Determination of the Nusselt versus Graetz correlation for heat transfer in channels of sinusoidal cross-section

Metallic catalytic converters are composed of monoliths through which pass hundreds or thousands of parallel channels. Their mode of manufacture is such that each channel has the cross-section of a sinusoidal curve closed off by a nearly straight edge. During the operation of such converters, heat generated at the channel walls is transferred to the gases flowing through the channels. To understand the overall heat transfer characteristics of the monolith, it is necessary to understand the heat transfer rate between the channel walls and the fluid contained within them. With the use of the computational fluid dynamics package FIDAP, a three-dimensional model of a single channel was used to determine the local Nusselt number (Nu) versus Graetz number (Gz) correlation for heat transfer between the fluid and the walls of the channel. Flow through the channel was laminar and developing from a flat velocity profile at the channel inlet to the fully developed flow towards the outlet. Three different models were developed which corresponded to sinusoid height to width aspect ratios of 5 : 2, 3 : 2, and 1:1, respectively. The Nu vs. Gz correlations for the straight edge, curved sinusoidal edge and entire perimeter were calculated and are reported.     

Short-channel structured reactor: Experiments versus previous theoretical design

The study deals with the so-called short-channel structures (very short monoliths) that were introduced by Ko?odziej and ?ojewska [1] on the theoretical basis. The structures are very short so the entrance (mixing) section occupies majority of the channel length. This leads to highly enhanced heat and mass transfer, but the flow resistance is increased as well. The article presents experimental results of flow resistance, heat and mass transfer for short-channel structures of sine and triangular cross-sectional channel shape. The results presented in terms of dimensionless quantities confirmed theoretical consideration presented in [1]. The efficiencies of the structure have been derived that show the structures as more efficient when compared to classic monoliths or packed bed reactors.     

Effect of the Re number on heat and mass transport in a catalytic monolith

The numerical investigation of inter-phase heat transfer in a catalytic combustor, under laminar flow regime at different values of the Re number, is performed by means of a 2D axi-symmetric model of a single monolith channel. Numerical results highlight that axial diffusion comes to play an important role in the ignition region also at high convective fluxes (high Re) due to the strong flow perturbation accompanying light-off, with the consequences that: (a) the ignition position is not a linear function of contact time, as it would be expected at high Re; (b) the heat and mass transfer between surface and bulk gas phase are non-linearly affected by Re, especially in the entrance and in the ignition regions.A previously developed correlation for Nu and Sh is, hence, extended to include the effect of the Re number on heat and mass fluxes, enabling the prediction of the local value of Nu and Sh in the main features, and in particular the enhancement in the ignition region and the dependence on Re.     

SIMULTANEOUS DEVELOPMENT OF VELOCITY AND TEMPERATURE PROFILES IN THE ENTRANCE REGION OF A PARALLEL PLATE CHANNEL: LAMINAR FLOW WITH UNIFORM WALL HEAT FLUX

Available boundary layer type solutions to the combined hydrodynamic and thermal entrance region problem are known to exhibit a discontinuity in the gradients of the velocity and temperature distributions in the entrance region. A new solution is presented which alleviates this shortcoming. The new solution is based on the hydrodynamic inlet-filled region concept originally proposed by Ishizawa (1966) and later adopted by Mohanty and Das (1982) to hydrodynamically developing flow in a channel. This concept is extended to the combined entry length problem by dividing the thermal entrance length into two lengthwise regions, a thermal inlet region and a thermally filled region. In the former, the effect of heat transfer between fluid and wall is confined within the thermal boundary layer developing along the wall. At the end of the thermal inlet region, the thermal boundary layers meet at the duct axis but the temperature profile is not yet developed. In the thermally filled region, the heat effects propagate throughout the entire cross section and the temperature profile undergoes adjustment in a fully thermal region to finally attain the fully developed form. A thermal shape factor is also introduced in the thermally filled region which ensures that all thermal quantities attain their fully developed values asymptotically. The new model is used to obtain solutions to the combined entry length problem for laminar flow through a parallel plate channel under the constant wall heat flux boundary condition. The analysis gives considerably better results for the local Nusselt number and thermal entrance length than previously available.     

ANALYTICAL STUDY OF HEAT TRANSFER TO LAMINAR FLOW OF NON-NEWTONIAN FLUIDS IN ANNULUS

An analytical study is made of the problem of laminar flow heat transfer to pseudoplastic fluids in a concentric circular tube annulus. The solution is obtained for simultaneously developing velocity and temperature profiles and constant wall heat flux. Constant property results are presented for different values of flow behavior index, n, and several inner to outer tube radius ratios and Prandtl numbers. Variable property solutions, with strongly temperature-dependent consistency index are obtained. The effect of viscous dissipation on the results of heat transfer is also presented.     

