Heat transfer to submerged surfaces in coarse-grained pressurized fluidized beds and fixed beds

This contribution reports on the theory underlying a uniform representation of heat transfer to submerged surfaces in fixed bed reactors and of gas convective part of heat transfer in fluidized beds with coarse-grained bulk solids and/or at elevated pressure. Based on an analysis of the pressure drop behaviour of fixed bed percolation at different gas pressures and with different bulk solids, a new dimensionless pressure drop parameter was developed. Fixed bed heat transfer data are very well correlated by this new dimensionless number. As soon as fluid throughput is in excess of minimum fluidization velocity, the pressure drop parameter transforms into the well-known Archimedes number. These two dimensionless numbers are connected by the condition of equilibrium for pressure drop and mass of practices in a fluidized bed. This equilibrium is fulfilled as soon as fluidization commences. Up to now, the Archimedes number has been generally accepted as the significant parameter, determining the gas convective part of heat transfer in fluidized beds; however, without any physical interpretation of this parameter. Introduction of the pressure drop number, which is consistent with the Archimedes number, reduces the heat transfer behaviour to pressure drop characteristics. The usefulness of this concept is proven by the comparison of experimental results and prediction.     

Heat transfer mechanisms in gas fluidized beds. Part 1: Maximum heat transfer coefficients

This contribution presents the prediction of maximum heat transfer coefficients in bubbling fluidized beds, which takes into account thermal and fluid-dynamic properties of particulate material and fluidizing agent. The analysis suggests that heat transfer between heating or cooling surfaces and bubbling fluidized beds consists mainly in a particular manifestation of convective heat transfer. Another feature is an appropriate modelling of the particle convective component leading to a two-phase Prandtl number.     

Heat transfer in fluidized beds: design methods

Large-scale fluidized beds for commercial processes commonly require heat transfer surfaces. Design then demands that heat transfer coefficients be specified. Empirical correlations are unable to cover the wide range of variables and conditions encountered. Mechanistic models are more reliable, but must be chosen carefully. For bubbling beds, the packet model approach gives reasonable predictions for the convective component of transfer, but further work is required to provide reliable estimates of two required time constants, dependent on the hydrodynamics. For industrial-scale circulating beds, a mechanistic model that incorporates the key factors influencing heat transfer, assumes fully developed transfer, and utilizes results from large-scale units is recommended.     

Particle scale study of heat transfer in packed and bubbling fluidized beds

The approach of combined discrete particle simulation (DPS) and computational fluid dynamics (CFD), which has been increasingly applied to the modeling of particle‐fluid flow, is extended to study particle‐particle and particle‐fluid heat transfer in packed and bubbling fluidized beds at an individual particle scale. The development of this model is described first, involving three heat transfer mechanisms: fluid‐particle convection, particle‐particle conduction and particle radiation. The model is then validated by comparing the predicted results with those measured in the literature in terms of bed effective thermal conductivity and individual particle heat transfer characteristics. The contribution of each of the three heat transfer mechanisms is quantified and analyzed. The results confirm that under certain conditions, individual particle heat transfer coefficient (HTC) can be constant in a fluidized bed, independent of gas superficial velocities. However, the relationship between HTC and gas superficial velocity varies with flow conditions and material properties such as thermal conductivities. The effectiveness and possible limitation of the hot sphere approach recently used in the experimental studies of heat transfer in fluidized beds are discussed. The results show that the proposed model offers an effective method to elucidate the mechanisms governing the heat transfer in packed and bubbling fluidized beds at a particle scale. The need for further development in this area is also discussed. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Heat transfer within dynamically structured bubbling fluidized beds subject to vibration: A two-fluid modeling study

Bubbling fluidized beds are often used to achieve a uniform particle temperature distribution in industrial processes involving gas and particles. However, the chaotic bubble dynamics pose significant challenges in scale-up. Recent work (Guo et al., 2021, PNAS 118, e2108647118) has shown that using vibration can structure the bubbling pattern to a highly predictable manner with the characteristic bubble properties independent of system width, opening opportunities to address key issues associated with conventional bubbling fluidized beds. Herein, using two-fluid modeling simulations, we studied heat transfer characteristics within the dynamically structured bubbling fluidized bed and compared to unstructured bubbling fluidized beds and packed beds. Simulations show that the structured bubbling fluidized bed can achieve the most uniform particle temperature distribution because it can achieve the best particle mixing while maintaining a global heat transfer coefficient similar to that of a freely bubbling fluidized bed.     

