Momentum Equations Research Articles

A Simplified Two-Fluid Model Based on Equilibrium Closure for a Dilute Dispersion of Small Particles

Two-fluid formalisms that fully account for all complex inter-phase interactions have been developed based on a rigorous ensemble-averaging procedure. Here, we apply equilibrium approximation to particle velocity to simplify two-phase flow equations for the case of a dilute dispersion of particles much smaller than the flow scales. First, we extend an earlier approach to consider the rotational motion of the particles and seek an equilibrium approximation for the angular velocity of the particulate phase. The resulting explicit knowledge of the particulate phase translational and rotational velocities in terms of fluid velocity eliminates the need to consider the momentum equations for the particulate phase. The equilibrium approximations also provide precise scaling for various terms in the governing equations of the two-fluid model, based on which a simplified set of equations is obtained here. Three different regimes based on the relative strength of gravitational settling are identified, and the actual form of the simplified two-phase flow equations depends on the regime. We present two simple examples illustrating the use of the simplified two-fluid formalism.

Fractional modeling of bioconvection in Jeffrey nanofluids with gyrotactic organisms

AbstractThe current study examines how mass and heat transfer affect mobility of Jeffrey fluid while taking sun radiation across the vertical plate into account. In the polyvinyl alcohol water base fluid, the study combines gyrotactic organisms with copper nanoparticles. Microorganisms classified as gyrotactic respond to gravitational and viscous forces by swimming and orienting themselves, which results in the formation of patterns known as bioconvection, which is the result of the collective movement of these microorganisms. The main goals are to address the growing uses of solar plates by creating a unique mathematical model for flow and thermal properties of the parabolic trough solar collector (PTSC) installed on solar panel. Sunlight is directed onto a single focal line by curved mirrors in PTSCs, which heat the fluid moving over the plate at this focused line. The momentum, heat, and mass equations are solved by the model using Fourier and Fick's laws. The Laplace transform is then used to convert the solution sets into dimensionless Partial differential equation (PDE) for the velocity, energy, and mass fields. The innovative aspect of this model is its in‐depth examination of non‐Newtonian nanofluids, which are boosted by the addition of gyrotactic organisms and copper nanoparticles to increase heat transfer efficiency. The impacts of several factors on flow characteristics, including the Lewis number, mass Grashof number, Grashof number for bioconvection, magnetic and electric parameters, Peclet number, chemical reaction parameter, and Prandtl number, are shown graphically. Increasing the radiation parameters and volume fraction results in a noticeable improvement in the temperature profile. By demonstrating the superior heat transfer capabilities of non‐Newtonian nanofluids in solar energy applications, this work advances the field. It is particularly relevant to microchip cooling, solar energy systems, and thermal energy systems. In summary, the work provides a comprehensive model that advances our understanding of heat and mass transfer in non‐Newtonian nanofluids that are exposed to ambient sunlight. In the future, the model will be employed in real solar energy systems to confirm its effectiveness. The findings have applications in the development of temperature control technology and solar energy systems that are more effective.

Investigation of activation energy of Maxwell fluid with Newtonian heat effect at the boundary layer flow on a cylinder embedded in porous medium

AbstractThis study aims to investigate the flow of Maxwell fluid over a cylinder, considering the combined effect of Magnetohydrodynamics (MHD) and a permeable medium on the equation of momentum, heat generation and radiation on the equation of energy, and activation energy on the mass equation. The objective is to analyze the impression of these physical phenomena on the fluid's behavior under specified boundary conditions, with a focus on the Newtonian heating effect. The significance of this research lies in its application to industrial processes, polymer extrusion and cooling of metallic sheets, where controlling fluid flow and energy transfer is crucial. The dimensional governing equations for momentum, energy, and concentration were changed into non‐dimensional forms via appropriate dimensionless quantities and similarity variables. The subsequent nonlinear ordinary differential equations were answered using the BPV4C inbuilt MATLAB solver. The key findings reveal that the concentration outline rises with a rise in activation energy and declines with a rise in the chemical reaction. Additionally, the Nusselt number profile displays an increasing trend with the Newtonian heating parameter, indicating enhanced heat transfer. Quantitative results demonstrate that the occurrence of a porous medium and MHD significantly impacts the velocity and temperature contours. The real‐time application of this study can be seen in the optimization and design of chemical reactors, heat exchangers, and cooling systems in engineering processes. The developed model delivers insights into controlling energy and mass transfer in processes involving stretching surfaces, which is essential for improving efficiency and product excellence in various industrial applications.

