High Biot Numbers Research Articles

Thermally developing flow inside a porous-filled channel in the presence of internal heat generation under local thermal non-equilibrium condition: A perturbation analysis

In the present work, a perturbation analysis is performed to study the thermally developing forced convection heat transfer inside a channel filled with a porous material under local thermal non-equilibrium (LTNE) condition. Internal heat generations within the solid and fluid phases are considered. Channel walls are subjected to a constant heat flux. It is assumed that there is a small temperature difference between the fluid and the solid phases of the porous material. So, performing a perturbation analysis enables us to avoid utilizing models for the constant wall heat flux boundary condition to investigate the hydrothermal behavior of the system. Therefore, analytical solutions are developed for temperature difference between the solid and the fluid phases as well as the local Nusselt number in the porous medium. Effects of pertinent parameters such as dimensionless axial length, Biot number, effective thermal conductivity ratio and dimensionless heat generation parameters on the Nusselt number are discussed. To further clarify the validity of the solution provided, the obtained results are compared with the solutions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. Results show that both the Nusselt number and the thermal entry length increase with the increase of thermal conductivity ratio. The Nusselt number and the thermal entry length are found to decrease with the increase of the internal heat generation of the solid phase. It is further observed that the Nusselt number and the thermal entry length are less sensitive to the solid internal heat generation at high Biot numbers. Finally, it is found that when the effective thermal conductivity ratio tends to infinity, the thermal entry length tends to zero for high Biot numbers.

Peristaltic flow of a reactive viscous fluid through a porous saturated channel and convective cooling conditions

This article addresses the heat transfer in a peristaltic flow of a reactive combustible viscous fluid through a porous saturated medium. The flow here is induced because of travelling waves along the channel walls. It is assumed that exothermic chemical reactions take place within the channel under the Arrhenius kinetics and the convective heat exchange with the ambient medium at the surfaces of the channel walls follows Newton’s law of cooling. The analysis is carried out in the presence of viscous dissipation and without consumption of the material. The governing equations are formulated by employing the long-wavelength approximation. Closed-form solutions for the stream function, axial velocity, and axial pressure gradient are obtained. It is found that the temperature decreases at high Biot numbers, and the Nusselt number increases with increasing reaction parameter. The Biot number and reaction parameter produce the opposite effects on the Nusselt number.

Comparison between Two Local Thermal Non Equilibrium Criteria in Forced Convection through a Porous Channel

Two criteria are used and compared to investigate the local thermal equilibrium assumption in a forced convection through a porous channel. The first criterion is based on the maximum local temperature difference between the solid and fluid phases, while the second is based on the average of the local differences between the temperature of the solid phase and the fluid phase. For this purpose, the momentum and energy equations based on the Darcy-Brinkman-Forchheimer and the local thermal non equilibrium models are solved numerically using the finite volume method. The analysis focused on searching thermophysical parameters ranges which validate local thermal equilibrium hypothesis. Thus, by using the two criteria, the obtained results mainly revealed that this local thermal equilibrium assumption is verified for low thermal conductivity ratio and Reynolds number values and for high interstitial Biot number and porosity, while it is unfavorably affected by the high values of Prandtl number. However, it is also found that the parameters ranges corresponding to the local equilibrium validity depends on the selected local thermal non equilibrium criterion.

Fluid Flow and Heat Transfer with Phase Change and Local Thermal Non-equilibrium in Vertical Porous Channels

This article investigates the multiphase flow in a porous channel, heated at one section of the wall. For non-isothermal flow with phase-change, a finite-volume solver in MATLAB® is used with a two-phase mixture model with local thermal non-equilibrium , i.e., allowing for different solid- and fluid temperatures locally. The effect of gravity is examined for aiding and opposing flow. The effects of Stanton number of evaporation, Rayleigh number, Peclet number and Biot number on fluid flow and heat transfer for steady case are discussed. For these characteristic numbers and aiding flow, the largest influence on the local shift of the biphasic zone is with Stanton number, Peclet number and Biot number. The minimal saturation in the domain is correlated with increasing Stanton number, Peclet number or Biot number. The Rayleigh number, however, has an opposite effect on minimal saturation. The displacement effect of generated vapor is discussed for high Stanton numbers, Peclet numbers and Biot numbers. Also, we show that for sufficiently high heat input, opposing flow yields large differences in position of the biphasic zone and minimal saturation. Applicability of local thermal equilibrium for cases with phase change is shown to introduce a large error, not only for small Biot numbers or large Reynolds numbers.

