Product inhibition in packed-bed immobilized enzyme reactors for complex processes: Modelling of two-substrate processes

An analytical model was developed for describing the performance of packed-bed enzymic reactors operating with two cosubstrates, and when one of the reaction products is inhibitory to the enzyme. To this aim, the compartmental analysis technique was used. The relevant equations obtained were solved numerically, and the effect of the main operational parameters on the reactor characteristics were studied.Notation C _infa,i ^sup* local concentration of products in the pores of stage i - C _j,i concentration of substrate j in the pores of stage i - D _infa ^sup* internal (pore) diffusion coefficient for the reaction product a - D _j internal (pore) diffusion coefficient of substrate j - J _infa,i ^sup* net flux of product a, taking place from the pores of stage i into the corresponding bulk phase - J _j,i net flux of substrate j, taking place from the bulk phase of stage i into the corresponding pores - K _b inhibition constant - K _m,1, K _m,2 Michaelis constants for substrate 1 and 2, respectively - K _q inhibition constant - n total number of elementary stages in the reactor - Q volumetric flow rate throughout the reactor - R _j,i, R _infa,i ^sup* local reaction rates in pores of stage i, in terms of concentration of substrate j and product a respectively - S _infa,i ^sup* , S _infa,i-1 ^sup* bulk concentration of the reaction product a, in the stages i and i — 1, respectively - S _j,0 concentration of substrate j in the reactor feed - S _j,i-1, S _j,i concentration of substrate j in the bulk phase leaving stages i — 1 and i, respectively - V total volume of the reactor - V _m maximal reaction rate in terms of volumetric units - y axial coordinate of the pores - y ₀ depth of the pores - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ₁ dimensionless parameter, defined in Equation (6) - ₂ dimensionless parameter, defined in Equation (6) - ^* dimensionless parameter, defined in Equation (22) - ^* dimensionless parameter, defined in Equation (22) - volumetric packing density of catalytic particles (dimensionless) - porosity of the catalytic particles (dimensionless) - V _infi ^sup* dimensionless concentration of reaction product in pores of stage i, defined in Equation (17) - j,i dimensionless concentration of substrate j in pores of stage i; defined in Equation (6) - _j,i-1, _j.i dimensionless concentration of substrate j in the bulk phase of stage i; defined in Equation (6) - dimensionless position along the pore; defined in Equation (6)     

The theory of a vibrating-rod densimeter

The paper presents a theory of a device for the accurate determination of the density of fluids over a wide range of thermodynamic states. The instrument is based upon the measurement of the characteristics of the resonance of a circular section tube, or rod, performing steady, transverse oscillations in the fluid. The theory developed accounts for the fluid motion external to the rod as well as the mechanical motion of the rod and is valid over a defined range of conditions. A complete set of working equations and corrections is obtained for the instrument which, together with the limits of the validity of the theory, prescribe the parameters of a practical design capable of high accuracy.Nomenclature A, B, C, D constants in equation (60) - A _j , B _j constants in equation (18) - a _j ⁺ , a _j ^– wavenumbers given by equation (19) - C _f drag coefficient defined in equation (64) - C _f ^/0 , C _f ^/1 components of C _f in series expansion in powers of - c speed of sound - D _b drag force of fluid b - D ₀ coefficient of internal damping - E extensional modulus - force per unit length - F _j ⁺ , F _j ^– constants in equation (24) - f, g functions of defined in equations (56) - G modulus of rigidity - I second moment of area - K constant in equation (90) - k, k constants defined in equations (9) - L half-length of oscillator - Ma Mach number - m _a mass per unit length of fluid a - m _b added mass per unit length of fluid b - m _s mass per unit length of solid - n _j eigenvalue defined in equation (17) - P power (energy per cycle) - P _a , P _b power in fluids a and b - p pressure - R radius of rod or outer radius of tube - R _c radius of container - R _i inner radius of tube - r radial coordinate - T tension - T _visc temperature rise due to heat generation by viscous dissipation - t time - v _r , v radial and angular velocity components - y lateral displacement - z axial coordinate - dimensionless tension - _a dimensionless mass of fluid a - _b dimensionless added mass of fluid b - _b dimensionless drag of fluid b - dimensionless parameter associated with - ₀ dimensionless coefficient of internal damping - dimensionless half-width of resonance curve - dimensionless frequency difference defined in equation (87) - spatial resolution of amplitude - R, , , _s , increments in R, , , _s , - dimensionless amplitude of oscillation - dimensionless axial coordinate - ratio of to - _a , _b ratios of to for fluids a and b - angular coordinate - parameter arising from distortion of initially plane cross-sections - _f thermal conductivity of fluid - dimensionless parameter associated with - viscosity of fluid - _a , _b viscosity of fluids a and b - dimensionless displacement - _j jth component of - density of fluid - _a , _b density of fluids a and b - _s density of tube or rod material - density of fluid calculated on assumption that * - dimensionless radial coordinate - * dimensionless radius of container - dimensionless times - _{rr rr}, _r radial normal and shear stress components - spatial component of defined in equation (13) - _j jth component of - dimensionless streamfunction - ⁰, ¹ components of in series expansion in powers of - phase angle - _r phase difference - _ra , _rb phase difference for fluids a and b - streamfunction - _j jth component defined in equation (22) - dimensionless frequency (based on ) - _a , _b dimensionless frequency in fluids a and b - _s dimensionless frequency (based on _s ) - angular frequency - ₀ resonant frequency in absence of fluid and internal damping - _r resonant frequency in absence of internal fluid - _ra , _rb resonant frequencies in fluids a and b - dimensionless frequency - dimensionless frequency when _a vanishes - dimensionless frequencies when _a vanishes in fluids a and b - dimensionless resonant frequency when _a , _b, _b and ₀ vanish - dimensionless resonant frequency when _a , _b and _b vanish - dimensionless resonant frequency when _b and _b vanish - dimensionless frequencies at which amplitude is half that at resonance     

