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

∂ cA = ∇∇∇⋅∇ (DAB ∇∇∇cA) (2.1-1) ∂t

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

∂ ∂2c cA A = DAB (2.1-2) ∂t ∂x2

hm , c A,inf hm , c A,inf

-L x=0 L 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

∂ ∂ cA cA B. C. x = 0, = 0; x = L, − D = h (c − c ∞) ∂ AB ∂ m Af A, x x=0 x x= L

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)

2-1 The differential equation and its boundary conditions are usually changed to the dimensionless forms to simplify the solutions. We define the following dimensionless variables

− * cA K' c A ,∞ * Dimensionless concentration: θ = ⇒ c =K’c ∞ + θ (c − K’c ∞) − A A, A,i A, cA, i K' c A , ∞

x Dimensionless distance: x* = ⇒ x = L x * L

D t L2 * AB ⇒ Dimensionless time or Fourier number: t = Fo = 2 t = Fo L DAB

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

∂θ * ∂2θ * 1 DAB 1 (c − c ∞) = ( c − c ∞) A,i A, 2 ∂ A,i A, 2 ∂ 2 DAB L Fo L x *

∂θ * ∂2θ * = (2.1-3) ∂Fo ∂x *2

Similarly, the initial and boundary conditions can be transformed into dimensionless forms

θ*(x*, 0) = 1

∂θ * ∂θ * h L = 0; = − Bi *θ*(1, t*), where Bi = m ∂ * ∂ * m m x x*=0 x x*=1 K' D AB

* * Therefore θ = f(x , Fo, Bi m)

* * The dimensionless concentration depends θ only on x , Fo, and Bi m. The mass transfer Biot number, Bi m, 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* −ζ 2 ζ = ∑Cn exp( n Fo) cos( nx* ) (2.1-4) n=1 where the coefficients Cn are given by

4sin ζ C = n n ζ + ζ 2 n sin( 2 n )

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

2-2

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

∞ θ* −ζ 2 ζ Table 2.1-1 Matlab program to evaluate and plot = ∑Cn exp( n Fo ) cos( nx* ) n=1

% 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

2-3 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.

Temperature distribution in a slab for Bi = 0.5 1 Fo=0.1

0.9

Fo=0.5 0.8

0.7 Fo=1

0.6

0.5 Theta* Fo=2 0.4

0.3

0.2

0.1

Fo=10 0 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 x* Figure 2.1-2 Dimensionless concentration distribution at various Fourier number.

For the roots of equation ζn tan( ζn) = Bi m, let

f = ζ tan( ζ) − Bi m

Then f’ = tan( ζ) + ζ(1 + tan( ζ)2);

2-4 The differential conduction equation for mass transfer in the radial direction of an infinite cylinder with radius R is

∂ 1 ∂ ∂  cA cA = DAB r  (2.1-5) ∂t r ∂r ∂ r 

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

∂ ∂ ∂  cA 1 2 cA = DAB r  (2.1-6) ∂t r 2 ∂r ∂ r 

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

∂ ∂ cA cA B. C. r = 0, = 0; r = R, − D = h (c − c ∞) ∂ AB ∂ m Af A, r r=0 r r= R

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

∞ θ* −ζ 2 ζ = ∑Cn exp( n Fo) J0( nx* ) (2.1-7) n=1

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

∞ sin( ζ r*) θ* = C exp( −ζ 2 F ) n (2.1-8) ∑ n n o ζ n=1 n r *

lim sin( ζ r*) lim ζ cos( ζ r*) Since n = n n = 1, it should be noted that at r* = 0 → ζ → ζ r* 0 n r * r* 0 n

∞ θ* −ζ 2 = ∑Cn exp( n Fo) n=1

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

∂ ∂2c cA A = DAB ∂t ∂x2

2-5 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 ∂c B. C.: − D A = N , c (x → ∞, t) = c AB ∂ A0 A Ai x x=0

Case 3: I. C.: cA(x, 0) = cAi ∂ cA B. C.: − D = h (c − c ∞), c (x → ∞, t) = c AB ∂ m Af A, A Ai x x=0

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* −ζ 2 ζ = ∑Cn exp( n Fo) cos( nx* ) (2.1-4) n=1

