Verlauf der Fluidtemperaturen im Querstromrohrbündel

A new analytical method is presented for the determination of temperature distribution and effectiveness of heat transfer in different cross-flow arrangements. As an improvement over the known analytical solutions simplified energy balance equations are used. This simplification consists only in their notation but not in representation of the exact energy balance. The demanded accuracy of the thermal analysis is achieved by proper selection of a mean value of fluid temperature outside the tube. For each tubei in the bundle a mean value ? _i of the dimensionless fluid temperature outside the tube is introduced according to $$\vartheta _i = \omega \vartheta _{i + {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + (1 - \omega )\vartheta _{i - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$ where ? _i?1/2 and ? _i+1/2 are the local temperatures in front and behind thei-th tube, respectively. With symbol ω a weight coefficient is denoted $$\omega = {1 \mathord{\left/ {\vphantom {1 {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}}} \right. \kern-\nulldelimiterspace} {(1 - e^{ - NTU_{{ \bot \mathord{\left/ {\vphantom { \bot n}} \right. \kern-\nulldelimiterspace} n}} } )}} - {n \mathord{\left/ {\vphantom {n {NTU_ \bot }}} \right. \kern-\nulldelimiterspace} {NTU_ \bot }}$$ The number of tube rows in the bundle isn and the number of transfer units of the outside stream is NTU_⊥. Through the introduction of the weight coefficient ω, the mathematical operations related to calculation of the temperature field are radically simplified. This enabled the development of the procedure, valid for three codirected cross-flow arrangements (Fig. 1) and any numbern of the rows.     

Pressure drop in alternating curved tubes

Pressure drop measurements in the laminar and turbulent regions for water flowing through an alternating curved circular tube (x=h sin 2πz/λ) are presented. Using the minimum radius of curvature of this curved tube in place of that of the toroidally curved one in calculating the Dean number (ND=Re(D/2R _c )², it is found that the resulting Dean number can help in characterizing this flow. Also, the ratio between the height and length of the tube waves which represents the degree of waveness affects significantly the pressure drop and the transition Dean number. The following correlations have been found:

1. For laminar flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.03,\operatorname{Re}< 2000.$$
2. For turbulent flow: $$F_w \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} = F_s \left( {\frac{{2R_c }}{D}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} + 0.005,2000< \operatorname{Re}< 15000.$$
3. The transition Dean number: $$ND_{crit} = 5.012 \times 10^3 \left( {\frac{D}{{2R}}} \right)^{2.1} ,0.0111< {D \mathord{\left/ {\vphantom {D {2R_c }}} \right. \kern-\nulldelimiterspace} {2R_c }}< 0.71.$$

     

A－HIGH－ORDER ACCURACY EXPLICIT DIFFERENCE SCHEME FOR SOLVING THE EQUATION OF TWO－DIMENSIONAL PARABOLIC TYPE

Ａ－ＨＩＧＨ－ＯＲＤＥＲＡＣＣＵＲＡＣＹＥＸＰＬＩＣＩＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＳＯＬＶＩＮＧＴＨＥＥＱＵＡＴＩＯＮＯＦＴＷＯ－ＤＩＭＥＮＳＩＯＮＡＬＰＡＲＡＢＯＬＩＣＴＹＰＥＭａＭｉｎｇｓｈｕ（马明书）（ＲｅｃｅｉｖｅｄＪｕｎｅ２,１...     

Remnant functions

Remnant functions are defined, with \(\kappa = \sigma + \tau + \tfrac{1}{2}\) , by $$R_{\sigma \tau } (z) = [{{\Gamma (\sigma - [\kappa ])} \mathord{\left/ {\vphantom {{\Gamma (\sigma - [\kappa ])} {\Gamma (\sigma )}}} \right. \kern-\nulldelimiterspace} {\Gamma (\sigma )}}]\sum\limits_{r = 1}^\infty {r^{2\tau } \left[\kern-0.15em\left[ {(r^2 + z)^{\sigma - 1} } \right]\kern-0.15em\right]_\kappa }$$ where \(\left[\kern-0.15em\left[ \right]\kern-0.15em\right]_\kappa\) denotes subtraction of sufficiently many terms of the Taylor series in powers of z to yield a convergent sum; for integral σ a factor \([1 + ({z \mathord{\left/ {\vphantom {z {r^2 }}} \right. \kern-0em} {r^2 }})]\) may also enter. These functions arise in various contexts, in particular, in the calculation of uniform remainder terms for the approximation by integrals of sums with singular summands. Differential recurrence relations, Taylor expansions, and various integral representations are obtained. The full asymptotic expansions for ¦z¦→∞ with ¦arg z¦ <π are derived, and it is shown that for integral τ these converge exponentially fast.     

