Dimensional Analysis Theory Research Articles

Dimensional analysis approach to dominant three-pole placement in delayed PID control loops

PID control loops with time delay are characterized by infinite number of poles but the pole assignment technique for adjusting the controller parameters can be applied to placing three poles only. The dominance of these poles is therefore an essential condition for this application. A novel approach to this problem involves applying dimensional analysis theory to obtain a generalized model of the control loop and then to perform a parameter tuning for its dimensionless representation. A one-row dimensional matrix results from the assumption of the usual dimensionless interpretation of both control error and actuating signals of the controller. Dimensionless similarity numbers of the so-called swingability and laggardness are introduced to specify the plant dynamics in the controller synthesis. A trio of numbers is assigned to become the dominant zeros of the characteristic quasi-polynomial of the control loop and the corresponding PID parameter adjustment is derived in the form of uniform formulae. The tuning of the proper damping and the real pole position ratios is provided by means of an IAE optimization technique. A dominance degree notion is introduced and an argument increment criterion is proposed to check the dominance of any of the pole placement cases. The quality of the disturbance rejection response is taken as the general criterion in the design of the time delay plant control.

Verification of pollutant creation model at dendromass combustion

Results of verification of two mathematical models presenting the creation of three pollutants at wood combustion in a low power pellet boiler are published in this article. The type of boiler used in the experiment was EKO-KARBON USPOR-AUTOMAT with nominal power of 18 kW. Nitrogen oxides, carbon oxide, and total organic carbon were tested. The models elaborated on the basic theory of dimensional analysis. In the first model, the emissions produced in the wood combustion process were described under the condition that the combustion product temperature in boiler outlet is one relevant parameter in the mathematical model. Combustion air volume flow, combustion air temperature, combustion product temperature, boiler power, and wood calorific power are other parameters in the model. The second model describes the mutual coherence of the same pollutants, provided that flame temperature, instead of combustion product temperature, is incorporated into the model. The outputs obtained from both models are subjected to the same combustion conditions.

Test Design on Minicar Transmission System Efficiency Using Dimensional Analysis

Based on formulating the minicar transmission efficiency test standard,the test was designed by using dimensional analysis theory. Selected the power loss of transmission system as evaluation index, the effects of parameters such as oil temperature,oil depth of stir oil gear, line speed of stir oil gear, transmission system ratio, input speed and input torque were experimented. It has the advantages of less test times and more design freedom. The test design provides theory basis for the empirical formula which indicates the relationship between power loss and test parameters.

The Similar Experimental Study on the Concentration Field in Construction Period of the Storage of Salt Rock in Solution Mining

The Jin-tan underground energy storage of salt rock in Jiangsu Province as a demonstration project, Established a similar model experimental platform of concentration filed in construction period of salt caverns, which based on the similar theory of dimensional analysis. To analysis the concentration distribution and the concentration field variation in cavity. The study shows that: the brine concentration in the same horizontal plane is uniformly distributed beyond the boundary layer where has a higher concentration than other locations, but a continuous gradient distribution of brine concentration increases with depth in the vertical direction. In the vertical direction, the brine concentration gradient firstly increases and then decreases with the increase of the freshwater flow in the reverse circulation method. And in the vertical direction, the brine concentration gradient also firstly increases and then decreases with the increase of the distance between the two wash pipes in the reverse circulation method. Comparing with the reverse circulation method, the brine concentration gradient in the vertical direction is very insignificant with direct circulation. During the solution of mining of the salt cavity, the brine concentration increases as time goes on, and the rate of increase continues to decrease until the concentration field to stabilize.

Modelling of Nitrogen Oxides Production during Wood Combustion

The process of elaboration and consequent verification of mathematical model of NOx production during wood combustion based on the dimensional analysis theory is presented in this paper. The model is based on selected physical quantities which can be measured continually during boiler operation. Primarily, the model was elaborated for boiler type VERNER V210 of 20 kW performances and its validity was also verified for the same construction type of boiler of lower performance (14 kW). Lump wood of lower calorific value approximately 15 MJ∙kg-1 was used as fuel in the boilers. The results showed that the dimensional analysis can be applied to describing such a complicated phenomenon as the NOx production during wood combustion is.

Application of dimensional analysis and similarity theory for simulation of electrode kinetics described by the Marcus–Hush–Chidsey formalism

Similarity theory and dimensional analysis were shown to be applicable for a kinetic analysis of an electrochemical reaction described by the Marcus–Hush–Chidsey (MHC) formalism. The corresponding dimensionless quantities (similarity criteria) were derived and approximating interrelations between them were analyzed. A convenient expression was obtained which connected the values of electrochemical rate constant, electrode polarization and reorganization energy in dimensionless complex coordinates.

