞, L, DAB, hm)

2-2

The differential equation and its boundary conditions are usually changed to the dimensionless forms to simplify the solutions. We define the following dimensionless variables

Dimensionless concentration: θ* = ,

, ,

'

'A A

A i A

c K c

c K c∞

∞

−−

⇒ cA =K’c A,∞ + θ*(cA,i − K’cA,∞)

Dimensionless distance: x* = L

x ⇒ x = L x*

Dimensionless time or Fourier number: t* = Fo = 2ABD t

L⇒ t =

2

AB

L

DFo

K’ is the equilibrium distribution coefficient. Substituting T, x, and t in terms of the dimensionless quantities into equation (2.1-2) yields

(cA,i − cA,∞)1

ABD 2ABD

L Fo∂∂ *θ

= (cA,i − cA,∞) 2

1L 2

*2

*x∂∂ θ

Fo∂

∂ *θ = 2

*2

*x∂∂ θ

(2.1-3)

Similarly, the initial and boundary conditions can be transformed into dimensionless forms θ*(x*, 0) = 1

0

*

*

* =∂∂

xx

θ = 0;

1

*

*

* =∂∂

xx

θ= − Bim*θ*(1, t*), where Bim =

'm

AB

h L

K D

Therefore θ* = f(x*, Fo, Bim) The dimensionless concentration depends θ* only on x*, Fo, and Bim. The mass transfer Biot number, Bim, denotes ratio of the internal resistance to mass transfer by diffusion to the external resistance to mass transfer by convection. Equation (2.1-3) can be solved by the method of separation of variables to obtain

θ* = ∑∞

=1nnC exp(− 2

nζ Fo) cos(ζnx*) (2.1-4)

where the coefficients Cn are given by

Cn = )2sin(2

sin4

nn

n

ζζζ

+

and ζn are the roots of the equation: ζn tan(ζn) = Bim.

2-3

Table 2.1-1 lists the Matlab program that evaluates the first ten roots of equation ζn tan(ζn) = Bim and the dimensionless concentrations given in equation (2.1-4). The program use Newton’s method to find the roots (see Review).

Table 2.1-1 Matlab program to evaluate and plot θ* = ∑∞

=1nnC exp(− 2

nζ Fo) cos(ζnx*)

% Plot the dimensionless concentration within a slab % % The guess for the first root of equation z*tan(z)=Bi depends on the Biot number % Biot=[0 .01 .1 .2 .5 1 2 5 10 inf]'; alfa=[0 .0998 .3111 .4328 .6533 .8603 1.0769 1.3138 1.4289 1.5707]; zeta=zeros(1,10);cn=zeta; Bi=1; fprintf('Bi = %g, New ',Bi) Bin=input('Bi = '); if length(Bin)>0;Bi=Bin;end % Obtain the guess for the first root if Bi>10 z=alfa(10); else z=interp1(Biot,alfa,Bi); end % Newton method to solve for the first 10 roots for i=1:10 for k=1:20 ta=tan(z);ez=(z*ta-Bi)/(ta+z*(1+ta*ta)); z=z-ez; if abs(ez)<.00001, break, end end % Save the root and calculate the coefficients zeta(i)=z; cn(i)=4*sin(z)/(2*z+sin(2*z)); fprintf('Root # %g =%8.4f, Cn = %9.4e\n',i,z,cn(i)) % Obtain the guess for the next root step=2.9+i/20; if step>pi; step=pi;end z=z+step; end % % Evaluate and plot the concentrations hold on Fop=[.1 .5 1 2 10]; xs=-1:.05:1; cosm=cos(cn'*xs); for i=1:5

2-4

Fo=Fop(i); theta=cn.*exp(-Fo*zeta.^2)*cosm; plot(xs,theta) end grid xlabel('x*');ylabel('Theta*') Bi = .5 Root # 1 = 0.6533, Cn = 1.0701e+000 Root # 2 = 3.2923, Cn = -8.7276e-002 Root # 3 = 6.3616, Cn = 2.4335e-002 Root # 4 = 9.4775, Cn = -1.1056e-002 Root # 5 = 12.6060, Cn = 6.2682e-003 Root # 6 = 15.7397, Cn = -4.0264e-003 Root # 7 = 18.8760, Cn = 2.8017e-003 Root # 8 = 22.0139, Cn = -2.0609e-003 Root # 9 = 25.1526, Cn = 1.5791e-003 Root # 10 = 28.2920, Cn = -1.2483e-003 Figure 2.1-2 shows a plot of dimensionless concentration θ* versus dimensionless distance x* at various Fourier number for a Biot number of 0.5.

