Experiment and mathematical model for the heat transfer in water around 4 C

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1 Eropean Jornal of Physics PAPER Experiment and mathematical model for the heat transfer in water arond 4 C To cite this article: Naohisa Ogawa and Fmitoshi Kaneko 2017 Er. J. Phys View the article online for pdates and enhancements. Related content - Thermal properties of a cylindrical YBa2C3Ox spercondctor in a levitation system: Triggered y the nonlinear dynamics Yi Hang, Xingyi Zhang and Yo-He Zho - Simple model of a photoacostic system as a CR circit Akiko Fkhara, Fmitoshi Kaneko and Naohisa Ogawa - Evaporation of Low-mass Planet Atmospheres: Mltidimensional Hydrodynamics with Consistent Thermochemistry Lile Wang and Fei Dai This content was downloaded from IP address on 29/09/2018 at 00:33

2 Eropean Jornal of Physics Er. J. Phys. 38 (2017) (13pp) doi: / /38/2/ Experiment and mathematical model for the heat transfer in water arond 4 C Naohisa Ogawa 1,3 and Fmitoshi Kaneko 2 1 Hokkaido University of Science, 7-15 Maeda, Teine, Sapporo Japan 2 Department of Macromoleclar Science, Gradate School of Science, Osaka University, Toyonaka Osaka Japan ogawanao@hs.ac.jp and toshi@chem.sci.osaka-.ac.jp Received 31 Agst 2016, revised 2 Novemer 2016 Accepted for plication 23 Novemer 2016 Plished 22 Decemer 2016 Astract Water, which is the haitat for a variety of living creatres, has a maximm density at 4.0 C. This crcial property is considered to play a very important role in the iology of a lake and also has a close relationship with the areas of environmentology and geoscience. It wold e desirale for stdents to confirm this important property of water themselves y carrying ot simple experiments. However, it is not easy to detect the maximm density at 4.0 C ecase the temperatre dependence of the water density is very small close to its freezing point. For example, the density of water is g cm 3 at 4.0 C and g cm 3 at 0.1 C. The aim in this manscript is to demonstrate a simple experiment to detect 4.0 C as the temperatre of maximm density, in which the time dependence of the water temperatre is measred at several different depths y chilling the water srface. This is a simple experiment that can also e performed y high school stdents. We also present a mathematical model that can explain the reslts of this experiment. Keywords: water, convection, heat diffsion, 4 degrees Celsis (Some figres may appear in color only in the online jornal) 1. Introdction In spite of its significance, the temperatre dependence of water density has not een adopted as an experimental sject for stdents, which might e for the following two reasons. First, it is not easy for stdents to measre the water density precisely at varios temperatres, and 3 Athor to whom any correspondence shold e addressed /17/ $ Eropean Physical Society Printed in the UK 1

3 Density ρ[g/cm 3 ] Temperatre [ C] Figre 1. Temperatre dependence of the water density nder atmospheric pressre. The open circles show experimental data (CRC Handook of Chemistry and Physics, [1]), while the solid line shows the approximate fnction given y eqation (1). The maximm difference etween the CRC data and eqation (1) is 0.001% among the 33 data points. second it may e difficlt to attract mch interest from ordinary stdents only y otaining nmerical data of the water density. We considered that these isses can e overcome if the temperatre dependence of the water density can e connected to the convection of water and related phenomena. Along these lines, we have developed two simple experiments for stdents, oth of which demonstrate that a marked change in the mechanism of water cooling occrs at the specific temperatre of 4.0 C; one involves the time-dependent temperatre measrement of water at different depths, and the other involves the visalization of water flow. The density change in water with temperatre is shown in figre 1 [1] and its crve can e well approximated y the qadratic fnction r ( T) [ kg m- 3 ] = ( T ) , ( 1) where T is the temperatre of water in C. Althogh the density change is small, many common phenomena are related to this property. For example, the cooled oxygen-rich srface water of a lake in winter indces strong convection and spplies oxygen to the ottom of the lake. As a reslt of this phenomenon, creatres living at the ottom of the lake can srvive. This convection contines ntil the whole lake reaches aot 4.0 C. After that, the strong convection stops, and frther atmospheric cooling decreases the pper water temperatre withot affecting the ottom water temperatre, ecase the water elow 4.0 C is less dense than that at 4.0 C. Note that even when the atmospheric temperatre is lower than 0 C, the srface freezing of a lake does not start ntil the water temperatre in the whole lake ecomes 4.0 C. Therefore, a long winter is reqired for the occrrence of srface freezing in a deep lake [2]. In this way, the ehavior of water arond 4.0 C is related to a wide range of natral sciences inclding physics, chemistry, iology, environmentology, and geoscience. In this article, we will descrie edcational materials that can e sed to detect 4.0 C as the temperatre of maximm density and to stdy the physics of convection and heat diffsion y analyzing experimental data. Althogh the concept of this experiment was 2

