Linear Temperature Profile Research Articles

Incorporating device and experimental loss mechanisms in AMR modelling

Active magnetic regenerators need magnetocaloric materials of sufficient size, shape, and quantity to create a matrix for characterization in a test apparatus. Limited availability of novel magnetocaloric materials in a suitable form leads to regenerator beds which are smaller than desired. When implemented in a test apparatus, result in unwanted loss mechanisms such as void spaces between the regenerator and heat exchangers. The loss of cooling power due to dead volume losses and thermal interactions with the ambient environment when using small regenerators is investigated with experimental and 1D numerical methods. Due to uncertainty regarding the casing thermal state, a range of cases for casing temperature are numerically tested including: a fixed temperature and a linear temperature profile from hot to cold end using the simulation boundary conditions. Including dead volume sections improves simulation results while casing heat leaks impact simulations with a net cooling power up to 2.5 W.

WKB solutions to the quasi 1-D acoustic wave equation in ducts with non-uniform cross-section and inhomogeneous mean flow properties – Acoustic field and combustion instability

Linear modal analysis based on the spatio-temporal harmonic representation of acoustic fluctuations is a well-known reduced order method to predict combustion instability frequencies. However, a limitation of this method is that it is only applicable to domains with homogeneous mean properties. The Wentzel-Kramers-Brillouin (WKB) method facilitates the development of approximate analytical solutions to the one-dimensional (1-D) acoustic wave equation in domains with inhomogeneous mean properties. The “classical” form of the WKB solution to the wave equation is based on the assumption of high frequency (or, short wavelength), and is applicable to uniform cross-section ducts with a mean temperature gradient, but no mean flow. Subsequently, the WKB method was extended to include uniform mean velocity as well. In the current study, we derive a WKB-type solution to the generalized Helmholtz equation in quasi 1-D ducts with non-uniform cross-sectional areas and inhomogeneities in mean flow properties such as the velocity, temperature, density, and pressure. The WKB solution is applicable under the assumptions of high frequency and slowly varying mean properties. In deriving this solution, a novel approach has been developed to relate the first and second spatial derivatives of the axial velocity fluctuations to the pressure fluctuations (and its derivatives).A second WKB-type solution (referred to as the WKB2 solution) is also obtained by relaxing the slowly-varying mean property assumption, but retaining the high frequency approximation. Thirdly, the Helmholtz equation for pressure is solved numerically without invoking the two WKB assumptions. The WKB, WKB2 and numerical solutions are compared for a number of cases involving combinations of non-uniform duct areas and axially varying mean properties. Finally, we apply the WKB solution to predict the longitudinal combustion instabilities in a 1-D combustor characterized by a sudden expansion, with uniform mean flow in each of the two ducts comprising the combustor, and a linear temperature profile downstream of a compact, planar flame.

Thermo-elastic analysis of multilayered plates and shells based on 1D and 3D heat conduction problems

Abstract The present work shows a generic 3D exact shell solution for the thermo-mechanical analysis of a heterogeneous group of one- and multi-layered isotropic, composite and sandwich structures. Plates, cylinders, cylindrical and spherical shells can be investigated using orthogonal mixed curvilinear coordinates. The 3D equilibrium equations for spherical shells automatically degenerate in those for simpler geometries. The elastic part of the proposed 3D model is based on a consolidated layer-wise exact solution which uses the exponential matrix method to solve the equilibrium differential equations through the thickness direction. The closed-form solution is obtained assuming simply-supported boundary conditions and harmonic forms for displacement and temperature fields. The temperature amplitudes are imposed at the top and bottom external surfaces in steady-state conditions. Therefore, the temperature profile can be evaluated through the thickness direction in three different ways: – calculation of the temperature profile via the steady-state 3D Fourier heat conduction equation; – evaluation of the temperature profile using the steady-state simplified 1D version of the Fourier heat conduction equation; – a priori assumed linear temperature profile through the entire thickness direction ranging from the bottom temperature value to the top temperature value. Once the temperature profile is defined at each thickness coordinate, it is considered as a known term in the 3D differential equilibrium equations. The obtained system consists in a set of non-homogeneous second order differential equilibrium equations which can be solved introducing appropriate mathematical layers. After a reduction to a first order differential equation system, the exponential matrix method is used to calculate both the general and the particular solutions. The effects of the temperature field on the static response of plates and shells are evaluated in terms of displacements and stresses. The proposed solution will be validated using reference results available in the literature. Then, new analyses will be presented for different thickness ratios, geometries, lamination schemes, materials and temperature values at the external surfaces. Results will demonstrate the importance in the 3D shell model of both the correct definition of the elastic part and the appropriate evaluation of the temperature profile through the thickness of the structure. .

