Axial Dispersion Of Heat Research Articles

Simulation of CO2 capture from natural gas by cyclic pressure swing adsorption process using activated carbon

This work presented modeling and simulation of CO2 from natural gas. One of the most promising technologies is Pressure Swing Adsorption (PSA), which is an energy-efficient and cost-effective process for separating and capturing CO2 from industrial processes and power plants. This paper provides an overview of the PSA process and its application for CO2 capture, along with a discussion of its advantages, limitations, and future research directions. This process is pressure swing adsorption (PSA) with four adsorption beds. The adsorption bed columns fill with activated carbon as adsorbent. In this simulation momentum, mass and energy balance are solved simultaneously. The process was designed with two beds in adsorption conditions and the other two beds in desorption conditions. The desorption cycle includes blow-down and purge steps. The linear driving force (LDF) estimates the adsorption rate in modeling this process. The extended Langmuir isotherm is used for the equilibrium between solid and gas phases. The temperature changes by heat transfer from the gas phase to solid and axial heat dispersion. The set of partial differential equations is solved using implicit finite difference.

Stationary spatially periodic and aperiodic solutions in membrane reactors

The formation of stationary spatially patterns is analysed for a detailed reaction mechanism of the oxidative dehydrogenation of ethane. In the first step, a simple steady-state model of an ideal plug flow membrane reactor is investigated by numerical bifurcation analysis. The model shows a complex nonlinear behaviour including period doubling bifurcations and aperiodic spatial patterns. In the next step, the influence of axial heat dispersion is studied. Finally, a more detailed model of a fixed bed membrane reactor is considered. It is found that pattern formation is possible under operation conditions realisable in a laboratory reactor.

An experimental and theoretical investigation into the hyperbolic nature of axial dispersion in packed beds

The concept of hyperbolic axial dispersion of heat in a flowing fluid which is known as `third sound wave' has been examined taking packed bed as an example. This technique analyse fluid flow and heat transfer by introducing an axial dispersion term in the one-dimensional energy equation to take care of flow maldistribution and backmixing. The present approach models this dispersion in terms of two parameters which are proposed to follow hyperbolic conduction law. A regenerator bed consisting of stainless steel wire mesh packing has been used to carry out experiments for the purpose of validating the concept. The analytical model presented uses a Laplace transform technique for the solution of simplified energy equation. The computed outlet fluid temperature is compared with experimental output and the two model parameters, dispersive Peclet number ( Pe) and its propagation velocity ( c *) are estimated. The present model, the parabolic (Fourier) dispersion model and the non-dispersive plug flow model are compared with the experimental result which brings out the closeness of the present proposition to reality compared to other models. A standard experimental technique is suggested which can be used for the measurement of parameters related to axial dispersion. The present study is the first experimental evidence of `third sound wave' in fluids and lays foundation of this proposition.

A CAPE tool for evaluation of adsorber-reactor systems for treatment of exhausts from mobile sources

Software for the simulation of dynamic operation and adaptive control of combined systems of monolithic catalytic reactors and adsorbers (catalytic traps) for treatment of exhausts from mobile sources is described. Monolithic reactors are modelled by a spatially 1-D two-phase model consisting of a system of PDE's (axial heat dispersion included) and for the adsorbers a spatially 1-D model (pseudohomogeneous for the enthalpy balance) is used. The system of PDE's is solved by the combination of the Crank-Nicolson method (reactors) and by the method of lines (adsorbers). Results of the application of the software to two examples — the use of adsorber-reactor systems with mutual heat exchange for the solution of the ‘cold-start’ problem and an adaptive dynamically operated system of coupled monoliths for NO x reduction using feed-forward neural network for predictive control are discussed in detail.

Transient response of multipass plate heat exchangers with axial thermal dispersion in fluid

To predict the transient response of multipass plate heat exchanger, an analysis is presented based on an axial heat dispersion model in the fluid which takes deviation from ideal plug flow into consideration. The method can be utilised to simulate the exit temperatures for both the fluids for any given arbitrary temperature transient at inlets. The methods takes the ‘phase-lag’ inside the distribution port into consideration and analyse the multipass plate heat exchanger by considering it as an assembly of single pass plate heat exchangers. The solution technique for equal number of passes on both the sides is different from that for unequal number of passes on the two sides. Examples of computation for the response of 1–2 and 2–2 pass configuration have been presented. The 1–2 pass exchanger is solved by analysing successive modules of the heat exchanger. The 2–2 pass heat exchanger is required to be simulated by iterating the responses of two smaller heat exchangers to which it has been divided. The method involves a Laplace transformation with the temporal variable, solution in space domain and numerical inversion of Laplace transform back to the time domain. The results indicate the strong effect of axial dispersion in fluid on the transient response. The effect of number of plates and number of transfer unit has also been studied which brings out a comprehensive picture of responses of a multipass plate heat exchanger in transient regime.

