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A thermoelectric generator (TEG) cell is a solid-state semi-conductor capable of converting thermal energy directly into electrical energy. Studies on specific designs for TEG modules are needed for an effective industrial waste heat recovery. Typical TEG systems apply direct heating or cooling without being assisted by heat transfer devices which makes it difficult to maintain high temperature difference between TEG cells surfaces for higher electrical power generation. To suit the TEG system with the industrial condition, a waste heat recovery (WHR) module was developed consisting of a Bismuth Telluride TEG cell and another module consisting of two TEG cells in a stacked configuration. The TEG cells were sandwiched between two copper blocks and connected to two plate-finned heat sinks and eight heat pipes. The two modules were tested on a dedicated test bench to study its performance based on an industrial WHR setup. The hot stream waste heat temperature was constant at 80 ℃ while the cooling air streams were set based on ambient conditions between 22 ℃ to 30 °C. The maximum power performance (MPP) varied from 10 mW to 153 mW where the effect of cooling is very significant on the power outputs. Under forced cooling condition, the MPP generated from the double stacking TEG configuration is significantly higher compared to single cell configuration, by approximately 250%.

The Einstein functions in generalized Puiseux and Taylor series are used as forcing functions in Airy’s inhomogeneous equation, and particular and general solutions are obtained. Comparison are made with solutions obtained using the Nield-Kuznetsov functions’ approach. For each of the Einstein’s functions, the standard Nield-Kuznetsov function of the second kind is expressed in terms of Bessel functions. Computations and graphs in this work were produced using Wolfram Alpha.

Higher derivatives and associated polynomials of the standard Nield-Kuznetsov function of the second kind are investigated in this work. Two approaches are introduced in this work. The first, is the direct method of differentiation and generalization of the nth derivative. This approach is dependent on higher derivatives of the Nield-Kuznetsov function of the first kind. The second is the transform method in which integral transforms associated with the Nield-Kuznetsov function of the second kind are introduce first, and higher derivatives are then obtained. The transform method is independent of the direct higher derivatives of the Nield-Kuznetsov function of the first kind. Both approaches are important in practical and theoretical mathematical analysis, and both give rise to associated Airy polynomials, discussed in this work.

Taylor and Maclaurin series, and polynomial approximations of the Standard Nield-Kuznetsov function of the second kind are obtained in this work. Convergence and error criteria are developed. The obtained series represent alternatives to existing asymptotic and ascending series approximations of this integral function, and are expected to provide an efficient method of computation.

In this work, particular and general solutions to Airy’s inhomogeneous equation are obtained when the forcing function is one of Airy’s functions of the first and second kind, and the standard Nield-Kuznetsov function of the first kind. Particular solutions give rise to special integrals that involve products of Airy’s and Nield-Kuznetsov functions. Evaluation of the resulting integrals is facilitated by expressing their integrands in asymptotic series. Corresponding expressions for the Nield-Kuznetsov function of the second kind are obtained.

In this work, the problem of obtaining particular and general solutions to Airy’s inhomogeneous equation when the forcing function is one of Einstein’s functions is examined. Expressions for the particular solutions provide connections between the Nield-Kuznetsov and Einstein functions. Computations have been carried out using Wolfram Alpha.

Analysis and solution of Airy’s inhomogeneous equation, when its forcing function is the sigmoid neural network activation function, are provided in this work. Relationship between the Nield-Kuznetsov, the Scorer, the sigmoid, the polylogarithm and Airy’s functions are established. Solutions to initial and boundary value problems, when the sigmoid function is involved, are obtained. Computations were carried out using Wolfram Alpha.

Taylor and Maclaurin series and polynomial approximations of the Standard Nield-Kuznetsov function of the first kind are obtained in this work. Convergence and error criteria are developed. The obtained series represent alternatives to the existing asymptotic and ascending series approximations of this integral function, and are expected to provide an efficient method of computation that is valid for all values of the argument.