Heat transfer coefficient measurement of LN2 and GN2 in a microchannel at low Reynolds flow

The heat transfer coefficients of single-phase fluids in the laminar flow regime have been studied for decades. However, inconsistent results are found in the literature. The common finding is that the Nusselt number is dependent on the Reynolds number in the laminar flow regime, which is contrary to laminar flow heat transfer theory. Recently, researchers indicated that axial conduction in the wall of the microchannel can affect the measurement. However, there have not been thorough studies that demonstrate consistency or lack thereof between experiment and theory. This study provides an experimental investigation on heat transfer performance of gaseous and liquid nitrogen flow through microchannels with hydraulic diameters of 110 μm and 180 μm. A model has been developed to investigate heat transfer in a microchannel from which analysis shows that the temperature profile of the fluid and wall change non-linearly along the length of the microchannel when the flow rate is low (e.g., Re < 1000). The non-linear temperature profile conflicts with the assumption of a linear temperature profile commensurate with the traditional Nusselt number estimation method, which leads to dependency on the Reynolds number. Comparison between the experiment and numerical model of the present work validates the conclusion that the heat transfer coefficient is uniform within the laminar flow regime (Re < 2000) for microchannels.

Simplified expressions of the transfer coefficients on a partially wet absorber tube

A simplified analytical solution of the governing differential equation for falling film absorption around a smooth horizontal tube is developed and compared to numerical results and experimental data from literature. Besides the assumption of a linear temperature profile, linear interfacial equilibrium together with the hypotheses of small mass-penetration are applied at the film interface to reach closed analytical expressions of the temperature and mass fraction fields, which can be used to estimate the local and average heat and mass transfer coefficients within falling film absorbers and generators. The noteworthy agreement with numerical results supports the simplifying assumptions introduced, and the inclusion of partial-wetting effects brings the results closer to the experimental data from previous literature, widening the applicability of this formulation.

Analysis of Infrared Radiation at an Air-Water Interface

The problem of determining the strength of the infrared radiation from an air-water interface has been addressed analytically. The approach taken here is to express the Planck spectrum as a linear function of the temperature, an approximation valid for small variations of the temperature from the surface temperature, and to assume a linear temperature profile across the thermal boundary layer. The main result shows that the deviation of the surface radiation intensity from the Planck spectrum due solely to thermal stratification, is linearly proportional to the temperature change across the thermal boundary layer and the optical depth, but is inversely proportional to the thermal boundary layer thickness. This signal was shown to be about one order of magnitude greater than the noise level expected from modern CCD IR sensors at a wavelength of about 3.8 microns. It is suggested that controlled laboratory experiments be conducted to verify these theoretical estimates.

Development and validation of an analytical formulation of the Nusselt and Sherwood numbers on a partially wetted absorber tube

With reference to a simplified analytical solution of the governing differential equation for falling film absorption, the corresponding formulation of the average Nusselt and Sherwood numbers obtained around a smooth horizontal tube is validated through a dedicated experimental investigation and data from previous literature. The assumption of a linear temperature profile together with the average interfacial mass fraction gradient and the hypotheses of small penetration for physical absorption are applied at the film interface to reach closed analytical expressions of average Nusselt and Sherwood numbers within falling film absorbers and generators. The inclusion of partial-wetting effects brings the results closer to the measurements and widens the applicability of these expressions.