Investigation of spatially patterned catalytic reactors

The theme of this work is to enhance the performance of a primary reaction system by introducing dual-functionality into a packed-bed reactor via a spatial pattern. This paper focuses on a generic system in which the primary reaction is equilibrium-limited and an auxiliary reaction is incorporated to act as a drain-off mechanism alleviating the equilibrium constraint. It is found that patterns in which the two catalysts are spatially layered (and maintained at different temperatures) or well-mixed (and operated at a common temperature) can both offer significant improvements over an unpatterned reactor. Operating the well-mixed pattern at high temperature is generally superior to switching between high- and low-temperature layers in the segmented configuration. The exception to this rule occurs when the mixed bed is punished for being hot as either the equilibrium ceiling on the primary reaction drops sharply or a competitive reaction dominates at high temperature. The extent to which the layered pattern’s improvements are compromised by axial heat dispersion between zones is examined in terms of a backflow mixing cell model.

Rotary heat exchanger performance with axial heat dispersion

A model of the rotary heat exchanger accounting for axial heat dispersion and longitudinal matrix conduction is developed. The heat conservation equations are solved using a finite difference approach. The heat exchanger effectiveness is then predicted and the effect of axial heat dispersion on the exchanger performance is studied.

Second law analysis of a plate heat exchanger with an axial dispersive wave

A second law analysis is presented for thermally dispersive flow through a plate heat exchanger. It is well known that in plate or plate fin type heat exchangers the backmixing and other deviations from plug flow contribute significantly to the inefficiency of the heat exchanger, which is of importance to heat exchangers working in the cryogenic regime. The conventional axial heat dispersion model which is used so far is found to be better than `plug flow' model but still unsatisfactory where the timescale related to heat transfer is comparable with the thermal relaxation time for the propagation of dispersion. The present work therefore considers dispersion as a wave phenomenon propagating with a finite velocity. The study discusses the nature of variation of different contributions to total exergy loss in the heat exchanger with respect to dispersion parameters of the Peclet number and propagation velocity of the dispersive wave. The practical example of the single-pass plate heat exchanger demonstrates how a second law optimization can be carried out for heat transfer equipment under such conditions.

Classification of steady-state and dynamic behavior of distributed reactor models

Singularity theory is combined with the Liapunov-Schmidt reduction and the shooting method to present a systematic procedure that may be used to classify the steady-state and dynamic behavior of distributed reactor models. The method is applicable to problems of diffusion-reaction, convection-reaction and diffusion-convection-reaction, dependent on one spatial coordinate and involves the numerical computation of various singularities (hysteresis, isola, double zero and double or degenerate Hopf loci). The usefulness of the present approach is demonstrated by applying it to four classical examples from chemical reactor theory. The first example deals with the problem of diffusion and reaction in a porous catalyst particle (slab geometry) with no external heat and mass transfer resistances (Dirichlet model). Here, the different loci divide the parameter space into six regions corresponding to six different bifurcation diagrams. It is observed that in the physically feasible parameter region, the catalyst particle model is more likely to exhibit oscillatory behavior when the reacting fluid is a liquid. The second example is the convection-reaction model of a cooled tubular reactor with recycle. It is shown that for the non-adiabatic case, the hysteresis, isola, double-zero and double Hopf loci divide the parameter space into thirty three different regions, each corresponding to a different bifurcation diagram. The cooled recycle reactor can exhibit both low and high temperature oscillations for realistic parameter values. The third example considers a diffusion-convection-reaction model of a tubular reactor with axial dispersion of heat and mass. For the adiabatic case, the parameter space of the model is divided, as in the catalyst particle model, into six regions corresponding to different bifurcation diagrams. The steady-state model of cooled tubular reactor is shown to be described by a single intrinsic state variable only in the limiting case of negligible axial dispersion of mass. This limiting model is examined and it is shown that oscillatory solutions are possible on the ignited high conversion branch for realistic values of the parameters. Finally, two autothermal reactor models with internal and external heat exchange are considered. Once again the parameter space is divided into six regions and for realistic parameter values no oscillatory behavior is exhibited.

