Biot Number Bi Research Articles

Prediction of the discharging time of a latent heat thermal energy storage system with a UA approach

Designing latent heat thermal energy storage systems is a cumbersome task and the estimation of the performance of such a storage system normally involves experiments and detailed numerical simulations. Analytical, empirical and simplified numerical models are much faster but subject to large uncertainties. Even the prediction of the performance of an existing latent heat thermal energy storage system under different boundary conditions is often not possible in an easy way. Therefore, we present an analytical method – the UA approach – to predict the discharging (solidification) time of a flat plate latent heat thermal energy storage system. A special feature of the UA approach is that one can incorporate experimental or numerical results to improve the prediction of the performance under a variety of boundary conditions or material properties. The UA approach was tested for a variation of the Stefan number (Ste), the Biot number (Bi), the number of transfer units (NTU) and the heat transfer fluid and was compared to the results of a validated numerical model. The results are promising, especially for small Ste. In addition, the prediction of performance for a high thermal heat conductivity of the phase change material based on a numerical reference solution with a low thermal conductivity worked remarkable well.

Energy mechanism for the instability of liquid jets with thermocapillarity

Xu and Davis [J. Fluid Mech. 161, 1–25 (1985)] examined the stability of long axisymmetric liquid jet subjected to an axial temperature gradient, finding capillary, surface-wave, and hydrodynamic modes. They showed that capillary breakup can be retarded or even suppressed for a small Prandtl number (Pr < 1) and a large Biot number (Bi ≥ 1). In the present work, the energy mechanism is carried out for these three kinds of flow instabilities, and the mechanism of suppressing capillary breakup is clarified. When the Reynolds number (RB) is not large, the work done by the pressure on the free surface (PS) is the main energy source of the capillary instability. At small Pr and large Bi, the phase difference between the radial velocity and surface deformation increases with RB, leading to the decrease in PS, which prevents the occurrence of capillary breakup. Meanwhile, the work done by thermocapillary force becomes the main energy source, making hydrodynamic modes unstable. The perturbation flow fields are displayed, which shows that the temperature fluctuations of three modes differ from each other.

Effect of surface heat exchange on phase change materials melting with thermocapillary flow in microgravity

This paper presents a numerical analysis of the melting dynamics of n-octadecane in microgravity. The phase change material (PCM) is held in a rectangular container of aspect ratio Γ=L/H=1.5, and the upper boundary is open to a layer of air, which both exchanges heat with the PCM and generates thermocapillary convection in the liquid phase via the Marangoni effect. This study extends the analysis conducted by Martinez et al. [“Effect of surface heat exchange on phase change materials melting with thermocapillary flow in microgravity,” Phys. Fluids 33, 083611 (2021)] in which the air temperature was assumed to vary linearly between the temperatures applied at the lateral walls. Two different scenarios are analyzed here. In the first case, the air temperature is assumed to be homogeneous and equal to the mean value of the temperatures applied at the lateral walls throughout the melting process. In the second case, the air temperature is similarly taken to be constant but with a value of 23 °C, which is representative of a laboratory environment, including many microgravity platforms. The investigation reveals the effect of key dimensionless parameters, including the Marangoni number (Ma), which quantifies the heat transport due to the thermocapillary flow, and the Biot number (Bi), which characterizes the heat exchanged across the PCM/air interface. In contrast to previous analyses of pattern selection, only oscillatory standing (pulsating) waves are observed under these boundary conditions. The results in each case are presented via stability maps in terms of Bi and Ma.

Thermal boundary condition studies in large aspect ratio Rayleigh–Bénard convection

We study the influence of thermal boundary conditions on large aspect ratio Rayleigh–Bénard convection by a joint analysis of experimental and numerical data sets for a Prandtl number Pr=7 and Rayleigh numbers Ra=105−106. The spatio-temporal experimental data are obtained by combined Particle Image Velocimetry and Particle Image Thermometry measurements in a cuboid cell filled with water at an aspect ratio Γ=25. In addition, numerical data are generated by Direct Numerical Simulations (DNS) in domains with Γ=25 and Γ=60 subject to different idealized thermal boundary conditions. Our experimental data show an increased characteristic horizontal extension scaleλ̃ of the flow structures for increasing Ra , which due to an increase of the convective heat transfer also leads to an increase of the Biot number (Bi) at the cooling plate. However, we find the experimental flow structure size to range in any case in between the ones observed for the idealized thermal boundary conditions captured by the simulations: On the one hand, they are larger than in the numerical case with applied uniform temperatures at the plates. On the other hand, they are smaller than in the case of an applied constant heat flux, the latter of which leads to a structure that grows gradually up to the horizontal domain size. We are able to link this observation qualitatively to theoretical predictions for the onset of convection. Furthermore, we study the effect of the asymmetric boundary conditions on the heat transfer. Contrasting experimental and numerical data reveals an increased probability of far-tail events of reversed heat transfer. The successive decomposition of the local Nusselt number Nuloc traces this effect back to the sign of the temperature deviation Θ̃, eventually revealing asymmetries of the heating and cooling plate on the thermal variance of the generated thermal plumes.

