Values Of Biot Number Research Articles

Transport of Cu–H2O nanofluid through a channel with wavy walls under velocity slip and connective boundary conditions

Present study examines the mixed convective peristaltic transport of Cu–H2O nanofluid with velocity slip and convective boundary conditions. Analysis is performed using the two-phase model of the nanofluid. Viscous dissipation and heat generation/absorption effects are also taken into account. Problem is formulated using the long wavelength and low Reynolds number approach. Numerical solutions for the pressure rise per wavelength, pressure gradient, axial velocity, temperature and heat transfer rate at the boundary are obtained and studied through graphs. Results show that the area of peristaltic pumping decreases with an increase in the nanoparticles volume fraction. Increase in the velocity slip parameter shows an increase of the pressure gradient in the occluded part of the channel. Further, addition of copper nanoparticles reduces both the axial velocity and temperature of the base fluid. Temperature of the nanofluid also decreases sufficiently for an increase in the value of Biot number.

Transient Heat Transfer Solution Using Improved Lumped Model

In present study, transient state one-dimensional heat conduction is analyzed using polynomial approximation method. In comparison with the classical lumped model, an improved lumped model have been employed as the classical lumped model is applicable for biot number of less than 0.1 and it cannot be used for high temperature gradient. An improved lumped model has been employed for sphere to calculate average temperature as a function of time for higher and lower values of biot number. The study shows that the lumped model provides better accuracy as compared to finite difference model.

MHD 3D flow of nanofluid in presence of convective conditions

An analysis has been carried out for the magnetohydrodynamic (MHD) flow of viscous nanofluid saturating porous medium. The flow is induced by a convectively heated permeable shrinking surface. Appropriate transformations reduce the nonlinear partial differential system to an ordinary differential system. Flow and heat transfer characteristics are computed by HAM solutions. The results of velocity, temperature and Nusselt number are analyzed for various parameters of interest. It is noted that higher nanoparticle volume fraction decreases the velocity field. Also the temperature and heat transfer rate are enhanced for larger values of Biot number.

Kinetics and mass transfer models for sorption of titanium industry effluent in activated carbon

The batch adsorber data obtained for removal of dissolved solids in titanium industry effluent by sorption in activated carbon was found to fit Freundlich, Langmuir, and Dubinin–Radushkevich isotherm models and the various non-linear error functions evaluated have low error values for these adsorption models. The applicability of pseudo-second-order models in adsorption kinetics has been explained and the predictive models at various dilutions were obtained. The external mass transfer coefficient obtained from various models, such as Mathews–Weber and Furusawa–Smith, was used to calculate the Biot number. The Biot number values were less than 100, which suggest that the external film coefficients control the adsorption process. The change in enthalpy of the adsorption process was −48 kJ mol−1, suggesting the release of exothermic heat during the sorption process. The analysis of various functional groups in the effluent was done using Fourier transform infrared spectroscopy.

Unsteady Boundary Layer Flow Induced by a Stretching Sheet in a Rotating Fluid with Thermal Radiation

The MHD unsteady flow heat and mass transfer of a viscous incompressible fluid induced by a stretching surface in a rotating fluid taking the effects of thermal radiation, heat absorption and first order chemical reaction into account with convective boundary condition. The rotation parameter is found to reduce the velocities and the magnetic parameter prevents the flow reversal in the x – direction. The increasing values of Biot number generate thicker thermal boundary layers resulting in the rise of temperature. The Schmidt number and chemical reaction parameter are found to have a strong influence on the species concentration resulting in small values. The rate of heat transfer is enhanced by magnetic field, thermal radiation parameter and rotation parameter. The rotation parameter and time are observed to enhance the rate of mass transfer.