Figure 2.1-2 Dimensionless concentration distribution at various Fourier number.

For the roots of equation ζn tan(ζn) = Bim, let f = ζ tan(ζ) − Bim Then f’ = tan(ζ) +ζ(1 + tan(ζ)2);

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x*

The

ta*

Temperature distribution in a slab for Bi = 0.5

Fo=1

Fo=2

Fo=10

Fo=0.1

Fo=0.5

2-5

The differential conduction equation for mass transfer in the radial direction of an infinite cylinder with radius R is

Ac

t

∂∂

= DABr

1 Acr

r r

∂ ∂ ∂ ∂

(2.1-5)

The differential conduction equation for mass transfer in the radial direction of a sphere with radius R is

Ac

t

∂∂

= DAB 2

1r

2 Acr

r r

∂ ∂ ∂ ∂

(2.1-6)

Equations (2.1-5) and (2.1-6) can be solved with the following initial and boundary conditions I. C. t = 0, cA(r, 0) = cA i

B. C. r = 0, 0

A

r

c

r =

∂∂

= 0; r = R, − DABA

r R

c

r =

∂∂

= hm(cAf − cA,∞)

The solution of equation (2.1-5) for the infinite cylinder is given as

θ* = ∑∞

=1nnC exp(− 2

nζ Fo) J0(ζnx*) (2.1-7)

where J0(ζnx*) is Bessel function of the first kind, order zero. The coefficient Cn are not the same as those in a slab. The solution of equation (2.1-6) for a sphere is given as

θ* = ∑∞

=1nnC exp(− 2

nζ Fo)**)sin(

r

r

n

n

ζζ

(2.1-8)

Since 0*

lim

→r **)sin(

r

r

n

n

ζζ

= 0*

lim

→r n

nn r

ζζζ *)cos(

= 1, it should be noted that at r* = 0

θ* = ∑∞

=1nnC exp(− 2

nζ Fo)

For one-dimensional mass transfer in a semi-infinite solid as shown in Figure 2.1-3, the differential equation is the same as that in one-dimensional mass transfer in a slab

Ac

t

∂∂

= DAB

2

2Ac

x

∂∂

2-6

x

Semi-Infinite Solid

Figure 2.1-3 One-dimensional mass transfer in a semi-infinite solid.

We consider three cases with the following initial and boundary conditions Case 1: I. C.: cA(x, 0) = cAi

B. C.: cA(0, t) = cAs, cA(x → ∞, t) = cAi Case 2: I. C.: cA(x, 0) = cAi

B. C.: − DAB 0

A

x

c

x =

∂∂

= NA0, cA(x → ∞, t) = cAi

Case 3: I. C.: cA(x, 0) = cAi

B. C.: − DAB0

A

x

c

x =

∂∂

= hm(cAf − cA,∞), cA(x → ∞, t) = cAi

All three cases have the same initial condition cA(x, 0) = cAi and the boundary condition at infinity cA(x → ∞, t) = cAi. However the boundary condition at x = 0 is different for each case, therefore the solution will be different and will be summarized in a table later. 2.2 Approximate Solutions The summation in the series solution for transient diffusion such as equation (2.1-4) can be terminated after the first term for Fo > 0.2. The full series solution is

θ* = ∑∞

=1nnC exp(− 2

nζ Fo) cos(ζnx*) (2.1-4)

The first term approximation is *θ = C1exp(- 2

1ζ Fo) cos(ζ1x*) (2.2-1)

where C1 and ζ1 can be obtained from Table 2.2-1 for various value of Biot number. Table 2.2-2 lists the first term approximation for a slab, an infinite cylinder, and a sphere. Table 2.2-3 lists the solution for one-dimensional heat transfer in a semi-infinite medium for three different boundary conditions at the surface x = 0. Table 2.2-4 shows the combination of one-dimensional solutions to obtain the multi-dimensional results.

