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SCIENTIFIC JOURNAL OF THE TECHNICAL UNIVERSITY OF CIVIL ENGINEERING

Mathematical Modelling in Civil Engineering

BUCHAREST

No. 4 - December – 2014

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Disclaimer With respect to documents available from this journal neither U.T.C.B. nor any of its employees make any warranty, express or implied, or assume any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed. Reference herein to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.T.C.B. The views and opinions of authors expressed herein do not necessarily state or reflect those of U.T.C.B., and shall not be used for advertising or product endorsement purposes

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CONTENTS

VORTEX RINGS - EXPERIMENTS AND NUMERICAL SIMULATIONS ....................................................... 5

IULIA - RODICA DAMIAN, NICOLETA OCTAVIA TĂNASE, ȘTEFAN - MUGUR SIMIONESCU, MONA MIHĂILESCU

R LANGUAGE: STATISTICAL COMPUTING AND GRAPHICS FOR MODELING HYDROLOGIC TIME SERIES ........................................................................................................................................................... 12

GABRIELA-ROXANA DOBRE

COMPARISON OF TWO-PHASE PRESSURE DROP MODELS FOR CONDENSING FLOWS IN HORIZONTAL TUBES ........................................................................................................................................... 22

ALINA FILIP, FLORIN BĂLTĂREŢU, RADU-MIRCEA DAMIAN

APPROPRIATE CFD TURBULENCE MODEL FOR IMPROVING INDOOR AIR QUALITY OF VENTILATED SPACES .......................................................................................................................................... 31

CĂTĂLIN TEODOSIU, VIOREL ILIE, RALUCA TEODOSIU

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Mathematical Modelling in Civil Engineering, no. 4/2014 5

VORTEX RINGS - EXPERIMENTS AND NUMERICAL SIMULATIONS

IULIA - RODICA DAMIAN - PhD Student, Reorom Laboratory, “Politehnica” University of Bucharest, Power Engineering Faculty, e-mail: [email protected] NICOLETA OCTAVIA TĂNASE - Assistant, PhD, Reorom Laboratory, “Politehnica” University of Bucharest, Power Engineering Faculty, e-mail: [email protected] ȘTEFAN - MUGUR SIMIONESCU - PhD Student, Reorom Laboratory, “Politehnica” University of Bucharest, Power Engineering Faculty, e-mail: [email protected] MONA MIHĂILESCU - Lecturer, PhD, “Politehnica” University of Bucharest, Physics Department, e-mail: [email protected]

Abstract: The present paper was concerned with the experimental study of the time evolution of a single laminar vortex ring generated at the interface between water and isopropyl alcohol. The experiment was performed by the submerged injection of isopropyl alcohol in a water tank of 100 100 150 mm. A constant rate of Q0 = 2 ml/min was maintained using a PHD Ultra 4400 Syringe Pump with a needle having the inner diameter 0.4 mm. The dynamics of the vortex formation was recorded with a Photron Fastcam SA1 camera at 1000 fps equipped with an Edmund Optics objective VZM1000i. The numerical simulations were performed on a 2D geometry using the ANSYS-FLUENT code with the Volume of Fluid multiphase model and the viscous-laminar solver. The numerical flow patterns were found to be in good agreement with the experimental visualizations.

Keywords: flow visualization, isopropyl alcohol, buoyancy, CFD.

1. Introduction

Vortex rings are one of the most fundamental and interesting phenomena and have fascinated many authors, starting from the classical work of H. Helmholtz [1] up to nowadays scientists.

The practical applications of the vortex ring formation are found in the field of hydrodynamics such as flow control, fluid mixing, heat transfer and propulsion, but also in the biological and biomedical field like animal locomotion or internal flows.

Almost all the previous research was focused on vortex rings generated by immersed jets using the same liquids or following up the impact of buoyancy acting on the same direction as the momentum [1, 5, 6, 9, 12, 14].

A. B. Olcay and P.S. Krueger [8] used the same fluid for both the impinging jet and the ambiental one. Other representative experiments, such as those in which the ejected fluid was more dense, with upper or lower injection were conducted by Camassa et al. [14], and O. J. Myrtroeen and G. R. Hunt respectively [12]. An example regarding the injection from the bottom of the tank of a less dense fluid than the ambiental one was analyzed by D. Bond and H. Johari [9].

The original contribution of our research is emphasized by the vortex ring formation imposing the buoyancy to act in the opposite direction of the momentum, as seen in Fig.1. However this subject of studies remains one of the most important in the fluid mechanics field.

Fig.1 - The analyzed case

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Buoyant intrusions are common in natural and engineered systems like volcanic eruptions, piston engine fuel injection and microscale heat exchange in microelectromechanical systems [2].

The aim of this article is to perform a numerical and experimental study on the dynamics of vortex rings. Qualitative direct visualizations of the flow were created by injecting a less viscous and dense fluid than the ambiental one, from the top of the tank, in order to observe the influence of buoyancy upon their formation and characteristics. The numerical simulations were performed in a 2D geometry with the laminar solver implemented in FLUENT, using the VOF code for the calculation of the interface between the two liquid phases.

Theoretical, numerical and experimental researches about the vortex rings have characterized the movement as a function of dimensional parameters: radius , core size , circulation Γ, and non-dimensional parameters [3,4,5]:

The formation number, ⁄ , where is the stroke length and is the inner orifice diameter,

The Reynolds number, ∙ V ∙ ⁄ , where V is the velocity of the injected fluid.

In laboratory, vortex rings are typically generated by forcing a slug of fluid through a tube or orifice. The vortex sheet on the fluid slug rolls up into a vortex core [9]. Consequently, both ejected fluid, which comes from the noozle, and ambient fluid, which is pulled from the vicinity of the needle outlet, must be accelerated as the ring forms. The structure of the vortex ring immersed in the tank, generated by the jet is shown in Fig. 2.

Fig.2 - The structure of the vortex ring immersed in the tank

The shape and size of a vortex ring depend on the ratio between the radius of cross section of the ring core and the ring radius ⁄ . For thick vortex rings, 0.0116, the atmosphere of the vortex represents a deformed sphere, for 0.0116 the atmosphere of the vortex ring in its cross section represents a “figure of eight” and finally for thin vortex rings, 0.0116, the atmosphere of the vortex ring represents a torus embracing the thin core of the vortex ring [2].

The circulation that a vortex ring could attain is finite. There is a maximum amount of fluid vorticity that could be contained within a ring [7]. Initially, it was found that for values smaller than 4⁄ , a solitary vortex ring was formed, while for larger values of ⁄ , a leading vortex followed by a trailing jet and secondary vortices were observed. The morphology changes from a shape resembling an oblate hemisphere, to a mushroom-shaped structure.

Further investigations revealed that the formation number may depend on various factors and the studies showed that the ratio ⁄ may reach values up to 8.

Buoyant vortex rings form a different class of vortices where the density difference between the ring and ambient fluid profoundly alters the flow dynamics [9]. A buoyant jet can be broadly

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Mathematical Modelling in Civil Engineering, no. 4/2014 7

classified as positively buoyant, if the buoyancy force and momentum fluxes are aligned in the same direction or negatively buoyant if the buoyant force and momentum flux are opposed [13].

2. Experimental results

The experiment was performed by the injection of isopropyl alcohol in a water tank of 100 100 150 mm at a constant flow rate of 2 ml/min, obtaining vortex rings with lower density than the ambient fluid.

Fig.3 -Vortex ring evolution

The characteristic parameter of the flow was the Reynolds number of 106. The flow was maintained constant using a PHD Ultra 4400 Syringe Pump with a needle having the inner diameter 0.4 mm. The dynamics of the vortex formation was recorded with a Photron Fastcam SA1 camera at 1000 fps equipped with an Edmund Optics objective VZM1000i, as shown in Fig. 4.

Fig.4 - Experimental setup: Harvard Apparatus 4400 Ultra Syringe Pump, PhotronFastcam SA1, Edmund

Microscopic Objective VZM1000i, glass tank, flat needle tip with inner diameter 0.4 mm.

250 ms 300 ms 350 ms 400 ms 450 ms 500 ms

550 ms 600 ms 650 ms 700 ms 750 ms 800 ms

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The working fluids have the following properties: the densities of isopropyl alcohol 786 kg/m3 and water 1000 kg/m3. The viscosities are 2.5mPa ∙ s, 1mPa ∙s respectively.

After the PHD Ultra 4400 Pump was turned on and started to inject the fluid, the stream of isopropyl alcohol slowly separated and started to roll-up. The ejected fluid entrained the ambient fluid, and it was observed a symmetrical buoyant vortex ring with an elongated structure. In figure 3 are presented the experimental results for the shape of the vortex ring at different time steps.

By analizing the length of the obtained vortical structures, we observed a time dependence, with three different behaviours. Initially, the vortex ring shape was dominated by the inertia force. It followed a transition period after which the buoyancy force became preponderent, as observed in Fig.5.

0 500 1000 1500 20001.0

1.5

2.0

2.5

3.0

3.5

4.0

Vor

tex

ring

leng

th [m

m]

Time [ms]

Fig.5 -Vortex length evolution, l (measured from the tip of the needle to the vortex lower boundary).

3. Numerical results

The aim of the numerical study was to establish the shape of the vortex rings, according to the experiments. The numerical computations of the vortex ring were performed using the VOF model implemented in FLUENT, the mixture of water and isopropyl alcohol being solved simultaneously with the same laminar solver in a 2D geometry. In the VOF model of mixture, the fluids were considered incompressible and immiscible (no real diffusion between the isopropyl alcohol and water being allowed), see for details [13]. The separation interface was obtained by connecting the cells of equal volume fraction using the Geo - Reconstruct scheme. From the computational point of view, Fluent solves the equation system consisting of the continuity equation, the movement equation, and the mass conservation equation of the immiscible fluid in motion.

To achieve the numerical calculation using the FLUENT code, a geometry that reproduces the actual flow was built and then a discretization of the domain into a spatial grid ("mesh") consisting of finite elements.

