Determination of a heat source in porous medium with convective mass diffusion by an inverse method

A formulation is given of the inverse natural convection problem by conjugate gradient with adjoint equations in a porous medium with mass diffusion for the determination, from temperature measurements by sensors located within the medium, of an unknown volumetric heat source which is a function of the solute concentration. The direct, sensitivity and adjoint set of equations are derived for a Boussinesq fluid, over an arbitrary domain in two dimensions. Solutions by control volumes are presented for a square enclosure subjected to known temperature and concentration boundary conditions, assuming a source term depending on average vertical solute concentration. Reasonably accurate solutions are obtained at least up to Ra=10⁵ with the source models considered, for Lewis numbers ranging from 0.1 to 10. Noisy data solutions are regularized by stopping the iterations according to the discrepancy principle of Alifanov, before the high frequency components of the random noises are reproduced.     

Solution of the inverse steady state convection problem in a porous medium by adjoint equations

The inverse free convection problem in a porous medium is solved by adjoint equaitons and conjugate gradient. A derivation of the set of adjoint equations in the steady case is provided. Numerical solutions are obtained for the case of a rectangular enclosure subjected to an unknown heat flux on one side, and to known conditions on the remaining sides. Results are presented for different flux profiles with a Rayleigh number ranging from 0 to 10⁴ and the effect of noise in the input data is also examined.     

Adjoint based optimisation of reactive compressible flows

We derive adjoint equations for the reactive compressible Navier–Stokes equations. Therein the reaction rate is modelled by an Arrhenius approach. The adjoint equations are validated by means of a comparison between the adjoint solution and a finite difference expression. An adjoint based optimisation framework for reactive compressible flows is presented. Formulations for different target functions are shown. One- and two-dimensional laminar flame configurations are presented. We find, that the adjoint approach works well despite the strong non-linearity of the reaction terms.     

Entropy Generation Due to Natural Convection in a Partially Open Cavity with a Thin Heat Source Subjected to a Nanofluid

This article presents a numerical study of natural convection cooling of a heat source mounted inside the cavity, with special attention being paid to entropy generation. The right vertical wall is partially open and is subjected to copper–water nanofluid at a constant low temperature and pressure, while the other boundaries are assumed to be adiabatic. The governing equations have been solved using the finite volume approach, using SIMPLE algorithm on the collocated arrangement. The study has been carried out for a Rayleigh number in the range 10³ < Ra < 10⁶, and for solid volume fraction 0 <? <0.05. In order to investigate the effect of the heat source and open boundary location, six different configurations are considered. The effects of Rayleigh numbers, heat source and open boundary locations on the streamlines, isotherms, local entropy generation, Nusselt number, and total entropy generation are investigated. The results indicate that when open boundary is located up, the fluid flow augments and hence the heat transfer and Nusselt number increase and total entropy generation decreases.     

Unusual behavior of dynamic hydrogen generation from sodium borohydride

In this study, for the first time, hydrogen gas generation rate is dynamically measured by transferring required amount of sodium borohydride solution to maintain 1 or 2 normal liters of hydrogen gas generation per minute. Three different catalysts (iron, platinum and ruthenium) are used in order to investigate the critical catalyst concentration, catalyst amount and the rate relation and as well as the influence of the sodium borohydride concentration on the rate equations. It has been demonstrated that the linear approximation from the initial hydrogen gas generation data usually reported in the literature is not valid. Moreover, the actual rates reported here are higher and lower than the linear approximations depending on the type of the catalyst. It is about 60 percent lower than linear approximation when iron catalyst is used. Therefore, calculating hydrogen gas generation rate as NL min⁻¹ g⁻¹_catalyst is invalid. On the other hand, it has been observed that the rate equation may not be dependent on the amount of the sodium borohydride concentration if the highest catalytic activity is achieved. Hence, most of the rate equations in the literature are only valid for a range, which is limited to the highest catalytic activity. Additionally, it has been concluded that the amount of catalyst and sodium borohydride concentration may be optimized to observe almost 100% efficiencies.     

Component analysis of a steady inverse convection problem solution in a porous medium

A steady-state inverse free convection problem is solved by conjugate gradient and adjoint equations in a porous medium confined within a square enclosure subjected to an unknown heat flux on one side, and to isothermal and adiabatic conditions on the other sides. Numerical solutions for sinusoidal flux profiles are examined on the basis of a Fourier decomposition for Rayleigh numbers ranging from 0 to 10⁴. Analytical perturbation results for the first iteration are provided, showing the initial effects of convection on the inverse solution. It is found that the problem involves scales of different magnitudes, slowing down convergence, mainly through the step size, as convection develops.     

