ENTROPY GENERATION AND THERMAL PERFORMANCE OF A PULSATING HEAT PIPE. A Thesis. presented to. the Faculty of the Graduate School

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1 ENTROPY GENERATION AND THERMAL PERFORMANCE OF A PULSATING HEAT PIPE A Thesis resented to the Faculty of the Graduate School at the University of Missouri-Columbia In Partial Fulfillment of the Requirements for the Degree Master of Science by SEJUNG KIM Dr. Yuwen Zhang, Thesis Suervisor DECEMBER 2012

2 The undersigned, aointed by the dean of the Graduate School, have examined the [thesis or dissertation] entitled resented by Sejung Kim, ENTROPY GENERATION AND THERMAL PERFORMANCE a candidate for the degree of master of science, OF A PULSATING HEAT PIPE and hereby certify that, in their oinion, it is worthy of accetance. Professor Yuwen Zhang Professor Chung-Lung Chen Professor Stehen Montgomery-Smith

3 ACKNOWLEDGEMENTS I would like to exress the deeest areciation to my advisor, Dr. Zhang for his continued suort and teaching throughout the roject. Without his guidance and ersistent hel this thesis would not have been ossible. I would also like to areciate the members of my thesis committee, Dr. Chen and Dr. Montgomery-Smith for taking time to rovide helful suggestions and comments. I would also like to thank all my colleagues, Hamid R. Seyf, Nazia Afrin, Yijin Mao, Yuneng Ren, Pengfei Ji, and Daofei Li. It is thankful for everything and really good time to work with them. I would like to exress my gratitude to my arents, Young-seung Kim and Hyo-Sook Ahn, who suort and give me the strength that I need to comlete this work. I would not been able to finish my master s degree. Most imortantly, I would like to thank my wonderful wife, Minsun who stand by me and suort me in the difficult time and believed me even when I did not believe in myself. Without all the love, sacrifice and devotion I have received these 2 years from her, I would have never been finished to successfully reach my study. There are many eole and others I should mention. I would also like to thank them for raying and encouraging me. I really want to say them: Thank you all. Above all, I would like to thank Almighty God who made this thesis ossible and an unforgettable exerience with good eole for me ii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS... ii LIST OF FIGURES... vi LIST OF TABLE... ix NOMENCLATURES...x ABSTRACT... xiii CHAPTER 1 INTRODUCTION Background Mechanism of Pulsating Heat Pies Analysis and Modeling of Pulsating Heat Pie Thesis Objectives Entroy Generation with Initial Temerature and Pressure Loss at Bends Random Noise of Wall Temerature Volume Fractions and Particle Diameters of Nanofluids...5 CHAPTER 2 ANALYSIS OF ENTROPTY GENERATION IN A PULSATING HEAT PIPE Physical model Problem Descrition Governing Equations for Pulsating Flow Latent and sensible heat transfer Entroy Generation of Two Vaor Plugs Entroy generation of liquid slug Entroy generation of latent heat and sensible heat in heating and cooling sections Numerical Alication...12 iii

5 2.3 Results and Discussions The effect of the entroy generation with the initial temerature The effect of the ressure loss at the bend on the entroy generation Conclusions...34 CHAPTER 3 Random Noise of Wall Temerature Physical model Problem Descrition Governing Equations and Random noise inut of wall temerature for Pulsating Flow Result and Discussion Effect of random temeratures, TH and TC, with different amlitudes of the heating and the cooling section Effect of random temeratures, TH and TC, for differnet frequency Effect of changes for different standard deviation Conclusions...54 CHAPTER 4 NANOFLUID EFFECT ON THE THERMAL PERFORMANCE Physical model Problem Descrition Governing Equations for Pulsating Flow Latent and sensible heat transfer Numerical Procedure Results and Discussion Conclusion...76 CHAPTER 5 CONCLUSIONS...77 iv

6 REFERENCES...78 PUBLICATIONS...81 v

7 LIST OF FIGURES Figure... Page Figure 1 Configuration of ulsating heat ie...6 Figure 2 Entroy generation in two vaor lugs...15 Figure 3 Entroy generation in liquid slugs...16 Figure 4 Entroy generation due to latent heat in vaor lug Figure 5 Entroy generation due to latent heat transfer in vaor lug Figure 6 Entroy generation due to sensible heat in liquid slug...20 Figure 7 Entroy generation due to friction in liquid slug...21 Figure 8 Total entroy generation at different initial temeratures...22 Figure 9 Entroy generation with or without ressure loss at bend in the vaor lug Figure 10 Entroy generation with or without ressure loss at bend in the vaor lug 2.25 Figure 11 Entroy generation with or without ressure loss at bend in the liquid slug...26 Figure 12 Entroy generation due to evaoration with or without ressure loss in vaor lug Figure 13 Entroy generation due to condensation with or without ressure loss in vaor lug Figure 14 Entroy generation due to evaoration with or without ressure loss in vaor lug Figure 15 Entroy generation due to condensation with or without ressure loss in vaor lug Figure 16 Entroy generation due to sensible heat into liquid slug with or without ressure loss...31 Figure 17 Entroy generation due to sensible heat out of liquid slug with or without ressure loss...32 vi

8 Figure 18 Entroy generation due to friction in liquid slug with or without ressure loss...33 Figure 19 Total entroy generation with or without ressure loss at bend...34 Figure 20 Fluctuation of heating and cooling section temeratures for different A H and A C...38 Figure 21 Liquid slug dislacements for different A H and A C...39 Figure 22 Temerature of vaor lug 1 for different A H and A C...40 Figure 23 Pressure of vaor lug 1 for different A H and A C...41 Figure 24 Latent heat transfer of vaor lug 1 for different A H...42 Figure 25 Latent heat transfer of vaor lug 1 for different A C...43 Figure 26 Sensible heat transfer of the liquid slug for different A H...45 Figure 27 Sensible heat transfer of the liquid slug for different A C...46 Figure 28 Temeratures of the heating and cooling section for different frequencies...47 Figure 29 Liquid slug dislacements for different frequencies...48 Figure 30 Temerature and ressure of vaor lug 1 for different frequencies...49 Figure 31 Latent heat transfer of vaor lug 1 for different frequencies...50 Figure 32 Sensible heat transfer of the liquid slug for different frequencies...51 Figure 33 Temerature of heating and cooling sections for different for σ H and σ C...52 Figure 34 Liquid slug dislacements for different for σ H and σ C...53 Figure 35 Configuration of U-shae ulsating heat ie with nanofluid with adding nanofluid...56 Figure 36 Effect of nanoarticles on liquid slug dislacement for d =38.4 nm...62 Figure 37 Effect of nanoarticles on temerature of two vaor lugs for d = 38.4 nm...63 Figure 38 Effect of nanoarticles on ressure of two vaor lugs for d =38.4 nm...64 vii