The construction of the geometry and mesh of the flow field have a great influence on the obtained results. For the present study, the construction of the flow field and its mesh were

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carried out using the pre-processor Gambit 2.4, which provides a common set of CAD functions for creating domains, and features implemented specifically for rapid creation of predefined and structured geometries.

The appearance and development of the vortex ring at the interface between isopropyl alcohol in water was followed using the Multiphase module of ANSYS-FLUENT. The VOF multiphase model can be used for two or more immiscible fluids, aiming to determine the surface between the two fluids by solving a single set of motion equations for all phases of fluid occupying the whole field analyzed.

For the approximation of the interface between two phases (a, b), FLUENT calculates at first the volume fraction of the fluid (noted ) present in each part of the domain, thus setting whether or not the respective fluid occupies the volume element:

0 → thevolumeelementdoesn′tcontainthe luid 1 → the luid illsthevolumeelemententirely

0 1thevolumeelementcontainstheinterfacebetween andanother luidphase

In order to determine the interface between the two phases, the code solves a particular solution of the equation of continuity for the determination of the volume of fluid fraction, which contains the mass transfer between the two phases ( , ). This solution has the following form:

1∙

The volume fraction equation can be solved explicitly or implicitly, based on the calculation method used to determine the interface.

In the studied case of the vortex ring formation, more work areas were used. The mesh of the flow domain (with 309967 elements and 311151 nodes) was made using quadrilateral elements. All simulations were performed using the same mesh and have run on a 64-bit Quad-core computer with 16 GB RAM memory.

The initial distribution of the phases and the corresponding boundary conditions are shown in figure 6: (i) at the entrance was the first phase, isopropyl alcohol with uniform and constant velocity (v v ), (ii) the flow took place at atmospheric pressure, at the outlet the condition was

; and (iii) adherence conditions on the wall: v 0.

Fig.6 - Numerical working domain, initial phases configurations and the boundary conditions implemented in the

code FLUENT; detail of the structured mesh in vicinity of the needle.

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In figure 7 are presented the results for the shape of the vortex rings corresponding to the laminar model and at the different time steps.

Fig.7 - Numerical simulations of vortex ring evolution

4. Comparison of experimental with numerical results

The Target was to establish the experimental procedure and to correlate the results with the numerical simulations.

We analized first the numerical results, by representing the vortex length evolution in time and we obtained the curves for different Re numbers. After wich we compared them with the values of this same parameter, again at different Re numers (48 106). We performed several simulations for a series of velocities, because we wanted to observe the quantitative data. There are obvious differences and one reason may be the fact that between the 2 fluids wasn't considered the surface tension and diffusion at the interface, the buoyancy force isn't preponderent.

In figure 8a are presented the results for the length of the vortex rings corresponding to the laminar model at different time steps and numbers, while in figure 8b it is shown the comparison of the numerical results with the experimental ones.

Fig.8 - Vortex length evolution l (measured from the tip of the needle to the vortex lower boundary); 8a numerical

results of the length of the vortex rings corresponding at the different time steps and numbers; 8b the comparison of the numerical results with the experimental ones for various inlet velocities.

250 ms 300 ms 350 ms 400 ms 450 ms 500 ms

550 ms 600 ms 650 ms 700 ms 750 ms 800 ms

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5. Conclusions

The paper was dedicated to the experimental and numerical investigations of the vortex rings formation with the buoyancy acting on the opposite direction as the momentum, due to the difference in density and viscosity of the chosen two working fluids. The numerical flow patterns were found to be qualitatively in good agreement with the experimental manifestations. The experimental investigations were limited up to now only to the direct visualization of the vortex rings. The author’s intentions to progress with this study is to obtain the quantitative data of velocities, so to compare directly the experimental results with the numerical simulations. In the future the numerical model will be perfected by testing various tipes of meshes, taking under consideration the diffusion coefficients between the two fluids, performing also experiments with other fluids and of course creating a 3D model.

The present result is very promising and offers possibilities to develop in the future more detailed studies of the vortex rings.

Acknowledgement

Iulia – Rodica Damian's work has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132397.

The work of Ștefan – Mugur Simionescu and Nicoleta – Octavia Tănase has been funded by the Sectoral Operational Programme Human Resources Development 2007-2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132395.

The authors acknowledge the support and advice of professor Corneliu Bălan (Reorom Laboratory, “Politehnica” University of Bucharest, Power Engineering Faculty).

This paper was presented at EENVIRO 2014 Conference.

References

[1] Sullivan I. S., Niemela J. J., Hershberger R. E., Bolster D., Donnelly R. J., "Dynamics of thin vortex rings", Cambridge University Press, Journal of Fluid Mechanics, vol. 609, 2008, 319 - 347

[2] R.J. Wang, A Wing-Keung Law., E.E. Adams, O. B. Fringer, “Buoyant formation number of a starting buoyant jet”, Theoretical Physics of Fluids, 2009.

[3] Meleshko V. V., Gourjii A. A., Krasnopolskaya T. S., “Vortex rings: history and state of the art”, Journal of Mathematical Sciences, vol. 42, No. 6, December 2012.

[4] Akhmetov D. G., “Formation and basic parameters of vortex rings” , Journal of Applied Mechanics and Technical Physics, vol. 42, No. 5, 2001.

[5] Maxworthy T., "Some experimental studies of vortex rings", Journal od Fluid Mechanics, vol. 81, part. 3, 1977. [6] Laursen T. S., Rasmussen J. J., Stenum B., Snezhkin E. N., "Formation of a 2D pair and its 3D breakup: an

experimental study", Experiments in Fluids, 1997. [7] Palacios-Morales C., Zenit R., "Vortex ring formation for low Re numbers", Acta Mechanica, 2013 [8] Olcay A. B., Krueger P. S., " Momentum evolution of ejected and entrained fluid during laminar vortex ring

formation", Theoretical Computational Fluid Dynamics, 2010. [9] Bond D., Johari H., "Impact of buoyancy on vortex ring development in the near field", Experiments in Fluids, 2010. [10] Mohseni K., Ran H., Colonius T., "Numerical experiments on vortex ring formation", Journal of Fluid

Mechanics, 2001. [11] Nitsche M., Krsny R., "A Numerical study on vortex ring formation at the edge of a circular tube", Journal of

Fluid Mechanics, 1994. [12] Myrtroeen O.J., Hunt G.R., "Negatively bouyant projectiles - from weak fountains to heavy vortices", Journal

of Fluid Mechanics, 2010. [13] Wang R.J., Wing-Keung Law A., Adams E.E., "Pinch - off and formation number of negatively bouyant jet ",

Phisics of Fluids, 2011. [14] *** Fluent Inc., Fluent 6.3 user's guide, 2006. [15] Camassa R., Khatri S., McLaughlin R., Mertens K., Nenon D., Smith C., Viotti C., "Numerical [16] simulations and experimental measurements of dense-core vortex rings in a sharply stratified environment",

Computational Sicence and Discovery (2013)   

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R LANGUAGE: STATISTICAL COMPUTING AND GRAPHICS FOR MODELING HYDROLOGIC TIME SERIES

GABRIELA-ROXANA DOBRE - Assistant Professor, Technical University of Civil Engineering Bucharest, Mathematics and Computer Science Department, e-mail: [email protected]; [email protected]

Abstract: The analysis and management of Hydrology time series is used for the development of models that allow predictions on future evolutions. After identifying the trends and the seasonal components, a residual analysis can be done to correlate them and make a prediction based on a statistical model. Programming language R contains multiple packages for time series analysis: ‘hydroTSM’ package is adapted to the time series used in Hydrology, package ‘TSA’ is used for general interpolation and statistical analysis, while the ‘forecast’ package includes exponential smoothing, all having outstanding capabilities in the graphical representation of time series. The purpose of this paper is to present some applications in which we use time series of precipitation and temperature from Fagaras in the time period 1966-1982. The data was analyzed and modeled by using the R language.

Keywords: R programming language; Hydrology; Graphics; forecast; Holt-Winters

1. Introduction

The analysis of temperature and precipitation is very important when we study climate changes. Due to the increasing concentrations of greenhouse gases in the atmosphere, the global temperature rise has been accompanied by changes in weather and climate. Climate change studies show an increase of precipitation of 0.5 - 1% per decade in most of the Northern Hemisphere mid and high latitudes [1] and an average global surface temperature increase of about 0.30C and 0.60C between the late 19th century and 1994.[2]

As for Romania, [3] from data over the period 1961-2007 in 94 meteorological stations, we have

an increase of about of the average temperature during summer, winter and spring, and a slight trend of decrease of the average temperature in autumn. The amount of precipitation shows a trend of decrease of the average in summer and winter, and a trend of increase of the amount of precipitation in autumn.

Fagaras Depression is situated in the Southern part of the Transylvanian Basin that is separated from the Romanian Plain in the south by the Southern Carpathians. The climate in the Fagaras area is influenced by the presence of the mountains, which prevent the passage of cold air masses through the South and stop the hot air entering from the South. In Fagaras the annual average temperature is 8.20C and the recorded rainfall has annual average values between 600 - 800mm. [4]

The data studied in this article is the daily precipitation and temperature series collected in the period 1966-1982 in the Fagaras area. We consider the problem of modeling the time series of precipitation and temperature using R language.

A seasonal time series consists of a trend component, a seasonal component and an irregular component [11]. The aim of the paper is to analyze time series of precipitations and temperatures from the Fagaras area. After we find the components: the trend component, the seasonal component and the irregular component we make predictions using exponential smoothing with the Holt-Winters method.

2. The R environment

R is a free software environment for statistical computing and graphics, having some advantages in front of other classical statistic programs like SPSS, SAS, STATISTICA, etc. [5]. A few of these advantages are: R is a programming language, an Open Source that can work on Multi-

C2

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Mathematical Modelling in Civil Engineering, no. 4/2014 13

platforms (Windows, Linux, MacOS), and an extendable language that contains over 5300 packages. R Development Core Team updates R and provides support for a large community of R users that share their knowledge.

