The dissociation of a substance in hot water can be well described by the diffusion equation:

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1 In this article we explore and present our solution to problem no. 15 of the 2017 International Physicist Tournament, Tea with honey. First we present our idea for the solution to the problem, then our experimental findings and finally we conclude with some remarks. I. THEORETICAL INTRODUCTION The solution to this problem is composed of two key parts: the design of the device to stir the tea, and a proposal for a measuring instrument that will give us a criterion for determining the homogeneity of the stirred solution. We begin with some background on sugar, bee honey and how they behave in water. where C( r, t) is the concentration of the substance and k is a constant which generally depends on temperature. In order to obtain uniform concentration of honey throughout the body of water as soon as possible, the honey should spread around the cup as much as possible. To get the sense of the optimal shape of the cup allowing this, we must look at its flow characteristics. Since water is considered an incompressible fluid, the flow is well described by the Navier-Stokes equations: A. Theory of mixing Table sugar is sucrose (C 12 H 22 O 11 ), while bee honey is composed mainly of glucose and fructose (both C 6 H 12 O 6 ) and of water. All of them are soluble in water, but their solubility depends on the temperature of water. Table sugar is usually in crystal form and is held that way by acetyl bonds between the molecules. These bonds are generally weak, and when in water, sucrose forms hydrogen bonds with water because of the highly electronegative oxygen atoms in the water molecules. The formed hydrogen bonds are strong and are able to brake apart the crystal structure of the sucrose crystal. If the temperature of the water and sugar solution is increased, the bonds in the sugar become relatively weaker which makes it easier to brake apart. Further, with increased temperature the water molecules are moving faster (on average) and there is a greater possibility for them to hit the crystalline sugar structure and break away sugar molecules. So, from the onset, it is clear that the high temperature of the tea allows a great degree of solubility of sugar. This conclusion holds for bee honey as well, since the molecules behave in a very similar way. Still, the question remains on how the sugar should be mixed with water and what kind of container would best suite this purpose. Fig. 1. The chemical composition of glucose and fructose. Sucrose is a simple combination of these two. The dissociation of a substance in hot water can be well described by the diffusion equation: C( r, t) t k 2 C( r, t) = 0, (1) v t + ( v ) v µ ρ 2 v = 1 P + g, ρ (2) v = 0, (3) where v( r, t) is the velocity field and P ( r, t) the local pressure in the body of the water. The full set of solutions of these equations for a given cup geometry proves to be very complicated and requires numerical simulations and special software. However, we can still draw some conclusions and use the equations in simple computer models. It can easily be shown that if the cup we plan to use for mixing doesn t have smooth edges (e.g. a cylinder shaped cup at the circle joining the base with the sidewall) then during the stirring one or more vortices are produced near these edges. The water that flows in these vortices almost doesn t mix well with the rest of the water in the cup. Thus, the honey diffuses much more slowly and it takes longer to reach a uniform concentration throughout the cup. Therefore, to ensure faster mixing, the cup surface should be as smooth as possible. A simulation in the next sections confirms this prediction. Even though mixing of the solution can be achieved in many ways, we decided to use a standard battery powered motor with a propeller attached to it. The motor is placed below the cup and fed through its base, so that the propeller blades sit near the bottom of the cup. To determine the optimal size of the blades we used several of them with varying lengths and shapes and chose the one that worked the best. We concluded that the blades shouldn t be too large in order to decrease the size of the flow lines and to ensure a large enough gap between the wall of the container and the blades. We assume that the fluid velocity is zero on the cup walls and the blade (noslip condition). We also found that if the gap between the blades and the walls is small the fluid velocity in the gap becomes too large (because the flow is solenoidal) and the kinetic energy of the fluid is dissipated much faster due to viscosity of the fluid.

2 2 B. The device We used two devices with the aim of improvement and comparison of our results. The first was just a household glass cup in whichan electric motor stirred the liquid. After a few experiments we decided to design a cup that will better suit our measuring needs and improve the stirring process. The cup was designed in SolidWorks, a 3D design software and printed on a 3D printer. It is composed of several parts and shown in Fig. 2. The full 3D computer model can be found in the attachment. Fig. 3. The circulation of the water in the designed cup. Note the false bottom that holds the propeller. The water can flow around and under the false bottom, while the honey can only sit on top if it. Fig. 2. The 3D design for the stirring cup. The base holds the electric motor that feeds into the body of the cup. The body is composed of two parts which separate the honey from the water. A filter (coloured brown) which has a number of 0.5mm holes drilled in it serves to let the water, but not the honey, pass through. The top part of the cup has 4 slots - 2 to hold laser diodes and 2 to hold photoresistors. The holes are separated from the water by a thin foil. The device that stirs the cup is, in a way, the cup itself. The electric motor is placed in the slot at the bottom with a propeller feeding into the body of the cup. Inside the cup there is a false bottom on top of which which there is a filter that stops the sugar and honey from going through. The propeller is placed inside the inner element of the false bottom. This way we make sure that the motor provides radial as well as tangential circulation of the water. The water circulation is shown in Fig. 3. The kind of stirring achieved ensures that the mixture is homogeneous radially. Also, because of the false bottom, the sugar and honey are immediately propelled upwards to the top of the cup where they would usually end up only after a prolonged period of stirring. The filter prevents the honey from dropping directly on the propeller so that it doesn t slow it down due to its high viscosity. The four slots on the sides of the cup are see- through. We pass laser lights through one pair of them, and using photoresistors mounted in the other pair, we detect the light. One pair is placed near the top of the cup and the other just above the false bottom to be able to compare the readings in different parts of the cup. Using specialized software, we simulated the tea inside this kind of cup being propelled by the motor. We also ran a simulation on an average household mug with the exact same parameters describing the motor motion. The results are displayed in Fig. 4 and Fig. 5 and represent the distribution of the velocity of the water solution in the cross sections of the cups. There are some obvious differences between the two cases. In the average cup the speed is highest on the vertical axis passing through the center of the cup and then drastically falls off as we move towards the rim. In the designed cup, the speed is more or less equally distributed along the radius of the cup as we previously predicted. This means that our cup will be better at dispersing the sugar and honey than the average one. The vertical speed of the water in the average cup seems to be greater than the speed in our cup, but that is not an important factor since we are only interested in the homogeneity of the distribution, and not the relative sizes of the velocities in it. C. Measurement To design an instrument capable of measuring the homogeneity of the solution, we used the effects of sugar