A technique for evaluation of three-dimensional behavior in turbulent boundary layers using computer augmented hydrogen bubble-wire flow visualization

A system is described which allows the recreation of the three-dimensional motion and deformation of a single hydrogen bubble time-line in time and space. By digitally interfacing dualview video sequences of a bubble time-line with a computer-aided display system, the Lagrangian motion of the bubble-line can be displayed in any viewing perspective desired. The u and v velocity history of the bubble-line can be rapidly established and displayed for any spanwise location on the recreated pattern. The application of the system to the study of turbulent boundary layer structure in the near-wall region is demonstrated.List of Symbols Reynolds number based on momentum thickness u /v - t⁺ nondimensional time - u shear velocity - u local streamwise velocity, x-direction - u ⁺ nondimensional streamwise velocity - v local normal velocity, -direction - x ⁺ nondimensional coordinate in streamwise direction - ⁺ nondimensional coordinate normal to wall - ₊ ^wire wire nondimensional location of hydrogen bubble-wire normal to wall - z ⁺ nondimensional spanwise coordinate - momentum thickness - v kinematic viscosity - _W wall shear stress     

Fluid flow and heat transfer in microchannels with rectangular cross section

Forced convective heat transfer coefficients and friction factors for flow of water in microchannels with a rectangular cross section were measured. An integrated microsystem consisting of five microchannels on one side and a localized heater and seven polysilicon temperature sensors along the selected channels on the other side was fabricated using a double-polished-prime silicon wafer. For the microchannels tested, the friction factor constant obtained are values between 53.7 and 60.4, which are close to the theoretical value from a correlation for macroscopic dimension, 56.9 for D _h  = 100 μm. The heat transfer coefficients obtained by measuring the wall temperature along the micro channels were linearly dependent on the wall temperature, in turn, the heat transfer mechanism is strongly dependent on the fluid properties such as viscosity. The measured Nusselt number in the laminar flow regime tested could be correlated by which is quite different from the constant value obtained in macrochannels.     

Unsteady flow in an annulus between two concentric rotating spheres

An analysis is presented for the unsteady laminar flow of an incompressible Newtonian fluid in an annulus between two concentric spheres rotating about a common axis of symmetry. A solution of the Navier-Stokes equations is obtained by employing an iterative technique. The solution is valid for small values of Reynolds numbers and acceleration parameters of the spheres. In applying the results of this analysis to a rotationally accelerating sphere, a virtual moment of intertia is introduced to account for the local inertia of the fluid.Nomenclature R _i radius of the inner sphere - R _o radius of the outer sphere - radial coordinate - r dimensionless radial coordinate, - meridional coordinate - azimuthal coordinate - time - t dimensionless time, - Re _i instantaneous Reynolds number of the inner sphere, _i R _k ² / - Re _o instantaneous Reynolds number of the outer sphere, _o R _o ² / - radial velocity component - V _r dimensionless radial velocity component, - meridional velocity component - V dimensionless meridional velocity component, - azimuthal velocity component - V dimensionless azimuthal velocity component, - viscous torque - T dimensionless viscous torque, - viscous torque at surface of inner sphere - T _i dimensionless viscous torque at surface of inner sphere, - viscous torque at surface of outer sphere - T _o dimensionless viscous torque at surface of outer sphere, - externally applied torque on inner sphere - T _p,i dimensionless applied torque on inner sphere, - moment of inertia of inner sphere - Z _i dimensionless moment of inertia of inner sphere, - virtual moment of inertia of inner sphere - Z _i,v dimensionless virtual moment of inertia of inner sphere, - virtual moment of inertia of outer sphere - _i instantaneous angular velocity of the inner sphere - _o instantaneous angular velocity of the outer sphere - density of fluid - viscosity of fluid - kinematic viscosity of fluid,/ - radius ratio,R _i/R _o - swirl function, - dimensionless swirl function, - stream function - dimensionless stream function, - _i acceleration parameter for the inner sphere, - _o acceleration parameter for the outer sphere, - shear stress - _r dimensionless shear stress,      