The R programming environment is freely available through CRAN and can be used for solving practical problems. The R programming environment has many statistical methods and techniques available either built-in or through packages.

In order to use it for the analysis of hydrological time series of the precipitations and the temperatures from the Fagaras area, we use the packages:

- HydroTSM developed for modeling of hydrological time series. We used version 0.4-2-1 (2014) for the management, analysis and plot of hydrological time series;

- TSA version 1.01 (2013) is used for time series decomposition and forecast;

- Forecast version 5.3 (2014) contains the exponential smoothing Holt–Winters method for the analysis and forecast of time series.

Regarding the graphical part, R has good capabilities for representing the time series plots, boxplots and histograms. The boxplot shows the median, first and third quartiles, the data extremes and outliers with the horizontal width of the box proportional to the square root of the size of the group. The histogram is a graphical representation of the data distribution.

3. Methods

In the following, we shortly present the procedures to find components of the time series and, by using the exponential method, to make a forecast for a time series.

3.1. Time series analysis

A stochastic process is a sequence of random variables that serve as a model

for an observed time series. [6]

For a stochastic process we define:

the mean function

(1)

the autocovariance function

(2)

the autocorrelation function

(3)

3.2. Decomposition of the time series

When the series have constant variability relative to their lengths, we can use an additive model for series decomposition:

(4)

where is a deterministic function and is a white noise process with . [6]

,...2,1,0, tYt

tY

,..2,1,0),( tYE tt

..2,1,0,,)()])([(),(, stYYEYYEYYCov ststssttstst

,..2,1,0,,)()(

),(),(

,,

,, st

YVarYVar

YYCovYYCorr

sstt

st

st

ststst

ttt XY

t tX 0)( tXE

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14 Mathematical Modelling in Civil Engineering, no. 4/2014

Many authors use the classical decomposition model:

(5) where is the trend, is the seasonal effect, and is the residual series. [7]

Many authors use the word trend only for a slowly changing mean function, and use the terms seasonal component for a mean function that varies cyclically. We did not make such distinctions here. Deterministic models describe the components: linear, seasonal means and cosine trends. [6]

The trend is the result of long-term factors and the time trend equations can be fit to the data by using the method of least squares.

The linear trend is expressed as

(6) where the slope and intercept, are unknown parameters.

In order to find a seasonal trend we take the mean function periodic with period 12:

(7) The model for the cosine trend has the form:

(8) where is the frequency, the amplitude, the phase of curve and

.

3.2. The Holt-Winters exponential smoothing

Another way to find the trend and seasonality is to use the Holt-Winters exponential smoothing. Three parameters control the smoothing: alpha is a smoothing constant for the level, beta for the estimate of the slope of the trend and gamma is a smoothing constant for seasonality estimate. [5], [7]

The level estimate:

(9) The trend estimate:

(10) The seasonality estimate

(11) where and is the length period of time series.

We can use the Holt-Winters exponential smoothing to make short-term forecasts:

(12)

where is the period to be forecast.

To see if the Holt-Winters model is correctly specified we have to see if the residuals are independently distributed by using the Ljung–Box test. [8]

Shortly, the Ljung–Box test can be described as follows.

tttt XsmY

tm ts tX

tt 10

01,

,...2,1,0,12 ttt

)sin()cos( tftft 22 210

f )sin();cos( 21

))(1()( 11 ttpttt basYa

11 )1()( tttt baab

ptttt saYs )1()(

)1,0(,, p

phptttht shbaY mod)1(1 h

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Mathematical Modelling in Civil Engineering, no. 4/2014 15

The null hypothesis of the Ljung–Box test is:

0H : The data are independently distributed.

The test statistic is

(13)

where:

- is the sample size,

- is the sample autocorrelation at lag k,

- is the number of lags tested.

The significance level is and the hypothesis of randomness is rejected if

(14)

where is the p-quantile of the chi-squared distribution with degrees of freedom.

4. Results and discussion

4.1. Main capabilities of the hydroTSM package

In order to make a basic analysis of the monthly values of the precipitation and temperature at the Fagaras station, from 01/Jan/1966 up to 31/Dec/1982, we convert the daily data to monthly data and obtain the data from table 1. [5]

Table 1

The summary statistics of the time series of monthly precipitation and temperature from Fagaras with data from 01/Jan/1966 up to 31/Dec/1982

Index Precipitation Temperature Observation

Min 0.10 -9.53 Minimum

1st Qu. 26.98 0.60 First quartile

Median 45.95 7.81 Median

Mean 61.61 7.64 Mean value

3rd Qu. 88.75 15.32 Third quartile

Max. 230.60 19.24 Maximum

IQR 61.77 14.72 Interquartile Range= 3rdQu-1stQu

sd 47.22 7.99 Standard deviation

cv 0.77 1.04 Coefficient of variation

Skewness 1.29 -0.26 Skewness

Kurtosis 1.71 -1.20 Kurtosis

n 204 204 Total number of elements

To highlight the characteristics of the monthly time series of precipitation and temperature from Fagaras, we made different graphs: time series plots, boxplots and histograms.

h

k

k

knnnQ

1

2^

)2(

n^

k

h

p

2,1 hpQ

hp ,1 h

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16 Mathematical Modelling in Civil Engineering, no. 4/2014

Fig.1-a) Precipitation time series with day, month and year frequency; b) Temperature time series with day, month and year frequency

The monthly boxplots from figures 1a and 1b describe the seasonal distributions of precipitation and temperature. We can see an increasing trend from December to July and after, a progressive decrease until December.

According to Table 1 and Figure 1a, the precipitation has a positive skewness, so we have an asymmetrical distribution with a long tail to the right and the positive kurtosis shows a distribution more peaked than the Gaussian distribution while in Figure 1b the temperature series has left a longer tail and a flatter distribution.

For identifying the dry/wet months from the monthly precipitation and the hot/cold months from the time series of monthly temperature, we plotted a matrix of the monthly values for each year.

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Mathematical Modelling in Civil Engineering, no. 4/2014 17

Fig. 2-a) Matrix plot of the monthly precipitations at Fagaras;

b) Matrix plot of monthly temperatures at Fagaras

The package hydroTSM contains many functions that can make the conversion between daily, monthly, annual and seasonal data. Regarding the seasonal data, we obtained the following values:

Table 2

Seasonal analysis for precipitation and temperature

Season Average seasonal values of

precipitation Average seasonal

values of temperature

DJF 87.8 -2.97

MAM 190.4 8.48

JJA 309.5 17.10

SON 151.6 7.94

Analyzing table1, table 2 and figures 1a and 2a, the monthly precipitation repartition reveals that: for the smaller quantities registered in the winter period, from December to February, an average seasonal value of precipitation of 87.8 mm resulted with a minimum of 0.1 mm on February 1976, while in the summer period the average seasonal value of precipitation was 309mm, with a maximum of 230 mm in July 1975.

Analyzing table1, table 2 and figures 1b and 2b the monthly temperature repartition reveals that: smaller temperatures are registered in the winter period, from December to February when we obtain an average seasonal value of temperature -2.970C with a minimum of -9.530C in January 1969, while in the summer period the average seasonal temperature was 17.10C with a maximum of 19.240C in July 1967. The seasonal values of precipitation and temperature show four generally recognized calendar-based seasons.

4.2 Estimate trends using TSA

In order to find the pattern of the time series we make graphical decomposition to see the component for the precipitation time series: trend, seasonal and irregular part. [5] Regarding the linear model for the monthly precipitation and temperature time series given by

equation (6), the coefficient of determination is almost 0, so that a fitting line to these data is not appropriate. In order to see the seasonal variation given by equation (7) we create a vector which contains the twelve parameters of the expected average monthly temperature and precipitation.

2R

1221 ,...,,

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18 Mathematical Modelling in Civil Engineering, no. 4/2014

Table 3 Seasonal Trends

Month Estimate temperature

Estimate precipitation

January -5.2869 29.912 February -1.6348 27.571 March 3.3480 31.259 April 8.3927 60.676 May 13.6636 98.494 June 16.3559 100.541 July 17.7641 128.318 August 17.1630 80.641 September 13.3702 59.800 October 7.9176 52.053 November 2.5594 39.729 December -1.8854 30.376

The calculated R-squared for the average monthly temperature is 0.9765 so we have a very good fitting of data points with this statistical model and one can explain this by the inclination change toward the sun of the Northern Hemisphere. The calculated R-squared for the average monthly precipitation is 0.8019, approximately 80 % of the variation of the precipitation series explained by the seasonal component. The seasonal component [6] does not take into account the differences or similarities between two close periods so we try to model the cyclical component with cosine curves given by equation (8) that incorporates the smooth change expected from one period to the next, while the seasonality is still preserved.

Table 4

Cosine Trend Model

Multiple R-squared for the average monthly temperature is 0.9448 and for the average monthly precipitation is 0.3993. This can be seen in the graphics of the fitting of data.

Fig.3- Cosine trends for the temperature and precipitation

Coefficients Temperature Precipitation

0.1866 7.224

7.644 61.614

-10.9485 -41.4640

1

2

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The best model for precipitation and temperature is the model with the seasonal component.

4.3. Forecasts Using Holt-Winters Exponential Smoothing

For our forecasting experiments we used the first 15 years (1966-1980) of data values as history by using monthly values of precipitation. [11] The estimated values of the parameters with the Holt-Winters method are:

Table 5 Holt-Winters exponential smoothing constants

Smoothing parameters alpha 0.0124 beta 0.0456

gamma 0.1700 Coefficients

a 61.5734 b 0.1589 s1 -25.0646 s2 -28.4599 s3 -22.8383

Coefficients s4 18.7979 s5 39.4857 s6 48.9741 s7 77.9200 s8 38.2382 s9 9.0780

s10 -5.0105 s11 -17.6685 s12 -26.7912

Estimated parameters alpha and beta are very close to zero, so the forecast for the level and the trend are based on further observations in the past. Gamma is relatively low, so this indicates that the estimate of the seasonality at the current point in time is based upon both recent observations and some observations in a more distant past. If the values of parameters are high (close to 1) the estimate of the components are based upon very recent observations.

