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In order to successfully study the condensation and separation of a steam–CO2 mixture, a boundary layer model was applied to the mixture condensation of steam and CO2 on horizontal and vertical plates. The modified condensation boundary layer model of steam and CO2, given the CO2 solubility in the condensate, was established, numerically solved, and verified with existing experimental data. Different condensation data of steam–air and steam–CO2 mixtures were compared, and the effect of CO2 solubility on the mixed gas condensation was analyzed under multiple pressure conditions (1 atm–10 MPa). The simulation data show that the presence of CO2 will deteriorate the condensation heat transfer, just like air. Given that CO2 is slightly soluble, some CO2 can pass through the gas–liquid interface to enter the condensate film and reduce the accumulated CO2 on the gas–liquid interface, which improves the condensation. However, the solubility of CO2 is only significant under high-pressure conditions, inducing its effects on condensation. A comparison of the condensation coefficients of the steam–CO2 mixture shows the lower impact of CO2 condensation on the horizontal plate compared to that on the vertical plate. For most conditions, the steam–CO2 mixture gas condensation heat transfer coefficient on the vertical plate surface is still larger than that on the horizontal plate surface, and the improvement in the condensation heat transfer coefficient caused by low CO2 solubility (2 or 10%) at 10 MPa on the vertical plate is also larger than that of the horizontal plate. [ABSTRACT FROM AUTHOR]

In order to successfully study the condensation and separation of a steam–CO2 mixture, a boundary layer model was applied to the mixture condensation of steam and CO2 on horizontal and vertical plates. The modified condensation boundary layer model of steam and CO2, given the CO2 solubility in the condensate, was established, numerically solved, and verified with existing experimental data. Different condensation data of steam–air and steam–CO2 mixtures were compared, and the effect of CO2 solubility on the mixed gas condensation was analyzed under multiple pressure conditions (1 atm–10 MPa). The simulation data show that the presence of CO2 will deteriorate the condensation heat transfer, just like air. Given that CO2 is slightly soluble, some CO2 can pass through the gas–liquid interface to enter the condensate film and reduce the accumulated CO2 on the gas–liquid interface, which improves the condensation. However, the solubility of CO2 is only significant under high-pressure conditions, inducing its effects on condensation. A comparison of the condensation coefficients of the steam–CO2 mixture shows the lower impact of CO2 condensation on the horizontal plate compared to that on the vertical plate. For most conditions, the steam–CO2 mixture gas condensation heat transfer coefficient on the vertical plate surface is still larger than that on the horizontal plate surface, and the improvement in the condensation heat transfer coefficient caused by low CO2 solubility (2 or 10%) at 10 MPa on the vertical plate is also larger than that of the horizontal plate.

The rheology of different materials at the micro and macro levels is an area of great interest to many researchers, due to its important physical significance. Past experimental studies have proved the efficiency of the utilization of nanoparticles in different mechanisms for the purpose of boosting the heat transportation rate. The purpose of this study is to investigate heat and mass transport in a pseudo-plastic model past over a stretched porous surface in the presence of the Soret and Dufour effects. The involvement of tri-hybrid nanoparticles was incorporated into the pseudo-plastic model to enhance the heat transfer rate, and the transport problem of thermal energy and solute mechanisms was modelled considering the heat generation/absorption and the chemical reaction. Furthermore, traditional Fourier and Fick’s laws were engaged in the thermal and solute transportation. The physical model was developed upon Cartesian coordinates, and boundary layer theory was utilized in the simplification of the modelled problem, which appears in the form of coupled partial differential equations systems (PDEs). The modelled PDEs were transformed into corresponding ordinary differential equations systems (ODEs) by engaging the appropriate similarity transformation, and the converted ODEs were solved numerically via a Finite Element Procedure (FEP). The obtained solution was plotted against numerous emerging parameters. In addition, a grid independent survey is presented. We recorded that the temperature of the tri-hybrid nanoparticles was significantly higher than the fluid temperature. Augmenting the values of the Dufour number had a similar comportment on the fluid temperature and concentration. The fluid temperature increased against a higher estimation of the heat generation parameter and the Eckert numbers. The impacts of the buoyancy force parameter and the porosity parameter were quite opposite on the fluid velocity.