Modeling of radiative heat transfer between high-temperature fluidized beds and immersed walls

This paper presents a theoretical model for predicting the radiative heat transfer rate between high-temperature fluidized bed and immersed walls, which can be used upon the base of emulsion packet model of heat transfer in bubbling fluidized bed. The model adopted radiative flux computation method to calculate radiative heat transfer between fluidized disperse phase contacting to the wall and immersed walls, in which the absorption and back-scattering coefficients was obtained from the reflectivity and the absorptivity of a layer of disperse media of a single particle thickness. In such a model, many factors, such as particle size, particle emissivity, bed void fraction, fluidized bed and wall temperatures, and so on, are included theoretically to calculate radiative heat transfer between immersed walls and fluidized beds. As a result, the model results provide a reasonable explanation of the experimental observation of that radiative heat transfer rate in fluidized beds increases with the increases of the superficial fluidizing velocity. In addition, the modeling prediction for the trend of radiative heat transfer rate between the fluidized bed and its immersed surface on the variation of wall temperature conforms to the classical experimental trend.     

Heat transfer in three-phase (G/L/S) circulating fluidized beds with low surface tension media

Characteristics of heat transfer were investigated in a three-phase circulating fluidized bed whose diameter and height were 0.102 m (ID) and 2.5 m, respectively. Effects of gas and liquid velocities, particle size (0.5–3.0 mm), solid circulation rate (2.0–6.5 kg/m² s), and surface tension (47.53–72.75×10⁻³ N/m) of liquid phase on the heat transfer coefficient were examined. It was found that the heat transfer coefficient (h) between the immersed vertical heater and the riser proper of the three-phase circulating fluidized bed increased with increase in gas and liquid velocities, but did not change considerably with a further increase in liquid velocity, even in the higher range. The value of heat transfer coefficient increased gradually with increase in the size of fluidized solid particles without exhibiting the local minimum, which represented that there was no bed contraction in three-phase circulating fluidized beds due to the higher liquid velocity. The heat transfer system could attain a stabilized condition more easily with increase in particle size. The value of heat transfer coefficient increased with increase in solid circulation rate in all the cases studied due to the increase of solid holdup in the riser. The value of heat transfer coefficient decreased with increase in surface tension of liquid phase, due to the decrease of bubbling phenomena and bubble holdup. The decrease in liquid surface tension could lead to an increase in elapsed time from which the temperature difference between the heater surface and the riser became an almost constant value. The experimentally obtained values of heat transfer coefficient were well correlated in terms of dimensionless groups as well as operating variables.     

On the heat transfer mechanism in three phase fluidized beds

A two resistance model is proposed for the heat transfer between a coaxially mounted heater and a three phase fluidized bed. Effects of gas and liquid velocity and particle size on individual heat transfer resistances in the heater and in the fluidized bulk zones have been determined. The optimum bed porosity at which the maximum heat transfer coefficient occurred coincided with the bed porosity at which the boundary layer thickness around the heater attained a minimum value. The fluidized bed resistance attained its minimum value when the maximum heat transfer coefficient is achieved in two and three phase fluidized beds. The heat transfer in the zone adjacent to the healer is found to be the rate controlling step since the contribution of fluidized bed resistance was found to be less than 10% of the heater zone resistance in two and three phase fluidized beds. The heat transfer resistances in liquid and three-phase fluidized beds have been represented by a modified Stanton and Peclet numbers based on the heat transfer resistances in the heater zone and in the fluidized bulk zone in series.     

Heat transfer in three-phase fluidized beds

Heat transfer coefficients h have been measured in two-phase (water—air, water—glass beads) and three-phase (water—air—glass beads) fluidized beds. Experiments were performed over a wide range of liquid and gas flowrates in a 0.24 m diam. column fitted with an axially mounted cylindrical heater. Four solids were employed ranging in size from 0.5 to 5 mm.Typical maximum values of h in the three-phase, liquid—gas, liquid—solid and liquid beds were approximately 4800, 4300, 3800 and 1300 W/m²K respectively. In the three-phase beds h generally increased with liquid and gas velocity and with particle size. Correlations are presented to calculate h in the different beds.     