The forecast were made for the two remaining years (1981-1982) and then the resulting values are compared with those observed for the same period [12]. We obtain 0.77 as a correlation coefficient for the generated and measured data. Since we obtain a strong uphill linear relationship, we can make a prediction for the next 2 years (1983-1984), which is not included in the original time series.

Table 6 The forecast values and prediction interval for the precipitation time series, which corresponds to the

January 1981-December 1982 period

Month Observed Forecast Lo80 Hi80 Lo95 Hi95 Jan-81 31.0 36.7 -13.8 87.2 -40.5 113.9 Feb-81 20.9 33.4 -17.1 83.9 -43.8 110.6 Mar-81 31.8 39.2 -11.3 89.7 -38.0 116.4 Apr-81 52.7 81.0 30.5 131.5 3.8 158.2 May-81 124.0 101.9 51.3 152.4 24.6 179.1 Jun-81 66.3 111.5 61.0 162.0 34.3 188.7 Jul-81 200.4 140.6 90.1 191.1 63.3 217.9

Aug-81 40.2 101.1 50.6 151.6 23.8 178.4 Sep-81 120.1 72.1 21.6 122.6 -5.2 149.4 Oct-81 44.0 58.2 7.6 108.7 -19.1 135.4 Nov-81 51.3 45.7 -4.9 96.2 -31.7 123.0 Dec-81 38.0 36.7 -13.9 87.2 -40.6 114.0 Jan-82 39.9 38.6 -12.9 90.0 -40.1 117.2 Feb-82 27.1 35.3 -16.1 86.8 -43.3 114.0 Mar-82 23.4 41.1 -10.3 92.6 -37.6 119.8 Apr-82 68.3 82.9 31.5 134.4 4.2 161.6 May-82 32.9 103.8 52.3 155.2 25.0 182.5 Jun-82 119.8 113.4 61.9 164.9 34.7 192.1 Jul-82 141.1 142.5 91.0 194.0 63.8 221.3

Aug-82 81.3 103.0 51.5 154.5 24.2 181.8 Sep-82 23.0 74.0 22.5 125.5 -4.8 152.8 Oct-82 16.9 60.1 8.5 111.6 -18.8 138.9 Nov-82 24.4 47.6 -4.0 99.1 -31.3 126.4 Dec-82 22.7 38.6 -13.0 90.2 -40.3 117.5

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20 Mathematical Modelling in Civil Engineering, no. 4/2014

Fig.4 - The graphics of the monthly precipitation time series from January 1966 up to December 1980 (black); Holt-

Winters exponential smoothing (red) for the precipitation time series from January 1966 up to December 1980; Holt-Winters exponential smoothing (green) for the precipitation time series from January 1981 up to December

1982; Holt-Winters exponential smoothing (blue) for the precipitation time series from January 1983 up to December 1984; Shaded areas show 80% (dark grey) and 95% (light grey) prediction intervals.

In order to see if the Holt-Winters exponential smoothing provides an adequate predictive model for the precipitation time series, we make a correlogram in order to check if the residuals show non-zero autocorrelations at lags 1-20. Autocorrelation function indicates how a time series is related to itself over time. [11]

Fig. 4 - The autocorrelation function for the residuals of precipitation time series

The correlogram shows that the autocorrelations for the in-sample forecast errors do not exceed the significance bounds for lags 1-20. From the Ljung-Box test we obtain a p-value of 0.9686 for the lag = 20 so it is above 0.05, which indicates non-significance of autocorrelation values. To see if our forecast is good enough for our purposes and that it cannot be improved, we check if the forecast errors are normally distributed using a histogram. [11] Holt-Winters exponential smoothing method provides an adequate predictive model for the precipitation time series from Fagaras, because the distribution of the forecast errors seems to be normally distributed with mean and zero constant variance over time and the Ljung-Box test showed that there is little evidence of non-zero autocorrelations.

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Mathematical Modelling in Civil Engineering, no. 4/2014 21

Fig. 5- The normal distribution of the forecast errors of the precipitation time series

5. Conclusion

The purpose of this article is to present some results related to the modeling of the hydrologic time series of precipitation and temperature from Fagaras meteorological station in the period 1966-1982 by using the R language. The paper presents the main capabilities of the three R packages: hydroTSM, TSA and forecast with applications. We used the hydroTSM package for the management, analysis and the plots that capture the information about the central tendency, distribution and frequency. In order to describe some deterministic components: linear, seasonal, means and the cosine trends components, we estimate parameters and investigate the efficiency of these regression methods by using the TSA package. By using the forecast package, we make predictions of future events related to precipitation in the Fagaras area based on the Holt-Winters method. We conclude that this method provides an adequate predictive model, which probably cannot be improved more. In a future study, we will try to make a better predictive model by trying other different methods: ARIMA, Markov, or by using artificial intelligence methods.

References

[1].Sayemuzzaman, M., Jha, MK. (2014). Seasonal and annual precipitation time series trend analysis in North Carolina, Atmospheric Research, Volume 137, pp 183–194. http://dx.doi.org/10.1016/j.atmosres.2013.10.012 [2].IPCC Fifth Assessment Report: Climate Change 2013 , http://www.ipcc.ch/report/ar5/ [3].Climate change Romania, http://www.climateadaptation.eu/romania/climate-change/ [4].Raport de mediu–Plan Urbanistic General Municipiul Fagaras, http://www.primaria-fagaras.ro/urbanism/PUG-2013/raport%20mediu%20revizuit%20mai%202013.pdf [5].The Comprehensive R Archive Network. http://cran.r-project.org/ [6].Cryer, J. D., Chan, K-S. (2008).Time Series Analysis with Applications in R, Springer [7].Brockwell, P. J., Davis R. A. (2002). Introduction to Time Series and Forecasting. Springer-Verlag New York, Inc [8].Ljung, G. M.; Box, G. E. P. (1978). On a Measure of a Lack of Fit in Time Series Models. Biometrika 65 (2), pp 297–303. [9].Barbulescu A., Deguenon, J. (2011). Mathematical models for extreme monthly precipitation, Ovidius University Annals, Series: Civil Engineering, issue 13, pp. 93 – 104, http://revista-constructii.univ-ovidius.ro/doc/anale/2011.pdf [10].Cowpertwait, P. S.P. (2006). Introductory Time Series with R, Springer Science+Business media [11].Coghlan, A. (2014). Using R for Time Series Analysis, https://media.readthedocs.org/pdf/a-little-book-of-r-for-time-series/latest/a-little-book-of-r-for-time-series.pdf [12].Miroiu, M., Petrehus, V., Zbaganu G. (2008-2011): Initiere in R pentru persoane cu pregatire matematica, POSDRU/56/1.2/S/32768

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COMPARISON OF TWO-PHASE PRESSURE DROP MODELS FOR CONDENSING FLOWS IN HORIZONTAL TUBES

ALINA FILIP - Lecturer, Dr. Eng., Technical University of Civil Engineering Bucharest, Thermal Engineering Department, e-mail: [email protected] FLORIN BĂLTĂREŢU - Associate Professor, Dr. Eng., Technical University of Civil Engineering Bucharest, Thermal Engineering Department, e-mail: [email protected] RADU-MIRCEA DAMIAN - Professor, Dr. Eng., Technical University of Civil Engineering Bucharest, Department of Hydraulics and Environmental Protection, e-mail: [email protected], [email protected]

Abstract: An important parameter in the hydraulic design of refrigeration and air-conditioning systems is the two-phase flow pressure drop. In this paper, the authors compare the numerical results obtained by using seven two-phase pressure-drop models with the experimental results found in the scientific literature, for the condensation of R600a and R717 (Ammonia = NH3) in horizontal tubes. Different mass flow rates and different conditions have been considered in order to see which correlation is applicable under specific operation conditions.

Keywords: two-phase pressure drop, refrigerant, condensation

Nomenclature

Bo Bond number, – CChisholm parameter, – Dinternal tube diameter or equivalent hydraulic diameter, m fFanning friction factor, – FrFroude number, – ggravitational acceleration, m2·s-1 Gtotal mass flow rate per unit area, kg·s-1·m-2 jsuperficial velocity, m·s-1 ReReynolds number, – WeWeber number, – xvapour mass quality, kg vapour/kg two-phase mixture Greek-Symbols density, kg·m-3 surface tension, N·m-1 dynamic viscosity, kg·s-1·m-1 (= Pa·s) Darcy-Weisbach friction factor, – 2two-phase friction multiplier, –

Subscripts Ggas GChomogeneous gas core GOgas only Lliquid LOliquid only Htwo-phase homogeneous mixture wwall

1. Introduction

In many thermal systems (like power steam or geothermal plants), an important parameter in the design is the pressure drop for the two-phase flow [1]. Another important area of applications concerns the refrigeration and the air-conditioning systems.

The present study uses the two-phase frictional pressure-drop correlations that are frequently used in the corresponding scientific literature [2]. In this respect, the: 1) homogeneous, 2) Lockhart-Martinelli, 3) Friedel, 4) Chen et al., 5) Cavallini, 6) Müller-Steinhagen & Heck and 7) Jung & Radermacher models are used, in order to compare them, to see which correlation is applicable under specific operation conditions.

2. The friction factor

All the models and correlations used for predicting the pressure drop in tubes use the friction factor formulation.

The most common friction factor is the Fanning friction factor, which is defined as the fraction:

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Mathematical Modelling in Civil Engineering, no. 4/2014 23

22uf w

, where w is unit stress at the wall , N·m-2 (1)

It must be mentioned that in some research papers, especially in the heat-transfer literature, the Darcy-Weisbach friction factor is used instead of the Fanning friction factor. In this respect, the fact that the Darcy-Weisbach friction factor is related to the Fanning friction factor must be taken into account, as:

fu

w 42

42

. (2) The equations must be further modified accordingly.

