Experimental study on the heat transfer and pressure drop of a cross-flow heat exchanger with different pin–fin arrays

In this study, convective heat transfer and pressure drop in a cross-flow heat exchanger with hexagonal, square and circular (HSC) pin–fin arrays were studied experimentally. The pin–fins were arranged in an in-line manner. For the applied conditions, the optimal spacing of the pin–fin in the span-wise and stream-wise directions has been determined. The variable parameters are the relative longitudinal pitch (S _L /D = 2, 2.8, 3.5), and the relative transverse pitch was kept constant at S _T /D = 2. The performances of all pin–fins were compared with each other. The experimental results showed that the use of hexagonal pin–fins, compared to the square and circular pin–fins, can lead to an advantage in terms of heat transfer enhancement. The optimal inter-fin pitches are provided based on the largest Nusselt number under the same pumping power, while the optimal inter-fin pitches of hexagonal pin–fins are S _T /D = 2 and S _L /D = 2.8. Empirical equations are derived to correlate the mean Nusselt number and friction coefficient as a function of the Reynolds number, pin–fin frontal surface area, total surface area, and total number. Consequently, the general empirical formula is given in the present form.

A study on heat transfer of the optimization in circular fins of parabolic profile with variable thermal parameters

Circular fins are used extensively in heat exchange devices to increase the heat transfer. For economic purposes, the traditional approach to the optimization of fins consists of minimizing the comsumption (investment) of fin material for the excution of a specified heat transfer task. The minimum weight cooling fin has optional profile to be a concave parabola. Therefore, the optimum geometric dimensions of circular fins of parabolic profile with variable thermal parameters are studied. The effect of the two pertinent physical parameters-thermal conductivity variation parameter α and the index of the heat transfer coefficient variationm upon the optimum geometric dimensions is also studied. The results pressented can be used as the design guideline for engineering practice.     

Boiling on a straight pin fin with variable thermal conductivity

This work theoretically investigated the thermal performance and stability characteristics of a straight pin fin subject to boiling considering a temperature-dependent thermal conductivity of fin, k=k _sat(1+b(T−T _sat)). Steady-state temperature distribution and the associated fin base heat flow were for the first time analytically found, whose stability characteristics were evaluated by linear stability analysis. A positive temperature coefficient b will raise both the fin's temperature and base heat flow. The corresponding stability for stable fin boiling was enhanced. A negative b results in an opposite trend. The use of a mean thermal conductivity in fin boiling calculations is discussed. Received on 3 November 1997     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

Laminar mixed convection adjacent to vertical, continuously stretching sheets

The analysis of laminar mixed convection in boundary layers adjacent to a vertical, continuously stretching sheet has been presented. The velocity and temperature of the sheet were assumed to vary in a power-law form, that is, u _w (x)=Bx ^m and T _w (x)−T _∞=Ax ⁿ . In the presence of buoyancy force effects, similarity solutions were reported for the following two cases: (a) n=0 and m=0.5, which corresponds to an isothermal sheet moving with a velocity of the form u _w =Bx ^0.5 and (b) n=1 and m=1, which corresponds to a continuous, linearly stretching sheet with a linear surface temperature distribution, i.e. T _w −T _∞=Ax. Formulation of the present problem shows that the heat transfer characteristics depends on four governing parameters, namely, the velocity exponent parameter m, the temperature exponent parameter n, the buoyancy force parameter G ^*, and Prandtl number of the fluid. Numerical solutions were generated from a finite difference method. Results for the local Nusselt number, the local friction coefficient, and temperature profiles are presented for different governing parameters. Effects of buoyancy force and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Received on 17 July 1997     

An experimental study of dynamic tensile properties of glass fiber at low-temperatures

Tensile impact experiments of EC8.0−24×7 glass fiber bundles at different low temperaturesT(14°C, −40°C and −10°C) and strain rates ɛ were carried out, and complete stress-strain curves were obtained. Within the range of the experiment temperatures and strain rates, it is found that the initial modulusE, the ultimate strength σ_max and the unstable strain ɛ _b of the glass fiber bundles all increase with ɛ at an identicalT. At an identical ɛ, with the decrease ofT, E and σ_max increase; but ɛ _b increases when 10°C>T>−40°C and decreases when −40°C>T>−100°C. The strain-rate- and temperature-dependent bimodal Weibull statistical constitutive theory was adopted for the statistical analysis of the experimental results, and the Weibull parameters of single fiber were obtained. The results show that the bimodal Weibull distribution function is suitable to represent the strength distribution of the glass fiber at low temperature and different strain rates. The differences in the mechanical properties between EC8.0−24×7 and EC5.5−12 ×14 glass fiber bundles were also discussed. Project supported by the National Natural Science Foundation of China (No. 19772058).     