Enhancing Simulation Efficiency and Quality of Transient Conjugate Thermal Problems by Using an Advanced Meta-modeling Approach

<div>In the field of thermal protection, detailed three-dimensional computational fluid dynamics (3D-CFD) simulations are widely used to analyze the thermal behavior on a full vehicle level. One target is to identify potential violations of component temperature limits at an early stage of the development process. In battery electric vehicles (BEVs), transient load cases play an increasing role in evaluating components and vehicle systems close to real-world vehicle operation. The state-of-the-art 3D simulation methodologies require significant time and computational effort when running transient load scenarios. One main reason is the conjugate characteristic of the problem, meaning that conduction within the component and convection into the surrounding air occur simultaneously. This requires a detailed consideration of both the fluid and structural domains.</div> <div>Therefore, this article derives a time-efficient simulation methodology for transient component temperatures in electric vehicles. The approach is to extract heat transfer coefficients and reference temperatures from sample flow simulations and to construct convective meta-models. Solid component temperatures are then transiently computed whereby the low-dimensional meta-models provide the convective heat transfer. Dimensional analysis determines the smallest possible parameter space for the meta-modeling. Two different types of meta-models, a scalar regression model and a vector proper orthogonal decomposition (POD) approach, are tested and compared.</div> <div>The study examines at first the applicability of the heat transfer formulation under different flow and component temperature conditions using a generic flat plate test case. A low Biot number (Bi) is crucial to receive accurate temperature predictions as heat transfer coefficients are derived on uniform temperature walls. The methodology is subsequently applied to a sample component in the motor compartment. Measurements on a test rig and a transient load case comparison with a coupled simulation prove the validity of the numerical procedure. Scaling to full-vehicle applications is feasible. The new methodology delivers a highly accurate temperature prediction and increases computation efficiency, especially for sensitivity studies.</div>

Thermoporoelastic response of a semi‐permeable wellbore subjected to convective cooling and non‐hydrostatic in situ stresses

AbstractExisting models of wellbore stability idealize the borehole wall as either a perfectly permeable or an entirely impermeable surface. The widespread observation that a shale borehole allows solvent molecules to pass through but impedes solutes suggests that the borehole wall should be regarded as a non‐ideal semi‐permeable medium. To address the magnitude and rate of fluid penetration into the formation when a semi‐permeable wellbore is exposed to a non‐isothermal drilling fluid quantitatively, this work develops analytical solutions for a semi‐permeable borehole undergoing convective cooling and far‐field non‐hydrostatic in situ stresses in the framework of fully coupled thermoporoelasticity. Integral transform and load decomposition techniques are employed to facilitate the derivation of analytical solutions. The results show that, in contrast to the permeable or impermeable borehole models, the semi‐permeable model predicts significantly different stress and pore pressure fields. The transient evolution of temperature, pore pressure, and stresses is predominantly governed by two non‐dimensional numbers: the Biot number Bi and a newly identified number πf that characterizes the capability of shale to transport fluid across the solid‐fluid interface.

Modeling of Core-shell Catalytic Pellets of Inert Shell with Various Morphologies for Series Reactions