Void fraction and pressure fluctuations of bubbly flow in a vertical annular channel

In this investigation some hydrodynamic characteristics of two phase, two component, air water bubbly flow in a vertical annulus were studied. In particular, the void fraction profiles, and the pressure fluctuations were measured by the electrical resistivity probe and a capacitive type differential transducer respectively. These measurements were assessed under various system parameters, viz the air and water flux, the perforation ratio (Area of holes/channel cross sectional area) and the dimensionless axial distance. In addition, the pressure drop calculated from the void fraction measurements was in very good agreement with the corresponding one measured by the pressure transducers.List of symbols D _eq equivalent diameter of the annular channel (m) - j flux (discharge/channel cross sectional area) (m/s) - m mass flow rate (kg/s) - P pressure (Pa) - AP static pressure difference along the test section (Pa) - P pressure fluctuations (Pa) - P ^* dimensionless pressure (P _m/P _S.P. ) - P dimensionless pressure fluctuations (P _max /P _T.P. ) - r radius (m) - z axial distance (m) Greek symbols void fraction - dimensionless axial distance (Z/Dimeq) - perforation ratio (area of holes/channel cross sectional area) - density (kg/m3) - time (s) - dimensionless radial distance (r–r _i )/(r _o-r _i ) Suffix g gas - i inner - L liquid - m mean - Max Maximum - O outer - S.P. single-phase - T.P. two-phase     

Translatory accelerating motion of a circular disk in a viscous fluid

The exact solution of the equation of motion of a circular disk accelerated along its axis of symmetry due to an arbitrarily applied force in an otherwise still, incompressible, viscous fluid of infinite extent is obtained. The fluid resistance considered in this paper is the Stokes-flow drag which consists of the added mass effect, steady state drag, and the effect of the history of the motion. The solutions for the velocity and displacement of the circular disk are presented in explicit forms for the cases of constant and impulsive forcing functions. The importance of the effect of the history of the motion is discussed.Nomenclature a radius of the circular disk - b one half of the thickness of the circular disk - C dimensionless form of C ₁ - C ₁ magnitude of the constant force - D fluid drag force - f(t) externally applied force - F() dimensionaless form of applied force - F ₀ initial value of F - g gravitational acceleration - H() Heaviside step function - k magnitude of impulsive force - K dimensionless form of k - M a dimensionless parameter equals to (1+37#x03C0;_s/4_f) - S displacement of disk - t time - t ₁ time of application of impulsive force - u velocity of the disk - V dimensionless velocity - V ₀ initial velocity of V - V _t terminal velocity - parameter in (13) - parameter in (13) - (t) Dirac delta function - ratio of b/a - () function given in (5) - dynamical viscosity of the fluid - kinematic viscosity of the fluid - _f fluid density - _s mass density of the circular disk - dimensionless time - _i dimensionless form of t _i - dummy variable - dummy variable     

Axial dispersion in fluid flow through porous plates in parallel and through a porous tube

Summary Effects of axial diffusion on liquid-liquid displacement in fluid flow through porous plates in parallel and through a porous tube are considered as problems of two zones in unsteady state mass transfer. The solutions of the differential equations of the system in terms of the Laplace transformed variable contain an infinite number of essential singularities in a complicated form. Therefore approximate solutions are obtained by numerical inversion of the Laplace transform. Some of the numerical results are presented and discussed.Nomenclature C ₁ concentration of solute in Zone 1 - C ₂ concentration of solute in Zone 2 - C ₀ initial concentration of solute in Zone 2 - D _e effective diffusivity - D* axial dispersion (mixing) coefficient - K ratio D*/D _e - P _e Péclet number, Xv/D _e, Rv/D _e - P _e * longitudinal Péclet number, Xv/D*, Rv/D* - R inner radius of a porous tube - t time - v average velocity of fluid flow through Zone 1 - W width of a porous plate - Y length of a porous plate (tube) - porosity - ₁ dimensionless concentration of solute in Zone 1, C ₁/C ₀ - ₂ dimensionless concentration of solute in Zone 2, C ₂/C ₀ - Laplace transform of ₁ - Laplace transform of ₂ - ₁ dimensionless distance in porous plate, x/X - ₂ dimensionless distance in a porous tube, r/R - ₁ dimensionless axial distance in porous plate, y/X - ₂ dimensionless axial distance in a porous tube, y/R - ₁ dimensionless time in porous plate, tD _e/X ² - ₂ dimensionless time in a porous tube, tD _e/R ² - Units CGS system     

