Dielectric properties of the composite system polyurethane — sodium chloride

Summary The dielectric properties of the composite system polyurethane-sodium chloride have been measured at frequencies between 10^–4 Hz and 3 · 10⁵ Hz in the temperature range from –150 °C up to +90 dgC. Three dielectric loss mechanisms have been found; they are indicated by ₁, ₂ and. The activation energy of the ₁-transition is 35 kcal/mole, that of the-transition 6.7 kcal/mole. The ₂-loss peak was only observed at frequencies of 10³ Hz and higher, forming one broad peak with the ₁-loss peak at lower frequencies. In the composite materials, the- and ₂-loss peaks measured at fixed frequencies were found at the same temperature. The ₂-loss peak, however, was shifted to a lower temperature, due to the sodium chloride filler. Comparison of experimental data of and tan with theoretical predictions concerning the dielectric properties of composite systems showed only partial agreement. The difference mainly consisted in. the temperature shift in the tan-peak of the ₁-transition.

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Zusammenfassung Die dielektrischen Eigenschaften des Verbundssystems Kochsalz-Polyurethankautschuk wurden im Frequenzgebiet zwischen 10^–4 Hz und 3.10⁵ Hz und im Temperaturbereich von –150 °C bis +90 °C gemessen. Es wurden drei dielektrische Verlustmaxima gefunden, die mit ₁, ₂ und angedeutet werden. Die Aktivierungsenergie des ₁-Überganges beträgt 35 kcal/Mol, die des-Überganges 6.7 kcal/Mol. Das ₂-Maximum konnte nur bei Frequenzen höher als 10³Hz vom ₁-Maximum gesondert erfaßt werden. Die Lage der ₂- und-Maxima war vom Füllgrad unabhängig. Das ₁-Maximum verschiebt sich mit steigendem Füllgrad zu niedrigeren Temperaturen. Die gemessenen Werte von und tan stimmen nur teilweise mit den Aussagen einer Theorie der dielektrischen Eigenschaften von Mischkörpern überein.

     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

On the measurement of normal stress differences in steady shear flow

Summary Compared to the similar pressure-distribution cone-and-plate apparatus of Adams and Lodge (4), the new apparatus' improvements include: temperature control of the cone (as well as the plate); increased stiffening of the frame; four (not three) pressuremeasuring holes in the cone/plate region; inclusion of a pressure-measuring hole on the axis under the cone truncation; exclusive use of a vertical free liquid boundary at the cone rim (without a sea of liquid). Temperature control of the rotating cone and of the fixed plate leads to acceptable temperature uniformity in the test liquid for working temperatures within 10°C or 20°C of ambient; the corresponding interval is about 1°C if the cone temperature control is abandoned. Pressure gradients measured using a Newtonian liquid agree with those calculated using Walters' eq. (3). For a viscoelastic liquid, after subtracting inertial contributions, pressure distributions measured at a given shear rate in the cone/plate region do not change when the gap angle is changed from 2° to 3°, showing that the effects of secondary flow are negligible. Values ofN ₃ =N ₁ + 2N ₂ obtained from the gradients of these distributions are believed to be in error by not more than ±1 Pa, in favorable cases. The present most useful ranges are: 10 to 5000 Pa forN ₃; 0.1 to 200 sec^–1 for shear rate; up to 5 Pa s for viscosity; and 5 to 40°C for temperature. As an application, it is shown that adding 0.1% of a high molecular weight polyisobutylene to a 2% polyisobutylene solution doublesN ₃ and has no detectable effect on the viscosity measured at low shear rates with a Ferranti-Shirley viscometer.

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Zusammenfassung Im Vergleich zu dem ähnlichen Kegel-Platte-Gerät von Adams und Lodge (4) zur Messung der Druckverteilung wurden an dem neuen Gerät die folgenden Verbesserungen vorgenommen: Temperaturregelung an Kegel und Platte, Versteifung des Rahmens, vier (anstatt drei) Druckmeßlöcher im Kegel-Platte-Bereich, ein zusätzliches Druckmeßloch auf der Achse unter der Kegelstumpf-Deckfläche, ausschließliche Verwendung einer vertikalen freien Grenzfläche der Flüssigkeit am Kegelrand (ohne umgebenden Flüssigkeitssee). Die Temperaturregelung des rotierenden Kegels und der festen Platte führt zu einer ausreichenden Temperaturgleichförmigkeit in der Testflüssigkeit für Betriebstemperaturen, die höchstens um 10–20°C von der Umgebungstemperatur abweichen. Dieses Intervall beträgt dagegen nur etwa 1°C, wenn auf die Temperaturregelung am Kegel verzichtet wird. Für newtonsche Flüssigkeiten entsprechen die gemessenen Druckgradienten den mittels der Gleichung von Walters (3) berechneten. Für viskoelastische Flüssigkeiten zeigen sich bei der Änderung des Spaltwinkels von 2° auf 3° nach Abzug der Trägheitsbeiträge keine Änderungen der bei einer bestimmten Schergeschwindigkeit gemessenen Druckverteilung. Dies zeigt, daß Sekundärströmungseffekte vernachlässigbar sind. Es darf angenommen werden, daß die Werte vonN ₃ =N ₁ + 2N ₂, die man aus den Gradienten dieser Verteilungen erhält, unter günstigen Umständen mit einem Fehler von nicht mehr als ±1 Pa behaftet sind. Gegenwärtig liegen die günstigsten Bereiche bei 10 bis 5000 Pa fürN ₃, 0,1 bis 200 s^–1 für die Schergeschwindigkeit, unterhalb von 5 Pa s für die Viskosität und 5 bis 40°C für die Temperatur. Als Anwendung wird gezeigt, daß ein Zusatz von 0,1% hochmolekularen Polyisobutylens zu einer 2%igen Polyisobutylenlösung den Wert vonN ₃ verdoppelt, aber keinen erkennbaren Einfluß auf die (bei geringen Schergeschwindigkeiten mit einem Ferranti-Shirley-Viskosimeter gemessen) Viskosität hat.

