Tesis sobre el tema "TWO-PHASE EXPANSION"

Sustainable Built Environments Senior Capstone Project<br>In a society growing towards alternative modes of transportation, this study explores the phase 2 streetcar expansion for the Tucson area. Using case studies and research articles, a criteria of factors were created to best judge three predetermined expansion routes. A case study was then done on these routes to determine which route would be best for the area. In this study, the factors that were looked at and used to rank the three routes included; population density and average annual income of the surrounding area of the route (400 meter buffer), accessibility and connectivity, which includes; bus stops, bike lanes, walkability, traffic counts and possible park & ride development. The last factor used in the case study was proximity to destination, specifically how many ‘big destinations’ (destinations that experience high volumes of people) each route had. The proximity to destination factor also determined how good of a balance each route had with housing, work and amenities/leisure destinations.

The transport and separation of oil and water is an essential process to the oil and chemical industries. Although transporting the mixtures is often necessary due to few reasons, it is generally beneficial to separate out the phases in order to reduce installation and maintenance costs, at the same time, avoiding safety problems. Thus, separation of liquid-liquid flows is a necessary part of many industrial processes. Hence, knowledge of two-phase flow dynamics is important for the design optimisation of separators. Therefore, the aim of this research is to investigate the feasibility of a sudden pipe expansion to be used as phase separator because it compact in design and capable for converting dispersed flow to stratified flow. In the test section, spatial distribution of the liquid-liquid phases in a dynamics flow system was visualised for the first time for by means of capacitance Wire Mesh Sensor (CapWMS), providing instantaneous information about the interface shapes, waves and phase layer evolution of oil-water flow. Visual assessment and analysis of the WMS data showed three distinct layers: an oil layer at the pipe top; a water layer at the pipe bottom and a mixed layer between them. The interfaces that form between the separated phases (oil or water) and the mixed layer were classified as oil interface or water interface. Results showed interface shapes were initially concave or convex near to the inlet of the test section and became flat further downstream the expansion, especially for water interfaces. There were no waves observed for horizontal and downward pipe orientations at all flow conditions and axial position downstream of the expansion. As for the upward inclined pipe orientation, waves were found, and they formed at position close to the inlet at all input oil volume fraction except at 0.2 OVF. The amplitude of the waves was: ~ 0.29D for 0.8 OVF; ~ 0.22D for 0.6 OVF and ~ 0.26D for 0.4 OVF. The higher the input oil volume fraction, the larger the waves become. In conclusion, the WMS results demonstrated that spatial distributions are strongly dependent on the mixture velocity, input oil fraction and inclination angles for the far position. In this present work, droplets were found to be larger near the interface. Drops were large nearer to the interface at the near position (10D) for all pipe orientations and throughout the test section for horizontal flow. The drops size decreased when the distance from the interface increased for these pipe configurations. As for the furthest position from the expansion for upward and downward inclined pipe orientation, larger droplets could also be seen at distance away from the interface and vice versa. The gravity or buoyant force is one of the contributing factors to the settling of the droplets. These forces are acting simultaneously on the droplets i.e. if the buoyant force which tends to spread the droplets throughout the pipe cross-section, is not large enough to overcome the settling tendency of gravity settling of the droplets occurs. Hence, the droplets that are non-uniformly scattered within the continuous phase begin to coalesce as they flow further downstream the pipe, producing larger drops. In addition, as the distance from expansion increased, the mixed layer becomes narrow and more drops begin to coalescence to form large drop due to increased droplet-droplet collision. Owing to these factors, results indicate that the mechanisms of coalescence occurred faster at the bottom, for water droplets and at the top, for oil droplets than the other locations in a pipe cross-section. For a better separation design, the coalescence process should occur at the aforementioned (bottom for water and top for oil) locations within the expansion pipe. However, at higher mixture velocities the mixed layer would be responsible for the smaller droplet size for horizontal and both inclinations of pipe orientation. The mixed layer dominated almost entirely in the pipe cross-section.