Consecutive chemical reactions in an annular reactor with non-newtonian flow

The purpose of this paper is to analyze the homogeneous consecutive chemical reactions carried out in an annular reactor with non-Newtonian laminar flow. The fluids are assumed to be characterized by a Ostwald-de Waele (powerlaw) model and the reaction kinetics is considered of general order. Effects of flow pseudoplasticity, dimensionless reaction rate constants, order of reaction kinetics and ratio of inner to outer radii of reactor on the reactor performances are examined in detail.Nomenclature c _A concentration of reactant A, g.mole/cm³ - c _B concentration of reactant B, g.mole/cm³ - c _A0 inlet concentration of reactant A, g.mole/cm³ - C ₁ dimensionless concentration of A, c _A/c _A0 - C ₂ dimensionless concentration of B, c _B/c _A0 - C ₁ dimensionless bulk concentration of A - C ₂ dimensionless bulk concentration of B - D _A molecular diffusivity of A, cm²/sec - D _B molecular diffusivity of B, cm²/sec - k _A first reaction rate constant, (g.mole/cm³)^1–m /sec - k _B second reaction rate constant, (g.mole/cm³)^1–n /sec - K ₁ dimensionless first reaction rate constant, k _A r ₀ ² c _A0 ^m–1 /D _A - K ₂ dimensionless second reaction rate constant, k _B r ₀ ² c _A0 ^n–1 /D _B - K apparent viscosity, dyne(sec) ^m /cm² - m order of reaction kinetics - n order of reaction kinetics - P pressure, dyne/cm² - r radial coordinate, cm - r _i radius of inner tube, cm - r _max radius at maximum velocity, cm - r _o radius of outer tube, cm - R dimensionless radial coordinate, r/r _o - s reciprocal of rheological parameter for power-law model - u local velocity, cm/sec - u _max maximum velocity, cm/sec - u bulk velocity, cm/sec - U dimensionless velocity, u/u - z axial coordinate, cm - Z dimensionless axial coordinate, zD _A/r ₀ ² /u - ratio of molecular diffusivity, D _B/D _A - ratio of inner to outer radius of reactor, r _i/r _o - ratio of radius at maximum velocity to outer radius, r _max/r _o     

A power-law fluid flow past a porous sphere

Summary A model has been developed for the flow of a non-Newtonian fluid past a porous sphere. The drag force exerted on a porous sphere moving in a power-law fluid is obtained by an approximate solution of equations of motion in the creeping flow regime. It is predicted that the effect of the pseudoplastic anomaly on the drag force is more pronounced at large porosity parameters.

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Zusammenfassung Es wird ein Modell für die Strömung einer nichtnewtonschen Flüssigkeit längs einer porösen Kugel entwickelt. Die auf die in einer Ostwald-DeWaele-Flüssigkeit bewegte Kugel ausgeübte Reibungskraft wird durch eine Näherungslösung der Bewegungsgleichungen für schleichende Strömung gewonnen. Man findet, daß der Einfluß der Abweichung vom newtonschen Verhalten um so ausgeprägter wird, je größer die Porosität ist.

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A, B, C, D a, b, c, d coefficients in eqs. [10] and [18] - F _D drag force - K consistency index in power-law model - k ₁ ,k ₂ coefficients defined by eq. [18] - m porosity parameter - n flow index in power-law model - P pressure - P ^* dimensionless pressure defined by eq. [4] - P pressure difference - R radius of porous sphere - r radial distance from the center of the sphere - U velocity of uniform stream - u _i velocity component - u _i ^* dimensionless velocity component defined by eq. [4] - Y drag force correction factor defined by eq. [27] - _ij rate of deformation tensor - _ij ^* dimensionless rate of deformation tensor defined by eq. [4] - , spherical coordinates - dimensionless radial distance defined by eq. [4] - second invariant of rate of deformation tensor - ^* dimensionless second invariant of rate of deformation tensor defined by eq. [4] - _ij stress tensor - _ij ^* dimensionless stress tensor defined by eq. [4] - stream function - ^* dimensionless stream function defined by eq. [4] - i inside the surface of the sphere - o outside the surface of the sphere With 1 figure and 1 table     

Estimate of the transient conduction of heat in materials with linear thermal properties based on the solution for constant properties

A method of analysis is described which yields quasianalytical solutions for one and multidimensional unsteady heat conduction problems with linearly dependent thermal properties, such as thermal conductivity and volumetric specific heat. The method accomodates rather general thermal boundary conditions including arbitrary variations in surface temperature or in surface heat flux or a convective exchange with a fluid having even varying temperature. Once the solution for the identical problem but with constant properties has been developed, its practical realization is rather direct, being facilitated by a reduced number of iterations. The four applied examples given in this work show that a wide variety of nonlinear heat conduction problems can be tackled by this procedure without much difficulty. These simple solutions compare favorably with more laborious results reported in the archival heat transfer literature.