Mixing and Bimolecular Reaction Kinetics in a Plane Poisseulle Flow

Mixing and a nonlinear bimolecular chemical reaction (reactant A + reactant B → product; reaction rate r?=?κc ₁ c ₂) in laminar shear flow are investigated. It is found that asymptotically the dominant balance between the rates of production and dissipation of the mean-squared concentration fluctuations \((\sigma_{c_1 }^2 ,\sigma_{c_2 }^2)\) and cross-covariance of concentration fluctuations \((\overline {c_1 c_2 })\) occurs under nonreactive and reactive conditions. Longitudinal dispersion of the cross-sectional averages (C ₁, C ₂), and variances and the cross-covariance of reactant concentrations can be asymptotically quantified by the classic Taylor dispersion coefficient (D) even under reactive conditions. The characteristic time-scale (τ) over which molecular diffusion dissipates concentration variance and the cross-covariance of reactant concentrations is also shown to be the same under nonreactive and reactive conditions. A variational estimate of τ is shown to be close to the values inferred from detailed numerical simulation. The production-dissipation balance implies that the cross-sectional averaged reaction rate follows \(\overline r =\kappa_{eff} C_1 C_2 \) and \(\kappa _{eff} \approx \kappa \left[ {1+2D\tau \left( {{\partial \ln C_1 } \mathord{\left/ {\vphantom {{\partial \ln C_1 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)\left( {{\partial \ln C_2 } \mathord{\left/ {\vphantom {{\partial \ln C_2 } {\partial x}}} \right. \kern-\nulldelimiterspace} {\partial x}} \right)} \right]\). The effective reaction rate parameter (κ _eff ) is higher than that of well-mixed batch test reaction rate constant (κ) for initially overlapping species and κ _eff is smaller than κ for initially non-overlapping species.     

Damping of plasma waves

The mechanism of formation of a plateau of the electron velocity distribution function in monochromatic. plasma wave damping is discussed. It is shown that the distribution function is subject to strong modulation in a neighborhood of the phase velocity of the wave and that the steady state is established as a result of collisions. The collisionless damping obtained in the linear approximation in [1] is caused by resonance particles and depends on the electron velocity distribution function in the region

$$\upsilon _p - \sqrt {{{2e\Phi _0 } \mathord{\left/ {\vphantom {{2e\Phi _0 } m}} \right. \kern-\nulldelimiterspace} m}} \lesssim \upsilon _{p^| } \lesssim \upsilon + \sqrt {{{2e\Phi _0 } \mathord{\left/ {\vphantom {{2e\Phi _0 } m}} \right. \kern-\nulldelimiterspace} m}}$$     

Forced convection heat transfer and friction characteristics in multi-exit -rectangular package with inclined tube bundles

An experiment was carried out to investigate the characteristics of the heat transfer and pressure drop for forced convection airflow over tube bundles that are inclined relative to the on-coming flow in a rectangular package with one outlet and two inlets. The experiments included a wide range of angles of attack and were extended over a Reynolds number range from about 250 to 12,500. Correlations for the Nusselt number and pressure drop factor are reported and discussed. As a result, it was found that at a fixed Re, for the tube bundles with attack angle of 45 ° has the best heat transfer coefficient, followed by 60, 75 and 90 °, respectively. This investigation also introduces the factors which can be used for finding the heat transfer and the pressure drop factor on the tube bundles positioned at different angles to the flow direction. Moreover, no perceptible dependence of Cand C on Re was detected. In addition, flow visualizations were explored to broaden our fundamental understanding of the heat transfer for the present study.     

Forced convective mass transfer studies from spheroids

In order to obtain information on the effect of shape on mass transfer, overall mass transfer rates were measured from naphthalene spheroids suspended in a wind tunnel (Schmidt number 2.4). Spheroidal shapes which included spheres, oblate spheroids and spheroids with composite halves were employed for the study. The ratio of the minor to major axes of the spheroids ranged from 1∶1 to 1∶4. The data obtained from one-hundred and fifty six experimental runs were best correlated by the use of Pasternak and Gauvin's characteristic dimension defined as total surface area of the body divided by maximum perimeter normal to flow. The correlations for the ranges 200 < Re < 2000 and 2000 < Re < 32000 are as follows. $$\begin{gathered} Sh = 0.62 (Re)^{0.5} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ Sh = 0.26 (Re)^{0.6} (Sc)^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ \end{gathered}$$ which correlated the data with standard deviation of 3.75% and 3.50% respectively.     