NUMERICAL ANALYSIS OF HEAT EXCHANGE IN A POROUS CHANNEL WITH HEAT GENERATION AND LOCAL THERMAL NONEQUILIBRIUM

We numerically investigate forced convection heat transfer with heat generation and local thermal nonequilibrium in a porous channel bounded by parallel plates. Macroscopic continuity, momentum, and energy equations are presented and solved. Local thermal nonequilibrium is considered by means of independent equations for the solid matrix and the working fl uid. The used numerical methodology is based on the control-volume approach. The eff ects of heat generation rate, thermal conductivity ratio, and Biot interstitial number on the local thermal equilibrium are presented. The main results obtained particularly show that lower values of the temperature diff erence (local thermal equilibrium) between the solid and the fl uid phases are obtained at low values of conductivity ratio and high values of Biot number, while large local temperature diff erences (local thermal nonequilibrium) are obtained for low Biot numbers values and high thermal conductivity ratios. Moreover, increasing heat generation rate generally leads to thermal nonequilibrium accentuation, particularly and most signifi cantly for low Biot number values.

Effect of Induced Magnetic Field on Natural Convection with Newtonian Heating/Cooling in Vertical Concentric Annuli

This paper is aimed to investigate the effect of induced magnetic field and Newtonian heating/cooling on fully developed laminar natural convective flow of a viscous incompressible and electrically conducting fluid in the presence of radial magnetic field. The governing equations of the model are transformed into non-dimensional simultaneously ordinary differential equations and solved analytically. We have analyzed the influence of the Hartmann number and Biot number on the fluid velocity, induced magnetic field and induced current density by the graphs while the values of skin-friction and mass flux are given in the tabular form. We observed that the values of the velocity, induced magnetic field and induced current density have decreasing tendency with increasing the values of Hartmann number. Also, the results show that on increasing the value of Biot number leads to increase the velocity, induced magnetic field, induced current density and skin-friction in case of the Newtonian heating and decrease the velocity, induced magnetic field, induced current density and skin-friction in case of the Newtonian cooling.

MHD flow of Jeffrey nanofluid with convective boundary conditions

This article investigates the effects of convective heat and concentration conditions in magnetohydrodynamic flow of non-Newtonian fluid with nanoparticles. Convective type boundary conditions are utilized for heat and nanoparticles concentration. The nonlinear partial differential equations are reduced into the nonlinear ordinary differential equations by suitable similarity variables. Homotopy analysis method is employed to obtain the dimensionless velocity, temperature and nanoparticles concentration expressions. Graphical results for temperature and nanoparticles concentration are plotted and examined. Numerical values of skin friction coefficient are computed and discussed. Heat transfer and nanoparticles concentration transfer rates at the wall are examined by plotting the graphs of different governing physical parameters. We noticed that the temperature and nanoparticles concentration profiles are enhanced when the values of Biot numbers are increased. An increase in thermophoresis parameter leads to an enhancement in the temperature and nanoparticles concentration. On the other hand the increasing values of Brownian motion parameter has reverse effects on the temperature and nanoparticles concentration fields.

Mixed convective peristaltic transport in a vertical channel with Robin’s condition

This paper describes the mechanism of the mixed convective peristaltic flow of a viscous fluid in an asymmetric vertical channel. A new type of boundary conditions is introduced in which one wall is kept at a constant temperature, while a convective heat flux is taken on the other wall. The Boussinesq’s and long wavelength approximations are employed in the process of modelling. The problems associated with the fluid motion and heat transfer are solved and exact solutions have been obtained in the wave frame of reference. Effects of Biot number, Grashof number and source/sink parameter have been analyzed through graphs. It is observed that increasing Biot number decreases the temperature, whereas temperature increases with an increase in the values of source/sink parameter. The pressure drop decreases with ascending values of Biot number and descending values of source/sink parameter and Grashof number. The pressure gradient is less in magnitude in absence of buoyancy forces, convective heat exchange at the boundary and source/sink. An increase in Biot number and a decrease in source/sink parameter and Grashof number results an increase in the size of the left part of the bolus.