Flow instabilities in long wells and risers under certain flow conditions, is well known. Expansion driven flow instability (EDI) which is relatively little known, refers to occurrence of flow instabilities in long wells and risers as a result of entrapment of gas upstream of the well or riser base. EDI can also occur in gas-lift systems at low pressure and low gas injection rate.This work was initiated to tackle flow instability problem related to deep water production operations where long wells and risers are extensively in use. The aim of this thesis is to investigate a type of flow instability known as &#147;Expansion Driven Flow Instability (EDI) in Long Wells and Risers&#148;. This involves experimental investigation as well numerical modelling of expansion driven flow instability in long wells and risers. Finally results of the experimental investigations are compared with numerical model data.An experimental flow loop was setup to verify EDI at varying pipe geometry, inlet flow pressure and gas flow rate to examine the effect of EDI in long wells and risers. The laboratory experiment was conducted using air and water at atmospheric conditions, in a flowline-riser system consisting of a 32mm diameter and 9.12m long riser. The expansion driven flow cycle was captured in video recording.Variation of inlet flow pressures was achieved by varying the height of the overflow tank. Three cases were considered, each at a different inlet pressure. Each inlet pressure of the fluid was examined against varying inclination angles of the horizontal pipe to the riser inlet. Different gas flow rates were tested at different inclination angles. It was observed that inclination angle has the greatest impact on EDI.Experimental result of one of the cases was modelled using OLGA and the results of the experimental compared against simulation results output. Discrepancies in the two sets of results were observed in some cases. These may be attributed to simplifications and assumptions made during the simulation model build.Both results of the experimental investigation and numerical simulation demonstrated that expansion driven flow instability can occur in laboratory setup and can probably occur in deep water natural-lift wells and risers as well as gas-lifted wells and risers under certain flow conditions and pipe inclinations.&amp;#8195;

Two-phase flow, boiling, and condensation in microchannels have received considerable attention in the recent past due to the growing interest in the high heat fluxes made possible by these channels. This dissertation presents a study on the condensation of refrigerant R134a in small hydraulic diameter (100 < Dh < 160 mm) channels. A novel technique is used for the measurement of local condensation heat transfer coefficients in small quality increments, which has typically been found to be difficult due to the low heat transfer rates at the small flow rates in these microchannels. This method is used to accurately determine pressure drop and heat transfer coefficients for mass fluxes between 300 and 800 kg/m2-s and quality 0 < x < 1 at four different saturation temperatures between 30 and 60oC. The results obtained from this study capture the effect of variations in mass flux, quality, saturation temperature, hydraulic diameter, and channel aspect ratio on the observed pressure drop and heat transfer coefficients. Based on the available flow regime maps, it was assumed that either the intermittent or annular flow regimes prevail in these channels for the flow conditions under consideration. Internally consistent pressure drop and heat transfer models are proposed taking into account the effect of mass flux, quality, saturation temperature, hydraulic diameter, and channel aspect ratio. The proposed models predict 95% and 94% of the pressure drop and heat transfer data within ±25%, respectively. Both pressure drop and heat transfer coefficient increase with a decrease in hydraulic diameter, increase in channel aspect ratio and decrease in saturation temperature. A new non-dimensional parameter termed Annular Flow Factor is also introduced to quantify the predominance of intermittent or annular flow in the channels as the geometric parameters and operating conditions change. This study leads to a comprehensive understanding of condensation in microchannels for use in high-flux heat transfer applications.

