Stefan tube

In Example 2.1.1 we described the experiments of Carty and Schrodt (1975) who evaporated a binary liquid mixture of acetone(l) and methanol(2) in a Stefan tube. Air(3) was used as the carrier gas. Using an effective diffusivity method calculate the composition profiles. 

SOLUTION The composition profiles in the Stefan tube are given by Eq. 8.6.2. Before we can compute the profiles we must determine the rates of evaporation of acetone and methanol. Since the evaporating species are present in low concentrations at the top of the tube (although not at the bottom) we shall use the dilute solution limit for the effective diffusivities 

Solution of Eq. 8.6.5 together with the determinacy condition = 0 yields the following values for the fluxes  

It is worth noting that simple repeated substitution of the fluxes is not effective for solving this particular problem if the calculations are started with 3/eff = 1 (corresponding to a null estimate of the fluxes). The oscillations in the fluxes that result with simple repeated substitution can be avoided by using an average of the last two computed estimates of the fluxes in the evaluation of the mass transfer rate factors. In this case, however, we used Newton s method to solve a single function of the total flux (cf. Algorithm 8.5). 

Using the converged set of fluxes, the rate factors have the values Oi.eff 2.427 0)2,eff = 1-672 

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EXAMPLE 14-9 Measuring Diffusion Coefficient fay the Stefan Tube... 

A 3-cm-diameter Stefan tube is used to measure the binary diffusion coefficient of water vapor irr air at 20°C at an elevation of 1600 m where the atmospheric... 

SOLUTION The amount of water that evaporates from a Stefan tube at a specified temperature and pressure over a specilied time period is measured. The diffusion coefficient of water vapor in air is to be determined. 

A 2.5-cm-diametei Stefan tube is used to measure the binary diffusion coefficient of water vapor in air at 25°C and 95 kPa. The tube is partially filled with water wiUi a distance from the water surface to the open end of the mbe of 25 cm. Dry air is blown over the open end of the tube so that water vapor rising to the top is removed immediately and the concentration of vapor at the lop of the tube is zero. During 10 days of continuous operation at constant pressure and temperature, the amount of water that has evaporated is measured to be 0.0011 kg. Determine the diffusion coefficient of water vapor in air at 25 C and 95 kPa. 

The Stefan tube, depicted schematically in Figure 2.4, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of diffusing vapor there to essentially nothing. The mole fraction of the vapor at the vapor-liquid interface is its equilibrium value. 

Figure 8.11. Composition profiles in a Stefan tube. Lines are computed from effective diffusivity model. Data of Carty and Schrodt (1975).

Repeat Example 8.6.1 (diffusion in a Stefan tube) using an exact matrix method of determining the fluxes. 

Repeat Example 8.6.1 (diffusion in a Stefan tube) using the Toor-Stewart-Prober method of determining the fluxes. Compare the profiles computed from the linearized equations to the profiles obtained with the exact method and the experimental data given in Example 2.2.1. 

A classical method for measuring the diffusion coefficient or the vapor of a volatile liquid in air or other gas (e.g,. toluene in N2) employs the Stefan tube shown in Fig. 2.3-8. a long tube of narrow diameter (to suppress convection) partially filled with a pure volatile liquid A and maintained in a constanl-tempeiaUire bath. A gcotle flow of air is sometimes established acmes the top of the tube to sweep away the vapor reaching the top of the lube. The fall of the liquid level with lime is observed. 

Example 1.17 Multicomponent, Steady-State, Gaseous Diffusion in a Stefan Tube (Taylor and Krishna, 1993)... 

The Stefan tube, depicted schematically in Figure 1.8, is a simple device sometimes used for measuring diffusion coefficients in binary vapor mixtures. In the bottom of the tube is a pool of quiescent liquid. The vapor that evaporates from this pool diffuses to the top of the tube. A stream of gas across the top of the tube keeps the mole fraction of the diffusing vapors there to essentially zero. The compositon of the vapor at the vapor-liquid interface is its equilibrium value. Carty and Schrodt (1975) evaporated a binary liquid mixture of acetone (1) and methanol (2) in a Stefan tube. Air (3) was used as the carrier gas. In one of their experiments the composition of the vapor at the liquid interface was yx - 0.319, y2 - 0.528, and y3 = 0.153. The pressure and temperature in the gas phase were 99.4 kPa and 328.5 K, respectively. The length of the diffusion path was 0.24 m. The MS diffusion coefficients of the three binary pairs are ... 

For a multicomponent mixture of ideal gases such as this, the Maxwell-Stefan equations (1-35) must be solved simultaneously for the fluxes and the composition profiles. At constant temperature and pressure the total molar density c and the binary diffusion coefficients are constant. Furthermore, diffusion in the Stefan tube takes place in only one direction, up the tube. Therefore, the continuity equation simplifies to equation (1-69). Air (3) diffuses down the tube as the evaporating mixture diffuses up, but because air does not dissolve in the liquid, its flux N3 is zero (i.e., the diffusion flux J3 of the gas down the tube is exactly balanced by the diffusion-induced bulk flux x3N up the tube). 

For a Stefan tube problem where a liquid containing n-1 components evaporating into a gas space containing the n-th component which is insoluble in liquid, the flux of the n-th component is simply zero, that is ... 

Open system Closed system Stefan tube... 

Having presented the flux equations for a multicomponent system, we will apply the Stefan-Maxwell s approach to solve for fluxes in the Stefan tube at steady state. Consider a Stefan tube (Figure 8.2-3) containing a liquid of species 1. Its vapour above the liquid surface diffuses up the tube into an environment in which a species 2 is flowing across the top, which is assumed to be nonsoluble in liquid. 

Figure 8.2-3 Concentration profile and flux directions in Stefan tube...

Very often the Stefan tube experiment is conducted with very high flow of gas above the tube, thus molecules of the component 1 are swept away very quickly by the gas stream, implying that the mole fraction of the component 1 can be effectively assumed as zero at the top of the tube. Thus, the evaporation written explicitly in terms of temperature and pressure is ... 

We have considered the Stefan tube with pure liquid in the tube. Now we consider the case whereby the liquid contains two components. These two species will evaporate and diffuse along the tube into the flow of a third component across the top of the tube. The third component is assumed to be non-soluble in the liquid. What we will consider next is the Maxwell-Stefan analysis of this ternary system, and then apply it to the experimental data of Carty and Schrodt (1975) where they used a liquid mixture of acetone and methanol. The mole fractions of acetone and methanol just above the liquid surface of the tube are 0.319 and 0.528, respectively. 

The mole fraction profiles of acetone and methanol after the convergence has been achieved are shown in Figure 8.2-4. Alternatively, one could use the vector analysis presented in Appendix 8.2 to solve for the fluxes without using the shooting method. We will present the method in the next section when we deal with a Stefan tube containing n" components, that is we have "n-1" components in the liquid and the n-th component is the non-soluble gas flowing across the tube. 

We illustrate below an example of solving the Stefan tube with n diffusing species by the method of Newton-Raphson. 

Example 8.2-8 Newton-Ralphson method for n-component system in a Stefan tube... 

This shows that the fluxes N in a special way (LHS of eq. 8.2-113) are related to the mole fractions of the n-th component at the end points of the Stefan tube. 

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