Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

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Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

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Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

Laminar heat transfer in the entrance region of ducts

A finite difference technique is used for the evaluation of the rate of heat transfer in the thermal entrance region of ducts with axial conduction. The velocity profile is fully developed and flow in a tube and between parallel plates is studied. Local and average Nusselt numbers and mixing temperatures are presented as a function of the Péclet number. A criterion is also established which proves useful for predicting the conditions under which axial conduction may be ignored.Nomenclature C transformation constant - c _v specific heat, constant volume - D _h hydraulic diameter - h local convective film coefficient, Eq. (15) - h* local convective film coefficient, Eq. (16) - h _m ^* mean convective film coefficient, Eq. (17) - k thermal conductivity - Nu local Nusselt number, hD _h/k - Nu* local Nusselt number, h*D _h/k - Nu _m ^* mean Nusselt number, hQD _h/k - Pe Péclet number, D _h v _m/ - q rate of heat transfer - r radial coordinate - r _o tube radius - R nondimensional radial coordinate, r/r _o - S transformed axial coordinate, Eq. (10) - T temperature - T _e entrance temperature - T _m mixing temperature, Eq. (18) - T _w wall temperature - v _z axial velocity - v _m mean axial velocity - V nondimensional axial velocity, v _z /v _m - y transverse coordinate in parallel plate flow - y _o half width of parallel plate duct - Y nondimensional transverse coordinate, y/y _o - z axial coordinate - Z nondimensional axial coordinate, z/r _o or z/y _o - Z ⁺ nondimensional axial coordinate divided by Peclet number, Z/Pe - thermal diffusivity - nondimensional temperature, (T–T _w)/(T _e–T _w) - mean nondimensional temperature, - _m nondimensional mixing temperature, Eq. (22) - density - i axial position index - j radial or transverse position index     

Viscous dissipation effects on thermal entrance region heat transfer in pipes with uniform wall heat flux

The analytical solution to Graetz problem with uniform wall heat flux is extended by including the viscous dissipation effect in the analysis. The analytical solution obtained reduces to that of Siegel, Sparrow and Hallman neglecting viscous dissipation as a limiting case. The sample developing temperature profiles, wall and bulk temperature distributions and the local Nusselt number variations are presented to illustrate the viscous dissipation effects. It is found that the role of viscous dissipation on thermal entrance region heat transfer is completely different for heating and cooling at wall. In the case of cooling at wall, a critical value of Brinkman number, Br _c=−11/24, exists beyond which (−11/24<Br<0) the fluid bulk temperature will always be less than the uniform entrance temperature indicating the predominance of cooling effect over the viscous heating effect. On the other hand, with Br < Br _c the bulk temperature T _b will approach the wall temperature T _w at some downstream position and from there onward the bulk temperature T _b becomes less than the wall temperature T _w with T _w > B _b > T ₀ indicating overall heating effect for the fluid. The numerical results for the case of cooling at wall Br < 0 are believed to be of some interest in the design of the proposed artctic oil pipeline.     

Laminar flow heat transfer in the entrance region of circular tubes

Summary The asymptotic solution of laminar convective heat transfer in the entrance region of a circular conduit where velocity and temperature profiles are developing simultaneously, is obtained for fluids with high Prandtl numbers. Numerical values of local and average Nusselt numbers as functions of Pr and dimensionless longitudinal distances have been evaluated and presented in graphical forms.Nomenclature A ₀, A ₁ ... A _k coefficients defined by (40) - B ₀, B ₁ ... B _k coefficients defined by (39) - C _p heat capacity of fluid - I _n (x) = i ^–n J _n (ix) where J _n is the n ^th order Bessel function - k thermal conductivity of fluid - Nu _z local Nusselt number defined by (41) - Nu _av average Nusselt number defined by (44) - P pressure - Pr Prandtl number of fluid defined as C _p /k - q heat flux - Re Reynold number, defined as PR/ - R radius of pipe - r radial distance - r ⁺ dimensionless radial distance defined by (8) - T temperature of fluid - T ₀ initial temperature of fluid - T _w wall temperature - T ⁺ dimensionless temperature defined by (11) - T ₀ ⁺ , T ₁ ⁺ , ... T _k ^/+ ... functions related to T ⁺ by (22). - u dimensionless variables defined by (20) - v _r radial component of velocity - v _z z-component of velocity - v ⁺ dimensionless velocity defined by (10) - y ⁺ dimensionless distance defined by (8) - X dimensionless parameter defined by (38) - z longitudinal distance - z ⁺ dimensionless longitudinal distance defined by (9) - thermal diffusivity - dimensionless parameter defined by (12) - a parameter appearing in (46) - (x) gamma function - density - dimensionless variable defined by (28) - parameter defined by (19) - dimensionless variable defined by (32) - viscosity of fluid - kinematic viscosity of fluid     

Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts

The laminar convective flow and heat transfer in a duct with a trapezoidal cross-sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three-dimensional geometries using co-located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no-slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross-section are considered.     