Properties and appropriate conditions of stress reduction factor and thermal shock resistance parameters for ceramics

Through introducing the analytical solution of the transient heat conduction problem of the plate with convection into the thermal stress field model of the elastic plate, the stress reduction factor is presented explicitly in its dimensionless form. A new stress reduction factor is introduced for the purpose of comparison. The properties and appropriate conditions of the stress reduction factor, the first and second thermal shock resistance (TSR) parameters for the high and low Biot numbers, respectively, and the approximation formulas for the intermediate Biot number-interval are discussed. To investigate the TSR of ceramics more accurately, it is recommended to combine the heat transfer theory with the theory of thermoelasticity or fracture mechanics or use a numerical method. The critical rupture temperature difference and the critical rupture dimensionless time can be used to characterize the TSR of ceramics intuitively and legibly.

Scaling of Thermal Positioning in Microscale and Nanoscale Bridge Structures

Heat transfer in a thermally positioned doubly clamped bridge is simulated to obtain a universal scaling for the behavior of microscale and nanoscale bridge structures over a range of dimensions, materials, ambient heat transfer conditions, and heat loads. The simulations use both free molecular and continuum models to define the heat transfer coefficient, h. Two systems are compared: one doubly clamped beam with a length of 100 μm, a width of 10 μm, and a thickness of 3 μm, and a second beam with a length of 10 μm, a width of 1 μm, and a thickness of 300 nm, in the air at a pressure from 0.01 Pa to 2 MPa. The simulations are performed for three materials: crystalline silicon, silicon carbide, and chemical vapor deposition (CVD) diamond. The numerical results show that the displacement and the response of thermally positioned nanoscale devices are strongly influenced by ambient cooling. The displacement depends on the material properties, the geometry of the beam, and the heat transfer coefficient. These results can be collapsed into a single dimensionless center displacement, δ* = δk/q″αl2, which depends on the Biot number and the system geometry. The center displacement of the system increases significantly as the bridge length increases, while these variations are negligible when the bridge width and thickness change. In the free molecular model, the center displacement varies significantly with the pressure at high Biot numbers, while it does not depend on cooling gas pressure in the continuum case. The significant variation of center displacement starts at Biot number of 0.1, which occurs at gas pressure of 27 kPa in nanoscale. As the Biot number increases, the dimensionless displacement decreases. The continuum-level effects are scaled with the statistical mechanics effects. Comparison of the dimensionless displacement with the thermal vibration in the system shows that CVD diamond systems may have displacements that are at the level of the thermal noise, while silicon carbide systems will have a higher displacement ratios.

Spatial-resolution, lumped-capacitance thermal model for cylindrical Li-ion batteries under high Biot number conditions

The time-efficient yet accurate thermal modeling of the battery cells for electric and hybrid electric vehicles is essential improving the performance, safety, and lifetime of the battery system. This paper presents a spatial-resolution, lumped-capacitance (LC) thermal model for cylindrical battery cells under high Biot number (Bi⩾1) conditions where the classical LC thermal model is generally inapplicable because of a significant temperature variation in the cell volume. The spatial-resolution LC model was formulated using zero- and first-order Hermite integral approximations. For model validation, a one-dimensional, transient analytical (exact) solution using Green functions was obtained for a cylindrical Li-ion battery cell with uniform volumetric battery heat generation of Joule and entropic heating under convective cooling boundary conditions. It was found from the comparison of the results that the spatial-resolution LC thermal model can accurately and quickly predicts the cell temperatures (core, skin and area-averaged) under various dynamic battery duty cycles even for high Biot numbers due to highly convective conditions such as liquid cooling.