Optimization of a vacuum freezing cool‐thermal storage coupled with wastewater treatment process

Purification of wastewater by freezing under vacuum in a cool‐thermal storage process is investigated. Mathematical models of heat transfer and mass transfer in this system are developed to describe the thickness and the solute distribution of ice as the cool‐thermal storage process proceeds. A mathematical treatment for determining the optimum solidification thickness and operating time to remove the maximum amount of solute is presented. For the system of constant volumetric rate of vacuum, the suction rate is an important factor in determining the efficiency of the process. The suction rate should be kept within the range of 10,000 to 20,000 m³/m²hr when the solute distribution coefficient is less than 1. When the coefficient is greater than 1, the suction rate should be run within 5000 to 10,000 m³/m²hr. Over the proper range of operation, the optimal operating time is a weak function of suction rate. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(3): 189–200, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10134     

A semi-global reaction rate model based on experimental data for the self-hydrolysis kinetics of aqueous sodium borohydride

This work studied the self-hydrolysis kinetics of aqueous sodium borohydride (NaBH₄) for hydrogen generation and storage purposes. Two semi-global rate expressions of sodium borohydride and hydrogen ion consumption were derived from an extensive series of batch process experiments where the following parameters were systematically varied: solution temperature (298 K–348 K), NaBH₄ concentration (0.5 wt% to 25.0 wt%), and sodium hydroxide (NaOH) concentration (0.0 wt% to 4.0 wt%). Transient hydrogen generation rates and transient solution pH were measured during the hydrolysis experiments. Given initial conditions (temperature, NaBH₄ concentration, and H⁺ concentration), the two coupled semi-global rate equations can be integrated to obtain the transient time history of H₂ generation (or NaBH₄ consumption) and solution pH (or H⁺ concentration). Comparing analytical results of transient hydrogen generation rate and transient solution pH with experimental data, good agreement was reached for many conditions, especially for elevated solution pH values, levels at which NaBH₄ solutions are used practically.     

Study on adjoint-based optimization method for multi-stage turbomachinery

Adjoint-based optimization method is a hotspot in turbomachinery.First,this paper presents principles of adjoint method from Lagrange multiplier viewpoint.Second,combining a continuous route with thin layer RANS equations,we formulate adjoint equations and anti-physical boundary conditions.Due to the multi-stage environment in turbomachinery,an adjoint interrow mixing method is introduced.Numerical techniques of solving flow equations and adjoint equations are almost the same,and once they are converged respectively,the gradients of an objective function to design variables can be calculated using complex method efficiently.Third,integrating a shape perturbation parameterization and a simple steepest descent method,a frame of adjoint-based aerodynamic shape optimization for multi-stage turbomachinery is constructed.At last,an inverse design of an annular cascade is employed to validate the above approach,and adjoint field of an Aachen 1.5 stage turbine demonstrates the conservation and areflexia of the adjoint interrow mixing method.Then a direct redesign of a 1+1 counter-rotating turbine aiming to increase efficiency and apply constraints to mass flow rate and pressure ratio is taken.     

Recovering a general space-time-dependent heat source by the coupled boundary integral equation method

As to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.     

Convective instability in binary nanofluids with an internal heat source in the presence of thermodiffusion and nanoparticles

This paper investigates the effect of internal heat source on the convective instability under the effect of thermodiffusion of nanoparticles and solute in binary nanofluids theoretically using stability criterion based on linear stability theory. A horizontal layer of binary nanofluid at constant and different temperatures is considered, and the problem is modeled by the system of highly nonlinear partial differential equations. These coupled equations are transformed into ordinary differential equations using nondimensional variables. To study the convective instability of a binary nanofluid, the Rayleigh numbers are derived analytically for stationary and oscillatory convections and the addition factor F_a is proposed. The comparison of the obtained results is favorable with previously published results. The Brinkman model for viscosity and the Bruggeman model for thermal conductivity are used to study the effect of nanoparticle on the system. The effects of various parameters, namely internal Rayleigh number, volume fraction of nanoparticles, Soret coefficients of nanoparticles, and solute on the system are shown through graphs. To check the variation in stability, we have considered NH₃/H₂O+Ag binary nanofluid and the effect of addition factor on the concentration profiles are explained with the help of a graph.     

Matched asymptotic solution for the solute boundary layer in a converging axisymmetric stagnation point flow

A novel boundary-layer solution is obtained by the method of matched asymptotic expansions for the solute distribution at a solidification front represented by a disk of finite radius R₀ immersed in an axisymmetric converging stagnation point flow. The detailed analysis reveals a complex internal structure of the boundary layer consisting of eight subregions. The development of the boundary layer starts from the rim region where the concentration, according to the obtained similarity solution, varies with the radius r along the solidification front as ∼ln^1/3(R₀/r). At intermediate radii, where the corresponding concentration is found to vary as ∼ln(R₀/r), the boundary layer has an inner diffusion sublayer adjacent to the solidification front, an inner core region, and an outer diffusion sublayer which separates the former from the outer uniformly mixed region. The inner core, where the solute transport is dominated by convection, is characterized by a logarithmically decreasing axial concentration distribution. The logarithmic increase of concentration along the radius is limited by the radial diffusion becoming effective in the vicinity of the symmetry axis at distances comparable to the characteristic thickness of the solute boundary layer.     