Rheo-dielectric behavior of low molecular weight liquid crystal 2. Behavior of 8CB in nematic and smectic states

Rheo-dielectric behavior was examined for 4−4^′−n-octyl-cyanobiphenyl (8CB) having large dipoles parallel to its principal axis (in the direction of the C≡N bond). In the quiescent state at all temperatures (T) examined, orientational fluctuation of the 8CB molecules was observed as dielectric dispersions at characteristic frequencies ω_c>10⁶ s⁻¹. In the isotropic state at high T, no detectable changes of the complex dielectric constant ɛ^*(ω) were found under slow flow at shear rates ˙γ≫ω_c. In the nematic state at intermediate T, the terminal relaxation intensity of ɛ^*(ω) was decreased under such slow flow. In the smectic state at lower T, the flow effect became much less significant. These results were related to the flow-induced changes of the liquid crystalline textures in the nematic and smectic states, and the differences of the rheo-dielectric behavior in these states are discussed in relation to a difference of the symmetry of molecular arrangements in the nematic and smectic textures. Received: 1 October 1998 Accepted: 13 January 1999     

Equivalence between Rank-One Convexity and Polyconvexity for Some Classes of Elastic Materials

Let W(F) = φ(λ ₁ ^s + λ ₂ ^s + λ ₃ ^s ) + ψ(λ ₁ ^r λ ₂ ^r + λ ₁ ^r λ ₃ ^r + λ ₂ ^r λ ₃ ^r ) + f(λ ₁ λ ₂ λ ₃) be a stored energy function. We prove that, for this function, rank-one convexity is equivalent to polyconvexity.under suitable assumptions on φ, ψ and f.     

On a Free Convection Problem Over a Vertical Flat Surface in a Porous Medium

The problem of the free convection boundary-layer flow over a semi-infinite vertical flat surface in a porous medium is considered, in which the surface temperature has a constant value T₁ at the leading edge, where T₁ is above the ambient temperature, and takes a value T₂ at a given distance L along the surface, varying linearly between these two values and remaining constant afterwards. Numerical solutions of the boundary-layer equations are obtained as well as solutions valid for both small and large distance along the surface. Results are presented for the three cases, when the temperature T₂ is greater, equal or less than the ambient temperature T_∞. In the first case, T₂ > T_∞, a boundary-layer flow develops along the surface starting with a flow associated with the temperature difference T₁ − T_∞ at the leading edge and approaching a flow associated with the temperature difference T₂ − T_∞ at large distances. In the second case, T₂ = T_∞, the convective flow set up on the initial part of the surface drives a wall jet in the region where the surface temperature is the same as ambient. In the final case, T₂ < T_∞, a singularity develops in the numerical solution at the point where the surface temperature becomes T_∞. The nature of this singularity is discussed.     

Crack linkup: An experimental analysis

TheT _ɛ ^* integral was used to assess stable crack growth and crack linkup in 0.8 mm thick 2024-T3 aluminum tension specimens with multiple site damage (MSD) under monotonic and cyclic loads. TheT _ɛ ^* values were obtained directly from the recorded moiré fringes on the fracture specimens with and without MSD. TheT _ɛ ^* resistance curves of these fracture specimens of different geometries were in excellent agreement with each other. The results suggest thatT _ɛ ^* is a material parameter which can be used to characterize crack growth and linkup in the absence of large overloading.T _ɛ ^* based crack growth and net-section-yield based crack linkup criteria for MSD specimens are proposed. The crack tip opening angle (CTOA) criterion can also be used to correlate crack growth larger than 2 mm.     