Le mouvement d'une bulle d'azote de Taylor dans des solutions mixtes glycérol-eau s'élevant à travers différents types d'expansions et de contractions est étudié par une approche numérique. La procédure CFD est basée sur un solveur open-source Basilisk, qui adopte la méthode du volume de fluide (VOF) pour capturer l'interface gaz-liquide. Les résultats des expansions/contractions soudaines sont comparés aux résultats expérimentaux. Les résultats montrent que les simulations sont en bon accord avec les expériences. La vitesse de la bulle augmente dans les expansions soudaines et diminue dans les contractions soudaines. Le modèle de rupture des bulles est observé dans les expansions soudaines avec de grands taux d'expansion, et un modèle de blocage des bulles est observé dans les contractions soudaines avec de petits rapports de contraction. De plus, la contrainte de cisaillement de la paroi, l'épaisseur du film liquide et la pression dans les simulations sont étudiées pour comprendre l'hydrodynamique de la bulle de Taylor montant par expansions/contractions. Le processus transitoire de la bulle de Taylor passant par une expansion/contraction soudaine est ensuite analysé pour trois singularités différentes: graduelle, parabolique convexe et parabolique concave. Une caractéristique unique de la contraction concave parabolique est que la bulle de Taylor passe par la contraction même pour de petits rapports de contraction. De plus, un modèle de changement de phase est développé dans le solveur Basilisk. Afin d'utiliser la méthode VOF géométrique existante dans Basilisk, une méthode VOF géométrique générale en deux étapes est implémentée. Le flux de masse n'est pas calculé dans les cellules interfaciales mais transféré aux cellules voisines autour de l'interface. La condition aux limites de température saturée est imposée à l'interface par une méthode de cellule fantôme. Le modèle de changement de phase est validé par évaporation de gouttelettes avec un taux de transfert de masse constant, le problème de Stefan unidimensionnel, le problème d'aspiration de l'interface et un cas d'ébullition à film plan. Les résultats montrent un bon accord avec les solutions analytiques ou les corrélations<br>The motion of a nitrogen Taylor bubble in glycerol-water mixed solutions rising through different types of expansions and contractions is investigated by a numerical approach. The CFD procedure is based on an open-source solver Basilisk, which adopts the volume-of-fluid (VOF) method to capture the gas-liquid interface. The results of sudden expansions/contractions are compared with experimental results. The results show that the simulations are in good agreement with experiments. The bubble velocity increases in sudden expansions and decreases in sudden contractions. The bubble break-up pattern is observed in sudden expansions with large expansion ratios, and a bubble blocking pattern is found in sudden contractions with small contraction ratios. In addition, the wall shear stress, the liquid film thickness, and pressure in the simulations are studied to understand the hydrodynamics of the Taylor bubble rising through expansions/contractions. The transient process of the Taylor bubble passing through sudden expansion/contraction is further analyzed for three different singularities: gradual, parabolic convex and parabolic concave. A unique feature in parabolic concave contraction is that the Taylor bubble passes through the contraction even for small contraction ratios. Moreover, a phase change model is developed in the Basilisk solver. In order to use the existed geometric VOF method in Basilisk, a general two-step geometric VOF method is implemented. Mass flux is calculated not in the interfacial cells but transferred to the neighboring cells around the interface. The saturated temperature boundary condition is imposed at the interface by a ghost cell method. The phase change model is validated by droplet evaporation with a constant mass transfer rate, the one-dimensional Stefan problem, the sucking interface problem, and a planar film boiling case. The results show good agreement with analytical solutions or correlations

PETROBRAS<br>O presente trabalho aborda o estudo do escoamento líquido-gás na câmara de expansão típica de um separador ciclônico cilíndrico. Para entender o funcionamento do separador, analisa-se primeiro, o campo de velocidades da fase líquida dentro do equipamento. A análise é realizada em função de simulações numéricas e medições experimentais. Em função dos resultados é desenvolvido um modelo para o cálculo da velocidade e a espessura do escoamento de líquido ao longo do separador. O modelo desenvolvido apresenta resultados com boa precisão para uma ampla faixa de condições de operação e dimensões do separador. Na segunda etapa, uma formulação para o seguimento de bolhas é acoplada no modelo monofásico, com a finalidade de avaliar, de modo teórico, a separação de bolhas de um meio líquido contínuo. O estudo de escoamento bifásico permite identificar o comprimento ótimo do separador para garantir a separação das bolhas e também permite entender a influência das condições de operação sobre o processo de separação. Os resultados alcançados devem contribuir para o estado da arte no assunto e fornecer a engenheiros que atuam na área uma ferramenta relativamente simples para o dimensionamento desse tipo de separadores.<br>This work approaches the study of gas-liquid flow in the expansion chamber of a typical cylindrical cyclonic separator. To understand its operation, first is analyzed the velocity field of the liquid phase inside the equipment. The analysis is based on numerical simulations and experimental measurements. From the results, is developed a model that determines the velocity and the thickness of the liquid film flow along the separator. The model developed produces results with good accuracy for a wide range of operating conditions and dimensions of the separator. In the second step, a formulation for bubble tracking is coupled in the single phase model, in order to evaluate its theoretical separation from a continuous liquid medium. The study of two-phase flow identifies the optimum length of the separator to separate bubbles and also allows understand the influence of operating conditions on the separation process. The results achieved should contribute to the state of the art in the subject and provide engineers working in the area a relatively simple tool for the design of these separators.