Modeling of core-shell catalytic pellets with inert shell was performed assuming three different particle shapes such as sphere, cylinder, and slab. For isothermal irreversible first order reaction, A→B, analytical solutions of intra-particle concentration of reactant were derived by solving reaction-diffusion equations, considering thickness of catalytically active core (r2), when the pellets were immersed in infinitely large medium. Unlike conventional pellets or core-shell pellets with inert core, effectiveness factor (η) was affected by r2 as well as ratio of effective diffusivity in inert shell and active core (Υ), and η could be enhanced by increasing Υ. η increased in the order of sphere > cylinder > slab due to surface area per unit volume of medium, and η increased with increasing Biot number (Bi), because of decreased external film resistance. For series reaction, A→B→C, transient change of the concentration of starting reactant (CA) and intermittent product (CB) in bulk fluid of reactor could be predicted by solving material balance equations assuming pseudo-steady state approximation and equal value of Thiele modulus for both reactions in batch reactor and CSTR. When B is desirable product, CB increased by increasing Υ and decreasing r2. Similar to conventional pellets, CB in batch reactor decreased by increase of distribution coefficient of B on surface of pellet, KB, due to increased adsorption, and increased in the order of sphere < cylinder < slab, similar to conventional pellets. In CSTR, CB at steady state could be enhanced by decreasing amount of catalyst and Thiele modulus (Φ), implying that performance of reactor can be controlled by adjusting various reaction parameters. From numerical solutions of reaction diffusion equations for series reactions, the effect of r2 and Υ were also confirmed from transient value of CA and CB in exit stream calculated by finite element method.

Investigation of Nanofluid Natural Convection Inside a Square Cavity for Two Orientations Using Lattice Boltzmann Method

In this study, natural convection heat transfer of a water based nanofluid inside a square cavity is numerically investigated for two different orientations of a wall-heated cavity. The enclosure is heated by applying a constant heat flux while cooled at ambient conditions. Lattice Boltzmann method (LBM) is used to simulate nanofluid natural convection. The Brownian motion of nanoparticles is considered. LBM simulation is validated by comparison with experimental and numerical results of the literature. The effect of Rayleigh number (Ra = 103, 104, 105, 106), Biot number (Bi = 0.1, 1, 10, 100) and volume fraction of nanoparticles (Φ = 0, 1, 3 and 5%) on the isotherms, streamlines, velocity components, local and Nusselt number is analyzed for two oriented cavities. The bottom-heated cavity shows higher heat transfer rate than that of the cavity heated from the sidewall. The average Nusselt number increases by up to 6.81%. Furthermore, Biot number, Rayleigh number, and volume fraction of nanoparticles show significant effects on the heat transfer rate.

Conjugate heat transfer and flow around tandem tubes: Effect of thermophysical parameters and tube arrangement

Thermodynamic calculation and geometry parameters are key issues in thermo-equipment design. The flow and conjugate heat transfer around the tandem tubes are numerically investigated, in which the examined parameters include thermal conductivity ratio (kr), volumetric heat capacity ratio (φr), and Biot number (Bi) which are varied from 200 to 800, 1000 to 3000, and 0.1 to 1.0, respectively. Further, the effect of the space ratio (L/D = 1.0–10.0) between the tandem tubes on the flow pattern and heat transfer is also studied to obtain a proper tube arrangement. In particular, a difference method for lattice Boltzmann algorithm is validated and then adopted on the curve fluid-solid interface to calculate the conjugate heat transfer. Results demonstrate that the thermal conductivity ratio significantly affects the heat transfer performance of the tube while the volumetric heat capacity ratio only affects the rate of heating process. The growth in thermal conductivity ratio leads to a more uniform temperature distribution and higher thermal performance, and this effect becomes weaker as the kr grows. Meanwhile, the increase of Bi strengthens the convective heat transfer inside the tube, further changing the dominant way of heat transfer. Finally, the flow varies considerably with the space ratio, classifying into four typical regimes. It is found that the Nu is sensitive to the flow regime and the case with L/D = 4.0 provides the best thermal performance of tandem tubes. These results can provide more realistic predictions and indications in thermo-equipment design.

Passive heating process enhancement of a new structure room using porous‐covering heat‐emitting channels

AbstractThis manuscript presents a numerical study of a passive heating process of a new structure room using porous‐covering heat‐emitting channels as small discrete radiators. The governing equations are discretized using the finite volume method and the SIMPLER algorithm is employed to handle the pressure–velocity coupling. The effect of porous‐cover to air thermal conductivity ratio (Kr), Darcy number (Da), Rayleigh number (Ra), thickness of the porous‐cover (Ep), and Biot number (Bi) on the heating process of the room is examined. The principal results of this investigation indicate that in the case with Kr &gt; 1, numerical simulations have enabled us to find an optimal value of Ep (Epcr = 0.125) for which the average room temperature reaches a maximum value. In addition, an enhancement in the heating process in the room can reach 77.77% by increasing Ep, Ra, Da, and Kr. Also, a maximum increase in the average room temperature up to 97.77% is achieved at better thermal insulation (Bi = 10).