Effect of the wall Nusselt number on the simulation of catalytic fixed bed reactors

Literature correlations for the apparent wall heat transfer coefficient (h_w) in fixed bed catalytic reactors are compared. At low to moderate values of the Reynolds number (Re), different correlations can produce estimates of the dimensionless wall Nusselt number (Nu_w = h_wd_p/k_f) that differ by an order of magnitude or more. Some correlations give Nu_w as a function of Re only, others allow for the effects of tube-to-particle diameter ratio and particle and fluid thermal conductivities. The value of Nu_w that is used in a simulation of a fixed bed catalytic reactor can have a strong effect on the predicted behavior. Two examples of fixed bed reactors are simulated and show that the more general correlations for Nu_w are to be preferred.     

Re-evaluation of the Nusselt number for determining the interfacial heat and mass transfer coefficients in a flow-through monolithic catalytic converter

The two-equation porous medium model has been widely employed for modeling the flow-through monolithic catalytic converter. In this model, the interfacial heat and mass transfer coefficients have been usually obtained using the asymptotic Nusselt and Sherwood numbers with some suitable assumptions. However, previously it seemed that there existed some misunderstanding in adopting these Nusselt and Sherwood numbers. Up to now, the Nusselt number based on the fluid bulk mean temperature has been used for determining the interfacial heat and mass transfer coefficients. However, the mass and energy balance formulations in the two-equation model indicate that the Nusselt number should be evaluated based on the fluid mean temperature instead of the fluid bulk mean temperature. Therefore, in this study, to correctly model the heat and mass transfer coefficients, the Nusselt number based on the fluid mean temperature was newly obtained for the square and circular cross-sections under two different thermal boundary conditions (i.e., constant heat flux and constant temperature at the wall). In order to do that, the present study employed the numerical as well as analytical method.     

Convective heat and mass transfer coefficient measurements and correlations for particle beds using temperature and humidity step changes with air flow through a test bed

In this paper, transient responses of a porous urea particle bed subject to a step change in the inlet temperature or humidity for a forced convective air flow through the particle bed are investigated to determine the convective heat and mass transfer coefficients inversely by comparing the measured time constant with the predicted characteristic time constant, which is a function of the convection coefficients and Reynolds number. The experimental results show, that although both the time constants for temperature and humidity step changes are dependent on Reynolds number, the temperature response time constant (35-1300 s) is much larger than the humidity response time constant (4-25 s) for the Reynolds number range of 300-5. The surface adsorption of water vapor is very rapid but the absorption inside the porous urea particle is slowed by a very low internal effective diffusion coefficient within the particles whereas the very low Biot number for heat transfer in the particles implies a complete thermal interaction with the air flow throughout each particle and a much larger time constant. Empirical correlations of the Chilton-Colburn j-factor and Nusselt number versus Reynolds number are compared with the correlations of other researchers. These new correlations, which include an uncertainty analysis, imply much lower convective coefficients than those reported previously in the literature.     

石蜡乳状液高温潜热输送材料的传热特性

石蜡乳状液高温潜热输送材料是一种在80～90℃之间存在相变的流体。由于相变潜热的存在,石蜡乳状液在相变区间内的储热、载热密度远大于水,且集储热与潜热输送于一体,适用于余热利用、太阳能利用和集中供热等领域,具有广阔的应用前景。研究了定热流条件下石蜡乳状液在管内的强制对流换热特性。结果表明,与纯水相比,虽然石蜡乳状液的有效比热大,但其对流传热系数低,且浓度越高,对流传热系数越低。实验发现入口温度对石蜡乳状液对流传热系数的影响不大,但随着入口温度升高,对流传热系数有增大的趋势。最后获得了石蜡乳状液对流换热量纲1关系式。     

气体流动条件下钴基催化剂固定床的传热

在气体流量4~8 Nm3/h、气体分布器进口温度190~210℃、加热管壁温约240℃的条件下,对气体流动时活性组分呈蛋壳型分布的钴基催化剂固定床的传热进行了实验研究,建立了二维拟均相传热模型,利用正交配置法和Levenberg-Marquardt法对其求解,得到了钴基催化剂床层径向有效导热系数及壁给热系数的关联式,并将传热参数与由气体处于静态时固定床的有效导热系数计算而得的固定床传热参数值进行了比较,在气体入口温度范围内考察了其对固定床传热参数的影响. 结果表明,实验所得传热参数与文献值的最大偏差绝对值均在15%以内.     