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udsf unidirectional shear flow - TCP truncated-cone and plate - N ₁,N ₂ 1st and 2nd normal stress differences in udsf - N ₃ N ₁ + 2N ₂ - : = A is defined by the equationA := B - P ^* hole pressurePw – Pm; Pw, Pm = pressures measured by flush transducer and by hole-mounted transducer - t time - , strain rate, shear rate - (P,t) covariant body metric tensor at particleP and timet - ⁱ , _i covariant and contravariant udsf body base vectors (i = 1, 2, 3) - ^–1 inverse of - R, plate radius, cone/plate gap angle - r ₀,h ₀ radius and height of cone truncation - r,, spherical polar coordinates; cone axis = 0; plate surface = /2 - physical components of stress; for a tensile component - cone angular velocity - p on the plate = /2 - ,T, density, absolute temperature, viscosity - P 0.15 ²(r ² –R ²) (inertial contribution) [2.7] - P _ve contribution [2.8] from flow perturbations of viscoelastic origin - r _i i = 1,2,3,4; values ofr at centers of holes in cone/plate region - P _i () pressure change recorded by transducerTi when cone angular velocity goes from zero to - 1/2 {P _i ()+ P _i (–)} (average for 2 senses of rotation) - rim pressure, from least-squares line through four points - Re Reynolds' number:R ²/ - (P,t)/t With 11 figures and 2 tables     

Theoretical investigation of the effect of an internal heat source on turbulent liquid metal heat transfer in a channel with uniform heat flux

Summary The effect of an internal heat source on the heat transfer characteristics for turbulent liquid metal flow between parallel plates is studied analytically. The analysis is carried out for the conditions of uniform internal heat generation, uniform wall heat flux, and fully established temperature and velocity profiles. Consideration is given both to the uniform or slug flow approximation and the power law approximation for the turbulent velocity profile. Allowance is made for turbulent eddying within the liquid metal through the use of an idealized eddy diffusivity function. It is found that the Nusselt number is unaffected by the heat source strength when the velocity profile is assumed to be uniform over the channel cross section. In the case of a 1/7-power velocity expression, the Nusselt numbers are lower than those in the absence of internal heat generation, and decrease with diminishing eddy conduction. Nusselt numbers, in the absence of an internal heat source, are compared with existing calculations, and indications are that the present results are adequate for preliminary design purposes.Nomenclature A hydrodynamic parameter - a half height of channel - a ₁ a constant, 1+0.01 Pr Re ^0.9 - a ₂ a constant, 0.01 Pr Re ^0.9 - C _p specific heat at constant pressure - D _h hydraulic diameter of channel, 4a - h heat transfer coefficient, q _w/(t _w–t _b) - I ₁ integral defined by (17) - I ₂ integral defined by (18) - k diffusivity parameter, (1+0.01 Pr Re ^0.9)^1/2 - m exponent in power velocity expression - Nu Nusselt number, hD _h/ - Nu ₀ Nusselt number in absence of internal heat generation - Pr Prandtl number, / - Q heat generation rate per volume - q _w wall heat flux - Re Reynolds number for channel, 2/ - s ratio of heat generation rate to wall heat flux, Qa/q _w - T dimensionless temperature, (t _w–t)/(t _w–t _b) - t fluid temperature, t _w wall temperature, t _b fluid bulk temperature - u fluid velocity in x direction, , fluid mean velocity - x longitudinal coordinate measured from channel entrance - x ⁺ dimensionless longitudinal coordinate, 2(x/a)/Pr Re - y transverse coordinate measured from channel centerline - z transverse coordinate measured from channel wall, a–y - molecular diffusivity of heat, /C _p - dummy variable of integration - dummy variable of integration - _H eddy diffusivity of heat - _M eddy diffusivity of momentum - dummy variable of integration - fluid thermal conductivity - _T dimensionless diffusivity, Pr ( _H/) - fluid kinematic viscosity - dummy variable of integration - fluid density - dummy variable of integration - ratio of eddy diffusivity for heat transfer to that for momentum transfer, _H/ _M - average value of - dimensionless velocity distribution, u/     

Calculation of a wall-stabilized argon arc with account for radiative energy transfer