Studying the effect of the force convection boundary condition and radius magnetic field on fluid flow and heat transfer between coaxial pipes

AbstractAn investigation of the heat transfer of Newtonian fluid flow through coaxial two pipes with variable radius ratio has been conducted with the boundary conditions of forced convection on the inner pipe walls and a radius magnetic field. This paper presents an exact analytical solution to the momentum equation and a novel semi‐analytic collocation method for solving the full‐term energy equation that takes Joule heating into account as well as viscous dissipation. Based on the results of the numerical fourth‐order Runge–Kutta method, it was found that increasing the magnetic parameter decreased the amount of friction on the surface of the pipe walls and the rate of heat transfer. As the radius ratio of the two pipes increases, so does the skin friction and heat transfer rate on the internal pipe walls. As Eckert (Ec) and Prandtl (Pr) numbers increase, the mean temperature as well as the dimensionless temperature between the two pipes increases. The increase in Biot number (Bi) has the opposite impact on the mean temperature. As Ec, Pr, and Bi increase, so does the rate of heat transfer on the inner wall of the pipe.

Irreversibility analysis with hybrid cross nanofluid of stagnation point and radiative flow ([formula omitted]) based on engine oil past a stretchable sheet

The present study addressed the physical significance of the entropy generation for the mixed convection time-dependent flow of cross-hybrid nanoliquid due to the stretched surface at a stagnation point. The Plot for heat transport is discoursed by applying the role of thermal radiation under convective conditions. For hybrid nanofluid, engine oil is used as a base liquid with copper (II) oxide CuO and titanium dioxide TiO2 nanoparticles. The existing model is framed in the highly partial differential equation system. The governing equations have been transformed into a set of ODS's using a similar scaling operation. Following this, the resulting ODEs are solved numerically through the BVP4c. The primary goal of this research is to analyze the results of varying the stretching ratio parameter (λ), Weissenberg parameter (We), thermal radiation (Rd), and Biot number (Bi) for both pure TiO2 and CuO + TiO2/ EO hybrid nanofluid, on the velocity, temperature, drag force, heat transfer as well as entropy generation, and Bejan number was studied. A drop in velocity is observed with increasing values of the We and upsurge in velocity for rising value of unsteady parameter (A), while increasing values of both of these parameters are associated with rising temperatures. Copper and titanium oxide nanoparticles are used to increase Engine oil (EO) thermal enactment, making it a more useful base fluid. Further, some significant industrial and engineering applications are related to the present problem discourse.

Adsorption mechanism and mass transfer modeling of cesium and europium ions adsorption onto hematite-loaded polyacrylamide-gel composite: Pore Volume and Surface Diffusion Model Approach

The overall adsorption rate of cesium (Cs+) and europium (Eu3+) ions onto hematite-loaded polyacrylamide-gel composite (HPAC) in single and binary pollutant systems was explained with diffusional model that takes into consideration both external mass transfer and intra-particle diffusion. The influence of changing system temperature on the overall adsorption rate of Cs+ and Eu3+ onto HPAC was studied. The fitting process of the experimental data with the three studied isotherm models (Langmuir, Freundlich and Prausnitz–Radke (P-R)) showed that Prausnitz–Radke model has the highest correlation coefficients with the smallest average relative error percentage. The maximum P-R adsorption capacity of HPAC decreased from 2.06 and 3.35 L/g in single pollutant solution to 1.02 and 1.51 L/g in binary pollutant solution at 298 K for Cs+ and Eu3+,respectively. The values of surface diffusion coefficient (DS) ranged within 10−8 for Cs+ and 10−8-10−9 cm2/s for Eu3+ in both studied single and binary pollutant systems. The external mass transfer coefficient (KF) was in the range of 10−4 for Cs+ and 10−3 cm/s for Eu3+ in both studied systems. Lastly, the Biot number (Bi) exposed that the diffusion inside the particle controlled the studied cations's adsorption onto HPAC material.