A fifth-order accurate finite difference scheme for the first derivative in von Mises coordinates is developed in this work. The scheme is tested in the computation of boundary vorticity in the study of two-dimensional flow of a viscous fluid in a curvilinear channel. Results obtained show an improvement in the computed solution over fourth-order accurate scheme

A collection of popular variable permeability functions is presented and discussed in this work. The functions have been used largely in Brinkman’s equation which governs the flow through a porous domain in the presence of solid, macroscopic boundaries on which the no-slip condition is imposed, and has been used in transition layer modelling. A convenient classification of permeability functions is also provided. The sigmoid logistic function is presented in this work in a modified form that is suitable for variable permeability modelling, and is used in obtaining solution to Poiseuille flow through a Brinkman porous channel.

Flow of a fluid with pressure-dependent viscosity through a composite of two porous layers is considered in this work in an attempt to validate velocity and shear stress continuity conditions at the interface, and are popular in the study of flow over porous layers and through composite layers when viscosity of the fluid is constant. For the current problem, conditions at the interface between the porous layers reflect continuity assumptions of velocity and shear stress, with additional continuity assumptions on pressure and viscosity. Viscosity is assumed to vary continuously and exponentially across the layers as a function of pressure. Analytical solutions are obtained to illustrate the effects of flow and media parameters (Darcy numbers, layer thicknesses, angle of inclination, and viscosity adjustment parameter) on the dynamic behaviour of pressure-dependent viscosity fluids in porous structures. All computations, simulations and graphs in this work have been carried out and obtained using Maple 2020 software package.

Forward finite difference expressions of first-order accuracy for boundary vorticity on a solid boundary are evaluated in this work when the physical coordinates are clustered and mapped using von Mises coordinates. Results show that schemes using in-field grid points do not improve solutions obtained. Results also show that the finer the grid used in the physical domain, and the more clustered it is, improves the boundary vorticity values in the computational domain. The “best” expressions forward finite difference expressions are identified when two, three, four and five grid points are used.

Polar fluid through a porous structure is considered in an attempt to derive expressions for its effective viscosity under Poiseuille flow. Expressions for the velocity field and micro-rotation are obtained and compared with the corresponding expressions for flow in free space in order to illustrate the effects of the porous medium parameters on the traversing fluid. This work might be of applicability in the study of lubrication problems in configurations involving porous linings, and in the study of the effects of viscosity changes on oil well yield.

In this work, we consider flow of a fluid with pressure-dependent viscosity down an inclined porous plane with variable permeability that is incorporated in the pressure-dependent drag coefficient. We provide a solution to a recently developed flow model, and study the effects of flow and domain parameters (viscosity control parameter, permeability proportionality constant, and angle of inclination) on the flow characteristics. Suitability of a variable permeability model that considers permeability proportional to the flow velocity is investigated. Results show that large values of the permeability proportionality constant have little or no effects on flow characteristics.

Coupled parallel flow of fluid with pressure-dependent viscosity through an inclined channel underlain by a porous layer of variable permeability and variable thickness is initiated in this work. Conditions at the interface between the channel and the porous layer reflect continuity assumptions of velocity, shear stress, pressure and viscosity. Viscosity is assumed to vary in terms of a continuous pressure function that is valid throughout the channel and the porous layer. Model equations are cast in a form where the pressure as an independent variable and solutions are obtained to illustrate the effects of flow and media parameters on the dynamics behaviour of pressure-dependent viscosity fluid. A permeability and a viscosity adjustable control parameters are introduced to avoid unrealistic values of permeability and viscosity. This work could serve as a model for flow over a mushy zone.

In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s equations. Derivations of these functions are provided, together with their methods of computations

Inhomogeneous Airy’s and Generalized Airy’s equations with initial and boundary date are considered in this work. Solutions are obtained for constant and variable forcing functions, and general solutions are expressed in terms of Standard and Generalized Nield-Kuznetsov functions of the first- and second-kinds. Series representations of these functions and their efficient computation methodologies are presented with examples.

