Dimensionless Parameter Groups Research Articles

Fluid Dynamics and Mass Transfer in a Perfusion Bioreactor with a Porous Wall

A numerical study is carried out on the fluid dynamics and mass transfer in a microchannel perfusion bioreactor. The bioreactor channel has a porous wall for the co-culture of two types of cells which are distributed randomly and uniformly. A group of dimensionless parameters is proposed, which can be applied to correlate the numerical data and characterize the mass transfer in the bioreactor. The normalized numerical data, for the concentration at the porous-fluid interface and concentration difference between the interface and the base, show satisfactory correlation when presented as a function of the effective distance parameter. Index Terms—Transport phenomena in porous media, numerical methods in fluid flow, mass transfer.

Regional scale impact of tidal forcing on groundwater flow in unconfined coastal aquifers

summary This paper considers the impact of tidal forcing on regional groundwater flow in an unconfined coastal aquifer. Numerical models are used to quantify this impact for a wide range of hydrogeological conditions. Both a shallow and a deep aquifer are investigated with regard to three dimensionless parameter groups that determine the groundwater flow to a large extent. Analytical expressions are presented that allow for a quick estimate of the regional scale effect of tidal forcing under the same conditions as used in the numerical models. Quantitatively, the results in this paper are complementary to previous studies by taking into account variable density groundwater flow, dispersive salt transport and a seepage face in the intertidal area. Qualitatively, the results are in line with previous investigations. The time-averaged hydraulic head at the high tide mark increases upon a decrease of each of the three considered dimensionless parameter groups: R (including the ratio of the hydraulic conductivity and the precipitation excess), a (the slope of the intertidal area) and AL (the ratio of the width of the fresh water lens and the tidal amplitude). The relative change of the location and the hydraulic head of the groundwater divide, which together characterize regional groundwater flow, increase as a and AL decrease, but decrease as R decreases. The difference between the analytical solutions and numerical results is small. Therefore, the presented analytical solutions can be used to estimate the bias that is introduced in a numerical model if tidal forcing is neglected. The results should be used with caution in case of significant wave forcing, as this was not considered.

A novel dimensionless form of unreacted shrinking core model for solid conversion during chemical looping combustion

In this paper, a novel dimensionless integral form of the unreacted shrinking core model (USCM) was derived from its original one to describe the redox reaction of oxygen carrier (OC) particles during chemical looping (CL) processes. Four dimensionless groups of the relevant parameters were introduced to correlate solid conversion with other parameters. The analysis revealed that three dimensionless groups are important for characterizing solid conversion, including Fo′, Bi′ and Rc. The group Bi′ has less effect on the solid conversion if Bi′ is larger than 10. The intrinsic reaction rate exerts a significant influence on the conversion when Rc is less than 10. The conversion rate is more sensitive to Rc while Rc is less than 1.The reaction front σ is related to the above three dimensionless groups of parameters as well. The dimensionless velocity of reaction front (dσ/d Fo′) is obtained to characterize the CL processes.

Cutting through the complexity of cell collectives

Via strength in numbers, groups of cells can influence their environments in ways that individual cells cannot. Large-scale structural patterns and collective functions underpinning virulence, tumour growth and bacterial biofilm formation are emergent properties of coupled physical and biological processes within cell groups. Owing to the abundance of factors influencing cell group behaviour, deriving general principles about them is a daunting challenge. We argue that combining mechanistic theory with theoretical ecology and evolution provides a key strategy for clarifying how cell groups form, how they change in composition over time, and how they interact with their environments. Here, we review concepts that are critical for dissecting the complexity of cell collectives, including dimensionless parameter groups, individual-based modelling and evolutionary theory. We then use this hybrid modelling approach to provide an example analysis of the evolution of cooperative enzyme secretion in bacterial biofilms.