9 Figure 39 Effect of nanoarticle on sensible heat transferred into the liquid slug for d = 38.4 nm...66 Figure 40 Effect of nanoarticles on latent heat of vaor lug 1 for different volume fraction at d = 38.4 nm...67 Figure 41 Effect of nanoarticles on latent heat of vaor lug 2 for different volume fraction at d = 38.4 nm...68 Figure 42 Effect of nanoarticle size on liquid slug dislacement for volume fraction of 4%...69 Fig.43 Effect of nanoarticle size on temerature of vaor lugs for volume fraction of 4%...70 Figure 44 Effect of nanoarticle size on ressure of vaor lugs for volume fraction of 4%...71 Figure 45 Effect of nanoarticle size on sensible heat of the liquid slug...73 Figure 46 Effect of nanoarticle size on latent heat vaor lug Figure 47 Effect of nanoarticle size on latent heat of vaor lug viii

10 LIST OF TABLE Table Page Table 1. Total heat transort rate and contribution of latent and sensible heat transfer...76 ix

11 NOMENCLATURES Area, m A 2 A H A C c c v d f g h( h lsen, hlv) Amlitude of heating section Amlitude of cooling section Secific heat at constant ressure, J/kg-K Secific heat at constant volume, J/kg-K Diameter of heat ie, m Frequency, Hz Gravity, m/s 2 Convection heat transfer coefficient of liquid slug, W/m 2 -K k ( K ) Thermal conductivity, W/m 2 -K K L L m m Nu Q Q in,v1 Q out,1 v Q in, v 2 Q out, v 2 Q in, s, l Q out, s, l r Loss coefficient Length, m Length of liquid slug, m Mass of vaor lugs, kg Mass flow rate, kg/s Nusselt number Vaor ressure, Pa Heat caacity, kj Evaoration heat transfer rate of left vaor lug, W Condensation heat transfer rate of left vaor lug, W Evaoration heat transfer rate of right vaor lug, W Condensation heat transfer rate of right vaor lug, W Sensible heat transfer into liquid slug, W Sensible heat transfer out of liquid slug, W Curvature radius of bend, mm x

12 R Re RN s S t T v x Gas constant of vaor, kj/kg-k Reynolds number Random number Secific entroy, kj/kg-k Entroy, kj/k Time, s Temerature of liquid slug, K Velocity of liquid slug, m/s Dislacement of liquid slug, m Greek symbols Thermal diffusivity, m 2 /s Ratio of secific heats Difference Density, kg/m 3 Standard deviation Angular frequency Phase angle (shift) and Volume fraction Ratio of a circle's circumference Normal distribution Mean value Shear stress, N/m 2 Subscrits 0 B c(c) Initial condition Bend Condensation(Cooling section) xi

13 e(h) eff f l v W Evaoration(Heating section) Effective nanofluid Base fluid Liquid Nanoarticle Vaor lug Wall xii

14 ABSTRACT A U-shaed ulsating heat ie is an excellent heat transfer erformance device. This study has been investigated ste by ste. a) The entroy generation is based on the second law of thermodynamics. In the resent study, the entroy generation in a U-shaed Pulsating Heat Pie (PHP) is numerically investigated. The following five arameters, which are vaor mass, liquid temerature, latent heat, sensible heat, and friction, determine the entroy generation. The results show that the entroy generation is significantly affected by the initial temerature in the PHP. Particularly, the variation of the vaor mass is a rimary factor of the entroy generation. On the other hand, the amlitude of the entroy generation is barely related with the ressure loss at the bend in the PHP. However, the frequency of the entroy generation with the ressure loss is faster than that without the ressure loss at the bend. b) Pulsating heat ie is a two-hase heat transfer device that transfers heat from heating section to cooling section via oscillatory liquid-vaor two-hase flow. The temeratures of heating and cooling sections are extremely imortant arameters, and lay significant roles for the erformance of ulsating heat ies. The objective of this work is to study the effects of fluctuations of heating and cooling section temeratures on the oscillatory flow, temerature and ressure of the vaor lugs, as well as latent and sensible heat transfer of a ulsating heat ie. The fluctuations of wall temeratures include a eriodic comonent and a random comonent. The eriodic comonent is characterized by the amlitude and frequency, while the random comonent is described by the standard deviation. The erformance of the ulsating heat ie is evaluated at various amlitudes, frequencies and standard deviations of the fluctuations. xiii

15 c) A numerical study is erformed to investigate heat transfer erformance and effect of nanofluids on a ulsating heat ie (PHP). Pure water is emloyed as the base fluid while Al 2 O 3 with two different article sizes, 38.4 and 47 nm, is used as nanoarticle. Different arameters including dislacement of liquid slug, vaor temerature and ressure, liquid slug temerature distribution, as well as sensible and latent heat transfer in evaorator and condenser are calculated numerically and comared with the ones for ure water as working fluid. The results show that nanofluid has significant effect on heat transfer enhancement of the system and with increasing volume fraction and decreasing articles diameter the enhancement intensifies. xiv