4 Figre 2. Experimental setp. Seven thermocoples were set on an acrylic pole at heights from 0 to 60 mm with 10 mm spacing. The whole system was placed in a chest freezer set to 15 C. Figre 3. Left: time dependence of water temperatre at each height. The temperatres at heights of 0, 10, 20, 30, and 40 mm are represented y solid lines with different densities, whereas those at heights of 50 and 60 mm are represented y lack roken and gray dotted lines, respectively. A platea appears at 4.0 C in three lines (0, 10, 20 mm), whereas no platea appears for heights 30 and 40 mm. Right: height dependence of temperatre at different times. At t = 100 min, the temperatre is almost flat in the region of 0 40 mm. However, at t = 160 min, the temperatres at heights of 30 and 40 mm clearly deviate from those at 0, 10, and 20 mm. preliminary presented as a simple research report in Japanese [3], here we present the edcational materials with an improved experimental procedre and data analysis for a mch wider readership. We also add a discssion of the mathematical model of the dynamical ehavior of water arond 4.0 C. In section 2, we show that a simple experiment allows s to oserve the same phenomenon as that occrring in a lake in winter, even in a classroom. In section 3, we introdce a mathematical model that explains the experimental reslt in section 2. In section 4, we introdce an additional experiment that enales the visalization of the convectional ehavior of water. From this experiment, we find that the convection of water stops at 4.0 C. 3

5 Figre 4. Assmed temperatre distrition ased on the experimental reslts in figre 3 (right). The existence of pper and ottom layers with temperatres T (t) and T (t), respectively, is assmed. Between the two layers, there is a temperatre gradient layer (TGL) withot a dynamical degree of freedom. 2. Experiment showing the significance of water temperatre of 4.0 C The simple experimental system shown in figre 2 was prepared from an open polystyrene foam ox. The interior dimensions of this heat inslation ox are 200 mm 135 mm 128 mm height. The thickness of the walls is 40 mm for the side faces and 60 mm for the ottom face. Water was added to the ox to a depth of 60 mm, then stirred to give a niform temperatre of arond 15 C. Then the ox was placed in a chest freezer set to 15 C. The water temperatres at heights of 0, 10, 20, 30, 40, 50, and 60 mm from the ottom were measred every minte sing a data logger with thermocople thermometers (Keyence NR calirated y a Testo TESTO177-T4 data logger). It was fond that the temperatres at heights of 0, 10, and 20 mm decrease at a similar rate, and reach a platea at 4.0 C, then slowly decrease at a similar rate as shown in figre 3. At heights 30 and 40 mm, there is no clear platea. These two sets of heights exhiit almost the same temperatre ntil the temperatre reaches 4.0 C, after which a ifrcation occrs and the temperatres of the two sets diverge. On the other hand, at heights of 50 and 60 mm, the heat is rapidly removed throgh the air and so the temperatres are mch lower than at the other heights. We next riefly explain the physical meaning of this phenomenon. First we divide water into for regions: the ottom layer (0 20 mm), the temperatre gradient layer (TGL; mm), the pper layer (30 40 mm), and the heat diffsion region (40 60 mm). Up to 134 min, the cooled pper layer has a larger mass density than the ottom layer. Therefore, convection occrs and the temperatres in the ottom, TGL, and pper layers are almost the same and decrease at a similar rate. When the temperatre reaches 4.0 C, water density ecomes the maximm in these three layers. Frther cooling redces the water density in the pper layer, while the density of ottom layer remains almost nchanged. Then the convection stops and the temperatre of the pper layer decreases, diverging from that of the ottom layer. Therefore the temperatre of the ottom layer remains at 4.0 C for a while, corresponding to the platea. After that, the temperatre of the ottom layer is very slowly decreased y thermal diffsion. In this way, we can riefly explain why the ottom layer shows a platea at 4.0 C and the ifrcation of the temperatre etween the ottom and pper layers. These phenomena are 4