It was shown experimentally in [1, 2] and in a study by E. I. Asinovskii and A. V. Kirillin reported at the Scientific Technical Conference of the High-Temperature Scientific Research Institute held in 1964 that the heat transfer mechanism in a wall-stabilized argon arc was not purely purely conductive at gas temperatures greater than 11 000° K. Asinovskii and Kirillin also showed that radiative energy transfer is the reason why similarity is lost when the current-voltage characteristics are constructed in dimensionless form. The radiation of an argon arc has been studied experimentally by a number of authors [3–5], Dautov [6] calculated an argon arc without allowing for radiation.In this article an argon arc stabilized by the cooled duct walls is calculated with account for radiation using theoretically computed relationships describing the transport properties of argon plasma. A large portion of the radiated energy pertains to spectral lines whose role has been studied by L. M. Biberman. The authors have used I. T. Yakubov's data on argon radiation published in the journal Optics and Spectroscopy. A method of calculation and data on argon plasma radiation are also given in [7].Reference [8] deals with the problem of the role of radiation in an arc burning in nitrogen. In particular, the above-mentioned loss of similarity follows from the results of this work. However, the relationships used in this article to describe the transport properties of nitrogen plasma were obtained experimentally in [9].Notation r₀ arc radius (cm) - r variablesradius (cm) - T temperature (°K) - heat transfer coefficient (ergcm^–1sec^–1deg^–1) - E electric field intensity (g^1/2cm^–1/2sec^–1) - electrical conductivity (sec^–1) - q₁ heat flux density due to conduction - q₂ heat flux density due to radiation - u divergence of radiative energy flux density in the transparent part of the spectrum (ergcm^–3sec^–1) - u₂ same for part of the spectrum where reabsorption is taken taken into account - m₀ atomic mass - m_e electronic mass - Stefan-Boltzmann constant - h Planck constant - k Boltzmann constant - e electronic charge - p pressure - degree of ionization - N_e electron concentration (cm^–3) - n₀ neutral atom concentration - Q_0e electron-neutral collision cross section - Q_ie electron-ion collision cross section (cm²) - ₀ line center frequency (sec^–1) - ₊ line halfwidth (distance from line center to contour for ) due to effects giving dispersion contour - k _v absorption coefficient (cm^–1) - energy radiated by a hemispherical volume - emissivity of hemispherical volume - radius of hemispherical volume - S line intensity The authorS thank I. T. Yakubov for allowing them to use his data on arc plasma radiation.     

Non-isothermal flow of polymer melts in a curved tube

The results of a numerical study (using finite differences) of heat transfer in polymer melt flow is presented. The rheological behaviour of the melt is described by a temperature-dependent power-law model. The curved tube wall is assumed to be at constant temperature. Convective and viscous dissipation terms are included in the energy equation. Velocity, temperature and viscosity profiles, Nusselt numbers, bulk temperatures, etc. are presented for a variety of flow conditions. Br — Brinkman number - c specific heat, J/kg K - De — Dean number - E dimensionless apparent viscosity, eq. (14d) - G dimensionless shear rate, eq. (19) - k parameter of the power-law model, °C^–1, eq. (7) - mass flow rate, kg/s - m ₀ parameter of the power-law model, Pa · s ⁿ , eq. (7) - n parameter of the power-law model, eq. (7) - Nu 2r _p/ — Nusselt number, eqs. (28,31) - p pressure, Pa - Pe — Péclet number - P —(p/)/r _c — pressure gradient, Pa/m - dissipated energy, W, eq. (29) - total energy, W, eq. (30) - r radial coordinate, m - r _c radius of tube-curvature, m, fig. 1 - r _p radius of tube, m, fig. 1 - r _t variable, m, eq. (6) - R dimensionless radial coordinate, eq. (14a) - R _c dimensionlessr _c, eq. (14a) - R _t dimensionlessr _t, eq. (14a) - t temperature, °C - bulk temperature, °C, eq. (27) - t ₀ inlet temperature of the melt, °C - t _w tube wall temperature, °C - T dimensionless temperature, eq. (14c) - T _w dimensionless tube wall temperature - T dimensionless bulk temperature - u ₁ variable, s^–1, eq. (4) - u ₂ variable, s^–1, eq. (5) - U ₁ dimensionlessu ₁, eq. (18) - U ₂ dimensionlessu ₂, eq. (18) - v velocity in-direction, m/s - average velocity of the melt, m/s - V dimensionlessv, eq. (14b) - dimensionless , eq. (15) - z r _c — centre length of the tube, m - Z dimensionlessz, eq. (14e) - heat transfer coefficient, W/m² K - shear rate, s^–1, eq. (8) - — shear rate, s^–1 - apparent viscosity, Pa · s, eq. (7) - ₀ — apparent viscosity, Pa · s - angular coordinate, rad, fig. 1 - thermal conductivity, W/m K - melt density, kg/m³ - axial coordinate, rad, fig. 1 - rate of strain tensor, s^–1, eq. (8) - (—p) pressure drop, Pa     

The effect of velocity distribution on forced convection laminar flow heat transfer in a pipe at constant wall temperature

In this paper, a theoretical study of heat transfer to a fluid of vanishing viscosity in laminar flow in a pipe is made. The constant wall temperature boundary condition is considered in order to facilitate comparison with other classical solutions. Using velocity profiles of simple geometrical shape, the dependence of the heat transfer on velocity distribution is illustrated. Because of the nature of the idealised flow and heat transfer models, the theoretical results are applicable to all axisymmetric flows. Accordingly, some account of the possible effects of swirl on heat transfer in real flows is given.

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Zusammenfassung Es handelt sich um eine theoretische Untersuchung des Wärmeübergangs in laminarer Rohrströmung bei verschwindender Viskosität. Zum Vergleich mit anderen klassischen Lösungen wurde konstante Wandtemperatur als Randbedingung vorgegeben. Unter Benutzung von Geschwindigkeitsprofilen einfacher Geometrie wurde deren Einfluß auf den Wärmeübergang ermittelt. Diese Ergebnisse sind wegen der gewählten Strömungs- und Wärmeübergangsmodelle auf alle achsensymmetrischen Strömungen anwendbar. Die mögliche Wirkung einer Wirbelströmung auf den Wärmeübergang wird diskutiert.