16 CHAPTER 1 INTRODUCTION 1.1 Background Problems of cooling electronic devices such as deskto and lato comuters are accounted for a significant roortion in design. This is because the roducts could have a malfunction when the imortant art like CPU generates a lot of heat. In addition, the smaller the size of the electronics is, the greater the imortance of heat removal becomes. More heat must be removed out of the area as the size of the area becomes reduced with the same amount of heat. The field of heat transfer has been evolved a lot as the electronics industries have been growing. Many researchers began to develo a two-hased control device, with rising interest in heat transfer. In the 1990 s, Akachi [1] has registered a atent for a new tye of heat ie called the ulsating heat ie (PHP). The features of PHP are that, unlike the conventional heat ie, there is no wick structure and this leads to much lower costs. 1.2 Fundamental of the U-shaed Pulsating Heat Pie on this Thesis Figure 1 shows the configuration of the U-shaed PHP under consideration. The evaorator and the condenser are located in the to and the bottom of the U-shaed PHP, resectively; the vaor lugs 1 and 2 are located on the left and the right sides as shown in the Fig. 1 (a), resectively. There is a liquid slug between the two vaor lugs, and the two ends of the PHP are sealed. With ressure at the bend considered emirically [18], the U-shaed PHP can be considered as an equivalent straighten heat ie in the modeling (see Fig. 1(b)). The dislacement of liquid slug is reresented by x, which can be negative or ositive when the liquid slug moves to the left side or the right side. The driving force of the oscillatory flow is the ressure difference between the two vaor lugs. The liquid slug is considered to be incomressible, and the vaor is assumed to behavior like ideal gas. The heat conduction of the liquid slug is assumed to be one dimension along the axial direction. 1

17 1.3 Analysis and Modeling of Pulsating Heat Pies In the ast two decades, alications of heat ie were focused on the urose of cooling an electronic system. The research on the heat ie begun in 1960 s as several scientists started the invention of the conventional heat ie. Trefethen [2] grafted the idea of the heat ie onto his sace rogram. Grover et al. [3, 4] revealed that the sodium heat ie is very effective in the heat transort with the high erformance. They reresented the exerimental results by using the working fluids such as sodium, silver, and lithium in the stainless steel heat ie with the screen wick. The results roduced the develoment of the heat ie as the effective thermal device. Also, Leefer [5] and Judge [6] led the research on the heat ie and showed that the maximum oerating temerature of the heat ie can be as high as 1,650 C. Dobson et al. [7] showed that the motion of the PHP with the oen-end mathematically. They assumed the heat transfer coefficient between the wall and the vaor, and ignored the effect of the surface tension and the heat transfer of the surrounding liquid. Cai et al. [8] visualized the rocesses of the heat transfer and the motion of the working fluid by using the lastic or the glass PHP. They also showed that the ulsating motion takes lace better if the working fluid has the low latent heat. Hosoda et al. [9] comuted the temerature and the ressure of the PHPs by solving the momentum and the energy equations in the two-dimensional two-hase flow. Zhang and Faghri [10] considered the thin film evaoration and condensation and showed that the heat transfer of PHP is dominated by the sensible heat regardless of the surface tension. Groll et al. [11] resented the generation and the collase of the vaor bubble due to the total fixed volume of the PHP. Rittidech et al. [12] investigated the energy thrift as an air re-heater using the oen-looed coer PHP. They used variable conditions such as 60 degree, 70 degree and 80 degree of temerature in heating section and 3.3 m/s of gas velocity with water and R-123 as the working fluid. The result was that the heat transfer rate increased a little when the heat section temerature climbed from 60 degree to 80 degree. Similarly, the heat transfer rate using R-123 was more increased than the one using water. Shao and Zhang [18] analyzed the movement and the effect of liquid, vaor and heat 2

18 transfer with constant temeratures of the heating and the cooling section in the U-shaed PHP. Kim et al. [13] also analyzed the effect of entroy generation at the initial vaor temeratures of 70 degree and 20 degree. Likewise, the temeratures in the heating and the cooling section aeared to affect the heat transfer of the PHPs. On the other hand, the entroy is generated by the thermodynamic cycle when the working fluid moves in the PHP. The entroy generation is also caused by the friction loss and the heat transfer of the working fluid [14]. Furthermore, the entroy generation is directly related with the irreversible rocess, and the quantity of the entroy generation is determined by the entroy and the lost work during the rocess. Al- Zaharnah [15] conducted the research on the effect of the wall temerature and the Reynolds number on the entroy distribution in the ie system. Moreover, Al-Zaharnah et al. [16] studied the effect of the fluid viscosity on the entroy generation for each different wall temerature. It was resented that the rate of the entroy generation is increased by the high temerature of the ie wall. Sahin et al. [17] researched the effect of the develoing laminar viscous fluid flow on the entroy generation in the circular ie. They showed that the entroy generation increases near the wall and decreases gradually toward the axial direction. Thermal conductivity and viscosity are the key roerties of nanofluids that have been reorted more in the last decade. Many theoretical models are resented in the literature for calculating thermal conductivity and viscosity of nanofluids. However, there are no reliable theoretical model for rediction of anomalous viscosity and effective thermal conductivity of nanofluids. The viscosity and thermal conductivity data of nanofluids are still contradictory in various research ublications. Therefore, it is still unclear to choose the best models to use for viscosity and thermal conductivity of nanofluids. The convectional models for thermal conductivity [26, 27] cannot describe accurately the thermal conductivity of nanofluids because they do not consider the effect of temerature and some of enhancement mechanisms such as effect of nanoarticle clustering, Brownian motion [24, 28], the nature of heat transort in nanoarticles, and molecular-level layering of the liquid at the liquid/article interface [29]. The 3

19 conventional understanding of the effective thermal conductivity of mixtures is useful for largersize solid/fluid systems. These theories originates from continuum formulations which assume diffusive heat transfer in both fluid and solid hases and tyically involve only the article size/shae and volume fraction. Exerimental results showed that thermal conductivity of nanofluids deends on different arameters including volume fraction, surface area, temerature, thermal conductivity of nanoarticles and base fluid, shae of nanoarticles and etc. Based on the traditional models, some dynamical models have been roosed to address random motion of nanoarticles. Khanafer and Vafai [21] analyzed thermohysical characteristics of nanofluids and their role in heat transfer enhancement. Based on ertinent exerimental data, they develoed some general correlations in terms of temerature, thermohysical roerties of base fluid, volume fraction and article diameter for calculating effective thermal conductivity and viscosity as well as density of nanofluids. Their models aeared to be accurate and comrehensive so in this aer we used these models for calculating effective roerties of nanofluids. 1.4 Thesis Objectives As I mentioned above so far, many studies of PHPs had investigated the effect of heat transfer by testing exeriments and using comuter rograms. Some studies have focused on characterizing the heat transfer erformance. Other studies have aroached theoretically by the details of the modeling such as different working fluids, random wall temerature and different volume fractions and nanoarticle diameters of nanofluids. The objective of this study is to examine the entroy generation and thermal erformance of a U-shaed ulsating heat ie. All simulations of this study are figured out to investigate the entroy generations with the initial temerature, the effect of the ressure loss at the bend on the entroy generation, random noise of wall temerature and volume fractions and nanoarticle diameters of nanofluids. 4