The Role of Heater Size and Location in Modulating Natural Convection Behavior in Cu-Water Nanofluid-Loaded Square Enclosures

Enhancing the energy efficiency of thermal systems reduces their consumption, lowers costs, and reduces undesired environmental impact, thus making these systems more sustainable. The current work introduces a passive method for heat transfer enhancement that is carried out using natural convection by nanofluid. This work introduces a computational study of the process of natural convection within a square cavity containing Cu/H2O nanofluid. The cavity wall on the left side undergoes partial isothermal heating, while the opposing side is fully cooled isothermally, with all other boundaries maintained adiabatic. A mathematical model formulated based on a 2-D model was used to provide the solution for the system of governing equations of mass, momentum, and energy conservation, employing the finite element technique. A commercial CFD package is utilized to perform the computational simulation. The present investigation delves into the impact of the Rayleigh number, nanoparticle concentration, heater length, and heater location on the flow field and heat transfer characteristics. The model outcomes were displayed for a wide range of the pertinent parameters as 103 ≤ Ra ≤ 106, 0.25 ≤ lh ≤ 1.0, 0.125 ≤ hc ≤ 0.875, and 0.02 ≤ ϕ ≤ 0.10. Also, correlation equations relating the average Nusselt number to these crucial parameters are derived. These equations are simple and can be applied in practice easily in many fields, such as electric and electronic equipment cooling and thermal management of heat sources. Also, these equations gather all the parameters that affect the heat transfer process. They are shedding light on the intricate interplay between these parameters in the natural convection heat transfer process.

Calendering of non-isothermal viscoelastic sheets of finite thickness: A theoretical study

Abstract In this article, the sheeting of pseudoplastic material under non-isothermal conditions has been studied. It is an excellent forecasting instrument for sheeting, where the thickness of the sheet is relatively thin in relation to the roll size, according to theoretical study based on the lubrication theory. At the calendering process, the detachment point is determined by taking the material parameter’s impact into account. The governing equations are derived for the constitutive equation of the Carreau fluid with momentum and energy equations. Suitable non-dimensional parameters are used to develop the non-linear partial differential expressions into ordinary differential systems. The maximum pressure, roll separating force, normal stress effect, and power delivered to the fluid by rolls are among the engineering-relevant quantities that are determined. Moreover, a graphic analysis is conducted to examine the impact of different material parameters on the temperature profile, pressure gradient, velocity profile, and pressure distribution. The results specific mechanism is explained in depth.

Numerical Investigation of Turbulent Convection Flow in a Rectangular Closed Cavity

Natural turbulent convection in closed cavities has many practical applications in the field of engineeringsuch as the design of electronic computer chips, atomic installation and industrial cooling among others. Inparticular, it enables in achieving a desired micro-climate and efficient ventilation in a building. Recent studiesshow that turbulent flow is affected by variations in Rayleigh numbers, aspect ratio, and heater positionamong others. Temperature is kept constant in all these studies hence inadequate literature on the effectsof temperature on a turbulent flow. In this study, aspect ratio and Rayleigh numbers are kept constant at2 and 1012 respectively and natural turbulent convection flow in a closed rectangular cavity is investigatednumerically as the operating temperature is varied from 285.5K to 293K. The rectangular cavity’s lower wallwas heated and cooling done at the top face wall while the rest of the vertical walls were kept in adiabaticcondition. Material properties such as density of the fluid kept on changing at any given temperature. Thethermal profile data generated influenced the nature of the turbulent flow. The non-linear averaged continuity,momentum, and energy equation terms were modeled by the SST k − ω model to generate streamlines,isotherms, and velocity magnitude for a different operating temperature and presented graphically. The finitedifference method and FLUENT were used to solve two SST k − ω model equations, vortices, and energy withboundary conditions. It was discovered that, as the operating temperature increased turbulence decreaseddue to a decrease in the velocity of the elements and vortices became more parallel and smaller.

