Vacuum-assisted air gap membrane distillation (V-AGMD) is a thermal process in which heat and mass transfer occur simultaneously through a porous, hydrophobic membrane under a partial vacuum. The main objective of this study is to theoretically analyze the factors influencing heat and mass transport phenomena and their effects on the water productivity and energy efficiency of the V-AGMD process. Two-dimensional momentum, and heat and mass transfer models for analyzing the V-AGMD process in commercial modules were proposed and solved using the Generalized Integral Transform Technique (GITT). Validation with 164 experimental data points with four types of commercial modules under different operating conditions ensured that the methodology and the proposed models could be used to evaluate the performance of V-AGMD spiral modules. The heat transfer analysis identified that the flow occurs mostly under thermally developing conditions in the feedwater channel, which explain and generalize reports on the strong effect of the residence time on energy efficiency. Based on this analysis, a power law relationship was identified, including the Graetz number and the dimensionless equivalent permeability as the main predictors for the energy efficiency of the V-AGMD process. The evaluation of the effects of membrane area and pressure in the air gap revealed that concentration polarization is less intense with the decrease in driving force. In contrast, the reduction in pressure increases both the permeate flux and the energy efficiency at the cost of an increase in the phenomena of temperature and concentration polarizations, requiring particular attention in the treatment of concentrated solutions.

The physics and modelling of cooling and freezing of droplets in contact with a colder substrate are of interest in various engineering applications. This work provides experimental results of this process employing infrared thermography for temperature measurements at the droplet's surface. Also, a high-speed camera is employed to observe the recalescence period and measure the freezing front movement and the droplet shape change. Three substrates are prepared with distinct wettability ranges, i.e. one hydrophilic and two hydrophobic surfaces. From the experimental observation of a solidification front parallel to the substrate plane, a mixed lumped-differential model of the heat transfer process based on the Coupled Integral Equations Approach is proposed, reformulating the two-dimensional partial differential formulation in cylindrical coordinates into a one-dimensional transient energy equation for the droplet external surface temperature. Direct comparisons of the experimental and theoretical results for the supercooling period show excellent agreement for the droplet surface temperatures at different heights and for different values of the substrate-droplet contact angle. It is also shown that the classical partial lumped system analysis does not provide adequate predictions in the present problem. Finally, the dynamics of the recalescence and freezing stages are experimentally evaluated and physically interpreted.This article is part of the theme issue 'Heat and mass transfer in frost and ice'.

Membrane distillation (MD) is an evolving thermal separation technique most frequently aimed at water desalination, compatible with low-grade heat sources such as waste heat from thermal engines, solar collectors, and high-concentration photovoltaic panels. This study presents a comprehensive theoretical–experimental evaluation of three commercial membranes of different materials (PE, PVDF, and PTFE), tested for two distinct MD modules—a Direct Contact Membrane Distillation (DCMD) module and an Air Gap Membrane Distillation (AGMD) module—analyzing the impact of key operational parameters on the performance of the individual membranes in each configuration. The results showed that increasing the feed saline concentration from 7 g/L to 70 g/L led to distillate flux reductions of 12.2% in the DCMD module and 42.9% in the AGMD one, averaged over the whole set of experiments. The increase in feed temperature from 65 °C to 85 °C resulted in distillate fluxes up to 2.36 times higher in the DCMD module and 2.70 times higher in the AGMD one. The PE-made membrane demonstrated the highest distillate fluxes, while the PVDF and PTFE membranes exhibited superior performance under high-salinity conditions in the AGMD module. Membranes with high contact angles, such as PTFE with 143.4°, performed better under high salinity conditions. Variations in operational parameters, such as flow rate and temperature, markedly affect the temperature and concentration polarization effects. The analyses underscored the necessity of a careful selection of membrane type for each distillation configuration by the specific characteristics of the process and its operational conditions. In addition to experimental findings, the proposed heat and mass transfer-reduced model showed good agreement with experimental data, with deviations within ±15%, effectively capturing the influence of operational parameters. Theoretical predictions showed good agreement with experimental data, confirming the model’s validity, which can be applied to optimization methodologies to improve the membrane distillation process.