Buoyancy induced flow in a nanofluid filled enclosure partially exposed to forced convection

A numerical study was performed on natural convection for water–CuO nanofluid filled enclosure where the top surface was partially exposed to convection. The cavity has a square cross-section and differentially heated. Except exposed convection part on the top, all sides are adiabatic on horizontal walls. Effects of Rayleigh number (103⩽Ra⩽105), Biot number (0⩽Bi⩽∞), length of partial convection (0.0⩽L⩽1.0) and volume fraction of nanoparticles (0.0⩽φ⩽0.1) on heat and fluid flow were investigated. The results showed that for the case of high Biot number that heat transfer along the heated was enhanced by increasing the Rayleigh number mainly at the upper portion of the heated wall. When the top wall was totally exposed to convection, the results prevail that the heat transfer was more effective at high Biot number especially at the upper portion of the heated wall. For the case of high Biot number, the results prevailed that the heat transfer at the upper portion of the heated wall increases considerably at high exposed length to convection (L); however, for L⩽0.75 the effect of L was less pronounced. Contour maps for percentage of heat transfer enhancement were presented and it was shown that the location of maximum enhancement in heat transfer was sensitive to Ra, φ and L.

Characterization and Optimization of Fluid Flow in a High Biot Number System

ABSTRACTExperiments and analysis have been conducted to characterize flow separators used in applications where heated fluid passes between layers of solid material such as in the manufacturing of gelatinous materials. The Biot number of the configuration is the key parameter, and must be taken into account when optimizing performance. It is shown that most prior work was for low Biot number systems, and the particular configurations under consideration operate at high Biot number. Existing designs developed for lower Biot number (such as membrane filter spacers) are shown to perform poorly for this application. An experimental apparatus was designed and fabricated to quantitatively assess pressure drop through the system using different separation strategies. These results were compared with a simplified two-term model based on the physics of viscous drag in these devices. Channels without separators behave like classical Poiseuille flow. Channels with separators can be modeled with a two-term equation: a baseline Poiseuille term and a form drag term. A variety of separator designs are compared and their overall performance is discussed. We also illustrate the high sensitivity to gap height in all configurations.

Explicit equations for transient heat conduction in finite solids for Bi > 2

A set of correlations is developed for transient heat conduction in finite solids (plates, cylinders and spheres) exposed to high Biot number convection boundary conditions. For Bi ≥ 3, the error of the approximate solutions is well below 1% of the initial temperature difference driving the transient.

Mathematical modelling of wheat kernel drying with input from moisture movement studies using magnetic resonance imaging (MRI), Part II: Model comparison with published studies

The predicted moisture data, using the three-dimensional (3D) single wheat drying model (Part I) considering actual kernel geometry, were compared with two published semi-empirical thin-layer wheat drying models, and one published axisymmetric heat and mass transfer model for a single wheat kernel. The published axisymmetric model assumed a uniform initial moisture distribution within the whole kernel of ellipsoidal geometry and a single value of the water diffusion coefficient. Our model predictions agreed well with the observations obtained from the published models. Further, our model can predict the moisture distribution pattern within individual wheat components with drying time. Moisture and temperature distributions inside the wheat kernel and the effect of drying temperature on wheat quality were explained by calculating the Biot and Lewis numbers. High Biot numbers for moisture transfer (0.89×106–3.04×106 for endosperm and 1.98×106–3.37×106 for germ) indicated that moisture transfer inside the wheat structural components was governed by high internal resistance, whereas low Biot numbers for heat transfer (0.12–0.20 for endosperm and 0.02–0.10 for germ) indicated that heat transfer was controlled by external resistance and, hence, temperature within the wheat kernels was uniform at any given time during drying. Lewis numbers indicated that the rate of heat transfer was very high compared to moisture transfer in both the endosperm and the germ. Sensitivity analysis was performed for selecting proper model parameters which indicated that values of water diffusion coefficients in the germ and endosperm are the most sensitive parameters for a grain drying process.