The first term approximation is

θ * ζ 2 ζ * = C1exp(- 1 Fo) cos( 1x ) (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. 2-6 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 C C Bi m ζ1(rad) 1 ζ1(rad) 1 ζ1(rad) C1 0.01 0.0998 1.0017 0.1412 1.0025 0.1730 1.0030 0.02 0.1410 1.0033 0.1995 1.0050 0.2445 1.0060 0.03 0.1732 1.0049 0.2439 1.0075 0.2989 1.0090 0.04 0.1987 1.0066 0.2814 1.0099 0.3450 1.0120 0.05 0.2217 1.0082 0.3142 1.0124 0.3852 1.0149 0.06 0.2425 1.0098 0.3438 1.0148 0.4217 1.0179 0.07 0.2615 1.0114 0.3708 1.0173 0.4550 1.0209 0.08 0.2791 1.0130 0.3960 1.0197 0.4860 1.0239 0.09 0.2956 1.0145 0.4195 1.0222 0.5150 1.0268 0.1 0.3111 1.0160 0.4417 1.0246 0.5423 1.0298 0.15 0.3779 1.0237 0.5376 1.0365 0.6608 1.0445 0.2 0.4328 1.0311 0.6170 1.0483 0.7593 1.0592 0.25 0.4801 1.0382 0.6856 1.0598 0.8448 1.0737 0.3 0.5218 1.0450 0.7465 1.0712 0.9208 1.0880 0.4 0.5932 1.0580 0.8516 1.0932 1.0528 1.1164 0.5 0.6533 1.0701 0.9408 1.1143 1.1656 1.1441 0.6 0.7051 1.0814 1.0185 1.1346 1.2644 1.1713 0.7 0.7506 1.0919 1.0873 1.1539 1.3525 1.1978 0.8 0.7910 1.1016 1.1490 1.1725 1.4320 1.2236 0.9 0.8274 1.1107 1.2048 1.1902 1.5044 1.2488 1.0 0.8603 1.1191 1.2558 1.2071 1.5708 1.2732 2.0 1.0769 1.1795 1.5995 1.3384 2.0288 1.4793 3.0 1.1925 1.2102 1.7887 1.4191 2.2889 1.6227 4.0 1.2646 1.2287 1.9081 1.4698 2.4556 1.7201 5.0 1.3138 1.2402 1.9898 1.5029 2.5704 1.7870 6.0 1.3496 1.2479 2.0490 1.5253 2.6537 1.8338 7.0 1.3766 1.2532 2.0937 1.5411 2.7165 1.8674 8.0 1.3978 1.2570 2.1286 1.5526 2.7654 1.8921 9.0 1.4149 1.2598 2.1566 1.5611 2.8044 1.9106 10.0 1.4289 1.2620 2.1795 1.5677 2.8363 1.9249 20.0 1.4961 1.2699 2.2881 1.5919 2.9857 1.9781 30.0 1.5202 1.2717 2.3261 1.5973 3.0372 1.9898 40.0 1.5325 1.2723 2.3455 1.5993 3.0632 1.9942 50.0 1.5400 1.2727 2.3572 1.6002 3.0788 1.9962 100.0 1.5552 1.2731 2.3809 1.6015 3.1102 1.9990 500.0 1.5677 1.2732 2.4000 1.6020 3.1353 2.0000 1000.0 1.5692 1.2732 2.4024 1.6020 3.1385 2.0000 ∞ 1.5708 1.2732 2.4048 1.6020 3.1416 2.0000

2-7 Table 2.2-2 Approximate solutions for diffusion and conduction (valid for Fo>0.2)

D t D t c− K' c ∞ Fo = AB = AB , θ * = A A , , θ * = C exp(-ζ 2 F ) 2 2 − 0 1 1 o L r0 cA, i K' c A , ∞ 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. M sin( ζ ) θ * = θ * cos( ζ x*) ; t = 1 − 1 θ * 0 1 ζ 0 M ∞ 1 Diffusion in an infinite cylinder M 2θ * θ * = θ * J (ζ r*) ; t = 1 − 0 J (ζ ) 0 0 1 ζ 1 1 M ∞ 1 Diffusion in a sphere 1 M 3θ * θ * = θ * sin( ζ r*) ; t = 1 − 0 [sin( ζ ) − ζ cos( ζ )] ζ * 0 1 ζ 3 1 1 1 1r M ∞ 1

If the concentration at the surface cA,s is known K’cA, ∞ will be replaced by cA,s ζ 1 and C1 will be obtained from table at Bi m = ∞

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’c A, ∞ = 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)

hm L Bi m = = ratio of internal resistance to mass transfer by diffusion to external mass K' D AB transfer by convection

hm = kc = mass transfer coefficient L = L for a slab with thickness 2 L 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

2-8 Table 2.2-3 Semi-infinite medium

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

c− c x  ∂c D( c− c , ) A A, s = erf   ; N = − D A = AB As, Ai −   A0 AB ∂ π cAi, c As , 2 DAB t  x x=0 DAB t

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

t −x2  N x x  c (x, t) − c = 2 N exp − A0 erfc   A A,i A0 π     DAB 4DAB t  DAB 2 DAB t  The complementary error function, erfc (w), is defined as erfc (w) = 1 – erf (w)

∂ cA Surface Convection: − D = h (c − c ∞) AB ∂ m Af A, x x=0 2  c− c x  hx h  t  x h t   A A, i −  m+ m   + m = erfc   exp   erfc    K' c∞ − c 2 D t KD' K ' D   2 D t K' D A, A , i AB  AB  AB   AB AB  

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’c A, ∞ = 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

2-9 Table 2.2-4 Multidimensional Effects

ro ro L (r,x) x L

L L r

c (r,x,t) A

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 2 L ]

[ parallelpiped ] = [ slab 2 L1 ] × [ slab 2 L2 ] × [ slab 2 L3 ]

c(,) xt− Kc ' S(x, t) ≡ A A ,∞ − cA, i K' c A , ∞ Semi-infinite solid

c(,) xt− Kc ' P(x, t) ≡ A A ,∞ − cA, i K' c A , ∞ Plane wall

c(,) rt− Kc ' C(r, t) ≡ A A ,∞ − cA, i K' c A , ∞ Infinite cylinder

2-10