7.1. UNSTEADY-STATE DIFFUSION

7.1A. Derivation of Basic Equation

In Chapter 6 we considered various mass-transfer systems where the concentration or partial pressure at any point and the diffusion flux were constant with time, hence at steady state. Before steady state can be reached, time must elapse after the mass-transfer process is initiated for the unsteady-state conditions to disappear.

In Section 2.3 a general property balance for unsteady-state molecular diffusion was made for the properties momentum, heat, and mass. For no generation present, this was

Equation 2.13-12


In Section 5.1 an unsteady-state equation for heat conduction was derived,

Equation 5.1-10


The derivation of the unsteady-state diffusion equation in one direction for mass transfer is similar to that performed for heat transfer in obtaining Eq. (5.1-10). We refer to Fig. 7.1-1, where mass is diffusing in the x direction in a cube composed of a solid, stagnant gas, or stagnant liquid and having dimensions Δx, Δy, Δz. For diffusion in the x direction we write

Equation 7.1-1


Figure 7.1-1. Unsteady-state diffusion in one direction.


The term ∂cA/∂x means the partial of cA with respect to x or the rate of change of cA with x when the other variable, time t, is kept constant.

Next we make a mass balance on component A in terms of moles for no generation:

Equation 7.1-2


The rate of input and rate of output in kg mol A/s are

Equation 7.1-3


Equation 7.1-4


The rate of accumulation is as follows for the volume Δx Δy Δz m3:

Equation 7.1-5


Substituting Eqs. (7.1-3), (7.1-4), and (7.1-5) into (7.1-2) and dividing by Δx Δy Δz,

Equation 7.1-6


Letting Δx approach zero,

Equation 7.1-7


The above holds for a constant diffusivity DAB. If DAB is a variable,

Equation 7.1-8


Equation (7.1-7) relates the concentration cA with position x and time t. For diffusion in all three directions a similar derivation gives

Equation 7.1-9


In the remainder of this section, the solutions of Eqs. (7.1-7) and (7.1-9) will be considered. Note the mathematical similarity between the equation for heat conduction,

Equation 5.1-6


and Eq. (7.1-7) for diffusion. Because of this similarity, the mathematical methods used for solution of the unsteady-state heat-conduction equation can be used for unsteady-state mass transfer. This is discussed more fully in Sections 7.1B, 7.1C, and 7.7.

7.1B. Diffusion in a Flat Plate with Negligible Surface Resistance

To illustrate an analytical method of solving Eq. (7.1-7), we will derive the solution for unsteady-state diffusion in the x direction for a plate of thickness 2x1, as shown in Fig. 7.1-2. For diffusion in one direction,

Equation 7.1-7


Figure 7.1-2. Unsteady-state diffusion in a flat plate with negligible surface resistance.


Dropping the subscripts A and B for convenience,

Equation 7.1-10


The initial profile of the concentration in the plate at t = 0 is uniform at c = c0 for all x values, as shown in Fig. 7.1-2. At time t = 0 the concentration of the fluid in the environment outside is suddenly changed to c1. For a very high mass-transfer coefficient outside the surface resistance will be negligible and the concentration at the surface will be equal to that in the fluid, which is c1.

The initial and boundary conditions are

Equation 7.1-11


Redefining the concentration so it goes between 0 and 1,

Equation 7.1-12


Equation 7.1-13


The solution of Eq. (7.1-13) is an infinite Fourier series and is identical to the solution of Eq. (5.1-6) for heat transfer:

Equation 7.1-14


where

, dimensionless
c=concentration at point x and time t in slab
Y=(c1-c)/(c1-c0) = fraction of unaccomplished change, dimensionless
1-Y=(c-c0)/(c1-c0) = fraction of change

Solution of equations similar to Eq. (7.1-14) are time-consuming; convenient charts for various geometries are available and will be discussed in the next section.

7.1C. Unsteady-State Diffusion in Various Geometries

1. Convection and boundary conditions at the surface

In Fig. 7.1-2 there was no convective resistance at the surface. However, in many cases when a fluid is outside the solid, convective mass transfer is occurring at the surface. A convective mass-transfer coefficient kc, similar to a convective heat-transfer coefficient, is defined as follows:

Equation 7.1-15


where kc is a mass-transfer coefficient in m/s, cL1 is the bulk fluid concentration in kg mol A/m3, and cLi is the concentration in the fluid just adjacent to the surface of the solid. The coefficient kc is an empirical coefficient and will be discussed more fully in Section 7.2.

In Fig. 7.1-3a the case where a mass-transfer coefficient is present at the boundary is shown. The concentration drop across the fluid is cL1cLi. The concentration in the solid ci at the surface is in equilibrium with cLi.

Figure 7.1-3. Interface conditions for convective mass transfer and an equilibrium distribution coefficient K =cLi/ci: (a) K = 1. (b) K > 1, (c) K < 1, (d) K > 1 and kc = ∞.


In Fig. 7.1-3a the concentration cLi in the liquid adjacent to the solid and ci in the solid at the surface are in equilibrium and are equal. However, unlike heat transfer, where the temperatures are equal, the concentrations here are in equilibrium and are related by

Equation 7.1-16


where K is the equilibrium distribution coefficient (similar to Henry's law coefficient for a gas and liquid). The value of K in Fig. 7.1-3a is 1.0.

In Fig. 7.1-3b the distribution coefficient K > 1 and cLi > ci, even though they are in equilibrium. Other cases are shown in Fig. 7.1-3c and d. This was also discussed in Section 6.6B.

