Theoretical Mixing Model Research Articles

Microwave Dielectric Behavior of Wet Soil-Part II: Dielectric Mixing Models

This paper is the second in a series evaluating the microwave dielectric behavior of soil-water mixtures as a function of water content and soil textural composition. Part II draws upon the data presented in Part 1 [13] to develop appropriate empirical and theoretical dielectric mixing models for the 1.4-to 18-GHz region. A semiempirical mixing model based upon the index of refraction is presented, requiring only easily ascertained soil physical parameters such as volumetric moisture and soil textural composition as inputs. In addition, a theoretical model accounting explicitly for the presence of a hydration layer of bound water adjacent to hydrophilic soil particle surfaces is presented. A four-component dielectric mixing model treats the soil-water system as a host medium of dry soil solids containing randomly distributed and randomly oriented disc-shaped inclusions of bound water, bulk water, and air. The bulk water component is considered to be dependent upon frequency, temperature, and salinity. The soil solution is differentiated by means of a soil physical model into 1) a bound component and 2) a bulk soil solution. The performance of each model is evaluated as a function of soil moisture, soil texture, and frequency, using the dielectric measurements of five soils ranging from sandy loam to silty clay (as presented in Part I [13]) at frequencies between 1.4 and 18 GHz. The semiempirical mixing model yields an excellent fit to the measured data at frequencies above 4 GHz. At 1.

Microwave Dielectric Behavior of Wet Soil-Part 1: Empirical Models and Experimental Observations

This is the first paper in a two-part sequence that evaluates the microwave dielectric behavior of soil-water mixtures as a function of water content, temperature, and soil textural composition. Part I presents the results of dielectric constant measurements conducted for five different soil types at frequencies between 1.4 and 18 GHz. Soil texture is shown to have an effect on dielectric behavior over the entire frequency range and is most pronounced at frequencies below 5 GHz. In addition, the dielectric properties of frozen soils suggest that a fraction of the soil water component remains liquid even at temperatures of -24° C. The dielectric data as measured at room temperature are summarized at each frequency by polynomial expressions dependent upon both the volumetric moisture content m and the percentage of sand and clay contained in the soil; separate polynomial expressions are given for the real and imaginary parts of the dielectric constant. In Part II, two dielectric mixing models will be presented to account for the observed behavior: 1) a semiempirical refractive mixing model that accurately describes the data and requires only volumetric moisture and soil texture as inputs, and 2) a theoretical four-component mixing model that explicitly accounts for the presence of bound water.

On the Correlation of Analytical and Experimental Free Shear Layer Similarity Profiles by Spread Rate Parameters

Analysis of turbulent isobaric free jet mixing normally requires the introduction of suitably formulated viscosity models. Similarity solutions can then be established which contain one empirical parameter. Such a parameter, however, not only describes the rate of spread of the mixing region, but also determines in detail the structure of the entire flow field. It is pointed out that this “spread rate parameter” σ depends on the selected viscosity model, the method of theoretical analysis, and the definition of profile matching. A comparison of different theoretical profiles can only be accomplished after these factors are properly recognized. Any attempts to contribute to the rather incomplete knowledge of the spread parameter must be cognizant of its dependence on the theoretical mixing model employed. This paper also establishes theoretical relations which allow comparison and consolidation of information based on different analytical concepts.

Polymer reactors and molecular weight distribution. IV. Free‐radical polymerization in a steady‐state stirred‐tank reactor train

AbstractThis paper reports the simulation of steady‐state styrene polymerization in a single, continuous, stirred‐tank reactor (CSTR) at high viscosity, a train of three CSTR's with and without recycle, and a single, imperfectly mixed CSTR. Monomer conversion and polymer molecular weight distribution (MWD), measured experimentally by gel permeation chromatography, were used as criteria of simulation. Earlier results for the single CSTR were used to modify the catalyst efficiency f and termination rate constant kt, to account for the effect of viscosity, μ. The regression equations obtained were log μ = −1.06 −0.522 log (S + 1) −8.34 log (1 − PF) + 0.565 logr̄w, log (f/fi) = 0.022 log (1 + μ) −0.0285 [log (1 + μ)]2, and log (kt/kti) = 0.060 log (1 + μ) −0.0934 [log (1 + μ]2, where μ is reactor viscosity in centipoise, S is solvent concentration in g‐mole/1., PF is polymer weight fraction, r̄W is weight‐average chain length, and fi and kti are values at zero conversion. The theoretical models for the single CSTR and train of CSTR's simulated the behavior of the real, bench‐scale CSTR's over a range of experimental conditions. The model for the CSTR system was applied to a theoretical mixing model made up of perfectly mixed CSTR's.