Boundary-value problems involving the linear differential equation εy" - xy' + y = 0 have surprising properties as ε → 0. We examine this equation from eight points of view, showing how it sheds light on aspects of numerical analysis (backward error analysis and illconditioning), asymptotics (boundary layer analysis), dynamical systems (slow manifolds), ODE theory (Sturm--Liouville operators), spectral theory (eigenvalues and pseudospectra), sensitivity analysis (adjoints and SVD), physics (ghost solutions), and PDE theory (Lewy nonexistence). [ABSTRACT FROM AUTHOR]

Geometrically, the upper pointed surface of an aircraft and bonnet of a car are examples of upper horizontal surfaces of a paraboloid of revolution ( uhspr ). The motion of these objects strongly depends on the boundary layer that is formed within the immediate space on it. However, each of these surfaces is neither a horizontal/vertical nor cone/wedge and neither a cone nor a wedge. This article presents the motion of 2-dimensional Blasius flow of Carreau fluid on the surface of such object. The case in which the reaction between the Carreau fluid and catalyst at the surface produces significant temperature differences which consequently set up buoyancy-driven flows within the boundary layer is investigated. Single first-order Arrhenius kinetics is adopted to model the reaction on the surface of the catalyst situated on uhspr which initiates the free convection. Suitable similarity variables are applied to non-dimensionalized, parameterized and reduce the governing partial differential equations to a coupled ordinary differential equations (BVP). The BVP is solved numerically using the shooting technique. Temperature distribution in the flow of viscoelastic Carreau fluid is greater than that of a Newtonian fluid. Local heat transfer rate decreases faster when the Carreau fluid is characterized as shear-thinning. Maximum concentration is guaranteed at a small value of power-law index n and large value of thickness parameter. [ABSTRACT FROM AUTHOR]

We study singularly perturbed time dependent convection–diffusion equations in a circular domain. Considering suitable compatibility conditions, we present convergence results and provide as well approximation schemes and error estimates. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via a specific boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical solution using a quasi-uniform mesh, that is without refinement of the mesh in the boundary layer. [ABSTRACT FROM AUTHOR]

The steady flow of an incompressible Burger fluid near a stagnation-point over a linearly stretching surface is investigated. The two-dimensional flow equations are modelled and then simplified by employing boundary layer analysis. The solution to the arising nonlinear problem is computed in the whole spatial domain (0 ≤ η ≤ ∞). Interpretation of various emerging parameters is given through graphs for velocity and temperature fields. Furthermore tables are constructed in order to make a comparative study with the previous published results. Comparison shows an excellent agreement with the previous limiting investigations in the field. [ABSTRACT FROM AUTHOR]

Abstract: Study of laminar boundary layer in mixed convection from vertical plates is carried out. The surface temperature along the vertical plate is assumed to vary arbitrarily with vertical distance. Perturbation technique is used to solve the governing boundary layer equations. The differentials of the wall temperature are used as perturbation elements, which are functions of vertical distance, to obtain universal functions. The universal functions are valid for any type of vertical wall temperature variation. Heat transfer rates and fluid velocity inside the boundary layer can be expressed and calculated using these universal functions. Heat transfer rates are obtained for the special cases of power-law variation of the wall temperature. The effect of the governing parameter and the power index of the power-law wall temperature variation on heat transfer rates is studied. For the purpose of validation, the mixed convection results obtained by the present technique pertaining to the special cases of isothermal vertical wall are compared with those obtained by similarity analysis reported in literature, and the agreement is found to be good. [Copyright &y& Elsevier]