Thermal conductivity effects on mixed convection flow of electrically conducting fluid along vertical magnetized plate embedded in porous medium with convective boundary condition

The temperature dependent thermal conductivity, surface heat flux and magnetohydrodynamic effects on electromagnetic fluid across the magnetically vertical surface placed in porous material have been performed numerically. To reduce excessive heating, the aligned magnetic field acts like a coating material to insulate the heat which is a very important mechanism in modern technologies. To solve this problem, the primary goal of the current analysis is to magnetize the surface. The developed partial differential equations of the present magnetic mechanism are changed in nonlinear coupled ordinary differential equations with defined stream functions and similarity variables for smooth algorithm and integration. The changed ODEs are again converted in similar form for numerical outcomes by applying Keller Box approach. The numerical outcomes are deduced in graphs and tabular form with the help of MATLAB program. The distinct parameters like thermal-conductivity number ξ, Biot number Bi, and porous number Ω in the flow model are affected by physical features such skin friction, heat transfer, and magnetic flux for velocity profile, magnetic field profile, and temperature profile together with their slopes. After obtaining the velocity graph, magnetic-field graph, and temperature graph, their gradients are numerically examined across the magnetically vertical surface. The current thermal and magneto hydrodynamics problem is very important in the fields of MRI resonance sequences, artificial heart wolves, internal heart cavities, and nanoburning technologies.

Improved modeling and correlation for heat transfer through a condensate droplet

In dropwise condensation, the heat transfer performance is determined by the heat transfer through a single droplet and the droplet size distribution on the cooled condensing surface. The conduction resistance inside a droplet is an important part of all heat transfer resistances, and it also affects the size distribution of small droplets further, such as in the popular population balance model. The heat transfer of a single droplet can be measured by the Nusselt number (Nu), and internal conduction resistance can be measured by the dimensionless shape factor (f). Previous studies have shown that Nu and f are functions of droplet contact angle (θ) and Biot number (Bi). Based on this, some fitting correlations of Nu or f in a certain range of Bi and θ are established. However, previous results show unreasonable values when Bi is very large or very small, the effective ranges of these fitting correlations are not broad enough, and the fitting accuracy is not gratifying enough. In this paper, the heat transfer characteristics of condensing droplets are investigated by a CFD method which is validated by comparing the simulation results and theoretical solution for hemisphere droplets. By adopting this method, f and Nu are obtained for droplets with 10° ≤θ≤ 170° and 10−3≤Bi≤ 104. Considering the relationship between current numerical results and the model of Kim and Kim (2011), an improved general fitting correlation covering all 374 simulation cases is proposed, and the root mean square (RMS) errors of f and Nu are 3.6% and 2.6%, respectively.

Effect of Magnetic Field on the Developing Thermal Field in a Duct Filled with Porous Media under Local Thermal Non-Equilibrium with a Nonlinear Flow Model

In this article, the numerical study of the influence of a magnetic field at the laminar forced convection in a thermally developing region coming under the influence of local thermal non-equilibrium (LTNE) of parallel plate channels completely immersed in the porous material is investigated. Constant wall heat flux boundary conditions are applied to the walls of the channel. In the nonlinear flow model, the Darcy-Brinkman-Forchheimer equation governs the flow field in the porous region, which is assumed to be unidirectional. The system is defined by certain well-known parameters, these being Darcy number (Da), thermal conductivity ratio (κ), Forchheimer number (F), Hartmann number (M), and Biot number (Bi). Numerical solutions have been obtained by applying a successive accelerated replacement (SAR) scheme. Exact solutions for the dimensionless temperature and the fully developed Nusselt number in the absence of the Forchheimer number (F = 0), for the fully developed thermal field, are obtained for the linear flow model, the Darcy-Brinkman model. Plots are given for the dimensionless temperature profiles in the fluid and solid phases, wall temperature, as well as the local Nusselt number at the parallel plate channel, which has been displayed. The effect of the magnetic field and the thermal conductivity ratio has a significant effect on the local Nusselt number.

Rayleigh-Taylor instability of viscous liquid film under an inclined substrate with time modulated temperature

The Rayleigh-Taylor instability of viscous liquid film under a temperature-modulated inclined substrate is studied. Long wave approximation and weighted residual model are used respectively to derive the nonlinear evolution equations of liquid film, and it is shown that the two evolution equations by the above two approaches are equivalent. The recurrence relation and damped Mathieu equation are obtained by employing Floquet theory and linear analysis, respectively. When amplitude a and oscillation frequency ω of the temperature exceed critical values, the film surface loses its stability. The stability diagram is obtained by linear analysis based on Floquet theory. Moreover, based upon the line approach, the nonlinear evolution equation is numerically simulated to give the spatiotemporal evolution of the free surface. Heated capillary tension might accelerate the production of suspended droplets due to the appearing of Marangoni number Ma produced by heating inclined wall. The results reveal that the larger Reynolds number Re, Marangoni number Ma and Biot number Bi will result in the instability of free surface. Moreover, the liquid film becomes more unstable at the smaller Froude number Fr and Weber number We.