Nonsimilarity solutions for non-Darcy mixed convection from horizontal surfaces in a porous medium

Non-Darcy mixed convection in a porous medium from horizontal surfaces with variable surface heat flux of the power-law distribution is analyzed. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ζ _f =Ra^* _x /Pe² _x is found to characterize the effect of buoyancy forces on the forced convection with K ^′ U _∞/ν characterizing the effect of inertia resistance. The second region covers the natural convection dominated regime where the dimensionless parameter ζ _n =Pe _x /Ra^*1/2 _x is found to characterize the effect of the forced flow on the natural convection, with (K ^′ U _∞/ν)Ra^*1/2 _x /Pe _x characterizing the effect of inertia resistance. To obtain the solution that covers the entire mixed convection regime the solution of the first regime is carried out for ζ _f =0, the pure forced convection limit, to ζ _f =1 and the solution of the second is carried out for ζ _n =0, the pure natural convection limit, to ζ _n =1. The two solutions meet and match at ζ _f =ζ _n =1, and R ^* _h =G ^* _h . Also a non-Darcy model was used to analyze mixed convection in a porous medium from horizontal surfaces with variable wall temperature of the power-law form. The entire mixed convection regime is divided into two regions. The first region covers the forced convection dominated regime where the dimensionless parameter ξ _f =Ra _x /Pe _x ^3/2 is found to measure the buoyancy effects on mixed convection with Da _x Pe _x /ɛ as the wall effects. The second region covers the natural convection dominated region where ξ _n =Pe _x /Ra _x ^2/3 is found to measure the force effects on mixed convection with Da _x Ra _x ^2/3/ɛ as the wall effects. Numerical results for different inertia, wall, variable surface heat flux and variable wall temperature exponents are presented. Received on 8 July 1996     

Heat transfer for laminar flow in internally finned pipes with different fin heights and uniform wall temperature

An analysis is presented for fully developed laminar convective heat transfer in a pipe provided with internal longitudinal fins, and with uniform outside wall temperature. The fins are arranged in two groups of different heights. The governing equations have been solved numerically to obtain the velocity and temperature distributions. The results obtained for different pipe-fins geometries show that the fin heights affect greatly flow and heat transfer characteristics. Reducing the height of one fin group decreases the friction coefficient significantly. At the same time Nusselt number decreases inappreciably so that such reduction is justified. Thus, the use of different fin heights in internally finned pipes enables the enhancement of heat transfer at reasonably low friction coefficient.Nomenclature A_f dimensionless flow area of the finned pipe, Eq. (8) - a_f flow area of the finned pipe - C_p specific heat at constant pressure - f coefficient of friction, Eq. (12) - H₁, H₂ dimensionless fin height h₁/r_o h₂/r_o - h₁, h₂ fin heights - average heat transfer coefficient at solid-fluid interface - KR fin conductance parameter, k_s/k_f - k_f thermal conductivity of fluid - k_s thermal conductivity of fin - l pipe length - mass flow rate - N number of fins - Nu Nusselt number, Eqs. (15) and (16) - P pressure - Q total heat transfer rate at solid fluid interface - Q_f1, Q_f2 heat transfer rate at fin surface - q_w average heat flux at pipe-wall, Q/(2 r_ol) - R dimensionless radial coordinate r/r_o - Re Reynolds Number, Eq. (13) - r radial coordinate - r_o radius of pipe - r₁, r₂ radii of fin tips - T temperature - T_b bulk temperature - U dimensionless velocity, Eq. (2) - U_b dimensionless bulk velocity - u_z axial velocity - z axial coordinate - angle between the flanks of two adjacent fins - half the angle subtended by a fin - angle between the center-lines of two adjacent fins - angular coordinate - dynamic viscosity - density - dimensionless temperature, Eq. (6) - _b dimensionless bulk temperature     

Control of Flow and Heat Transfer in a Porous Enclosure due to an Adiabatic Thin fin on the Hot Wall

Natural convection flow in a differentially heated square enclosure filled with porous matrix with a solid adiabatic thin fin attached at the hot left wall is studied numerically. The Brinkman–Forchheimer-extended Darcy model is used to solve the momentum equations, in the porous medium. The numerical investigation is done through streamlines, isotherms, and heat transfer rates. A parametric study is carried out using the following parameters: Darcy number (Da) from 10⁻⁴ to 10⁻², dimensionless thin fin lengths (L _p) 0.3, 0.5, and 0.7, dimensionless positions (S _p) 0.25, 0.5, and 0.75 with Prandtl numbers (Pr) 0.7 and 100 for Ra = 10⁶. For Da = 10⁻³ and Pr = 0.7, it is observed that there is a counter clock-wise secondary flow formation around the tip of the fin for S _p = 0.5 for all lengths of L _p. Moreover when Da = 10⁻² the secondary circulation behavior has been observed for S _p = 0.25 and 0.75 and there is another circulation between the top wall and the fin that is separated from the primary circulation. However, these secondary circulations features are not observed for Pr = 100. It is also found that the average Nusselt number decreases as the length of the fin increases for all locations. However, the rate of decrease of average Nusselt number becomes slower as the location of fin moves from the bottom wall to the top wall. The overall heat transfer rate can be controlled with a suitable selection of the fin location and length.     