On the Lagrangian/Eulerian modeling of dispersed droplet inertia: Internal circulation transition

This article addresses a limitation of Lagrangian methods for droplet tracking, when approaching the transition point of internal circulation within droplets. Laminar multiphase flow with dispersed droplets in a co-flowing airstream is considered. Analytical and numerical formulations of droplet motion are developed based on a Lagrangian finite difference method of droplet tracking. Cases of both high and low relative Reynolds numbers are formulated. The role of interfacial drag in cross-phase momentum exchange increases at higher relative Reynolds numbers. A new transition criterion is developed to characterize conditions leading to shear-driven non-uniformities of velocity within a droplet. This criterion entails a momentum Biot number, in analogy with the Biot number criterion for conjugate heat transfer problems involving conduction and convection. At sufficiently high momentum Biot numbers, appreciable changes of velocity within a droplet imply that Lagrangian methods become unsuitable and transition to Eulerian volume averaging is needed. Predicted results of Lagrangian modeling of droplet motion in a co-flowing airstream are presented and discussed.

EFFECT OF MASS TRANSFER RESISTANCE ON ELECTROCHEMICAL BIOSENSORS WITH ASYMMETRICAL SANDWICH MEMBRANES

A mathematical model of a sandwich membrane biosensor with an enzyme layer between two membranes is considered. The electrochemically active product of the enzymatic reaction is detected by a catalytic electrode on one side of the sandwich membrane. A finite but different mass transfer resistance across the solution– membrane interface was assumed for the substrate and product. The steady state solution depended on the Biot numbers and the diffusivities of the substrate and electrochemically active product. At low Biot numbers the signal depended on the mass transfer coefficient and was independent of the membrane thickness. At high Biot numbers the signal depended on the position of the enzyme layer and the relative diffusivity of the electrochemically active product in the two membranes.

Verification of a formula for estimating average quality factor degradation in thermally processed foods in cylindrical containers

Summary A formula suggested by Jen et al. (1971) may be suitable for estimating the average retention of quality factors in conduction heated foods. This hypothesis was tested by comparing the results obtained for thiamine retention with calculations based on analytical and finite difference solutions of transient heat conduction in cylindrical containers. For a high Biot number, agreement was within 10.8% over a wide range of processing conditions (can size, retort temperature, initial temperature, etc).

A method for the decrease of phenolic content in commercial canola meal using an enzyme preparation secreted by the white-rot fungus Trametes versicolor.

An enzymatic process for upgrading the quality of canola meal (CM) by decreasing its phenolic content was investigated. The new method was based on the addition of the enzyme preparation from white-rot fungus Trametes versicolor to the meal-buffer slurry. A 98% decrease in the concentration of SAE was observed after 1 h of the treatment. The following process variables were considered for optimizing the process: pH, temperature, enzyme, meal, and oxygen concentrations. It was found that: (1) the natural buffering capacity of CM resulted in a negligible effect of the pH of the buffer, which was used as the continuous phase in the process, on the extent of decrease in sinapic acid esters (SAE); (2) the system was saturated with the enzyme when its concentration was 4 nkat/mL of the continuous phase; and (3) the optimum temperature was 50 degrees C. The process could be carried out even at higher temperatures due to the protective action of CM, which resulted in an increase in the thermal stability of the enzyme. The particle size influenced the extraction of the SAE from the meal, indicating that, at lower SAE concentrations, the process became diffusion limited. This result, together with those showing no effect of the intensity of agitation, indicated that the enzymatic process can be characterized by high Biot numbers. During the enzymatic process, the molar concentration of available oxygen can become a limiting factor when it is more than four times lower than the molar concentration of phenolics in the treated meal. The new enzymatic method was compared with other methods reported in the literature for the decrease in the phenolic content of rapeseed meals. It was found that, among the methods tested, the enzymatic treatment was the most effective, followed by the lime treatment. The enzymatic process did not reduce the quality of the protein isolates prepared from the CM. After the addition of a simple acetone-washing step, the isolate from the enzymatically treated meal had even better properties. Copyright 1999 John Wiley & Sons, Inc.

