In-line Square Research Articles

NEW DESIGN RECOMMENDATIONS FOR FLUIDELASTIC INSTABILITY IN HEAT EXCHANGER TUBE BUNDLES

Design equations are presented to determine the critical velocities for the occurrence of fluidelastic instability in uniform single-phase cross-flow. These equations are an essential part of a new guideline in the “VDI-Wärmeatlas” for estimating vibration excitation in real tube bundle heat exchangers. Six existing guidelines for fluidelastic instability were tested and compared with about 300 experimental data from 34 papers (eight of them being not yet considered in a guideline before). New equations for the stability factor K as a function of the pitch ratio for different tube configurations were derived with statistical methods by a variation of the reference definitions for the structural parameters and the exponents of either the dimensionless mass and the damping or the mass-damping parameter. The criterion used here was first to be on the safe side with a minimum number, and minimum deviation, of experimental data below the recommended threshold line, and second a minimum r.m.s. error for all data considered. The pitch ratio has the strongest influence for the normal triangular array and the influence becomes less for the rotated as well as for the in-line square configuration. This significant gradation of the stability constants as a function of the pitch ratio and the tube bundle configuration enables a reasonable interpolation for non-standard configurations. The exponent of the dimensionless mass-damping parameter for gas cross-flow depends on the tube configuration and is 0·5 for the 30°- and 45°-configuration, and 0·4 for the 60°- and 90°-configuration. For liquid flow, an average exponent of 0·15 has been observed.

Thermal boundary condition effects on forced convection heat transfer

We propose a numerical solution of an adjoint problem of forced convection heat transfer to evaluate the mean heat transfer characteristics under arbitrary thermal boundary conditions. Using the numerical solution of the adjoint problem under the Dirichlet condition, which can be computed by slightly modifying a conventional heat transfer code, we obtain an influence function of local surface temperature on total heat transfer. As a result, the total heat transfer for arbitrary surface temperature distributions can be calculated by the influence function. Similarly, using the numerical solution of the adjoint problem under the Neumann condition, we can also obtain an influence function of the local heat flux on the mean surface temperature. The influence functions for a circular cylinder and for an in-line square rod array are presented to illustrate the capability of this method. © 1999 Scripta Technica, Heat Trans Asian Res, 28(3): 227–238, 1999

Thermal Boundary Condition Effects on Forced Convection Heat Transfer. Application of Numerical Solution of Adjoint Problem.

A numerical approach based on the adjoint formulation of forced convection heat transfer is proposed, to evaluate the total or mean heat transfer rate under arbitrary thermal boundary conditions. Using the numerical solution of the adjoint problem under the uniform Dirichlet condition, which can be computed by a slightly modified CFD code, an influence function of the local surface temperature can be obtained. As a result, the total heat transfer for arbitrary surface temperature distributions can be calculated by the influence function. In a similar fashion, using the numerical solution of the adjoint problem under the Neumann condition, the mean surface temperature can also be expressed as an influence function of the local heat flux distributions. The influence functions for a circular cylinder and for an in-line square rod array are presented to illustrate the present method.

NUMERICAL STUDY OF FLUID FLOW AND HEAT TRANSFER IN TUBE BANKS WITH STREAM-WISE PERIODIC BOUNDARY CONDITIONS

A numerical procedure for the calculation of laminar fully-developed cross-flow and heat transfer in tube-banks heat exchangers is presented. Boundary conditions are introduced in a form similar to the treatment at conventional inlets and outlets. Calculations are performed for in-line square, rotated square and equilateral triangle geometries, with pitch-to-diameter ratio 1.25-2, Reynolds number 101-103 and Pradtl number 1-100, under both constant heat flux and constant wall temperature boundary conditions. Overall pressure drop and heat transfer factors are compared to previous numerical work, empirical correlations and experimental data. Overall pressure drop calculations correlate with experimental data, while at low Reynolds number, heat transfer calculations are shown to depend upon the choice of reference bulk temperature in the rate equation. Graphs of local pressure, drag, and heat transfer coefficients are provided and discussed in detail.

THE INFLUENCE OF TRIP-WIRES ON THE FLUID-DAMPING-CONTROLLED INSTABILITY OF A FLEXIBLE TUBE IN A BUNDLE

This paper reports on experimental investigations concerning the influence of disturbed flow on the stability behaviour of normal and rotated triangular tube arrays of different pitch-to-diameter ratios as well as of in-line square arrays. The excitation is due to air cross-flow in a wind tunnel. The experimental apparatus allows the measurement of the stability boundaries of a fully flexible bundle as well as of single flexibly mounted tubes with variable equilibrium position in otherwise fixed arrays as a function of the mass-damping parameterδrand the reduced gap velocityVr. A disturbance of the flow channels between the tubes is achieved by application of thin Prandtl trip-wires with a diameter of180of the tube diameter on the top and the bottom of each tube. In most of the cases this leads to a shift of the stability boundaries to higher reduced gap velocities or even to stability in the entire parameter region, depending on the bundle configuration and equilibrium position. To understand these effects a simultaneous flow visualization based on the laser-light-sheet procedure has been carried out.