Dynamic Formulation of the Double Multiple Stream Tube Model of Offshore Vertical Axis Wind Turbines

Abstract The paper proposes a novel algorithm to simulate Vertical Axis Wind Turbines in floating motion based on a low-fidelity approach. The conventional Double Multiple Stream Tube Model is formulated as a steady state model to deal with the inherent unsteady aerodynamics of VAWTs. The Double Multiple Stream Tube Model makes use of an average contribution of the blade passages over a revolution to solve the Blade Element Momentum equation. The model reduces the dependence of the solution (the induction field) to a space variable. Floating Vertical Axis Wind Turbines undergoes a variation of aerodynamic quantities according to wave periods, also different from the rotor period. This paper provides a new formulation of the Double Multiple Stream Tube Model to solve the turbine rotation in time and space, introducing the variation of the platform velocity and a revised approach to the downstream actuator disc. The most impactful parameter is found to be the wave frequency: wave periods at full-scale are lower than the ones of the turbine rotor, featuring optimal operating points at low tip speed ratios. The increase of the wave frequency highlights the differences of the average torque prediction for a single blade up to 10% between the unsteady and the standard formulation. In this context, the development of fast low-fidelity computational tools play a key role in the assessment of the preliminary designs of Floating Offshore Vertical Axis Wind Turbines and it will be used to optimise the aerodynamic design of the laboratory scaled model under surge motion.

Mathematical modeling of electroosmotically driven peristaltic propulsion due to transverse deflections of two periodically deformable curved tubes of unequal wavelengths

The present study aims to investigate the viscid fluid propulsion due to the electroosmosis and transverse deflections of the sinusoidally deformable tubes of unequal wavelengths in the presence of electro-kinetic forces. This situation is estimated from the physical model of physiological fluid flow through a tubular structure in which an artificial flexible tube is being inserted. In this model, both peristaltically deforming tubes are taken in a curved configuration. The flow-governing momentum equations are simplified by the approximation of the long wavelength as compared to the outer tube's radius, whereas the Debye–Hückel approximation is used to simplify the equations that govern the electric potential distribution. Here, the authors have used the DSolve command in the scientific computing software MATHEMATICA 14 to obtain the expressions for electric potential and axial velocity of viscid fluid. In this work, the authors have analyzed the impact of various controlling parameters, such as the electro-physical parameters, curvature parameter, radius ratio, wavelength ratio, and amplitude ratios, on the various flow quantities graphically during the transport of viscid fluid through a curved endoscope. Here, contour plots are also drawn to visualize the streamlines and to observe the impacts of the control parameters on fluid trapping. During the analysis of the results, a noteworthy outcome extracted from the present model is that an increment in electro-physical parameters, such as Helmholtz–Smoluchowski velocity and the Debye–Hückel parameter, are responsible for enhancement in the shear stress at the inner tube's wall and the axial velocity under the influence of electro-kinetic forces. This is because of the electric double layer (EDL) thickness, which gets reduced on strengthening the Debye–Hückel parameter. This reduced EDL thickness is responsible for the enhancement in the axial velocity of the transporting viscid fluid. The present model also suggests that the axial velocity of viscid fluid can be reduced by enhancing the ratio of wavelengths of waves that travel down the walls of the outer curved tube and the inner curved tube. The above-mentioned results can play a significant role in developing and advancing the endoscopes that will be useful in many biomedical processes, such as gastroscopy, colonoscopy, and laparoscopy.

The results of a magnetic field on flow past an inclined isothermal vertical plate with heat and varying mass diffusion

This work describes the influence of a magnetic environment on the circulation of matter and thermal energy over an accelerating advanced inclined isothermal sectional surface. The temperature is elevated to . The proximity intensity is increased to . The Laplace-transform approach is applied to solve the dimensionless analytical equation. In this research work different physical variables including thermal grashof number (Tg), mass grashof number (Tm), Schmidt number (Sc), Prandtl number(Pr), time(t), velocity profile ( ), temperature ( ), and intensity ( ) are investigated. The output of the graph produced by the Matlab software commands an energy equation, momentum equation, and concentration equation. The values are shown in an aligned pattern for a variety of flow variables. Diagrammatic representations of the fluid velocity profiles are provided. The velocity rises corresponding to different Tg and the velocity rises corresponding to different Tm. As the angle is decreased, the velocity increases differently. , As the different magnetic environment values lower, the velocity rises. Elevated values of angle ( ), Sc, Pr, and contribute to an enhancement in local skin friction, and an upsurge in Gr, Gc, and t leads to a reduction. A tabulated presentation of Nusselt quantities reveals a positive correlation with increasing Pr. Similarly, Sherwood quantities, as tabulated for various parameters, demonstrate a proportional increase with rising Sc.