Abstract: The motion of a naturally straight inextensible flexible elastic hanging rod is formulated and then linearized about the straight solution. To solve this equation by separation of variables, an eigenvalue problem is derived. When the stiffness of the rod is small, the eigenvalue equation is a singular perturbation problem. This paper is devoted to solving this eigenvalue problem by boundary layer analysis when the stiffness is suitably small, especially on the analytic approximate solutions of the first several eigenvalues and eigenfunctions. The first three eigenvalues are also compared with the numerical results computed by a finite difference method. The excellent agreement shows the efficiency of the boundary layer analysis. [Copyright &y& Elsevier]

We consider the flow of a viscous Newtonian fluid in a bifurcation of thin pipes with a diameter-to-length ratio of order O(ℇ). The model is based on the stationary Navier-Stokes equations with pressure conditions on the outflow boundaries. Existence and local uniqueness is established under the assumption of small data represented by a Reynolds number Reℇ of order O(ℇ). We construct an asymptotic expansion in powers of ℇ and Reℇ for the solution consisting of Stokes flow in the junction part of the bifurcation and Poiseuille flow in the pipes. We introduce a correction to Kirchhoff's law of the balancing fluxes in O(ℇ) which allows to establish error estimates for the gradient of velocity. These estimates result from the analysis of the decay properties of the flow in the layer near the bifurcation. [ABSTRACT FROM AUTHOR]

Theoretical analyses of mass/heat transfer from a neutrally buoyant particle in simple shear flow indicate that mass/heat must diffuse across a region of closed streamlines of finite thickness at zero Reynolds number, whereas spiraling streamlines allow the formation of a thin mass transfer boundary layer at small but non-zero Reynolds numbers (Subramanian and Koch, Phys Rev Lett. 2006;96:134503; Subramanian and Koch, Phys Fluids. 2006;18: 073302). This article presents the first numerical results for mass/heat transfer at finite Reynolds and Peclet numbers. The simulations indicate that fluid particles in the flow-gradient plane spiral away from the particle for Reynolds numbers smaller than about 2.5 while they spiral toward the particle for higher Reynolds numbers. Solutions of the Navier-Stokes equations coupled with a boundary layer analysis of mass transfer yield predictions for the rate of mass transfer at asymptotically large Peclet numbers and Reynolds numbers up to 10. Simulations of mass transfer for zero Reynolds number and finite Peclet numbers confirm Acrivos' (Acrivos, J Fluid Mech. 1971;46:233-240) prediction that the Nusselt number approaches a finite value with increasing Peclet number. Simulations at finite Reynolds numbers and Peclet numbers up to 10,000 confirm the theoretical predictions for the concentration gradient at the particle surface at angular positions away from the flow-gradient plane. However, the wake near the flow-gradient plane remains too large at this Peclet number to yield a quantitative agreement of the overall rate of mass transfer with the theory for asymptotically large Peclet number. © 2010 American Institute of Chemical Engineers AIChE J, 2011 [ABSTRACT FROM AUTHOR]

The rheology of different materials at the micro and macro levels is an area of great interest to many researchers, due to its important physical significance. Past experimental studies have proved the efficiency of the utilization of nanoparticles in different mechanisms for the purpose of boosting the heat transportation rate. The purpose of this study is to investigate heat and mass transport in a pseudo-plastic model past over a stretched porous surface in the presence of the Soret and Dufour effects. The involvement of tri-hybrid nanoparticles was incorporated into the pseudo-plastic model to enhance the heat transfer rate, and the transport problem of thermal energy and solute mechanisms was modelled considering the heat generation/absorption and the chemical reaction. Furthermore, traditional Fourier and Fick's laws were engaged in the thermal and solute transportation. The physical model was developed upon Cartesian coordinates, and boundary layer theory was utilized in the simplification of the modelled problem, which appears in the form of coupled partial differential equations systems (PDEs). The modelled PDEs were transformed into corresponding ordinary differential equations systems (ODEs) by engaging the appropriate similarity transformation, and the converted ODEs were solved numerically via a Finite Element Procedure (FEP). The obtained solution was plotted against numerous emerging parameters. In addition, a grid independent survey is presented. We recorded that the temperature of the tri-hybrid nanoparticles was significantly higher than the fluid temperature. Augmenting the values of the Dufour number had a similar comportment on the fluid temperature and concentration. The fluid temperature increased against a higher estimation of the heat generation parameter and the Eckert numbers. The impacts of the buoyancy force parameter and the porosity parameter were quite opposite on the fluid velocity. [ABSTRACT FROM AUTHOR]