The thermal shock resistance of solids

The thermal shock resistance of a brittle solid is analysed for an orthotropic plate suddenly exposed to a convective medium of different temperature. Two types of plate are considered: (i) a plate containing a distribution of flaws such as pores, for which a stress-based fracture criterion is appropriate, and (ii) a plate containing a single dominant crack aligned with the through-thickness direction, for which a critical stress intensity factor criterion is appropriate. First, the temperature and stress histories in the plate are given for the full range of Biot number. For the case of a cold shock, the stress field is tensile near the surface of the plate and gives rise to a mode I stress intensity factor for a pre-existing crack at the surface of the plate. Alternatively, for the case of hot shock, the stress field is tensile at the centre of the plate and gives rise to a mode I stress intensity factor for a pre-existing crack at the centre of the plate. Lower bound solutions are obtained for the maximum thermal shock that the plate can sustain without catastrophic failure according to the two distinct criteria: (i) maximum local tensile stress equals the tensile strength of the solid, and (ii) maximum stress intensity factor for the pre-existing representative crack equals the fracture toughness of the solid. Merit indices of material properties are deduced, and optimal materials are selected on the basis of these criteria, for the case of a high Biot number (high surface heat transfer) and a low Biot number (low surface heat transfer). The relative merit of candidate materials depends upon the magnitude of the Biot number, and upon the choice of failure criterion. The effect of porosity on thermal shock resistance is also explored: it is predicted that the presence of porosity is generally beneficial if the failure is dominated by a pre-existing crack. Finally, the analysis is used to develop merit indices for thermal fatigue.

Transfer function method to calculate moisture absorption and desorption in buildings

A transfer function approach has been described to calculate dynamic moisture absorption and desorption by building materials in the hygroscopic range, and the dynamic moisture absorption and desorption processes have been theoretically modeled. Their surface vapor pressure and moisture absorption and desorption flux can be evaluated by this approach. It has turned out mathematically that the material surface attains instantaneous moisture equilibrium with the surroundings at high Biot number (Bi → ∞), and that the material moisture behavior can be described through a lumped-parameter modeling at low Biot number (Bi → 0). For building materials at intermediate Biot number, comparisons with experimental results and numerical solutions have shown satisfactory agreement with the proposed approach. Using this approach, the dynamic response of moisture absorption and desorption in interior building materials to the variation of indoor air humidity and the dynamic effect of moisture absorption and desorption by building interior materials on indoor air humidity and space latent cooling load can be simultaneously calculated.

New logarithmic technique in the flash method

The logarithmic transformation of the rear face temperature rise issued from a flash experiment can be interpreted as adding up a heat loss and radial effect component to the one for the adiabatic model. Two data reduction methods based on this transformation are proposed with the common main goal to identify one parameter depending upon the sample's thickness and thermal diffusivity, not upon the other experimental conditions. They allow one to determine thermal diffusivity with good accuracy for nonuniform pulses and high Biot numbers.

A comparison between the steady state performance of particulate and conventional tubeless collectors

In order to compare the performance of a particulate collector with that of a conventional type, two mathemetical models which describe the steady state behaviour of these collectors are presented. From the results of these models, it is concluded that the efficiency of the particulate collector is higher than that of the conventional type by 5–100%. This improvement in the efficiency by the particulate collector is due to its much lower surface temperature compared to the surface temperature of the conventional tubeless type. It is found that the improvement in performance, resulting from using the particulate collector, is more significant at high Biot and low Peclet numbers. Also, the improvement becomes more apparent as the length of the collector increases.

Thermocapillary flow with evaporation and condensation at low gravity. Part 2. Deformable surface

The free surface behaviour of a volatile wetting liquid at low gravity is studied using scaling and numerical techniques. An open cavity model, which was applied in part 1 to investigate fluid flow and heat transfer in non-deforming pores, is used to evaluate the influence of convection on surface morphology with length scales and subcooling/superheating limits of 1 ≤ D ≤ 102 μm and ∼ 1 K, respectively. Results show that the menisci shapes of highly wetting fluids are sensitive to thermocapillary flow and to a lesser extent the recoil force associated with evaporation and condensation. With subcooling, thermocapillarity produces a suction about the pore centreline that promotes loss of mechanical equilibrium, while condensation exerts an opposing force that under some conditions offsets this destabilizing influence. With superheating, thermocapillarity and evaporation act in the same direction and mutually foster surface stability. All of these trends are magnified by high capillary and Biot numbers, and the stronger circulation intensities associated with small contact angles. These phenomena strongly depend on the thermal and interfacial equilibrium between the liquid and vapour, and have important ramifications for systems designed to maintain a pressure differential across a porous surface.