The flow regime in the turbulent near wake of a hemisphere

The work presented is a wind tunnel study of the near wake region behind a hemisphere immersed in three different turbulent boundary layers. In particular, the effect of different boundary layer profiles on the generation and distribution of near wake vorticity and on the mean recirculation region is examined. Visualization of the flow around a hemisphere has been undertaken, using models in a water channel, in order to obtain qualitative information concerning the wake structure.List of symbols C _p pressure coefficient, - D diameter of hemisphere - n vortex shedding frequency - p pressure on model surface - p ₀ static pressure - Re Reynolds number, - St Strouhal number, - U, V, W local mean velocity components - mean freestream velocity inX direction - U ^* shear velocity, - u, v, w velocity fluctuations inX, Y andZ directions - X Cartesian coordinate in longitudinal direction - Y Cartesian coordinate in lateral direction - Z Cartesian coordinate in direction perpendicular to the wall - it* boundary layer displacement thickness, - diameter of model surface roughness - elevation angleI - O boundary layer momentum thickness, - _w wall shearing stress - dynamic viscosity of fluid - density of fluid - streamfunction - _x longitudinal component of vorticity, - _y lateral component of vorticity, - _z vertical component of vorticity, This paper was presented at the Ninth symposium on turbulence, University of Missouri-Rolla, October 1–3, 1984     

Optimal cupolas of uniform strength: Spherical M-shells and axisymmetric T-shells

Summary The first part of this paper is concerned with the optimal design of spherical cupolas obeying the von Mises yield condition. Five different load combinations, which all include selfweight, are investigated. The second part of the paper deals with the optimal quadratic meridional shape of cupolas obeying the Tresca yield condition, considering selfweight plus the weight of a non-carrying uniform cover. It is established that at long spans some non-spherical Tresca cupolas are much more economical than spherical ones.

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Optimale Kuppeln gleicher Festigkeit: Kugelschalen und axialsymmetrische Schalen

Übersicht Im ersten Teil dieser Arbeit wird der optimale Entwurf sphärischer Kuppeln behandelt, wobei die von Misessche Fließbewegung zugrunde gelegt wird. Fünf verschiedene Lastkombinationen werden untersucht. Der zweite Teil befaßt sich mit der optimalen quadratischen Form des Meridians von Kuppeln, die der Fließbedingung von Tresca folgen.

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List of Symbols a_k, b_k, c_k, A_k, B_k, C_k coefficients used in series solutions - A, B constants in the nondimensional equation of the meridional curve - normal component of the load per unit area of the middle surface - meridional and circumferential forces per unit width - radial pressure per unit area of the middle surface, - skin weight per unit area of the middle surface, - vertical external load per unit horizontal area, - base radius, - R radius of convergence - s - cupola thickness, - u, w subsidiary functions for quadratic cupolas - vertical component of the load per unit area of middle surface - resultant vertical force on a cupola segment - structural weight of cupola, - combined weight of cupola and skin, - distance from the axis of rotation, - vertical distance from the shell apex, - z auxiliary variable in series solutions - specific weight of structural material of cupola - radius of the middle surface, - uniaxial yield stress - meridional stress, - circumferential stress, - _a, _b, _c, _d, _e subsidiary variables used in evaluating the meridional stress - auxiliary function used in series solutions This paper constitutes the third part of a study of shell optimization which was initiated and planned by the late Prof. W. Prager     

Analysis of thermally activated formation and breakdown of Kear-Wilsdorf barriers in Ni<Subscript>3</Subscript>Ge single crystals of various orientations

The orientation dependence of the yield stress in Ni₃Ge single crystals has been examined both theoretically and experimentally. The positive temperature dependence of the yield stress in the low temperature region is attributed to formation of Kear-Wilsdorf barriers. The forces driving the formation and breakdown of barriers are calculated within the framework of the Hirsch scheme. A distinctive feature of the model proposed is that the barrier is considered on the screw component of the a/2[ 01](111) superdislocation in the primary octahedral plane. The major role in barrier formation belongs to anisotropy of elastic moduli, energy of antiphase boundaries in the octahedral plane, shear stresses in the cubic and octahedral planes, and friction-induced stress in the cubic plane. A comparison of predicted values of the driving force of barrier formation and breakdown with experimental values reveals their good agreement. An analysis of the orientation dependence of the driving force of barrier formation in the temperature range T = 77–293 K shows that the dependence (T) has an extremum for crystals deformed along the [ 39] crystallographic direction, which is confirmed experimentally.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 116–125, January–February, 2005.     