Simulation of non-Newtonian fluid-food particle heat transfer in the holding tube used in aseptic processing operations

In aseptic food processing, a hot carrier fluid, usually non-Newtonian, is used to thermally sterilize the food particles suspended in a holding tube. The effects of a spherical food particle's diameter relative to the holding tube diameter on the heat transfer rates are investigated using Computational Fluid Dynamics (CFDs) simulations. As the particle to holding tube diameter ratio (blockage-ratio) increases, higher particle heating rates were usually observed when compared to the heating rates of an unconfined particle. Variations in the non-Newtonian fluid viscosities with shear rate and temperature played important roles in affecting the local Nusselt numbers. Significant effect of the blockage ratio was found on the integrated lethality of the thermal treatment at low particle Reynolds numbers (Rep). For such cases, conventional steady state fluid–particle heat transfer coefficient correlations, applicable when the particle is immersed in an unbounded stream of fluid, may lead to erroneous predictions of integrated lethality of treatment inside the holding tube. The thermal processing of the food particle was compared using two approaches. In the first rigorous approach, the transient and spatial fluid–particle heat flux variations around the sphere were accounted while in the second approach, a constant heat transfer coefficient value was specified as the boundary condition. Even at intermediate Biot number values (4–17), considerable differences between the two approaches could be observed in the conductive heating patterns inside the sphere as well as in the integrated lethalities.

Generalization of the integral transform method to nonlinear heat-conduction problems in multilayered spherical media

A method of solving heat conduction problems in multilayered spherical media is presented; it is based on producing an alternative formulation in which the inner layer of the composite sphere is lumped and the transient temperature distribution in the outer layer is obtained, in this, the nonlinear boundary conditions are treated as a source. The temperature distribution of the alternative formulation is then solved analytically. Different parametric studies are worked out and plotted to compare the two formulations for different values of Biot number and different thermal characteristic ratio with those obtained from a numerical solution developed using an explicit finite difference method, and to find the limiting criterion where breakdown of the approximation occurs.

Second-Order Model for Transient Heat Conduction in a Sphere

T HE use of simple mathematical formulations to solve the heat conduction problems is of great interest for engineering calculations. Though the traditional analytical models are widely used because of their numerous industrial applications [1–4], they sometimes do not provide satisfaction for large values of Biot number. In addition, these models do not predict the temperature inside the solid (at the center, for example). In the case of spherical geometries, the analytical solution is expressed as an infinite series which necessitates the numerical solution of a characteristic equation for each value of theBiot number [5]. Moreover, this analytical method sometimes requires the calculation of several hundred terms in the series in order to reach the expected accuracy [6]. A perturbation method has been used in [7] to develop first-order models which can be viewed as improved lumped models for the slab, infinite cylinder, and spherical geometries. A second-order model has been examined in [8] for the slab and in [9] for the cylinder. The aim of this paper is to extend the ideas of [7–9] to the case of a sphere subjected to convection to a surrounding fluid. It is shown that this model gives accurate enough results even for an infinite Biot number. The validity of the model is discussed and the region of the sphere where it can be used is determined.

Modeling of arsenic adsorption kinetics of synthetic and contaminated groundwater on natural laterite

A simple shrinking core model is applied to predict the adsorption kinetics of arsenite and arsenate species onto natural laterite (NL) in a stirred tank adsorber. The proposed model is a two-resistance model, in which two unknown parameters, external mass transfer coefficient ( K f ) and pore diffusion coefficient ( D e ) are estimated by comparing the simulation concentration profile with the experimental data using a nonlinear optimization technique. The model is applied under various operating conditions, e.g., initial arsenic concentration, NL dose, NL particle size, temperature, stirring speed, etc. Estimated values of D e and K f are found to be in the range of 2.2–2.6 × 10 −11 m 2/s and 1.0–1.4 × 10 −6 m/s at 305 K for different operating conditions, respectively. D e and K f values are found to be increasing with temperature and stirrer speed, respectively. Calculated values of Biot numbers indicate that both external mass transfer and pore diffusion are important during the adsorption. The model is also applied satisfactorily to predict the arsenic adsorption kinetics of arsenic contaminated groundwater–NL system and can be used to scale up.