博士<br>國立臺灣大學<br>材料科學與工程學研究所<br>93<br>In the present study, a unit cell model is proposed to predict the elastic constants (Poisson’s ratio, Young’s, bulk and shear moduli) and thermal expansion coefficient of two-phase composites. The feasibility of the present model is verified by comparing the model predictions with experimental model. Comparing with many monolithic materials, many more factors can affect the elastic and thermal properties of two-phase composites. For examples, the shape of the second phase, composition, the size and distribution of pores, interface integrity, etc., can all affect the resulting properties of composites. Apart from these microstructural complexities, the transfer of stress within the composite is also unknown. Therefore, it is almost impossible to deliver a precise estimation on the elastic constants and thermal expansion coefficient of composites. Instead, a pair of upper and lower bounds to cover the possible variation on the elastic and thermal constants is proposed in the present study. We further assume that the model is applied only to the fully dense composites in which the components are strongly bonded. Therefore, it is not possible for the two components to separate at their interfaces when the composite is loaded or heated. The discussion also restricts itself to macro-composites, namely, to those in which the scale of the second phase is large to microstructure so that composite properties can be modeled by using continuum mechanics without resort to atomic and dislocation mechanics. Furthermore, the size of specimen is much larger than the size of second phase so that the properties are an appropriate average of those of the components in the composites. The present model tend to offer predictions over the entire composition range of the two-phase composites, namely, the amount of the second phase varies from 0 to 100 vol.%. The experimental data for the composites with the composition covers the entire composition range are sparse. Therefore, the Al2O3-NiAl composites with the NiAl content varied from 0 to 100 vol.% are prepared by hot-pressing. The elastic constants of the composites are measured by employing an ultrasonic technique, the thermal expansion coefficient by using a thermal mechanical analyzer. As the amount of second phase is large, the second phase particle tends to interconnect with each other to form an interpenetrating microstructure. To reflect such microstructural feature, a unit cell model with elongated second phase is proposed in the present study. The stress-strain coupling is also taken into account. The model prediction on elastic modulus and Poisson’s ratio shows strong dependence on the ratio of elastic modulus of matrix to that of second phase. The thermal expansion coefficient of composite also shows strong dependence of the elastic modulus ratio. It demonstrate that the thermal expansion of a composite as it is under a temperature change is in-fact an elastic problem. The model predictions are compared with the experimental data of the Al2O3-NiAl composites. A comprehensive collection on the experimental data for other two-phase composites has also been conducted. The comparison between the model predictions and all available experimental data demonstrates the validation of the model. The model prediction is also compared with other theoretical models, for example, the Hashin-Shtrikman model for elastic constants and Kerner and Schapery models for thermal expansion coefficient. The model proposed in the present study shows similar precision on the properties of composites as the elastic modulus ratio is lower than 10. However, the elastic modulus ratio is higher than 20, the model proposed in the present study shows improved prediction on the properties of two-phase composites.

碩士<br>國立高雄第一科技大學<br>機械與自動化工程研究所<br>100<br>This research main purpose is obtains of thermal-expansion coefficient a two-phase compound materials theory of by the compound materials theory of by the compound materials mechanics And double-phase of compound materials&apos;&apos; using the ellipse spheroid medium theoretical formula heat expansion coefficient value, case study&apos;&apos;s result makes the comparison again by way of literature experiment second. And the case study (1) take the resin as a parent metal, the textile fiber is a medium; Of theoretical formula estimate shuttle fabric compound materials quotation present paper of thermal-expansion coefficient each different medium content. And compares of thermal-expansion coefficient with the medium of theoretical formula as well as the literature experimental result. The case study (2) piece take the mortar as a parent metal, the bone material is a medium. Also theory of the quotation present paper forecasts its thermal-expansion coefficient. Also compares of thermal-expansion coefficient with the medium of theoretical formula as well as the literature experimental result.The prediction on the concretes compound materials of thermal-expansion coefficient each different medium content. By theoretical formula and literature experimental result knowing. Of this article two research cases respectively be the textile fiber and the bone material and so on two medium for the horizontal isotropism material arrangement forecast that of thermal-expansion coefficient and the literature actual value the shuttle fabric compound materials and the concretes compound materials are close really.