SIMULTANEOUSLY DEVELOPING LAMINAR FLOW AND HEAT TRANSFER IN THE ENTRANCE REGION OF A CIRCULAR TUBE WITH CONSTANT WALL TEMPERATURE

The developing flow and heat transfer in the entry region of a heated circular tube is analyzed for the case of constant wall temperature. An integral or boundary-layer solution is presented which has a number of advantages over earlier Karman-Pohlhausen integral analyses. Thus, in the present analysis, the velocity and temperature distributions, the local and mean drag coefficients, and the local and mean Nusselt numbers approach their fully-developed values asymptotically. The new analysis is based on the hydrodynamic inlet-filled region concept originally proposed by Ishizawa (1966) and later adopted by Mohanty and Asthana (1978) to flow through a circular tube. This concept is extended to the combined entry-length problem by introducing a thermal transition region, herein called the thermally-filled region, between the thermal inlet boundary-layer region and the thermally fully-developed region. A thermal shape factor is also introduced which ensures smooth transition of all pertinent thermal quantities from the entrance region to the fully-developed region. Results for the variation of the local and mean Nusselt numbers with axial distance along the tube for Pr = 0.1,0.5,0.7,1,5, and 10 are presented. These results agree well with the numerical solutions of Hombeck (1965), Manohar (1969), and Hwang and Sheu (1974) and also with the correlations of Churchill and Ozoe (1973).     

Ignition, extinction, and thermal hysteresis of a heterogeneous exothermic reaction

Critical conditions of ignition and extinction are studied theoretically for the case of a heterogeneous exothermic reaction proceeding on the uniformly accessible surface of a channel or a pore. Particular emphasis is placed on the thermal-hysteresis effect of the reaction (an ignition temperature in excess of the extinction temperature), which ensures stability of the reaction against changes in the external conditions. A model for the process is proposed and substantiated. It permits one to analyze the main regularities of the thermal reaction regime with allowance for the distribution of heat flows from the reaction zone to the reaction medium and the channel wall. The dependence of the critical conditions of ignition and extinction on the rate constants of heat release, the reaction order, and the thermal and geometric characteristics of the system is established. A rule is proposed that permits evaluating the effect of heat losses on the critical phenomena and the thermal hysteresis of a reaction on a cooled surface in the case where the critical conditions for the reaction on an adiabatic surface are known. Translated fromFizika Goreniya i Vzryva, Vol. 34, No. 2, pp. 51–58, March–April, 1998.     

Propagation and quenching of premixed flames in narrow channels

The shape and propagation of unsteady premixed flames in narrow channels with adiabatic and isothermal walls are numerically investigated in the present study. The flame chemistry is modelled by an one-step overall reaction, which simulates the reaction of a stoichiometric acetylene-air mixture. The numerical results show that both ignition methods and thermal boundary conditions affect flame formation and its shape. The flame keeps a single tulip shape in the whole process of propagating through the channel if plane ignition is used and a single mushroom shape if spark ignition is used. For isothermal cold walls, the flame cannot keep a single tulip shape or mushroom shape all the time. Under plane ignition, a transition from a tulip-shaped flame to a mushroom-shaped flame occurs as flames propagate from one end of channel to the other. Under spark ignition, the process is just the contrary. It is also shown that the heat loss at the cold walls has a dual effect on the formation of a tulip-shaped flame. Flame propagation and quenching in narrow channels of different heights are analyzed systematically, and a criterion is proposed to judge the flame states of partial extinction and total extinction. It is called the criterion of flame propagation and quenching in narrow channels. __________ Translated from Fizika Goreniya i Vzryva, Vol. 42, No. 3, pp. 27–36, May–June, 2006.     

超疏水表面有滑移时的层流换热分析

在考虑速度滑移的前提下,对于圆管形微通道内温度边界层充分发展的恒热流量对流传热,推导了速度和温度分布表达式,并进一步得到对流传热系数和Nusselt数计算式。此外,还针对超疏水表面不同结构参数下的滞留空气层,提出了超疏水表面的有效导热模型,推导出超疏水表面不同结构参数下的热阻。最后将传热系数与超疏水表面热阻进行耦合,得出超疏水表面有效传热系数及其与超疏水表面结构参数的关系。计算结果表明:超疏水表面上流体的滑移使得管内温度更加均匀,传热系数或Nusselt数有所增加,恒热流条件下最多可以增加1.8倍;超疏水表面热阻随肋间距和肋高的增加而增大;超疏水表面的表观传热系数随肋间距或孔宽的增加而显著降低,随肋高或孔深的增加,表观传热系数也降低,其幅度与肋间距以及肋高与肋间距之比有关;各种结构参数条件下均存在表面滞留空气层的临界厚度,在此厚度以下表面有效传热系数不低于普通表面无滑移时的数值。因此,需要综合考虑超疏水表面的结构参数,包括肋高、肋间隙等,才能使超疏水表面有利于传热。     

Effect of Thermal Asymmetry on Laminar Forced Convection Heat Transfer in a Porous Annular Channel

The effect of thermal asymmetry on laminar forced convection heat transfer in an annular porous channel with a Darcy dissipation of fluid kinetic energy was investigated numerically. The cylindrical surfaces making the channel boundaries were kept at constant but different temperatures. The thermal asymmetry thus imposed on the system results in an asymmetric temperature field and different heat fluxes across the channel boundaries. Depending on the Darcy, Péclet and Reynolds numbers, the thermal asymmetry may lead to a reversal of the heat flux along the channel at least at one of the channel walls. The corresponding Nusselt number becomes zero and subsequently experiences a discontinuity, thereby jumping from infinite negative to infinite positive, or vice versa. This feature is observed in the region of thermal development. In the fully developed heat transfer region, the Nusselt numbers can be positive or negative for the same inlet conditions, depending on the heat source strength. In the case of a plug flow, the analytical expressions for the Nusselt numbers have been obtained.     

