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Analysis of. Transport. Phenomena. SECOND EDITION. William M. Deen. MASSACHUSETTS INSTITUTE OF TECHNOLOGY. NewYork. Oxford. OXFORD . Transport Phenomena Deen Zapallitojeldres pdf, Free Analysis Of Transport Phenomena Deen Zapallitojeldres Download Pdf, Free Pdf Analysis Of. Transport Phenomena Deen Solution Manual pdf, Free Analysis Of Transport Phenomena Deen Solution Manual Download Pdf, Free Pdf Analysis Of.
These relationships, obtained for remarkably good fitting, the proposed relationship Eq. Furthermore, it is actually possible to reduce the allow for variation in rheological properties of the fluid number of dimensionless characteristic groups to two, with temperature.
For power law fluids, the relationship, expressing the local Sherwood number in terms of a sort similar to Eq.
The flow behavior index, n, is assumed constant removed. Therefore, a second 3D non-linear interpola- because it can be considered roughly independent of tion is performed according to the following complete temperature over considerable ranges. The obtained results are reported in Table 3.
Therefore, Eqs. Starting from the above assumptions, more than different values of local mass transfer coefficient, calcu- Table 2 lated by numerical program every 1 mm of the mem- Model parameters obtained after 3D non-linear fitting of Eq.
Then, parameter these values of Shloc have been interpolated, by means of Value 6. As a first attempt, Model parameters obtained after 3D non-linear fitting of Eq. Conclusions and zero flux conditions have been applied, the follow- ing boundary conditions have been employed: The transport phenomena involved in the concentra- Interface between the membrane and the channel: The momentum and mass balance equations, Momentum balance: The presented theoretical model is absolutely general, Osmotic pressure model applies: The results obtained from each simulation allow also the estimation of local mass Module outlet section: It has Momentum balance: Therefore, the calculated values of local mass Membrane outlet section: This relationship, obtained from theoretical considerations based only on fundamental principles, References shows a remarkable agreement with the empirical corre- Agashichev, S.
Modelling temperature and concentration lations reported in the literature for the study of heat polarization phenomena in ultrafiltration of non-newtonian fluids transfer in laminar flow of non-newtonian fluids, mod- under non-isothermal conditions.
Separation and Purification eled by power law equation. Technology, 25, — Bernardi, M. Effects of prolonged ingestion of graded doses of licorice by healthy-volunteers.
Life Appendix A Sciences, 55 11 , — Bhattacharya, S. Concentration polarization, separation factor, and Peclet number in membrane processes. Transport equation within the channel: Journal of Membrane Sciences, , 73— Bird, R. Transport Momentum balance and continuity equation: Bowen, W. Chemical Engi- Kimura, Y. Effects of flavonoids neering Science, 53 5 , — A calculation of viscous force exerted by metabolism and aggregation in human platelets.
Phytotherapy flowing fluid on a dense swarm of particles. Applied Scientists Research, 7 5 , — Research Series A, 1 1 , 27— Massarotto A. General Manager of Nature Med s. Curcio, S.
Fruit juice Personal communication. Unit operations of concentration polarization phenomena. Journal of Food Engineer- chemical engineering 4th ed. McGraw-Hill Book Company. Mi, L. Correlation of concentration polar- Curcio, S. A FEM analysis of ization and hydrodynamic parameters in hollow fiber modules.
Modelling of combined Navier—Stokes and Darcy modelling in chemical and food engineering pp. Chemical Engineering Eloot, S. Science, 53 6 , — Computational flow modeling in hollow-fiber dialyzers. Artificial Ruszymah, B. Organs, 26 7 , — Effects of glycyrrhizic acid on right atrial pressure and Gabriele, D. Optimal design pulmonary vasculature in rats. Clinical and Experimental Hyper- of single-screw extruder for liquorice candy production: In a multicomponent mixture, there are as many indepen- 'Adolph Fick In addition to his law far m s i o n , he made pioneering theoretical and experimental conoibutions in biomechanics.
Reference velocity Molar units Mass units Flux rehrionships: Moreover, the fluxes and concen- tration gradients of all species are interdependent see Chapter Accordingly, Fick's law is not applicable to multicomponent systems, except in special circumstances e.