A note on the heat transfer in convective fins

In this paper a generalized approach to the problem of heat transfer through convective fins is given. The proper dimensionless variables, which specify the general problem are identified, and upper bounds of the values of the dimensionless number N_r defined as “the ratio of the heat transferred by the fin to that of the corresponding bare surface” are derived. It was shwon that these limiting values of the N_r are 1/√B₁ and √2/B₁ for longitudinal fins and spines respectively, where B₁ is the Biot number hb/k, while for annular fins of constant thickness and hyperbolic profile, N_r? K(β)/√B_i, where K(β) is a number determined by the profile of the fin and the ratio β=x₂/x₁ of the outside to the inside radii. It was also shown that for longitudinal fins and spinces the possible adverse insulating effect by the use of the fin is avoided, if one selects the value of √hA/KC < 1, which is a rather stricter criterion than the one reported in the literature, namely that of hA/kC < 1 [2–5]. An example is given to show how one may utilize the appropriate value of N_r and the fin effectiveness e, to obtain the dimensions of the fin.     

Viscosity of concentrated noncolloidal bidisperse suspensions

At the same solid volume fraction (Φ) the relative viscosity (η _r ) of a concentrated noncolloidal bidisperse suspension of hard spherical particles is lower than that of a monodisperse suspension. In this paper a semi-analytical viscosity model of noncolloidal bidisperse suspensions is derived using an integration method. In this model the random loose packing density obtained by computer simulation is taken as the limit of solid volume fraction Φ _m which depends upon both the diameter ratio (λ) of large to small particles and the volume fraction of large particles (ξ=Φ _l /Φ). This model shows that at high solid volume fraction, Φ > 0.40, both λ and ξ significantly influence η _r . For example, at Φ=0.5, it predicts that for monodisperse suspensions η _r =70, while for bidisperse suspensions (λ=2 and ξ=0.7) η _r =40. Comparison shows that, at high solid volume fraction (0.4–0.5), the relative viscosity predicted by this model is in good agreement with that predicted by the work of Shapiro and Probstein (1992) and of Patlazhan (1993), but is higher than that predicted by the work of others. Received: 27 February 2001 Accepted: 25 April 2001     

Maximum density effects on convective instability of horizontal plane poiseuille flows in the thermal entrance region

A linear stability analysis is used to study the conditions marking the onset of secondary flow in the form of longitudinal vortices for plane Poiseuille flow of water in the thermal entrance region of a horizontal parallel-plate channel by a numerical method. The water temperature range under consideration is 0∼30°C and the maximum density effect at 4°C is of primary interest. The basic flow solution for temperature includes axial heat conduction effect and the entrance temperature is taken to be uniform at far upstream location jackie=−∞ to allow for the upstream heat penetration through thermal entrance jackie=0. Numerical results for critical Rayleigh number are obtained for Peclet numbers 1, 10, 50 and thermal condition parameters (λ ₁, λ ₂) in the range of −2.0≤λ ₁≤−0.5 and −1.0≤λ ₂≤1.4. The analysis is motivated by a desire to determine the free convection effect on freezing or thawing in channel flow of water.     

