Gas and Electrolyte Diffusion Experiment

Chemical Engineering 310Dr. David Keffer, PIC Group #1

Brad JaquithJohn Robert Yates

Performed 11/09/2004 Luke StewartPresented 11/30/2004

Objectives

Determine the diffusion coefficient of a gas by evaporation from a liquid surface

Determine the diffusion coefficient of a 2M solution of NaCl in distilled water

Compare experimental values with correlation equations

Introduction/Background

Diffusion is a process of mass transfer in which a fluid moves from an area of higher concentration to an area of lower concentration, as a result of the fluid’s kinetic properties, until an equilibrium is established.

Experimental Setup/Data Acquisition - Gas Diffusion

Acetone is placed in a capillary tube which is then inserted into the apparatus (see Figure 1).

Maintained at a constant temperature of 40oC by the heater, measurements of the level of the acetone remaining in the capillary are taken every 3600 seconds (or one hour) for use in determining the diffusion coefficient.

Figure 1 -Gaseous Diffusion Apparatus

(source: ChE 310, Gaseous Diffusion Coefficients Apparatus Instruction Manual, 1999)

Experimental Setup/Data Acquisition - Electrolyte Diffusion

A 2M solution of NaCl is poured into the cell and the bubbles are removed (see Figure 2).

The cell is placed inside a CSTR to induce a dynamic equilibrium at the point of the measurements.

Conductivity measurements are made by the computer-controlled data acquisition system for the determination of the diffusion coefficient.

Figure 2-Electrolyte Diffusion Apparatus

(source: ChE 310, Gaseous Diffusion Coefficients Apparatus Instruction Manual, 1999)

Determination of Diffusion Coefficient for a Gas (Acetone) from Experimental Data

Assuming constant T and P, the total concentration at the top of the capillary is given by

cT,1 = N/V = p/RT.

The concentrations of acetone and air are

cB,1 = xB,1 * cT,1

cA,1 = xA,1 * cT,1

Then at the gas/liquid interface, the total concentration is

cT,1 = N/V = p/RT = cT,2

Cont’d Raoult’s Law states:

zApAvap(T) = xA,2p

Assuming pure acetone, zA is 1, so that the concentration of acetone and air at the gas/liquid interface is

cB,2 = xB,2 * cT,2 = cT,2[p-pAvap(T)]/p

cA,2 = xA,2 * cT,2 = cT,2[pAvap(T)/p]

The equations for the molar flux due to diffusion of A at point 2 and for the molar flux due to the evaporation of A at point 2 are:

NA,2= cT,2*DAB*(cA,2 – cA,1)/(cB,Im*L)

NA,2 = (ρA/MWA)*(dL/dt)

Cont’d Since the rate of diffusion of a gas and the rate of evaporation of the liquid are

the same, these two equations can be equated. Rearranging into the form of the integral, integrating from to to t and Lo to L, and rewriting:

2cT,2*DAB*MWA*(cA,2 – cA,1)*(t-to)/(ρA*cB,lm) = (L-Lo)2 + 2(L-Lo)Lo

Further rearranging leads to:

(t-to)/(L-Lo) = S(L-Lo) + 2SLo,

where S = ρA*cB,lm/(2cT,2*DAB*MWA*(cA,2 – cA,1)

As the equation above indicates, this is a linear function of the form y=mx+b, and when (t-to)/(L-Lo) is plotted as a function of L-Lo, the slope is S. Thus, the diffusion coefficient can be derived from the equation for S above to give:

DAB= ρA*cB,lm/(2ScT,2*MWA*(cA,2 – cA,1)

Determination of Diffusion Coefficient of an Electrolyte (NaCl) from Experimental Data Fick’s Law describes mass transfer due to diffusion:

For a binary mixture, the mole fractions sum to 1, so the gradients are inversely proportional:

Therefore DAB=DBA=D and Fick’s Law is written for one diffusivity:

The mole fraction gradient can be written as a differential, and assuming constant molar concentration,

J*A = -D(∂cA/∂z)

Cont’d Assuming the measured conductivity is linearly

proportional to the concentration of salt:k = CMcsalt

Rearranging,csalt = (1/CM)(k)

From the mass flow balance on salt:

V(∂csalt/∂t) = in

Differentiating the csalt equation and substituting into the above equation gives:

J*A = in/area = V(1/CM)*(∂k/∂t)*(4/πd2N)

Cont’d Approximating the concentration gradient as:

(cA,2 – cA,1)/(z2-z1)

Considering point 1 on the salt-rich side and point 2 on the salt-lean side, the distance is the membrane thickness L.

Substituting in L in the above equation gives:

J*A = -D(-cA,1/L)

Cont’d

Equating the previous equation with the J*A equation on the previous

slide gives:

V(1/CM)*(∂k/∂t)*(4/πd2N) = -D(-cA,1/L)

Solving for D yields:

D = 4LV/(cA,1CMπd2N) * (∂k/∂t)

which is of the form y=mx, such that when conductivity (k) is plotted against time (t), the slope is equal to (∂k/∂t), and D can be evaluated.

Determination of Diffusion Coefficient for Acetone from Correlation Equations Diffusivity in Gas Mixtures is given by the Fuller, Schettler, and

Giddings Correlation:

DAB = (0.00143T1.75)/(PMAB1/2[(ΣV)A

1/3 + (ΣV)B1/3]2)

where MAB = 2/((1/MA)+(1/MB)).

ΣV = summation of atomic and structural diffusion volumes from Table 3.1 of Separation Process Principles, Seader and Henley.

Determination of Diffusion Coefficient in Electrolyte Solutions The diffusion coefficient of a single salt in an aqueous solution can

be estimated from the Nernst-Haskell equation:

(DAB)∞ = (RT[(1/n+)+(1/n-)])/(F2[(1/λ+)+(1/λ-)])

Where n+ and n- = valences of the cation and anion, respectivelyλ+ and λ- = limiting ionic conductances given in Table 3.5, Seader and

HenleyF = Faraday’s Constant = 96,500 C/g-equivT = Temperature, KR = gas constant = 8.314 J/mol-K

Table of ResultsComparison of Experimental and Correlated Results

in the Electrolyte Diffusion Experiment

Type Diffusion Coefficient Standard Deviation Percent Error (m2/s) (m2/s) (%) ------------ --------------------- ------------------- --------------- Experimental 2.5136e-009 1.0991e-011 56.12 Correlated 1.6100e-009

Comparison of Experimental and Correlated Results in the Gaseous Diffusion Experiment

Type Diffusion Coefficient Standard Deviation Percent Error (m2/s) (m2/s) (%) ------------ --------------------- -------------------- ---------------- Experimental 1.1142e-005 2.6872e-007 4.56 Correlated 1.1674e-005

Conclusions

Diffusion Coefficient in the gaseous diffusion experiment compared very well to the correlation – 4.56% error

Liquid Diffusion Coefficient did not compare as well – 56.12% error

Possible Explanation of Error in the Electrolyte Diffusion Experiment

The diffusion coefficient is dependent on the position of the magnetic stir bar. Magnetic Stir bar causes diffusion by convection when we are measuring the diffusion by conduction. In our experimental setup, the stir bar position was at the 8 o’clock position on a clock face.