Unsteady conjugated forced convective heat transfer in a duct with convection from the ambient

A numerical finite difference solution is found for the problem of unsteady, laminar, forced convection heat transfer in a parallel plate duct with finite thermal capacity walls which interact with an ambient medium outside the duct. Response functions are presented for the duct wall temperature, fluid bulk mean temperature, and inside wall surface heat flux as a function of position down the duct and time for a range of the parameters involved. Comparisons are made with the zero thermal capacity wall solution and with quasi-steady results.     

Transient conjugated forced convection in ducts with periodically varying inlet temperature

Transient forced convection for slug flow inside parallel-plate channels and circular ducts including conjugation to the walls is solved analytically and exactly for periodic variation of the inlet temperature. The periodic solution to the problem involved eigenfunctions and eigenvalues of a complex eigenvalue problem. The complex eigenvalue problem is solved by modifying the recently advanced Count Method, and benchmark results are presented for the eigenvalues in tabular form. The amplitude and phase lag of oscillations with respect to the conditions at the inlet are determined for the wall temperature, fluid bulk temperature and heat flux. The results for the cases of both parallel-plate channels and circular ducts are presented in the graphical form as a function of the axial position for different values of the parameters signifying the rate of energy storage in the walls. The effects of walls on damping the amplitude and altering the phase of temperature and heat flux oscillations along the duct are investigated.     

A 2-D INVERSE METHOD FOR SIMULTANEOUS ESTIMATION OF THE INLET TEMPERATURE AND WALL HEAT FLUX IN A LAMINAR CIRCULAR DUCT FLOW

A two-dimensional inverse analysis is presented for the estimation of the inlet temperature of the fluid flow and wall heat flux in a thermally developing hydrodynamically developed laminar flow in a duct. The inverse analysis is based on the temperature reading located at a stream inside the duct at several different points. At the beginning of the study, finite difference methods are employed to discretize the problem, and then a linear inverse model is constructed to identify the unknown conditions. The present approach is to rearrange the matrix forms of the differential governing equation and estimate the inlet temperature of the fluid and unknown surface conditions of the duct. The linear least squares method is adopted to find the solution. The advantage of applying this method in inverse analysis is that no prior information is needed on the functional form of the unknown quantities, no initial guess is required, and the number of iterations in the calculation process is limited to one. The effects of sensor position, magnitude of measurement error, and number of measurements on the accuracy of estimates are examined. The results show that the preferred position of the sensor is closer to the inlet region and only few measuring points are sufficient to estimate the wall heat flux and inlet temperatures of the fluid when the measurement errors are neglected. When the measurement errors an considered, more measuring points are needed in order to increase the congruence of the estimated results to exact solutions.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

Time periodic flow boiling heat transfer of R-134a and associated bubble characteristics in a narrow annular duct due to flow rate oscillation

An experiment is conducted here to investigate the effects of the imposed time periodic refrigerant flow rate oscillation in the form of nearly a triangular wave on refrigeriant R-134a flow boiling heat transfer and associated bubble characteristics in a horizontal narrow annular duct with the duct gap fixed at 2.0 mm. The results indicate that when the imposed heat flux is close to that for the onset of stable flow boiling, intermittent flow boiling appears in which nucleate boiling on the heated surface does not exist in an entire periodic cycle. At somewhat higher heat flux persistent boiling prevails. Besides, the refrigerant flow rate oscillation only slightly affects the time-average boiling curves and heat transfer coefficients. Moreover, the heated wall temperature, bubble departure diameter and frequency, and active nucleation site density are found to oscillate periodically in time as well and at the same frequency as the imposed mass flux oscillation. Furthermore, in the persistent boiling the resulting heated wall temperature oscillation is stronger for a longer period and a larger amplitude of the mass flux oscillation. And for a larger amplitude of the mass flux oscillation, stronger temporal oscillations in the bubble characteristics are noted. The effects of the mass flux oscillation on the size of the departing bubble and active nucleation site density dominate over the bubble departure frequency, causing the heated wall temperature to decrease and heat transfer coefficient to increase at reducing mass flux in the flow boiling, opposing to that in the single-phase flow. But they are only mildly affected by the period of the mass flux oscillation. However, a short time lag in the wall temperature oscillation is also noted. Finally, a flow regime map is provided to delineate the boundaries separating different boiling regimes for the R-134a flow boiling in the annular duct.     