NEURAL NETWORK–BASED HEAT AND MASS TRANSFER COEFFICIENTS FOR THE HYBRID MODELING OF FLUIDIZED REACTORS

The complex flow patterns induced in fluidized bed catalytic reactors and the competing parameters affecting the mass and heat transfer characteristics make the design of such reactors a challenging task to accomplish. The models of such processes rely heavily on predictive empirical correlations for the mass and heat transfer coefficients. Unfortunately, published empirical-based correlations have the common shortcoming of low prediction efficiency compared with experimental data. In this work, an artificial neural network approach is used to capture the reactor characteristics in terms of heat and mass transfer based on published experimental data. The developed ANN-based heat and mass transfer coefficients relations were used in a conventional FCR model and simulated under industrial operating conditions. The hybrid model predictions of the melt-flow index and the emulsion temperature were compared to industrial measurements as well as published models. The predictive quality of the hybrid model was superior to other models. This modeling approach can be used as an alternative to conventional modeling methods.     

微槽道内单相流动阻力与传热特性

对去离子水在微槽道中的单相流动阻力和传热特性进行了实验研究.结果表明,微槽道中的流动阻力特性及传热特性与常规尺度通道相比存在明显的差异,实验的fRe值高于传统理论预测值,并且随Reynolds数的增加而增大,而传热系数则明显小于传统理论预测值,分析表明微槽道的表面粗糙度对流动和传热有重要的影响.在热阻相同的情况下,提高工质的入口温度或增加热流通量,可减小实验段的压降.同时给出了层流区的流动阻力计算关系式.     

On the Prandtl or Schmidt number dependence of the turbulent heat or mass transfer coefficient

Numerical experiments using a direct numerical simulation (DNS) of turbulent flow between two parallel plates in conjunction with Lagrangian scalar tracking (LST) of trajectories of thermal markers in the flow field are conducted for Prandtl or Schmidt numbers between 0.01 and 50,000. The LST methodology is used to generate mean temperature profiles as a function of the entry distance in the case of a step change in heat or mass flux at the walls of the channel. The heat transfer coefficient and the Nusselt number ratio, Nu(x)/Nu(x→∞), downstream from the step change in the wall flux are determined for the range of Pr or Sc fluids examined. Relations between the heat or mass transfer coefficient at the fully developed part of the channel and Pr or Sc are proposed for low and high Pr or Sc cases. Finally, unified correlations, which provide the heat or mass transfer coefficient for all Pr or Sc, in the Reynolds number range examined, are proposed. Also, the exponent of the asymptotic dependence of the eddy diffusivity close to the wall is obtained.     

Flow and heat transfer characteristics of structural viscosity fluids

Laminar friction loss and heat transfer characteristics for a non-Newtonian fluid described by a general viscosity function of the structural type have been computed for tube and slit geometries. The model includes most other models for fluids without a yield stress as special cases. The Graetz problem for constant wall temperature and temperature independent properties has been solved, including viscous dissipation, for the transverse and axial temperature distribution. Both momentum and energy transport characteristics are shown to be strongly dependent upon a dimensionless fluid time constant, which characterizes the transition from Newtonian to non-Newtonian shear thinning behavior. A direct and exact analogy between the momentum and energy transport mechanisms is demonstrated for hydrodynamically and thermally fully developed flow, provided the heat transfer rate is represented by a suitably defined Nusselt number.     

CFD Study of Effects of Module Geometry on Forced Convection in a Channel with Non‐Conducting Fins and Flow Pulsation

CFD simulations were carried out to investigate the effects of the module geometry on forced convection in a rectangular channel containing series of regularly spaced non‐conducting baffles with flow oscillation. The simulations were performed at constant wall temperature. Steady‐flow Reynolds numbers Re in the range of 200 and 600 were studied. The results of the CFD simulations show that, for the effect fin spacing to be significant on heat transfer enhancement in finned system with oscillating flow, the oscillating flow velocity must be higher than the mean flow velocity. Superposition of oscillation yields increasing heat transfer performance with increasing fin height. Fin geometry with pyramidal shape yields highest performance in terms of the heat transfer effectiveness.