碩士<br>雲林科技大學<br>機械工程系碩士班<br>96<br>This study investigates the pressure change and flow pattern subject to the influence of sudden contraction and sudden expansion. The air and water mixture flows from small rectangular channels (2×4 mm, 2×6 mm, 4×4 mm and 4×6 mm, respectively) connect to a 2 mm diameter tube and rectangular channels (3×6 mm, 3×9 mm, respectively) connect to a 3 mm diameter tube. The total mass flux (G) ranges from 100 to 700 kg/m2·s with gas quality (x) being varied from 0.001 to 0.8. In general the contraction pressure change increases with the rise of mass flux, and vapor quality but an unique deflection of contraction pressure change pertaining to liquid vena contracta at a very low gas quality is encountered at a very low gas quality in the 4×6 mm and 3×9 mm test section. It is found that the influence of surface tension and tube size, or quivalently the Bond number plays a major role for the departure of various models/correlations. Hence by taking account the influences of Bond number, Weber number and area contraction ratio into the homogeneous model, a modified homogeneous correlation is proposed. The mean deviation is 29.3% to the present data and the available literature data. For the expansion pressure change, an unique deflection of contraction pressure change at a very low gas quality is also observed. The phenomenal observation is related to the liquid jet-like flow pattern. By contrast, an appreciable increase of pressure difference is seen when the liquid jet-like flow pattern is completely gone. Finally, the predictions of the available literature correlations are compared to all literature and the present data. The lowest mean deviation is Richardson’s prediction with the mean deviation of 71.80%.

碩士<br>國立雲林科技大學<br>機械工程系碩士班<br>95<br>針對目前高效率電子設備為了發展散熱技術，常會使用密集式熱交換器，在密集式熱交換器等此類的相關相關元件中，因應逐漸微小化的電子設備本體，會於有限的空間內前後對應的長方形流道通常用一個突擴、突縮段來做連接，然而目前只有一些單相流壓力變化和熱傳的論文發表過，關於微小流道的壓力驟變相關的研究卻很少被提出討論，甚至於兩相流體流過突擴、突縮段的研究都沒有發現，因此本研究的目的就是要探討在此突擴段內的兩相流摩擦壓力變化關係和流型與其相對的變化。 此實驗將用透明的壓克力板做成帶有突擴段的長方形流道以方便觀察兩相流流型的型態，這些測試段將具有類似的幾何型狀和結構，其流道的窄度皆為3㎜，而寬度分別是6、9㎜（ ＝4、4.5㎜）以顯現其突擴段表現的特徵性，測試的空氣和水之質通率從100到 700 kg/m2.s和乾度則從0.001到 0.1，在一般國內常溫下的實驗參數下測試。 進而根據這長方形流道所參照公認理論的空泡比值加以比較，配合流型和摩擦壓力變化的數據將用來互相比對現有模式預測的正確度，解釋與其相關實驗的性別差異所在，另外在突擴段所量得的兩相流摩擦壓力變化數據將用來整理成一個實用的數學經驗式用來預測通過突擴段的摩擦壓力變化，且通過這突擴段兩相流流型的結構和變化，解釋在接頭內壓力變化變化的原因。

Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2007.<br>Source: Dissertation Abstracts International, Volume: 69-02, Section: B, page: 1279. Adviser: Predrag S. Hrnjak. Includes bibliographical references (leaves 187-198) Available on microfilm from Pro Quest Information and Learning.

In recent years, heat pumps and refrigeration systems are widely used in both residential and industrial applications. The possibility of recovering the large throttling losses by using an expander could give a substantial contribution to the performance improvement. In this thesis, a reciprocating expander developed from a hydraulic motor was numerically and experimentally analyzed. A numerical model was developed to identify the needed small modifications to be made on the expander without change its architecture. Successively, an extensive experimental activity on the modified expander has been carried out to characterize it in detail and evaluate the effective performance. With this aim, a dedicated test rig and a measurement system have been developed. The expander was tested in a R134a heat pump cycle and in a CO2 refrigeration cycles. Despite of the mechanical losses due to the different original application of the machine, the thermodynamic cycles showed very promising results with the adoption of this solution. For this reason, a redesign and manufacturing of the machine was done in order to improve the efficiency in HFC cycles and to decrease the mechanical losses. The new version of the expander was tested in a "hot-gas bypass cycle", which has been designed and manufactured. The aim of this cycle is to obtain high stability and flexibility, and lower size due to the lack of the evaporator. The results showed improvement in both the thermodynamic behavior and mechanical losses. Finally, a 1D thermal conduction model has been developed to study the two-phase expansion with R134a with an improved accuracy.