Stress and Momentum Flux The linear momentum of a fluid element of mass m and velocity v is mv. In a pure fluid, v is defined merely by selecting a fixed reference frame; in a mixture, v must be inter- preted as the mass-average velocity, as given by Eq.
The local rates of momen- tum transfer in a fluid are determined in part by the stresses. Indeed, just as a force represents an overall rate of transfer of momentum, a stress force per unit area repre- sents a fiux of momentum. Consider some point within a fluid, through which passes an imaginary surface having an orientation described by the unit normal n. The stress at that point, with reference to the orientation of that test surface, is given by the stress vector; s n.
The stress vector is defined as the force per unit area on the test surface, exerted by the fluid towad which n points. As shown in Chapter 5, an equal and opposite stress is exerted by the fluid on the opposite side of the test surface. As shown in Fig. The normal component represents a normal stress or "pressure" on the surface, and the tangential components are shear stresses. As shown in Chapter 5, the stress vector is reIated to the stress tensor; u,by The stress tensor, which may be represented in component form as gives the information needed to compute the stress vector for a surface with any orienta- tion.
This is illustrated most simply by considering a surface which is normal to the y axis of a rectangular coordinate system, as shown in Fig. In that case so that, for example, uYxIS seen to be theforce per unit area on a plane perpendicular to the y axis, acting in the x direction, and exerted by the fluid at greater y. Note that the definition of uvcontains three parts, which specify a reference plane, a direction, and a sign convention, respehively.
The orientation of the reference plane is specified by the first subscript, and the direction of the force is indicated by the second subscript. A stress may be interpreted as a flux of momentum, as already noted. Because we have defined uyxin terms of the force exerted on a plane of constant y by fluid at greater y, positive values of uyxcorrespond to transfer of x momentum in the -y direction.
The sign convention adopted here is the one used most commonly in the fluid mechanics literahue; the opposite sign convention as used by Bird et al.. Representation of the s-ss vector s n for a a surface with arbitrary orientaton, and b a normal to the y nmis. In rhe lattn case the scalar components of s n equd components of the stress tensor u. As described in more detail in Chapter 5, the part of the stress which is caused solely by fluid motion is termed the viscous stress.
This contribution to the stress tensor is denoted as I. For a Newtonian Jluid with constant density, it is related to velocity gradients by where p is the viscosi ,Vv is the velocity gradient tensor, and Vv 'is the transpose of Vv. There is a qualitative analogy among Eq. That is, the only nonvanishing component of the velocity is in the x direction, and it depends only on y.
For that special case, Eq. The correspondence between Eq. If we had chosen the opposite sign convention for the stress, then a minus sign would have appeared in Eq. Gases and low-molecular-weight liquids are ordinarily Newtonian; everyday ex- amples are air and water. Various high-molecular-weight liquids e. Such fluids are characterized by structural features which influence, and also are influenced by, their flow. Although non-Newtonian fluids do not satisfy Eq.
The experi- mental and modeling studies which seek to establish constitutive equations for complex fluids comprise the field of rheology; we return to this subject briefly in Chapter 5.
The analogies among the constitutive equations are strengthened by expressing each as a flux that is proportional to the gradient of a concentration. The proportionality constant in each case is a type of diffusivity.
In this context, "concentration" is the amount per unit volume of any quantity, not only chemical species. In systems where thermal effects are more important than mechanical forms of energy, a useful measure of energy is the enthalpy. Considering variations in temperature only i. For constant cpand p, Eq.
If the density p of a mixture is constant, they component of Eq. B of Table becomes 1. Either is a good approximation for a dilute liquid solution, but constant Cis often more accurate for gases e.
The rate of viscous momentum transfer relative to heat conduction is given by the Prandtl number, 1. The purpose of this section is to summa- rize some representative data.
Many methods are available for estimating the values of transport coefficients in materials where specific data are limited or absent, but a discus- sion of those methods is beyond the scope of this book. A good source of information on property estimation is Reid et al. Figure shows the approximate ranges of viscosity seen for gases and ordinary nonpolymeric liquids, at room temperature and pressure.