Simulations of Damage, Crack Initiation, and Propagation in Interlayer Dielectric Structures: Understanding Assembly-Induced Fracture in Dies

Performance enhancement by lowering the dielectric constant of interlayer dielectric (ILD) materials often compromises the mechanical integrity of the dielectric stack. At the present time, fracture in the ILD stacks induced by assembly to either an organic substrate or a die stack (3-D) is an important reliability consideration. These interactions include what is popularly referred to as the chip-package interactions. In this paper, we develop insights on the potential crack initiation site within the ILD, die-substrate geometrical parameters that cause most damage, as well as insights on the manufacturing process that is critical to failure. Towards this end, we utilize analytical models based on classical elasticity theory as well as sophisticated numerical techniques that are capable of nucleating and propagating cracks at arbitrary locations within the structure without remeshing. Specifically, we analytically estimate the strength of singularities at all the possible multimaterial corners in the ILD stack to provide insight on the likely damage nucleation sites for various material configurations in the ILD stack. Two novel numerical approaches are used for fracture simulation. In the first, cracks are modeled as discontinuous enrichments over an underlying continuous behavioral approximation. In the second approach, the underlying material description is enriched with a cohesive damage description whose stiffness is evolved according to a prescribed damage law. Multilevel finite-element models are used to determine the load imposed on the ILD structure by the substrate. Maximum damage induced in the ILD stack by the above load is used as an indicator of the reliability risk. Parametric simulations are conducted by varying ILD material, die size, die thickness, as well as the solder material. Through analytical models of bonded assemblies, we identify groups of relevant dimensionless parameters to relate the numerically estimated damage in ILD stacks to the die/substrate material and geometrical parameters. We demonstrate that the damage in the ILD stack is least when the flexural rigidity of the die is matched to that of the assembled substrate. We also demonstrate that ILD damage is only weakly correlated to shear deformation on the die surface due to assembly. We generalize the above observations into mathematical fits (for use as design rules) relating damage in ILD stacks to ILD material choice, relative substrate flexural rigidity, and die size.

Determination of Thermal Response on Mold Surface and Recommendation of Feasible Heating Time and Gas Flow Rate for a Circular Cavity by Gas Preheating

AbstractThis paper studies the thermal response on the mold surface of a circular cavity in the preheating process by high-temperature gas via a numerical approach, where the heat and mass transfer inside the cavity and mold are predicted by a two-fluid model. The dimensionless parameter group is identified from a dimensional analysis of the gas preheating problem. The correlations among dimensionless parameters are established by a least square method applied to the numerical predictions. The upper limit of the temperature rise on the cavity surface at a given gas flow rate is found approximately proportional to the square root of Q∗and h∗, respectively. The lower limit of gas flow rate to achieve a given temperature rise on the cavity surface is inversely proportional to h∗but grows with (Q∗)1.5. The first limiting line to describe the dependence of Q∗on t∗is independent of h∗and T∗, but the second one is governed by Q∗c. For reaching a specified temperature rise on the cavity surface, the growth in the thickness of thermal resistive layer reduces the required preheating time but increases the corresponding gas flow rate. The dependence of (t∗f, Q∗f) on the parameter group (T∗, h∗) is given in this study to obtain the reasonable combinations of (t∗f, Q∗f) based on the balance between the allowable preheating time and heating capacity.

Model-based neural network correlation for refrigerant mass flow rates through adiabatic capillary tubes

A capillary tube is a common expansion device widely used in small-scale refrigeration and air-conditioning systems. A generalized correlation of refrigerant mass flow rate through adiabatic capillary tubes covering both subcooled and two-phase inlet conditions is expected for multiple purposes. Based on the homogeneous equilibrium flow model, a new group of dimensionless parameters has been proposed. To express the nonlinear relationship between the mass flow rate and the associated parameters, the multi-layer perceptron neural network is employed as a universal function approximator. Simulated data from a validated homogeneous equilibrium model are used for the neural network training and testing. A 5-6-1 network trained with the simulated data of R600a and R407C shows good generality in predicting the simulated data of R12, R22, R134a, R290, R410A, and R404A. Also, the deviations between the trained neural network and the experimental data from the open literature fall into ±10%.

