Empirical relationships

Freundlich isotherm The empirical relationship between the amount of a substance adsorbed and the concentration of the solute... 

The flash point depends closely on the distillation initial point. The following empirical relationship is often cited ([, 1 ... 

The Kraft point (T ) is the temperature at which the erne of a surfactant equals the solubility. This is an important point in a temperature-solubility phase diagram. Below the surfactant cannot fonn micelles. Above the solubility increases with increasing temperature due to micelle fonnation. has been shown to follow linear empirical relationships for ionic and nonionic surfactants. One found [25] to apply for various ionic surfactants is ... 

Enthalpy of Vaporization (or Sublimation) When the pressure of the vapor in equilibrium with a liquid reaches 1 atm, the liquid boils and is completely converted to vapor on absorption of the enthalpy of vaporization ISHv at the normal boiling point T. A rough empirical relationship between the normal boiling point and the enthalpy of vaporization (Trouton s rule) is ... 

Accuracy For macro-major samples, relative errors of 0.1-0.2% are routinely achieved. The principal limitations are solubility losses, impurities in the precipitate, and the loss of precipitate during handling. When it is difficult to obtain a precipitate free from impurities, an empirical relationship between the precipitate s mass and the mass of the analyte can be determined by an appropriate standardization. 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

Rate of Mass Transfer in Bubble Plates. The Murphree vapor efficiency, much like the height of a transfer unit in packed absorbers, characterizes the rate of mass transfer in the equipment. The value of the efficiency depends on a large number of parameters not normally known, and its prediction is therefore difficult and involved. Correlations have led to widely used empirical relationships, which can be used for rough estimates (109,110). The most fundamental approach for tray efficiency estimation, however, summarizing intensive research on this topic, may be found in reference 111. 

Process Systems. Because of the large number of variables required to characterize the state, a process is often conceptually broken down into a number of subsystems which may or may not be based on the physical boundaries of equipment. Generally, the definition of a system requires both definition of the system s boundaries, ie, what is part of the system and what is part of the system s surroundings and knowledge of the interactions between the system and its environment, including other systems and subsystems. The system s state is governed by a set of appHcable laws supplemented by empirical relationships. These laws and relationships characterize how the system s state is affected by external and internal conditions. Because conditions vary with time, the control of a process system involves the consideration of the system s transient behavior. 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

The characteristics of soluble sihcates relevant to various uses include the pH behavior of solutions, the rate of water loss from films, and dried film strength. The pH values of sihcate solutions are a function of composition and concentration. These solutions are alkaline, being composed of a salt of a strong base and a weak acid. The solutions exhibit up to twice the buffering action of other alkaline chemicals, eg, phosphate. An approximately linear empirical relationship exists between the modulus of sodium sihcate and the maximum solution pH for ratios of 2.0 to 4.0. 

The thermal glass-transition temperatures of poly(vinyl acetal)s can be determined by dynamic mechanical analysis, differential scanning calorimetry, and nmr techniques (31). The thermal glass-transition temperature of poly(vinyl acetal) resins prepared from aliphatic aldehydes can be estimated from empirical relationships such as equation 1 where OH and OAc are the weight percent of vinyl alcohol and vinyl acetate units and C is the number of carbons in the chain derived from the aldehyde. The symbols with subscripts are the corresponding values for a standard (s) resin with known parameters (32). The formula accurately predicts that resin T increases as vinyl alcohol content increases, and decreases as vinyl acetate content and aldehyde carbon chain length increases. 

The two steps in the removal of a particle from the Hquid phase by the filter medium are the transport of the suspended particle to the surface of the medium and interaction with the surface to form a bond strong enough to withstand the hydraulic stresses imposed on it by the passage of water over the surface. The transport step is influenced by such physical factors as concentration of the suspension, medium particle size, medium particle-size distribution, temperature, flow rate, and flow time. These parameters have been considered in various empirical relationships that help predict filter performance based on physical factors only (8,9). Attention has also been placed on the interaction between the particles and the filter surface. The mechanisms postulated are based on adsorption (qv) or specific chemical interactions (10). 

Moreover, the receptrode has an extremely rapid response time, requiring only two to three milliseconds to fully respond to a target concentration change. The response times of conventional chemical sensors (qv) are typically from several seconds to several minutes. The receptrode exhibits a response that follows the empirical relationship... 