6 Figre 5. Exchange of heat energy and matter etween the pper and ottom layers throgh the TGL. cased y the change in the cooling mechanism of water at 4.0 C. On the other hand, at heights of 50 and 60 mm, the effect of the convection appears to e less from the start. The heat transfer is considered to e mainly governed y heat diffsion. At t = 163 min, a phase transition from water to ice occrs at the water srface. Then the interface etween ice and water takes a temperatre of 0 C, and ths the temperatre nder 50 mm height slowly approaches 0 C (see figre 3, left). 3. Model of ifrcation and platea nder water cooling In this section, we present a simple mathematical model that explains the ifrcation and the platea. It has een shown y simlations and experiments that the srface cooling of water generates nstale convection [4]. However, to or knowledge, there has een no discssion of the ifrcation and platea of the water temperatre arond 4.0 C. Since a system with a phase transition from water to ice cannot e treated with a simple theoretical model, we confine or discssion to the temperatre change of the system p to time t = 160 min (from the starting time to the platea region). From figre 3, we may assme the temperatre distrition figre 4. Of corse, sch a temperatre distrition is not the soltion of a heat diffsion eqation, althogh it represents the experimental reslts well. Accordingly, we analyze the energy transportation in the asis of this assmed temperatre distrition. Using this distrition, we consider the dynamics shown in figre 5. In figre 5, the pper layer has temperatre T and the ottom layer has temperatre T. Between the two layers there are a downward mass flow at a rate of q and an pward mass flow at a rate of q that case convection. The internal energy transferred y the convection per nit time from the pper layer to the ottom layer is E, and that from the ottom layer to the pper layer is E. The heat flows y thermal diffsion are defined y Q i, where Q 1 is that from the ottom layer to the pper layer and Q 2 is that from the pper layer to the heat 5