Conjugate thermal creep flow in a thin microchannel

In the present work, we analyze asymptotically and numerically the conjugate heat transfer between a rarified gas flow and the lower wall of a thin horizontal microchannel. The laminar motion of the gas is originated only by the thermal creep or transpiration effect on the lower wall of the microchannel. It is well known the need to impose, in general, a variable temperature regime at the lower wall to induce the transpiration effect. Usually, it can be reached by setting a linear temperature profile as a boundary condition. However, in our case, we prefer to avoid this simplification taking into account that in practical applications, the temperature profile at the lower wall can be unknown. This case can occur, for instance, in a heat sink or a similar device with a well defined heat dissipation rate. Under this physical configuration, we can assume then that the bottom or external face of this heat sink with finite thermal conductivity is exposed to a uniform heat flux. On the other hand, the upper wall of the microchannel is subjected to a prescribed condition. The above conditions are sufficient to consider the simultaneous or conjugate heat transfer analysis of the heat conduction equation for the heat sink and the mass, momentum and energy equations for the gaseous-phase. Resulting governing equations are written in dimensionless form, assuming that the Reynolds number associated with the characteristic velocity of the thermal creep and the aspect ratio of the microchannel, are both very small. The velocity and temperature profiles for the gas phase and the temperature profiles for the solid wall are predicted as functions of the involved dimensionless parameters and the main results confirm that the phenomenon of conjugate thermal creep exists whenever the temperature of the lower wall varies linearly or nonlinearly.

Finite element computation of multi-physical micropolar transport phenomena from an inclined moving plate in porous media

Non-Newtonian flows arise in numerous industrial transport processes including materials fabrication systems. Micropolar theory offers an excellent mechanism for exploring the fluid dynamics of new non-Newtonian materials which possess internal microstructure. Magnetic fields may also be used for controlling electrically-conducting polymeric flows. To explore numerical simulation of transport in rheological materials processing, in the current paper, a finite element computational solution is presented for magnetohydrodynamic, incompressible, dissipative, radiative and chemically-reacting micropolar fluid flow, heat and mass transfer adjacent to an inclined porous plate embedded in a saturated homogenous porous medium. Heat generation/absorption effects are included. Rosseland’s diffusion approximation is used to describe the radiative heat flux in the energy equation. A Darcy model is employed to simulate drag effects in the porous medium. The governing transport equations are rendered into non-dimensional form under the assumption of low Reynolds number and also low magnetic Reynolds number. Using a Galerkin formulation with a weighted residual scheme, finite element solutions are presented to the boundary value problem. The influence of plate inclination, Eringen coupling number, radiation-conduction number, heat absorption/generation parameter, chemical reaction parameter, plate moving velocity parameter, magnetic parameter, thermal Grashof number, species (solutal) Grashof number, permeability parameter, Eckert number on linear velocity, micro-rotation, temperature and concentration profiles. Furthermore, the influence of selected thermo-physical parameters on friction factor, surface heat transfer and mass transfer rate is also tabulated. The finite element solutions are verified with solutions from several limiting cases in the literature. Interesting features in the flow are identified and interpreted.

Unstable buoyant flow in a vertical porous layer with convective boundary conditions

The instability of natural convection in a vertical porous layer is analysed. The plane parallel boundaries of the vertical layer are modelled as open and subject to Robin-type temperature conditions. The latter conditions describe heat transfer to the external environment, with a finite conductance. The basic state is given by a stationary fully-developed flow with linear velocity and temperature profiles. Instability arises when the Darcy-Rayleigh number exceeds its critical value. This value depends on the Biot number associated with the temperature boundary conditions. The most unstable normal modes turn out to be transverse. By solving numerically the stability eigenvalue problem, it is shown that the critical Darcy-Rayleigh number is a decreasing function of the Biot number when the Biot number is sufficiently small. For larger Biot numbers, a minimum is attained, and then the critical Darcy-Rayleigh number becomes an increasing function of the Biot number.