Exact solution of a problem on heat exchange in channels under unsteady conditions at low biot number values

A number of exact solutions of a problem on heat exchange in channels under unsteady conditions at the low Biot numbers and sufficiently low Brune numbers with and without consideration for a heat carrier stay period in a pipeline are obtained.

Performance and optimization analysis of SRC profile fins subject to simultaneous heat and mass transfer

Fin material near the tip of a uniform cross sectional (UC) fin does not participate actively in transferring heat. This effect may seem to have progressed much with the increase in fin length. A uniform cross sectional fin with a step reduction in local cross section (SRC) not only increases the effective utilization of fin material near the tip but also it promotes the ease of fabrication. In this study, an effort has been devoted to determine analytically the overall fin performance of both longitudinal and pin fins of SRC profile under fully dry, partially wet and fully wet conditions. The effect of various design and psychometric parameters on the fin performance of SRC fins has been investigated and compared it is with the corresponding UC fin. A scheme for optimizing SRC fins has also been demonstrated in the present work. From the result, it can be highlighted that the optimum values of Biot number and aspect ratio of SRC fins increase with the increase in relative humidity for the same fin volume. In comparison with the UC fin for the identical fin volume, the SRC fin transfers more rate of heat and consequently, this difference in heat transfer rate increases slowly with the relative humidity.

Spontaneous spreading of surfactant-bearing drops in the sorption-controlled limit

Axisymmetric spreading of a liquid drop containing a soluble surfactant on a smooth solid substrate is numerically investigated for the case in which surfactant mass transfer between the interface and the bulk liquid is sorption/kinetic controlled. The fastest spreading rate is achieved by drops with O ( 1 ) values of Biot number for which the rate of surface convection is comparable to the sorption rate, and the surfactant molecules transferred to the interface are effectively convected to the contact line region.

Dynamics and stability of a thin liquid film on a heated rotating disk film with variable viscosity

A theoretical analysis of the thermal effects on the dynamics of a thin nonuniform film of a nonvolatile incompressible viscous fluid on a heated rotating disk has been considered and the effects of temperature-dependent viscosity and surface tension have been analyzed. A nonlinear evolution equation describing the shape of the film interface has been derived as a function of space and time and its stability characteristics have been examined using linear theory. It has been observed that the infinitesimal disturbances decay for small wave numbers and are transiently stable for large wave numbers, for both zero and nonzero values of Biot number.

RETRACTED: Critical radius for radial heat conduction: a necessary criterion but not always sufficient