20 1.4.1 Entroy Generation with Initial Temerature and Pressure Loss at Bends The entroy generation is based on the second law of thermodynamics. In the resent study, the entroy generation in a U-shaed Pulsating Heat Pie (PHP) is numerically investigated. The following five arameters, which are vaor mass, liquid temerature, latent heat, sensible heat, and friction, determine the entroy generation. The results show that the entroy generation is significantly affected by the initial temerature in the PHP. In the resent study, the entroy generation will be numerically analyzed with considering two vaor lugs, a liquid slug, evaoration and condensation in the PHP. The effects of initial temerature and the ressure loss in the bend on the entroy generation in the PHP will also be studied Random Noise of Wall Temerature Most researchers had investigated the heat transfer in PHPs based on the assumtion that the heating and cooling section temeratures are constant. However, heating and cooling section temeratures are usually not constant in the real oeration as it can fluctuate by changing of heating load, cooling condition or even oscillation of liquid slugs. Effects of fluctuations of heating and cooling section temeratures on the erformance of a PHP are investigated in this aer. Effects of amlitude, frequency and standard deviation of the temerature fluctuations on the erformance of the PHP are investigated Volume Fractions and Particle Diameters of Nanofluids. It aears that there is no theoretical model for redicting the thermal erformance of PHPs with nanofluid as working fluid. Clearly, it is necessary to do further research to determine the effect of nanofluids on different arameters including dislacement of liquid slug, vaor temerature and ressure, liquid slug temerature distribution, as well as sensible and latent heat transfer in evaorator and condenser. Therefore, the objective of this aer is to conduct a theoretical analysis redicting the heat transfer enhancement of PHPs and investigating the effect of nanoarticles volume concentration and size on thermal erformance of the system. 5

21 CHAPTER 2 ANALYSIS FO ENTROPY GENERATION IN A PULSATING HEAT PIPE 2.1 Physical Model Problem Descrition Figure 1 Configureation of ulsatign heat ie Figure 1 shows the configuration of the U-shaed PHP [18]. The evaorator and the condenser are located in the to and the bottom of the U-shaed PHP, and the vaor lug 1 and the vaor lug 2 are dislayed in the left side and the right side as shown in the Fig. 1 (a), resectively. There is also liquid slug between two vaor lugs, and the end of the U-shaed PHP is already sealed. Figure 1 (b) is considered as the straighten heat ie, not the U-shaed one. L indicates the length of the liquid slug, and 6 L e and Lc demonstrate the lengths of the evaorator and the condenser, resectively. Furthermore, the dislacement of liquid slug is indicated by x, and the diameter of heat ie and the radius of the bend are resented by d and r, resectively. x can be negative or ositive when the liquid slug moves the left side or the right side, deending on the movement of the liquid slug.

22 The following assumtions are made in order to investigate the entroy generation in the U-shaed PHP: 1) The ressure dro in the vaor due to the friction is neglected. 2) The vaor lug is assumed to be always in the saturation condition and behave as an ideal gas. 3) The entroy is assumed to be generated by the heat transfer and the friction. 4) The liquid is assumed to be incomressible, and the sub-cooling in the liquid is considered Governing Equations for Pulsating Flow The acceleration of the liquid slug is related with ressure, gravity and shear stress. Thus, the momentum equation is [18]: 2 dx l v1 v2 b l AL [( ) ] A 2 gax dl 2 dt (1) where b is the ressure loss at the bend [19] (Rohsenow et al., 1985): 2 Kl v 2 b 2 v Kl v v (2) and is the shear stress: 2 c / 2 16 / Re, Re 2200 l, c l [ Re, Re 2200 (3) The energy equations of the two vaor lugs are [18]: d( m c T ) dm dx dt dt 4 dt v1 v v1 v1 2 ctv 1 v 1 d (4) d( m c T ) dm dx dt dt 4 dt v2 v v2 v2 2 ctv 2 v2 d (5) 7

23 The vaor lug can be assumed to be an ideal gas. Thus, the equation of state can be resented by: 2 v 1( Le x ) d mv 1RTv1 (6) 4 2 v2( Le x ) d mv 2RTv 2 (7) 4 Considering the initial conditions, the masses and the temeratures of the two vaor lugs can be obtained: 1 2 d 0 v1 v1 e 4RT0 0 m ( L x ) (8) 1 2 d 0 v2 v2 e 4RT0 0 m ( L x ) T T v1 v1 T v2 v2 T0 0 (9) (10) (11) where T 0 and P 0 are the initial temerature and the ressure, resectively Latent and sensible heat transfer The heat transfer in the PHP means that the heat moves from the evaorator to the condenser. There are two tyes of the heat transfer in the PHP. One is the latent heat due to the hase change of the working fluid, and the other is the sensible heat because of the heat transfer between the wall and the liquid slug. The heat transfer rates in the two vaors are related with the mass flux, and can be obtained by: Q m h (12) in, v1 ev, v1 lv 8

24 Q m h (13) out, v1 con, v1 lv Q m h (14) in, v2 ev, v2 lv Q m h (15) out, v2 con, v2 lv equation: The temerature distribution of the liquid slug can be obtained by solving the energy 1 dtl dt l d T dx 2 l 2 hlsend ( Tl T k A l w ) (16) The initial and boundary conditions for Eq. (16) are: T T0, t 0, 0 x L (17) T T v, x 0 (18) 1 T T, x L (19) v2 The sensible heat transfer in and out of the liquid slug can be obtained by integrating the heat transfer over the length of the liquid slug. Q in, s, l L dh( Te Tl ) dxl, x 0 Lx x dh( T ), 0 0 e Tl dxl x (20) Q out, s, l x dh( T ), 0 0 l Tc dxl x L dh( Tl Tc ) dxl, x 0 x (21) where the sensible heat transfer coefficient can be obtained from h Nuk / d [23]. l 9