Internal Gravity Waves in the Lower Thermosphere with Linear Temperature Profile: Theory and Experiment

The problem of internal gravity waves in a medium with linear altitude profile of equilibrium temperature as applied to the experiments related to the creation of artificial periodic irregularities and diagnostics of the neutral atmosphere at the altitudes of the E region is considered. A solution to the initial linearized system of equations for weak disturbances of the thermospheric parameters, such as pressure, density, temperature, and velocity of the medium, is obtained. Among a large array of experimental data on the temperature profiles of the neutral component, the sessions in which the altitude dependence of the equilibrium temperature was well approximated by a linear function were selected. The characteristics of internal gravity waves obtained for such a temperature profile are compared with the results of measuring the atmospheric parameters. Satisfactory agreement between the theoretical and experimental values of the quantities being determined is established, which confirms the validity of the chosen model.

Modeling and Validation of Temperature and Void Effects on Reactivity Experiments at the Missouri S&T Research Reactor

To validate an MCNP5 model of the Missouri S&T Research Reactor (MSTR), temperature and void effects on reactivity experiments were simulated and performed. We compared the keff of the modeled reactor mirroring the position of all control rods to the actual critical reactor (keff = 1.00000). In the simulation we modeled three different scenarios. In the first two scenarios, the reactor is modeled as isothermal at two different temperatures (measured experimentally near the core), and in the third scenario, we split the core into bottom and top parts and used interpolated values for the temperatures of both halves. The model predicted keff’s for the “critical reactor” between 1.00234 and 1.00248 (±0.00018) when using as temperature the experimental thermocouple readings at the top of the core and keff’s between 1.00296 to 1.00383 (±0.00018) when using the temperature of thermocouple readings at the bottom of the core. In the third experiment, a linear vertical temperature profile was included in the model (only top and bottom of the core), and the model predicted keff’s between 1.00218 and 1.00302 (±0.00018). The keff modeled and experimental values differed by as much as 0.40%. A void coefficient of the reactivity experiment was also simulated introducing a void tube in the model and the control rods made to mirror the critical experimental reactor with an identical void.

The one-dimensional acoustic field in a duct with arbitrary mean axial temperature gradient and mean flow

This paper presents an analytical solution for the one-dimensional acoustic field in a duct with arbitrary mean temperature gradient and mean flow. A wave equation for the pressure perturbation is derived which relies on very few assumptions. An analytical solution for this is derived using an adapted WKB approximation. The solution is a superposition of waves travelling in either direction and thus provides physical insight. It is also very easy to calculate. The proposed solution is applied to ducts with a mean temperature profile which varies axially with (i) a linear and (ii) a partial sine wave profile: predictions are compared to those obtained by numerically solving the linearised Euler equations (LEEs). The analytical solution reproduces the acoustic field very accurately across a wide range of flow conditions which span both low and moderate-to-high subsonic Mach numbers. It always performs well when the frequency exceeds a certain value; when the mean temperature profile is linear, it also performs well to very low frequencies. This increased frequency range for linear mean temperature profiles leads to its application to more complicated profiles in a piecewise linear manner, axially segmenting the temperature profile into regions that can be approximated as linear. The acoustic field is predicted very accurately as long as enough segmentation points are used and the condition for the linear mean temperature profile is satisfied: |k0|>|α|, where k0 is the local wave number when there is no mean flow and α is the normalised mean density gradient. The proposed solution is extensively compared to previous analytical solutions, and is found to be more accurate and reliable, especially at higher Mach numbers. The entropy wave generated by communication between the acoustic waves and the distributed mean temperature zone is calculated using the LEEs. It is found to remain very small across all operating conditions, such that both the entropy wave and its impact on the acoustic field can be neglected.

