STUDY OF A SINGLE SCREW COMPRESSOR WITH A CONICAL TEETH GATE ROTOR

2008 ◽  
Vol 32 (3-4) ◽  
pp. 333-352
Author(s):  
Yang Shyue-Cheng ◽  
Tsang-Lang Liang

From a geometric viewpoint, a mathematical model of a single screw compressor with a conjugate pair of meshing conical teeth gate rotor is a conjugate problem. Coordinate transformation and envelope theory are applied to determine the sets of spatial points of the contacting surfaces that define the main rotor of a single screw compressor. Envelope theory and analytical procedure are used to derive mathematical models of a gate rotor and a main rotor. Stress analysis for the single screw compressor mechanism is performed. PowerMILL software package is used to simulate the manufacture of a main rotor. A numerical example with a compressor ratio of 11:6 is presented to demonstrate the application of the mathematical models developed in this paper.

Author(s):  
S-C Yang

In this paper, a method is proposed for determining a basic profile of a cc-type single-screw compressor including the gate rotor and the screw rotor. The cc-type has a cylindrical screw and two cylindrical gate rotors. Based on this method, a mathematical model of the meshing principles of a cc-type screw rotor meshed with a gate rotor, that has either straight edge teeth or conical teeth, is presented. The inverse envelope concept is used to determine the cutting-edge curve of a gate rotor. Based on this concept, the required cutter for machining a cc-type screw rotor can be obtained by the envelope of a one-parameter family. The obtained screw rotor is an envelope to the family of the gate rotor's surfaces. The obtained envelope becomes the generating surface. The inverse envelope can be used to obtain the envelope to the family of generating surfaces. Then the profile of a gate rotor cutting-edge curve can be easily obtained. The surface analysis including contact lines is shown for the design and manufacture of a screw compressor. As an example, the cc-type single-screw compressor with a compressor ratio of 11:6 was determined with the aid of the proposed mathematical model. Using rapid prototyping (RP) and manufacturing technology, a cc-type single-screw rotor with a gate rotor was designed. The RP primitives provide an actual full-size physical model that can be analysed and used for further development. Results from these mathematical models should have applications in the design of cc-type single-screw compressors.


Author(s):  
S-C Yang

This paper presents a method for determining the basic profile of a single-screw compressor including a gate rotor and a screw rotor. The inverse envelope concept for determining the cutting-edge curve of the gate rotor is presented. Based on this concept, the required cutter for machining the screw rotor can be obtained by an envelope of the one-parameter family of obtained screw rotors. The obtained screw rotor is an envelope of the family of gate rotor surfaces. Let the obtained envelope of the one-parameter family of gate rotor surfaces become the generating surface. The inverse envelope can be used to obtain the envelope of the family of generating surfaces. Then, the profile of the gate rotor with the cutting-edge curve can be easily obtained. The proposed method shows that the gate rotor and the screw rotor are engaged along the contact line at every instant. This is essential to reduce the effect of leakage on compressor performance. In this paper, a mathematical model of the meshing principle of the screw rotor with the gate rotor is established. As an example, the single-screw compressor for a compressor ratio of 11:6 is determined with the aid of the proposed mathematical model. Results from these mathematical models should have applications in the design of single-screw compressors.


2007 ◽  
Vol 31 (2) ◽  
pp. 219-234 ◽  
Author(s):  
Yang Shyue-Cheng ◽  
Tsang-Lang Liang

A geometric model and a mathematical model of a PP-type single screw rotor with planar gate rotor are derived from the gate-rotor generation process and gear theory. The teeth of gate rotor are planar. Based on the inverse envelope concept, the cutter required for machining the single screw rotor can be obtained using an inverse envelope of a one-parameter family of screw surfaces. The surface of the proposed screw rotor is analyzed using the developed mathematical model. A surface analysis, including stress analysis, of the design and manufacture of the screw compressor is presented. Finally, a numerical example demonstrates the geometric model of the PP-type single screw rotor with a compression ratio of 11:6.


2018 ◽  
Vol 15 (1) ◽  
pp. 39-55
Author(s):  
V. B. Rudakov ◽  
V. M. Makarov ◽  
M. I. Makarov

The article considers the problem of determining the rational plans of the input sampling reliability and technical parameters of components of space technology, the totality of which is supplied to the Assembly plants for the manufacture of complex products of space technology. Problem statement and mathematical model based on the minimization of the economic costs of control and losses related to the risks of taking wrong decisions, are given in the article. The properties of the mathematical models are investigated, the algorithm for its optimization is developed. The result is an optimal plan for the sampling of sets of components, which includes: an optimal product mix subject to mandatory control of the aggregate and optimum risks of first and second kind, when acceptance number of statistical plan is zero. The latter circumstance is due to the high requirements of reliability and technical parameters of products of space technology.