Implications of rainfall temporal resolution for soil-moisture and transpiration modeling

Dimensionless groups of parameters characterizing an ecosystem are valuable indicators for the a priori assessment of the effect of rainfall data resolution on predictions of soil moisture and transpiration. Knowledge of these dimensionless groups enables identification of appropriate levels of rainfall data resolution, when using historical rainfall directly or when using it to derive rainfall model parameters for use in models of soil–plant–climate systems. Detailed simulation studies of the soil, plant, and climate systems in Colorado and Texas, highly resolved in time and vertical space, show that historical rainfall data resolved at the daily level allow accurate prediction of soil-moisture and transpiration dynamics for smaller time resolutions. These results support inferences based on the dimensionless groups. Furthermore, no significant improvement in the prediction of soil-moisture and transpiration dynamics is attained, when representing rainfall through a more complex Neyman–Scott model rather than the simple rectangular pulses Poisson model.

Design optimization by numerical characterization of fluid flow through the valveless diffuser micropumps

Valveless piezoelectric micropumps are in wide practical use due to their ability to conduct particles with absence of interior moving mechanical parts. In this paper, an extended numerical study on fluid flow through micropump chamber and diffuser valves is conducted to find out the optimum working conditions of micropump. In order to obtain maximum generality of the reported results, an analytical study along with a dimensional analysis is presented primarily, to investigate the main dimensionless groups of parameters affecting the micropump net flux. Consequently, the parameters appeared in the main dimensionless groups have been changed in order to understand how the pump rectification efficiency and optimum diffuser angle depend on these parameters. A set of characteristic curves are constructed which show these dependencies. The application of these curves would have far reaching implications for valveless micropumps design and selection purposes.

Generalized neural network correlation for flow boiling heat transfer of R22 and its alternative refrigerants inside horizontal smooth tubes

The correct prediction of refrigerant boiling heat transfer performance is important for the design of evaporators. A generalized neural network correlation for boiling heat transfer coefficient of R22 and its alternative refrigerants R134a, R407C and R410A inside horizontal smooth tubes has been developed in this paper. Four kinds of dimensionless parameter groups from existing generalized correlations are selected as the input of neural network, while the Nusselt number is used as the output. Three-layer perceptron is employed as the universal approximator to build the relationship between the input and output parameters. The neuron number of hidden layer is determined by the performance of model accuracy and the standard sensitivity analysis. The experimental data of the four refrigerants in open literatures are used for correlation. The results show that the input parameter group based on the Gungor–Winterton correlation is better than the other three groups. Compared with the experimental data, the average, mean and root-mean-square deviations of the trained neural network are 2.5%, 13.0% and 20.3%, respectively, and approximately 74% of the deviations are within ±20%, which is much better than that of the existing generalized correlations.

Determination of multiple aquifer parameters using generated water level disturbances

A method was developed to determine bulk values of aquifer and sediment resistance parameters in a coupled canal‐aquifer system using generated sinusoidal hydraulic disturbances. The method is based on analytical solutions to the canal‐aquifer interaction problem derived assuming small perturbations, linear analysis, and spectral methods. Results were obtained in terms of dimensionless parameters so that they can be applied to many practical problems. Numerical models were used to verify the solutions in selected test problems. The method was applied to determine physical parameters of the L‐31W canal in Miami‐Dade County, Florida, near the Everglades in both dimensionless and dimensional forms. The field experiment was carried out using operational control features of the south Florida conveyance system. Results of the analysis show that various dimensionless parameter groups determine various dynamic behaviors in both confined and unconfined aquifers. Which flow component dominates in the system depends on the range of a particular dimensionless parameter. The analytical solutions derived during the analysis are useful in calculating parameters in regional systems where the historical data are noisy or questionable. Calculation of parameters is accomplished by designing field experiments in which input and output signals of a particular frequency can be used with analytical equations to obtain parameters.

Numerical simulation of natural convection heat transfer in a porous cavity heated from below using a non-Darcian and thermal non-equilibrium model

The present paper investigates the numerical simulation of steady laminar incompressible natural convection heat transfer in an enclosed cavity that is filled with a fluid-saturated porous medium. The bottom wall is subjected to a relatively higher temperature than the top wall while the vertical walls are considered to be insulated. The flow field is modeled upon incorporating different non-Darcian effects, such as the convective term, Brinkman effect and Forchhiemer quadratic inertial effect. Moreover the two-equation model is used to separately account for the local fluid and solid temperatures. The numerical solution is obtained through the application of the finite volume method. The appraisals of the sought objectives are performed upon identifying key dimensionless groups of parameters. These dimensionless groups along with their operating domains are: Rayleigh number 1 ⩽ Ra ⩽ 400, Darcy number 10 −4 ⩽ Da ⩽ 10 −3, effective fluid-to-solid thermal conductivity ratio 0.1 ⩽ κ ⩽ 1.0, and the modified Biot number 1 ⩽ χ ⩽ 100. The non-Darcian effects are first examined over a broad range of Rayleigh number. Next, the implications of the group of parameters on the flow circulation intensity, local thermal non-equilibrium (LTNE) and average Nusselt number are highlighted and pertinent observations are documented.