2-7

Table 2.2-1 Coefficients used in the one-term approximation to the series

solutions for transient one-dimensional conduction or diffusion

PLANE WALL INFINITE CYLINDER SPHERE

Bim ζ1(rad) C1 ζ1(rad) C1 ζ1(rad) C1 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 20.0 30.0 40.0 50.0 100.0 500.0 1000.0 ∞

0.0998 0.1410 0.1732 0.1987 0.2217 0.2425 0.2615 0.2791 0.2956 0.3111 0.3779 0.4328 0.4801 0.5218 0.5932 0.6533 0.7051 0.7506 0.7910 0.8274 0.8603 1.0769 1.1925 1.2646 1.3138 1.3496 1.3766 1.3978 1.4149 1.4289 1.4961 1.5202 1.5325 1.5400 1.5552 1.5677 1.5692 1.5708

1.0017 1.0033 1.0049 1.0066 1.0082 1.0098 1.0114 1.0130 1.0145 1.0160 1.0237 1.0311 1.0382 1.0450 1.0580 1.0701 1.0814 1.0919 1.1016 1.1107 1.1191 1.1795 1.2102 1.2287 1.2402 1.2479 1.2532 1.2570 1.2598 1.2620 1.2699 1.2717 1.2723 1.2727 1.2731 1.2732 1.2732 1.2732

0.1412 0.1995 0.2439 0.2814 0.3142 0.3438 0.3708 0.3960 0.4195 0.4417 0.5376 0.6170 0.6856 0.7465 0.8516 0.9408 1.0185 1.0873 1.1490 1.2048 1.2558 1.5995 1.7887 1.9081 1.9898 2.0490 2.0937 2.1286 2.1566 2.1795 2.2881 2.3261 2.3455 2.3572 2.3809 2.4000 2.4024 2.4048

1.0025 1.0050 1.0075 1.0099 1.0124 1.0148 1.0173 1.0197 1.0222 1.0246 1.0365 1.0483 1.0598 1.0712 1.0932 1.1143 1.1346 1.1539 1.1725 1.1902 1.2071 1.3384 1.4191 1.4698 1.5029 1.5253 1.5411 1.5526 1.5611 1.5677 1.5919 1.5973 1.5993 1.6002 1.6015 1.6020 1.6020 1.6020

0.1730 0.2445 0.2989 0.3450 0.3852 0.4217 0.4550 0.4860 0.5150 0.5423 0.6608 0.7593 0.8448 0.9208 1.0528 1.1656 1.2644 1.3525 1.4320 1.5044 1.5708 2.0288 2.2889 2.4556 2.5704 2.6537 2.7165 2.7654 2.8044 2.8363 2.9857 3.0372 3.0632 3.0788 3.1102 3.1353 3.1385 3.1416

1.0030 1.0060 1.0090 1.0120 1.0149 1.0179 1.0209 1.0239 1.0268 1.0298 1.0445 1.0592 1.0737 1.0880 1.1164 1.1441 1.1713 1.1978 1.2236 1.2488 1.2732 1.4793 1.6227 1.7201 1.7870 1.8338 1.8674 1.8921 1.9106 1.9249 1.9781 1.9898 1.9942 1.9962 1.9990 2.0000 2.0000 2.0000

2-8

Table 2.2-2 Approximate solutions for diffusion and conduction (valid for Fo>0.2)

Fo = 2ABD t

L=

20

ABD t

r, *θ = ,

, ,

'

'A A

A i A

c K c

c K c∞

∞

−−

, *0θ = C1exp(- 2

1ζ Fo)

Diffusion in a slab

L is defined as the distance from the center of the slab to the surface. If one surface is insulated,

L is defined as the total thickness of the slab.