Use of the critical radius for radial heat conduction in thermal insulation systems has been widely reported in the literature. When it is desirable to increase heat dissipation in these systems, the critical radius can be used in a definitive manner to maximize the heat dissipation. However, if it is desirable to decrease heat gain or heat loss, the critical radius only serves as a necessary condition, but it is not sufficient. To address design issues of such thermal systems, the crossover radius is utilized. The crossover radius is defined as a radius greater than the critical radius such that the heat transfer with the corresponding amount of insulating material is equal to that of the bare thermal system. Both cylindrical and spherical systems are considered. Justification: It will be pointed out later in this paper that the concept of the crossover insulation radius is applicable when the Biot number is less than 1 in a cylindrical system or when it is less than 2 for the spherical system. Once the Biot number is larger than these respective values, the heat transfer reduction is immediate with application of any amount of insulation. However, the justification for the present work is threefold. First, all the available heat transfer textbooks discuss only the concept of the critical insulation radius and the subsequent increase in the relative heat transfer as compared to the bare radial system. The present work extends the discussion beyond the critical insulation radius along with the impact on the relative heat transfer and thus completing the discussion. Secondly, even today there are smaller, cylindrical thermal systems that are employed in the refrigeration and cryogenics industry which have rather small value of the Biot numbers. Third, with the current trends in the compact heat exchangers, as thermal systems get smaller and smaller, it is natural that the smaller values of Biot number will be encountered more frequently where the concept of the crossover insulation radius is applicable. At times such systems are improperly insulated resulting actually in an energy penalty due to increased heat transfer. With adequate knowledge about the crossover insulation radius, these systems can be properly insulated while improving their energy effectiveness. Such present-day smaller thermal systems are discussed next. In this work the Biot number is defined as the ratio of convective heat transfer coefficient times the inner radius of the insulation divided by the thermal conductivity of the insulating material. Thus all the three parameters will be addressed first before discussing the smaller Biot numbers. As reported in one cryogenic heat transfer study, the convective heat transfer coefficient varies almost linearly with the prevailing temperature difference [Adv. Cryog. Eng. 6 (1960) 499]. The value varies from 8.1 to 11.6 W/m2 K over the temperature difference of 100–300 K. According to the most recent developments, the smallest pipe or tubing diameter that is employed in the industry is 2/16 in. for aluminum and 3/16 in. for copper [[12]; [13] ; [14], http://www.johnstonesupply.com/main/]. The capillary tubing is available even in smaller sizes. The thermal conductivity values of foam insulation at cryogenic temperatures range from 0.033 to 0.055 W/m K [R.H. Kropschot, Low-Temperature Insulation, Applied Cryogenic Engineering, John Wiley, 1962, pp. 152–169]. The interstitial gas contained in the cells plus internal radiation and a contribution due to solid conduction determines the thermal conductivity of foam. Evacuation of foam provides the initial apparent low thermal conductivity. But many Freon-blown types of foam will be penetrated by air over a period of time, which may increase their conductivity by as much as 30%. Based on the available data for refrigeration tube diameters (2/16 in. for aluminum and 3/16 in. for copper) and the insulation thermal conductivity values, the current range of smaller Biot numbers for the refrigeration systems turns out to be 0.23–0.84, which makes the present work relevant. It should be noted that as the thermal conductivity increases with age, the Biot number will become even smaller. Further, with the current trends in miniaturization, it is expected that the smaller Biot numbers will be encountered more frequently in the future. Representative energy penalties are analyzed next.

Heat and mass transfer models in convection drying of clay slabs

Abstract Drying kinetics data for convection drying of industrially prepared roof tile clay slab were approximated with different mathematical models. Applied conventional model analysis enables evaluation of main transport properties: effective diffusion coefficient, mass and heat transfer coefficients, thermal conductivity, drying constant, and exponential model parameters. A neural network-based drying model was established using backpropagation algorithm for dynamics modelling of moisture content and temperature of thin clay sample. Obtained results confirm the assumption that both, the heat and mass transfers, are under external conditions. Very small values of Biot numbers confirm that fact. Drying air temperature and initial moisture content of clay strongly influence the drying kinetics and transport properties. The dependence between the drying air temperature and evaluated transport properties shows an exponential trend. Tomas and Skansi exponential model parameter, n , is independent from temperature. At lower values of initial moisture content of clay higher drying rates are achieved, which results with higher values of calculated transport properties. It was shown that neural network as an alternative method has potential for modelling the drying process and predicting drying dynamics based on experimental data.

Improved lumped models for asymmetric cooling of a long slab by heat convection

Unsteady cooling of a long slab by asymmetric heat convection is analyzed within the framework of lumped parameter model. As the application of classical lumped model is limited to values of Biot numbers less than 0.1075, we propose improved lumped models that can be applied in transient heat conduction with larger values of Biot numbers. The proposed lumped models are obtained through two point Hermite approximations for integrals. Closed form analytical solutions are given for the lumped models. It is shown by comparison with a reference finite difference solution of the original distributed parameter model that the proposed higher order lumped model ( H 1,1/ H 0,0 approximation) yields significant improvement of average temperature prediction over the classical lumped model.