Analysis of the effect of viscous dissipation for laminar flow in stadium-shaped ducts

The fully-developed laminar forced convection of a Newtonian fluid in a duct with stadium-shaped cross section has been analyzed. The effect of viscous dissipation has been taken into account. Three different thermal boundary conditions have been considered: (T) uniform wall temperature distribution; (H1) axially uniform wall heat flux distribution with peripherally uniform wall temperature distribution; (H2) axially and peripherally uniform wall heat flux distribution. The adiabatic-wall boundary condition has also been analyzed as a special case of the H2 boundary condition. The velocity and temperature distributions in the fluid, as well as the Fanning friction factor and the Nusselt number, have been evaluated numerically, by employing a Galerkin finite element method. As expected, the numerical evaluation of the dimensionless temperature distribution and of the Nusselt number reveals that increasing discrepancies between the H1 and H2 boundary conditions exist if the stadium-shaped duct is gradually flattened.     

Heat Transfer of Bingham Fluids in an Annular Duct with Viscous Dissipation

ABSTRACT

Analytical expressions for the velocity and temperature profiles in a fully-developed laminar Poiseuille flow through a concentric annular duct of a Bingham fluid with constant wall heat flux at the inner and outer wall, in the presence of viscous dissipation are deduced and presented. It is found that the proportion of the heat generated by viscous dissipation near the outer wall increases with an increase of the dimensionless flow parameter, and a decrease of the duct radius ratio. The Nusselt numbers are first calculated based on a single bulk temperature for the entire duct cross section. The possibility of performing calculations of the relevant parameters discussed in this work is available via the Supplementary Material as an Excel file. Also in this work a new approach is employed, where two different bulk temperatures are used, one for each side of the radial location in the temperature profile whose derivative is zero. With this new approach the Nusselt number behavior is free of either unphysical discontinuities or negative values. As a consequence, the Nusselt number values better reflect the actual heat transfer coefficient at the walls and are more comparable with the heat transfer inside ducts when the temperature profile is symmetric.     

The effect of uniform lateral mass flux on free convection about a vertical cone embedded in a saturated porous medium

The effect of uniform lateral mass flux on natural convection about a cone embedded in a saturated porous medium is numerically analyzed. The surface is maintained at a uniform wall temperature (UWT) or uniform heat flux (UHF). The transformed governing equations are solved by Keller box method. Numerical data for the dimensionless temperature profiles and the local Nusselt number are presented for a wide range of the mass flux parameter. In general, it has been found that the local surface heat transfer rate increases owing to suction of fluid. This trend reversed for blowing of fluid. The mass flux parameter is found to have a more pronounced effect on the local Nusselt number for the case of UWT than it does for the case of UHF.     

A way to explain the thermal boundary effects on laminar convection through a square duct

A way using the reformulation of the energy conservation equation in terms of heat flux to explain the thermal boundary effects on laminar convective heat transfer through a square duct is presented. For a laminar convection through a square duct, it explains that on the wall surface, the velocity is zero, but convection occurs for uniform wall heat flux (UWHF) boundary in the developing region due to the velocity gradient term; for uniform wall temperature (UWT) boundary, only diffusion process occurs on the wall surface because both velocity and velocity gradient do not contribute to convection; for UWHF, the largest term of the gradient of velocity components (the main flow velocity) on the wall surface takes a role in the convection of the heat flux normal to the wall surface, and this role exists in the fully developed region. Therefore, a stronger convection process occurs for UWHF than for UWT on the wall surface. The thermal boundary effects on the laminar convection inside the flow are also detailed.     

Forced convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.     

Combined forced and free flow in a vertical circular duct subjected to non-axisymmetric wall heating conditions

The fully developed mixed convection flow in a vertical circular duct is investigated analytically, under the assumption of laminar parallel flow. A wall heat flux uniform in the axial direction and dependent on the angular coordinate is considered. As a consequence, the fluid temperature is three dimensional, since it changes in the radial, axial and angular directions. An analytical method based on Fourier series expansions of temperature and velocity fields is adopted to determine the velocity and the temperature distributions as well as the friction factor and the average Nusselt number. The general solution, expressed in terms of Bessel functions, is applied to study a case that has a special importance in technical applications: a duct whose wall is half subject to a uniform heat flux and half adiabatic. The positive and negative threshold values of the ratio between the Grashof number Gr and the Reynolds number Re for the onset of the flow reversal phenomenon are determined. A comparison between the average Nusselt number for the considered non-axisymmetric case and that for the case of a duct subject to a uniform wall heat flux is performed.     