Heat transfer in laminar and turbulent flows in the thermal entrance region of concentric annuli: Axial heat conduction effects in the fluid

The present analytical study investigates the influence of axial heat conduction within the flow on the heat transfer in the thermal entrance region of a concentric annular duct with laminar and turbulent internal flow. The solution is based on a decomposition of the elliptic energy equation into a pair of first order partial differential equations. By using a new defined vector norm it is possible to obtain a selfadjoint eigenvalue problem for the extended Graetz problem even though the original convective diffusion operator is non-selfadjoint. The obtained exact analytical solutions for the Graetz problem with axial heat conduction are as simple to compute as the related solutions of the parabolic problem. Received on 28 October 1996     

Heat transfer augmentation and pumping power in double-pipe heat exchangers

One of the criteria for evaluating the performance of a heat exchanger with extended surfaces is the pumping power required for a specified heat duty. The results of an experimental project to relate the pumping power to heat transfer augmentation in a double-pipe heat exchanger are reported. The inner, electrically heated pipe was provided with external, rectangular, axial extended surfaces with interruptions. Heat transfer augmentation and friction factors were determined for different configurations with air as the fluid. Starting with continuous fins, cuts were introduced in the fins to give four ratios of the finssegment length to the gap between the segments, and finally all the fins were removed, which resulted in smooth pipes. Five different mass flow rates in two different inner pipes were employed. Lengths, surface areas, and pumping powers for finned pipes are compared with those for smooth pipes. The average heat transfer coefficient increases with an increase in the frequency of the interruptions. For equal heat transfer rates a significant reduction in the lengths can be achieved by interrupted fins. In many cases the reduction in the length is also accompanied by a reduction in the pumping power.     

Convective heat transfer in laser-irradiated graphite

The gas dynamic and thermal processes developing near the surface of graphite after exposure to a 20-nsec laser pulse with an energy E- 0.1–1 J and a wavelength of 0.6943 m are investigated experimentally and by mathematical modeling. The times required for the shock wave to degenerate into an acoustic wave are also considered. Typical density profiles over the axial section of the inhomogeneity are presented for various moments of time. It is noted that the rate of ascent of the thermal inhomogeneity is much higher than the free convection velocity. The convective heat-transfer processes are studied in detail through numerical solution of the system of two-dimensional Navier-Stokes equations. The results of the calculations are in satisfactory agreement with the experimental data.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 180–182, May–June, 1989.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

Finite difference analysis of forced-convection heat transfer in entrance region of a flat rectangular duct

Summary The laminar forced convection heat transfer in the entrance region of a flat rectangular duct is studied. In this region temperature and velocity profiles are simultaneously developed. The basic governing equations of momentum, continuity, and energy are expressed in finite difference form and solved numerically by use of a high speed computer for a mesh network superimposed on the flow field. All fluid properties are assumed to be constant. The cases of uniform constant wall temperature and of uniform constant heat flux from wall to fluid are considered. Nusselt numbers are reported for Prandtl numbers in the range of 0.01 to 50. The exact solution of the energy equation obtained by means of the numerical method is compared with the results of approximate solutions.Nomenclature A surface area of channel walls through which heat is being transferred - a duct half-height - C _p specific heat at constant pressure - D _e equivalent diameter for a duct, 4a - G _Z Graetz number, Re _d Pr/(x/D _e ) - h heat-transfer coefficient, Q/{A(t)} - k thermal conductivity of the fluid - Nu _m average Nusselt number, h _m D _e /k - Nu _x local Nusselt number, h _x D _e /k - Pr Prandtl number, C _p /k - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, (p–p ₀)/u ₀ ² - Q heat-transfer rate - Re _a Reynolds number, u ₀ a/ - Re _d diameter Reynolds number, u ₀ D _e /=u ₀4a/ - t temperature - t ₀ temperature of fluid at entrance section of channel - t ₁ constant wall temperature - t _w wall temperature - u fluid velocity in x-direction - u ₀ fluid velocity at inlet - U dimensionless u velocity, u/u ₀ - v fluid velocity in y-direction - V dimensionless velocity, av/ - x coordinate along channel - X dimensionless x-coordinate, x/(a ² u ₀)=(x/a)/Re _a - X dimensionless x-coordinate defined as x/(D _e ² u ₀)=(x/D _e )/Re _d =X/16 - y coordinate across channel - Y dimensionless y-coordinate, y/a - thermal diffusivity of fluid, k/C _p - kinematic viscosity of fluid - fluid density - dynamic viscosity of fluid - dimensionless temperature, defined by (8), (t–t ₀)/(t ₁–t ₀) for constant wall temperature, k(t–t ₀)/(ag) for constant heat flux case - _b,x dimensionless bulk temperature at any location x, defined by (15) - _w dimensionless wall temperature defined by (8)     