7 diffsion region. Frthermore, the TGL is also cooled as well as the pper and ottom layers, and a heat flow DQ is released from the TGL to the pper layer. We assme the temperatre dependence of the mass flow rate to have the following form q = a( r( T ) - r( T )) q( r( T ) - r( T )), ( 2) where r ( T ) is the mass density of water at temperatre T. θ is a step fnction satisfying q () x = 1 when x 0 and q () x = 0 when x < 0. Accordingly, q takes a non-zero vale only when r ( T) > r ( T ), as reqired. q is proportional to the flow rate, which is determined y the alance etween the gravitational force and the oyant force and viscosity. Therefore, we assme that q is proportional to r ( T) - r ( T) with an nknown proportional constant a. Frthermore, from the conservation law of flows, we have q = q. Next, we consider the energy transfer. Denoting the specific heat of water as c, the energy transfer E from the pper layer to the ottom layer per nit time is given y E = qct ( + U. ) ( 3) In the same way, the energy transfer from the ottom layer to the pper layer per nit time E is given y E = qct ( + U, ) ( 4) where U is the internal energy of water per nit mass at 0 C. Since the constant U in eqations (3) and (4) disappears at the end of the calclation after applying the law of flow conservation, we exclde this qantity hereafter. Next, we discss the heat diffsion from the ottom layer to the pper layer. The heat transfer is proportional to the temperatre difference; ths, we can write Q = ( T - T ), ( 5) 1 where is the heat transfer coefficient of water, which can e estimated as kws = = 1.57 [ W K ], ( 6) D l where k w is the heat condctivity of water, S is the cross-sectional area of water, and D l is the distance etween the pper and ottom layers (the width of the TGL) [5]. The physical qantities sed in the calclations employing this model are listed in the following tale. Physical qantity Notation Nmerical vale Unit Specific heat of water c Jkg 1 K 1 Heat condctivity of water k w Wm 1 K 1 Cross-sectional area of water container S m 2 Thickness of TGL D l m Heat capacity (1 cm thickness) C JK 1 The heat transfer from the pper layer to the heat diffsion region Q 2 cannot e controlled in or system. Ths, we se the oserved vale in or experiment. This is discssed in the appendix and we otain Q 2 = t ( 0 < t < 134 ), t ( 134 < t < 160 ). ( 7) 6

8 Figre 6. Reslts of nmerical calclation ( ã = 100 ~ 10000). This is written in the nit of W, and the nit of t is min. Next we discss DQ, which is the heat emission from the TGL ( 2 < h < 3), where h = h Dl is a dimensionless height parameter. By tilizing we otain T( h, t) = ( T - T ) h + 3T - 2T ( 2 < h < 3 ), 3 D =-r D T( h, t) 1 d Q cs lò h =- C T d + 2 t 2 dt dt. ( 8) dt The heat capacities are expressed as C for the pper layer and C for the ottom layer. They are constants and proportional to the thickness. From or assmption, we otain C = C = C 2, ( 9) where C is the heat capacity with 1 cm thickness as shown in the tale. Then we otain the following differential eqation for each layer d C T dt = ( cq + )( T - T ) - Q + DQ, ( 10) 2 2C dt dt =- ( cq + )( T -T ). ( 11) By dividing oth sides y and changing the parameters as cq C q º, t º, t* º t t, ( 12) we otain dt * = 5 ( + q )( T t - T ) - Q d ( dt * =- 1 ( + q t )( T - T d 2 1 ). ( 14) Now we consider the vales of the parameters. The time parameter τ is given as C t = D l = [] s» 12[ min ]. ( 15) k S For the qantity q, we have w 7

9 Figre 7. Comparison of the theory ( ã = 1000) and the experimental reslt. The temperatres of the pper and ottom layers are indicated y the thick dashed line and thick solid line, respectively. cq ca q = = ( r( T) - r( T)) q( r( T) - r( T)) = a (( T - 4) 2 - ( T - 4) 2) q (( T - 4) 2 - ( T - 4 ) 2), ( 16) where = - ca a [ K -2]. ( 17) This qantity ã represents the strength of convection; however, it cannot e determined from or theory. Ths, we control this parameter and compare the time dependence of the temperatre of each layer otained experimentally and sing the model. We solve eqations (13) and (14) with the initial conditions T() 0 = T() 0 = 15.3 and ã = 100 ~ 10000, and we otain the reslts shown in figre 6. The magnitde of ã represents the strength of the convection that reslts in the niform temperatre distrition for T > 4.0 C. When ã = 100, the two lines (pper and ottom temperatres) are slightly apart from each other, t they converge for larger ã. At T = 4.0 C, the convection stops and the ifrcation starts. The theoretical reslt shows good agreement with the experimental data (figre 7). Note that the ifrcation originates from the step fnction appearing in q, which explains the cessation of convection at T < 4.0 C. On the other hand, the step change at t = 134 min appearing in Q 2 shows the redced heat transfer from the pper layer to the heat diffsion region, and is not essential for the ifrcation. However, it determines the rate of decrease in T at t > 134 min. To close this section, we smmarize or assmptions and findings otained from the experimental data. First, we employed three assmptions. The temperatre dependence of the mass flow rate (convection flow) is given y eqation (2). The spatial distrition of the temperatre is given y figre 4. The dynamics of each layer is given y figre 5. Second, the findings otained from the experimental data are as follows. 8