Analysis of the Difference between Static and Dynamic Performance of Copper Cylinder Pressure Measurement and the Correcting Method

The serious static-dynamic discrepancy will be caused when the copper cylinders statically calibrated are used to measure the dynamic pressure. In this paper, the reasons for static-dynamic error caused are analyzed in detail. Taking φ3.5×8.15 copper cylinders for instance, the static-dynamic error’s change law of this type of copper cylinders is analyzed. Dimensional analysis and analogical theory are used to set up the correction equation which can be used to diminish the static-dynamic error. The correction equation to diminish static-dynamic error is testified to be effective through combining with the actual experimental data. This paper gives an effective way to decrease the measuring error of copper cylinders statically calibrated.

Fluid-to-fluid modeling of two-phase flow critical heat flux in horizontal helically coiled tubes

The new similarity laws for fluid-to-fluid modeling of two-phase flow critical heat flux (CHF) in horizontal helically coiled tubes were derived based on the dimensional analysis and similarity theory considering the effect of the geometrical parameters on CHF. A generalized factor D n was introduced to the new similarity laws, and all the new dimensionless numbers were derived from the classical theorem of Buckingham π for dimensional analysis. The obtained dimensionless parameter sets were a reasonable extension to Ahmad's compensated distortion model, which may be considered as a special case of the new dimensionless parameter sets when the variable n is equal to unity. Based on the experimental data, the specific similarity numbers were determined for CHF phenomena in horizontal helically coiled tubes. A new equivalent characteristic parameter D e-helix was developed, which could reflect the influence of complex flow channels on the occurrence of CHF. The equivalent characteristic parameter consists of the essential geometrical parameters of tubes and the fluid thermophysical properties. The new fluid-to-fluid modeling methods were proposed for CHF of R134a-water in horizontal helically coiled tubes, which could be used readily to derive the CHF data of water through the CHF data of R134a at the corresponding experimental conditions.

Dimensional Analysis to Ozone Generation by Pulsed Streamer Discharge in Dry Air

The generation of ozone is efficient by using short high-voltage combined with a dielectric layer placed adjacent to the outer electrode in dry air. Ozone concontration depend on the peak pulsed voltage, pulse repetition rate, gas flow rate, reactor length and so on. For ozone generation by pulsed streamer discharge in dry air, a mathematical model, which describes the relation between ozone concentration and these parameters that are of importance in its design, is developed according to dimensional analysis theory. The presence of nitrogen on ozone formation in dry air cannot only consume high energy electron but also form nitrogen oxides (NOx) in discharge plasma. Nitrogen oxides (NOx) play an important role in the process of ozone generation in air discharge, where the catalytic cycle for ozone destruction takes place. So the nitrogen influencing factor should be considered. Formula is derived for predicting the characteristics of the ozone generation, within the corona inception voltage to the gap breakdown voltage. The predicted results are shown to be in good agreement with experimental findings.

Hydraulics of clear and submerged overfall weirs with bottom circular-openings

Many standard weirs in Fayoum district (Egypt) have been perforated by 1, 2 or 3 pipes, to convey much water in the downstream direction. Measurements of discharge with these structures are complicated due to existence of the openings. In this research, an experimental study was carried out to investigate the flow over clear over fall weir with bottom circular opening. Nine weir models with different heights were tested in a horizontal laboratory flume. Three weir heights were used with each one, three pipes with different diameters were tested. The results of the flow over the weir with a circular opening were compared with those of the weir without an opening having the same dimensions. It was found that there is a large difference between them. Multiple regression equations based on energy principal and dimensional analysis theory were developed for computing the discharge over clear over fall weir with a bottom opening.

Deducing the USLE mathematical structure by dimensional analysis and self-similarity theory

The Universal Soil Loss Equation (USLE) was originally deduced by a statistical analysis of a large data set of soil loss measurements. The multiplicative structure of the model has been criticised due to the considerable interdependence between the variables. Using the soil erosion representative variables and the reference condition adopted in the USLE, the aim of this paper was to apply dimensional analysis and self-similarity theory to deduce the functional relationship among the selected variables. The analysis yielded a multiplicative equation, similar to the USLE. Therefore, this study suggested that the USLE has a logical structure with respect to the variables used to simulate the physical soil erosion process and the adopted reference condition.