2. Relation between mass- and heat-transfer parameters

In order to use the unsteady-state heat-conduction charts in Chapter 5 for solving unsteady-state diffusion problems, the dimensionless variables or parameters for heat transfer must be related to those for mass transfer. In Table 7.1-1 the relations between these variables are tabulated. For K ≠ 1.0, whenever kc appears, it is given as Kkc, and whenever c1 appears, it is given as c1/K.

Table 7.1-1. Relation Between Mass- and Heat-Transfer Parameters for Unsteady-State Diffusion[*]
 Mass Transfer
Heat TransferK = cL/c = 1.0K = cL/c ≠ 1.0

[*] x is the distance from the center of the slab, cylinder, or sphere; for a semi-infinite slab, x is the distance from the surface, c0 is the original uniform concentration in the solid, c1 the concentration in the fluid outside the slab, and c the concentration in the solid at position x and time t.

3. Charts for diffusion in various geometries

The various heat-transfer charts for unsteady-state conduction can be used for unsteady-state diffusion and are as follows:

  1. Semi-infinite solid, Fig. 5.3-3.

  2. Flat plate, Figs. 5.3-5 and 5.3-6.

  3. Long cylinder, Figs. 5.3-7 and 5.3-8.

  4. Sphere, Figs. 5.3-9 and 5.3-10.

  5. Average concentrations, zero convective resistance, Fig. 5.3-13.

EXAMPLE 7.1-1. Unsteady-State Diffusion in a Slab of Agar Gel

A solid slab of 5.15 wt % agar gel at 278 K is 10.16 mm thick and contains a uniform concentrations of urea of 0.1 kg mol/m3. Diffusion is only in the x direction through two parallel flat surfaces 10.16 mm apart. The slab is suddenly immersed in pure turbulent water, so the surface resistance can be assumed to be negligible; that is, the convective coefficient kc is very large. The diffusivity of urea in the agar from Table 6.4-2 is 4.72 × 1010 m2/s.

  1. Calculate the concentration at the midpoint of the slab (5.08 mm from the surface) and 2.54 mm from the surface after 10 h.

  2. If the thickness of the slab is halved, what would be the midpoint concentration in 10 h?

Solution: For part (a), c0 = 0.10 kg mol/m3, c1 = 0 for pure water, and c = concentration at distance x from center line and time t s. The equilibrium distribution coefficient K in Eq. (7.1-16) can be assumed to be 1.0, since the water in the aqueous solution in the gel and outside should be very similar in properties. From Table 7.1-1,


Also, x1 = 10.16/(1000 × 2) = 5.08 × 103 m (half-slab thickness), x = 0 (center), X = = (4.72 × 1010)(10 × 3600)/(5.08 × 103)2 = 0.658. The relative position n = x/x1 = 0/5.08 × 103 = 0, and relative resistance m = DAB/Kkcx1 = 0, since kc is very large (zero resistance).

From Fig. 5.3-5 for X = 0.658, m = 0, and n = 0,


Solving, c = 0.0275 kg mol/m3 for x = 0.

For the point 2.54 mm from the surface or 2.54 mm from center, x = 2.54/1000 = 2.54 × 103 m, X = 0.658, m = 0, n = x/x1 = 2.54 × 103/5.08 × 103 = 0.5. Then from Fig. 5.3-5, Y = 0.172. Solving, c = 0.0172 kg mol/m3.

For part (b) and half the thickness, X = 0.658/(0.5)2 = 2.632, n = 0, and m = 0. Hence, Y = 0.0020 and c = 2.0 × 104 kg mol/m3.


EXAMPLE 7.1-2. Unsteady-State Diffusion in a Semi-infinite Slab

A very thick slab has a uniform concentration of solute A of c0 = 1.0 × 102 kg mol A/m3. Suddenly, the front face of the slab is exposed to a flowing fluid having a concentration c1 = 0.10 kg mol A/m3 and a convective coefficient kc = 2 × 107 m/s. The equilibrium distribution coefficient K = cLi/ci = 2.0. Assuming that the slab is a semi-infinite solid, calculate the concentration in the solid at the surface (x = 0) and x = 0.01 m from the surface after t = 3 × 104 s. The diffusivity in the solid is DAB = 4 × 109 m2/s.

Solution: To use Fig. 5.3-3, use Table 7.1-1.


For x = 0.01 m from the surface in the solid,


From the chart, 1 − Y = 0.26. Then, substituting into the equation for (1 − Y) from Table 7.1-1 and solving,


For x = 0 m (i.e., at the surface of the solid),


From the chart, 1 − Y = 0.62. Solving, c = 3.48 × 102. This value is the same as ci, as shown in Fig. 7.1-3b. To calculate the concentration cLi in the liquid at the interface,


A plot of these values will be similar to Fig. 7.1-3b.


4. Unsteady-state diffusion in more than one direction

In Section 5.3F a method was given for unsteady-state heat conduction, in which the one-dimensional solutions were combined to yield solutions for several-dimensional systems. The same method can be used for unsteady-state diffusion in more than one direction. Rewriting Eq. (5.3-11) for diffusion in a rectangular block in the x, y, and z directions,

Equation 7.1-17


where cx,y,z is the concentration at the point x, y, z from the center of the block. The value of Yx for the two parallel faces is obtained from Fig. 5.3-5 or 5.3-6 for a flat plate in the x direction. The values of Yy and Yz are similarly obtained from the same charts. For a short cylinder, an equation similar to Eq. (5.3-12) is used, and for average concentrations, ones similar to Eqs. (5.3-14), (5.3-15), and (5.3-16) are used.

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