10 Figre 8. Apparats for oserving convection. The temperatre dependence of the density is given y the approximate fnction in eqation (1). The time dependence of T w for 0 < t < 134 [min] is approximately expressed y eqation (19) in the appendix and is sed to determine Q2 () t. The time dependences of T and T for 134 < t < 160 [min] are expressed y eqations (23) and (24) in the appendix, respectively, and are sed to determine Q () t Oservation of convection It is very significant that the cooling process of water changes at 4.0 C dring cooling from its srface. When T > 4.0 C, the cooling is mainly governed y convection; however, when T 4.0 C, it is governed only y heat diffsion. To oserve this change in the cooling mechanism of water, we have developed the apparats shown in figre 8. The pper region of water is cooled y a metal heat sink connected to a Peltier device. Then the temperatre of the pper region T decreases and we oserve convection in the water ath ntil the ottom temperatre T reaches 4.0 C. The temperatre of the pper region T is measred at the side of the metal heat sink nder the water, and T is measred at the ottom of water. The temperatre is measred y sing a thermistor thermometer logger (T&D Corp. TR52, TR52S). The dimensions of the plastic aqarim are 85 mm 85 mm 70 mm height. To oserve the convection clearly, the water is illminated y a slit light prepared sing three le LEDs, a light condenser, and a slit hole. The light condenser is a cylindrical lens made of acrylic resin, and the slit has a height of 25 mm and 3 mm width. A drop of aqeos florescein soltion (3 5 mgcm 3 ) is placed on the water srface, and then images of the florescein distrition in the water are taken at 10 s intervals, some of which are shown in figres 9(a), (), and (c). The convection slowly disappears as the ottom temperatre T decreases toward 4.0 C. The green area in the pictres corresponds to the region with a high florescein concentration. The green vortices or lines that can e oserved at t = 0 represent the distrition of florescein formed dring the first 10 s after adding a drop of the florescein soltion. 9

11 Figre 9. Temporal changes in florescein pattern nder different conditions. (a) T = 9.8 C, T = 11.3 C. Strong convection occrs. () T = 5.3 C, T = 6.6 C. There is even less convection than efore. (c) T = 1.4 C, T = 4.0 C. Convection stops. The florescein soltion diffses while slowly sinking on the right side. The images taken at each time reflect how the water in the aqarim flows at different times after the aqeos florescein soltion has een added. Figre 9(a) shows the case of T = 9.8 C, and T = 11.3 C, figre 9() shows the case of T = 5.3 C, and T = 6.6 C, figre 9(c) shows the case of T = 1.4 C, and T = 4.0 C. As descried in the previos section, when T > 4.0, convection occrs, which cools the whole region of water. On the other hand, when 4.0 T > T, the convection disappears and the heat diffsion cools the whole region of water very slowly. In this way, the two types of cooling mechanism, convection and heat diffsion, cool the water with the dominant mechanism changing at T = 4.0 C. This reslts in the temperatre distrition shown in figre 3. 10