BEHAVIOUR OF AN ADIABATIC PACKED BED REACTOR PART 2: MODELLING

Abstract The steady state and dynamic behaviour of an adiabatic packed bed reactor for the selective hydrogenation of mixtures of ethyne and ethene is studied. A heterogeneous model with axial dispersion of heat is solved numerically by means of a fully implicit discretisation scheme. From an analysis of the inlet and outlet boundary conditions, it follows that in order to avoid an influence of the type of boundary condition on the reactor behaviour, sufficiently long inert zones before and after the active bed should be present. Using preliminary kinetic expressions adapted from Mcn'shchikov ct al. (1975), the model calculations show a reasonable agreement with the experiments performed in a laboratory scale reactor and demonstrate the importance influence of small amounts of carbon monoxide both on the steady state as well as on the transient behaviour of the reactor. However, using the rate expressions derived from our own kinetic experiments in a Berty reactor, the behaviour under runaway conditions a...

Significance of axial heat dispersion for the description of heat transport in wall‐cooled packed beds

AbstractA two‐dimensional pseudo‐homogeneous model with axial dispersion of heat has been solved numerically with different boundary conditions at the inlet and outlet of the packed bed. The model solutions are fitted to experimental temperature profiles, determined in a wall‐cooled packed bed in which a hot gas is cooled down, and best fit values of the effective axial and radial thermal conductivities and the wall heat transfer coefficient are obtained. In the range of Reynolds numbers employed, Re > 50, the axial dispersion of heat is found to be of no significance for the description of heat transport in wall‐cooled packed beds without reaction, provided that the inlet boundary conditions are chosen appropriately. If a radially flat inlet temperature profile is assumed, while the actual profile is curved, an apparent improvement in the description of heat transport is observed when axial dispersion is incorporated into the heat balance and high effective axial thermal conductivities are obtained. If a Danckwerts type inlet boundary condition is used, assuming a flat temperature profile immediately in front of the inlet, an apparent improvement is also found on incorporation of axial dispersion of heat. This is caused by the temperature jump at the inlet, compensating for the overestimation of inlet temperature, in the case of cooling. The latter also explains why the inclusion of axial dispersion may eliminate the so‐called “length effect”, often related to the effective radial thermal conductivity and the wall heat transfer coefficient. It is shown for the outlet boundary condition that deletion of the axial dispersion term from the heat balance at the outlet is a convenient boundary condition for the model being solved numerically.

Heat transfer in packed beds with fluid flow: remarks on the meaning and the calculation of a heat transfer coefficient at the wall

In order to describe heat transfer in diabatic packed tubes a heat transfer coefficient at the wall is usually used. The present work shows that this approach is adequate only for large values of the molecular Péclet number. In this region heat transfer at the wall is inhibited by a layer of unmixed fluid. For small values of the molecular Péclet number no resistance at the rigid wall exists. In this region experimental results should be plotted in the Nusselt-Graetz diagram, using the effective properties of the bed in the definitions of the dimensionless numbers. In the course of the analysis theoretical arguments and numerical calculations, as well as experimental data from the literature, are discussed. The influence of lateral maldistribution of fluid velocity and/or thermal conductivity, of axial dispersion of heat, and of small, systematic errors in measurement is investigated. Finally, recommendations for practical use are given.

Wrong‐way behavior of packed‐bed reactors: Influence of interphase transport

AbstractLarge transient temperature excursions may be caused by a sudden reduction in the feed temperature to a packed‐bed reactor operating at an intermediate conversion. When a unique steady state exists for all feed temperatures, the magnitude of the wrong‐way behavior predicted by a two‐phase model is very close to that predicted by a pseudohomogeneous model if PeH is equal to the dimensionless heat transfer coefficient H. The two‐phase model enables a more efficient numerical simulation in such cases. The predictions of these two models may be rather different when steady‐state multiplicity exists for some feed temperatures. In such cases, a two‐phase model, which accounts for the axial dispersion of heat, should be used to simulate the transient behavior. The wrong‐way behavior may lead to an ignition of a low‐temperature state or an upstream propagation of a transient temperature wave.