Thermal Boundary Condition Effects on Forced Convection Heat Transfer. Application of a Numerical Solution of an Adjoint Problem.

We propose a numerical solution for the adjoint operator of a forced convection heat transfer problem to evaluate mean heat transfer characteristics under arbitrary thermal conditions. Using the numerical solutions of the adjoint problems under Dirichlet and Neumann conditions, both of which can be computed using a conventional CFD code, the influence function of the local surface temperature on the total heat transfer and that of the local surface heat flux on the mean surface temperature are obtained. As a result, the total heat fluxes for arbitrary surface temperature distributions and the mean surface temperatures for arbitrary surface heat flux distributions can be calculations using these influence functions. The influence functions for a circular cylinder and for an in-line square rod array are presented.

A Lumped-Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media

Based on a lumped-parameter method, algebraic expressions for the stagnant thermal conductivity of some two-dimensional and three-dimensional spatially periodic media are obtained. The geometries under consideration include arrays of touching and non-touching in-line square and circular cylinders (two-dimensional), as well as touching and nontouching in-line cubes (three-dimensional). A comparison of results based on these algebraic expressions with existing numerical solutions and experimental data shows that they are in excellent agreement.

Stability Behaviour of a Single Flexible Cylinder in Rigid Tube Arrays of Different Geometry Subjected to Cross-Flow

This paper reports on experimental investigations of fluidelastic instability in tube arrays subjected to air cross-flow. The vibration behaviour of one flexibly mounted tube within otherwise fixed bundles with different geometries has been examined. The applied experimental apparatus allows to realize four tube patterns (normal and rotated triangular, in-line and rotated square) which are usual in heat exchangers. The influence of the pitch-to-diameter ratio is investigated for the triangular arrays. The free tube has a linear iso-viscoelastic suspension. This means, neither stiffness nor damping of the tube mounts is dependent on the amplitude and the direction of motion. Using eddy current dampers the mass-damping parameter of the free tube can be varied continuously over a wide range (11·7 ≤ δ, ≤ 500). Adjustable tube mounts of the flexible cylinder enable to maintain the ideal tube array geometry during all experiments. Results are presented showing the stability boundaries of the flexibly mounted cylinder positioned in each of the first three or four rows of the different arrays. These stability thresholds of the equilibrium position of the free tube depend on the array pattern as well as on the tube distance. This shows the limitations of the standards stability diagram, common for different pitch-to-diameter ratios.

A generalized constrained-mode analysis for cylinder arrays in cross-flow

For cylinder arrays subject to cross-flow, the fluid dynamic and mechanical coupling between cylinders has an important effect on the fluidelastic stability of the array, especially for high values of non-dimensional mass-damping, mδ / ρd 2 . Due to the large number of cylinders in heat exchanger tube banks, this coupling results in coupled matrices prohibitively large from a computational point of view. To overcome this problem, the constrained-mode method of analysis was developed, which allows a very large array of cylinders to be represented in a stability analysis by a kernel of a small group of cylinders. The disadvantage of this constrained-mode method is that it assumes adjacent cylinders to be moving either in-phase or out-of-phase with each other, but nothing intermediate. In the present paper the above restriction is relaxed, such that adjacent cylinders can have any phase angle between their respective displacements. Using this form of the constrained-mode analysis, the stability of in-line square, rotated square and rotated triangular arrays has been investigated. It is shown that for sufficiently high values of mδ / ρd 2 (the exact value depending on the type of array) the phase angle between the displacements of adjacent cylinders can have an important effect on the stability. Not only can it change the flow velocity at which instability will occur, but for some arrays it can also change the inter-cylinder mode shape from being predominantly in the cross-flow direction to being predominantly in the in-flow direction.