25 2.1.4 Entroy Generation of Two Vaor Plugs The entroy is generated when the liquid slug becomes the vaor lug due to the evaoration in the PHP. According to the second law of thermodynamics, the rate entroy generation due to hase change can be calculated by: ds dt v minsv m outsv (22) The entroy generation in the vaor lug 1 is d( m s dt dmv sv 1 (23) dt v1 v1) 1 since the change of secific entroy with resect to time is zero, i.e., ds / 0 v1 dt. Thus, the entroy generation in the vaor lug 1 is: dsv 1 dmv 1 sv 1 (24) dt dt Similarly, the entroy generation in the vaor lug 2 is dsv 2 dmv 2 sv2 (25) dt dt Therefore, the equations of the entroy generation in the vaor lug 1 and 2 can be rearranged as: S m s (26) v1 v1 v1 S m s (27) v2 v2 v2 Also, these equations can be written as following because the entroy of the two vaor lugs is equal in the reference state. Sv 1 mv 1s0 (28) 10

26 Sv2 mv2s0 (29) Finally, the entroy generations in the two vaor lugs can be obtained by: S ( s (30) v1 mv 1 mv 1,0) 0 S ( s (31) v2 mv 2 mv 2,0) Entroy generation of liquid slug Since the liquid hase is incomressible, the following entroy equation is valid: Tds c dt (32) ds c dt (33) dt T dt Integrating Eq. (33) yields s l s (34) Tl l, 0 c ln T 0 The entroy generation of the liquid slug can be obtained by integrating Eq. (34) over the length of the liquid slug: S l 2 Tl sldm l d sl,0 c ln dx (35) 4 0 T ml L 0 0 L 2 Tl Sl S, l0 l d c ln dx (36) 4 0 T 0 S l L 2 Tl l d c ln dx (37) 4 0 T 0 The friction by the liquid slug is related to area, velocity and shear stress. The entroy generation due to the friction on the liquid slug can be gained by: S f S f,0 S 11 f L 1 ldv dx (38) 0 T l

27 2.1.6 Entroy generation of latent heat and sensible heat in heating and cooling sections When the working fluid is ulsating, the system is affected by the latent heat and the sensible heat. By using the latent heat from Eqs. (12)-(15), the entroy generations of the latent heat for the evaoration and the condensation are as follows: Qe, lat Qc, lat Se, lat and Sc, lat (39) T T e c The entroy generations of the sensible heat transfer are obtained by using the sensible heat from: S e, sen Q e, sen T e and Q c, sen Sc, sen (40) Tc As a result, the total entroy generation in the U-shaed PHP can be obtained by adding all the terms of the entroy generation. S S S S ( S e, lat + S e, tot, gen v1 v2 l sen ) + ( Sc, lat + Sc, sen ) S f (41) 2.2 Numerical Alication The iteration method and the imlicit finite difference method are used to solve the hysical model of the vaor lugs and the liquid slug numerically. Also, the equation of the heat conduction is solved by TDMA (Tridiagonal Matrix Algorithm) in order to get the temerature distribution of the liquid slug. The numerical rocedures to obtain the entroy generation are as follows: 1) Temeratures of the two vaor lugs, and, are assumed, and the thermal and hysical Tv1 T v 2 roerties of the liquid slug are calculated according to. 2) The vaor ressures, and 2, are solved by Eqs. (10) and (11). v1 v 3) The dislacement of the liquid slug,, is calculated by Eq. (1). x T l 12

28 4) The new masses of the two vaor lugs, and, are obtained by accounting for the change of the vaor masses from Eqs. (8) and (9). 5) The ressures of the two vaor lugs, and, are calculated by Eqs. (6) and (7). 6) and are solved by Eqs. (9) and (10). Tv1 T v 2 7) and are comared, which are obtained in ste 6, with the assumed values in the ste 1. Tv1 T v 2 If the differences meet the small tolerance, go to the ste 8; otherwise, the above rocedures are reeated the stes 2-6 until a converged solution is obtained. 8) The temerature distribution of the liquid slug is solved by Eq. (16) and then the sensible heat is calculated. 9) The latent heat is obtained by Eqs. (12)-(15). 10) The entroy generations in the vaor 1 and 2, S v1 and S v 2, are calculated by Eqs. (30) and (31). 11) The entroy generations of the liquid slug and the friction, and S, are calculated by Eqs. (37) and (38). 12) The entroy generations of the latent heat and the sensible heat, S e, lat, S e, sen, S c, lat and S c, sen, are calculated by Eqs. (39) and (40). The sensible heat and the latent heat are obtained from the ste 8 and 9. 13) The total entroy generation of the U-shaed PHP is calculated by Eq. (41). 2.3 Results and Discuss mv1 m v 2 v1 v 2 The following arameters are used in the numerical calculation, which are obtained from Zhang et al [18]: L =0.1m, L =0.1m, L =0.2m, d =3.34mm, Te =123.4, Tc =20, and e c S l f h e h c 200W/m 2 4 K, t

29 2.3.1 The effect of the entroy generation with the initial temerature. When the initial temeratures in the U-shaed PHP are 70 o C and 20 o C, resectively, the entroy generations with the initial temerature are shown in Figs Figure 2 (a) shows the entroy generations with time in the vaor lugs 1. Although the value of the entroy generation is suosed to be ositive, the entroy generation in the vaor lug 1 has the negative value because the initial mass is larger than the new mass. The entroy generation in the vaor lug 1 with the initial temerature of 20 o C shows the relatively large amlitude and the constant attern with time. Unlike the initial temerature of 20 o C, the entroy generation in the vaor lug 1 with the initial temerature of 70 o C resents the relatively small amlitude. Also, the amlitude and the frequency of the entroy generation in the vaor lug 2 with the initial temerature of 20 o C are relatively larger than those with the initial vaor temerature of 70 o C as shown in Fig. 2 (b). The reason of this henomenon is that the difference between the initial and the current masses at 20 o C is relatively larger than those at 70 o C. Figure 3 resents the entroy generations of the liquid slug with time when the initial temeratures are 70 o C and 20 o C. The entroy generations at the initial temeratures of 70 o C and 20 o C maintain about kj/k and kj/k, resectively. The entroy generation of the liquid slug is almost uniform without the difference of amlitude and not significantly affected by the initial temerature. 14