Multiphysics simulations of thermoelectric generator modules with cold and hot blocks and effects of some factors

Transient and steady-state multiphysics numerical simulations are performed to investigate the thermal and electrical performances of a thermoelectric generator (TEG) module placed between hot and cold blocks. Effects of heat radiation, leg length and Seebeck coefficient on the TEG thermal and electrical performances are identified. A new correlation for the Seebeck coefficient with temperature is proposed. Radiation effects on the thermal and electric performances are found to be negligible under both transient and steady-state conditions. The leg length of the TEG module shows a considerable influence on the electrical performance but little on the thermal performance under transient conditions. A nearly linear temperature profile on a leg of the TEG module is identified. The temperature profile of the substrate surfaces is non-uniform, especially in the contacted areas between the straps (tabs) and the substrates.

Numerical analysis on transient behaviors of slag layer in an entrained-flow coal gasifier

In an entrained-flow coal gasifier with water-cooled walls, it is important to maintain an appropriate range of slag thickness on the wall for protecting the refractory and preventing blockage at the slag tap. Owing to the difficulty in measurement within the gasifier, the heat absorption on the wall and downstream syngas composition are monitored as indicators of slag thickness and gas temperature. However, there is a certain delay between these parameters that can be highlighted by the dynamic modeling of slag behaviors on the wall and reactions/flow in the gas regime. In this study, a new dynamic model for the slag layer was developed, which solves for the energy, heat transfer, and flow of the solid and liquid slag layers including the transformation between the two phases. The model was applied to a commercial gasifier varying operating parameters, and the transient behaviors of the slag layers were analyzed focusing on the thickness at the slag tap and the heat absorption on the wall. When the operating conditions changed abruptly, the slag layer required 10min or longer to reach the new steady state. The thinning of the slag layer was faster than its thickening. In addition, the response of heat absorption was significantly faster than that for the slag thickness at the slag tap. The characteristic times of the slag layer were shorter with changes in gas temperature than with changes in the mass rate and temperature of the slag deposition. During the transient states, the temperature profile within the liquid slag became non-linear, which suggests that existing models assuming a linear temperature profile may not be appropriate for dynamic simulations.

Heat and Mass Transfer on the Unsteady MHD Flow of Chemically Reacting Micropolar Fluid with Radiation and Joule Heating

To explore numerical simulation of transport in rheological materials processing, in the current paper, a finite element computational solution is presented for magnetohydrodynamic (MHD), incompressible, radiative and chemically-reacting micropolar fluid flow, heat and mass transfer adjacent to a vertical porous plate embedded in a saturated homogenous porous medium. Rosseland’s diffusion approximation is used to describe the radiative heat flux in the energy equation. A Darcy model is employed for the porous medium. The homogeneous chemical reaction of first order is accounted for in the mass diffusion equation. The numerical solutions of the system of non-linear partial differential equations which are rendered into non-dimensional form are obtained using a Galerkin formulation with a weighted residual scheme. The impact of Eringen coupling number, radiation-conduction number, chemical reaction parameter, plate moving velocity parameter, magnetic parameter, thermal Grashof number, species (solutal) Grashof number, permeability parameter, Eckert number on linear velocity, micro-rotation, temperature and concentration profiles. Furthermore, the influence of selected thermo-physical parameters on friction factor, surface heat transfer and mass transfer rate is also tabulated. The finite element solutions are verified with solutions from several limiting cases in the literature. Interesting features in the flow are identified and interpreted.