1998 ◽  
Vol 84 (6) ◽  
pp. 2154-2162 ◽  
Author(s):  
Cord Sturgeon ◽  
Albert D. Sam ◽  
William R. Law

Rapid measurement of glomerular filtration rate (GFR) by an inulin single-bolus technique would be useful, but its accuracy has been questioned. We hypothesized that reported inaccuracies reflect the use of inappropriate mathematical models. GFR was measured in 14 intact and 5 unilaterally nephrectomized conscious male Sprague-Dawley rats (mean weight 368 ± 12 g) by both single-bolus (25 mg/kg) and constant-infusion techniques (0.693 mg ⋅ kg−1 ⋅ min−1). The temporal decline in plasma inulin concentration was analyzed through biexponential curve fitting, which accounted for renal inulin loss before complete vascular and interstitial mixing. We compared our mathematical model based on empirical rationale with those of other investigators whose studies suggest inaccuracy of single-bolus methods. Our mathematical model yielded GFR values by single bolus that agreed with those obtained by constant infusion [slope = 0.94 ± 0.16 (SE); y intercept = 0.23 ± 0.64; r = 0.82]. In comparison to the data obtained by constant inulin infusion, this method yielded a very small bias of −0.0041 ± 0.19 ml/min. Two previously reported models yielded unsatisfactory values (slope = 1.46 ± 0.34, y intercept = 0.47 ± 1.5, r = 0.72; and slope = 0.17 ± 1.26, y intercept = 17.15 ± 5.14, r = 0.03). The biases obtained by using these methods were −2.21 ± 0.42 and −13.90 ± 1.44 ml/min, respectively. The data indicate that when appropriate mathematical models are used, inulin clearance after single-bolus delivery can be used to measure GFR equivalent to that obtained by constant infusion of inulin. Attempts to use methods of analysis for simplicity or expediency can result in unacceptable measurements relative to the clinical range of values seen.


1970 ◽  
Vol 19 (1-2) ◽  
pp. 141-141
Author(s):  
L. Gedda ◽  
G. Brenci ◽  
M. T. Lun

The theoretical relationship between the distribution of a given trait in a population of twin pairs and several genetic parameters has been examined. In particular, a series of mathematical models has been worked out, that, when applied to a twin population, nonselected for the occurrence of a given trait and nondiagnosed as to zygosity, leads to an estimate of:1) The MZ: DZ ratio in the population;2) The frequency of the genotype responsible for a given trait;3) The probability of manifestation of the trait;4) The value of epistatic factors.A further mathematical model affords the estimate of linkage in the hypothesis of simultaneous recording of more than one trait.


Author(s):  
T-S Lai

This article presents a mathematical model and geometric design algorithm for a new type of roller drive. The pinion has conical teeth in two circular arrays instead of one. This work is based on coordinate transformation and envelope theory, from which the equation of meshing of the cycloid drive is derived. The pinion profiles are the equidistant curves of the epicycloid profiles except the contour of the pinion conical tooth holes. Although there are twice as many pinion teeth as conventional rollers, their speed ratios are identical. This approach can design roller drives in which the pinion has two circular arrays of conical and cylindrical rollers. On the basis of these results, the corresponding solid modelling is constructed by CAD. Four examples are presented to demonstrate the feasibility of this approach. These examples can be a useful reference as a design case for other tooth profiles.


Author(s):  
Андрей Геннадьевич Деменков ◽  
Геннадий Георгиевич Черных

С применением математической модели, включающей осредненные уравнения движения и дифференциальные уравнения переноса нормальных рейнольдсовых напряжений и скорости диссипации, выполнено численное моделирование эволюции безымпульсного закрученного турбулентного следа с ненулевым моментом количества движения за телом вращения. Получено, что начиная с расстояний порядка 1000 диаметров от тела течение становится автомодельным. На основе анализа результатов численных экспериментов построены упрощенные математические модели дальнего следа. Swirling turbulent jet flows are of interest in connection with the design and development of various energy and chemical-technological devices as well as both study of flow around bodies and solving problems of environmental hydrodynamics, etc. An interesting example of such a flow is a swirling turbulent wake behind bodies of revolution. Analysis of the known works on the numerical simulation of swirling turbulent wakes behind bodies of revolution indicates lack of knowledge on the dynamics of the momentumless swirling turbulent wake. A special case of the motion of a body with a propulsor whose thrust compensates the swirl is studied, but there is a nonzero integral swirl in the flow. In previous works with the participation of the authors, a numerical simulation of the initial stage of the evolution of a swirling momentumless turbulent wake based on a hierarchy of second-order mathematical models was performed. It is shown that a satisfactory agreement of the results of calculations with the available experimental data is possible only with the use of a mathematical model that includes the averaged equations of motion and differential equations for the transfer of normal Reynolds stresses along the rate of dissipation. In the present work, based on the above mentioned mathematical model, a numerical simulation of the evolution of a far momentumless swirling turbulent wake with a nonzero angular momentum behind the body of revolution is performed. It is shown that starting from distances of the order of 1000 diameters from the body the flow becomes self-similar. Based on the analysis of the results of numerical experiments, simplified mathematical models of the far wake are constructed. The authors dedicate this work to the blessed memory of Vladimir Alekseevich Kostomakha.


2021 ◽  
Author(s):  
Ed Rutgers Durner

Abstract Plants are studied to understand their growth and development so that their quality and productivity can be optimised. Models are developed that can be simple and descriptive, or quite complex with numerous mathematical equations; their level of complexity is linked to their purpose. This summary serves as an introduction to mathematical models in horticulture. It is not a manual for modelling itself, but rather an overview of how important mathematical models are in horticultural production. Mathematical models are used extensively in horticulture both extrinsically, i.e. when calculating chilling hour accumulations and intrinsically, i.e. when applying fertilizer to a crop. In chilling calculations, developed models are used directly. Fertilizer recommendations were probably developed using a mathematical model. The first part of this article discusses models in general and reviews general characteristics of mathematical models. The second part outlines the major uses of mathematical modelling in modern horticultural production. Presentations of specific models are limited in order to present a general discussion of models with examples that will interest most horticulturists.


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