The S. The range of liquid viscosities is even wider than the four orders of magnitude suggested by Fig. Depending on the forces and time scales of interest, materials ordinarily considered to be solids e.
Thus, liquid viscosities do not have an upper bound. Ranges of thermal conductivities are depicted in Fig. Most common organic liquids have thermal conductivities within a very narrow range, 0. The largest values of k are for pure metals, where conductivities for heat parallel those for electric current. This correspondence between electrical and thermal conductivity arises from the fact that, in pure metals, free electrons are the major carriers of heat; in other materials, the concentration of free electrons is low, and energy is transmitted primarily by atomic or molecular motions.
Nonmetallic solids range from good thermal insulators to good heat conductors. Values of k, and Pr for several gases and liquids are shown in Tables and For gases, J. L and k both increase with temperature, whereas the opposite tempera- ture dependence is seen for liquids except k for water.
For gases or liquids at moderate pressures, the effect of pressure on IL and k is generally negligible. The Prandtl numbers for gases are typically near unity, indicating that the intrinsic rates of heat conduction Gases. Approximate ranges for the viscosity of gases and nonpolymeric liquids, at ambient temperature and pressure.
Approximate ranges for the thermal conductivities of various classes of materials. Values of Pr for liquids typically fall between 1 and The exceptions include liquid metals e.
Ranges of binary diffusivities characteristic of gases, liquids, and so1ids are illus- trated in Fig. The values for air are from Gebhart et al. The Prandl] numbers for the other gases. All Prandtl numbers shown are for ooc. The data for liquid sodium and engine oil were taken from Rohsenow and Hartnett The values for mollen silicon are from Touloukian and Ho and Glazov et al.
The property values for molten LDPE are order-of-magnitude estimates from Tadmor and Oogos and depend on the flow conditions. This excludes porous mate- rials e.
Values of DAB for several gas pairs are shown in Table As shown in Table , DAB for dilute liquid solu- tions also increases with temperature. The pressure dependence of DAB in liquids is negligible in most appli- cations.
For gases, DAB is normally assumed to be independent of mole fraction. Also, for Approximate ranges for binary diffusivities in gases, liquids, and solids. The dependence of DAB on composition Jeads to certain distinctions among mea- sured values of diffusivity.
The binary diffusivity DAB defined in Table is termed the mutual diffusion coefficient; it is the coefficient measured in the presence of macroscopic concentration gradients.
Using radioisotopes, certain light scattering techniques, or other methods, it is also possible to measure diffusion within a mixture of macroscopically uniform composition; a diffusivity determined in this way is termed a tracer diffusion coefficient. In a mixture containing only A and B, DA Likewiset DAB Serum albumin, which has a molecular weight of 7 X 10", is the predominant protein in blood plasma. As discussed in Chapter 11, an activity-coefficient correction can be introduced into Fick's law to greatJy reduce the dependence of DAB on composition.
This is actually a special kind of tracer diffusivity, referring to conditions where species A is surrounded by itself. A careful interpretation of such results requires a multicomponent diffu- sion formulation see Chapter Representative vaJues of the Schmidt number can be estimated from the data in the tables. For gases at room temperature and pressure, typical viscosities and densities are J. For common liquids, p..
For solutes in water at room temperature, Sc ranges from about for H2 Gebhart et aL, to for large proteins. Even larger values of Sc result for polymeric solutes in very viscous liquids, such as polymer melts. In summary, species diffusion and viscous transfer of momentum have compara- ble intrinsic rates in gases, but diffusion of chemical species is by far the slower process in liquids.
The models discussed here are chosen for their illustrative value, rather than for their quantitative accuracy in predicting viscosi- ties, thermal conductivities, or diffusivities. The objective is to understand the orders of magnitude of the transport coefficients, and to some extent their dependence on tempera- ture, pressure, and composition.
For a more rigorous treatment of the molecular interpre- tation of transport coefficients in gases or liquids, a standard reference is Hirschfelder et al. Lattice Model The "molecular" or "diffusive" fluxes of energy, species, and momentum are based ulti- mately on the random motions of molecules. An elementary model to relate random motions of molecules to diffusive fluxes is developed by assuming molecular move- ments to be confined to a cubic lattice, as shown in Fig.