Extension of k–ε models for a turbulent falling film

An extension applicable to various k–ε models is proposed to account for the damping of turbulence due to the surface tension. It consists of a sink in the equation for $ {{\overline{D} k} \mathord{\left/ {\vphantom {{\overline{D} k} {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ and a source in the equation for $ {{\overline{D} \varepsilon } \mathord{\left/ {\vphantom {{\overline{D} \varepsilon } {\overline{D} t}}} \right. \kern-0em} {\overline{D} t}} $ both derived from dimensional analysis. First numerical experiments are undertaken with a commercial CFD software. Reasons are given why the tuning of the model should be performed with thermal experiments. The proposed model is practical, since it only needs the programming of source terms. On account of it’s mathematical structure it needs a very small closure coefficient. The intention of this article is to stimulate the numerical turbulence research in a way that is applicable to the engineering practice. A comparison to experimental data is not done here.     

Diffusion of the dicumyl peroxide in molten polymer probed by rheology

The mutual coefficient of diffusion of the dicumyl peroxide (DCP), which diffuses into an ethylene–octene copolymer above its T _g was measured from an innovative rheological experiment. The experiments were carried out on a parallel plate geometry rheometer. The method is based on the cross-linking of a two-layer sample; the upper layer contains 2 wt% of DCP, and the lower layer is free of DCP. Actually, this experiment is based on the competition between the reaction of cross-linking and the diffusion of DCP in the lower layer. Comparing this rheological behavior with the rheological kinetic of cross-linking of an homogeneous sample with 1 wt% DCP, we are able, from an inverse fitting procedure, to calculate the mutual coefficient of diffusion. Our hypothesis is that the diffusion of DCP in the copolymer above T _g, can be described by Fick’s classical law. Using Fick’s law, the concentration of the DCP was established for any given point of the thickness of the two-layer sample at any time. Using a one-dimensional grid to solve continuous equations that describe the different rheological contributions of each abscissa, we determined the linear viscoelastic response of the whole sample. Comparing the experimental storage modulus of the two layer sample to the values measured from an homogeneous sample, we found the values of the mutual coefficient of diffusion. Finally, a simple relation, which describes the mutual coefficient of diffusion of DCP into melt ethylene–octene copolymer was established according to an Arrhenius law as:

Improved theoretical model for wavy filmwise condensation

This paper presents a numerical solution for wavy laminar film-wise condensation on vertical walls. Integral method is achieved based on the recently developed simple wave equations. Solutions are obtained for ranges of dimensionless groups as follows: $$1.5 \leqslant \left( {Pr = \frac{{^{\mu C} p}}{k}} \right) \leqslant 6.0$$ $$10 \leqslant \left( {G = \frac{{^h fg}}{{^{C_p \Delta T} }}} \right) \leqslant 400$$ $$100 \leqslant \left( {S = \left( {\frac{{\sigma ^2 \rho }}{{g_\rho \mu ^4 }}} \right)^{{1 \mathord{\left/ {\vphantom {1 5}} \right. \kern-\nulldelimiterspace} 5}} } \right) \leqslant 400$$ $$1000 \leqslant \left( {L = \frac{{{\rm H}_t }}{{^\delta cr}}} \right) \leqslant 10000$$ . Such ranges cover the expected situations in industrial applications. It is found that the Reynolds number (Re=h_LΔTH_t/h_fg) is a linear function of L on the log-log plane. It is also relatively insensitive to small variations of Pr at high values of this number. At situations where G less than 200 the Re appears to be dependent on S. Agreement with experimental observation is improved over that obtained from previous analytical theories.     

Visualization and acoustic sounding of the fine structure of a stratified flow behind a vertical plate

High-resolution shadow visualization and high-frequency sonar detection are applied to separate out the density wake and a fine streaky structure in the vicinity of a vertical plate in motion in salt-stratified water. The length of the sounding acoustic wave is taken to be approximately equal to the universal microscale $\delta _N^v = \sqrt {{v \mathord{\left/ {\vphantom {v N}} \right. \kern-0em} N}}$ , where ν and N are the kinematic viscosity and the buoyance frequency. In the spectra of the vertical oscillations of the acoustic contrast some characteristic frequencies ω are separated out and used to calculate the local Stokes microscales $\delta _\omega ^v = \sqrt {{v \mathord{\left/ {\vphantom {v \omega }} \right. \kern-0em} \omega }}$ in the density wake region. The scales determined from the data of independent optical and acoustic measurements are in agreement with each other.