The present work addresses biomolecular interactions in Surface Plasmon Resonance (SPR)-based biosensors, explicitly focusing on mass transport and binding kinetics. The Generalized Integral Transform Technique (GITT) is employed to solve the nonlinear system of partial differential equations describing mass transport, while the Markov Chain Monte Carlo (MCMC) method is adopted for accurately estimating the kinetic constants of the model. The outcomes were corroborated with simulated measurements and validated against experimental data related to the binding of the receptor-binding domain (RBD) of the spike protein of SARS-CoV-2 bound to the cell receptor angiotensin-converting enzyme 2 (ACE2) in the Biacore system. Our findings demonstrate the efficacy of the GITT in describing the dynamics of average concentrations of the free analyte and of the bound analyte-receptor complex, aligning with results obtained in prior studies. Furthermore, our results demonstrate that the MCMC method is a robust tool for estimating model kinetic constants, with estimates closely approximating the exact values and falling within a 99 % confidence interval. The estimated average concentrations concurred with simulated measurements, even when accounting for Gaussian noise. The experimental validation results strengthen our conclusions, aligning the model parameter estimates with reference values from the literature. Therefore, this study suggests that the adopted mathematical model and numerical methodology hold significant potential for analyzing and comprehending biomolecule binding data, representing a valuable tool for studying complex biomolecular interactions.

Therapeutic ultrasound heating is a very common technique in the treatment of muscular and joints injuries. However, insufficient or excessive heating delivered to living tissues can be either innocuous or hazardous to the patient. The present work deals with the heat transfer process in living tissues, with particular concern on the behavior at the muscle-bone interface, when ultrasound irradiation is applied. Based on the Generalized Integral Transform Technique (GITT), a hybrid numerical-analytical solution of the Pennes’ equation using the open-source UNIT code (“UNified Integral Transforms”) is proposed, considering different boundary conditions. The influence of physical properties and parameters, such as beam incidence angle, acoustic attenuation coefficients for refracted longitudinal and shear waves in the bone, and blood perfusion in the muscle, were more closely analyzed. Numerical results of temperature profiles showing variations of up to 2 °C for different situations are presented, illustrating a marked dependence of heating patterns on the perfusion rates as well as on the effect of shear waves. The conclusions are that such type of more detailed analysis is essential for the accurate identification of physical properties in living tissues and phantoms and can contribute to the treatment planning for therapeutic ultrasound.

The film condensation heat transfer problem in the presence of a heavy non-condensable gas is revisited. To provide an accurate and robust hybrid numerical-analytical treatment, a systematic framework for solving heat and fluid flow described by boundary layer formulations via integral transforms has been proposed. A physically inspired change of variables stemming from the scale analysis of the governing equations is employed, aiming at facilitating algebraic manipulations and improving the convergence behavior of the Generalized Integral Transform Technique (GITT) hybrid solution. The boundary layer model for condensation processes with non-condensing substances, along with the developed solution method, are verified through comparisons with the Karman-Pohlhausen integral method and experimental results from the literature. The agreement is overall satisfactory, building confidence on the proposed methodology and computational code. In addition, a physical analysis further confirms the marked effect the non-condensable gas has on the condensation process, in some cases, decreasing the heat transfer rate by more than 80% when compared to the case with pure vapor. Interestingly, the heat transfer rate is determined to hold the same scaling with the height of the wall predicted by the classical Nusselt model for condensation of saturated pure vapor. In contrast, the temperature difference between the gas mixture and the wall, which is the driving force of the process, partially loses the strong effect predicted by the Nusselt model due to the build-up of non-condensable gas at the interface and the associated decrease in local vapor pressure.

This is an in-memoriam honoring Professor Darrell W. Pepper as an exceptional researcher, educator, and engineer.

Natural convection inside horizontal concentric annular cavities is dealt with through the generalized integral transform technique (GITT), offering a hybrid numerical-analytical solution of the continuity, Navier-Stokes, and energy equations in cylindrical coordinates. The flow is in steady-state, laminar regime, two-dimensional, buoyancy-induced, and the governing equations are written in the streamfunction-only formulation. Two strategies of integral transformation are adopted to verify the best computational performance, namely, the usual one with eigenvalue problems for both streamfunction and temperature defined in the radial variable, and a novel alternative with eigenvalue problems defined in the azimuthal angular coordinate. First, the eigenfunction expansions convergence behavior is analyzed to critically compare the two integral transform solution strategies. Then, test cases for different aspect ratios and Rayleigh numbers are validated with experimental data from the classical work of Kuehn and Goldstein. A maximum relative deviation of 5% is found comparing the GITT results for the average Nusselt number against the experimental data, while an 8% maximum relative deviation is found comparing against an empirical correlation by the same authors. It is concluded that the GITT solution with the eigenvalue problem in the angular coordinate yields better convergence rates than the more usual eigenfunction expansion in the radial variable. This is due to the originally homogeneous boundary conditions in the angular direction, which do not require filtering for convergence enhancement, as opposed to the required filter in the radial direction that introduces a source term in the filtered equation for the streamfunction. Nusseltxteta