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Nomenclature =(k/c) Thermal diffusivity - C, C ₁, C₂, C₃, C_n Constants - c Specific heat at constant pressure - D=(2r_w) Diameter - k Thermal conductivity - M _n Root of Bessel Equation,J ₀(M_n)=0 - r Radius - T Temperature - u, Velocity, average velocity - x Axial distance - X, R Function ofx, (r) alone - _n (= 2M _n/r _w ² ) Eigen value - Dynamic viscosity - (=/) Kinematic viscosity - Density - (=(T-T _w)/(T₁-T_w)) Dimensionless temperature - (=(T–T _w)/(T ₁–T _w)) Nusselt number - Pe (=Re·Pr) Péclet number - Pr (= c/k) Prandtl number - Re(=2r_w·v) Reynolds number Suffixes b Bulk - 1 Inlet - w wall     

The rheology of polymeric liquid crystals

The molecular theory of Doi has been used as a framework to characterize the rheological behavior of polymeric liquid crystals at the low deformation rates for which it was derived, and an appropriate extension for high deformation rates is presented. The essential physics behind the Doi formulation has, however, been retained in its entirety. The resulting four-parameter equation enables prediction of the shearing behavior at low and high deformation rates, of the stress in extensional flows, of the isotropic-anisotropic phase transition and of the molecular orientation. Extensional data over nearly three decades of elongation rate (10^–2–10¹) and shearing data over six decades of shear rate (10^–2–10⁴) have been correlated using this analysis. Experimental data are presented for both homogeneous and inhomogeneous shearing stress fields. For the latter, a 20-fold range of capillary tube diameters has been employed and no effects of system geometry or the inhomogeneity of the flow-field are observed. Such an independence of the rheological properties from these effects does not occur for low molecular weight liquid crystals and this is, perhaps, the first time this has been reported for polymeric lyotropic liquid crystals; the physical basis for this major difference is discussed briefly. A Semi-empirical constant in eq. (18), N/m² - c rod concentration, rods/m³ - c ^* critical rod concentration at which the isotropic phase becomes unstable, rods/m³ - C interaction potential in the Doi theory defined in eq. (3) - d rod diameter, m - D semi-empirical constant in eq. (19), s^–1 - D _r lumped rotational diffusivity defined in eq. (4), s^–1 - rotational diffusivity of rods in a concentrated (liquid crystalline) system, s^–1 - D _ro rotational diffusivity of a dilute solution of rods, s^–1 - f distribution function defining rod orientation - F tensorial term in the Doi theory defined in eq. (7) (or eq. (19)), s^–1 - G tensorial term in the Doi theory defined in eq. (8) - K _B Boltzmann constant, 1.38 × 10^–23 J/K-molecule - L rod length, m - S scalar order parameter - S tensor order parameter defined in eq. (5) - t time, s - T absolute temperature, K - u unit vector describing the orientation of an individual rod - rate of change ofu due to macroscopic flow, s^–1 - v fluid velocity vector, m/s - v velocity gradient tensor defined in eq. (9), s^–1 - V mean field (aligning) potential defined in eq. (2) - x coordinate direction, m - Kronecker delta (= 0 if = 1 if = ) - _r ratio of viscosity of suspension to that of the solvent at the same shear stress - _s solvent viscosity, Pa · s - ^* viscosity at the critical concentrationc ^*, Pa · s - v ₁, v₂ numerical factors in eqs. (3) and (4), respectively - deviatoric stress tensor, N/m² - volume fraction of rods - ⁰ constant in eq. (16) - ^* volume fraction of rods at the critical concentrationc ^* - average over the distribution functionf(u, t) (= d ²u f(u, t)) - gradient operator - d ²u integral over the surface of the sphere (|u| = 1)     

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a vertical heated plate

The effect of surface mass transfer on buoyancy induced flow in a variable porosity medium adjacent to a heated vertical plate is studied for high Rayleigh numbers. Similarity solutions are obtained within the frame work of boundary layer theory for a power law variation in surface temperature,T _Wx and surface injectionv _Wx(–1/2). The analysis incorporates the expression connecting porosity and permeability and also the expression connecting porosity and effective thermal diffusivity. The influence of thermal dispersion on the flow and heat transfer characteristics are also analysed in detail. The results of the present analysis document the fact that variable porosity enhances heat transfer rate and the magnitude of velocity near the wall. The governing equations are solved using an implicit finite difference scheme for both the Darcy flow model and Forchheimer flow model, the latter analysis being confined to an isothermal surface and an impermeable vertical plate. The influence of the intertial terms in the Forchheimer model is to decrease the heat transfer and flow rates and the influence of thermal dispersion is to increase the heat transfer rate.

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Der Effekt des Oberflächenstoffaustausches bei auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt

Zusammenfassung Es wird der Effekt des Oberflächenstoffaustausches in auftriebsinduzierter Strömung in einem variablen porösen Medium, das an eine vertikale, beheizte Platte angrenzt, für große Reynoldszahlen untersucht. Ähnliche Lösungen werden im Rahmen der Grenzschicht-Theorie, durch Variation des Potenzansatzes der Oberflächentemperatur,T _Wx , und der Oberflächengeschwindigkeit,v _Wx(–1/2), erreicht. Die Analyse vereinigt sowohl den Ausdruck, der Porösität und Permeabilität verbindet, als auch den Ausdruck, der Porösität und Wärmeleitfähigkeit miteinander verbindet. Der Einfluß der Temperaturverteilung auf Strömung und Wärmeübergangskennzahlen wird ebenfalls im Detail analysiert. Als Ergebnis der vorliegenden Untersuchung ergibt sich die Tatsache, daß variable Porösität Wärmeübertragungsrate und Betrag der Geschwindigkeit in Wandnähe steigert. Die bestimmenden Gleichungen, sowohl für das Darcysche Strömungsmodell als auch für das Forchheimersche Strömungsmodell, werden mit Hilfe eines implizierten Differenzenschemas gelöst. Die Berechnung wird für die beiden Fälle, isotherme Oberfläche und undurchlässige vertikale Platte, angewandt. Der Einfluß der Terme für die Trägheitskräfte im Forchheimerschen Modell senkt Wärmeübergangs- und Durchgangsrate, wogegen die Wärmeübergangsrate durch den Einfluß der Temperaturverteilung erhöht wird.