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Berechnung nichtstationärer Wärmeleitvorgänge mit linear temperaturabhängigen Stoffwerten aus der Lösung für konstante Stoffwerte

Zusammenfassung Es werden quasi-analytische Lösungen für ein- und mehrdimensionale nichtstationäre Wärmeleitprobleme mit linear temperaturabhängigen Stoffwerten, wie Wärmeleitfähigkeit und volumetrische Wärmekapazität, mitgeteilt. Die Methode gilt für recht allgemeine Randbedingungen wie beliebige Veränderungen der Oberflächentemperatur, der Wärmestromdichte oder auch konvektiven Wärmeaustausch mit veränderlicher Fluidtemperatur. Ist die Lösung für das identische Problem mit konstanten Stoffwerten bekannt, kann die Methode direkt mit einer begrenzten Zahl von Iterationen angewandt werden. Die vier hier mitgeteilten Beispiele zeigen, daß eine große Zahl nichtlinearer Wärmeleitprobleme auf diese Weise ohne Schwierigkeit angepackt werden können. Die einfachen Lösungen stimmen befriedigend mit komplizierteren Ergebnissen aus der Literatur überein.

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Nomenclature a side of square bar - B _i0 reference Biot number,hR/k₀ - B _i0 ^T transformed Biot number, equation (16) - c geometric parameter, equation (8) - h convective coefficient - k thermal conductivity - k ₀ value ofk atT ₀ - K dimensionless thermal conductivity,k/k ₀ - K _i value ofK at _i - K _i+1 value ofK at _i+1 - m _k slope of theK- line, equation (3) - m _s slope of theS- line, equation (4) - R characteristic length - s volumetric specific heat - s ₀ value of s at T₀ - S dimensionless volumetric specific heat, s/s₀ - S _i value ofS at _i - S _i+1 value of S at _i+1 - t time - T temperature - T ₀ reference temperature - x, y cartesian coordinates - X, Y dimensionless cartesian coordinates,x/a andy/a - thermal diffusivity - _k transformed time, equation (11) - _s transformed time, equation (37) - _k dimensionless time for variable conductivity, equation (8) - _s dimensionless time for variable specific heat, equation (34) - dimensionless temperature,T/T ₀ - dimensionless coordinate,r/R - ₀ value of at T₀ - _i lower value of the interval (_i, _i+1) - _i+1 upper value of the interval (_i, _i+1     

Unsteady convective diffusion in a rotating parallel plate channel

The paper presents an exact analysis of the dispersion of a passive contaminant in a viscous fluid flowing in a parallel plate channel driven by a uniform pressure gradient. The channel rotates about an axis perpendicular to its walls with a uniform angular velocity resulting in a secondary flow. Using a generalized dispersion model which is valid for all time, we evaluate the longitudinal dispersion coefficientsK _i (i=1, 2, ...) as functions of time. It is shown thatK ₁=0 andK ₃,K ₄, ... decay rapidly in comparison withK ₂. ButK ₂ decreases with increasing (the dimensionless rotation parameter) for values of upto approximately =2.2. ThereafterK ₂ increases with further increase in and its value gets saturated for large values of (say, 500) and does not change any further with increase in . A physical explanation of this anomalous behaviour ofK ₂ is given.

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Instationäre konvektive Diffusion in einem rotierenden Parallelplattenkanal

Zusammenfassung In dieser Untersuchung wird eine exakte Analyse der Ausbreitung eines passiven Kontaminierungsstoffes in einer zähen Flüssigkeit gegeben, die, befördert durch einen gleichförmigen Druckgradienten, in einem Parallelplattenkanal strömt. Der Kanal rotiert mit gleichförmiger Winkelgeschwindigkeit um eine zu seinen Wänden senkrechte Achse, wodurch sich eine Sekundärströmung ausbildet. Unter Verwendung eines generalisierten, für alle Zeiten gültigen Dispersionsmodells werden die longitudinalen DispersionskoeffizientenK _i (i=1, 2, ...) als Funktionen der Zeit ermittelt. Es wird gezeigt, daßK ₁=0 gilt und dieK ₃,K ₄, ... gegenüberK ₂ schnell abnehmen.K ₂ nimmt ab, wenn , der dimensionslose Rotationsparameter, bis etwa zum Wert 2,2 ansteigt. Danach wächstK ₂ mit bis auf einem Endwert an, der etwa ab =500 erreicht wird. Dieses anomale Verhalten vonK ₂ findet eine physikalische Erklärung.

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List of symbols C solute concentration - D molecular diffusivity - K _i longitudinal dispersion coefficients - 2L depth of the channel - P ₀ dimensionless pressure gradient along main flow - Pe Péclet number - q velocity vector - Q _x,Q _y mass flux along the main flow and the secondary flow directions - dimensionless average velocity along the main flow direction - (x, y, z) Cartesian co-ordinates Greek symbols dimensionless rotation parameter - the inclination of side walls withx-axis - kinematic viscosity - fluid density - dimensionless time - angular velocity of the channel - dimensionless distance along the main flow direction - dimensionless distance along the vertical direction - dimensionless solute concentration - integral of the dispersion coefficientK ₂() over a time interval     