DUFOUR, SORET AND DISSIPATION EFFECTS ON MAGNETOHYDRODYNAMIC FLOW PAST A STRETCHING PERMEABLE SHEET WITH RADIATIVE CHEMICAL REACTION ANDHEAT SOURCE

Present investigation explores the numerical solution of Magnetohydrodynamic laminar steady gray fluid flow with heat and mass transfer past a stretching permeable sheet, incorporating several factors such as heat source, viscous and joules dissipation effects, thermal radiation, chemical reactions, Soret and Dufour effects. The governing nonlinear PDEs are simplified into ODEs with the help of similarity transformations. Momentum equation is solved analytically. Non-Dimensional temperature and concentration calculated with the help of RK- shooting method with Nachtsheim-Swigert iteration method. Key results include the calculation of heat and mass transfer rates and Co-efficient of skin friction. The effects of various parameters such as magnetic field strength, suction, radiation, chemical reactions, internal heat generation and dissipative effects on the flow characteristics and other physical parameters are thoroughly investigated, highlighting their impact on the thermal and concentration profiles in the system. This comprehensive analysis provides insights into the behaviour of the fluid flow under the specified conditions, which can be useful for applications in engineering and industrial processes.

Three-dimensional bioconvection in a nanofluid film over a wedge surface with entropy generation and nonlinear radiation: A revised Buongiorno model

ABSTRACT The advances in the field of nanotechnology tends to improve the rate of heat extraction and transmission from foreseeable causes and has fascinated researchers and scientists. Furthermore, nanofluids are having a useful impact on various fields such as nuclear industries, powered lasers, chilling of microelectronic devices, solar energy capture, transformers oil cooling, and in cancer diagnosis. Hence, magnetohydrodynamic (MHD) three-dimensional bioconvective flow of nanofluid accommodating gyrotactic microorganisms over a wedge surface with zero nanoparticles mass flux boundary conditions is studied, using large Reynolds number approximation. The additional novelty of the study is the implementation of the revised Buongiorno model, which deliberates the effects of thermo-migration and Brownian diffusion of nanoparticles, together with passive control of nanoparticles fraction on the wedge surface. The self-similar form of conservation equations of energy, nanoparticle fraction, gyrotactic microorganisms, and momentum are handled numerically. T The results of the Blasius and the Hiemenz flows are retrieved as special cases of the model. It was found that the temperature of the nanofluid increases considerably when considering the nonlinear thermal radiation compared to the linear radiant heat. Increasing the strength of the Brownian diffusion increases the heat transport but reduces the mass transport.

Multi-Objective Numerical Analysis of Horizontal Rectilinear Earth–Air Heat Exchangers with Elliptical Cross Section Using Constructal Design and TOPSIS

This study presents a numerical evaluation of a Horizontal Rectilinear Earth–air Heat Exchanger (EAHE), considering the climatic and soil conditions of Viamão, Brazil, a subtropical region. The Constructal Design method, combined with the Exhaustive Search, was utilized to define the system constraints, degree of freedom, and performance indicators. The degree of freedom was characterized by the aspect ratio between the vertical and horizontal lengths of the elliptical cross-section duct (H/L). The performance indicators for the EAHE configurations were assessed based on thermal potential (TP) and pressure drop (PD). The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) was applied for multi-objective evaluation, and a methodology for EAHE is proposed. The problem was solved using FLUENT software (version 2024 R2), which employs the Finite Volume Method to solve the conservation equations for mass, momentum, and energy. The (H/L)T,o = 6.0 configuration showed a 16.4% increase in thermal performance for heating and 15.9% for cooling compared to the conventional circular duct. Conversely, the (H/L)F,o = 1.0 configuration reduced pressure loss by 65.33%. The integration of Constructal Design with TOPSIS facilitated the identification of optimized geometries that achieve a balance between performance indicators and those that specifically prioritize thermal or fluid dynamic aspects, being this approach an original scientific contribution of the present work.