*θ = *0θ cos(ζ1x

*) ; tM

M ∞

= 1 − 1

1)sin(

ζζ *

0θ

Diffusion in an infinite cylinder

*θ = *0θ J0(ζ1r

*) ; tM

M ∞

= 1 − 1

*02

ζθ

J1(ζ1)

Diffusion in a sphere

*θ = *

1

1

rζ*0θ sin(ζ1r

*) ; tM

M ∞

= 1 − 31

*03

ζθ

[sin(ζ1) − ζ1cos(ζ1)]

If the concentration at the surface cA,s is known K’cA,∞ will be replaced by cA,s ζ1 and C1 will be obtained from table at Bim = ∞

Notation:

cA = concentration of species A in the solid at any location at any time

cA,s = concentration of species A in the solid at the surface for t > 0

cA,i = concentration of species A in the solid at any location and at t = 0

cA,∞ = bulk concentration of species A in the fluid surrounding the solid

K’cA,∞ = cA* = concentration of species A in the solid that is in equilibrium with cA,∞

Mt = amount of A transferred into the solid at any given time

M∞ = amount of A transferred into the solid as t → ∞ (maximum amount transferred)

Bim = 'm

AB

h L

K D = ratio of internal resistance to mass transfer by diffusion to external mass

transfer by convection

hm = kc = mass transfer coefficient

L = L for a slab with thickness 2L or a slab with thickness L and an impermeable surface

L = ro for radial mass transfer in a cylinder or sphere with radius ro

K’ = equilibrium distribution coefficient

DAB = diffusivity of A in the solid

2-9

Table 2.2-3 Semi-infinite medium

Constant Surface Concentration: cA(0, t) = cA,s

,

, ,

A A s

A i A s

c c

c c

−−

= erf2 AB

x

D t

; NA0 = − DAB0

A

x

c

x =

∂∂

= ,( , )AB A s A i

AB

D c c

D tπ−

Constant Surface Flux: NA(x=0) = NA0

cA(x, t) − cA,i = 2NA0AB

t

Dπ

2

exp4 AB

x

D t

−

− 0A

AB

N x

D 2 AB

xerfc

D t

The complementary error function, erfc(w), is defined as erfc(w) = 1 – erf(w)

Surface Convection: − DAB0

A

x

c

x =

∂∂

= hm(cAf − cA,∞)

,

, ,'A A i

A A i

c c

K c c∞

−−

= 2 AB

xerfc

D t

− 2

exp' 'm m

AB AB

h x h t

K D K D

+

'2m

ABAB

x h terfc

K DD t

+

Notation:

cA = concentration of species A in the solid at any location at any time

cA,s = concentration of species A in the solid at the surface for t > 0

cA,i = concentration of species A in the solid at any location and at t = 0

cAf = concentration of species A in the liquid at the solid-liquid interface at any time

cA,∞ = bulk concentration of species A in the fluid surrounding the solid

K’cA,∞ = cA* = concentration of species A in the solid that is in equilibrium with cA,∞

hm = kc = mass transfer coefficient

K’ = equilibrium distribution coefficient

DAB = diffusivity of A in the solid

L

L

c (r,x,t)A

The concentration profiles for a finite cylinder and a parallelpiped

concentration profiles of infinite cylinder and slabs.

[ finite cylinder ] = [ infinite cylinder ] × [ slab 2[ parallelpiped ] = [ slab 2L1 ] × [ slab 2

S(x, t) ≡ ,

Semi-infinite, ,solid

( , ) '

'A A

A i A

c x t K c

c K c∞

∞

−−

P(x, t) ≡ ,

Plane, ,wall

( , ) '

'A A

A i A

c x t K c

c K c∞

∞

−−

C(r, t) ≡ ,

Infinite, ,cylinder

( , ) '

'A A

A i A

c r t K c

c K c∞

∞

−−

2-10

Table 2.2-4 Multidimensional Effects

x(r,x)

r

ro ro

c (r,x,t)

The concentration profiles for a finite cylinder and a parallelpiped can be obtained from the

concentration profiles of infinite cylinder and slabs.

[ finite cylinder ] = [ infinite cylinder ] × [ slab 2L ] ] × [ slab 2L2 ] × [ slab 2L3 ]

Semi-infinite

Infinitecylinder

L

L

can be obtained from the