The very slow flow of a Powell-Eyring fluid around a sphere

Summary The very slow flow of a Powell-Eyring type non-Newtonian fluid around a sphere is investigated by a variational technique. The result, a correction factor that is applied to the Stokes' equation, is given as a plot and as an equation which is empirically fit to the plot. Also, a comparison of the very slow flows of a simplified viscoelastic Oldroyd fluid and the Powell-Eyring fluid is made which indicates that in a certain restricted region of the very slow flows, both models give essentially the same results. The Oldroyd and Powell-Eyring model parameters are interrelated by forcing both models to fit the same tube flow viscosity data.Nomenclature B dimensionless quantity, v /R - C dimensionless second invariant - c ₁ constant determined by variational method - D dimensionless variational integral - D _2j , D _j+k position-independent variables used in specification of trial functions - E _2j , E _j+k position-independent variables used in specification of trial functions - f friction factor - f _corr friction factor correction - F _drag drag force on sphere - g, g ₀, g ₁ general trial function; first and second terms in the general trial function - G, H terms in the expression for C - j index - J variational integral - k index - K term in the expression for C - p, q integers - r integer, radial coordinate - R radius of sphere - Re Reynolds number - Re ₀ Reynolds number at point of zero shear rate - Re Reynolds number at infinite distance from sphere - Re _NN Reynolds number based on variable part of viscosity - u, v dimensionless position coordinates - V volume considered - v _i ith velocity component - v _r , v , v _z velocity components in the r, , and z-directions - v approach velocity of the fluid - x/ parameter in Powell-Eyring model - x _i i-position coordinate - parameter in Powell-Eyring model - rate of deformation - , _c , _N , ₀ coefficient of viscosity; cross viscosity; parameter in Powell-Eyring model; viscosity in limit of zero shear rate - spherical coordinate - , _ij rate of deformation tensor; ij-component of rate of deformation tensor - ₁, ₂ parameters in Oldroyd model - Newtonian viscosity - ₁, ₂ parameters in Oldroyd model - dimensionless radial coordinate, r/R - second invariant - fluid density - spherical coordinate - stream function     

Temperature distribution in a thin rotating disk

The problem of heat conduction in a thin rotating disk with heat input at a fixed point is considered. The disk is cooled by forced convection from its lateral surfaces. By defining a complex temperature, the temperature throughout the disk is presented as a series of Bessel functions of complex argument. Results are given for a range of rotational speeds.Nomenclature R radial coordinate - angular coordinate - a radius of disk - b thickness of disk - T temperature - T ambient temperature - rotational speed of disk - q heat flux into disk - k thermal conductivity of disk - density of disk - c specific heat of disk - h coefficient of convective heat transfer - r dimensionless radial coordinate, R/a - T* characteristic temperature, q ₀ a/ k - t dimensionless temperature, (T–T )/T* - C ₁, C ₂ dimensionless parameters defined in (3)     

On the cross-effects of coupled macroscopic transport equations in porous media

In this paper, the derivation of macroscopic transport equations for this cases of simultaneous heat and water, chemical and water or electrical and water fluxes in porous media is presented. Based on themicro-macro passage using the method of homogenization of periodic structures, it is shown that the resulting macroscopic equations reveal zero-valued cross-coupling effects for the case of heat and water transport as well as chemical and water transport. In the case of electrical and water transport, a nonsymmetrical coupling was found.Notations b mobility - c concentration of a chemical - D rate of deformation tensor - D molecular diffusion coefficient - D _ij ^eff macroscopic (or effective) diffusion tensor - electric field - E ₀ initial electric field - k _ij molecular tensor - j, j ^*, current densities - K _ij macroscopic permeability tensor - l characteristic length of the ERV or the periodic cell - L characteristic macroscopic length - L _ijkl coupled flows coefficients - n _i unit outward vector normal to - p pressure - q _t ,q _t ⁺ , heat fluxes - q _c ,q _c ⁺ , chemical fluxes - s specific entropy or the entropy density - S entropy per unit volume - t time variable - t _ij local tensor - T absolute temperature - v _i velocity - V ₀ initial electric potential - V electric potential - x macroscopic (or slow) space variable - y microscopic (or fast) space variable - _i local vectorial field - _i local vectorial field - electric charge density on the solid surface - , bulk and shear viscosities of the fluid - _ij local tensor - _ij local tensor - _i local vector - _ij molecular conductivity tensor - _ij ^eff effective conductivity tensor - homogenization parameter - fluid density - ₀ ion-conductivity of fluid - _ij dielectric tensor - _i ¹ , _i ² , _i ³ local vectors - ⁴ local scalar - _S solid volume in the periodic cell - _L volume of pores in the periodic cell - boundary between _S and _L - ^s rate of entropy production per unit volume - total volume of the periodic cell - _l volume of pores in the cell On leave from the Politechnika Gdanska; ul. Majakowskiego 11/12, 80-952, Gdask, Poland.     