Usually, for normal tubes, the Fanning friction factor can be predicted by using different equations for each regime, as the Hagen–Poiseuille correlation for laminar flow and the Blasius correlation for turbulent flow, with a transition region between the laminar and the turbulent flow:

3000Refor,Re0791.0

3000Re2300for ,Re

2300Refor,Re16

25.0

1

trff

(3)

Fig. 1. Fanning friction factor for smooth tubes

A simple linear approximation can be used as (see figure 1):

2300Re23003000

Re 2300Re,3000Re,2300Re,

lamturblamtr

ffff

(4)

However, it must be noted that each model or correlation uses a specific friction factor correlation or specific limits.

3. Models and correlations analyzed in this study

3.1. The homogeneous model [1, 2]

In the homogeneous model, the two-phase flow is treated as an equivalent single-phase flow, having the specific volume of the mixture (two-phase gas–liquid flow) defined as:

LGH vxxvv 1 (5)

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24 Mathematical Modelling in Civil Engineering, no. 4/2014

and, therefore, the density of the mixture: 1

11

LGHH

xx

v

. (6) The Reynolds number is determined using a homogeneous mixture dynamic viscosity:

H

dG

Re

(7) the dynamic viscosity of the mixture is, after McAdams:

11

LGH

xx

(8) The friction factor is considered as:

00020Refor,Re046.0

00020Re2000for ,Re079.0

2000Refor,Re16

2.0

25.0

1

f

(9)

3.2. The Lockhart-Martinelli separated flow model [3, 4]

Lockhart and Martinelli (1949) [3] proposed the concept of different two-phase friction multipliers (for gas and for liquid), identified as 2

G and 2L , respectively.

The pressure drop is then given by:

GG

LL z

p

z

p

z

p

d

d

d

d

d

d 22 (10)

D

xGf

z

p

L

L

L

22 12

d

d

(11)

D

xGf

z

p

G

G

G

222

d

d

(12) The two-phase friction multipliers were initially given in a graphical form, and then expressed by Chislom [4] as the following dependence:

22 1

1XX

CL

, (13) 22 1 XCXG (14)

where the Lockhart-Martinelli parameter X is defined as: G

L

zp

zpX

dd

dd2 (15)

The Chisholm parameter depends on the specific liquid/gas viscous/turbulent regime:

case)(1500Rand1500Refor,20

case)(1500Rand1500Refor,12

case)(1500Rand1500Refor,10

case)(1500Rand1500Refor,5

TTe

TVe

VTe

VVe

C

GL

GL

GL

GL

(17)

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Mathematical Modelling in Civil Engineering, no. 4/2014 25

The liquid and gas Reynolds numbers are determined by using the corresponding mass flow rate per unit area:

L

L

DxG

1

Re,

G

G

DxG

Re (16)

3.3. The Friedel model [5]

Friedel proposed a liquid-only multiplier for the frictional pressure drop:

2

d

d

d

dLO

LOz

p

z

p

(18) The multiplier is expressed as a function of gas/liquid properties, of vapour mass quality and of gravity and surface tension effects, by using the Froude and Weber numbers:

035.0045.0

7.019.091.0

224.078.0

222

WeFr1124.3

1

tptpL

G

L

G

G

L

LO

GO

G

LLO

xx

f

fxx

(19) The Froude and the Weber numbers are expressed using the homogeneous mixture density:

2

2

FrH

tp gD

G

, Htp

DG

2

We (20)

3.4. The Chen et al. model (2001) [6]

This model introduces the dependence on the Bond and Weber number, as a correction of the homogeneous model:

homhomd

d

d

d

z

p

z

p

(21) where the correction factor is:

.5.2for Bo,Boexp9.0BoexpWe1

,5.2for Bo,Boexp9.02.013.02.0hom

(22) The Bond number is expressed as:

22

BoD

g GL (23)

The Weber number is considered with the homogeneous mixture density:

H

DG

2

We (24)

3.5. The Cavallini model [7]

This model uses the form of the equation (18), however the two phase multiplier correlation is given as a function of vapour and liquid properties, of vapour quality, of reduced pressure Rp and of the entrained liquid fraction E :

WLO EHFZ 1595.32 (27)

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26 Mathematical Modelling in Civil Engineering, no. 4/2014

where:

RpW 398.1 , K

R p

pp (28)

2.0

221

L

G

G

LxxZ

, 414.09525.0 1 xxF (29) 542.344.0132.1

1

L

G

L

G

G

LH

(30)

The friction factor (for surfaces with negligible surface roughness) is calculated as:

2.0Re046.0 LOLOf , for any number L

LO

DG

Re . (26)

The liquid entrainment ratio E is calculated following Paleev and Filippovich (1966) :

4

2

10 10log44.0015.0

GL

L

GC jE

(31)

with the limitations:

95.0then95.0if

0then0if

EE

EE

(32)

The superficial gas velocity Gj is determined as:

GG

xGj

(33)

The homogeneous gas core density is defined also as a function of the liquid entrainment ratio E :

11

1

LGGC

ExxExx

(34)

which means that equations (31) and (34) must be solved together, as a nonlinear equation, using an iterative approach. The correlation applies only if the dimensionless gas velocity 5.2GJ , where the dimensionless

gas velocity is determined as:

(25)

If , the Friedel model is used instead.

3.6. The Müller-Steinhagen & Heck model [8]

The Müller-Steinhagen & Heck approach consider a combination of each phase only flow,

further identified by the subscript .

The corresponding Reynolds number is determined with the total mass flow rate per unit area:

(35)

GLG

GDg

xGJ

5.2GJ

LOGOk ,

kk

DG

Re

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Mathematical Modelling in Civil Engineering, no. 4/2014 27

The friction factor is calculated as:

(36)

The pressure drop for each phase results:

(37)

Finally, the pressure drop for the two-phase flow is evaluated as:

(38)

3.7. The Jung & Radermacher model [9]

This approach considers the pressure drop associated to the liquid phase only flow, corrected by a corresponding multiplier, as expressed by equation (18).

The liquid only flow pressure-drop multiplier is given as a function of the vapor mass quality and of the Lockart-Martinelli turbulent-turbulent parameter:

(39)

The expression for the Lockart-Martinelli turbulent-turbulent parameter is: 5.09.01.0

1

L

G

G

Ltt x

xX

(40)

4. Comparison with experimental data

4.1. Comparison with experimental data of Dalkilic (2010) [10]

First experimental data set used for comparison concerns the annular flow condensation of R600a in a horizontal tube at low mass flux [10].

Table 1

The experimental versus numerical frictional pressure drop, experimental data of Dalkilic (2010)

Parameter satt G avgx expdd zp L-M Friedel Cav MSH JR

Unit C 2ms

kg

- mPa mPa mPa mPa mPa mPa

Exp. # 1 30 85 0.85 850 500 718 753 646 854 Exp. # 2 30 85 0.70 700 654 625 700 546 920 Exp. # 3 30 75 0.90 775 427 598 597 535 610 Exp. # 4 30 75 0.60 625 519 450 508 378 691 Exp. # 5 43 115 0.85 875 785 883 891 785 1129 Exp. # 6 43 115 0.55 475 899 634 703 527 1072 Exp. # 7 43 95 0.85 800 462 640 639 562 808 Exp. # 8 43 95 0.45 450 598 397 427 318 648

L-M: Lockart-Martinelli, Cav: Cavallini, MSH: Müller-Steinhagen & Heck, JR: Jung & Radermacher

The specific conditions, the experimental data, and the numerical results are presented in Table 1.

1187Refor ,Re0791.0

1187Refor,Re1625.0

1

kk

kkkf

D

Gf

z

p

k

k

k

22

d

d

33

d

d1

d

d

d

d2

d

d

d

dx

z

pxx

z

p

z

p

z

p

z

p

GOLOGOLO

8.147.12 182.12 xX ttLO

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28 Mathematical Modelling in Civil Engineering, no. 4/2014

The calculation pressure drop versus the experimental pressure drop is represented in figures 3 and 4, where the ± 30 % error lines are shown.

Fig. 3. Calculation versus experimental pressure drop, R600A Lockart-Martinelli (“ ”) and Friedel (“ ”)

Fig. 4. Calculation versus experimental pressure drop, R600A Cavallini (“ ”) and MSH (“ ”)

Figures 5 and 6 present the variation of the pressure drop with the vapour quality, and with the total mass flow rate per unit area, respectively, giving the possibility to analyze the behavior of the models. A general observation can be made, that is, models usually follow the trend of the experimental data, however in a quite different slope. Another observation concerns the fact that in this application the models under-estimate the experimental pressure-drop. Finally, there are strong connections between parameters, so a multi-criterial approach should be used; however, for such analysis, supplementary parametric variations of the experimental data are needed.

Fig. 5. Variation of the pressure drop with the vapour quality, R600A

Experimental (“ ”), Lockart-Martinelli (“ + ”) and Friedel (“ ”)

Fig. 6. Variation of the pressure drop with the total mass flow rate per unit area, R600A

Experimental (“ ”), Lockart-Martinelli (“ + ”) and Friedel (“ ”)

4.2. Comparison with experimental data of da Silva Lima (2009) [11]

Another experimental data set used for comparison concerns the condensation of R717 (Ammonia = NH3), flowing in a horizontal smooth tube.

The comparisons of the experimental and the numerical results are shown in figures 7-10, for different condensation temperatures and total mass fluxes per unit area.

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Mathematical Modelling in Civil Engineering, no. 4/2014 29

Fig. 7. Calculation versus experimental

pressure drop, R717, case 1

Fig. 8. Calculation versus experimental

pressure drop, R717, case 2

Fig. 9. Calculation versus experimental

pressure drop, R717, case 3

Fig. 10. Calculation versus experimental

pressure drop, R717, case 4

For low values of the vapour mass quality (0.1…0.2 kg vapour/kg two-phase mixture), one can observe that the homogeneous and Chen models are the best suited models. Instead, for the range 0.25…0.35, the best suited models are Friedel, Müller-Steinhagen & Heck and Lockhart-Martinelli. For values of the vapour quality greater than 0.4, the Jung & Radermacher and Cavallini correlations provide the best behavior. On an overall basis in the 0.1…0.6 range, the Cavallini correlation (which completes the Friedel correlation) gives the best approximation of the da Silva experimental data. One can also observe that Lockhart-Martinelli also gives a good overall approximation.

