Small Biot Number Research Articles

Heat spreading effect on the optimal geometries of cooling structures in a manifold heat sink

Embedded cooling is a promising solution to address thermal challenges of power electronics. The present work investigates heat spreading effect on the optimal geometries of cooling structures within a single-phase manifold heat sink in laminar regime by integration of topology optimization (TO) and Bayesian optimization (BO). Such integration is for the manual parameter tuning issue in the density-based TO of conjugate heat transfer, and the TO process is accelerated by using a dual-level simulator and the optimality criteria optimizer combined with the stochastic gradient descent method. The heat spreading effect significantly alters the optimal geometries, due to the change of thermal resistance constitutions. Physical analyses on the BO-TO-generated geometries give two guidelines for improving the cooling structures here: (a) Heat source-coolant distance should increase properly for narrow heating area and small Biot number; (b) Highly-thermal-conductivity solid can be filled into low-velocity regions directly connecting to hot area. As an example, a fabrication-friendly cooling structure is devised in the case of concentrating heating following these guidelines, avoiding the computationally-expensive optimization process and the over-complicated geometries. The CFD simulations show this design can significantly reduce the thermal resistance with minor increase of pressure loss as the Reynold number from 50 to 300.

Thermal Runaway in Battery Cells: Guiding Principles for Continuum Scale Modeling

Lithium-ion batteries are sensitive to temperature. Low-temperature or high-temperature operation may lead to lithium plating and dendrites, SEI decomposition, and separator collapse. Such phenomena may cause internal short circuits, rapid heat generation and, subsequently, may trigger thermal runaway, resulting in fire or explosion [1][2]. Hence, accurate modeling and prediction of the thermal response of battery cells inside the pack is critical to preventing safety hazards as well as optimizing battery performance. Battery thermal management system (BTMS) plays a significant role in maintaining an optimal operation temperature and ensuring thermal safety for electric vehicles [3]. However, the large number of battery cells inside one module makes it challenging for BTMS algorithms to realize real-time simulations based on fully resolved physics-based models. The conventional lumped capacitance (LC) thermal model achieves reduced computation costs under the hypothesis that temperature is spatially uniform. Nevertheless, such a model works better for small Biot number conditions, usually under low C-rate operation and moderate cooling scenarios, and it lacks justification under a broad range of battery operating conditions and materials [4]. To overcome such limitations, rigorous multiscale models, based on mathematical coarse-graining theories, have been developed to predict the thermal response of batteries at the module scale [5] while not compromising on computational cost. In this work, we investigate the accuracy of the classical LC thermal model for twenty potential battery modules constructed with commercial batteries and packing materials. We show that the classical LC thermal model may be inaccurate under a broad range of design conditions. In the context of multiscale modeling, we identify which effective models should be instead used, and analyze the impact of SOC and the relative strength of heat insulation and heat dissipation on model selection and accuracy. This work provides guidance, based on first principles and rigorous multiscale theory, on best practices for model selection in practical applications involving thermal behavior and thermal runaway in battery cells at the module scale.

Modeling the stability of thin liquid film flows on a uniformly heated slippery inclined substrate: A realistic approach

We investigate the stability of gravity-driven, Newtonian, thin liquid film falling down a uniformly heated slippery rigid inclined wall. The authors of previous research works considered specified temperature (ST) boundary condition to study the effects of slip length. However, the ST boundary condition does not include the effects of heat fluxes at wall–air and wall–liquid interfaces and so fails to incorporate the real situation. Consequently, we consider heat flux/mixed-type boundary condition as the thermal boundary condition on the rigid plate. This boundary condition involves the heat flux from the rigid plate to the surrounding liquid and the heat losses from the wall to the ambient air. Using long-wave expansion method, we construct a highly nonlinear evolution equation in terms of the film thickness at any instant. Using normal mode approach, the linear study reveals the stabilizing (destabilizing) behavior of the wall film Biot number (dimensionless slip length). It is found that the destabilizing tendency of the slip length is more in the absence of thermocapillary stress. The linear study reveals that the destabilizing role of MB may be controlled to some extent by increasing the wall film Biot number Bw. Using asymptotic expansions of the flow variables in terms of the small wave number k, the Orr–Sommerfeld boundary value problem gives an onset of instability in terms of critical Reynolds number. It slightly differs from that of the same as obtained by Benney's long-wave expansion method, due to the consideration of small free surface Biot number [B=O(ϵ)]. For arbitrary wave numbers, using Chebyshev spectral collocation method, the effect of Marangoni number (Ma), slip length (δ), and wall film Biot number (Bw) on the H, S, P, and shear modes of instability are discussed in detail. Near the threshold, both Ma and δ show the destabilizing effect on H mode of instability, whereas Bw gives the stabilizing effect. Interestingly, their roles on H mode of instability becomes diametrically opposite far from the onset of instability. For S mode, both Ma and Bw display the destabilizing effect, whereas δ plays the dual role. For P mode, both Ma and δ show the destabilizing effect, whereas Bw plays the stabilizing role. The slip length (δ) plays the stabilizing role, in the case of shear mode. In the absence of thermocapillary effect, the vorticity balance at the liquid–air interface explains that the amplitude of the vorticity perturbation amplifies the surface deformation due to the presence of inertia and the slip length. In the absence of the slip length, a weakly nonlinear study transforms the evolution equation to the famous Kuramoto–Sivashinsky (KS) equation.