Membrane distillation (MD) has emerged as a promising thermal separation technology aimed mainly at water desalination, compatible with low-grade heat sources such as waste heat from thermal engines, solar collectors, and high concentration photovoltaic panels. This study presents a comprehensive theoretical-experimental evaluation of direct contact membrane distillation (DCMD) and air gap membrane distillation (AGMD), focusing on the influence of different membrane materials (PVDF, PTFE, and PE) and operational parameters on performance. Ex-periments were conducted in a flat-sheet MD module under varying NaCl concentrations (7 to 70 g/L), feed temperatures (65–85°C), and flow rates (0.2–0.8 L/min). The DCMD configuration achieved significantly higher permeate flux, with PTFE membranes achieving up to 33.59 kg/m²h at 85°C, attributed to the high hydrophobicity (contact angle: 143.4°) and porosity (85%). Con-versely, AGMD exhibited lower distillate fluxes (maximum 24.77 kg/m²h with PE) but outper-formed DCMD in specific energy consumption (SEC), achieving values as low as 1390.14 kWh/m³ with PTFE at 85°C, in a setup without any heat recovery from either the feed or distillate streams. The thermal efficiency of AGMD reached 0.54, 17% higher than DCMD with the same membrane. Salinity impacted performance differently: PTFE exhibited consistent flux across NaCl concen-trations, while PE showed a reduction of 31.3% at 70 g/L. The proposed heat and mass transfer reduced model showed good agreement with experimental data, with deviations within ±15%, effectively capturing the influence of operational parameters. These findings underscore the trade-offs between DCMD's higher fluxes and AGMD's higher energy efficiencies, offering quantitative insights to guide the design and optimization of MD systems for desalination and wastewater treatment.

This is an eulogy to late Prof. Mikhail Dimitrov Mikhailov

A fast and robust computational model of a spiral-wound vacuum-enhanced air gap membrane distillation (V-AGMD) module at the pilot-scale is proposed and implemented. In contrast with data-driven models available in the literature, a physics-based approach is adopted for more reliable generalization beyond the validation dataset. A total of 86 experimental results, of which 41 are described in this work and 45 come from independent sources available in the literature, are used in the validation effort with quite favorable results. With the confidence on the robustness of the methodology due to the wide range of operational parameters and the use of spiral-wound modules of four different sizes included in the validation comparisons, a physical analysis is conducted varying the air gap pressure, the number of feedwater channels, the feedwater flow rate, and the membrane area. Improvements of up to 60% in both water productivity and energy efficiency can be achieved by intensifying the vacuum in the air gap or decreasing the number of feedwater channels. These parameters achieve performance gains due to less resistance in the air gap for vapor to migrate through it, in the former case, and a reduced temperature polarization effect, in the latter case. Smaller flow rates favor energy efficiency at the expense of water productivity by simultaneously decreasing transport-phenomena-related irreversibility and the partial pressure difference across the membrane and the air gap. In addition, this tradeoff between energy efficiency and driving force is shown to lead to an optimum value for the membrane area beyond which the permeate flow rate through the membrane starts to fall due to the small driving force. An illustrative case is predicted to achieve energy efficiency metrics, such as a gain-output ratio of 12.7, competitive with multi-effect distillation.

The generalized integral transform technique (GITT) is advanced to deal with conduction heat transfer in anisotropic heterogeneous media. A formal solution for exact integral transformation of conduction in anisotropic media is extended to account for heterogeneities expressed as space variable equation coefficients and source terms. The proposed eigenfunction expansion is based on biorthogonal eigenvalue problems, which results in an exact integral transformation for a class of linear problems and in a coupled transformed ordinary differential system for nonlinear situations. An algorithm is proposed for the associated eigenvalue problems, also handled through the GITT, by considering simpler biorthogonal eigenvalue problems of known analytical solution, leading to transformed algebraic eigenvalue problems. A single domain reformulation strategy is adopted to merge the information from multiple regions and materials, either isotropic or anisotropic, into one single diffusion equation. A two-dimensional transient test case is considered that presents an abrupt transition between isotropic and anisotropic materials yielding a marked change in thermal behavior in a defined region of interest formed by the anisotropic inclusion. Convergence behavior of the integral transform solution is illustrated, and the fully converged results are employed as a benchmark to inspect the accuracy of a commercial finite element code for automatically defined mesh refinement levels.