5. Conclusions

The present study concerns the comparison of the numerical results regarding the two-phase pressure-drop in pipes with the experimental results found in the scientific literature, for the condensation of R600A and R717 in horizontal tubes, in different operation conditions. The study reveals the behavior of the considered models and the importance of the value of the vapour mass quality. The overall behaviour of the models has been also analyzed.

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References

[1] Popescu Fl., Andrei V. & Damian R.M. (2005). Dinamica fluidelor polifazice, Galaţi: Editura Fundaţiei Universitare “Dunărea de Jos”.

[2] Kim S.-M., Mudawar I. (2014). Review of databases and predictive methods for pressure drop in adiabatic, condensing and boiling mini/micro-channel flows, International Journal of Heat and Mass Transfer, 77, 74-97. DOI: 10.1016/j.ijheatmasstransfer.2014.04.035.

[3] Lockhart R.W., Martinelli R.G. (1949). Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem. Eng. Prog. 45(1), 39–48.

[4] Chisholm D. (1967). A theoretical basis for the Lockhard-Martinelli correlation for two-phase flow, International Journal of Heat and Mass Transfer, 10, 1767-1778. DOI: 10.1016/0017-9310(67)90047-6.

[5] Friedel L. (1979). Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow, In European Two-Phase Group Meeting, Ispra, Italy, Paper E2.

[6] Chen I.Y., Yang K.-S., Chang Y.J., Wang C.-C. (2001). Two-phase pressure drop of air-water and R-410A in small horizontal tubes, International Journal of Multiphase Flow, Volume 27, 1293-1299. DOI: 10.1016/S0301-9322(01)00004-0.

[7] Cavallini A., Rossetto L., Matkovic M., Del Col D. (2005). A model for frictional pressure drop during vapour–liquid flow in minichannels. In: IIR International Conference Thermophysical Properties and Transfer Processes of Refrigerants, 31 August–2 September, Vicenza, Italy , pp. 71–78.

[8] Müller-Steinhagen H., Heck K. (1986). A Simple Friction Pressure Drop Correlation for Two-Phase Flow in Pipes, Chem. Eng. Process., 20, 297-308. DOI: 10.1016/0255-2701(86)80008-3.

[9] Jung D.S., Radermacher R. (1989). Prediction of pressure drop during horizontal annular flow boiling of pure and mixed refrigerants, Int. J. Heat Mass Transfer, 32, 2435-2446. DOI: 10.1016/0017-9310(89)90203-2.

[10] Dalkilic A.S., Agra O., Teke I., Wongwises S. (2010). Comparison of frictional pressure drop models during annular flow condensation of R600a in a horizontal tube at low mass flux and of R134a in a vertical tube at high mass flux, International Journal of Heat and Mass Transfer, 53, 2052-2064. DOI: 10.1016/j.ijheatmasstransfer.2008.12.001.

[11] da Silva Lima R.J., Quiben J.M., Kuhn C., Boyman T., Thome J.R. (2009). Ammonia two-phase flow in a horizontal smooth tube: Flow pattern observations, diabatic and adiabatic frictional pressure drops and assessment of prediction methods, International Journal of Heat and Mass Transfer, 52, 2273-2288. DOI: 10.1016/j.ijheatmasstransfer.2008.12.001.

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APPROPRIATE CFD TURBULENCE MODEL FOR IMPROVING INDOOR AIR QUALITY OF VENTILATED SPACES

CĂTĂLIN TEODOSIU–Associate Professor, PhD, Technical University of Civil Engineering, Faculty of Building Services engineering, e-mail: [email protected] VIOREL ILIE – PhD Student, Technical University of Civil Engineering, Faculty of Building Services, e-mail: [email protected] RALUCA TEODOSIU – Lecturer, PhD, Technical University of Civil Engineering, Faculty of Building Services, e-mail:[email protected]

Abstract: Accurate assessment of air-flow in ventilated spaces is of major importance for achieving healthy and comfortable indoor environment conditions. The CFD (Computational Fluid Dynamics) technique is nowadays one of the most used approaches in order to improve the indoor air quality in ventilated environments. Nevertheless, CFD has still two main challenges: turbulence modeling and experimental validation. As a result, the objective of this study is to evaluate the performance of different turbulence models potentially appropriate for the prediction of indoor air-flow. Accordingly, results obtained with 6 turbulence models (standard k-ε model, RNG k-ε model, realizable k-ε model, LRN SST k-ω model, transition SST k-ω model and low Reynolds Stress-ω model) are thoroughly validated based on detailed experimental data. The configuration taken into account in this work corresponds to isothermal and anisothermal airflows produced by mixing ventilation systems in small enclosures at low room air changes per hour. In general, the transition SST k-ω model shows the better overall behavior in comparison with measurement values. Consequently, the application of this turbulence model is appropriate for air flows in ventilated spaces, being an interesting option to more sophisticated LES (Large Eddy Simulation) models as it requires less computational resources.

Keywords: turbulence model, CFD, anisothermal jet, ventilation, building

1. Introduction

The basic goal of conditioning enclosed spaces is to supply comfortable and healthy indoor conditions for human beings. This increasingly becomes a vital issue as people spend more time indoors at home, in addition to time spent in shopping malls, theaters, restaurants, vehicles, and other spare time facilities. In fact, recent studies in both Europe and the U.S. clearly show that people spend over 90 percent of their time indoors [1].

On the other hand, it is obvious nowadays that good air quality inside the ventilated spaces cannot be achieved without studies based on modern computational techniques. In line with this, the CFD (Computational Fluid Dynamics) approach is more and more used for analyses concerning: ventilation efficiency for different applications [2-6], indoor air quality for all kind of buildings (e.g. residential [7], offices [8], hospitals [9], museums [10], sport large enclosures [11] or ice skating rinks [12]) and thermal comfort in buildings [13], cars [14], trains [15] or planes [16]. All this is now possible as a result of the remarkable increase in computer hardware capacity in the last years [17].

Despite the fact that the CFD models became useful routinely tools in civil engineering for predicting air movement in ventilated spaces [18], there still are two major challenges. The first one is related to the proper choice of the turbulence model associated to the characteristics of the indoor airflow (e.g. transitional airflow regime, turbulence anisotropy and presence of adverse pressure gradients) [19]. The second major challenge in CFD is the validation of the numerical results [17].

As a result, the objective of this study is to bring new elements concerning the assessment of different turbulence models for airflows produced by mixing ventilation systems within small enclosures at low room air changes per hour. It is worthwhile to mention that the choice of

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this configuration is not random. In fact, this allows us to thoroughly study all the critical features mentioned above for the CFD application of turbulent indoor airflows (confined low Reynolds number airflow, with recirculation regions and boundary layer separation). In addition, the judgment of CFD results is based on experimental data validation using a full-scale test room. This responds to the second major challenge concerning the CFD modeling: the lack of verification and validation, particularly in the case of complex airflows.

Consequently, we first present in a succinct manner the experimental set-up, followed by the description of the turbulence models taken into account and their integration in the CFD modeling. We conclude with comprehensive experimental – numerical comparisons in terms of velocity and temperature fields within the ventilated enclosure.

2. Description of the experimental set-up

This work is entirely based on the experimental investigations fulfilled in [20] on indoor air quality in ventilated rooms. It must be said that this study was preferred as it makes available comprehensive experimental data:

- detailed descriptions of the boundary conditions (temperature and flow rate of the supply air, surface temperature on the inside the walls) - required for the numerical model

- velocity and temperature air distributions inside of the room in a vertical plane normal to the center line of the air terminal devices - required for the model validation.

In addition, the tests taken into consideration (see Table 1) correspond perfectly with our objective: study of airflow for mixing ventilation systems within small enclosures at low room air changes per hour. In order to methodically examine the pertinence of the turbulence models taken into account, the tests selected cover all the situations: cold jet, hot jet and isothermal jet (for different low air flow rates) – according to the mean air room temperature.

Table 1

Experimental configurations [20]

Test T0(0C) ACH (h-1) Ar0 Re0

B2 (hot jet) 31.4 2.06 0.0032 13566 B4 (cold jet) 9.7 2.02 0.0042 13999 B5 (isothermal jet) 22.3 1.06 0.0005 7106

The physical model is a full-scale test room. The air supply terminal is represented by a commercial diffuser (a grille having an aspect ratio of 12.5), which was placed after a plenum (Fig. 1). Detailed descriptions of the test room and the diffuser are given in [20].

3. CFD model

In view of the fact that this study is focused on the turbulence modeling rather than the CFD model itself, we present only the main characteristics of the numerical approach (see Table 2). All the numerical investigations presented in this work are based on a general-purpose, finite-volume, Navier-Stokes solver (Fluent 15.0.0).

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Fig. 1 - Test room and its mixing ventilation system

Table 2

Numerical model principal elements and hypothesis

Feature Description Air Newtonian fluid; incompressible; constant viscosity Flow Three-dimensional; steady; non-isothermal; turbulent Computational domain discretization

Finite volumes; unstructured mesh (tetrahedral elements); optimum mesh size (grid independent solutions); 1300000-1400000 cells

Turbulence model see section 4: Turbulent air flow modeling Near wall treatment k-ε models: two-layer model with enhanced wall functions; k-

ω models: no need for special treatment (low Reynolds number corrections)

Numerical resolution Segregated implicit solver; diffusion terms: second order central-difference scheme; convective terms: second order upwind schemes; velocity-pressure coupling: SIMPLE algorithm; convergence acceleration: algebraic multigrid (gradient method, Green-Gauss cell based)

Air supply boundary conditions Velocity – fixed value across the diffuser (ratio of the measured air flow rate to the diffuser free area); temperature – uniform value (based on experimental data); turbulence quantities – uniform specification, defining two parameters (turbulence intensity and hydraulic diameter) for all the turbulence models

Air exhaust boundary conditions Longitudinal exit velocity from mass balance; transverse velocity components are set to zero; gradients normal to flow direction of the other variables are also set to zero

Wall boundary conditions Velocity – no slip boundary conditions; temperature – fixed values at wall internal surfaces (based on experimental values)

Concerning the construction of the mesh, the size of the smallest cell in the domain is 9.7 x 10-10 m3 (positioned obviously in the first “chain” of cells within the boundary layer), while the biggest cell is 8.47 x 10-5 m3. Consequently, the ratio between the smallest and biggest cell in the computational domain is 1.15 x 10-5.