Diffusion of macromolecular solutions in the turbulent boundary layer of a cylindrical pipe

A model is presented describing the changes that occur in the diffusion boundary layer upon injection of a macromolecular solution (PEO) into a cylindrical pipe under turbulent flow conditions (Re 40,000). A shape parameter was introduced to describe the shape of the turbulent plume. The value of this parameter was found to be the same for water and various dilute PEO solutions. The proposed model gives a good approximation at low homogeneous concentrations. x downstream distance from the slot - y normal distance from the wall - R radius of the pipe - C concentration - C _w wall concentration - Q _i flow rate injection - Q _t flow rate - C _j =C _i *Q _i /Q _t equivalent homogeneous polymer concentration - L _tf characteristic length of the diffusion plume - characteristic height of the diffusion plume, i.e. the value ofy at whichC/C _w = 0.5 - thickness of the diffusion boundary layer - x ₀ characteristic distance from the slot, i.e. the value ofx at which/R = 1/2 - ⁺ shape parameter of the diffusion boundary layer - ⁺ —/R nondimensionalized variables - x ⁺ —x/L _tf nondimensionalized variables     

Multicomponent turbulent boundary layer on a chemically active surface

When a gaseous mixture flows past chemically active surfaces the boundary layer formed on the wetted body may contain a large number of components with different diffusion properties. This leads to the necessity for studying the diffusion of the components in the multicomponent boundary layer.The use of thebinary boundary layer concept in the general case cannot yield satisfactory results, since replacement of the mutual diffusion coefficients D_ij of the various pairs of components by a single diffusion coefficient D in many cases is a rough approximation.In the general case the number of different diffusion coefficients is equal to N(N–1)/2 (N is the number of components). Usually it is possible to identify groups of components with similar molecular weights. Then the number of different diffusion coefficients may be reduced without large error. However, even in the comparatively simple case when it is possible to divide all the components into two groups with similar molecular weights we must take account of three different diffusion coefficients (one diffusion coefficient in each group and also the diffusion coefficient for the components of one group relative to the components of the other group). Only in particular cases when the gaseous mixture consists of only two components with arbitrary molecular weights, or if all the components of the gaseous mixture have similar molecular weights, can we with justification introduce a single diffusion coefficient (if in this case there are no limitations on the direction of the diffusion).Studies have been published covering the laminar multicomponent boundary layer. An analytic method for solving the equations of the laminar multicomponent boundary layer was developed by Tirskii [1]. There are also studies in which concrete results were obtained by numerical methods with the use of computers (for example, [2, 3]).As far as the author knows, for turbulent flow there are studies (for example, [4, 5]) covering flow with chemical reactions only in the case when all the diffusion coefficients are equal (D_ij=D).The present paper presents a method for calculating the turbulent multicomponent boundary layer with account for several different diffusion coefficients.Notation x, y coordinates - u, v velocity components - density - T temperature - h heat content - H enthalpy - c_i mass concentration of the i-th component - c ₁ ⁽¹⁾ element concentrations in solid body - J_i diffusion flux of the i-th component - m molecular weight - dynamic viscosity coefficient - kinematic viscosity coefficient - heat conduction coefficient - c_p specific heat - adiabatic index - D_ij binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - S_t Stanton number - M Mach number - friction - q radiant thermal flux - boundary layer thickness - D rate of displacement of gas-solid interface - degree of gasification - r_ij weight fraction of element i in component j - _ij stoichiometric coefficients - K_i reaction equilibrium constants - l number of components for which I_i0 Indices i, j component number - w quantities for y=0 - * quantities on the edge of the laminar sublayer - ⁽¹⁾ quantities at the solid body - quantities at the outer edge of the boundary layer - molar transport coefficients     

Neue Näherungsgleichung zur einheitlichen Berechnung von Wärmeübertragern

Zusammenfassung Es wird eine für alle Stromführungen einheitliche Näherungsgleichung mit drei oder vier anpaßbaren Parametern zur Berechnung des Korrekturfaktors für die mittlere logarithmische Temperaturdifferenz angegeben. Die anpaßbaren Parameter wurden für etwa 50 verschiedene Stromführungen durch Ausgleichsrechnung bestimmt. Die Genauigkeit der Gleichung ist für die Berechnung im praktisch wichtigen Bereich mehr als ausreichend.

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New approximate equation for uniform heat exchanger design

An approximate equation with three or four empirical parameters for the uniform calculation of the LMTD-correction factor of all heat exchanger configurations is proposed. The empirical parameters have been determined for about 50 different flow configurations using least squares estimation. The accuracy of the equation is more than sufficient for practical design purposes.

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Formelzeichen A Übertragungsfläche - a, b, c, d Parameter der Näherungsgleichung - Wärmekapazitätsstrom - F Korrekturfaktor für die logarithmische mittlere Temperaturdifferenz - k Wärmedurchgangskoeffizient - m, n Zahl der Durchgänge oder Einzelapparate - NTU Anzahl der Übertragungseinheiten (number of transfer units); NTU_i=kA/ _i - P dimensionslose Temperaturänderung - R Wärmekapazitätsstromverhältnis;R ₁=₁/₂;R ₂=₂/₁ - relativer Fehler - Mittelwert von NTU₁ und NTU₂ Indizes 1, 2 Stoffstrom 1, 2 - G Gegenstrom - s Schätzwert Herrn Prof. Dr.-Ing. E.h. K. Stephan zum 65. Geburtstag gewidmet.     