Size, shape and diffusivity asymmetries in the transient mass and heat transfer of a mixture of bodies introduced in a well-stirred fluid

The transient concentration of a solute in a fluid is studied when a mixture of absorbing bodies are immersed in it. The fluid is perfectly stirred and the bodies may be spherical, cylindrical as well as thin slabs. The treatment presented in this paper constitutes a generalization of the classical problem of a sphere absorbing a solute with a finite presence in a well-stirred fluid. In spite of the possible presence of an unlimited number of differential equations, the nature of the problem lends to an analytical solution. Several observations are made, based on the equations and the numerical results. Among others, an important and strikingly simple result is found in relating the velocity of the fluid's concentration decay to the number of bodies, for monodisperse systems.     

Natural convection in an enclosure with a localized nonuniform heat source on the bottom wall

Natural convection in an air filled enclosure with a localized nonuniform heat source mounted centrally on the bottom wall is numerically investigated. The vertical walls are cooled while the top wall and the remaining portions of the bottom wall are insulated. The heat source is assumed to be isothermal with a linearly varying temperature. The governing equations were solved using finite volume method on a uniformly staggered grid system. The computational results are presented in the form of isotherm and streamline plots and Nusselt numbers. The effects of the source nonuniformity parameter, λ and the line source length, ε are investigated for the Grashof numbers Gr = 10⁶ and 10⁷. It is found that for Gr = 10⁶ nonuniform heating of the line source enhances the overall heat transfer rate markedly compared to uniform heating of the heat source whereas for Gr = 10⁷ its effect is marginal.     

Entropy generation analysis of convective airflow in a solar updraft tower power plant

The main objective of this numerical investigation is to interpret the entropy generation for free convection airflow in a solar tower updraft system. The ground surface is subjected to uniform hot temperature and the collector cover is maintained at lower constant temperature while the chimney wall is adiabatic. Two dimensionless equations of steady laminar free convective airflow are discretized using the finite volume approach. Numerical solutions were accomplished for different values of the Rayleigh number. Results are given in terms of isotherms, velocity magnitude, local entropy generation associated with thermal and fluid friction, local total entropy generation and local Bejan number contours for Rayleigh number ranging between 10³ and 10⁸. The reported results show that thermal and frictional irreversibilities are proportional to the Rayleigh number. Also, it was found that, at lower Rayleigh, total irreversibility is attributable to the thermal irreversibilities and occurs essentially in the collector section. At higher Rayleigh, frictional irreversibilities are increased significantly and become the dominant source of irreversibility in the solar tower, and the chimney section is the main contributor in the total irreversibility in the system.     

Entropy optimized flow of Darcy-Forchheimer viscous fluid with cubic autocatalysis chemical reactions

Background and objectiveThe dynamic of entropy generation phenomenon is important in industrial and engineering process and thermal polymer processing. In order to improve the thermal efficiency of industrial and systems, the main concern of scientists is to reduce the entropy generation. The optimized frame for the Darcy-Forchheimer flow accounted by curved surface has been worked out this continuation. The applications of the chemically reactive material are focused via heterogeneous and homogeneous chemical utilizations. The thermal and velocity slip constraints are imposed for investigating the flow phenomenon. Additionally, the determination of heating phenomenon is investigated by incorporating the heat source features. The importance of entropy generation and Bejan number is also signified.MethodologyNonlinear partial systems are reduced to dimensionless differential system through suitable variables. The problem consists of highly nonlinear equations are numerically worked out with appliances of ND-solve procedure.ResultsInfluence of fluid flow, thermal field, entropy rate, concentration and Bejan number via influential variables are examined. A slower velocity change due to implementation of slip is noticed. The applications of Brinkman number offer resistance to fluid particles while an enhancement in the Bejan number is claimed.ConclusionsFor an augmentation in curvature variable, the concentration and velocity show reverse effect. There is an increase in temperature distribution against heat generation parameter. Velocity field is reduced against higher porosity and slip parameters. Temperature has revers trends against radiation and thermal slip parameters. Larger Schmidt number decreases concentration distribution. Entropy rate is augmented versus larger radiation parameters. An augmentation in Brinkman number leads to improve the velocity filed whereas it reduces the Bejan number. Brinkman number influence on Bejan number is similar to that of homogenous reaction parameter on concentration. The comparative simulations against the reported results are performed.     