In addition, we present in Table 3 the values of non-dimensional wall distance (y+) for the boundary layers of the test room walls. This allows us to estimate the grid suitability for near wall treatment of the flow, in the case of k-ε turbulence models used in conjunction with the approach based on two-layer model. It should be said that the values in Table 3 are representative for all k-ε turbulence models taken into consideration and for all configurations (hot jet, cold jet and isothermal jet).

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Table 3 y+ values, B2 test (hot jet), realizable k-ε turbulence model

Wall* y+ minimum value y+ maximum value y+ mean value east 0.5 18.0 5.2

north 0.6 47.0 10.2 west 0.5 16.7 5.5 south 0.4 18.9 7.9

ceiling 0.3 54.8 9.1 floor 0.5 15.1 6.5

* The surface behind the air supply is considered to be the “south” wall of the test room (the identification of the other walls for the test room is based on this assumption).

4. Turbulent airflow modeling

As stated previously, the airflow within ventilated spaces is extremely complex. This imposes important challenges on turbulence modeling when one wants to use the CFD approach for predicting the convection indoor airflows. In fact, it is hard to have only one turbulence model able to manage all the characteristics of the airflow in ventilated enclosures in an optimal and efficient manner [18]. Consequently, the choice of a turbulence model for the precise calculation of the airflow in ventilated spaces is all the time a compromise between accuracy, hardware resources and computational time. On the other hand, the selection of the right turbulence model depends also on the objective of the analysis. For instance, it is known that for the design and improvement of the airflow in ventilated enclosures the mean air parameters are more useful than the turbulent characteristics of the airflow. Taking all these factors into account, the CFD prediction of the turbulent airflows that occur in enclosed environments can be theoretically performed through three main methods: direct numerical simulation (DNS), large-eddy simulation (LES) and Reynolds-averaged Navier-Stokes (RANS) – divided into two principal types, eddy-viscosity models and Reynolds-stress models [21]. It is worthwhile to note that all these approaches were / are / will be intensely used to predict the air distribution in ventilated enclosures. Nevertheless, the DNS application for complex indoor airflows is impossible now because it demands an extremely fine grid resolution, which leads to unreasonable calculations for the existing computers, in spite of recent advances in the field [22]. The LES approach has been increasingly applied to study airflows in enclosed environments in the last decade. However, the storage and execution time of the LES models are very expensive for real scale 3D indoor flows and their accuracy may not always be the highest [23].

Consequently, in order to accomplish the objective of our study, we performed numerical investigations taking into account several turbulence models based on the RANS approach. These models are detailed below, their integration within the CFD model being performed according to the data from Table 2. For all turbulence models taken into account in this study, the default model constants are used, which are not mentioned here for the sake of brevity.

4.1. RANS Eddy-Viscosity Models

4.1.1. k-ε two-equation model

This turbulence model was the most used and probably the most popular between 1980s and 2000s. The standard k-ε model [24] is based on transport equations for the turbulence kinetic energy (Eq. 1) and its dissipation rate (Eq. 2).

(1)

(2)

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The turbulent viscosity is calculated by using the turbulence kinetic energy and its dissipation rate as follows:

(3)

where Cμ is a constant (0.09).

The standard k-ε model is based on the assumption that the flow is fully turbulent [18], therefore this turbulence model was developed for high Reynolds number flows [22]. As a result, despite its success for numerous engineering applications, the use of standard k-ε model for low Reynolds airflows in enclosed environments leads to unsatisfactory results [25].

4.1.2. RNG k-ε two-equation model

This turbulence model is based on the renormalization group theory [26]. This results in different constants from those in the standard k-ε model. Moreover, there are additional terms and functions in the transport equations for the turbulence kinetic energy (Eq. 4) and its dissipation rate (Eq. 5).

The turbulent viscosity is computed in this case using a differential equation:

.72 6

where:

Eq. (6) allows the model to improve the prediction of the low Reynolds number and the near-wall flows as the turbulent viscosity varies effectively with the flow eddy scale, depending on the flow characteristics. It is worthwhile to mention that Eq. (6) becomes analogous to Eq. (3), with Cμ = 0.0845 (very close to the value of 0.09 used in the standard k-ε model) for high Reynolds number flows.

The RNG k-ε model has been extensively used for indoor airflows for different configurations [22]. The results agreed generally rather well with the experimental data but there are also several reports about weak performance [19].

4.1.3. Realizable k-ε two-equation model

According to numerous studies [27,28], the implementation of the realizable k- model [29] in comparison with the standard k- model for flows including boundary layers under strong adverse pressure gradients, separation or recirculation provided superior results. This is supposed to be caused by a new formulation concerning the eddy viscosity and a new model dissipation rate equation, too. In fact, the eddy viscosity is computed using the same equation as in other k- models (see Eq. 3) but the major difference is that the coefficient Cμ is no longer constant. Its

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value is a function of the mean strain and rotation rates, as well as of the turbulence parameters, the turbulence kinetic energy and its dissipation rate. The complete formulation is given in [29]. The modeled transport equation for the dissipation rate of the turbulent kinetic energy is based on the mean square vorticity fluctuation dynamic equation [29]:

7

The second term on the right hand side of Eq. (7) does not involve anymore the turbulence kinetic energy production as the other k- models. This can lead to more appropriate turbulence length scale descriptions. On the other hand, the transport equation for the turbulence kinetic energy is exactly the same compared to the classical k- model (see Eq. 1).

4.1.4. LRN SST k-ω two-equation model

The k-ω models are based on transport equations for the turbulence kinetic energy (Eq. 8) and the turbulence frequency, or specific dissipation rate (Eq. 9).

г 8

г 9

As a result, the turbulent viscosity is computed from these scalars (Eq. 10).

∗ 10

where the coefficient * introduces low Reynolds number (LRN) corrections.

The shear-stress transport (SST) k-ω model [30] is based on similar forms for the Eqs. (9) and (10). Nevertheless, this model introduces a damped cross-diffusion derivative term in the specific dissipation rate equation. In addition, there is a modified turbulent viscosity formulation to take into account the transport effects of the turbulent shear stress. These features make the SST k-ω model more appropriate for adverse pressure gradient flows than the standard k-ω model. Consequently, the LRN SST k-ω model has a good potential for predicting indoor environment flows [21].

4.1.5. Transition SST k-ω four-equation model

This recently developed turbulence model is based on the coupling of the SST k-ω model transport equations with two other transport equations [31]: one for the intermittency (Eq. 11) and one for the transition momentum thickness Reynolds number (Eq. 12).

11

12

Unfortunately, no study has been found in the literature on the use of this new turbulence model in indoor environments. As a result, this work makes available extremely valuable data on the application of the transition SST k-ω model for ventilated spaces.

4.2. RANS Reynolds-stress models

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The Reynolds-stress models (RSM) are generally based on 7 equations: six transport equations for the Reynolds stresses and one transport equation for a turbulent quantity (the dissipation rate of turbulence energy or the turbulence frequency). Consequently, the RSM models allow the “natural” development and transport of individual Reynolds stresses, which allow taking into account the anisotropy of turbulent flows. These anisotropic effects play an important role in flows with significant buoyancy, streamline curvature, swirl or strong circulation [21]. The correct prediction of these effects leads normally to more accurate results for complex indoor airflows compared with two-equation turbulence models [18].

4.2.1. Low Reynolds Stress-ω model

The RSM model selected in this study to solve transport equations for the individual Reynolds stresses is the low-Reynolds Stress-ω by Wilcox [32]. This model is based on the transport equation for the specific dissipation rate (Eq. 9) and the Launder-Reece-Rodi (LRR) stress-transport model. The model closure coefficients are identical to the k-ω model. However, the low-Reynolds Stress-ω model requires additional closure coefficients. The comprehensive physical-mathematical formulation of the model can be found in [32]. It is worthwhile to mention that this model was used with good results for the prediction of room air movement induced by a wall jet [19].

5. Results

The comparisons between numerical results and experimental values are exposed in terms of air mean velocity and temperature profiles in a median vertical plane for three sections located at different distances from the coordinate system presented in the Fig. 1. The exact position of these rakes is shown in Fig. 2.

It is worthwhile to mention that we focus our validation on air mean velocity and air temperature as these parameters represent the main issues to assess the efficiency of ventilation systems - deeply related to indoor air quality and thermal comfort in ventilated spaces.

We first present in Fig. 3 the data for the isothermal situation (velocity profiles). We notice that the jet region (including its spread) is correctly predicted by the k-ε realizable model in the first section (at x = 1 m) while there are 3 turbulence models (k-ε realizable model, transition SST k-ω and low Reynolds stress-ω model) with good performance at x = 1.8 m and x = 2.7 m.

Fig. 2 – Results sections

Concerning the comparisons between predicted and measured values for the hot jet, the velocity profiles and temperatures profiles are shown in Figs. 4 and 5, respectively. In the closest section

z

x

Air supply

(x = 0.42 m)

x = 1.0 m

x = 1.8 m

Median vertical plane

x = 2.7 m

o Air exhaust

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from the air supply (x = 1 m), the low Reynolds stress-ω model has slightly the best velocity predictions. Nevertheless, as a general remark, the k-ω models (low Reynolds number shear-stress transport – LRN SST and transition SST) show the best agreement with measurements both for velocity and temperature.