Wrong‐way behavior of packed‐bed reactors: II. Impact of thermal dispersion

AbstractA sudden reduction in the feed temperature of a packed‐bed reactor may lead to a transient temperature rise, referred to as a wrong‐way behavior. As expected, the axial dispersion of heat decreases the magnitude of the temperature excursion and prolongs the transient shift to a new steady state. In addition, the thermal dispersion may enable the wrong‐way behavior to ignite a low‐temperature steady state leading to a disastrous runaway of the reactor. Moreover, it may create a transient high‐temperature wave, which moves initially in the upstream direction. The axial dispersion of heat can lead to some behavioral features which are qualitatively different from those of a model which ignores it. The transient temperature excursion does not exceed a value, which can be estimated by a simple analytical expression.

THERMAL PARAMETER SENSITIVITY IN THE SIMULATION OF THE NON-ISOTHERMAL, NON-ADIABATIC FIXED BED CATALYTIC REACTOR—THE TWO-DIMENSIONAL HETEROGENEOUS MODEL

Utilizing a two phase (fluid-solid) heterogeneous (inter-intraphase gradient) model of the fixed bed catalytic reactor, the sensitivity of thermal transport in the axial and radial directions is numerically assessed in terms of axial and radial Peclet numbers for the fluid and solid phases, the fluid-wall and wall-solid transport coefficients and local interphase fluid-solid transport in the highly exothermic case of napthalene oxidation. Axial dispersion of heat and mass in both fluid and solid phases are shown to be of little consequence in both the steady and transient modes of operation for beds of length to particle diameter ratios normally encountered i.e. L/dp > 100. Radial temperature distribution is, in contrast, sensitive to the choice of bed solid phase effective conductivity and the fluid thermal Peclet number as well as to the wall coefficients governing fluid-wall and solid-wall heat transport at that boundary. Sensitivity of the radial temperature profiles, particularly the magnitude of the...

Mathematical and Experimental Modelling of Adsorption in Fixed Bed. I. Mathematical Model Development and Numerical Investigation of Nonisothermal Single Solute Adsorption

Mass and heat balances describing the process of a nonisothermal one-component adsorption in a fixed bed adsorber are formulated. The mathematical model permits simulation of bed performance under isothermal, adiabatic or nonisothermal-nonadiabatic conditions and incorporates heat and mass transport resistances within and around the adsorbent particle. Both heat and mass axial dispersion in the bed and moderate heat losses to the surroundings are considered. The adsorption equilibrium is described by a temperature-dependent Langmuir isotherm. The mathematical model is solved for different values of the physicochemical parameters. Numerical investigation results for different values of axial heat dispersion coefficient, effective pore diffusivity and overall heat transfer coefficient are compared with real adsorption data available from the literature.

The effect of electrolyte solutions on hydrodynamic and backmixing characteristics in bubble columnns

Gas holdup and axial heat dispersion coefficient measurements in the presence of an electrolyte solution are presented. The experiments were carried ou

On the observability of heat dispersion parameters of the pseudohomogeneous model of heat transfer in packed beds

Solution has been obtained of the pseudohomogeneous model of heat dispersion in a packed bed under simultaneous radial and axial dispersion of heat using Danckwerts' boundary condition. The obtained solution has been used to evaluate parameters of the model from experimental data based on the least sum of square deviations of the temperature profiles. Maps have been constructed of the joint confidence regions in the parameter domain. The results have shown the method to be suitable for the determination of radial thermal dispersion while suppressing the effects of structural disturbances of the bed near the containing walls.

Fixed bed catalytic reactor modelling: The heat transfer problem

The issue of radial heat transport in fixed beds is discussed and discrepancies between observed and computed hot spots are attributed to neglect of axial dispersion of heat, which neglect also accounts for observed length-dependent radial conductivities. The Nusselt number for heat transport at the wall is inferred to be Nu w = 5.73( D t d p ) 1 2 P r (0.11 Re p + 20.64)/ Re p 0.262 which is in accord with recently secured Sherwood numbers for mass transport at a packed wall. The recent success of unconventional reactor models is discussed.

Modeling of chemical reactors—XXXI: The one-phase backflow cell model used for simulation of tubular adiabatic reactors

The backflow cell model is used to simulate steady state operation of a tubular adiabatic reactor. The model proposed embraces different mechanism of axial dispersion of heat and mass. It will be shown that the backflow cell model may be used for approximation of the dispersion model. While there are differences in qualitative behavior of simple cell and dispersion models, the backflow cell model gives results which are in agreement with the dispersion model. The model may be used for simulation of steady-state behavior of tubular homogeneous and heterogeneous reactors.