Unidirectional Flow Through Time-Dependent Cross-Sectional Areas of a Compliant Tube and a Valve: A Nonlinear Model

This work investigates the conditions for net flow generation by a straight tube with a cross-sectional area harmonically varying in time that connects two tanks—a problem that is mainly found in the design of impedance pumps. By assuming a quasi-one-dimensional flow and applying continuity and momentum equations, a first-order differential equation with respect to the flow rate is derived and presented for the first time, including a nonlinear term that is responsible for net flow rate generation. Namely, the net flow rate is found to be nonzero (as is the nonlinear term) if the cross-sectional areas of the two tanks are unequal and one of them is smaller than that of the straight tube. In this case, the flow is directed from the smaller to the larger tank and the net flow rate increases with the frequency of the tube’s cross-sectional area variation. In contrast, when the tanks’ cross-sections are equal, the net flow is generated only if a valve is installed, e.g., at one end of the tube, due to the large asymmetries imposed in the hydraulic losses with respect to the tube mid-length. Compared with constant valve opening, the net flow rate is augmented significantly if the valve opening is time-dependent. By employing the same equation, the flow rate of an intra-aortic counter-pulsating balloon pump is also examined, in which the valve (representing the aortic valve) opens during the shrinkage of the tube, and it is shown that the net flow rate increases with the frequency and amplitude of the tube’s cross-sectional area. Conclusively, the harmonic oscillation in time of a tube’s wall can cause unidirectional flow only if asymmetric losses are present with respect to its mid-length.

A Review of Comprehensive Guidelines for Computational Fluid Dynamics (CFD) Validation in Solar Chimney Power Plants: Methodology and Manzanares Prototype Case Study

This review provides a comprehensive examination of CFD modeling procedures for SCPP, with an emphasis on the detailed methodologies and a case study of the Manzanares prototype in Spain. The introduction delineates the historical context and physical modeling principles of solar chimneys, while highlighting their potential in industrial applications. The governing equations are meticulously discussed, covering assumptions in both 2D and 3D CFD modeling, the continuity and momentum equations, and the selection and accuracy of turbulence models, particularly the k-ε equations. The review also delves into heat transfer modeling, encompassing the energy equation and radiation modeling. Analytical evaluations of turbine pressure drop ratios and performance metrics for power generation efficiency are critically analyzed. The establishment of boundary conditions in solar chimney applications, including sky temperature assessments and distinctions between 2D and 3D boundary conditions, is extensively explored. Mesh generation techniques for both 2D and 3D CFD models are presented, supported by case studies. Parametric studies and experimental investigations are scrutinized to elucidate their impact on the performance of solar chimneys. The temperature–entropy diagram for an idealized Brayton cycle is introduced as a conceptual framework for efficiency analysis. Validation of the CFD codes, both 2D and 3D, against experimental data is performed to ensure model accuracy. The review further examines energy balance approaches in modeling solar chimneys, presenting state-of-the-art CFD results and discussing their implications in both 2D and 3D contexts. The synthesis of these findings culminates in a comprehensive conclusion, offering insights into the future directions and potential advancements in the CFD modeling of solar chimneys. This work aims to serve as a definitive reference for researchers and practitioners in the field, providing a robust foundation for the development and optimization of SCPP technology.

Blood Based Magnetohydrodynamic Casson Hybrid Nanofluid Flow on Convectively Heated Bi-Directional Porous Stretching Sheet with Variable Porosity and Slip Constraints

Abstract Fluid flow through porous spaces with variable porosity has wide-range applications, notably in biomedical and thermal engineering, where it plays a vital role in comprehending blood flow dynamics within cardiovascular systems, heat transfer and thermal management systems improve efficiency using porous materials with variable porosity. Keeping these important applications in view, in current study blood based hybrid nanofluid flow has considered on a convectively heated sheet. The sheet exhibits the properties of a porous medium with variable porosity and extends in both the x and y directions. Blood has used as base fluid in which the nanoparticles of Cu and CuO have been mixed. Thermal radiation, space-dependent, and thermal-dependent heat sources have been incorporated into the energy equation, while magnetic effects have been integrated into the momentum equations. Dimensionless variables have employed to transform the modeled equations into dimensionless form and facilitating their solution using bvp4c approach. It has concluded in this study that, both the primary and secondary velocities augmented with upsurge in variable porous factor and declined with escalation in stretching ratio, Casson, magnetic and slip factors along x- and y-axes. Thermal distribution has grown up with upsurge in Casson factor, magnetic factor, thermal Biot number and thermal/space dependent heat sources while has retarded with growth in variable porous and stretching ratio factors. The findings of this investigation have been compared with the existing literature, revealing a strong agreement among present and established results that ensured the validation of the model and method used in this work.