Heat transfer from laminar flow in ducts with elliptic cross-section

Summary The cooling of a hot fluid in laminar Newtonian flow through cooled elliptic tubes has been calculated theoretically. Numerical data have been computed for the two values 1.25 and 4 of the axial ratio of the elliptic cross-section . For =1.25 the influence of non-zero thermal resistance between outmost fluid layer and isothermal surroundings has also been investigated. Special attention has been given to the distribution of heat flux around the perimeter; when increases the flux varies more with the position at the circumference. This positional dependence becomes less pronounced, however, as the (position-independent) thermal resistance of the wall increases.Flattening of the conduit, while maintaining its cross-sectional area constant, improves the cooling. Comparison with rectangular pipes shows that this improvement is not as marked with elliptic as with rectangular pipes.Nomenclature A _k =A _{m, n} coefficients of expansion (6) - a, b half-axes of ellipse, b<a - a _p =a _{r, s} coefficients of representation (V) - D hydraulic diameter, = 4S/P; S = cross-sectional area, P = perimeter - D _e equivalent diameter, according to (13) - n coordinate (outward) normal to the tube wall - T temperature of fluid - T _i temperature of fluid at the inlet - T _s temperature of surroundings - v ₀ mean velocity of fluid - v _z longitudinal velocity of fluid - x, y carthesian coordinates coinciding with axes of ellipse - z coordinate in flow direction - , dimensionless half-axes of ellipse, =a/D and =b/D - _t heat transfer coefficient from fluid at bulk temperature to surroundings; equation (11) - _w heat transfer coefficient at the wall; equation (3) - axial ratio of ellipse, = a/b = / - , , , dimensionless coordinates; =x/D, =y/D, =z/D, =n/D - dimensionless temperature, = (T–T _s)/(T _i–T _s) - ₀ cup-mixing mean value of ; equation (10) - thermal conductivity of fluid - _m,n = _k eigenvalue - c volumetric heat capacity of fluid - _{m, n} = _k = _k eigenfunction; equations (6) and (I) - Nu total Nusselt number, = _t D/ - Nusselt number at large distance from the inlet - Nu _w wall Nusselt number, = _w D/, based on _w - Pé Péclet number, = ₀ Dc/     

Flow of a multicomponent gas between parallel permeable surfaces

The present paper gives an exact solution of the equations describing the flow of a multicomponent gas between two parallel permeable planes, one of which moves relative to the other with constant velocity (i. e., we study a flow of the Couette type).Notation y coordinate - u, v velocity components - density - c_i mass concentration of i-th component - I_i diffusional flux of i-th component - H enthalpy - T temperature - m molecular weight - viscosity coefficient - heat conduction coefficient - c_p mixture specific heat - D_ij the binary diffusion coefficients - P Prandtl number - S_ij Schmidt number - N total number of components - n number of components in injected gas - l distance between planes Indices i, j component numbers - w applies to quantities for y=0 - ^* applies to quantities for y=l     

Darcy-Forchheimer natural,forced and mixed convection heat transfer in non-Newtonian power-law fluid-saturated porous media

The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.Nomenclature A cross-sectional area - b _i coefficient in the chosen temperature profile - B ₁ coefficient in the profile for the dimensionless boundary layer thickness - C coefficient in the modified Forchheimer term for power-law fluids - C ₁ coefficient in the Oseen approximation which depends essentially on pore geometry - C _i coefficient depending essentially on pore geometry - C _D drag coefficient - C _t coefficient in the expression forK ^* - d particle diameter (for irregular shaped particles, it is characteristic length for average-size particle) - f _p resistance or drag on a single particle - F _R total resistance to flow offered byN particles in the porous media - g acceleration due to gravity - g _x component of the acceleration due to gravity in thex-direction - Grashof number based on permeability for power-law fluids - K intrinsic permeability of the porous media - K ^* modified permeability of the porous media for flow of power-law fluids - l _c characteristic length - m exponent in the gravity field - n power-law index of the inelastic non-Newtonian fluid - N total number of particles - Nux,_D,F local Nusselt number for Darcy forced convection flow - Nux,_D-F,F local Nusselt number for Darcy-Forchheimer forced convection flow - Nux,_D,M local Nusselt number for Darcy mixed convection flow - Nux,_D-F,M local Nusselt number for Darcy-Forchheimer mixed convection flow - Nux,_D,N local Nusselt number for Darcy natural convection flow - Nux,_D-F,N local Nusselt number for Darcy-Forchheimer natural convection flow - pressure - p exponent in the wall temperature variation - _Pe c characteristic Péclet number - _Pe x local Péclet number for forced convection flow - _Pe x modified local Péclet number for mixed convection flow - _Ra c characteristic Rayleigh number - _Ra x local Rayleigh number for Darcy natural convection flow - _Ra x local Rayleigh number for Darcy-Forchheimer natural convection flow - Re convectional Reynolds number for power-law fluids - Reynolds number based on permeability for power-law fluids - T temperature - T _e ambient constant temperature - T w,_ref constant reference wall surface temperature - T _w(X) variable wall surface temperature - T _w temperature difference equal toT w,_ref–T _e - T ₁ term in the Darcy-Forchheimer natural convection regime for Newtonian fluids - T ₂ term in the Darcy-Forchheimer natural convection regime for non-Newtonian fluids (first approximation) - T _N term in the Darcy/Forchheimer natural convection regime for non-Newtonian fluids (second approximation) - u Darcian or superficial velocity - u ₁ dimensionless velocity profile - u _e external forced convection flow velocity - u _s seepage velocity (local average velocity of flow around the particle) - u _w wall slip velocity - U c _M characteristic velocity for mixed convection - U c _N characteristic velocity for natural convection - x, y boundary-layer coordinates - x ₁,y ₁ dimensionless boundary layer coordinates - X coefficient which is a function of flow behaviour indexn for power-law fluids - effective thermal diffusivity of the porous medium - shape factor which takes a value of/4 for spheres - shape factor which takes a value of/6 for spheres - ₀ expansion coefficient of the fluid - _T boundary-layer thickness - T ₁ dimensionless boundary layer thickness - porosity of the medium - similarity variable - dimensionless temperature difference - coefficient which is a function of the geometry of the porous media (it takes a value of 3 for a single sphere in an infinite fluid) - ₀ viscosity of Newtonian fluid - ^* fluid consistency of the inelastic non-Newtonian power-law fluid - constant equal toX(2 ^2–n )/ - density of the fluid - dimensionless wall temperature difference     