Transient thermal entrance heat transfer in laminar pipe flows with step change in pumping pressure

Analysis is made for the transient heat transfer phenomena in the thermal entrance region of laminar pipe flows. The transient results from both the change in flow field, a step change in pressure gradient from zero to a fixed value, and the change in thermal field, a step change in the inlet temperature. An exponential scheme has been employed to solve the energy equation with the presence of axial heat conduction in the fluid. In order to demonstrate the results more clearly, a modified Nusselt number is introduced. The unsteady axial variations of conventional Nusselt number, modified Nusselt number, bulk fluid temperature and pipe wall temperature are presented for water and air over a wide range of outside heat transfer coefficients. It is observed that the outside heat transfer coefficient has a significant influences on the transient heat transfer processes. The results can be comprehensively interpreted by the interactions among the axial convection, axial diffusion, and radial diffusion.     

Unsteady thermal entrance heat transfer in laminar pipe flows with step change in ambient temperature

A fully implicit upwind finite difference numerical scheme has been proposed to investigate the characteristics of thermal entrance heat transfer in laminar pipe flows subject to a step change in ambient temperature. In order to demonstrate the results more clearly, a modified Nusselt number is introduced. The unsteady axial variations of modified Nusselt number, bulk fluid temperature, and wall temperature and the transient temperature profiles at certain axial locations are presented graphically for various outside heat transfer coefficients. The effects of the outside heat transfer coefficient on the heat transport processes in the flow are examined in detail. The results can be comprehensively explained by the interaction between the upstream convective heat transfer and the diffusion heat transfer in the radial direction. Steady state is reached when the axial convection balances the radial diffusion.     

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.     

Convective heat transfer of nanofluids with correlations

To investigate the convective heat transfer of nanofluids, experiments were performed using silver–water nanofluids under laminar, transition and turbulent flow regimes in a horizontal 4.3 mm inner-diameter tube-in-tube counter-current heat transfer test section. The volume concentration of the nanoparticles varied from 0.3% to 0.9% in steps of 0.3%, and the effects of thermo-physical properties, inlet temperature, volume concentration, and mass flow rate on heat transfer coefficient were investigated. Experiments showed that the suspended nanoparticles remarkably increased the convective heat transfer coefficient, by as much as 28.7% and 69.3% for 0.3% and 0.9% of silver content, respectively. Based on the experimental results a correlation was developed to predict the Nusselt number of the silver–water nanofluid, with ±10% agreement between experiments and prediction.     

Convective heat transfer in a rotating annulus device

Convective heat transfer in a horizontal annulus device rotating around its horizontal axis has been examined. The results show that heat transfer in the annulus depends on the rotational speed. At a certain value of the rotational speed there is only conduction in the annulus. A criterium is given to calculate this rotational speed from the physical properties of the liquids. For the calculation of the heat transfer in the standstill of the annulus two equations are proposed.

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Konvektiver Wärmeübergang in einem rotierenden Ringspalt

Zusammenfassung Der Wärmeübergang bei Konvektion in einem um seine Horizontalachse rotierenden Ringspalt wurde untersucht. Wie die Ergebnisse zeigen, hängt der Wärmeübergang von der Drehzahl ab. Ab einer bestimmten Drehzahl wird Wärme nur noch durch Leitung übertragen. Es wird ein Kriterium angegeben, diese Drehzahl aus den physikalischen Daten der Flüssigkeiten zu berechnen. Zur Berechnung des Wärmeübergangs im Stillstand werden zwei Gleichungen vorgeschlagen.