12 5. Conclsion In this paper, we have shown the following points. First, the existence of the important temperatre of 4.0 C for water was confirmed y a simple experiment. When we cool water from its srface, the time dependences of the temperatres of the pper layer and ottom layer show a specific ehavior. At temperatres higher than 4.0 C, the temperatres of oth layers decrease at a similar rate, t at 4.0 C the ifrcation of the temperatre occrs etween the two layers, and a platea region appears for the temperatre of the ottom layer. These reslts show that 4.0 C is the temperatre at which the water density is maximm. Second, we constrcted a mathematical model to explain how the temperatres changes in the pper and ottom layers. The platea and ifrcation at 4.0 C can e expressed mathematically from the scenario given y figres 4 and 5. Third, we developed a simple device to oserve the convection in cooling water y sing a florescein soltion, which allowed s to confirm that convection occrs when T > 4.0 C and disappears when 4.0 C T > T. This is an essential reason why the platea and ifrcation occr. The experiment and model are considered to e sefl edcational tools for stdents stdying the important properties of water in relation to convection and heat diffsion. Acknowledgments The athors wold like to thank Dr S Yoshida and Dr K Toyota for teaching them aot the experiments on flids, Dr Y Frkawa for lending them the temperatre logger, and Dr S Nagasawa and Ms A Fkhara for their help. Appendix The time of the oserved ifrcation in the experiment is 134 min. Before the ifrcation ( t < 134 min), the temperatres at heights of 0, 10, 20, 30, and 40 mm have almost the same vale, which we write as T w (t) (mean vale). Then we can simply express the heat flow as d Q =- rcsh T w =- 4C dt w 2 ( ) dt d t, 18 where H is the water height ( H = 4cm. ) On the other hand, the experimental vale of T w is approximated y the qadratic fnction T = t t [ C ], ( 19) w where the nit of t is min. This is an approximate fnction etween times t = 0 and t = 134 min, meaning that the temperatre is slightly different at some points. However, the maximm error is only 2.7% (for example, the initial temperatre has an error of 1.1%) and has no significant effect on the overall reslts. Then we otain Q = t [ W]. ( 20) 2 After the ifrcation, Q 2 can e otained y the eqation 4 =-r D T( h, t) C T T Q cs lò dh =- + ( ) t 2 5 d 3 d 2, 21 0 dt dt where h = h Dl is a dimensionless height parameter sed for the integration variale, and T( h, t) [ C] is the temperatre at height h and time t, which is given y the following 11

13 Figre 10. T and T for 134 < t < 160 min and their approximate fnctions. fnction (see figre 4) T ( < 0 h < 2 ), T( h, t) = ( T - ) + - ( < T h 3T 2T 2 h < 3 ), ( 22) T ( 3 < h < 4 ). Using the experimental data (see figre 10), we otain an approximate fnction. The experimental vales of T (the mean temperatres at 0, 10, and 20 mm) can e approximated y a linear fnction: however, the experimental vales of T (mean temperatres at 30 and 40 mm) follow a smooth crve, which shold e approximated y a qadratic (or higher-order) fnction. Ths, we approximate T y a linear fnction and T y a qadratic fnction T = t t , ( 23) T = t ( 24) Then we otain dt dt dt dt = t , ( 25) = , ( 26) where oth are written in the nit of C min 1. Note that T has a non zero small slope. Ths, the platea refers to the very small slope compared to that of the time period t < 134 min. Then, from eqations (21), (25), and (26), we otain Q = t [ W]. ( 27) 2 for 134 < t < 160 min, where the nit of t is min. References [1] Haynes W M Standard density of water CRC Handook of Chemistry and Physics 95th edn (Boca Raton, FL: CRC Press) pp 6 8 [2] Hokkaido Research Organization, Environmental and Geological Research Department, Institte of Environmental Sciences [3] Ogawa N, Fkhara A, Kaneko F and Nagasawa S 2015 J. Phys. Edc. Soc. Japan (in Japanese) 12

14 [4] Horsch G M and Stefan H G 1988 Convective circlation in littoral water de to srface cooling Limnol. Oceanogr Spangenerg W G and Rowland W R 1961 Convective circlation in water indced y evaporating cooling Phys. Flids Bednarz T P, Lei C and Patterson J C 2008 An experimental stdy of nsteady natral convection in a reservoir model cooled from the water srface Exper. Therm. Flid Sci [5] Halliday D, Resnick R and Walker J 2001 Fndamentals of Physics 6th edn (New York: Wiley) 13