30 Figure 2 Entroy generation in two vaor lugs 15

31 Figure 3 Entroy generation in liquid slugs The entroy generations with time are shown in Figs. 4 and 5 due to the latent heat in the vaor lugs 1 and 2, resectively. Figures 4 (a) and 5 (a) reresent the entroy generations in the evaoration, and the entroy generations in the vaor lugs 1 and 2 increase sharly with the regular intervals at the initial temerature of 20 o C because of the noticeable difference in 16

32 temerature. The amlitude and the frequency at the initial temerature of 70 o C are steady, and the entroy generation with time is almost the same in the cycle. Also, the frequency at the initial temerature of 20 o C is longer than that at the initial temerature of 70 o C in the maximum entroy generation. Figures 4 (b) and 5 (b) show the entroy generations with time in the vaor lugs 1 and 2 when the vaor lug enters into the cooling section. Unlike the evaoration rocess, the entroy generation at the initial temerature of 70 o C is higher than that at the initial temerature of 20 o C because the difference between the cooling section s temerature and the initial temerature of 70oC is larger than that with the initial temerature of 20 o C. However, the frequency at the initial temerature of 20 o C is longer than that at the initial vaor temerature of 70oC. Figure 6 shows the entroy generation due to the sensible heat at the initial temeratures of 70 o C and 20 o C. When the liquid slug is ulsated, the entroy generation is related with the sensible heat as well as the latent heat. As seen from Fig. 6 (a), the grah at the initial temerature of 70 o C resents the entroy generation of the constant eriod and shae when the sensible heat transfers into the liquid slug. However, the change of the entroy generation at the initial temerature of 20 o C is gradually decreased with increasing time because the sensible heat is sensitive to the temerature difference. On the other hand, the entroy generation in the cooling section at the initial vaor temerature of 70 o C is larger than that at the initial temerature of 20 o C as shown in Fig. 6 (b). The amlitude of the entroy generation at the initial temerature of 70 o C is gradually decreased because the difference of temerature is reduced with increasing time. However, there is a little change in the entroy generation at the initial temerature of 20 o C. 17

33 Figure 4 Entroy generation due to latent heat in vaor lug 1 18

34 Figure 5 Entroy generation due to latent heat transfer in vaor lug 2 19

35 Figure 6 Entroy generation due to sensible heat in liquid slug Figure 7 resents the entroy generation due to the friction of the liquid slug at the initial temeratures of 70 o C and 20 o C. There is a little difference in the entroy generation because the velocity of the liquid slug is not nearly affected by the initial temerature. The values of the 20

36 entroy generation due to the friction are very small, which have the order of aroximately kj/k. Therefore, since the friction of the liquid slug does not affect the entroy generation, it can be ignored. Figure 7 Entroy generation due to friction in liquid slug Figure 8 shows the above mentioned entroy generations at the initial temeratures of 70 o C and 20 o C. The change in the vaor lugs 1 and 2 is noticeably larger than the others in the 21

37 entroy generation. In addition, the value of the entroy generation at the initial temerature of 20 o C is higher than that at the initial temerature of 70 o C. As a result, the initial vaor temerature has a significant effect on the change of the entroy generation in the U-shaed PHP. Figure 8 Total entroy generation at different initial temeratures 22

38 2.3.2 The effect of the ressure loss at the bend on the entroy generation Figures 9-19 demonstrate the effect of the ressure loss at the bend on the entroy generation. The ressure loss coefficient of 0.31 is used in the calculations when the curvature radius (r) of the bend is 5.83mm as shown in Fig. 1 [18]. Figures 9 and 10 show the entroy generations with or without the ressure loss at the bend in the vaor lugs 1 and 2, resectively. In the startu stage, the entroy generations have the same frequency. However, in the final stage, the frequency of the entroy generation with the ressure loss is faster than that without the ressure loss. The reason is that the ressure dro revents the liquid slug from moving in the PHP. Thus, the ressure loss at the bend has a significant effect on the frequency, while it does not affect the amlitude of the entroy generation. Unlike the cases for the two vaor lugs, Figure 11 shows the entroy generation of the liquid slug with time, and there is a little difference of the entroy generation with or without ressure loss at the bend. The amlitude difference is about kj/k, and the frequency difference is almost constant. Figures show the entroy generations due to the evaoration and the condensation with or without the ressure loss in the vaor lugs 1 and 2, resectively. In other words, they resent the effect of the latent heat on the entroy generation with or without the ressure loss. The frequencies of the entroy generations due to the evaoration and the condensation in the two vaor lugs are similar to those as shown in Figs. 4 and 5. The entroy generation in the startu stage is almost constant regardless of the existence of the ressure loss at the bend. However, the frequencies with the ressure loss are still faster than those without the ressure loss in the final stage. 23

39 Figure 9 Entroy generation with or without ressure loss at bend in the vaor lug 1 24

40 Figure 10 Entroy generation with or without ressure loss at bend in the vaor lug 2 25

41 Figure 11 Entroy generation with or without ressure loss at bend in the liquid slug 26

42 Figure 12 Entroy generation due to evaoration with or without ressure loss in vaor lug 1 27

43 Figure 13 Entroy generation due to condensation with or without ressure loss in vaor lug 1 28

44 Figure 14 Entroy generation due to evaoration with or without ressure loss in vaor lug 2 29

45 Figure 15 Entroy generation due to condensation with or without ressure loss in vaor lug 1 Figures 16 and 17 resent the entroy generation due to the sensible heat of the liquid slug with or without the ressure loss in the heating section and the cooling section, resectively. Likewise as shown in Figs , the frequencies of the entroy generation with the ressure loss are faster than those without the ressure loss in the final stage. 30

46 Figure 16 Entroy generation due to sensible heat into liquid slug with or without ressure loss 31

47 Figure 17 Entroy generation due to sensible heat out of liquid slug with or without ressure loss Figure 18 shows the entroy generation due to the friction of the liquid slug with or without the ressure loss at the bend. The entroy generation with the ressure loss is higher than that without the ressure loss. However, the difference of the entroy generations due to the friction is small. Therefore, the friction can be ignored in the entroy generation. Figure 19 32