Thermodynamic quantities of the low-density gas in the weakly nonequilibrium heat-conduction steady state in the linear temperature profile approximation

Thermodynamic quantities of the low-density gas in the weakly nonequilibrium heat-conduction steady state in the linear temperature profile approximation

A variable kinematic shell formulation applied to thermal stress of laminated structures

ABSTRACTIn this article, the thermoelastic static analysis of multilayered shell structure is performed using some advanced theories, obtained by expanding the unknown displacement variables along the thickness direction using equivalent-single-layer (ESL) models, layer-wise (LW) models, and variable-kinematic models. The variable-kinematic models permit to reduce the computational cost of the analyses grouping some layers of the multilayered structure with ESL models and keeping the LW models in other zones of the multilayer. This model is here extended for the static analysis of uncoupled thermomechanical problems. The results obtained with the classical assumed linear temperature profile along the thickness of the shell are compared with those achieved with the calculated temperature profile solving the Fourier heat conduction equation. The used refined models are grouped in the Carrera’s unified formulation (CUF), and they accurately describe the displacement field and the stress distributions along the thickness of the multilayered shell. The shell element has nine nodes, and the mixed interpolation of tensorial components method is used to contrast the membrane and shear locking phenomenon. The governing equations are derived from the principle of virtual displacement, and the finite element method is used to solve them. Cross-ply plates and shells with simply supported edges, subjected to bisinusoidal thermal load are analyzed. Various aspect ratios and radius to thickness ratios are considered. The results, obtained with different theories within CUF context, are compared with the elasticity solutions given in the literature. From the results, it is possible to conclude that the shell element based on the CUF is very efficient in the study of thermomechanical problems of composite structures. The variable-kinematic models combining the ESL with the LW models permit to have a reduction of the computational costs, with respect with the full LW models, preserving the accuracy of the results in localized layers.

Assessment with computational fluid dynamics of the effects of baffle clearances on the shell side flow in a shell and tube heat exchanger

This work presents an analysis of the shell side flow in a shell and tube heat exchanger using Computational Fluid Dynamics (CFD). The heat exchanger has been designed with the software HTRI Xchanger Suite®, while CFD simulations have been performed with the computational package ANSYS Fluent 15.0. Both k-ε and SST turbulence models have been assessed and the SST model has shown to provide more accurate results for temperature prediction. Instead of using an uniform temperature on the tubes’ walls, a linear temperature profile (provided by HTRI Xchanger Suite®) was used as boundary condition for the CFD simulations. Two single-segmental-baffle geometries have been considered: with and without baffle clearances. Simulations results with the SST turbulence model regarding mean velocity, temperature and pressure profiles in the shell side have been analysed and compared with those provided by HTRI Xchanger Suite®. The outlet temperature predicted with CFD for the geometry with clearances differed from that provided by HTRI Xchanger Suite® only by 0.12K. Although leakage streams represent a loss in thermal performance, the results clearly showed smaller recirculation zones and lower temperature peaks on the geometry with baffle clearances, when compared to the same heat exchanger without them. Such peaks could reach values way beyond the specified outlet temperature, in the case where the cold fluid is in the shell side. In the present case, for instance, the highest temperature inside the heat exchanger overpasses the outlet temperature about 8K. Moreover, pressure drop was about 40% smaller when clearances were considered. Therefore, the results highlight an important role of baffle clearances, not yet discussed, which might be crucial for processes with maximum temperature limits, such as those with crude oil.

Complex bifurcations in Bénard–Marangoni convection

We study the dynamics of a system defined by the Navier–Stokes equations for a non-compressible fluid with Marangoni boundary conditions in the two-dimensional case. We show that more complicated bifurcations can appear in this system for a certain nonlinear temperature profile as compared to bifurcations in the classical Rayleigh–Bénard and Bénard–Marangoni systems with simple linear vertical temperature profiles. In terms of the Bénard–Marangoni convection, the obtained mathematical results lead to our understanding of complex spatial patterns at a free liquid surface, which can be induced by a complicated profile of temperature or a chemical concentration at that surface. In addition, we discuss some possible applications of the results to turbulence theory and climate science.