According to this model, each discrete location lattice site is occupied at any instant by a collection of mole- cules, each of which is able to jump to any of the six adjacent sites.
The lattice spacing, f, corresponds to the distance moved by a molecule in one "jump. It is assumed further that the molecules leaving a given position carry amounts of energy or momentum representative of that position.
Implicit in this assumption is that exchanges of energy and momentum among molecules at a given position allow them to equilibrate rapidly with one another. Other restrictions are the same as those used to obtain Eqs. To exploit the analogies among transport of energy, species, and momentum, we use f to denote any of the flux vectors molecular contributions only and use b to represent any of the corresponding concentrations, as summarized in Table Cubic-lattice model for random molecular motion.
Molecules are assumed to make jumps of length f parallel to one of the coordinate axes, moving at speed u. Including the molecules moving along the y axis which originated at either of those sites, we obtain u! We assume now that the jump distance eis much smaller than any macroscopic or system dimension, L.
Substituting Eq. Application of Lattice Model to Gases Any attempt to relate this elementary model to the properties of real materials depends on the interpretation of u and e. The situation is simplest for gases where, according to kinetic theory, transport results from exchanges that occur during molecular collisions. The concept of a collision is meaningful when the time spent during encounters with other molecules is much smaller than that spent between such encounters.
For a mixture of A and B it is necessary to use average values of M and din Eqs. This value of DAB is of the same order of magnitude as most of the gas diffusivities in Table The prediction from Eq.
Moreover, Eqs. Finally, Eq. We conclude that this elementary model does remarkably well in predicting the main features of the transport properties of gases. More careful comparisons with measurements in gases reveal that the numerical coefficients in Eqs. Although precise power-law relationships are not obeyed, it is roughly true that t-Lockrx T rather than T and that DABrx.
T rather than T These discrepancies are largely corrected by the rigorous Molecular Interpretation of Transport Coefficients Figure Qualitative dependence of inter- molecular potential energy on separation dis- tance, according to Eq.
Potential Energy 'I' f'T ro Separation Distance r kinetic theory of Chapman and Enskog Chapman and Cowling, , which considers in some detail the effect of intermolecular potential energies on the interactions between colliding molecules. The Chapman-Enskog theory assumes that all collisions are binary and elastic and that molecular motion during collisions can be described by classical mechanics. It also treats the molecules as spherically symmetric.
It successfully de- scribes the transport properties of gases at low pressures and high temperatures, except for the thennal conductivities of polyatomic gases.
In that case a correction must be included to account for transfer of internal i. This function is depicted qualitatively in Fig.
Neglecting the time spent during a collision was one of several assumptions used to derive Eq. In essence. Diffusivities for Liquids A consideration of length scales for 1iquids yields a very different picture than for gases.
This indicates that significant inter- molecular forces are present at aU times, so that the concept of distinct molecular colli- sions is much less meaningful for liquids than it is for gases. The transport coefficient which is related most directly to molecular displacements is DAB, and a displacement distance or jump length f can be estimated using Eq. Lacking a kinetic theory for liquids comparable to that available for gases, we equate u with the speed of sound.
The observation that Pr and Sc in liquids often differ widely from unity Section 1. The absence of a common mechanism is suggested further by the markedly different dependencies of k, DAB, and J. L on temperature. Although DAB increases with temperature in liquids as it does in gases , k and J.
L generttllY decrease Section 1. The gravitational force acting on a rain drop counteracts the resistance or drag imparted by the surrounding air. Commonalities among phenomena[ edit ] Diffusion[ edit ] There are some notable similarities in equations for momentum, energy, and mass transfer  which can all be transported by diffusion , as illustrated by the following examples: Mass: the spreading and dissipation of odors in air is an example of mass diffusion.
Energy: the conduction of heat in a solid material is an example of heat diffusion. Momentum: the drag experienced by a rain drop as it falls in the atmosphere is an example of momentum diffusion the rain drop loses momentum to the surrounding air through viscous stresses and decelerates. The molecular transfer equations of Newton's law for fluid momentum, Fourier's law for heat, and Fick's law for mass are very similar. One can convert from one transfer coefficient to another in order to compare all three different transport phenomena.