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Fig. 3 –Velocity profiles (B5 - isothermal jet test)

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Fig. 4 – Velocity profiles (B2 – hot jet test)

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Fig. 5 – Temperature profiles (B2 – hot jet test)

For the more complex airflow, which occurs in the case of a cold jet supplied in the room (Figs. 6 and 7), there is now a turbulence model capable to predict the overall flow pattern in the enclosure. However, there are 3 models that lead to results in better agreement with experimental data: transition SST k-ω, low Reynolds stress-ω and k-ε realizable (the last one only in the section at x = 2.7 m).

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Fig. 6 – Velocity profiles (B4 – cold jet test)

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Fig. 7 – Temperature profiles (B4 – cold jet test)

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6. Conclusions

Despite the expectations, none of the turbulence models taken into consideration provide entirely superior results for the mean velocity and temperature fields in the ventilated test room. We include here as well the Reynolds-stress model although the results obtained with this model show in general fair agreement with experimental data. Nevertheless, the prediction of the airflow for all the 3 configurations taken into account based on the Reynolds-stress turbulence model deviates sometimes a lot from the measurements. The same comment is found in [19].

On the other hand, from the results it is noted that the k-ω models (especially the transition SST model) have the best average overall performance in comparison with the measurements, no matter the configuration (isothermal jet, hot jet or cold jet). This indicates that the k-ω models present a good potential to model indoor airflow in ventilated spaces even if the studied configuration is highly complex due to the presence of transitional flow, adverse pressure gradient, and a wall jet that is basically anisotropic. The results of this study show that the transition SST k-ω model clearly improves the predictions concerning the temperature distributions, based usually on k-ε models. This theory is confirmed also by [21, 33].

In conclusion, the new transition SST k-ω four-equation model should be methodically applied for indoor environments with complex airflows in future work in order to better validate its performance. This turbulence model can represent an interesting alternative to LES turbulence models for indoor airflow as it demands less computational resources.

Nomenclature

ACH air changes per hour, h-1 Ar0 Archimedes number (based on inlet dimensions) C1, C2, C1, C2, C3, C, C

k- models coefficients

E1, E2 transition SST k- source terms Gb, Gk, G k- and k- source terms k turbulent kinetic energy, m2/s2 P1, P2, Pt transition SST k- source terms Re0 Reynolds number (based on inlet dimensions) Ret transition momentum thickness Reynolds number R RNG k- model source term for RNG theory S, Sk, S, S k- and k- models user source terms t time, s T0 inlet air temperature (C) ui,j air velocity components, m/s xi,j coordinates YK, YM, Y k- and k- models source terms Greek symbols * LRN SST k- model LRN corrections coefficient K, RNG k- model – inverse effective Prandtl numbers for k and dissipation of turbulent kinetic energy, m2/s3 K, LRN SST k- model – effective diffusivity for k and intermittency laminar (molecular) viscosity eff effective viscosity t turbulent (eddy) viscosity cinematic viscosity density, kg/m3 K, , , t k- model and transition SST k- model coefficients turbulence frequency (specific turbulence dissipation rate), s-1

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Acknowledgments

This work was supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-II-ID-JRP-RO-FR-2012-0071.

References

[1] Hancock, T. (2002). Built Environment (Encyclopedia of Public Health). Retrieved June 19, 2014 from http://www.encyclopedia.com/doc/1G2-3404000130.html

[2] Rota R., Canossa L. & Nano G. (2001). Ventilation design of industrial premises through CFD modelling. Canadian Journal of Chemical Engineering. 79(1), 80-86.

[3] Kaji H., Akabayashi S.I. & Sakaguchi J. (2009). CFD analysis for detached house: Study on the ventilation efficiency on constantly ventilated house part 1. Journal of Environmental Engineering. 74(636), 161-168

[4] Papanikolaou E., Venetsanos A.G., Cerchiara G.M., Carcassi M. & Markatos N. (2011). CFD simulations on small hydrogen releases inside a ventilated facility and assessment of ventilation efficiency. International Journal of Hydrogen Energy. 36(3), 2597-2605

[5] Kwon K.S., Lee I.B., Han H.T., Shin C.Y., Hwang H.S., Hong S.W., Bitog J.P., Seo I.H. & Han C.P. (2011). Analysing ventilation efficiency in a test chamber using age-of-air concept and CFD technology. Biosystems Engineering. 110(4), 421-433.

[6] Zhai Z.Q. & Metzger I.D. (2012). Taguchi-Method-Based CFD Study and Optimisation of Personalised Ventilation Systems. Indoor and Built Environment. 21(5), 690-702.

[7] Yang L., Ye M. & He B.J. (2014). CFD simulation research on residential indoor air quality. Science of the Total Environment.472, 1137-1144.

[8] Zhuang R., Li X. & Tu J. (2014). CFD study of the effects of furniture layout on indoor air quality under typical office ventilation schemes. Building Simulation. 7(3), 263-275.

[9] Helmis C.G., Adam E, Tzoutzas J, Flocas H.A., Halios C.H, Stathopoulou O.I., Assimakopoulos V.D., Panis V., Apostolatou M. & Sgouros G. (2007). Indoor air quality in a dentistry clinic. Science of the Total Environment. 377(2), 349-365.

[10] Corgnati S.P. & Perino M. (2013). CFD application to optimise the ventilation strategy of Senate Room at Palazzo Madama in Turin (Italy). Journal of Cultural Heritage. 14(1), 62-69.

[11] Stathopoulou O. I. & Assimakopoulos V. D. (2008). Numerical Study of the Indoor Environmental Conditions of a Large Athletic Hall Using the CFD Code PHOENICS. Environmental Modeling & Assessment. 13(3), 449-458.

[1] Yang C., Demokritou P.,Chen Q., Spengler J. & Parsons A. (2000). Ventilation and air quality in indoor ice skating arenas. ASHRAE Transactions, 106, p. 338

[2] Cheng Y., Niu, J. & Gao N. (2012). Thermal comfort models: A review and numerical investigation. Building and Environment. 47(1), 13-22.

[3] Lombardi G., Maganzi M., Cannizzo F. & Solinas G. (2009). CFD simulation for the improvement of thermal comfort in cars. Auto Technology. 9(2), 52-56.

[4] Chen N., Liao S. & Rao Z. (2012). CFD evaluation on the temperature field and thermal comfort of coach in low atmospheric pressure passenger trains with oxygenation at high altitudes. China Railway Science. 33(4), 126-132.

[5] Sun H., An L., Feng Z. & Long Z. (2014).CFD simulation and thermal comfort analysis in an airliner cockpit. Journal of Tianjin University Science and Technology. 47(4), 298-303.

[6] Li Y. & Nielsen P.V. (2011).CFD and ventilation research. Indoor Air. 21, 442-453. [12] Sørensen D.N. & Nielsen P.V. (2003). Quality control of computational fluid dynamics in indoor environments.

Indoor Air. 13, 2-17. [13] van Hooff T. Blocken B. & van Heijst G.J.F. (2013). On the suitability of steady RANS CFD for forced mixing

ventilation at transitional slot Reynolds numbers. Indoor Air. 23, 236-249. [14] Castanet S. (1998). Contribution to the study of ventilation and indoor air quality. Doctoral dissertation, INSA

de Lyon, Villeurbanne, France. [15] Stamou A. & Katsiris I. (2006). Verification of a CFD model for indoor airflow and heat transfer. Building and

Environment. 41, 1171-1181. [16] Zhai Z., Zhang Z., Zhang W. & Chen Q.Y. (2007). Evaluation of Various Turbulence Models in Predicting

Airflow and Turbulence in Enclosed Environments by CFD: Part 1 – Summary of Prevalent Turbulence Models. HVAC&R Research. 13(6), 853-870.

[17] Zhang Z, Zhang W., Zhigiang J.W. & Chen Q.Y. (2007).Evaluation of Various Turbulence Models in Predicting Airflow and Turbulence in Enclosed Environments by CFD: Part 2—Comparison with Experimental Data from Literature. HVAC&R Research. 13(6), 871-886.

[18] Launder B.E. & Spalding D.B. (1972). Lectures in Mathematical Models of Turbulence. London, England: Academic Press

[19] Schälin A. & Nielsen P.V. (2004). Impact of turbulence anisotropy near walls in room air flow. Indoor Air. 14(3), 159-168.

Page 46: Modelling Nr4 2014

46 Mathematical Modelling in Civil Engineering, no. 4/2014

[20] Yakhot V. & Orszag S.A. (1986). Renormalisation group analysis of turbulence. Journal of Science Computing. 1(1), 3-51.

[21] Teodosiu C., Rusaouen G. & Hohotă R. (2003). Influence of boundary conditions uncertainties on the simulation of ventilated enclosures. Numerical Heat Transfer, Part A: Applications – An International Journal of Computation and Methodology. 44, 483-504.

[22] Hussain S. & Oosthuizen P.H. (2012). Validation of numerical modeling of conditions in an atrium space with a hybrid ventilation system. Building and Environment. 52, 152-161.

[23] Shih T.H., Liou A., Shabbir A., Yang Z. & Zhu J. (1995). A new k-ε Eddy-Viscosity Model for High Reynolds Number Turbulent Flows - Model Development and Validation. Computers Fluids. 24(3), 227-238.

[24] Menter F.R. (1994). Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA Journal. 32(8), 1598-1605.

[25] Langtry R.B. & Menter F.R. (2009). Correlation-Based Transition Modeling for Unstructured Parallelized Computational Fluid Dynamics Codes. AIAA Journal. 47(12), 2894-2906.

[26] Wilcox D.C. (1998). Turbulence Modeling for CFD. La Canada, California, USA: DCW Industries, Inc. [27] Hussain S., Oosthuizen P.H. & Kalendar A. (2012). Evaluation of various turbulence models for the prediction

of the airflow and temperature distributions in atria. Energy and Buildings. 48, 18-28.