Macrokinetic analysis of polarisation characteristics of gas-diffusion electrodes in contact with liquid electrolytes, Part II: Oxygen reduction as example for a higher order reaction

In this paper principles of gas-liquid chemical reaction engineering are applied to analyse the current-potential characteristics of gas-diffusion electrodes (GDE) in contact with liquid electrolytes. A macroscopic electrode model is formulated which accounts for mass transfer in the external diffusion films, in the gas layer and in the flooded layer. The set of model equations accounts for material balances, mass transport kinetics and Butler-Volmer polarization kinetics. Several dimensionless parameter groups are introduced which allow a compact reformulation of the proposed model. For first order reactions its solution can be derived analytically. The introduced parameter groups allow a classification of the different operating modes of a GDE, that is, slow reaction, fast reaction and instantaneous reaction.

Kinetic Modeling and Prediction of Polymer Properties for Ethylene Polymerization over Nickel Diimine Catalysts

A kinetic model is developed for predicting the polymerization rate, chain-branching characteristics, and molecular weight from ethylene polymerization over homogeneous and supported nickel diimine catalysts. Seven dimensionless parameter groups are able to describe the polymer properties and kinetic behavior over a broad range of temperatures and pressures. Relative to existing models in the literature, the proposed model is simpler, requires fitting of fewer kinetic parameters, and can be more easily extrapolated over a wide range of reaction conditions. As such, the model can be more readily incorporated into reactor process models for performing dynamic simulations. Experimental data from the literature are used to fit the model kinetic parameters. The model is able to describe the kinetic behavior of both homogeneous and heterogeneous nickel diimine catalysts, and the results compare favorably with existing models in the literature.

Modeling of Mass Transport into Immiscible Polymeric Blends

A nonlinear 3D model for mass transport into immiscible polymeric blends is developed by explicitly incorporating the interface dynamics into the transport equations. The interface is characterized, on a mesoscopic level of description, by a scalar Q(r,t) and a second-order tensor q(r,t) respectively describing the local size and anisotropy densities of the interfacial area. The newly obtained constitutive equation for the diffusion mass flux density extends Fick's first law by involving two additional terms accounting for the local changes of the interface morphology. The model provides an expression for the distribution of both isotropic (Laplace) and anisotropic stresses created by mass transport within the immiscible polymeric blend. The governing equations are parametrized by the free energy density that includes a mixing part and an excess energy term attributed to the presence of the interface. We investigate in more detail a one-dimensional sorption process of a solvent into a thin immiscible blend consisting of a matrix and a dispersed phase. Three dimensionless groups of physical parameters arise in the 1D dimensionless formulation; two are coupling constants that explicitly relate diffusion to the interface dynamic changes, and one is the diffusion Deborah number. Numerical results show that diffusion becomes non-Fickian for values of Deborah number approaching unity. The time evolution of the calculated mass uptake, swelling, stresses, and total size and anisotropy densities provides a good indication of the effects of diffusion-interface interaction on both mass transport and the morphology of the interface.

Dynamic Modeling and Simulation of a Countercurrent Molten Carbonate Fuel Cell (MCFC) with Internal Reforming

AbstractA dynamic mathematical model for a countercurrent MCFC with direct internal methane reforming is introduced. It allows for the simulation of the gas phase compositions and flows, the temperatures in gas and solid phases, the cell voltage, and the current density distribution. This state space model is fully formulated in terms of dimensionless parameter groups, which are derived from the underlying physical and chemical phenomena. The model assumptions and equations are outlined in detail and a numerical solution strategy is proposed. The model is used to predict the current‐voltage curve as well as the dynamic cell response to a stepwise load change.