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Nomenclature a constant defined by Eq. (12) - A constant defined by Eq. (12) - B constant defined by Eq. (3) - b _s/_f ratio of thermal conductivities - C constant defined by Eq. (1) - C _P specific heat of the convective fluid - d particle diameter - f dimensionless function defined by Eq. (14) - f _w lateral mass flux parameter - g acceleration due to gravity - k ₀ mean permeability of the mediumk ₀= ₀ ³ d ²/150 (1– ₀)² k ₀=1.75d/(1– ₀) 150 (Inertia parameter) - L length of the source or sink - m mass transfer - n constant defined in Eq. (12) - k (y) permeability of the porous medium - k (y) interial coefficient in the Ergun expression - Gr modified Grashof numberGr=(g k ₀ k ₀ (T _w–))/ ² - R _a Rayleigh number (g k ₀ x T _w–)/ - R _ad modified Rayleigh number (g k ₀ d|T _w–|)/ - N _u Nusselt number - s x/d - Q overall heat transfer rate - T temperature - T _w surface temperature - T ambient fluid temperature - u velocity in vertical direction - v velocity in horizontal direction - x vertical coordinate - y horizontal coordinate Greek symbols ₀ mean thermal diffusivity _f/ C_p - coefficient of thermal expansion - constant defined in Eq. (4) - ratio of particle to bed diameter - _e effective thermal conductivity - f thermal conductivity of fluid - _s thermal conductivity of solid - dimensionless similarity variable in Eq. (13) - value of at the edge of the boundary layer - constant defined in Eq. (1) - _e effective molecular thermal diffusivity - (y) porosity of the medium - ₀ mean porosity of the medium - viscosity of the fluid - ₀ density of the convective fluid - stream function - w condition at the wall - condition at infinity     

Toward a viscoelastic modelling of the injection molding of polymers

Summary By using theLeonov viscoelastic constitutive equation, an idealized problem has been solved for onedimensional, unsteady, non-isothermal flow of polymer between two parallel plates and the subsequent non-isothermal relaxation following cessation of flow. Numerical results are presented for the time dependence of the pressure gradient, the gapwise distribution of linear velocity, shear rate, shear stress and normalstress differences, together with the components of birefringence in different planes. Comparison of the present predictions for the pressure gradient with results based upon an inelastic model indicate close agreement whereas the corresponding predictions for normal-stress differences are found to be markedly different from those for the inelastic case.The model is applied to the injection-molding process which is treated in terms of a filling and a cooling stage. Final results are given in terms of the distribution of residual stresses and associated birefringences in the molded part, as influenced by the rheological and thermal properties of the polymer and the processing conditions. The theoretical predictions are compared with birefringence measurements in the literature. Reasonable agreement is obtained for the position and value of maximum birefringence in the 1–2 plane although the birefringence predictions in the 1–3 and 2–3 planes are found to be markedly smaller than the measured values. The present theory indicates that, for a given polymer, the main factors affecting residual stresses and birefringence are melt temperature and flow rate, both of which should be held at the highest permissible levels.

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Zusammenfassung Das idealisierte Problem einer eindimensionalen, instationären, nicht-isothermen Strömung eines Polymeren zwischen zwei parallelen Platten sowie das der nichtisothermen Relaxation, die auf das Anhalten dieser Strömung folgt, wird mit Hilfe der viskoelastischen Stoffgleichung vonLeonov gelöst. Numerische Ergebnisse werden für die Zeitabhängigkeit der folgenden Größen gegeben: des Druckgradienten, der Verteilung der linearen Geschwindigkeit, der Schergeschwindigkeit, der Schubspannung, der Normalspannungsdifferenzen sowie der Komponenten der Doppelbrechung in verschiedenen Ebenen. Die hier vorliegenden Voraussagen sind bezüglich des Druckgradienten in guter Ubereinstimmung mit denen, die auf dem inelastischen Modell beruhen, unterscheiden sich von diesen aber wesentlich bezüglich der Normalspannungsdifferenzen.Das Modell wird auf den Spritzgußprozeß angewandt. Dieser wird als zweistufiger Prozeß, bestehend aus einer Abfüll- und einer Kühlstufe, behandelt. Numerische Ergebnisse werden für die Verteilung der Restspannungen und der assoziierten Doppelbrechung im Formteil gegeben, so wie sie durch die rheologischen und thermischen Eigenschaften des Polymeren und der Prozeßbedingungen beeinflußt werden. Die theoretischen Voraussagen für die Doppelbrechung werden mit Meßergebnissen aus der Literatur verglichen. Gute Übereinstimmung wird für die Lage und den Wert der maximalen Doppelbrechung in der 1–2 Ebene erzielt, während die Voraussagen für die Werte der Doppelbrechung in den 1–3 und 2–3 Ebenen wesentlich kleiner als die gemessenen Werte ausfallen. Die vorliegende Theorie zeigt an, daß für ein gegebenes Polymer die Schmelzentemperatur und die Einspritzgeschwindigkeit als Hauptfaktoren zu werten sind, die die Restspannungen und die Doppelbrechung beeinflussen. Diese sollen auf dem höchstzulässigen Stand gehalten werden.

