Steady-state coefficient

Figure 8.19 Evolution of the liquid concentration at the interface with a solid growing at the constant rate v from a solution initially at C0. K is the solid-liquid partition coefficient. Steady-state takes longer to establish for incompatible elements.

Diffusion coefficient Steady state Non-steady state ... 

Sj = threshold coefficient steady-state substrate concentration that must be exceeded for a positive net biomass production rate same units as S. 

Capillary and extrudate diameter, respectively Droplet deformability in extensional flow Rotational Brownian diffusion coefficient Steady-state tensile compliance Volume-to-surface average particle diameter Elasticity of the interphase Interaction energy Tensile, or Young s, modulus Electron... 

The criterion Dt/i = 0.5 to reach steady-state permeation is also useful when measuring the permeability coefficient steady-state data wiU be reached after an experiment time t = fl2D. Other values of as a function of D and thickness i are presented in Fig. 11.21. 

If tire diffusion coefficient is independent of tire concentration, equation (C2.1.22) reduces to tire usual fonn of Pick s second law. Analytical solutions to diffusion equations for several types of boundary conditions have been derived [M]- In tlie particular situation of a steady state, tire flux is constant. Using Henry s law (c = kp) to relate tire concentration on both sides of tire membrane to tire partial pressure, tire constant flux can be written as... 

Furtliennore, since tlie bifurcation must occur from a stable homogeneous steady state we must have D ID < 1 i.e. tlie diffusion coefficient of tlie inhibitor is greater tlian tliat of tlie activator. The critical diffusion ratio at tlie bifurcation is... 

Assuming constant physical coefficients for simplicity, the steady-state energy equation is expressed as... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

Thermal Stresses. When the wak of a cylindrical pressure vessel is subjected to a temperature gradient, every part expands in accordance with the thermal coefficient of linear expansion of the steel. Those parts of the cylinder at a lower temperature resist the expansion of those parts at a higher temperature, so setting up thermal stresses. To estimate the transient thermal stresses which arise during start-up or shutdown of continuous processes or as a result of process intermptions, it is necessary to know the temperature across the wak thickness as a function of radius and time. Techniques for evaluating transient thermal stresses are available (59) but here only steady-state thermal stresses are considered. The steady-state thermal stresses in the radial, tangential, and axial directions at a point sufficiently far away from the ends of the cylinder for there to be no end effects are as fokows ... 

The assumption of equiUbrium between soHd and bulk melt is frequently violated because of lack of complete mixing ia the melt. A steady-state fictitious stagnant-film treatment may be employed to arrive at an effective distribution coefficient,... 

Table 10 contains some selected permeabiUty data including diffusion and solubiUty coefficients for flavors in polymers used in food packaging. Generally, vinyUdene chloride copolymers and glassy polymers such as polyamides and EVOH are good barriers to flavor and aroma permeation whereas the polyolefins are poor barriers. Comparison to Table 5 shows that the large molecule diffusion coefficients are 1000 or more times lower than the small molecule coefficients. The solubiUty coefficients are as much as one million times higher. Equation 7 shows how to estimate the time to reach steady-state permeation t if the diffusion coefficient and thickness of a film are known. 

Water Transport. Two methods of measuring water-vapor transmission rates (WVTR) ate commonly used. The newer method uses a Permatran-W (Modem Controls, Inc.). In this method a film sample is clamped over a saturated salt solution, which generates the desired humidity. Dry air sweeps past the other side of the film and past an infrared detector, which measures the water concentration in the gas. For a caUbrated flow rate of air, the rate of water addition can be calculated from the observed concentration in the sweep gas. From the steady-state rate, the WVTR can be calculated. In principle, the diffusion coefficient could be deterrnined by the method outlined in the previous section. However, only the steady-state region of the response is serviceable. Many different salt solutions can be used to make measurements at selected humidity differences however, in practice,... 

Carbon Dioxide Transport. Measuring the permeation of carbon dioxide occurs far less often than measuring the permeation of oxygen or water. A variety of methods ate used however, the simplest method uses the Permatran-C instmment (Modem Controls, Inc.). In this method, air is circulated past a test film in a loop that includes an infrared detector. Carbon dioxide is appHed to the other side of the film. AH the carbon dioxide that permeates through the film is captured in the loop. As the experiment progresses, the carbon dioxide concentration increases. First, there is a transient period before the steady-state rate is achieved. The steady-state rate is achieved when the concentration of carbon dioxide increases at a constant rate. This rate is used to calculate the permeabiUty. Figure 18 shows how the diffusion coefficient can be deterrnined in this type of experiment. The time lag is substituted into equation 21. The solubiUty coefficient can be calculated with equation 2. 

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

To illustrate, consider the hmiting case in which the feed stream and the two liquid takeoff streams of Fig. 22-45 are each zero, thus resulting in batch operation. At steady state the rate of adsorbed carty-up will equal the rate of downward dispersion, or afV = DAdC/dh. Here a is the surface area of a bubble,/is the frequency of bubble formation. D is the dispersion (effective diffusion) coefficient based on the column cross-sectional area A, and C is the concentration at height h within the column. 

Two types of interac tion, competition, and predation are so important that worthwhile insight comes from considering mathematical formulations. Assuming that specific growth-rate coefficients are different, no steady state can be reached in a well-mixed continuous culture with both types present because, if one were at steady state with [L = D, the other would have [L unequal to D and a rate of change unequal to zero. The net effect is that the faster-growing type takes over while the other dechnes to zero. In real systems—even those that approximate well-mixed continuous cultures—there may be profound... 

From the final value theorem given in equation (3.10) it is possible to define a set of steady-state error coefficients. 

The overall heat transfer coefficient, U, is a measure of the conductivity of all the materials between the hot and cold streams. For steady state heat transfer through the convective film on the outside of the exchanger pipe, across the pipe wall and through the convective film on the inside of the convective pipe, the overall heat transfer coefficient may be stated as ... 

Burgess et al." describe a study of gas storage cabinets. In the study, coefficient of entry (CJ for various inlet/outlet configurations was measured. A tracer gas study is also described. The tracer gas study involved releasing sulfur hexafluoride (SF ) at 0.032 L s" at a critical leak position in the cabinet and measuring SFg concentration in the exhaust stream. The tracer gas was turned off when a steady exhaust stream concentration was observed and the time for the concentration to decay to 5% of steady state was measured. 

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