Abstract: The main objective of this analytical study is to investigate heat transfer from tube banks in crossflow under isothermal boundary condition. For this purpose, a control volume is selected from the fourth row of a tube as a typical cell to study the heat transfer from an in-line or staggered arrangement. An integral method of boundary layer analysis is employed to derive closed form expressions for the calculation of average heat transfer from the tubes of a bank, that can be used for a wide range of parameters including longitudinal pitch, transverse pitch, Reynolds and Prandtl numbers. The models for in-line and staggered arrangements are applicable for use over a wide range of parameters when determining heat transfer from tube banks. [Copyright &y& Elsevier]

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Forward osmosis-based processes are low energy separation technology that can help in the recovery of fresh water. The design of such systems is approximated either from the simple mass conservations or using the analogy of heat exchanger design correlations. Such relationship has serious drawbacks, particularly at high recoveries, when the impact of mass transfer boundary layer is paramount. Unlike the reverse osmosis, the permeate flux (hence the recovery) is reduced with the convection (Peclet number, Pe). The membrane surface concentration and the thickness of concentration boundary layer on the draw side decreases with Pe. Since, permeate flux increases with the surface concentration, the recovery is dependent on Pe. Besides the relative concentration and flow-rates, the membrane area is influenced by the mass transfer boundary layer, manifested through Pe. The results reported in the Mondal and Field (2018) is a special case when Pe ≪ 1 (negligible convection), leading to maximum recovery. We evaluate the required membrane area and the performance of the forward-osmosis system as a function of the operating conditions, including the Pe. We have identified the range of operating conditions, for which the sizing of FO system is underpredicted in the case of negligible convection and the log-mean concentration difference. Unlabelled Image • The performance of a forward osmosis is affected by the boundary layer formation. • Membrane area is underpredicted without the consideration of convective transport. • The recovery of the process decreases with convective mass transport. • Flow configuration does not have much effect at high Peclet number operation. [ABSTRACT FROM AUTHOR]

• Study focuses on condensation over a vertical tube with noncondensable gas present. • Model predicts enhancement of condensation heat transfer due to tube curvature. • Predictions are in good agreement with previous analyses and experimental data. • A wide-ranging correlation for curvature correction is proposed for laminar flows. Filmwise vapor condensation in the presence of noncondensable gases (NCG) continues to attract interest due to its widespread range of applications. In studies of NCG effects on condensation over external surfaces, experimentalists and analysts alike usually employ correction factors to transfer tube condensation rates to flat plate equivalents or vice-versa. For laminar flows over vertical cylinders, these factors are estimated from pure convection correlations as a function of the Grashof number and tube curvature only. We show that this procedure is not rigorous and leads to significant errors, especially at low NCG fractions and small tube radii. Indeed, the usual way of correcting for curvature is strictly valid only in the limit where the NCG fraction is so large that convection heat transfer dominates condensation. On the other hand, as the NCG fraction becomes vanishingly small, the vapor condensation rates should tend to values predicted by the Nusselt film theory, and no enhancement due to tube curvature is to be expected. This paper focuses on vapor condensation as the gas mixture moves along a vertical tube in the laminar free convection regime. The gas and liquid film boundary layer equations are coupled at the gas/film interface and a marching solution procedure using local non-similarity is adopted. The methodology is first verified for near pure free convection flows as well as previous analyses of condensation over a flat surface. The model is subsequently validated against data on steam condensation on a square plate in the presence of air. Finally, a correlation is deduced from the parametric numerical results to allow a proper correction for curvature effects when vapor condenses in the presence of a NCG. The proposed correction spans the whole range of NCG fractions, and displays the expected behavior at the extremes of this range. [ABSTRACT FROM AUTHOR]