HYDRAULICS OF CLEAR OVERFALL WEIRS WITH BOTTOM-OPENINGS

Many Fayoum Standard Weirs have been perforated by one, two or three pipes, to convey much water in downstream direction. Measurements of discharge with these structures are complicated due existence of the opening. In this research, an experimental study was carried out to investigate the flow over clear over fall weir with bottom opening. Nine weir models with different heights were tested in horizontal laboratory flume of 17 m length, 0.3 m width, and 0.5 m depth. Weir height is changed three times. For each one, the diameter of opening is changed three times. The downstream depth was changed to cover all the expected flow regimes of the pipe and the weir. These flow regimes are; free pipe-free weir, submerged pipe-free weir and submerged pipe-submerged weir. The results of flow over weir with opening were compared with those of weir without opening having the same dimensions. It was found that there is a large deviation between them. Multiple regression equations based on energy principal and dimensional analysis theory were developed for computing discharge over clear over fall weir with bottom opening. Also, values of discharge coefficients were estimated for combined flow over the weir and through the pipe. Then the total discharge can be computed by multiplying the discharge coefficient by the summation of theoretical weir discharge plus theoretical orifice discharge. Equations for computing the discharge of combination are awarded.

HYDRAULICS OF SURFACE POLLUTED WATER - JET IN OPEN CHANNEL FLOW

Presented in this paper are the results of an intensive experimental program for studying the influence of jet inclination, velocity ratio, pollutant type and initial concentration on hydraulic characteristics of polluted surface water jet in open channel flow. Polluted jet is injected from a sided polluted source with different angles of inclination to mainstream direction. The polluted water is prepared by adding two types of chemicals (sodium chloride and pfennig) with different concentrations to fresh water. The initial jet velocity is varied to cover a range of velocity ratio R from 2.73 to 10.0. For each velocity ratio, the solute concentration is changed to take 800, 1050, 1382.23, 1440, 1780 and 2320 p.p.m .The centerline jet trajectory, velocity decay, pollutant dilution and the width of polluted area are measured. The velocity is measured by a calibrated current meter while the jet coordinates and water depths are measured using point gages. Samples are taken from jet centerline to measure the jet concentration. The concentrations are measured using Conductance /Total Dissolved Solids Model 76. The factors influencing the problem are normalized with the help of dimensional analysis theory. The analysis of the results revealed that the characteristics of a polluted water jet in cross-flow depend mainly on jet to main stream velocity ratio and jet inclination to the main stream. Also, the decay of the velocity and dilution of pollution concentration are mainly depending on the studied parameters. The type of pollution does not affect these characteristics. Equations have been developed for the correlation of the experimental results. A comparison between the results of present study and those from previous studies are carried out. These comparisons give a reasonable agreement.

Flow over Gabion Weirs

A conventional weir typically consists of an impermeable body constructed of concrete, since its primary functions are to heading up water and efficiently regulate flow. However, an impermeable body prevents the longitudinal movement of aquatic life and transportation of physical and chemical substances in water, eventually having a negative impact on the water environment. One of the advantages of gabions as a building material is that the motion of individual stones comprising the gabion is not of much concern. The wire mesh of the gabion basket serves to restrain any significant movement. Also, gabion weirs offer an alternative design that could be adopted for flash flood mitigation. In this study, a series of laboratory experiments was performed in order to investigate the flow over gabion weirs. For this purpose, two different gabion weir models were tested in two horizontal laboratory flumes of 10-m and 17-m length, 0.3-m width, and 0.3- and 0.5-m depth, respectively, for a wide range of discharge, upstream water depth, downstream water depth, weir height, weir length, and gabion filling gravel material size. The results of the gabion weir were compared with those of experiments carried out on solid weirs having the same dimension and it was found that there is a large deviation when the solid weirs equation is applied to gabion weirs (permeable weirs). So, using one of the existing solid weir flow formulas would lead to an erroneous calculated value. Multiple regression equations based on the dimensional analysis theory were developed for computing the discharge over gabion weirs for both free and submerged flow regimes. Also, equations were introduced for computing the discharge coefficient to be applied with the traditional solid weir equation.