Multi-fluid modeling of heat transfer in bubbling fluidized bed with thermally-thick particles featuring intra-particle temperature inhomogeneity

The multi-fluid model has been widely used to study heat transfer between gas and solid in bubbling fluidized beds. However, zero-dimensional (0D) model has been commonly used. The model assumes a uniform temperature distribution, which is only reasonable for thermally-thin particles (Biot number (Bi) < 1). However, one-dimensional (1D) model considering intra-particle temperature inhomogeneity inside particles is difficult to be implemented in multi-fluid model. To solve this issue, a corrected coefficient is introduced to quantitively feature the effects of intra-particle temperature inhomogeneity inside particles on external heat transfer, which forms a corrected 0D model. The corrected coefficient is correlated as a binary function of Bi and dimensionless temperature. The results of particle-scale modeling show that temperature profiles predicted by the corrected 0D model are the same as those of the 1D model for both thermally-thin and thermally-thick particles, while the 0D model overestimates heat transfer between particles and surrounding gas. The corrected 0D model is further implemented in the multi-fluid model to simulate particle cooling and heating process in bubbling fluidized beds. The results predicted by CFD simulations with both the 0D and the corrected 0D models are in good agreement with the experimental data of thermally-thin particles. For both the cooling and heating processes, a significant difference is observed for thermally-thick particles, indicating the importance of considering intra-particle temperature inhomogeneity in multi-fluid modeling. Consistent with the results of particle-scale modeling, the corrected 0D model predicts a smaller heat transfer rate between the gas and solid phases, as compared to the 0D model. Additionally, computational efficiency of the corrected 0D model is comparable to that of the 0D model.

Particle non-isothermality effect on the radiative thermal conductivity in dense particulate systems

Emerging high-temperature applications require a deeper understanding of radiative heat transfer in particulate systems, while existing theoretical tools are still limited in estimating the influence of the highly variable particle parameters (morphology, size, and optical properties) on heat transfer characteristics. In this work, a novel method is developed to determine radiative heat transfer in optically thick particulate media based on the radiation distribution factor. The important parameters of characterizing radiative heat transfer in particulate systems, i.e., transport coefficient and radiative thermal conductivity, are directly calculated by the Monte Carlo ray tracing method at particle scale. Two ideal cases of high and low thermal conductivity of solid particles, characterized by radiative Biot number Bi, are considered to investigate the effect of particle non-isothermality on the radiative thermal conductivity, which correspond to the cases of isothermal (Bi<<1) and non-isothermal particles (Bi>>1), respectively. The results show that radiative transfer is more intensive in isothermal particles due to the additional contribution of internal heat transfer in highly conductive particles. The difference between the limits of isothermal particles and non-isothermal particles is significant at high values of solid fraction and surface emissivity. In addition, the results are used to estimate the applicability domain of the analytic multiphase approach. This work brings new insights to the understanding of radiative heat transfer in particulate media.

Analytical solutions of reaction–diffusion phenomena by porous cylindrical pellets with finite length

AbstractAnalytical solutions were derived to predict transient change of intra‐particle concentration inside cylindrical pellets with finite length immersed in an infinitely large medium. After the derivation of steady‐state concentration by eigenfunction expansion, a transient solution was obtained by the separation of variables. Effects of the Thiele modulus (Φ) and aspect ratio of the pellet (H/R) were studied by calculating the average intra‐particle concentration at a stationary state. The effectiveness factor (η) was also affected by H/R and Φ, since irreversible first order kinetics was assumed. As Biot number (Bi) increased, both intra‐particle concentration and η increased due to the decrease of the mass transfer resistance of the surrounding film. Good agreement between analytical and numerical solutions was confirmed for both steady‐state and transient solution by comparing the analytical results with the numerical solutions by finite element method. To extend the results to a batch reactor with finite volume, Duhamel's theorem was applied by assuming time‐dependent boundary conditions to reflect the change in bulk concentration as a function of time. A method was proposed to measure intra‐particle diffusivity for batch adsorber by setting Φ = 0. To investigate the effect of H/R, η could be included in material balance of batch, continuously stirred tank reactor (CSTR), and fixed bed reactors to predict reactor performance assuming a pseudo‐steady state. First‐order reaction by egg‐shell catalysts as well as nonlinear reaction with two reactants could be solved numerically assuming finite cylindrical pellets. By solving coupled differential equations of material and energy balances, dynamic catalysis was confirmed from the enhancement of conversion, assuming forced oscillation of inlet temperature in CSTR.