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Nomenclature a _T temperature-shift factor - b thickness of channel or mold - C stress-optical coefficient - C _ij,k components of elastic strain tensor ink ^th relaxation mode - G storage modulus - G loss modulus - L length of channel or mold - n ₂₂ – n₃₃ birefringence in 2–3 plane - n ₁₁ – n₃₃ birefringence in 1–3 plane - n ₁₁ – n₃₃ gapwise-averaged birefringence in 1–3 plane - N number of modes retained in Leonov model, see eq. [9] - N ₁ first normal-stress difference, ₁₁ – ₂₂ - N ₂ second normal-stress difference, ₂₂ – ₃₃ - N ₃ normal-stress difference, ₁₁ – ₃₃ - p isotropic pressure - s dimensionless rheological parameter (0 <s < 1) - t time - t _fill filling time - T temperature - T _g glass-transition temperature - T ₀ inlet melt temperature - T _ref reference temperature, see eq. [21] - T _w mold temperature or channel wall temperature - u velocity inx direction - U average velocity inx direction - W width of channel or mold - x coordinate in flow direction - y coordinate in gapwise direction - z coordinate in width direction - thermal diffusivity of polymer melt - strain rate - , dimensionless strain rate - n birefringence in 1–2 plane - p pressure loss in channel or mold - apparent viscosity - _k viscosity ink ^th relaxation mode - ₀ zero-shear-rate viscosity - _k relaxation time ink ^th relaxation mode - = —p/x - µ shear modulus - µ _k shear modulus ink ^th relaxation mode - _ij components of deviatoric stress tensor - frequency With 14 figures and 2 tables     

Wärmeverhältnisse in Mehrstoff-Regenerativ-Wärmeaustauschsystemen

Zusammenfassung Das Referat bringt eine allgemeine Analyse der inneren Wärmeverhältnisse bei Regenerativ-Wärmeaustauschsystemen, die mit beliebig vielen wärmetragenden Medien arbeiten. Von dieser Analyse ausgehend, werden die Eigenschaften von Regenerativ-Wärmetauschern mit drei wärmetragenden Stoffen untersucht. Für die beiderseitige Gegenstromanordnung der aktiven Wärmetauschzonen wurden die Werte des Gesamtwärmeaustauschgrades tabellarisch festgehalten; zugleich wird deren Berechnungsverfahren gezeigt, das auch für andere technische Anwendungsbereiche geeignet ist.

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Heat transfer ratios in multifluid regenerative systems

The paper presents a general analysis of the internal thermal conditions in regenerative heat exchange systems using an arbitrary number of heat carrying media. This analysis serves as the base for a detailed investigation of the properties of three-fluid regenerative heat exchangers. The values of the overall thermal efficiency for the case of counterflow in both. active heat transfer zones are tabulated and the method of their calculation, suitable also for other cases, is presented.

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Bezeichnungen F (m²) Wärmeaustauschfläche - i Einheitsrichtungskoeffizient - k (Wm^–2K^–1) Wärmedurchgangszahl - kF/W dimensionslose Wärmeaustauschzahl - Q (W) Wärmemenge - q relativer Wärmeverlust - t (°C) Temperatur - W (WK^–1) Wärmekapazitätsstrom - (K) Eintrittstemperaturdifferenz - Korrekturfaktor - Wärmeaustauschgrad Indizes c Gesamtwert - G Gegenstrom - z Wärmeverlust - j, k, m, n 0,1,2 Hilfsindizes     

Natural convection in partitioned enclosed spaces of solar collector

The two-dimensional, steady, laminar natural convection phenomena in partitioned enclosure of solar collector has been studied numerically. Heat conduction along the partition is considered. An iterative finite-difference scheme is employed to solve the governing equations in the flow field. The effects of Rayleigh number, thermal conductivity ratio, partition angle, tilt angle, and aspect ratio on both the local and average heat transfer coefficients of the solar collector have been discussed. The range of Rayleigh number tested was up to 5 × 10⁴, the thermal conductivity ratio of 4.5 and 30, partition angle from 10 deg to 170 deg, tilt angle from 10 deg to 90 deg, and aspect ratio varied between 0.2 and 10. The results indicate that the convective heat transfer is strongly affected with the aspect ratio of the enclosures.

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Freie Konvektion in unterteilten Kammern von Solarkollektoren

Zusammenfassung Die zweidimensionale, stetige, laminare freie Konvektion in unterteilten Kammern von Solarkollektoren wurde numerisch untersucht. Die Wärmeübertragung entlang dieser Kammern wurde betrachtet. Ein iteratives Finite-Differenzen-Schema wurde angewandt um die Gleichungen, welche das Strömungsfeld beschreiben, zu lösen. Der Einfluß der Rayleigh-Zahl, der thermische Leitfähigkeit, des Kammerwinkels, des Neigungswinkels und der Längenverhältnisse auf die örtlichen und durchschnittlichen Wärmeübertragungskoeffizienten des Solarkollektors wurde diskutiert. Der Bereich der Rayleigh-Zahl erstreckte sich bis zu 5 × 10⁴, das Verhältnis der thermischen Leitfähigkeit betrug 4.5 und 30, der Kammerwinkel lag zwischen 10° und 170°, der Neigungswinkel zwischen 10° und 90° und das Längenverhältnis variierte zwischen 0.2 und 10. Die Ergebnisse beinhalten, daß die konvektive Wärmeübertragung sehr stark durch das Längenverhältnis der Kammern beeinflußt wird.