Asymptotic analysis of a two-phase Stefan problem in an annulus with the convective boundary

The Stefan problem (whether in its classical form or not) has assumed that the interface moves instantaneously with time as a prescribed initial condition. This assumption is valid in the problems subjected to an isothermal boundary. Yet, it may not represent the physics for a convective boundary since a certain amount of time is needed to cool or melt the convective surface to its fusion depending on the heat transfer coefficient (or Biot number in a dimensionless way). In this study, we formulate a two-phase Stefan problem in an annulus for outward solidification subjected to a convective or Robin boundary condition while not assuming that the interface moves right away at time t=0. A comprehensive asymptotic analysis is performed by expanding around a small Stefan number; four spatial and five temporal scales are characterized based on the Stefan problems involving a convective surface. The method of property averaging is also employed at the scale where the equilibrium freezing occurs. The developed asymptotic solution is verified with numerical data generated by the enthalpy method at various Biot and Stefan numbers. The results demonstrate the significance of abandoning the ansatz (i.e., the interface does not move immediately with time), especially towards small Biot numbers. Further, it is found that the presented asymptotic solution considerably extends the valid range of the Stefan number when compared with the conventional asymptotic technique.

Comment on the Marshall‐Hoare‐Henssge model for estimating the time since death

Timewise temperature variations in objects that are undergoing unsteady heating or cooling is a commonly encountered problem in the thermal sciences. One particular area of application is the cooling of a body post-death and the use of body temperatures to estimate the time of death. Here, a new approach based on the theory of transient heat transfer is formulated to allow efficient calculation of unsteady conduction problems. The theoretically derived unsteady temperature models are compared with experimentally based correlations (the Marshall-Hoare-Henssge model). The two approaches are found to agree very well. With this new theoretically based approach, timewise temperature variation can be calculated for both large and small Biot number transient problems.

Thermal analysis of oscillating thermomagnetic devices beyond the lumped approximation

The temperature (T) dependent magnetization of materials near the Curie temperature has been leveraged for thermomagnetic energy harvesting and thermal rectification. In these thermal devices, a translating shuttle containing a ferromagnet (e.g., gadolinium) experiences a T-dependent magnetic attraction to a stationary permanent magnet, leading to steady-periodic mechanical oscillations of the shuttle between the hot and cold surfaces. Prior analytical work investigating thermomagnetic devices has focused on the small-Biot (Bi) number regime in which the shuttle T is thermally lumped due to the small interfacial thermal contact conductances. Here, we use a separation of variables technique to derive analytical expressions to the governing heat conduction equation for the full spatiotemporal T profile considering a thin magnetic foil mounted on a thick shuttle. The results show that the lumped solution prediction for the time-averaged heat flow Q is accurate within <5% when either the hot or cold surface Bi is <0.1. In the moderate-to-large Bi regime, analytical solutions and numerical validations show that Q approaches a maximum value determined by conduction through the shuttle, while the oscillation frequency f increases with increasing Bi. The analytical solutions developed here can be used to design and evaluate the thermal performance of thermomagnetic devices proposed for use in energy scavenging and thermal rectification applications.