Measurement of Transient Fluid Temperature in a Pipeline

The aim of this paper is to develop a method of determining the heat transfer coefficients on the inner surface of the pipeline and outer surface of the thermometer used to measure the temperature of a fluid flowing under high pressure. The method is based on the solutions to the inverse heat conduction problems for the thermometer and the pipeline wall. The heat transfer coefficients are determined based on the measurement of the temperature of a cylindrical metal thermometer and the temperature of the wall of a cylindrical pipeline. The temperature sensor is located in the pipeline wall close to the inner surface. The correlations for the Nusselt numbers used to determine heat transfer coefficients on the outer surface of the thermometer and the inner surface of the pipeline contain unknown coefficients which are found using the least squares method. The unknown coefficients are selected so that the sum of the squares of differences between the fluid temperature determined based on the measurement of the temperature of the pipeline wall and the fluid temperature obtained from measurements inside the thermometer, calculated for several dozen set time points, is as small as possible.     

Convective heat transfer to water near the critical region in a horizontal square duct

Numerical analysis has been carried out to investigate forced convective heat transfer to water near the critical region in a horizontal square duct. Near the critical point convective heat transfer in the duct is strongly coupled with large variation of thermophysical properties such as density and specific heat. Buoyancy force parameter has also severe variation with fluid temperature and pressure in the duct. There is flow acceleration along the horizontal duct resulted from fluid density decrease due to the heat transfer from the wall. Local heat transfer coefficient has large variation along the inner surface of the duct section and it depends on pressure. Nusselt number on the center of the bottom surface also has a peak where bulk fluid temperature is higher than the pseudocritical temperature and the peak decreases with the increase of pressure. Flow characteristics of velocity, temperature, and local heat transfer coefficient with water properties are presented and analyzed. Nusselt number distributions are also compared with other correlations for various pressures in the duct.     

Subcooled flow boiling heat transfer and associated bubble characteristics of R-134a in a narrow annular duct

Experiments are conducted here to investigate how the channel size affects the subcooled flow boiling heat transfer and associated bubble characteristics of refrigerant R-134a in a horizontal narrow annular duct. The gap of the duct is fixed at 1.0 and 2.0 mm in this study. From the measured boiling curves, the temperature undershoot at ONB is found to be relatively significant for the subcooled flow boiling of R-134a in the duct. The R-134a subcooled flow boiling heat transfer coefficient increases with a reduction in the gap size, but decreases with an increase in the inlet liquid subcooling. Besides, raising the imposed heat flux can cause a substantial increase in the subcooled boiling heat transfer coefficient. However, the effects of the refrigerant mass flux and saturated temperature on the boiling heat transfer coefficient are small in the narrow duct. Visualization of the subcooled flow boiling processes reveals that the bubbles are suppressed to become smaller and less dense by raising the refrigerant mass flux and inlet subcooling. Moreover, raising the imposed heat flux significantly increases the bubble population, coalescence and departure frequency. The increase in the bubble departure frequency by reducing the duct size is due to the rising wall shear stress of the liquid flow, and at a high imposed heat flux many bubbles generated from the cavities on the heating surface tend to merge together to form big bubbles. Correlation for the present subcooled flow boiling heat transfer data of R-134a in the narrow annular duct is proposed. Additionally, the present data for some quantitative bubble characteristics such as the mean bubble departure diameter and frequency and the active nucleation site density are also correlated.     

Startup of a horizontal lithium-molybdenum heat pipe from a frozen state

Results of the simulation of the startup from a frozen state of a molybdenum heat pipe with lithium working fluid are presented and discussed. The 1.8-m-long heat pipe was tested in the horizontal position and had a liquid annular space between the porous wick and the wall. The 30-cm-long evaporator section was inductively heated and the 147-cm-long condenser was cooled by thermal radiation to the quartz tube enclosing the heat pipe and to the ambient. The space between the quartz tube and the heat pipe was evacuated in order to minimize heat losses by convection and conduction. Model results on the progression of the thaw front, liquid pooling at the end of the condenser, and the wall temperature along the heat pipe were found to be in good agreement with experimental measurements. Results showed that, as the heat pipe reached quasi-steady state operation at an evaporator wall temperature of 1550 K, the wall temperature near the end of the condenser dropped precipitously by 450 K, because of the formation of a 8.3-cm-long liquid plug and the end heat losses in the condenser.     