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Nomenclature a slope of the partition with respect to the horizontal - A H/L=cell aspect ratio - A _w t/L=wall aspect ratio - g acceleration due to gravity - h local heat transfer coefficient - average heat transfer coefficient - H cell length - k thermal conductivity of fluid within the cell - k _w thermal conductivity of the cell wall - L plate spacing - Nu _f h L/k=local cell Nusselt number - L/k=average cell Nusselt number - overall average Nusselt number - qL/k _w t(T _h–T _c)=average wall Nusselt number - Pr /=Prandtl number - q heat transfer in the cell wall from the hot to cold plate per unit depth - Ra g L ³ T/=Rayleigh number - R _k k _w/k=ratio of wall thermal conductivity to that of the fluid - t thickness of cell wall - T _c cold plate temperature - T _f temperature in cell - T _h hot plate temperature - T _w temperature in cell wall - u, U dimension and dimensionless velocities inx-direction - v, V dimension and dimensionless velocities iny-direction - x distance measured from the bottom of the cell (Fig. 1) - X x/L=normalized distance ofx - y distance measured from hot plate (Fig. 1) - Y y/L=normalized distance ofy - x ₁ distance measured in wall (Fig. 1) - X ₁ x/L=normalized distance ofx ₁ Greek symbols thermal diffusivity of fluid - coefficient of volumetric expansion of fluid - partition angle with respect to the hot plate - _f T _f–T _c/T _h–T _c=dimensionless temperature in cell - _w T _w–T _c/T _h–T _c=dimensionless temperature in cell wall - kinematic viscosity of fluid - enclosure tilt angle from horizontal - dimensional vorticity - L ²/=dimensionless vorticity - dimensionless streamline     

Landslide generated impulse waves. 2. Hydrodynamic impact craters

Landslide generated impulse waves were investigated in a two-dimensional physical laboratory model based on the generalized Froude similarity. Digital particle image velocimetry (PIV) was applied to the landslide impact and wave generation. Areas of interest up to 0.8 m by 0.8 m were investigated. PIV provided instantaneous velocity vector fields in a large area of interest and gave insight into the kinematics of the wave generation process. Differential estimates such as vorticity, divergence, and elongational and shear strain were extracted from the velocity vector fields. At high impact velocities flow separation occurred on the slide shoulder resulting in a hydrodynamic impact crater, whereas at low impact velocities no flow detachment was observed. The hydrodynamic impact craters may be distinguished into outward and backward collapsing impact craters. The maximum crater volume, which corresponds to the water displacement volume, exceeded the landslide volume by up to an order of magnitude. The water displacement caused by the landslide generated the first wave crest and the collapse of the air cavity followed by a run-up along the slide ramp issued the second wave crest. The extracted water displacement curves may replace the complex wave generation process in numerical models. The water displacement and displacement rate were described by multiple regressions of the following three dimensionless quantities: the slide Froude number, the relative slide volume, and the relative slide thickness. The slide Froude number was identified as the dominant parameter.List of symbols a wave amplitude (L) - b slide width (L) - c wave celerity (LT^–1) - d _g granulate grain diameter (L) - d _p seeding particle diameter (L) - F slide Froude number - g gravitational acceleration (LT^–2) - h stillwater depth (L) - H wave height (L) - l _s slide length (L) - L wave length (L) - M magnification - m _s slide mass (M) - n _por slide porosity - Q _d water displacement rate (L³) - Q _D maximum water displacement rate (L³) - Q _s maximum slide displacement rate - s slide thickness (L) - S relative slide thickness - t time after impact (T) - t _D time of maximum water displacement volume (L³) - t _qD time of maximum water displacement rate (L³) - t _si slide impact duration (T) - t _sd duration of subaqueous slide motion (T) - T wave period (T) - v velocity (LT^–1) - v _p particle velocity (LT^–1) - v _px streamwise horizontal component of particle velocity (LT^–1) - v _pz vertical component of particle velocity (LT^–1) - v _s slide centroid velocity at impact (LT^–1) - V dimensionless slide volume - V _d water displacement volume (L³) - V _D maximum water displacement volume (L³) - V _s slide volume (L³) - x streamwise coordinate (L) - z vertical coordinate (L) - slide impact angle (°) - bed friction angle (°) - x mean particle image x-displacement in interrogation window (L) - _x random displacement x error (L) - _tot total random velocity v error (LT^–1) - _xx streamwise horizontal elongational strain component (1/T) - _xz shear strain component (1/T) - _zx shear strain component (1/T) - _zz vertical elongational strain component (1/T) - water surface displacement (L) - density (ML^–3) - _g granulate density (ML^–3) - _p particle density (ML^–3) - _s mean slide density (ML^–3) - _w water density (ML^–3) - granulate internal friction angle (°) - _y vorticity vector component (out-of-plane) (1/T)     

An approximate theoretical analysis and experimental verification of turbulent entrance region flow of drag reducing fluids

Summary Entry lengths for pipe flows of moderately drag reducing fluids are determined using momentum integral technique. It is shown theoretically that the entry lengths for drag reducing fluids could be significantly larger than the Newtonian fluids flowing turbulently under otherwise identical conditions. The experimental data from the literature bear out the theoretical calculations.

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Zusammenfassung Mit Hilfe der Impuls-Methode wird die Einlauflänge in einer Rohrströmung für Flüssigkeiten mit mäßig starker Widerstandsverminderung berechnet. Es wird vorausgesagt, daß die Einlauflänge für derartige Flüssigkeiten erheblich größer sein kann als für newtonsche Flüssigkeiten unter sonst identischen Bedingungen. Aus der Literatur entnommene experimentelle Daten bestätigen diese theoretischen Berechnungen.

