scholarly journals Analytical Solution of the Time-Dependent Microfluidic Poiseuille Flow in Rectangular Channel Cross-Sections and Its Numerical Implementation in Microsoft Excel

Biosensors ◽  
2019 ◽  
Vol 9 (2) ◽  
pp. 67
Author(s):  
Patrick Risch ◽  
Dorothea Helmer ◽  
Frederik Kotz ◽  
Bastian E. Rapp
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Exact Analytical Solution ◽  
Rectangular Channel ◽  
Channel Cross Section ◽  
Analysis Tool ◽  
Complex Flow ◽  
Microsoft Excel ◽  
Navier Stokes Equation

We recently demonstrated that the Navier–Stokes equation for pressure-driven laminar (Poiseuille) flow can be solved in any channel cross-section using a finite difference scheme implemented in a spreadsheet analysis tool such as Microsoft Excel. We also showed that implementing different boundary conditions (slip, no-slip) is straight-forward. The results obtained in such calculations only deviated by a few percent from the (exact) analytical solution. In this paper we demonstrate that these approaches extend to cases where time-dependency is of importance, e.g., during initiation or after removal of the driving pressure. As will be shown, the developed spread-sheet can be used conveniently for almost any cross-section for which analytical solutions are close-to-impossible to obtain. We believe that providing researchers with convenient tools to derive solutions to complex flow problems in a fast and intuitive way will significantly enhance the understanding of the flow conditions as well as mass and heat transfer kinetics in microfluidic systems.

scholarly journals Numerical and Theoretical Study of Performance and Mechanical Behavior of PEM-FC Using Innovative Channel Geometrical Configurations

2021 ◽  
Vol 11 (12) ◽  
pp. 5597
Author(s):  
Hussein A. Z. AL-bonsrulah ◽  
Mohammed J. Alshukri ◽  
Ammar I. Alsabery ◽  
Ishak Hashim
Keyword(s):  
Fuel Cell ◽  
Cross Section ◽  
Rectangular Channel ◽  
Compressive Load ◽  
Proton Exchange ◽  
Channel Cross Section ◽  
Channel Height ◽  
Cross Sectional ◽  
Triangular Channel ◽  
Higher Power

Proton exchange membrane fuel cell (PEM-FC) aggregation pressure causes extensive strains in cell segments. The compression of each segment takes place through the cell modeling method. In addition, a very heterogeneous compressive load is produced because of the recurrent channel rib design of the dipole plates, so that while high strains are provided below the rib, the domain continues in its initial uncompressed case under the ducts approximate to it. This leads to significant spatial variations in thermal and electrical connections and contact resistances (both in rib–GDL and membrane–GDL interfaces). Variations in heat, charge, and mass transfer rates within the GDL can affect the performance of the fuel cell (FC) and its lifetime. In this paper, two scenarios are considered to verify the performance and lifetime of the PEM-FC using different innovative channel geometries. The first scenario is conducted by adopting a constant channel height (H = 1 mm) for all the differently shaped channels studied. In contrast, the second scenario is conducted by taking a constant channel cross-sectional area (A = 1 mm2) for all the studied channels. Therefore, a computational fluid dynamics model (CFD) for a PEM fuel cell is formed through the assembly of FC to simulate the pressure variations inside it. The simulation results showed that a triangular cross-section channel provided the uniformity of the pressure distribution, with lower deformations and lower mechanical stresses. The analysis helped gain insights into the physical mechanisms that lead to the FC’s durability and identify important parameters under different conditions. The model shows that it can assume the intracellular pressure configuration toward durability and appearance containing limited experimental data. The results also proved that the better cell voltage occurs in the case of the rectangular channel cross-section, and therefore, higher power from the FC, although its durability is much lower compared to the durability of the triangular channel. The results also showed that the rectangular channel cross-section gave higher cell voltages, and therefore, higher power (0.63 W) from the fuel cell, although its durability is much lower compared to the durability of the triangular channel. Therefore, the triangular channel gives better performance compared to other innovative channels.

A Comparison of Micro-PIV Experiments in a Mini-Channel to Numerical and Analytical Solutions

2003 ◽  
Author(s):  
D. Newport ◽  
D. Curtin ◽  
M. Davies
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Exact Analytical Solution ◽  
Measurement Tool ◽  
Channel System ◽  
Micro Piv ◽  
Software Packages ◽  
Micro Systems ◽  
Experimental Rig ◽  
Good Agreement

In this paper, measurements are presented of the velocity profile in a mini-channel at different locations. The channel is rectangular in cross-section, approximately 1.2mm wide, 1.4mm deep and 29mm long. A micro-PIV system was used to obtain the velocity profiles at the inlet, mid-length and exit of the channel. The raw image maps were processed using three different commercial PIV software packages, and compared to an exact analytical solution. The mini-channel system was also simulated using a commercial CFD code as a further check on the dataset, and the experimental rig itself. It was found that the different processing procedures had little influence on the micro-PIV data, and good agreement was found with theory, numerical prediction and experiment. This establishes confidence in micro-PIV as a measurement tool in micro-systems.

Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach

2014 ◽  
Vol 748 ◽  
pp. 416-432 ◽  
Author(s):  
Alexei Rybkin ◽  
Efim Pelinovsky ◽  
Ira Didenkulova
Keyword(s):  
Wave Equation ◽  
Shallow Water ◽  
Cross Section ◽  
Exact Analytical Solution ◽  
Auxiliary Function ◽  
Arbitrary Cross Section ◽  
Channel Cross Section ◽  
Linear Wave ◽  
Nonlinear Shallow Water Theory ◽  
Run Up

AbstractWe present an exact analytical solution of the nonlinear shallow water theory for wave run-up in inclined channels of arbitrary cross-section, which generalizes previous studies on wave run-up for a plane beach and channels of parabolic cross-section. The solution is found using a hodograph-type transform, which extends the well-known Carrier–Greenspan transform for wave run-up on a plane beach. As a result, the nonlinear shallow water equations are reduced to a single one-dimensional linear wave equation for an auxiliary function and all physical variables can be expressed in terms of this function by purely algebraic formulas. In the special case of a U-shaped channel this equation coincides with a spherically symmetric wave equation in space, whose dimension is defined by the channel cross-section and can be fractional. As an example, the run-up of a sinusoidal wave on a beach is considered for channels of several different cross-sections and the influence of the cross-section on wave run-up characteristics is studied.

Lateral Distributions of Depth-Averaged Velocity in Compound Channels with Submerged Vegetated Floodplains

2014 ◽  
Vol 641-642 ◽  
pp. 288-299
Author(s):  
Ming Wu Zhang ◽  
Chun Bo Jiang ◽  
He Qing Huang
Keyword(s):  
Experimental Data ◽  
Analytical Solution ◽  
Rectangular Channel ◽  
Navier Stokes ◽  
Compound Channel ◽  
Navier Stokes Equation ◽  
Compound Channels ◽  
Vegetated Floodplain ◽  
Vegetation Layer ◽  
Good Agreement

Lateral distributions of depth-averaged velocity in open compound channels with submerged vegetated floodplains are analyzed, based on an analytical solution to the depth-integrated Reynolds-Averaged Navier-Stokes equation with a term included to account for the effects of vegetation. The cases of open channels are: rectangular channel with submerged vegetated corner, and compound channel with submerged vegetated floodplain. The present paper proposes a method for predicting lateral distribution of the depth-averaged velocity with submerged vegetated floodplains. The method is based on a two-layer approach where flow above and through the vegetation layer is described separately. An experiment in compound channel with submerged vegetated floodplain is carried out for the present research. The analytical solutions of the three cases are compared with experimental data. The corresponding analytical depth-averaged velocity distributions show good agreement with the experimental data.

A modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform one-dimensional waveguides

2011 ◽  
Vol 225 (12) ◽  
pp. 2864-2880 ◽  
Author(s):  
M Khoshbayani-Arani ◽  
N Rasekh-Saleh ◽  
M Nikkhah-Bahrami
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Exact Analytical Solution ◽  
Mode Shapes ◽  
Transmission Matrix ◽  
Approximate Methods ◽  
One Dimensional ◽  
Wave Propagation Method ◽  
Number Of Segments

In this article, using the wave propagation method, the natural frequencies and mode shapes of an arbitrary non-uniform one-dimensional waveguide are calculated. The non-uniform rods and beams are partitioned into several continuous segments with constant cross-sections, for which there exists an exact analytical solution. At the end of each segment, waves in positive and negative directions are obtained in terms of waves at initial segment and subsequently, the calculations of the mode shapes become simple. By satisfying the boundary conditions, the characteristic equation is obtained and natural frequencies are calculated for both the arbitrary non-uniform rod and beam. Also, by adding waves in positive and negative directions at the end point of each segment, the mode shapes are obtained. To verify the modified wave method presented here, the frequencies and mode shapes of the rod and the beam with a polynomial cross-section having an exact analytical solution are compared and have been proven to be of high accuracy. Besides, comparisons of finite element method are also included. Therefore, this method can also be used to calculate the natural frequencies and mode shapes of rods and beams with any arbitrary variable cross-section for which no analytical solution is available. For the ‘Modified Wave Approach’ developed here, dimensions of transmission matrix remain constant if the number of segments is increased, while in general wave propagation method, dimensions of transmission matrix increase upon increasing the number of segments. Besides this novelty, this method has the advantage that it gives all the natural frequencies and mode shapes, unlike other approximate methods such as weighted residual, Rayleigh–Ritz, and finite difference methods which have their own shortcomings such as limited number of natural frequencies. Also, since each segment has an exact analytical solution, in contrast to other approximate methods, much higher accuracy is obtained even with only a few number of partitions.