Water content indirectly affects other lens characteristics. Water evaporation from the lens can result in a dry eye sensation and subsequent desiccative erosion of the cornea. Clinical studies have shown the incidence of corneal erosion as a result of lens desiccation to be a material-dependent and water-content-dependent phenomenon (25,26). The nature of water and sodium ions in hydrogels has been studied primarily by nmr and thermal techniques (27,28). An empirical relationship between water mobility in contact lens polymers and desiccative staining has been proposed (29). 

Neural nets can also be used for modeling physical systems whose behavior is poorly understood, as an alternative to nonlinear statistical techniques, eg, to develop empirical relationships between independent and dependent variables using large amounts of raw data. 

Circular Tubes Numerous relationships have been proposed for predicting turbulent flow in tubes. For high-Prandtl-number fluids, relationships derived from the equations of motion and energy through the momentum-heat-transfer analogy are more complicated and no more accurate than many of the empirical relationships that have been developed. 

Colburn relationship found that the optimum number of trays varies from 2 to 3 times the number at total reflux. Gilliland [Ind. Eng. Chem, . 32, 1220 (1940)] from the establishment of an empirical relationship between reflux ratio and theoretical trays based on a study of existing columns indicated that... 

Calvert et al. []. Air Pollut. Control Assoc., 22, 529 (1972)] obtained an explicit equation by making some simplifying assumptions and incorporating an empirical constant that must be evaluated experimentally the constant may absorb some of the deficiencies in the model. Although other models avoid direct incorporation of empirical constants, use of empirical relationships is necessary to obtain specific-estimates of scrubber collec tion efficiency. One of the areas of greatest uncertainty is the estimation of droplet size. 

The enU opy of activation may be estimated from experimental data for gaseous molecules, and in the special case of HI formation, which may be regarded as involving the intermediate formation of tire dimer H2I2, using the general empirical relationship... 

As in die case of die diffusion properties, die viscous properties of die molten salts and slags, which play an important role in die movement of bulk phases, are also very stiiicture-seiisitive, and will be refeiTed to in specific examples. For example, die viscosity of liquid silicates are in die range 1-100 poise. The viscosities of molten metals are very similar from one metal to anodier, but die numerical value is usually in die range 1-10 centipoise. This range should be compared widi die familiar case of water at room temperature, which has a viscosity of one centipoise. An empirical relationship which has been proposed for die temperature dependence of die viscosity of liquids as an AiTlienius expression is... 

Additional empirical relationships for approach to saturation and dynamic recovery are given by Follansbee and Kocks [14]. 

Solvent effects on chemical equilibria and reactions have been an important issue in physical organic chemistry. Several empirical relationships have been proposed to characterize systematically the various types of properties in protic and aprotic solvents. One of the simplest models is the continuum reaction field characterized by the dielectric constant, e, of the solvent, which is still widely used. Taft and coworkers [30] presented more sophisticated solvent parameters that can take solute-solvent hydrogen bonding and polarity into account. Although this parameter has been successfully applied to rationalize experimentally observed solvent effects, it seems still far from satisfactory to interpret solvent effects on the basis of microscopic infomation of the solute-solvent interaction and solvation free energy. 

Figure 2.15 Empirical relationships between (a) and Cp and (b) q and Cp (with 95% confidence limits)...

Figure 3.3 Empirical relationship between Cp and Cp showing degree of process shift expected for the components analysed...

A number of years ago an empirical relationship, now called Cram Jr rule, was recognized. When R, R, and R differ in size, and the molecule is oriented such that the largest group is anti to the carbonyl oxygen, the major product arises from addition of the nucleophile syn to the smaller substituent. ... 

In addition to flow regime, hold-up and pressure drop are two other important parameters in two-phase gas-liquid flows. Hold-up is defined as the relative portion of space occupied by a phase in the pipe. It can be expressed on a time or space average basis, with the actual method chosen depending on the intended use of the hold-up value, and the measurement method employed. There are numerous correlations in the literature for hold-up, but most are based upon a pressure drop-hold-up correlation. The following expression is a widely recognized empirical relationship between hold-up and pressure drop ... 

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