Über den Einfluß von Querkrümmung und Dichte bei inhomogenen turbulenten Freistrahlen

Zusammenfassung Zur Berechnung turbulenter Strömungen wird das k--Modell im Ansatz für die turbulente Scheinzähigkeit erweitert, so daß es den Querkrümmungs- und Dichteeinfluß auf den turbulenten Transportaustausch erfaßt. Die dabei zu bestimmenden Konstanten werden derart festgelegt, daß die bestmögliche Übereinstimmung zwischen Berechnung und Messung erzielt wird. Die numerische Integration der Grenzschichtgleichungen erfolgt unter Verwendung einer Transformation mit dem Differenzenverfahren vom Hermiteschen Typ. Das erweiterte Modell wird auf rotationssymmetrische Freistrahlen veränderlicher Dichte angewendet und zeigt Übereinstimmung zwischen Rechnung und Experiment.

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On the influence of transvers-curvature and density in inhomogeneous turbulent free jets

The prediction of turbulent flows based on the k- model is extended to include the influence of transverse-curvature and density on the turbulent transport mechanisms. The empirical constants involved are adjusted such that the best agreement between predictions and experimental results is obtained. Using a transformation the boundary layer equations are solved numerically by means of a finite difference method of Hermitian type. The extended model is applied to predict the axisymmetric jet with variable density. The results of the calculations are in agreement with measurements.

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Bezeichnungen Wirbelabsorptionskoeffizient - c_i Massenkonzentration der Komponente i - c_D, c_L, c, c₁, c₂ Konstanten des Turbulenzmodells - d Düsendurchmesser - E bezogene Dissipationsrate - f bezogene Stromfunktion - f Korrekturfunktion für die turbulente Scheinzähigkeit - j turbulenter Diffusionsstrom - k Turbulenzenergie - k_i Schrittweite in -Richtung - K dimensionslose Turbulenzenergie - L turbulentes Längenmaß - M_i Molmasse der Komponente i - p Druck - allgemeine Gaskonstante - r Querkoordinate - r_0,5 Halbwertsbreite der Geschwindigkeit - r_0,5c Halbwertsbreite der Konzentration - T Temperatur - u Geschwindigkeitskomponente in x-Richtung - v Geschwindigkeitskomponente in r-Richtung - x Längskoordinate - y allgemeine Funktion - Y_i diskreter Wert der Funktion y - Relaxationsfaktor für Iteration - turbulente Dissipationsrate - transformierte r-Koordinate - kinematische Zähigkeit - Exponent - transformierte x-Koordinate - Dichte - _k, Konstanten des Turbulenzmodells - Schubspannung - allgemeine Variable - Stromfunktion - Turbulente Transportgröße Indizes 0 Strahlanfang - m auf der Achse - r mit Berücksichtigung der Krümmung - t turbulent - mit Berücksichtigung der Dichte - im Unendlichen - Schwankungswert oder Ableitung einer Funktion - – Mittelwert Herrn Professor Dr.-Ing. R. Günther zum 70. Geburtstag gewidmet     

Hydrodynamical changes due to polymer migration in very dilute solutions

The slip hypothesis, based on thermodynamical arguments, has been extended to obtain the flow characteristics of polymer solutions flowing in a nonhomogeneous flow field. An asymptotic analysis, valid for both channel and falling film flows, is presented that predicts the flow enhancement due to polymer migration. Concentration-viscosity coupling is shown to be a critical factor in the hydrodynamic analysis. The analysis, which essentially provides an upper bound on flow enhancement, explicitly accounts for the influence of wall shear stress, initial polymer concentration etc. A comparison with the pertinent experimental data shows reasonable agreement. c concentration - c ₀ concentration in shear-free region - c _i initial concentration - d rate of deformation tensor - g acceleration due to gravity - g ₁ function defined in eq. [13] or [15] - g ₂ function defined in eq. [18] or [20] - H half-channel thickness or film thickness - K gas law constant - L length of the channel or film - q flow rate per unit width - q ^* normalized flow rate - T temperature - v velocity - V mean velocity - y transverse distance - y _c location of solvent layer - _w /µ _s - _w /c ₀ KT - /t convected derivative - dimensionless cenentration,c/c ₀ - _c dimensionless interface concentration - _w dimensionless wall concentration - relaxation time - µ _eff effective viscosity - µ _s solvent viscosity - dimensionless transverse distance,y/H - _c dimensionless interface location - density - stress tensor - _w wall shear stress - c _i KT/ _w - ns no slip NCL-Communication No. 3155