Random wave forces on horizontal cylinders

The random wave induced in-line and transverse forces on a smooth circular cylinder, placed horizontally near a plane boundary, are analyzed in the time domain. The time histories of intantaneous water surface elevation and the corresponding in-line and transverse forces are Fourier decomposed into various frequency components. The time histories of water particle kinematics for different frequency components are generated using the linear numerical transforms. Each frequency component of the in-line force is correlated with the corresponding wave component, and the in-line hydrodynamic coefficients of inertia and drag are evaluated through the use of the Morison equation and least-squares method. In addition, the in-line root mean square (r.m.s.) coefficients are also obtained. The component transverse forces are analyzed in terms of maximum and r.m.s. transverse force coefficients. All the in-line and transverse force transfer coefficients are correlated with the component Keulegan-Carpenter number or period parameter, gap ratio of the cylinder from the plane boundary and depth parameter. The component force transfer coefficients are also compared with those obtained from regular wave force experiments, and the coefficients obtained from the time domain analysis of the entire length of the instanteneous random water surface elevation and the corresponding force records. It is found that the component force transfer coefficients compare quite well with the coefficients determined from the regular wave force experiments, and show the same trend with all the parameters mentioned above as the regular wave force results under the same cylinder and water depth conditions.

The flow-induced response of a single flexible cylinder in an in-line array of rigid cylinders

The cross-flow induced vibration of a single flexible cylinder in a six-row, in-line square array of otherwise rigid cylinders, with pitch-to-diameter ratio of 1·5, has been investigated. Experiments have been done in both air- and water-flow; in air-flow with two levels of upstream turbulence. Measurements were made of the cylinder vibration amplitudes in the in-flow and cross-flow directions, for different positions of the flexible cylinder in the array and for different values of mechanical damping. Interstitial flow velocity measurements were also made in both rigid and flexible arrays of cylinders.

Void distributions in bubbly flow through yawed rod arrays

In this communication observations of void distribution in bubbly flow through an array of rods inclined relative to the average flow direction are presented. The experiments reveal a strong dependence of void distribution on bubble size and rod arrangement. It is postulated that this dependence is due to the entrapment and migration of bubbles in vortices behind the rods. Not surprisingly, attempts to simulate the experimental results with a six-equation porous body model, in which localized phenomenon such as vortices have been averaged out, were unsuccessful. 2. EXPERIMENTS Figure 1 is a sketch of the flow field considered in this study, an air-water bubbly mixture flowing vertically upward through an inclined rod array. The inclination angle of the rods relative to the vertical was 45 °. The flow through arrays of two rod arrangements, inline square and rotated square was considered. Rod arrangements are defined according to the average direction of the transverse (normal to the rod axis) component of flow relative to the layout of the rod array, as illustrated in figure 2. Characteristics of each rod array and the experimental conditions are listed in table 1. The superficial velocities JL and Jo are defined by rhk J* 7, Ax,' [1] where k represents phase (L for liquid, G for gas), mk is the mass flow rate of phase k, A,, is the total cross-sectional area of the bundle and Yv is the bundle porosity, defined as the ratio of fluid volume to total volume. 2.1. Inline square array Void distributions in the inline 45 ° test section were observed for a range of gas and liquid flow rates (0 <JG < 0.3 m/s, 0 <JL < 1 m/s). Details of the study are given in Ebeling-Koning (1983). Figures 3 and 4 show the flow development through the test section for two liquid flow rates. Figures 5 and 6 are photographs of the exit plenum for a range of gas and liquid flow rates. The most striking observation from these plates is the influence of bubble size on void distribution. At low liquid flow rates a wide distribution of bubble sizes is generated. The large bubbles drift to the right wall across the rods, as predicted by the porous body model. However, the smaller bubbles migrate along the rod axis to the left wall. At high liquid flow rates only small bubbles exist. These also move along the rod axis until they meet the left wall.

Strouhal numbers for flows in deep tube array models

The analysis of the forced vibration response of tubes in heat exchanger banks is based upon the premise that the forces experienced by the tubes due to random pressure fluctuations in the first few rows of the array are significantly different from those encountered deeper in the tube bundle. The basis for this assumption is that the turbulence spectrum arising from local velocity fluctuations contains narrow band peaks in the entrance region but is broadband with no significant peaks deeper in the bank. This paper reports on an experimental study in which the frequency spectrum of turbulence has been measured in three relatively deep in-line heat exchanger tube bank models. For each array, different characteristics are observed. Strouhal numbers associated with peaks in the turbulence spectrum are shown to be identifiable at significantly greater depths than reported heretofore and their values can be functions of the measurement position and the Reynolds number of the flow. The values reported are compared with the data available from the literature for in-line square arrays using a modified Strouhal number.