碩士<br>雲林科技大學<br>機械工程系碩士班<br>97<br>Recently, the advances of the high performance electronic devices have been resulted a great demand in developing efficient heat removal techniques for dissipating increasingly large heat flux from compact heat exchangers in small rectangular channels. In the limited space of compact heat exchangers, the adjacent channels can be connected by abrupt flow area changes. Due to the nature of the geometries involved, the flow would induce a region of flow separation and cause significant pressure loss at the abrupt flow area change. The knowledge of frictional pressure drop and flow characteristics at the abrupt flow area change is very needed for the design of such compact heat exchangers. However, very few data are available for the two-phase flow pressure drop at the sudden expansion and contraction in smaller rectangular channels. Therefore, this research is proposed to visualize the flow structure and measure the two-phase pressure drop at the sudden expansion in small rectangular channels. The effects of the aspect ratio of the rectangular test section, area ratio, flow rate and gas quality for flow across the abrupt flow area change will be considered in this study. The test sections will also be arranged in horizontal longitudinal and transverse orientations to see the influence of gravity force on the flow pattern and pressure drop. Seven test sections are made of transparent acrylic resin, so that the flow pattern and flow structure could be visualized. The test sections have the same overall structures with 6x2, 6x4, 4x2, 4x4. The four rectangular channels are connected with a 2 mm diameter tube, 9x3, 6x3, 3x6 are connected with a 3 mm diameter tube. The area of the tube is smaller than the area of the rectangular channels. As fluid flows into the tube and then enters the rectangular channel, it will observe a sudden expansion at the interconnection between the round tube and the rectangular channel. The total mass flux of the air-water mixture is ranged from 500 to 2000 kg/m2.s and quality is changed from 0.01 to 0.7. Based on the obtained data of void fraction, generated flow regime and frictional loss, the adequacy of the existing predictive methods will be examined and their differences will be discussed. In addition, the two-phase frictional pressure drop data at the sudden expansion will be correlated to propose correlations for the prediction of two-phase pressure drops at the sudden expansion. The flow separation and flow pattern change at the sudden expansion and contraction will be visualized for the interpretation of the large pressure change in the sudden expansion areas.

Dans ce mémoire, nous étudions le problème de l'estimation de la variance pour les estimateurs par double dilatation et de calage pour l'échantillonnage à deux phases. Nous proposons d'utiliser une décomposition de la variance différente de celle habituellement utilisée dans l'échantillonnage à deux phases, ce qui mène à un estimateur de la variance simplifié. Nous étudions les conditions sous lesquelles les estimateurs simplifiés de la variance sont valides. Pour ce faire, nous considérons les cas particuliers suivants : (1) plan de Poisson à la deuxième phase, (2) plan à deux degrés, (3) plan aléatoire simple sans remise aux deux phases, (4) plan aléatoire simple sans remise à la deuxième phase. Nous montrons qu'une condition cruciale pour la validité des estimateurs simplifiés sous les plans (1) et (2) consiste à ce que la fraction de sondage utilisée pour la première phase soit négligeable (ou petite). Nous montrons sous les plans (3) et (4) que, pour certains estimateurs de calage, l'estimateur simplifié de la variance est valide lorsque la fraction de sondage à la première phase est petite en autant que la taille échantillonnale soit suffisamment grande. De plus, nous montrons que les estimateurs simplifiés de la variance peuvent être obtenus de manière alternative en utilisant l'approche renversée (Fay, 1991 et Shao et Steel, 1999). Finalement, nous effectuons des études par simulation dans le but d'appuyer les résultats théoriques.<br>In this thesis we study the problem of variance estimation for the double expansion estimator and the calibration estimators in the case of two-phase designs. We suggest to use a variance decomposition different from the one usually used in two-phase sampling, which leads to a simplified variance estimator. We look for the necessary conditions for the simplified variance estimators to be appropriate. In order to do so, we consider the following particular cases : (1) Poisson design at the second phase, (2) two-stage design, (3) simple random sampling at each phase, (4) simple random sampling at the second phase. We show that a crucial condition for the simplified variance estimator to be valid in cases (1) and (2) is that the first phase sampling fraction must be negligible (or small). We also show in cases (3) and (4) that the simplified variance estimator can be used with some calibration estimators when the first phase sampling fraction is negligible and the population size is large enough. Furthermore, we show that the simplified estimators can be obtained in an alternative way using the reversed approach (Fay, 1991 and Shao and Steel, 1999). Finally, we conduct some simulation studies in order to validate the theoretical results.