Similarity between morphological characteristics of rills and ephemeral gullies in Sicily, Italy

AbstractThis paper reports the results of a field investigation aimed to establish morphological similarity between rills and ephemeral gullies. Rill measurements were made on 14 plots having a surface area of 22–352 m2 located on a 14·9% slope and on a plot 6·0 m wide and 22·0 m long having a uniform 22·0% slope. The plots are located on the experimental station for soil erosion measurements, ‘Sparacia’, of the Agricultural Faculty of Palermo University, in Sicily, Italy. All plots are subjected to natural rainfall. The measurements were made immediately following five events between November 2004 and December 2005. The ephemeral gully measurements were made on a cultivated area of about 120 ha, located in Central Sicily, which is representative of many soil‐crop conditions in the Mediterranean basin. The morphological similarity between rill and ephemeral gully was first tested. Then a power relationship between rill or gully volume and length, theoretically deduced by dimensional analysis and self‐similarity theory, was applied. This power relationship needs a different scale factor for rill and gully measurements. Finally, using two dimensionless groups representative of the channel morphology variables, the analysis showed that a single relationship can be applied to rill and gully measurements. Copyright © 2009 John Wiley & Sons, Ltd.

Generalized relative characteristics of limiting states of material under nonisothermal deformation

We describe a procedure for representing material characteristics as thermodynamic invariants. It has enabled us to put forward arguments of equations for limiting mechanical states under nonisothermal deformation for similar materials. The procedure is based on relations of a simple theory of fracture and on the concepts of the similarity theory and dimensional analysis theory.

Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables

Let ℱ(R) be a set of holomorphic functions on a Reinhardt domain R in a Banach sequence space (as e.g. all holomorphic functions or all m-homogeneous polynomials on the open unit ball of ). We give a systematic study of the sets dom ℱ(R) of all z ∈ R for which the monomial expansion of every ƒ ∈ ℱ(R) converges. Our results are based on and improve the former work of Bohr, Dineen, Lempert, Matos and Ryan. In particular, we show that up to any ε > 0 is the unique Banach sequence space X for which the monomial expansion of each holomorphic function ƒ converges at each point of a given Reinhardt domain in X. Our study shows clearly why Hilbert's point of view to develop a theory of infinite dimensional complex analysis based on the concept of monomial expansion, had to be abandoned early in the development of the theory.

Study of Similarity Law for Bird Impact on Structure

With dimensional analysis and similarity theory, the model similarity law of aircraft structures under bird impact load is investigated. Numerical calculations by means of nonlinear dynamic software ANSYS/LS-DYNA are conducted on the finite element models constructed with different scaling factors. The influence of strain rate on the model similarity law is found to be dependent on the strain rate sensitivity of materials and scale factors. Specifically, materials that are not sensitive to strain rate obey the model similarity law in the bird impact process. The conclusions obtained are supposed to provide a theoretical basis for the experimental work of bird impact on aircraft structure.

Analysis of blast wave propagation inside tunnel

The explosion inside tunnel would generate blast wave which transmits through the longitudinal tunnel. Because of the close-in effects of the tunnel and the reflection by the confining tunnel structure, blast wave propagation inside tunnel is distinguished from that in air. When the explosion happens inside tunnel, the overpressure peak is higher than that of explosion happening in air. The continuance time of the blast wave also becomes longer. With the help of the numerical simulation finite element software LS-DYNA, a three-dimensional nonlinear dynamic simulation analysis for an explosion experiment inside tunnel was carried out. LS-DYNA is a fully integrated analysis program specifically designed for nonlinear dynamics and large strain problems. Compared with the experimental results, the simulation results have made the material parameters of numerical simulation model available. By using the model and the same material parameters, many results were adopted by calculating the model under different TNT explosion dynamites. Then the method of dimensional analysis was used for the simulation results. As overpressures of the explosion blast wave are the governing factor in the tunnel responses, a formula for the explosion blast wave overpressure at a certain distance from the detonation center point inside the tunnel was derived by using the dimensional analysis theory. By comparing the results computed by the formula with experimental results which were obtained before, the formula was proved to be very applicable at some instance. The research may be helpful to estimate rapidly the effect of internal explosion of tunnel on the structure.

Walking on the Moon and on Other Extraterrestrial Bodies

AbstractIt will be shown that it is possible to predict the walking behavior of humans on extraterrestrial bodies – i.e. under different gravitational forces – by applying techniques from dimensional analysis to measurements obtained in a stationary terrestrial laboratory. These terrestrial experiments allow us to generate artificial data about extraterrestrial walking and are, therefore, of great importance for space travel as well as for extraterrestrial physiology. What at first glance appears to be an interesting application of dimensional analysis and model theory, proves in retrospect to be an elegant means for gaining important insights for space technology.