An Experimental Stability Analysis of a Single Flexible Cylinder Positioned in an Array of Rigid Cylinders and Subject to Cross-Flow

The fluidelastic response of a single flexible cylinder in an array of rigid cylinders, subject to cross-flow, has been investigated. Experiments were done in two wind tunnels on three different types of array: (i) a double row in-line (square) array, with pitch-to-diameter ratio P/d = 1.5; (ii) a triple row staggered (rotated-square) array, with P/d = 1.5 2; (iii) a double row version of (ii). The modal damping logarithmic decrement was varied in the range 0.01 to 0.46. Measurements of vibration were made in both the in-flow and cross-flow directions, taking not only the vibration amplitudes, but also the power spectral density of the response over a wide frequency range and over a broad range of flow velocities. This permitted a more detailed examination of system behavior, prior to, at, and beyond the threshold of fluidelastic instabilities, as well as a better definition of the onset of instability. Fluidelastic instability was observed for downstream cylinders of both the in-line and staggered arrays, the motion in both cases being predominantly in the cross-flow direction. However, in the in-line array, further increase in flow velocity tended to shift the instability to one involving mainly in-flow motion.

A NUMERICAL ANALYSIS OF LAMINAR FLOW AND HEAT TRANSFER OF AIR IN AN IN-LINE TUBE BANK

A numerical analysis of laminar flow and heat transfer in a tube bank is described. The two-dimensional Navier-Stokes and energy equations are solved numerically for successive subregions from the inlet to the exit of the bank by a procedure catted the “one step forward and half step backward iteration method” and by using a hybrid grid system. Calculations are carried out for an in-line square tube bank up to five rows deep, with pitch-to-diameter ratios 1.5 × 1.5, under the conditions of uniform tube wall temperature, Re = 60, 120, and 300, and Pr = 0.7. Stream, isovorticity, and isothermal lines through the whole tube bank, local and average Nusselt numbers, pressure distributions along the tube walls and in the flow directions, and friction factor are presented. The friction factor and average Nusselt number agree well with published experimental results. An analogous relation between an in-line tube bank and parallel plates with respect to developing heat transfer characteristics is presented.

Heat-Transfer and Flow-Friction Characteristics of Nine Pin-Fin Surfaces

Experimental results for convective heat-transfer and flow-friction characteristics are presented for pin-fin surfaces with in-line square pin spacing of 2D0, 3D0 and 4D0 and with plate spacings of 4D0, 8D0, 12D0.

Heat Transfer and Fluid Friction During Flow Across Banks of Tubes—IV: A Study of the Transition Zone Between Viscous and Turbulent Flow

Abstract The experimental results of a long-term research program on flow across tube banks have been extended from the zone of viscous flow through the transition zone and well into the zone of turbulent flow. Pressure drop and rate of heat transfer are reported for the flow of a light oil across five rectangular banks of ⅜-in-diam tubes. Equilateral triangle, staggered square, and in-line square arrangements on a 1.25 pitch ratio, and equilateral triangle and in-line square arrangements on a 1.50 pitch ratio are discussed. The flow rate of the oil covers a Reynolds number range of from 25 to 10,000. Heating, cooling, and isothermal tests are reported.

Heat Transfer and Fluid Friction During Viscous Flow Across Banks of Tubes—III: A Study of Tube Spacing and Tube Size

Abstract In the course of a research program on tubular heat exchangers, pressure-drop and heat-transfer data are reported for heating and cooling a medium-viscosity oil flowing across banks of vertical tubes in seven test exchangers. The apparatus variables include equilateral triangle, in-line square, and staggered square arrangements; tube sizes of ⅜ in, and ¾ in. OD; and pitch ratios of 1.25 and 1.50. The results are shown both in simple plots of pressure drop and coefficient of heat transfer versus the rate of flow, and in generalized correlations. Tentative correlations are provided for friction and heat transfer which bring the data for these seven tube banks somewhat closer together than previous correlations. In a comparison of the heat-transfer coefficient versus pumping-power loss per unit surface area, the smaller diameter tubes at the smaller pitch ratio give the best performance.

A Study of Three Tube Arrangements in Unbaffled Tubular Heat Exchangers

Abstract Pressure-drop and heat-transfer tests, using a medium-viscosity oil in viscous flow across vertical tubes in three unbaffled exchangers, are reported in the course of a research program on tubular exchangers. The three once-flow-through units have, respectively, equilateral triangular, in-line square, and staggered square tube arrangements, but, in all other respects, are as nearly identical as possible. Heat-transfer data for three bulk-oil temperatures and a constant, but lower, tube-wall temperature, and both isothermal and nonisothermal pressure-drop data at three temperature levels are reported. The two staggered-tube arrangements provide greater heat transfer at a given pumping power loss than the in-line arrangement in the region of viscous flow, but this superiority decreases at higher velocities and may disappear in turbulent flow as previous data have indicated. A correction is presented for the effect of viscosity gradient upon friction during heat transfer but this correction does not allow for free convection or nonuniform flow.