Finite element solution of low peclet number fluid flow in a round pipe with the cauchy boundary condition

This paper presents a finite element solution to the problem of low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed to be laminar and fully developed throughout the pipe and the fluid temperature is kept uniform atX = —∞. The pipe wall is adiabatic at X ≤ 0 and cooled convectively at X ≥ 0. The solutions include temperature distributions and Nusselt numbers for the parameters, Bi = 0.04, 0.4, 4, 20 and Pe = 1, 3, 5, 10, 20, 30, which are in excellent agreement with the existing analytic solution except in the region near the singular point Δ A temperature discrepancy in the analytic solution at this point is physically impossible. The finite element method overcomes this mathematical difficulty and shows a greater value in the Nusselt number due to a higher wall temperature at X ≥ 0.     

Low peclet number heat transfer in the thermal entrance region of parallel-plate channels with unequal wall temperatures

The problem of low-Peclet-number thermal entry heat transfer for plane Poiseuille flow in parallel-plate channels with uniform but unequal wall temperatures is approached by the eigenfunction expansion method utilizing the Gram-Schmidt orthonormalization procedure. The formulation considers axial heat conduction and allows upstream heat penetration through the thermal entrance. Numerical results are obtained for the case with entrance condition parameter θ₀ = 1 and Peclet number Pe = 1, 5, 10 and 50. The effect of Peclet number on temperature distributions in both upstream and downstream regions is studied. At Pe = 50, the concept of thermal boundary layer is applicable and the present series solution does not yield physically reasonable temperature distribution locally near the upper plate at the thermal entrance. The difficulty may be attributed to the nature of thermal boundary conditions at the thermal entrance and the transition from elliptic problem to parabolic problem with the increase of Peclet number.     

Low peclet number heat transfer in the thermal entrance region of parallel-plate channels with unequal wall temperatures

The problem of low-Peclet-number thermal entry heat transfer for plane Poiseuille flow in parallel-plate channels with uniform but unequal wall temperatures is approached by the eigenfunction expansion method utilizing the Gram-Schmidt orthonormalization procedure. The formulation considers axial heat conduction and allows upstream heat penetration through the thermal entrance. Numerical results are obtained for the case with entrance condition parameter θ₀ = 1 and Peclet number Pe = 1, 5, 10 and 50. The effect of Peclet number on temperature distributions in both upstream and downstream regions is studied. At Pe = 50, the concept of thermal boundary layer is applicable and the present series solution does not yield physically reasonable temperature distribution locally near the upper plate at the thermal entrance. The difficulty may be attributed to the nature of thermal boundary conditions at the thermal entrance and the transition from elliptic problem to parabolic problem with the increase of Peclet number.     

Liquid solidification in low peclet number pipe flows

The combined effects of axial conduction and solidification on heat transfer and pressure drop in pipe flows are investigated by the use of a modified Galerkin finite element method. To allow for the upstream heat conduction, the domain of study is extended from X = -∞ to X = -∞ as has been done in previous analyses. As a preliminary study on the effect of axial conduction, the present investigation assumes a superheat ratio that is sufficiently large (T_o > T_f or T_w ≈ T_f) such that solidification begins at a location near X = 0. For numerical convenience, the infinite domain -∞ ≤ X ≤ ∞ is transformed onto a finite domain -1 ≤ z ≤ 1. The energy equation for the liquid-phase is then solved by a modified Galerkin finite element method. For better numerical stability, a procedure is proposed for controlling the numerical error that might propagate from the singular point (X, R) = (O, R_o). The profile of the solid-liquid interface, the heat transfer rate and the pressure drop are presented for various values of Peclet number, Pe = 1, 3, 5, 10 and 30, and for the modified superheat ratio, c = 0.1, 0.5, 5.0, and ∞.     

RTD studies in a falling film graphite evaporator with internal spiral fins: Influence of the geometry

A hydrodynamic study of a new graphite evaporator has been performed. In this exchanger the fluid flows as a thin film on a spirally wound fin which forms a spiral staircase inside a tube. The slope θ_m of the spirally wound fin is modifiable. The residence time distribution (RTD) of the film was measured for different fin slopes θ_m and several film flowrates m_f. The flow pattern was then modelled thanks to the DTS software as a cascade of perfect mixed cells and the mean residence time τ_m was calculated. The evolution of τ_m as a function of θ_m and m_f is accurately represented by a power law relation.     

Combined effects of temperature and Reynolds number on drag‐reducing characteristics of a cationic surfactant solution

The drag‐reducing characteristics in the turbulent channel flow of dilute cationic surfactant solution, cetyltrimethyl ammonium chloride (CTAC)/sodium salicylate (NaSal) aqueous solution, were experimentally investigated in a closed loop fluid flow facility at different temperatures. The mass concentrations of the surfactant solution ranged from 75 to 200 ppm, and the temperatures ranged from 15 to 55°C. The cationic surfactant solution showed a great drag‐reducing ability, which was greatly affected by concentration, temperature, and Reynolds number. It was found that there existed a critical temperature T_c in each solution at different concentrations. Above T_c, drag‐reduction level decreases and reaches the behaviour of water flow without drag‐reducing ability. A new temperature parameters T_f, was proposed, and the difference between T_c and T_f can represent the effective temperature range for the drag reduction at a certain Reynolds number. The variation tendency of T_f and T_c with Reynolds numbers can give the guidance of selecting effective drag reduction range to the practical application in the district heating systems (DHS). It was supposed that temperature and shear stress are two kind of energy applied on the surfactant microstructure, which can be helpful to the surfactant network formation or dissociation depending on their values. © 2011 Canadian Society for Chemical Engineering     

Fluorous Surface‐Active Distannoxane Catalysts

Fluorous distannoxanes (XR^f₂SnOSnR^f₂X)₂⋅n H₂O (R_f=C₆F₁₃C₂H₄) ( 1 : X=C₈F₁₇SO₃, n=10; 4 : X=Cl, n=0) ( 1 ) catalyze the Mukaiyama aldol reaction and the allylation of aldehydes with tetraallyltin at room temperature in fluorous/organic biphasic solvent systems, in which the reactions proceed more rapidly than in a single organic or fluorous solvent. Due to the unique surface activity of 1 , the catalyst, organic substrate(s), and reagent(s) are distributed in both organic and fluorous phases to facilitate smooth reactions. Upon dilution with toluene after the reaction, the catalyst concentrates to the fluorous phase, while the organic substances migrate to the organic phase to effect facile catalyst recovery and recycling. By virtue of such a unique solvophilicity, a new version of fluorous biphase technology has been developed.     

倾斜管内热进口段自然强制复合对流传热

用数值分析方法研究了倾斜管内热进口段自然强制同向复合对流传热 ,分析了浮力、倾斜角及轴向导热对流动和传热特性的影响 ,得到了速度、温度、壁面剪切应力、局部及平均Nusselt数的分布和变化 研究结果显示Pe=71时平均Nusselt数的最大值位于 2 5°～ 45°之间 ,Pe=2 5时则在 90°处出现 .     

Separation of the System NaCl-NaBr-Nal by “Solventing Out” from Aqueous Solution

Abstract

The precipitation of NaX (X = Cl, Br, I) from their saturated aqueous solution by three miscible organic solvents (MOS)-acetone, acetonitrile and isopropylamine-was studied for various MOS-aqueous solution volumes ratios. It was found that the fraction precipitated f is given by f = K log V/V_c where V is the volume ratio and K and V _c are constants. For the case of NaCl this plot was found to be a broken line composed of two straight lines intersecting at the point of “salting out.” A scheme is suggested to separate the three electrolytes using isopropylamine to precipitate NaCl and acetone to precipitate NaBr.     

Free Convection Flow from a Vertical Isothermal Cone with Temperature-Dependent Viscosity

This paper reports a study of the effect of variable viscosity, expressed as an inverse linear function of temperature, on the free convective heat and fluid flow past a vertical isothermal cone. By similarity analysis together with the boundary layer assumptions, this complex problem is reduced to a system of coupled nonlinear ordinary differential equations. The resultant boundary layer equations are integrated numerically to obtain the flow field and the temperature distribution for selected influence parameters such as the viscosity parameter θ _r and the Prandtl number Pr. The viscosity parameter θ _r plays a prominent role in both flow field and heat transfer. In this paper, θ _r relates to the reciprocal of the temperature difference between the heated surface and the ambient. The calculated results indicate that some variations will be introduced in the heat transfer rate as well as the skin friction coefficient.     

Transversal thermal patterns in packed‐bed reactors with simple kinetics: Bifurcation criterion and simulations

We derive a new criterion for transversal instability of planar fronts based on the bifurcation condition dV_f/dK|_K=0 = 0, where V_f and K are the front velocity and its curvature, respectively. This refines our previously obtained condition, which was formulated as α = (ΔT_adPe_T)/(ΔT_mPe_C) > 1 to α > 1 + |δ|, where ΔT_ad and ΔT_m are the adiabatic and maximal temperature rise, respectively, Pe_C and Pe_T are the axial mass and the heat Pe numbers, respectively, and δ is a small parameter. The criterion is based on approximate relations for ΔT_m and V_f, which account for the local curvature of a propagating front in a packed bed reactor with a first‐order activated kinetics. The obtained relations are verified by linear stability analysis of planar fronts. Simulations of a simplified 2D model in the form of a thin cylindrical shell are in good agreement with the critical parameters predicted by dispersion relations. Three types of patterns were detected in simulations: “frozen” multiwave patterns, spinning waves, and complex rotating–oscillating patterns. We map bifurcation diagrams showing domains of different modes using the shell radius as the bifurcation parameter. The possible translation of the 2D cylindrical shell model results to the 3D case is discussed. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

A numerical study of particle wall-deposition in a turbulent square duct flow

The deposition of dense solid particles in a downward, fully developed turbulent square duct flow at Re_τ = 360, based on the mean friction velocity and the duct width, is studied using large eddy simulations of the fluid flow. The fluid and the particulate phases are treated using Eulerian and Lagrangian approaches, respectively. A finite-volume based, second-order accurate fractional step scheme is used to integrate the incompressible form of the unsteady, three-dimensional, filtered Navier-Stokes equations on an 80 × 80 × 128 grid. A dynamic subgrid kinetic energy model is used to account for the unresolved scales. The Lagrangian particle equation of motion includes the drag, lift, and gravity forces and is integrated using the fourth-order accurate Runge-Kutta scheme. Two values of particle to fluid density ratio (ρ_p/ρ_f = 1000 and 8900) and five values of dimensionless particle diameter (d_p/δ × 10⁶ = 100, 250, 500, 1000 and 2000, δ is the duct width) are studied. Two particle number densities, consisting of 10⁵ and 1.5 × 10⁶ particles initially in the domain, are examined.Variations in the probability distribution function (PDF) of the particle deposition location with dimensionless particle response time, i.e. Stokes number, are presented. The deposition is seen to occur with greater probability near the center of the duct walls, than at the corners. The average streamwise and wall-normal deposition velocities of the particles increase with Stokes number, with their maxima occurring near the center of the duct wall. The computed deposition rates are compared to previously reported results for a circular pipe flow. It is observed that the deposition rates in a square duct are greater than those in a pipe flow, especially for the low Stokes number particles. Also, wall-deposition of the low Stokes number particles increases significantly by including the subgrid velocity fluctuations in computing the fluid forces on the particles. Two-way coupling and, to a greater extent, four-way coupling are seen to increase the deposition rates.     

Contribution of interferometry and mechanical parameters in evaluating molecular orientation for drawn polyester

Mechanical and structural parameters related to the optical properties of polyester (PET) (woollen type) fibres drawn at room temperature have been investigated. The changes in the strain were evaluated to obtain the molecular orientation factors 〈P₂(cos θ)〉 and 〈P₄(cos θ)〉. From the optical orientation, the values of f₂(θ), f₄(θ) and f₆(θ) orientation parameters were calculated. The structure and properties of oriented PET have been studied in the light of the rubber elasticity theory. The dielectric constant, magnetic susceptibility, number N′ of chains between crosslinks per unit volume, optical configuration parameter and the segment anisotropy, were among the calculated parameters. The results of the extension were used to calculate the shrinkage factor. Relationships between the calculated parameters and the draw ratios, together with micro-interferograms, are given for illustration. The present study throws light on how the applied stress changes the molecular orientation factors and the structural parameters. © 1999 Society of Chemical Industry     

Study on the thermal degradation of epoxidized natural rubber

The thermal degradation and thermooxidative degradation of epoxidized natural rubber (ENR) were studied by thermogravimetry (TG). In the thermal degradation of ENR, the initial temperature of weight loss T₀ = 1.20B + 348, the temperature of maximum weight loss rate T_p = 1.07B + 392, and the final temperature of weight loss T_f = 0.77B + 445. The C_p, which corresponds to the degradation rate at temperature T_p, increases along with the heating rate B and its mean value is 43%, but C_f, which corresponds to the degradation rate at temperature T_f, is not affected by the heating rate, and its average value is close to 100%. As in the thermooxidative degradation, T₀ = 1.84B + 246, T_p = 0.30B + 378, and T_f = 2.27B + 584. The value of C_p increases along with the heating rate B and its mean value is 36%, but C_f is not affected by the heating rate and the average value approximately equals 100%. The thermal degradation in nitrogen could be a one-step reaction, whereas the thermooxidative degradation has a multiple-step reaction. The reactive environment has a great effect on the thermal degradation of ENR and the difference of the mechanisms of the two reaction systems is obvious. © 1998 John Wiley & Sons, Inc. J Appl Polym Sci 67:2207–2211, 1998     

Application of the molecular dynamics method and the excited state model to the investigation of the glass transition in argon

The glass transition in argon at a high cooling rate is simulated. At a temperature of 50 K (considerably below the melting temperature T _f = 83.8 K), the fluctuation volume fraction reaches the constant value f _g ? 0.03–0.05, which is close in the order of magnitude to the criterion for the glass transition in liquids f _g = const 0.02–0.03 within the excited state model. At this temperature, the second maximum of the radial distribution function is split as a result of the glass transition at the temperature T _g = 50 K. The approximate empirical “two-thirds” rule T _g = (2/3)T _f is reasonably satisfied for argon. The data obtained are interpreted in the framework of the excited state model.     

Effect of Ordering on the Microwave Dielectric Properties of Spinel‐Structured (Zn1−x(Li2/3Ti1/3)x)2TiO4 Ceramics

Spinel‐structured (Zn_1?x(Li_2/3Ti_1/3)_x)₂TiO₄ (x = 0–1) microwave dielectric ceramics were manufactured via a conventional mixed‐oxide method. The X‐ray diffraction and Raman spectra revealed that a disordered face‐centered cubic phase was found in the composition range of x < 0.5, and an ordered primitive cubic spinel solid solution was achieved as x was beyond 0.5. Such a disorder–order transition near x = 0.5 was accompanied by the variation of composition‐induced cation occupancy. The Q × f value first kept increasing up to ~160 000 (GHz) in disordered ceramics, and then sharply decreased as an ordered structure appeared at x ≥ 0.5. An obvious decrease in τ_f value was also accompanied by the appearance of an ordered structure. The minimum τ_f value (~ ?20 ppm/°C) was obtained in the x = 0.75 sample with the highest structural order degree. These results demonstrated that microwave dielectric properties of current spinel ceramics could be successfully modified by adjusting their structural order degree, which could be appropriately adopted for the design of spinel‐structured materials with favorable properties.     

Rheological interpretation of heat generation associated with fatigue of polycarbonate

Temperature change was measured of polycarbonate under monotonically increasing tensile and pulsating tensile loads. In the former case, the specimen temperature began to rise when an appreciable amount of viscoelastic strain was noticed on the stress—strain diagram. The rise, θ_v, could be formulated as a function of the viscoelastic strain, εv: In fatigue tests, the average temperature began to rise immediately after the decrease due to the thermoelastic effect. The amount of the heat generation, σ, was nearly constant for each cycle throughout the fatigue process and has a relation to the fatigue life, N_f, (α? a)·N_f = constant, where a is another adjustable constant. From a rheological aspect of dissipation energy, the equation is transformed to a relation between the viscoelastic strain and the fatigue life as εV^1/2 · N_f = constant, which is similar to the one for metals given by Manson and Coffin.⁶ The temperature rise in the fatigue was also related to the viscoelastic strain. The relation is of the same form as for static tension but less by a factor of one order.