Mass transfer in two-and three-phase fluidized beds

The effects of liquid (0.03-0.12 m/s) and as (0.04-0.20 m/s) velocities, and particle size (0-8.0 mm) on the volumetric mass transfer coefficients at the grid zone have been determined in a 0.152 mI.D. x 1.8 m high Plexiglas column. The volumetric mass transfer coefficient in the grid zone increases with increasing gas velocity and particle size. However, the coefficient exhibits a maximum value at an optimum bed porosity condition. The volumetric mass transfer coefficients in terms of the Sherwood number in three-phase fluidized beds have been correlated with the Schmidt number and particle Reynolds number which is related to the energy dissipation rate in the beds based on the local isotropic turbulence theory. Also, the coefficient has been correlated with the experimental variables.     

Wall-to-bed heat transfer in three-phase fluidized beds of low density particles

Bed-to-wall heat transfer was measured in three-phase fluidized beds under conditions typical of biochemical process applications. The thermal resistance of the fluidized bed, which was significant in the absence of gas, became negligible when gas was introduced. Decreasing the particle density at constant gas and liquid velocity increased the bed-to-wall heat transfer coefficient. Previously published heat transfer correlations were used and gave poor predictions of our data. A new correlation was developed which predicted very well all the heat transfer coefficient results in this paper.     

Wall-to-bed heat transfer coefficient in gas—liquid—solid fluidized beds

Wall to bed heat transfer has been studied in three-phase fluidized beds with a cocurrent up-flow of water and air. Six sizes of glass beads, two sizes of activated carbon beads and one size of alumina beads, varying in average diameter from 0.61 to 6.9 mm and in density from 1330 to 3550 kg/m³, were fluidized in a 95.6 mm diameter brass column heated by a steam jacket. Complementary heat transfer experiments have been performed also for a gas–liquid cocurrent column and liquid–solid fluidized beds. The wall-to-bed coefficient for heat transfer in the gas–liquid–solid fluidized bed is evaluated on the basis of the axial dispersion model concept. The ratio of the wall-to-bed heat transfer coefficient in the gas–liquid–solid fluidized bed to that in the liquid–solid fluidized bed operated at the same liquid flow rate is correlated in terms of the ratio of the velocity of gas to that of liquid and the properties of solid particles. A correlation equation for estimating the wall-to-bed heat transfer coefficient in the liquid–solid fluidized bed is also developed.     

Wall-to-bed heat transfer in liquid—solid and gas—liquid—solid fluidized beds part II: Gas—liquid—solid fluidized beds

Wall-to-bed heat transfer in gas—liquid—solid fluidized beds with a cocurrent upflow was analyzed on the basis of a series thermal resistance model. The effective radial thermal conductivity and the apparent wall heat transfer coefficient were determined over a wide range of experimental conditions. The behavior of the effective thermal conductivity strongly depends on the flow mode for the three-phase fluidized bed, directly indicating the trend of the radial liquid mixing. The modified Peclet number for the radial thermal diffusivity takes on a minimum with respect to the liquid velocity in a manner similar to that in a liquid—solid fluidized bed, but the value of the modified Peclet number decreases significantly with gas velocity. The apparent wall heat transfer coefficient can be correlated well with a Colburn type equation which at zero gas velocity reduces to the same equation as that proposed for liquid—solid fluidization, as follows: j′_H = 0.137 Re′_l.g^?0.271     

Heat and mass transfer characteristics in a gas-slurry-solid fluidized bed

The gas-slurry-solid fluidized bed is a unique operation where the upward flow of a liquid-solid suspension contacts with the concurrent up-flow of a gas, supporting a bed of coarser particles in a fluidized state. In the present study we measured the gas holdup, the coarse particle holdup, the cylinder-to-slurry heat transfer coefficient, and the cylinder-to-liquid mass transfer coefficient at controlled slurry concentrations. The slurry particles were sieved glass beads of 0.1 mm average diameter and their volumetric fraction was varied at 0, 0.01, 0.05 or 0.1. The slurry and the gas velocities were varied up to about 12 and 15 cm/s, respectively. The coarse particles fluidized were sieved glass beads of average diameters of 3.6 and 5.2 mm. The individual phase-holdup values were measured and served for use in correlating the heat and mass transfer coefficients. The heat and mass transfer coefficients in the slurry flow, gas-slurry transport bed, slurry-solid fluidized bed and gas-slurry-solid fluidized bed operations can be correlated well by dimensionless equations of a unified formula in terms of the Nusselt (Sherwood) number, the Prandtl (Schmidt) number and the specific power group including the energy dissipation rate per unit mass of slurry, with different numerical constants and exponent values, respectively, to the heat and mass transfer coefficients. The presence of an analogy between the heat and mass transfer from the vertically immersed cylinder in these slurry flow, gas-slurry transport bed and gas-slurry-solid fluidized bed systems is suggested.     