48 resents the above mentioned entroy generations with or without the ressure loss at the bend. The entroy generations are almost the same regardless of the ressure loss. However, the frequencies of the entroy generation with the ressure loss are quicker than those without the ressure loss at the bend. Figure 18 Entroy generation due to friction in liquid slug with or without ressure loss 33

49 Figure 19 Total entroy generation with or without ressure loss at bend 2.4 Conclusions The entroy generation in the U-shaed PHP has been investigated in consideration of the initial temeratures and the ressure loss at the bend. The entroy generations are calculated from the variations of vaor mass, the liquid temerature, latent heat, sensible heat, and friction. 34

50 In result, the entroy generations are sensitively affected by the initial temerature, and the variation of the vaor mass is the main cause of the entroy generation. The amlitude of the entroy generation is irrelevant to the ressure loss at the bend. However, the frequency of the entroy generation with the ressure loss is faster than those without the ressure loss. 35

51 CHAPTER 3 Random Noise of Wall Temerature 3.1 Physical Model Problem Descrition Most researchers had investigated the heat transfer in PHPs based on the assumtion that the heating and cooling section temeratures are constant. However, heating and cooling section temeratures are usually not constant in the real oeration as it can fluctuate by changing of heating load, cooling condition or even oscillation of liquid slugs. Effects of fluctuations of heating and cooling section temeratures on the erformance of a PHP are investigated in this aer. Effects of amlitude, frequency and standard deviation of the temerature fluctuations on the erformance of the PHP are investigated Governing Equations and Random noise inut of wall temerature for Pulsating Flow To consider the effects of fluctuations of the wall temeratures on the erformance of PHP, the following heating and cooling section temeratures are considered: T ( t) [ T A cos( t )](1 / T ) (42) H H 0 H H H 0 T ( t) [ T A cos( t)](1 / T ) (43) C C 0 C C C 0 The fluctuations of wall temeratures include a eriodic comonent and a random comonent. The eriodic comonent has an amlitude of A H or A C and an angular frequency of ω. The hase difference between the eriodic comonents of heating and cooling sections is. The random comonent is characterized by a random fluctuation ε H or ε C. The random fluctuation can have different atterns in grah according to mean value and standard deviation. The following normal distribution [20] is used to obtain ε H or ε C : n n ( RN i ) i1 2 n /12 (44) where RN i is a random number between 0 to 1, and n is an integer. 36

52 3.2 Results and Discussion Effect of random temeratures, T H and T C, with different amlitudes of the heating and the cooling section The arameters used in the simulation are: L e =0.1m, L c =0.1m, L =0.2m, d =3.34mm, 2 and he hc 200 W / m K. The base temeratures for heating and cooling sections are T H 0 =393.4 K and T C 0 =293 K, resectively. Figure 20 illustrate the changes of temerature of the heating and the cooling sections due to eriodic and random fluctuations. The frequency of the eriodic fluctuation was f = 9 Hz, and the standard deviations for the random fluctuation are σ H =σ C =2.0 K. The fluctuation of the temeratures of the heating and the cooling section increase when increasing amlitudes. Figure 21 shows the different liquid slug dislacements with different A H and A C. As shown in Fig. 21 (a), the liquid slugs have the similar movements when the amlitude of the heating section increases; however, the eriod of oscillatory flow slightly increases with increasing A H. The attern of the grah shows the very small high-frequency fluctuations with time due to the random fluctuation of the temerature of the heating section. Figure 21 (b) resents the change of the liquid slug dislacement for different A C. The liquid slug dislacements show very little difference for different amlitudes of eriodic fluctuation; however, very small high-frequency fluctuations are still observed. Figure 22 resents the temeratures of the vaor lug 1 for different A H and A C. The maximum temerature of vaor lug 1 gradually decreases when A H increases as shown in Fig. 22 (a). However, as A H increases, the temerature of the vaor lug 1 is more random during evaoration than that during condensation, because A C is equal to 0.0 for Fig. 22 (a). Figure 22 (b) shows the effect of amlitude of eriodic fluctuation of the cooling section temerature on the vaor temerature. As A C increases, the minimum vaor temerature decreases during condensation, while the vaor temerature during evaoration is not affected by A C. Figure 23 shows the ressure of the vaor lug 1 for different amlitudes of eriodic wall temerature fluctuation. The trends of vaor 37

53 ressure change with the increment of the amlitude in the heating and the cooling section are quite similar with those in Fig. 22. Figure 20 Fluctuation of heating and cooling section temeratures for different A H and A C 38

54 Figure 21 Liquid slug dislacements for different A H and A C 39

55 Figure 22 Temerature of vaor lug 1 for different A H and A C 40

56 Figure 23 Pressure of vaor lug 1 for different A H and A C 41

57 Figure 24 Latent heat transfer of vaor lug 1 for different A H 42

58 Figure 25 Latent heat transfer of vaor lug 1 for different A C Figure 24 (a) shows the change of latent heat in the vaor lug 1 with A H. Although the latent heat transfer at different A H showed different variations, its effects on the overall evaoration heat transfer is very insignificant. Figure 24 (b) shows the latent heat transfer in the 43

59 cooling section with increasing A H, and it is shown that there is delay with increasing A H. The latent heat transfer in the condenser section at different A H shows the similar atterns. Figure 25 shows the latent heat transfer at the vaor lug 1 with increasing A C. It can be seen that the atterns of Fig. 25 (a) and (b) are similar to those in Fig. 24 (a) and (b). Figure 26 and 27 resent the effect of wall temerature fluctuation on the sensible heat transfer by the liquid slug. Figure 8 shows the henomenon of the delay with increasing A H. The sensible heat is the highest when the amlitude is equal to 0.0; however, the value of the sensible heat changes randomly as deicted in Fig. 26 (a). The sensible heat transferred out of liquid slug shows similar attern for different A H because the amlitude is not alied to the cooling section (A C = 0) as shown in Fig. 26 (b). However, the sensible heat transferred out of the liquid slug shows delay with increasing A H. The sensible heat transfer by liquid slug at different A C is shown in Fig. 27. It can be seen from Fig. 27 (a) that sensible heat transfer increases with increasing A C for most of the times. Due to the eriodic comonents of the fluctuation, the sensible heat transfer exhibit some sikes at some instance and this behavior reeats eriodically. The sensible heat transfer shows more delay with the high amlitude than the low amlitude. Figure 27 (b) has almost same attern with Fig. 27 (a); however, the eak values of the sensible heat transferred out of the liquid slug are greater than those for sensible heat transferred into to the liquid slug. The random oscillation of the sensible heat transfer is receded by the eriodic fluctuation of the temeratures, T H and T C. Therefore, the fluctuations of the heating and cooling temeratures have significant effects on the erformance of the PHPs. 44