Falling liquid films on a slippery substrate with variable fluid properties

We investigate the stability of a gravity-driven, thin, Newtonian liquid flowing on a uniformly heated slippery inclined plane. We construct a mathematical model of the flow that comprises the Navier–Stokes equation coupled with the equation of energy. While the rest of the boundary conditions are standard for thin-film problems, we apply a Navier slip boundary condition at the solid–liquid interface. We assume that the fluid thermophysical properties — density, dynamical viscosity, surface tension, and thermal diffusivity vary linearly with temperature as long as the change in temperature is small. In the analysis part, we follow the standard long-wave theory and construct a nonlinear evolution equation for the film thickness. This is followed by a linear and weakly nonlinear stability analysis. The linear analysis allows us to compute the critical Reynolds number of our problem and from this study, we conclude that the slippery substrate destabilizes the film flow. The weakly nonlinear stability analysis finds a finer description of various stable/unstable zones. Finally, we perform a numerical simulation of the evolution equation in a periodic domain using spectral methods. Our numerical results support the analytical predictions of the instability threshold using the linear and weakly nonlinear theories. The influence of the small Biot number is also investigated in presence of the slip length together with the variable fluid properties.

Low Heat Capacity 3D Hollow Microarchitected Reactors for Thermal and Fluid Applications

Lightweight reactor materials that simultaneously possess low heat capacity and large surface area are desirable for various applications such as catalytic supports, heat exchangers, and biological scaffolds. However, they are challenging to satisfy this criterion originating from their structural property in most porous cellular solids. Microlattices have great potential to resolve this issue in directing transport phenomena because of their hierarchically ordered design and controllable geometrical features such as porosity, specific surface, and tortuosity. In this study, we report hollow ceramic microlattices comprising a 10 μm thick hollow nickel oxide beam in an octet-truss architecture with low heat capacity and high specific surface area. Our microarchitected reactors exhibited a low heat capacity for a rapid thermal response with a small Biot number (Bi &lt;&lt; 1) and large intertwined surface area for homogeneous flow mixing and chemical reactions, which made them ideal candidates for various energy applications. The hollow ceramic microlattice was fabricated by digital light three-dimensional (3D) printing, composite electroless plating, polymer removal, and subsequent thermal annealing. The transient thermal response and fluidic properties of the 3D-printed microstructures were experimentally investigated using a small-scale thermal and fluid test system, and analytically interpreted using simplified models. Our findings indicate that hollow microarchitected reactors provide a promising platform for developing multifunctional materials for thermal and fluid applications.

Effect of fluctuation on heat transfer enhancement of a spherical particle

Heat transfer of a particle in turbulent flows is essential, while the study on the effect of turbulent fluctuation on the heat transfer enhancement of a spherical particle is rarely reported. In this work, a 4.4 mm copper particle with a small Biot number was subject to a near homogenous and isotropic turbulent flow field to investigate the heat transfer behaviors. The detailed flow characteristics were measured with the Particle Image Velocimetry (PIV). As the fan speed increases, both the fluctuating velocity (urms) and the time-averaged velocity (uave) increase quickly at first and then slowly. When the furnace temperature increases, both urms and uave barely change. For all conditions, urms is apparently larger than uave. As for the particle heating experiment, the heating rate increases with the fan speed and the furnace temperature. Based on the measured results of the flow field and the particle temperature, a modified correlation for predicting Nusselt number was proposed to describe the effect of the turbulent fluctuation on the heat transfer of a single spherical particle. The newly proposed correlation was further validated by particle heating experiments of a 2 mm copper sphere in this work and direct numerical simulation results from the literature.

Buoyancy instabilities in a liquid layer subjected to an oblique temperature gradient

We investigate the temporal and spatio-temporal buoyancy instabilities in a horizontal liquid layer supported by a poorly conducting substrate and subjected to an oblique temperature gradient (OTG) with horizontal and vertical components, denoted as HTG and VTG, respectively. General linear stability analysis (GLSA) reveals a strong stabilizing effect of the HTG on the instabilities introduced by the VTG for Prandtl numbers $Pr&gt;1$ via inducing an extra vertical temperature gradient opposing the VTG through energy convection. For $Pr&lt;1$ , a new mode of instability arises as a result of a velocity jump in the liquid layer caused by cellular circulation. A long-wave weakly nonlinear evolution equation governing the spatio-temporal dynamics of the temperature perturbations is derived. Spatio-temporal stability analysis reveals the existence of a convectively unstable long-wave regime due to the HTG. Weakly nonlinear stability analysis reveals the supercritical type of bifurcation changing from pitchfork in the presence of a pure VTG to Hopf in the presence of the OTG. Numerical investigation of the spatio-temporal dynamics of the temperature disturbances in the layer in the weakly nonlinear regime reveals the emergence of travelling wave regimes propagating against the direction of the HTG and whose phase speed depends on $Pr$ . In the case of a small but non-zero Biot number, the wavelength of these travelling waves is larger than that of the fastest-growing mode obtained from GLSA.