Equations governing the flow of a fluid with pressure-dependent viscosity through an isotropic porous structure are derived using the method of intrinsic volume averaging. Viscosity of the fluid is assumed to be a variable function of pressure, and the effects of the porous microstructure are modelled and included in the pressure-dependent drag coefficient. Five friction factors relating to five different microstructures are used in this work

Equations governing the flow of a fluid-particle mixture with variable viscosity through a porous structure are developed. Method of intrinsic volume averaging is used to average Saffman’s dusty gas equations. A modelling flexibility is offered in this work by introducing a dust-phase partial pressure in the governing equations, interpreted as the pressure necessary to maintain a uniform particle distribution in the flow field. Viscosity of the fluid-particle mixture is assumed to be variable, with variations in viscosity being due to fluid pressure. Particles are assumed spherical and Stokes’ coefficient of resistance is expressed in terms of the pressure-dependent fluid viscosity. Both Darcy resistance and the Forchheimer micro-inertial effects are accounted for in the developed model

In this article, we discuss a class of functions known as the Nield-Kuznetsov functions, introduced over the past decade. These functions arise in the solutions to inhomogeneous Airy’s and Weber’s equations. Derivations of these functions are provided, together with their methods of computations

Gravity-driven flow through an inclined channel over a semi-infinite porous layer is considered in order to obtain a modification to the usual Beavers and Joseph slip condition that is suitable for this type of flow. Expressions for the velocity, shear stress, volumetric flow rates, and pressure distribution across the channel are obtained together with an expression for the interfacial velocity. In the absence of values for the slip parameter when the flow is over a Forchheimer porous layer, this work provides a relationship between the slip parameters of the Darcy and Forchheimer layers. Expressions for the interfacial velocities in both cases are obtained. This original work is intended to provide baseline analysis and a benchmark with which more sophisticated types of flow, over porous layers in an inclined domain can be compared.

Fully developed flow of a magnetorheological fluid through a non-isotropic porous medium bounded by parallel plates under the effect of an external magnetic field is considered. Permeability is taken as a periodic function of the transverse direction of the fluid flow. The Darcy-Brinkman-Lapwood-Lorentz equation for the fluid flow in porous media has been used and solved under non-slip boundary conditions by Power Series Method and the results validated by the Numerical Shooting Method. Finally, the analysis of results is made of the influence on the velocity, volumetric flow, and wall shear stress taking into account the Ochoa-Tapia and Whitaker’s effective viscosity relationship., Se ha considerado el flujo totalmente desarrollado de un fluido magneto-reológico a través de un medio poroso no isotrópico delimitado por placas paralelas bajo el efecto de un campo magnético externo. La permeabilidad se toma como una función periódica de la dirección transversal al flujo. La ecuación Darcy-Brinkman-Lapwood-Lorentz para el flujo de fluidos en medios porosos se ha utilizado y resuelto en condiciones de frontera no deslizante por el método de series de potencia y los resultados han sido validados por el método numérico del disparo. Por último, el análisis de los resultados se hace sobre la influencia en la velocidad, flujo volumétrico y el esfuerzo de deformación del fluido en la pared teniendo en cuenta la relación de viscosidad efectiva de Ochoa-Tapia y Whitaker.

Plane, transverse MHD flow through a porous structure is considered in this work. Solution to the governing equations is obtained using an inverse method in which the streamfunction of the flow is considered linear in one of the space variables. Expressions for flow quantities are obtained for finitely conducting and infinitely conducting fluids.

Flow-through and over a Brinkman porous layer of variable permeability, immersed in a fluid-filled channel, is modeled. The model governing equation through the porous layer inevitably gives rise to an inhomogeneous Weber's differential equation, solved in this work and solutions expressed in terms of parabolic cylindrical functions. Using state-of-the-art computational techniques and a body of knowledge, the parabolic cylindrical functions are evaluated for a range of flow and medium parameters in order to illustrate intrinsic characteristics of the flow quantities. The approach followed in this work is novel and sets precedent in the study of flow through general porous media configurations and flow domains with variable permeability.