Asymptotic and Exact Analysis for Constructal Optimization of Microchannel Heat Sink

2008 ◽  
Author(s):  
Omid Asgari ◽  
Mohammad Hassan Saidi
Keyword(s):  
Heat Transfer ◽  
Cross Section ◽  
Cross Sections ◽  
Rectangular Channel ◽  
Channel Cross Section ◽  
Geometrical Parameters ◽  
Approximate Methods ◽  
Maximum Heat ◽  
Circular Duct ◽  
Maximum Heat Transfer

Microchannel are at the fore front of today’s cooling technologies. They are widely being considered for cooling of electronic devices and in micro heat exchanger systems due to their ease of manufacture. One issue which arises in the use of microchannels is related to the small length scale of the channel or channel cross-section. In this work, the maximum heat transfer and the optimum geometry for a given pressure loss have been calculated for forced convective heat transfer in microchannels of various cross-section having finite volume for laminar flow conditions. Solutions are presented for 10 different channel cross sections, namely parallel plate channel, circular duct, rectangular channel, elliptical duct, polygonal ducts, equilateral triangular duct, isosceles triangular duct, right triangular duct, rhombic duct and trapezoidal duct. The model is only a function of Prandtl number and geometrical parameters of the cross-section, i.e., area and perimeter. This solution is performed with two exact and approximate methods. Finally, in addition to comparison and discussion about these two methods, validation of the relationship is provided using results from the open literature.

The Vaporization of Superheated Sodium in a Vertical Channel

1972 ◽  
Vol 94 (3) ◽  
pp. 300-304 ◽  
Author(s):  
Ralph M. Singer ◽  
Robert E. Holtz
Keyword(s):  
Cross Section ◽  
Thin Liquid Film ◽  
Rectangular Channel ◽  
Vertical Channel ◽  
Growth Patterns ◽  
Single Bubble ◽  
Bulk Liquid ◽  
Channel Cross Section ◽  
Channel Axis ◽  
Vapor Growth

Measurements of the vapor growth patterns and rates following the nucleation of superheated sodium in a vertical rectangular channel are presented and discussed. The vapor was found to grow as a single bubble for incipient bulk-liquid superheats greater than about 10 deg C, and this single bubble tended to completely fill the channel cross section (except for a thin liquid film on the walls) and to grow as a vapor slug for incipient bulk-liquid superheats greater than about 50 deg C. The temperature gradients in the liquid both normal and parallel to the channel axis prior to nucleation were found to have an important effect upon the dynamics of the vapor slug. Experimental data on the vapor growth and collapse rates and the associated pressure transients are presented for boiling pressures up to 1 atm and incipient superheats up to about 180 deg C.

A Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes of the Rods with Variable Cross Section

2011 ◽  
Vol 110-116 ◽  
pp. 2537-2547
Author(s):  
M. Khoshbayani Arani ◽  
N. Rasekh Saleh ◽  
M. Nikkhah Bahrami
Keyword(s):  
Analytical Solution ◽  
Cross Section ◽  
Natural Frequencies ◽  
Exact Analytical Solution ◽  
Finite Difference Methods ◽  
Variable Cross Section ◽  
Mode Shapes ◽  
Variable Cross ◽  
Approximate Methods ◽  
Modified Method

Analytical solutions for vibration analysis of the rods with variable cross section are in general complex and in many cases impossible. On the other hand, approximate methods such as the weighted residual, Rayleigh-Ritz and finite difference methods also have their own shortcomings such as a limited number of natural frequencies and accuracy. Using the wave propagation method, the structure is partitioned into several continuous segments with constant cross-section, for which there exists an exact analytical solution. Waves in positive and negative directions at the entrance of each segment are obtained in terms of waves at the initial segment. Then, by satisfying the boundary conditions, the characteristic equation is obtained and all natural frequencies are calculated. By adding waves in positive and negative directions at each point, the shape modes are obtained. To verify this modified method, the frequencies and mode shapes of a rod with polynomial cross section, which has an exact analytical solution, are compared and have proven to be of highly accuracy. Therefore, this method can also be used to calculate the natural frequencies and its mode shapes of the rods with variable cross section for which no analytical solution is available.

scholarly journals The exact analytical solution of the dual-phase-lag two-temperature bioheat transfer of a skin tissue subjected to constant heat flux

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Hamdy M. Youssef ◽  
Najat A. Alghamdi
Keyword(s):  
Heat Flux ◽  
Analytical Solution ◽  
Exact Analytical Solution ◽  
Constant Heat Flux ◽  
Bioheat Transfer ◽  
Skin Tissue ◽  
Phase Lag ◽  
Dual Phase ◽  
Constant Heat ◽  
Two Temperature

Abstract This work is dealing with the temperature reaction and response of skin tissue due to constant surface heat flux. The exact analytical solution has been obtained for the two-temperature dual-phase-lag (TTDPL) of bioheat transfer. We assumed that the skin tissue is subjected to a constant heat flux on the bounding plane of the skin surface. The separation of variables for the governing equations as a finite domain is employed. The transition temperature responses have been obtained and discussed. The results represent that the dual-phase-lag time parameter, heat flux value, and two-temperature parameter have significant effects on the dynamical and conductive temperature increment of the skin tissue. The Two-temperature dual-phase-lag (TTDPL) bioheat transfer model is a successful model to describe the behavior of the thermal wave through the skin tissue.

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