Numerical and experimental study of pool boiling heat transfer mechanisms in V-shaped grooved porous metals

In this study, pool boiling heat transfer mechanisms in V-shaped grooved porous metals are studied using our porous metal model. The model eliminates the small Biot number restriction of linear temperature-distribution assumption and thus extends to large Biot number conditions, such as boiling heat transfer of porous metals relating multi-bubble agitation and coalescence. The effects of groove number, groove distance and groove width on pool boiling heat transfer are investigated using lattice Boltzmann method. The experimental study is also conducted to reveal bubble escaping paths. The results show that bubble departure frequency on the porous metal with double V-shaped grooves is higher than those on the porous metal with single V-shaped groove and the uniform porous metal. The experimental result indicates that the venting bubbles in lateral grooves are entrained by the central departure bubbles, which validate the numerical results. At the low heat fluxes, the wall superheat increases with increasing groove distance. The lotus-shaped bubbles form in the central porous metal. The bubble escaping resistance reduction leads to the heat transfer performance enhancement for the porous metal with the wide V-shaped grooves.

Mitigating battery thermal runaway through mild combustion

With the fast expansion of electric vehicle market, thermal runaway of Li-ion battery has become a major concern for ground transportation and public safety. Fundamentally, thermal runaway initiation of a Li-ion battery bears intrinsic similarity to a gas-phase thermal explosion process, where the system response is controlled by the competition between the heat generation and heat loss, each with distinct temperature dependence. In this work, the fundamentals of mild combustion described by the stretched S-curve is revisited from one-step overall reaction with and without reactant depletion. The key dimensionless parameter that can lead to mild combustion are identified. Furthermore, the critical state of Li-ion battery thermal runaway in the small Biot number limit has been theoretically and computationally analyzed using detailed thermochemistry of battery thermal runaway. With insight from the stretched S-curve concept, the conditions corresponding to the loss of criticality are investigated, where in principle only mild heat release can occur. It is found that reducing the reaction heat associated with negative-electrode and electrolyte has the strongest effect to mitigate thermal runaway, where the reaction heat of electrolyte decomposition plays a minor role. Li-ion battery thermal runaway can be mostly prevented if the reaction heat of the cathode and anode with electrolyte are both reduced by an order of magnitude. The results stimulate future experiments seeking novel battery materials, and can provide useful guidance on the screening of safer battery materials to reduce the possibility of thermal runaway.

Enhancement of Convection Heat Transfer in Air Using Ultrasound

The use of ultrasound is a new method to enhance convection drying. However, there is little information in the literature on the improvement of convective heat transfer caused by ultrasound. Therefore, the heat transport during ultrasound-assisted convective heating of small samples in a hybrid dryer was experimentally examined. A small Biot number regime of heat transfer was considered. The results confirmed a great enhancement of heat transfer due to the application of ultrasound. Due to the use of ultrasound, the convective heat exchange coefficient increased from 45% to almost 250%. The enhancement is a linear function of applied ultrasound power. It was shown that the energy absorption of ultrasound existed, but the thermal effect of this absorption was very small.

Analysis of Transient Heat Conduction in a 2-D Rectangular Fin.(Dept.M)

In this work a new developed dimensionless finite difference technique for 2-D transient heat conduction is developed and presented. The technique is then applied to predict the time development of temperature profiles in a convectively cooled rectangular fin. In addition, a corrected analytical form that reduces the 2-D problem to a 1-D one is developed. Results prove that the numerical technique is simple and fast, while the 1-D reduced analytical solution is valid only for early time, small Biot number, and thin fins.