In this study, the characteristics of clicking-type noise and vibration occurring in the automotive heating, ventilation and air conditional (HVAC) systems are investigated. A lab-scale model of HVAC system is developed, and validation is carried out with a vehicle system. A fixed blower speed of 1 (at an airflow of 2.53 m/s) with alternated air conditional (AC) was implied in this study. Three different sensors namely as tachometer, accelerometer, and microphone were used to measure and prove the existing noise in the HVAC system. The study inferred that the compressor contributed significantly to the total vibration and noise in the HVAC system. Other components such as AC pipe, evaporator, and thermal expansion valve (TXV) also contributed to a slight extent. The clicking noise was observed in the operating frequency range of 200 ~ 300 Hz. This noise and vibration issues are partly influenced by the running conditions of the AC and the effect was significant when the AC was turned on. The validation of the findings in the model shows a good agreement with the results obtained in the vehicle system, whereby the clicking noise and vibration can be observed at a similar frequency range.

Flow through a channel bounded by a porous layer is considered when a transition layer exists between the channel and the medium. The variable permeability in the transition layer is chosen such that Brinkman’s equation governing the flow reduces to a generalized inhomogeneous Airy’s differential equation. Solution to the resulting generalized Airy’s equation is obtained in this work and solution to the flow through the transition layer, of the same configuration, reported in the literature, is recovered from the current solution.

The automotive heating and ventilating air condition (HVAC) system, when vibrating, can generate various types of noises such as humming, hissing, clicking and air-rushes. These noises can be characterised to determine their root causes. In this study, the humming-type noise is taken into consideration whereby the noise and vibration characteristics are measured from various HVAC components such as power steering pump, compressor and air conditional pipe. Four types of measurement sensors were used in this study - tachometer for rpm tracking; accelerometer for the vibration microphone for the noise; and sound camera for the visualization measurement. Two types of operating conditions were taken into consideration - they were “idle” (850 rpm) and “running” (850-1400 rpm) conditions. A constant blower speed was applied for both conditions. The result shows that the humming noises can be determined at the frequency range of 300-350 Hz and 150-250 Hz for both idle and running conditions, respectively. The vibration of the power steering pump shows the worst acceleration of 1.8 m/s2 at the frequency range of 150-250 Hz, compared to the compressor and air conditional pipe. This result was validated with the 3D colour order and sound camera analyses, in which the humming noise colour mapping shows dominance in this frequency range.

In this work, we provide a solution to a two-point boundary value problem that involves an inhomogeneous Airy’s differential equation with a variable forcing function. The solution is expressed in terms of the recently introduced Nield-Koznetsov integral function, Ni(x), and another conveniently defined integral function, Ki(x). The resulting expressions involving these integral functions are then evaluated using asymptotic and ascending series.Â

Third- and fourth-order accurate finite difference schemes for the first derivative of the square of the speed are developed, for both uniform and non-uniform grids, and applied in the study of a two-dimensional viscous fluid flow through an irregular domain. The von Mises transformation is used to transform the governing equations, and map the irregular domain onto a rectangular computational domain. Vorticity on the solid boundary is expressed in terms of the first partial derivative of the square of the speed of the flow in the computational domain, and the schemes are used to calculate the vorticity at the computational boundary grid points using up to five computational domain grid points. In all schemes developed, we study the effect of coordinate clustering on the computed results.

In this work, we consider the flow through composite porous layers of variable permeability, with the middle layer representing a porous core bounded by two Darcy layers. Brinkman’s equation is valid in the middle layer and has been reduced to an Airy’s inhomogeneous differential equation. Solution is obtained in terms of Airy’s functions and the Nield-Kuznetsov function.

In this work we consider coupled-parallel flow through a finite channel bounded below by a porous layer that is either finite or infinite in depth. The porous layer is one in which Darcy’s equation is valid under the assumption of variable permeability. A suitable permeability stratification function is derived in this work and the resulting variable velocity profile is analyzed. It will be shown that when an infinite porous layer is implemented, Darcy’s equation must be used with a constant permeability.