Models of soil moisture dynamics in ecohydrology: A comparative study

An accurate description of plant ecology requires an understanding of the interplay between precipitation, infiltration, and evapotranspiration. A simple model for soil moisture dynamics, which does not resolve spatial variations in saturation, facilitates analytical expressions of soil and plant behavior as functions of climate, soil, and vegetation characteristics. Proper application of such a model requires knowledge of the conditions under which the underlying simplifications are appropriate. To address this issue, we compare predictions of evapotranspiration and root zone saturation over a growing season from a simple bucket‐filling model to those from a more complex, vertically resolved model. Dimensionless groups of key parameters measure the quality of the match between the models. For a climate, soil, and woody plant characteristic of an African savanna the predictions of the two models are quite similar if the plant can extract water from locally wet regions to make up for roots in dry portions of the soil column; if not, the match is poor.

Modification of Canal Flow due to Stream-Aquifer Interaction

Unsteady canal flow in an integrated canal-flow–groundwater-flow system is analyzed by solving the coupled equations governing canal flow, groundwater flow and the seepage between them. Analytical solutions are obtained for the coupled system for small water-level disturbances using Fourier analysis methods and complex variables. Dimensionless parameter groups characterizing the aquifer, the canal, and the sediment layer are identified using the governing equations and the solution. The influence in the aquifer and the semipermeable bottom sediment layer due to disturbances in canal flow is studied. The analytical solutions are compared to numerical solutions obtained using the MODFLOW model and the Hydrologic Simulation Engine of the South Florida Regional Simulation Model. Results of the analysis are useful in determining the range of aquifer, sediment, and canal characteristics for which stream-aquifer interaction is important. The results can be used to determine the conditions for which the canal is hydraulically disconnected from the aquifer because of the sediment layer. The analytical solution is useful to understand the propagation characteristics of small-amplitude water-level disturbances in the canal and the aquifer. The characteristics studied include the amplitude decay constant and the speed. The solution can be used to design benchmark problems that can be used to evaluate integrated canal-flow–groundwater-flow models. The results of the study can be used to estimate the space and time steps needed in the canal and the aquifer when simulating stream-aquifer interaction.

Numerical simulation of non-linear heat conduction subjected to a laser source: the effects of variable thermal properties

This paper presents a numerical study of the temperature distribution in a body subjected to a spatially exponential decaying laser source. The governing heat conduction equation, the boundary conditions and the initial condition are presented in a dimensionless form as a function of a group of dimensionless parameters, namely, χ, τ, Bi, w, c 1, c 2 and c 3. Three different sets of thermal boundary conditions imposed at the “far” boundary, including convection, insulated and constant surface temperature conditions are investigated in the present study. The governing equation is discretized using a control volume approach, with a variable grid to increase the resolution of the domain near the boundary where the laser heat source is applied. The effects of the different parameters and the temperature dependent thermal properties are studied in detail. The calculated results are compared with previous analytical studies for constant thermal properties obtained for both semi-infinite and finite domains. Finally, the present numerical solutions are compared to existing experimental data.

A numerical model for density-and-viscosity-dependent flows in two-dimensional variably saturated porous media

We present a formulation for water flow and solute transport in two-dimensional variably saturated media that accounts for the effects of the solute on water density and viscosity. The governing equations are cast in a dimensionless form that depends on six dimensionless groups of parameters. These equations are discretized in space using the Galerkin finite element formulation and integrated in time using the backward Euler scheme with mass lumping. The modified Picard method is used to linearize the water flow equation. The resulting numerical model, the MARUN model, is verified by comparison to published numerical results. It is then used to investigate beach hydraulics at seawater concentration (about 30 g l −1) in the context of nutrients delivery for bioremediation of oil spills on beaches. Numerical simulations that we conducted in a rectangular section of a hypothetical beach revealed that buoyancy in the unsaturated zone is significant in soils that are fine textured, with low anisotropy ratio, and/or exhibiting low physical dispersion. In such situations, application of dissolved nutrients to a contaminated beach in a freshwater solution is superior to their application in a seawater solution. Concentration-engendered viscosity effects were negligible with respect to concentration-engendered density effects for the cases that we considered.