60 Figure 26 Sensible heat transfer of the liquid slug for different A H 45

61 Figure 27 Sensible heat transfer of the liquid slug for different A C Effect of random temeratures, T H and T C, for differnet frequency Figures show the effect of the frequency of the eriodic comonent of temerature fluctuation on the erformance of the U-shaed PHP. The amlitudes of the eroidic comoent 46

62 of the fluctuation are set as A H = 10 K and A C = 5 K, resectively. Figure 28 shows temeratures of the heating and the cooling sections for different frequencies. Figure 28 Temeratures of the heating and cooling section for different frequencies 47

63 Figure 29 illustrate the liquid slug dislacements for different frequency of wall temerature fluctuations. The liquid slug has larger dislacement with increasing frequency and the eriod of oscillation increases gradually. Figure 29 Liquid slug dislacements for different frequencies Figure 30 shows the temerature and ressure of vaor lug 1 for different frequencies. The magtitude of vaor temerature and ressure oscillations are different for different frequencies. The meximum (minuum) vaor temerature and ressure increase (decrease) with increasing frequency. Figure 31 shows latent heat transfer of the vaor lug 1 for differnet frequencies of the wall temeerature fluctuations. It can be seen from Fig. 31 (a) that the latent heat transfer in the heating section increases suddenly and then decreases raidly. The eak values of evaoration heat transfer is higher when the frequency is higher. Figure 31 (b) resents quite constant attern of the variation of the condensation heat transfer. The difference between the eak values of the latent heat transfer in the heating and the cooling sections is almost one order of magnitude. Figures 32 shows the sensible heat transfer by the liquid slug at different frequencies. It can be seen that the eak values of sensible heat transfer do not exhibit clear trend with the frequency. In other words, the maximum eak value amoung the three frequecies can be 48

64 achieved at any frequency, deending on the time. Thus, the frequency randomly affects the change and the cycle of ressures and temeratures of two vaor lugs, the dislacement of the liquid slug, as well as the latent and sensible heat transfer. Figure 30 Temerature and ressure of vaor lug 1 for different frequencies 49

65 Figure 31 Latent heat transfer of vaor lug 1 for different frequencies 50

66 Figure 32 Sensible heat transfer of the liquid slug for different frequencies 51

67 3.2.3 Effect of changes for different standard deviation Figure 33 Temerature of heating and cooling sections for different for σ H and σ C 52

68 Figure 34 Liquid slug dislacements for different for σ H and σ C Effects of standard deviations of the random comonents of the temerature fluctuation are then studied. Figure 33 shows the temerature of the heating and the cooling sections for different σ H and σ C, while the frequency of the eriodic comonent is ket at f = 9 Hz, and A H and A C are 10 K and 5 K, resectively. Figure 34 shows the effects of σ H and σ C on the 53

69 dislacements of the liquid slug. It can be seen that there is not any notable effect of the standard deviations of the random comonents of the temerature fluctuation on the oscillatory flow. Consequently, there is also not any notable difference on vaor temerature and ressure, as well as latent and sensible heat transfer. 3.3 Conclusion Heat transfer in the U-shaed PHP has been analyzed by alying eriodic and random noises on the temeratures of the heating and the cooling sections. Effects of amlitude and frequency of the eriodic comonent and standard deviations of the random comonents of the heating and cooling sections are investigated. The results showed that both amlitude and frequency of the eriodic comonent of the temerature fluctuation have effects on the liquid slug dislacements, temeratures and ressures of the two vaor lugs, as well as the latent and sensible heat transfer. The frequency of the liquid slug oscillation decreases with increasing amlitude and frequency of the eriodic fluctuation of the wall temerature. However, the change of different standard deviations did not have any effect on the erformance of the PHP. 54

70 CHAPTER 4 NANOFLUID EFFECT ON THE THERMAL PERFORMANCE 4.1 Physical Model Problem Descrition Figure 35 shows the configuration of a U-shaed ulsating heat ie with nanofluid and its two ends sealed which are considered as the building block of a PHP. The arameters and oerating conditions of this hysical model is the same as in Chater 2, excet adding nanoarticles. In this study, Pure water is emloyed as the base fluid while Al 2 O 3 with two different article sizes i.e., 47nm and 38.4 nm is used as nanoarticle. The following assumtions are made in order to model heat transfer and fluid flow in the heat ie [18, 30]: (1) Shear stress at the liquid-vaor interface is negligible. (2) Heat conduction in the liquid slug is assumed to be one-dimensional in the axial direction and exchange of heat between the liquid and wall is considered by a convective heat transfer coefficient. (3) The liquid is incomressible and the vaor is saturated and behaves as an ideal gas. (4) Evaorative and condensation heat transfer coefficients are assumed to be constants. (5) The U-shaed minichannel is assumed to be a straight ie and the effect of ressure loss in bend is considered using an emirical correlation. 55

71 Figure 35 Configuration of U-shae ulsating heat ie with nanofluid with adding nanofluid Governing Equations for Pulsating Flow The acceleration of the liquid slug is related with ressure, gravity and shear stress. Thus, the momentum equation is [18]: A L eff d 2 dt x 2 ) ] A 2 [( v 1 v2 b eff g A x d L (45) where b is the ressure loss at the bend [21]: b 2 v K eff 2 v K eff 2 2 v 0 v 0 (46) where K is the ressure loss coefficient, and eff is the effective density of nanofluid and can be calculated using the following correlation [21]: 56