Hydrogen generation by alcohol reforming in a tandem cell consisting of a coupled fuel cell and electrolyser

The performance of a novel electro-reformer for the production of hydrogen by electro-reforming alcohols (methanol, ethanol and glycerol) without an external electrical energy input is described. This tandem cell consists of an alcohol fuel cell coupled directly to an alcohol reformer, negating the requirement for external electricity supply and thus reducing the cost of operation and installation. The tandem cell uses a polymer electrolyte membrane (PEM) based fuel cell and electrolyser. At 80 °C, hydrogen was generated from methanol, by the tandem PEM cell, at current densities above 200 mA cm⁻², without using an external electricity supply. At this condition the electro-reformer voltage was 0.32 V at an energy input (supplied by the fuel cell component) of 0.91 kWh/Nm³; i.e. less than 20% of the theoretical value for hydrogen generation by water electrolysis (4.7 kWh/Nm³) with zero electrical energy input from any external power source. The hydrogen generation rate was 6.2 × 10⁻⁴ mol (H₂) h⁻¹. The hydrogen production rate of the tandem cell with ethanol and glycerol was approximately an order of magnitude lower, than that with methanol.     

The effect of surface blocking on mass transfer from a stagnant cap drop

Numerical results for single-drop exterior mass transport of a solute from a surfactant covered drop to the continuous phase are presented. In particular the effect of physicochemical surface blocking is determined by considering the case in which the adsorbed surfactant accumulates at the rear of the translating drop. The stagnant cap velocity profiles are used to describe convective transport. Surface blocking is incorporated through the choice of a zero flux boundary condition on those portions of the drop where surfactant is present. Finite element numerical results for the Sherwood numbers as a function of Peclet number (Pe 10⁴) and stagnant cap angle, φ, show that for surface coverages greater than 0.1π, the effect of surface blocking cannot be ignored. For a Peclet number equal to 10⁴ and φ = 0.5π, the mass transfer coefficients calculated under the assumption that the presence of surfactant reduced convection in the vicinity of the drop without inhibiting the interfacial transport of solute, are found to overestimate the rate of solute mass transfer by as much as 20%.     

Heat transfer and entropy generation for laminar forced convection flow of graphene nanoplatelets nanofluids in a horizontal tube

The results are reported of an investigation of the heat transfer characteristics and entropy generation for a graphene nanoplatelets (GNP) nanofluid with specific surface area of 750 m²/g under laminar forced convection conditions inside a circular stainless steel tube subjected to constant wall heat flux. The analysis considers constant velocity flow and a concentration range from 0.025 wt.% to 0.1 wt.%. The impact of the dispersed nanoparticles concentration on thermal properties, convective heat transfer coefficient, thermal performance factor and entropy generation is investigated. An enhancement in thermal conductivity for GNP of between 12% and 28% is observed relative to the case without nanoparticles. The convective heat transfer coefficient for the GNP nanofluid is found to be up to 15% higher than for the base fluid. The heat transfer rate and thermal performance for 0.1 wt.% of GNP nanofluid is found to increase by a factor of up to 1.15. For constant velocity flow, frictional entropy generation increases and thermal entropy generation decreases with increasing nanoparticle concentration. But, the total entropy generation tends to decrease when nanoparticles are added at constant velocity and to decrease when velocity rises. Finally, it is demonstrated that a GNP nanofluid with a concentration between 0.075 wt.% and 0.1 wt.% is more energy efficient than for other concentrations. It appears that GNP nanofluids can function as working fluids in heat transfer applications and provide good alternatives to conventional working fluids in the thermal fluid systems.     

Catalytic hydrolysis of sodium borohydride by a novel nickel–cobalt–boride catalyst

With the aim of designing an efficient hydrogen generator for portable fuel cell applications nickel–cobalt–boride (Ni–Co–B) catalysts were prepared by a chemical reduction method and their catalytic hydrolysis reaction with alkaline NaBH₄ solution was studied. The performance of the catalysts prepared from NaBH₄ solution with NaOH, and without NaOH show different hydrogen generation kinetics. The rate of hydrogen generation was measured using Ni–Co–B catalyst as a function of the concentrations of NaOH and NaBH₄, as well as the reaction temperature, in the hydrolysis of alkaline NaBH₄ solution. The hydrogen generation rate increases for lower NaOH concentrations in the alkaline NaBH₄ solution and decreases after reaching a maximum at 15 wt.% of NaOH. The hydrogen generation rate is found to be constant with respect to the concentration of NaBH₄ in the alkaline NaBH₄ solution. The activation energy for hydrogen generation is found to be 62 kJ mol⁻¹, which is comparable with that of hydrogen generation by a ruthenium catalyst.