Numerical investigation of gas-solid heat transfer in rotating fluidized beds in a static geometry

Gas-solid heat transfer in rotating fluidized beds in a static geometry is theoretically and numerically investigated. Computational fluid dynamics (CFD) simulations of the particle bed temperature response to a step change in the fluidization gas temperature are presented to illustrate the gas-solid heat transfer characteristics. A comparison with conventional fluidized beds is made. Rotating fluidized beds in a static geometry can operate at centrifugal forces multiple times gravity, allowing increased gas-solid slip velocities and resulting gas-solid heat transfer coefficients. The high ratio of the cylindrically shaped particle bed “width” to “height” allows a further increase of the specific fluidization gas flow rates. The higher specific fluidization gas flow rates and increased gas-solid slip velocities drastically increase the rate of gas-solid heat transfer in rotating fluidized beds in a static geometry. Furthermore, both the centrifugal force and the counteracting radial gas-solid drag force being influenced by the fluidization gas flow rate in a similar way, rotating fluidized beds in a static geometry offer extreme flexibility with respect to the fluidization gas flow rate and the related cooling or heating. Finally, the uniformity of the particle bed temperature is improved by the tangential fluidization and resulting rotational motion of the particle bed.     

Axial pressure profile in circulating fluidized beds

Design and operation of a circulating fluidized bed requires the knowledge of fluid mechanics. According to heat and mass transfer as well as chemical reactions, the effect of the set superficial gas velocity on the axial pressure profile is of particular interest. The axial pressure profile was measured for a variety of solids, as a function of the superficial gas velocity, in a cylindrical circulating fluidized bed with an inner diameter of 0.19 m and an overall height of 11.5 m. Depending on the solids content and superficial gas velocity, two or one sections can be observed in the plant where the pressure gradient is constant. A pressure profile with one pressure gradient exists only at high gas velocities, so long as the acceleration pressure drop immediately above the gas distributor is negligible. Comparison of measured pressure drops in circulating fluidized beds with those measured in vertical pneumatic conveying led to a state diagram for vertical gas-solid flows. The operation behaviour of different types of circulating fluidized bed plants can be explained with the aid of this diagram.     

Bed-to-wall heat transfer characteristics in a circulating fluidized bed

The bed-to-wall heat transfer coefficients were measured in a circulating fluidized bed of FCC particles (d_p = 65 μm). The effects of gas velocity (1.0–4.0 m/s), solid circulation rate (10–50 kg/m²s) and particle suspension density (15–100 kg/m³) on the bed-to-wall heat transfer coefficient have been determined in a circulating fluidized bed (0.1 m-ID x 5.3 rn-high). The heat transfer coefficient strongly depends on particle suspension density, solid circulation rate, and gas velocity. The axial variation of heat transfer coefficients is a strong function of the axial solid holdup profile in the riser. The obtained heat transfer coefficient in terms of Nusselt number has been correlated with the pertinent dimensionless groups     

Heat transfer in aggregative and particulate liquid-fluidized beds

Wall-to-bed heat transfer in liquid fluidized beds, particulately and aggregatively fluidized, was studied. Glass particles fluidized with water gave particulate fluidization and lead particles with water gave aggregative fluidization. Local heat transfer coefficients and bed temperature profiles were measured. Heat transfer coefficients were found to be strongly dependent on particle size and porosity and increased with increasing particle size, but were independent of the height of the heater surface from the grid. Any variations in local bed properties, such as porosity do not affect wall-to-bed heat transfer. The heat transfer coefficients show a characteristic, maximum at porosities near 0.7 for both systems. Bed temperature profiles deviate considerably from open-pipe values.A two-resistance model for the heat transfer resistance agrees well with the data. Bed resistance is modeled by a radial eddy diffusivity, which indicates the mixing effectiveness in the core of the bed. Glass beds (particulate) show a maximum mixing effectiveness at porosities near 0.7 and the mixing effectiveness increases with particle diameter. Lead beds (aggregative) show two maxima in mixing effectiveness, the first between porosities of 0.5 and 0.6, and the second between porosities of 0.7 and 0.8. Mixing is greatest at an intermediate particle size in the case of lead beds. In both systems the fraction of the total resistance in the bed core increases as porosity decreases towards packed bed conditions.