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Nomenclature A ₁ Coefficient in eq. [7] - A Slope of logarithmic velocity profile - a Exponent in eq. [10] - B Intercept function for logarithmic velocity profile - De Deborah number, - f Friction factor - F Function, eq. [30] - G Function given in eq. [11] - Static pressure, dynes/cm² - q Index of power law velocity profile - R Pipe radius, cm - r Radial distance, cm - R Core radius, cm - Re Reynolds number - Axial velocity, cm/s - u _c Core velocity, cm/s - u ⁺ Dimensionless velocity, eq. [5] - u ^* Friction velocity, , cm/s - Radial velocity, cm/s - V Average velocity, cm/s - x Axial distance, cm - x _e Entry length, cm - y Distance from the wall, cm - y ⁺ Dimensionless distance, eq. [5] - y _I ⁺ Dimensionless viscous sublayer thickness - coefficient in eq. [17] - exponent of Reynolds number in eq. [17] - shear rate, s^–1 - boundary layer thickness, cm - _fl fluid relaxation time, s - µ fluid viscosity, gm/cm s - v kinematic viscosity, cm²/s - _l laminar sublayer thickness, dimensionless - fluid density, gm/cm³ - shear stress, dynes/cm² - _w shear stress at the wall, dynes/cm² - ₁, ₂, ₃, ₄ functions in eq. [27] - ~ time averaged quantities - — dimensionless quantity With 3 figures and 1 table     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Dynamic crack resistance of acrylic plastic

The dependence of the stressintensity factor near the tip of a growing crack in an SO120 acrylic plastic on the crackpropagation velocity K_I(:v) within the range of 10^–5 –300 m/sec is determined. Specific features of crack propagation associated with the shape of the curve K_I(v), which has discontinuities and nonuniqueness intervals, are discussed.     

Diffusion of moisture in deformable porous media. Anisotropic fibrous materials

This paper presents a study on the deformation of anisotropic fibrous porous media subjected to moistening by water in the liquid phase. The deformation of the medium is studied by applying the concept of effective stress. Given the structure of the medium, the displacement of the solid matrix is not taken into account with respect to the displacement of the liquid phase. The transport equations are derived from the model proposed by Narasimhan. The transport coefficients and the relation between the variation in apparent density and effective stress are obtained by test measurements. A numerical model has been established and applied for studying drip moistening of mineral wool samples capable or incapable of deformation.Nomenclature D mass diffusion coefficient [L²t^–1] - e void fraction - g gravity acceleration [Lt^–2] - J mass transfer density [ML^–2t^–1] - K hydraulic conductivity [Lt^–1] - K _s hydraulic conductivity of the solid phase [Lt^–1] - K ^* hydraulic conductivity of the deformable porous medium [Lt^–1] - P pressure of moistening liquid [ML^–1 t^–2] - S degree of saturation - t time [t] - V speed [Lt^–1] - X horizontal coordinate [L] - Z vertical coordinate measured from the bottom of porous medium [L] - z z-coordinate [L] Greek Letters porosity - ₁ total hydric potential [L] - _g gas density [ML^–3] - ₁ liquid density [ML^–3] - ₀ apparent density [ML^–3] - _s density of the solid phase [ML^–3] - density of the moist porous medium [ML^–3] - external load [ML^–1t^–2] - effective stress [ML^–1t^–2] - bishop's parameter - matrix potential or capillary suction [L] Indices g gas - 1 moistening liquid - p direction perpendicular to fiber planes - s solid matrix - t direction parallel to fiber planes - v pore Exponent * movement of solid particles taken into account     

Binary laminar boundary layer on a vertical surface in the presence of combined free and forced convection

Heat and mass transfer at a vertical surface is examined in the case of combined free and forced convection. The boundary layer equations, transformed to ordinary differential equations, contain a parameter that determines the effect of free convection on the forced motion. Criteria are offered for differentiating the free-convection, forced-convection, and combined regimes.Notation x, y coordinates - u, v velocity components - g acceleration of gravity - T temperature - kinematic viscosity - coefficient of thermal expansion - a thermal diffusivity - ₁ partial vapor density - D diffusion coefficient - W₂ mass velocity of air - independent variable - _w shear stress at wall - thermal conductivity - r latent heat of phase transition - , dimensionless temperature and partial vapor density - m^* the complex (m ₁–m _1w )/(1–m(_1w ) - c_p specific heat at constant pressure - G Grashof number - R Reynolds number - P Prandtl number - S Schmidt number     

Transient temperature response of a porous matrix entering a planetary atmosphere

Summary An analysis is made of the transient temperature behavior of a transpiration-cooled porous matrix entering a planetary atmosphere with constant velocity and negative entry angle. The analysis is based on one dimensional heat conduction in a porous plate subjected to a time dependent heat flux at one side and cooled internally by mass injection from a constant temperature reservoir at the opposite side. An exact closed-form solution is obtained and temperature charts are presented for a wide range of Fourier number and coolant flow parameters.Nomenclature A surface area, ft² - C constant, 17,600 Btu/ft^3/2-sec - C _c constant pressure specific heat of coolant, Btu/lb_m-°F - g local gravitational acceleration, ft/sec² - g _c coolant flow parameter, defined by equation (15) - h height of entry above planet surface, ft - K ₁ ratio of local heat flux to stagnation point heat flux - K thermal conductivity of plate material, Btu/sec-ft-°F - L plate thickness, ft - m constant, 3.15 - m _c coolant mass flow rate, lb_m/sec - M _n roots of equation (33) - n constant, 0.50 - N defined by equation (30) - P porosity - q surface heat flux, Btu/ft²-sec - q ₀ surface heat flux at t=0, Btu/ft²-sec, defined by (6) - r distance from planet center, ft - R radius of curvature at stagnation point, ft - t time, sec - T temperature, °F - T _c coolant supply temperature, °F - V velocity, ft/sec - x normal coordinate through plate, ft - y altitude, ft - thermal diffusivity of plate, ft²/sec - atmospheric density decay parameter, 1/23500 ft^–1 - flight path angle relative to local horizontal direction, positive for climbing and negative for descent, deg - dimensionless temperature parameter, defined by (12) - dimensionless distance, defined by (13) - free stream atmospheric density, slug/ft³ - ₀ atmospheric density at reference state, slug/ft³ - Fourier number, defined by (14) - 1/sec, defined by (7) - flight entry parameter, defined by (16)     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence