Solidification of a low peclet number fluid flow in a round pipe with the boundary condition of the third kind

This paper presents a theoretical investigation on the solidification of a low Peclet number fluid flow in the thermal entrance region of a round pipe. The velocity is assumed laminar and fully developed throughout the pipe and the fluid temperature is taken to be uniform at X = ?∞. The pipe wall is adiabatic at X ≤ 0 and is cooled convectively at X ≥ 0. The exact solution in the liquid and solid phase are obtained by using the method of separation of variables and constructing two sets of orthonormal functions from the nonorthogonal eigenfunctious at X = ± 0. The solutions including temperature distributions, liquid-solid interfaces and Nusselt numbers for the parameters, Bi = 20, 4, 0.4, 0.04, Pe = 1, 3, 5, 10, 20, 30, ∞ and the superheat ratios, ξ = (1 ? θ_f)/θ_f, θ_f= 0.1, 0.2, …, 0.9 are presented in this paper. The length of ice-free zone x_f corresponding to Pe = ∞ are in excellent agreement with the existing solution.     

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 ∞.     

边界条件对圆管层流流动的影响分析

应用Fluent软件,分别就匀速度、抛物线形速度分布入口边界条件下模拟圆管层流流场,得到进出口截面的静压降,并与理论值进行比较。计算结果显示,匀速度入口条件下,进出口静压降的百分误差随雷诺数增加而急剧增大;而抛物线形速度入口条件下,进出口静压降与理论值基本吻合。     

Finite element analysis of flow of thermoplastic elastomer melt through axisymmetric die with slip boundary condition

Abstract

A finite element model for the flow of thermoplastic elastomers in extrusion dies has been developed. The rheological behaviour of the polymer melt is assumed to be described by the generalised Newtonian models and as a special case, the well known, power law equation was selected. Owing to the very low variation of the temperature field, the flow regime was considered to be isothermal. The set of governing equations are solved using the finite element method in a cylindrical (r, z) coordinate system. Slip–stick of the polymer melt on the solid wall, encountered in the flow of highly viscous fluids, is incorporated into the model by the use of Navier's slip condition. A new method based on a technique developed previously is described for the inclusion of this condition in the working equations. The applicability of the model was verified by a comparison between the results of the simulation of a polypropylene–nitrile/butadiene rubber thermoplastic elastomer with experimentally measured data. These comparisons show that there are very good agreements between the model predictions and actual data, provided that the slip of the polymer melt during the flow in extrusion die has been taken into consideration.     

Stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition

The problem of a steady laminar two‐dimensional stagnation point flow towards a stretching/shrinking sheet in a micropolar fluid with a convective surface boundary condition is studied. The governing partial differential equations are transformed into ordinary differential equations using a similarity transformation, before being solved numerically using the Runge–Kutta–Fehlberg method with shooting technique. The effects of the material parameter and the convective parameter on the fluid flow and heat transfer characteristics are disscussed. It is found that the skin friction coefficient and the heat transfer rate at the surface decrease with increasing values of the material parameter. Moreover, dual solutions are found to exist for the shrinking case, while for the stretching case, the solution is unique. © 2011 Canadian Society for Chemical Engineering     

Finite element collocation solution of tubular and packed-bed reactor two point boundary value problems

Nonlinear two point boundary value problems (TPBVP's) are frequently encountered in chemical engineering. TPBVP's describing tubular and packed-bed reactors are important and have received much attention in the literature. For certain ranges of parameter values of the reactors, the TPBVP's become stiff and pose problems in numerical solution. These problems have been efficiently solved by the finite element collocation method using Hermite and B-spline functions in conjunction with quasilinearization. Numerical results have been presented and discussed. The problems have been solved for a wide range of parameter values without encountering any numerical difficulty.     

Finite element analysis of mass transport through a viscous fluid with reaction

A finite element analysis is developed for transport of a chemical species through a viscous fluid with reaction at a solid boundary. In particular, we examine the relative effect of diffusion, convection and reaction on the quantity of mass transported through the fluid and reacting at the boundary. Numerical comparison studies with a global collocation approximation for a convection-dominated transport problem are given and mesh refinement studies performed to assess the effect of an entry singularity. A quantitative measure of the effectiveness of the reaction is derived using the asymptotic downstream solution. Finally, we consider the influence of a non-uniform fluid-solid interface on the transport process, examining the enhancement or depression of the reaction rate at corners.     

Finite element solution for gas–solid reactions: Application to the moving boundary problems

Gas–solid reactions are very important in the chemical and metallurgical process industries. Several models described these reactions such as volume reaction model, grain model, and nucleation model. These models give two coupled partial differential equations (CPDEs). In this work an integral transformation and subsequent finite element method is used for solving the coupled partial differential equations of these reactions. In each mesh the Rayleigh–Ritz method is applied. Finally the results of this work are compared with the existing numerical solutions and experimental data successfully.     

Mass transfer downstream of nozzles in turbulent pipe flow with varying Schmidt number

Local mass transfer rates at the wall of a pipe downstream of constricting nozzles have been measured using the electrochemical limiting diffusion current technique for different electrolyte Schmidt numbers. The familiar peaked axial distribution of mass transfer downstream of the nozzle was verified and the peak mass transfer values were found to agree well with the data of Tagget al. [1]. An overall correlation of the data in terms of both Reynolds number and nozzle expansion ratio produced the equation $$({{Sh_{2P} } \mathord{\left/ {\vphantom {{Sh_{2P} } {Sh_{2FD} }}} \right. \kern-\nulldelimiterspace} {Sh_{2FD} }})({{D_1 } \mathord{\left/ {\vphantom {{D_1 } {D_2 }}} \right. \kern-\nulldelimiterspace} {D_2 }})^{ - 0.7} = 14.39Re_2^{ - 0.182} $$ Limiting current-time traces produced evidence of the highly turbulent flow in the recirculation zone near the position of peak mass transfer.     

A low reynolds number reynolds stress modeling of turbulent pipe flow: Flow pattern and energy balance

The present paper addresses a comparative analysis of two different versions of low Reynolds number Reynolds stress turbulence models. The predictive capability of the models has been tested on the basis of flow patterns and energy balance. Numerical simulations were performed at the Reynolds numbers of 7400 and 22,000 respectively. The predicted mean axial velocity, three components of the r/m/s fluctuating velocities and the turbulent kinetic energy have been compared with the experimental data of Durst et al. (1995) Schildknecht et al. (1979). The predicted turbulent energy dissipation rate was validated with the experimental data of Schildknecht et al. (1979) and the direct numerical simulations of Eggles et al. (1994). The energy balance calculations were performed for the Reynolds numbers of 7400, 22,000 and 50,000. A comparison for the models has been presented.     

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.     

A boundary element analysis of flow in sheet molding compound

A new boundary element method has been developed for analyzing the flow of sheet molding compound (SMC) during compression molding. The boundary element equations can be used to determine the velocities on the perimeter of the charge. Successive flow front configurations are then generated by a simple explicit updating procedure. This approach was used to predict the flow front progression for elliptical, rectangular, and L-shaped charges. Comparisons with experimental data for elliptical and rectangular charges were encouraging. The fact that it was possible to obtain reasonable agreement for charges with different shapes and thicknesses lends support to the underlying flow model. Furthermore, valuable insight regarding knit line formation was acquired by analyzing the L-shaped charge. Results from the boundary element analysis showed that the initial thickness of the charge has a pronounced effect on knit line development. Even though there is considerable industrial experience in making SMC parts, the important role of charge thickness on knit line formation appears to have been largely overlooked. Prior analyses gave no indication of this effect because they were based on lubrication models that were independent of charge thickness.     

Finite element modelling of concentration profiles in flow domains with curved porous boundaries

Domains containing porous boundaries are a feature of important processes such as crossflow filtration commonly used in a variety of industries ranging from medical to oil and gas to water treatment. Prediction of concentration profiles of solids being carried by the fluid stream in these types of flow domains is a necessary step in the design of efficient systems used in such operations. The novel approach of the present paper is shown to provide a very convenient technique for simulating concentration profiles within geometrically complex domains. The main feature of this approach is the construction of a very simple technique for the handling of concentration boundary conditions along the porous walls. Therefore this model provides a necessary and useful step towards the development of a fast engineering technique for the quantitative analysis of crossflow filters. The developed algorithm is based on the Streamline Upwind Petrov Galerkin finite element scheme (SUPG) for the solution of flow and convective dispersion equations in two dimensional domains with solid and porous walls. Using iterative algorithms the model can take into account process dependent changes of physical parameters that inevitably occur as a portion of the fluid seeps through the porous wall in these domains. The numerical tests prove the stability of the developed scheme and show that the model can produce results that are consistent with theoretical considerations.     

起伏振动状态下倾斜管内两相流多尺度熵分析

将振动装置和气液两相流实验回路相结合,在起伏振动状态下对倾斜管内气液两相流进行实验研究,观察记录流型并使用多尺度熵方法对压差波动信号进行分析。研究结果表明振动条件下倾斜管内出现两种新流型为珠状流、起伏弹状流。同时对95种流动条件下的5种典型流型的差压波动信号进行多尺度熵分析。结果表明,不同流型在低尺度（前9个）下的熵值变化率有明显不同,高尺度下的熵值可以反映不同流型的动力学特性。