Numerical simulation of fully developed turbulent flow and heat transfer in annular-sector ducts

 The mixing length theory is employed to simulate the fully developed turbulent heat transfer in annular-sector ducts with five apex angles (θ₀=18,20,24,30,40^∘) and four radius ratios (R _o/R _i=2,3,4,5). The Reynolds number range is 10⁴10⁵. The numerical results agree well with an available correlation which was obtained in following parameter range: θ₀=18,20,24,30,40^∘, R _o/R _i=4 and Re=10⁴5×10⁴. The present work demonstrates that the application range of the correlation can be much extended. Apart from the mixing length theory, the kɛ model with wall function and the Reynolds stress model are also employed. None of the friction factor results predicted by the three models agrees well with the test data. For the heat transfer prediction the mixing length theory seems the best for the cases studied. Received on 17 July 2000 / Published online: 29 November 2001     

Near-field tip vortex behind a swept wing model

The near-field flow structure of a tip vortex behind a sweptback and tapered NACA 0015 wing was investigated and compared with a rectangular wing at the same lift force and Re=1.81×10⁵. The tangential velocity decreased with the downstream distance while increased with the airfoil incidence. The core radius was about 3% of the root chord c _r, regardless of the downstream distance and α for α<8°. The core axial velocity was always wake-like. The core Γ_c and total Γ_o circulation of the tip vortex remained nearly constant up to x/c _r=3.5 and had a Γ_c/Γ_o ratio of 0.63. The total circulation of the tip vortex accounted for only about 40% of the bound root circulation Γ_b. For a rectangular wing, the axial flow exhibited islands of wake- and jet-like velocity distributions with Γ_c/Γ_o=0.75 and Γ_o/Γ_b=0.70. For the sweptback and tapered wing tested, the inner region of the tip vortex flow exhibited a self-similar behavior for x/c _r≥1.0. The lift force computed from the spanwise circulation distributions agreed well with the force-balance data. A large difference in the lift-induced drag was, however, observed between the wake integral method and the inviscid lifting-line theory.     

Thermal convection in a porous layer: Effects of anisotropy and surface boundary conditions

The theory describing the onset of convection in a homogeneous porous layer bounded above and below by isothermal surfaces is extended to consider an upper boundary which is partly permeable. The general boundary condition p + λ ∂p/∂n = constant is applied at the top surface and the flow is investigated for various λ in the range 0 ⩽ λ < ∞. Estimates of the magnitude and horizontal distribution of the vertical mass and heat fluxes at the surface, the horizontally-averaged heat flux (Nusselt number) and the fraction of the fluid which recirculates within the layer are found for slightly supercritical conditions. Comparisons are made with the two limiting cases λ → ∞, where the surface is completely impermeable, and λ = 0, where the surface is at constant pressure. Also studied are the effects of anisotropy in permeability, ξ = K _H /K _V , and anisotropy is thermal conductivity, η = k _H /k _V , both parameters being ratios of horizontal to vertical quantities. Quantitative results are given for a wide variety of the parameters λ, ξ and η. In the limit ξ/η → 0 there is no recirculation, all fluid being converted out of the top surface, while in the limit ξ/η → ∞ there is full recirculation.     

Measurements of heat transfer coefficients between turbulent liquid jets and a radially finned heated disk placed at the bottom of an open vertical circular cavity

 Experiments have been performed to assess the impact of an extended surface on the heat transfer enhancement for axisymmetric, turbulent liquid jet impingement on a heated round disk. The disk, with an array of integral radial fins mounted on its surface, is placed at the bottom of an open vertical circular cavity. Hydrodynamic and heat transfer data were obtained for a dielectric fluorocarbon liquid FC-77. For a fixed circular heater of diameter D=22.23 mm, several geometric parameters were tested: the nozzle diameter (4.42≤d≤9.27 mm), the confining wall diameter of the vertical cavity (22.23≤D _c≤30.16 mm), and the nozzle-to-heater spacing (0.5≤S/d≤5.0). The FC-77 flow rates varied from =0.2 to 11.0 l/min producing Reynolds numbers in the wide interval 700≤Re _d ≤44,000. For d=4.42 mm, the heat transfer response to the separation distance S/d was small but increased gradually with increasing nozzle diameter up to d=9.27 mm. The thermal resistance R _th increased with the confining wall diameter D _c and also with the nozzle diameter d. A minimum value of the thermal resistance of R _th,min=0.4 cm² K/W was attained for a combination of d=4.42 mm, D _c=22.23 mm, S/d=1, and =7.5 l/min. Based on a simplified heat transfer model, reasonable agreement was obtained between measured values of the thermal resistance and the R _th-predictions. The total fin effectiveness ɛ_f was shown to increase with increasing nozzle diameter, but was invariant with the flow rate (or the jet exit velocity). More than a three-fold heat transfer enhancement was realized through the addition of the array of integral radial fins on the heated round disk. Received on 30 August 2000 / Published online: 29 November 2001