Passive Sampler Exchange Kinetics in Large and Small Water Volumes Under Mixed Rate Control by Sorbent and Water Boundary Layer

Exchange kinetics of organic compounds between passive samplers and water can be partly or completely controlled by transport in the sorbent. In such cases diffusion models are needed. A model is discussed that is based on a series of cosines (space) and exponentials (time). The model applies to mixed rate control by sorbent and water boundary layer under conditions of fixed aqueous concentrations (open systems, infinite water volumes, in situ sampling) and fixed amounts (closed systems, finite water volumes, ex situ sampling). Details on the implementation of the model in computational software and spreadsheet programs are discussed, including numerical accuracy. Key parameters are Biot number (ratio of internal/external transfer resistance) and sorbent/water phase ratio. Small Biot numbers are always indicative of rate control by the water boundary layer, but for large Biot numbers this may still be the case over short time scales. Application to environmental monitoring of nonpolar compounds showed that diffusion models are rarely needed for sampling with commonly used single-phase polymers. For determining sorption coefficients in batch incubations, the model demonstrated a profound effect of sorbent/water phase ratio on time to equilibrium. Application of the model to sampling of polar organic compounds by extraction disks with or without a membrane showed that moderate to major sorbent-controlled kinetics is likely to occur. This implies that the use of sampling rate models for such samplers needs to be reconsidered. Environ Toxicol Chem 2021;40:1241-1254. © 2021 SETAC.

Analysis of Cooling Efficiency Using the Biot Number of a Rectangular Fin

The performance of a rectangular fin, the relative value of the performance for L=0.1, and the ratio of the heat transfer from the fin tip to the total heat transfer from the rectangular fin were investigated using a two-dimensional analytical method. The variables are the Biot number and non-dimensional fin length. The relative resistance, effect, and efficiency increase as the non-dimensional length increases for a small Biot number (Bi≤0.1). On the other hand, the relative resistance and effect decrease and the relative efficiency increases as the non-dimensional length increases for a large Biot number (Bi≥0.7).

An Approximate Analysis of Insulation Thickness of Elliptic Pipe

The thermal insulation characteristics of elliptic pipe were analyzed preliminarily. Thermal resistances between elliptic pipe line to surrounding ambient were studied in dimensionless form. The expression for critical insulation thickness was obtained. The lower bound thickness and heat transfer rate were researched. Influences of different factors were discussed. It is found that for cases of small Biot number, the lower bound thickness is obviously greater than the critical thickness. And the addition of insulation material upon elliptic pipe line should be conducted enough so that the insulation layer exceeds the lower bound thickness in order the insulation investment play a positive role.

Kinetics, Mechanism, and Thermodynamics Studies of Vacuum Drying of Biomass from Taxus chinensis Cell Cultures

In this study, we investigated the kinetics, mechanism, and thermodynamics of vacuum drying and dehydration of plant cells of Taxus chinensis. The efficiency of vacuum drying increased with increases in the drying temperature (45–60°C). When the experimental data were fitted to typical kinetic models for drying, the Modified Page were the most appropriate. Thermodynamic parameters revealed the endothermic, irreversible, and spontaneous nature of the drying. The effective diffusion coefficient (3.6602 × 10−14−10.9810 × 10−14 m2/s) and mass transfer coefficient (2.2536 × 10−11−10.8110 × 10−11 m/s) increased with increasing drying temperature. The small Biot number (2.6167 × 10−2−4.1841 × 10−2) indicated that the process of mass transfer was externally controlled.

Non-local effects and size-dependent properties in Stefan problems with Newton cooling

We model the growth of a one-dimensional solid by considering a modified Fourier law with a size-dependent effective thermal conductivity and a Newton cooling condition at the interface between the solid and the cold environment. In the limit of a large Biot number, this condition becomes the commonly used fixed-temperature condition. It is shown that in practice the size of this non-dimensional number is very small. We study the effect of a small Biot number on the solidification process with numerical and asymptotic solution methods. The study indicates that non-local effects become less important as the Biot number decreases.

Characteristics and Mechanism of Microwave-assisted Drying of Amorphous Paclitaxel for Removal of Residual Solvent

We investigated the characteristics and mechanism of microwave-assisted drying of amorphous paclitaxel for the removal of residual solvent, acetonitrile. The removal efficiency of residual acetonitrile increased with increasing drying temperature from 35 to 55°C. When the experimental data were applied to typical kinetic models, the Page model was determined to be the most suitable. Thermodynamic parameters revealed the spontaneous and endothermic nature of microwave-assisted drying. The effective diffusion coefficient of acetonitrile (0.865 × 10−8∼ 1.553 × 10−8 m2/s) and the convective mass transfer coefficient (2.138 × 10−7∼7.656 × 10−7 m/s) increased with increasing drying temperature. The small Biot number (0.001975∼ 0.003939) indicated that the process of mass transfer was externally controlled.