Numerical investigation of the bubble growth in horizontal rectangular microchannels

To explore the mechanism of flow boiling in microchannels, the processes of a single-vapor bubble evaporating and two lateral bubbles merging in a 2D microchannel are investigated. The temperature recovery model based on volume of fluid method is adopted to perform the flow boiling phenomena. The effects of wall superheat, Reynolds number, contact angle, surface tension, and two-bubble merger on heat transfer are discussed. Wall superheat dominates the bubble growth and is roughly proportional to wall heat flux. The update of velocity and temperature fields’ distribution in the channel increases with increasing inflow Reynolds number, which improves the wall heat flux markedly. Besides, the area of thin liquid film between the wall and the bubble is enlarged by reducing the contact angle, thus, expanding the wall heat flux several times compared with the single-phase cross section. However, variation of surface tension (0.0589, 0.1?N/m) is found to be insignificant.     

Conjugate Heat Transfer Analysis for a Finite Thickness Cylinder in a Hypersonic Flow

Prediction of surface heating rates is of prime importance for the hypersonic flow regime. Experimental and conventional computational efforts overlook the heat transfer phenomenon in the solid due to the rigid assumptions involved in the solution methodologies. In order to address this fact, conjugate heat transfer (CHT) studies are carried out using various coupling techniques to examine their implementation abilities. Three types of solution methodologies are adopted, namely, decoupled, strongly coupled, and loosely coupled analysis. This study is also focused on looking into the effect of a hypersonic flow field on wall heat flux for a finite thickness insulating cylinder at moderately large time scales. Increase in wall temperature and decrease in surface heat flux have been noticed using strong and loose coupling techniques with an increase in simulation time. Decoupled fluid and solid domain analysis is found to be useful for typical shock tunnel test durations (～1 ms) while investigations with loose coupling techniques are advisable for time scales corresponding to flight testing (～1 s). Efforts are also made to reason the discrimination in prediction of stagnation point heat flux using conventional computational and experimental analysis.     

Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts

The present study concentrates on the effects of viscous dissipation and the yield shear stress on the asymptotic behaviour of the laminar forced convection in a circular duct for a Bingham fluid. It is supposed that the physical properties are constant and the axial conduction is negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial distributions of wall heat flux which yield a thermally developed region. It is shown that if the asymptotic value of wall heat flux distribution is vanishes, the asymptotic value of the Nusselt number is zero. The case of the asymptotic wall heat flux distribution non-vanishing giving a value of the Nusselt number dependent on the Brinkman number and on the dimensionless radius of the plug flow region was also analysed. For an infinite asymptotic value of wall heat flux distributions, the asymptotic value of the Nusselt number depends on the dimensionless radius of the plug flow region and on the dimensionless parameter which depends on the asymptotic behaviour of the wall heat flux. The condition of uniform wall temperature and convection with an external isothermal fluid were also considered. The comparison with other existing solutions in the literature in the Newtonian case is analysed.     

污垢对管内层流换热性能影响的热力学分析和评价

采用热力学第二定律，分别在恒壁温和恒热流两种典型工况下分析了污垢对管内层流换热性能的影响；引入单位传热量的熵增率对污垢管道的热力学性能进行了评价；讨论了管内流体雷诺数(无污垢时)、量纲为1的入口换热温差、量纲为1的热流密度和污垢层厚度等参数对单位传热量熵增率的影响；并把结果和紊流时的对应工况进行了比较。结果可为工程上换热设备的优化设计提供依据。     

Prandtl number scaling of the unsteady natural convection boundary layer adjacent to a vertical flat plate for Pr>1 subject to ramp surface heat flux

It is found in the literature that the existing scaling results for the boundary layer thickness, velocity and steady state time for the natural convection flow over an evenly heated plate provide a very poor prediction of the Prandtl number dependency of the flow. However, those scalings provide a good prediction of two other governing parameters’ dependency, the Rayleigh number and the aspect ratio. Therefore, an improved scaling analysis using a triple-layer integral approach and direct numerical simulations have been performed for the natural convection boundary layer along a semi-infinite flat plate with uniform surface heat flux. This heat flux is a ramp function of time, where the temperature gradient on the surface increases with time up to some specific time and then remains constant. The growth of the boundary layer strongly depends on the ramp time. If the ramp time is sufficiently long, the boundary layer reaches a quasi-steady mode before the growth of the temperature gradient is completed. In this mode, the thermal boundary layer at first grows in thickness and then contracts with increasing time. However, if the ramp time is sufficiently short, the boundary layer develops differently, but after the wall temperature gradient growth is completed, the boundary layer develops as though the startup had been instantaneous.