Computational analysis of unsteady oscillatory flow of nanofluid with variable electric conductivity: gear-generalized differential quadrature approach

Abstract This study numerically investigated the entropy production in nanofluids’ dissipative unsteady oscillatory flow characterized by variable electric conductivity and magnetic heating effects. The imposition of the non-isothermal boundary condition on the oscillatory stretching sheet plays a crucial role in establishing the self-similar solution in the presence of viscous heating. An external magnetic field (uniform in space and time) is imposed perpendicular to the plane of the oscillating stretched boundary. The energy equation, incorporating viscous dissipation effects and momentum equation, is reduced to nonlinear coupled partial differential equations and numerically solved using the Gear-generalized differential quadrature scheme.Additionally, to ensure the precision and reliability of the outcomes, the numerical code undergoes a thorough validation process that involves comparing its outputs to the findings of previous available studies. The Corcione model is implemented to describe the nanofluid’s effective viscosity and thermal conductivity. Furthermore, expressions for entropy production and relative irreversibility parameter (Bejan number), considering variable electric conductivity, are derived and computed based on solutions obtained from momentum and energy equations. The impacts of parameters such as magnetic parameter, variable electric conductivity parameter, Eckert number, Strouhal number, Prandtl number and temperature difference parameters on flow, heat transfer, entropy generation, and Bejan number are systematically illustrated and examined. We observed that increasing the variable electric conductivity parameters reduces the velocity profiles while improving the thermal fields. Similar behavior is found when the strength of a magnetic field is increased. The skin friction coefficient exhibits an augmentation in response to the Eckert number, dimensionless time, Strouhal number, nanoparticle volume fraction, magnetic parameter, and variable thermal conductivity parameter. Conversely, the Nusselt number increases concerning the Strouhal number and nanoparticle volume fraction. At the same time, it declines in association with the magnetic parameter, dimensionless time, Eckert number, and variable electric conductivity parameter.This comprehensive investigation enhances our understanding of nanofluid dynamics and provides valuable insights for optimizing thermal management systems across various engineering disciplines.

A fully-coupled algorithm with implicit surface tension treatment for interfacial flows with large density ratios

The stability of most surface-tension-driven interfacial flow simulations is governed by the capillary time-step constraint. This concerns particularly small-scale flows and, more generally, highly-resolved liquid-gas simulations with moderate inertia. To date, the majority of interfacial-flow simulations are performed using an explicit surface-tension treatment, which restrains the performance of such simulations. Recently, an implicit treatment of surface tension able to breach the capillary time-step constraint using the volume-of-fluid (VOF) method was proposed, based on a fully-coupled pressure-based finite-volume algorithm. To this end, the interface-advection equation is incorporated implicitly into the linear flow solver, resulting in a tight coupling between all implicit solution variables (colour function, pressure, velocity). However, this algorithm is limited to uniform density and viscosity fields. Here, we present a fully-coupled algorithm for interfacial flows with implicit surface tension applicable to interfacial flows with large density and viscosity ratios. This is achieved by solving the continuity and momentum equations in conservative form, whereby the density is treated implicitly with respect to the colour function, and the advection term of the interface-advection equation is discretised using the THINC/QQ algebraic VOF scheme, yielding a consistent discretisation of the advective terms. This new algorithm is tested by considering representative surface-tension-dominated interfacial flows, including the Laplace equilibrium of a stationary droplet and the three-dimensional Rayleigh-Plateau instability of a liquid filament. The presented results demonstrate that interfacial flows with large density and viscosity ratios can be simulated and energy conservation is ensured, even with a time step larger than the capillary time-step constraint, provided that other time-step restrictions are satisfied.

Part-load performance analysis of a modular biomass boiler with a combined heat and power industrial Rankine cycle and supplementary sCO2 Brayton cycle

Abstract The addition of a supplementary high efficiency cycle integrated with an existing steam power cycle may increase energy efficiency and net generation. In this paper, part load performance and operation of a modular biomass boiler with an existing industrial Rankine steam heat and power cycle and supplementary supercritical CO2 (sCO2) Brayton cycle is analysed. The aim is to leverage the high efficiency of the sCO2 cycle by retrofitting sCO2 heaters in the existing biomass boiler, increasing net power output and thermal efficiency. With nominal load performance previously investigated, understanding part-load performance and operation is vital to determine cycle feasibility. A quasi-steady-state one dimensional thermofluid network model was used to simulate the integrated cycle performance for loads ranging from 100% to 60%. The model solves the mass, energy, momentum, and species balance equations, capturing detailed component characteristics. Two control methodologies are explored for the sCO2 Brayton cycle, namely inventory control and inventory control combined with throttling valve control. Inventory control is selected as the better performing control strategy for load following, maintaining high thermal efficiency across partial loads. At 60% load, the sCO2 compressor operates near the pseudo-critical point, leading to a sharp decrease in sCO2 cycle capacity, which requires careful management of inventory control. Two sCO2 heater configurations are investigated, namely a single convective-dominant heater, and a dual heater configuration with a radiative and a convective heater. The single heater configuration is preferred to minimise adverse impacts on the Rankine cycle superheaters.

A numerical study of double flow focusing micro-jets

Purpose Double flow-focusing nozzles (DFFNs) form a coaxial flow of primary liquid with micro-crystalline samples, surrounded by secondary liquid and focusing gas. This paper aims to develop an experimentally validated numerical model and assess the performance of micro-jets from a DFFN as a function of various operating parameters for the water–ethanol–helium system, revealing the jet's stability, diameter, length and velocity. Design/methodology/approach The physical model is formulated in the mixture-continuum formulation, which includes coupled mass, momentum and species transport equations. The model is numerically formulated within the finite volume method–volume of fluid approach and implemented in OpenFOAM to allow for a non-linear variation of the fluid's material properties as a function of the mixture concentration. The numerical results are compared with the experimental data. Findings A sensitivity study of jets with Reynolds numbers between 12 and 60, Weber numbers between 4 and 120 and capillary numbers between 0.2 and 2.0 was performed. It was observed that jet diameters and lengths get larger with increased primary and secondary fluid flow rates. Increasing gas flow rates produces thinner, shorter and faster jets. Previously considered pre-mixed and linear mixing models substantially differ from the accurate representation of the water–ethanol mixing dynamics in DFFNs. The authors demonstrated that Jouyban–Acree mixing model fits the experimental data much better. Originality/value The mixing of primary and secondary liquids in the jet produced by DFFN is numerically modelled for the first time. This study provides novel insights into mixing dynamics in such micro-jets, which can be used to improve the design of DFFNs.

Darcy's law survival from no-slip to perfect-slip flow in porous media

A macroscopic model for perfect-slip flow in porous media is derived in this work, starting from the pore-scale flow problem and making use of an upscaling technique based on the adjoint method and Green's formula. It is shown that the averaged momentum equation has a Darcy form in which the permeability tensor, $\boldsymbol{\mathsf{K}}_{ps}$ , is obtained from an associated adjoint (closure) problem that is to be solved on a (periodic) unit cell representative of the structure. Similarly to the classical permeability tensor, $\boldsymbol{\mathsf{K}}$ , in the no-slip regime, $\boldsymbol{\mathsf{K}}_{ps}$ is intrinsic to the porous medium under consideration and is shown to be symmetric and positive. Integral relationships between $\boldsymbol{\mathsf{K}}_{ps}$ , the partial-slip flow permeability tensor, $\boldsymbol{\mathsf{K}}_{s}$ , and $\boldsymbol{\mathsf{K}}$ are derived. Numerical simulations carried out on two-dimensional model porous structures, together with an approximate analytical solution and an empirical correlation for a particular configuration, confirm the validity of the macroscopic model and the relationship between $\boldsymbol{\mathsf{K}}_{ps}$ and $\boldsymbol{\mathsf{K}}$ .