3 3 Fig. 4. The simulated distribution of the velocity field of the water solution in our specifically designed cup. as a result the intensity of the light passing through will rise. At some point virtually none of them will affect the intensity of light. When the intensity will have stabilized, we can say that our mixture is homogeneous. This gives us a radial measure of homogeneity. To ensure vertical homogeneity, we use two lasers, one at the bottom of the cup and one at the top. If the intensities of both lasers, as detected by the photoresistors, are the same and stabilized in time as described, we consider the solution to be homogeneous. When honey is used in tea it considerably changes the mixture. Sugar, when dissolved, does not change the opacity of the liquid by much if it is used in small amounts. Honey, on the other hand will change the opacity and as a result the measured intensity of light. As we stir the tea, the intensity will start to drop since there is more and more honey dissolved that blocks the light. The amount of intensity itself does not concern us, but the behaviour of it in time is essential. If the intensity is stable, no more honey is being moved around and the mixture is well-stirred. As with the sugar case, we use two lasers and compare the intensities in time. If they are the same and if they are stable we say that the solution is homogeneous. When added to the tea, the lemon slice will start to release its juice. The juice, just like honey, changes the opacity of the mixture. It takes a lot more time for lemon to fully release the juice than to dissolve the honey so we expect a slightly different, but time invariant, reading of the intensities of two lasers. Namely, one laser would shine through the area of greater lemon juice concentration and therefore lower opacity than the other. The time it takes for the lemon juice to uniformly distribute is comparable to the time it takes for the lemon to release its juice. The effect of the lemon on the solubility of honey should not be considerable. Since lemon juice is less dense than water, the slice will float on the top and it shouldn t affect the flow considerably. II. RESULTS Fig. 5. The simulated distribution of the velocity field of the water solution in an average household cup. dissolved in water on light that is passing through it. When the device is stirring, the tea, honey, or sugar particles are forced to move. Gradually, as the honey, or sugar, dissolve, there is less of it moving through the tea. Let us first examine the case for sugar. Suppose we shine a laser radially through one side of the cup and measure the intensity of light on the other side with a standard CdS photoresistor. The amount of sugar granules in the tea will affect the measured intensity of light, i.e. the fewer sugar particles there are, the stronger the measured intensity. Since the sugar granules dissolve in tea, in time there will be fewer and fewer of them and First we present the simple case of measurements with an average household glass cup and then the case with the 3D printed cup to be able to compare the two. A. The glass cup We used a glass cup, two lasers, two CdS photoresistors, an electric motor, tea, sugar and honey. The photoresistors were connected to an Arduino device that enabled us to track the intensity of light and graph it in real time. The schematic of the setup is given in Fig. 6. The electric motor is placed underneath the cup with its axle running through the bottom of the cup and the propeller blades placed just above the sugar (or honey) that had settled on the bottom. The Arduino actively

4 4 Laser 1 Arduino Laser 2 Arduino Electromotor 1.5V Fig. 6. A schematic view of the measuring setup. Fig. 7. Intensity over time for honey for an ordinary cup, measured the resistance of the photoresistors and drew a diagram that represented the intensity (for standard CdS photoresistors the intensity is approximately inversely proportional its resistance).the Arduino code and the setup schematic is included in the attachment. The graphs for sugar and honey are shown in Fig. 7 and Fig. 8, respectively. The orange curve is the intensity on the lower photoresistor and the blue curve on the upper one. The x-axis shows the time and the y-axis (given in arbitrary units) is directly proportional to intensity. We made multiple measurements with both honey and sugar. We used the graphs to establish the time needed for the honey/sugar to mix with the water. As mentioned earlier, we consider the mixture homogeneous when the curve is generally stabilized. First let s look at Fig. 7. The electric motor was turned on at t = 6s which is indicated by a sharp spike in the graph. The intensity of the lower resistor (orange) spiked up because the honey that was at the bottom was blocking the light so when the motor started spinning the honey went up and the intensity increased. As time went on the intensities dropped below a level present before the motor started to spin. At around t = 20s both intensities were stable. On Fig. 8 the motor was turned on at t = 5s as indicated by a sharp fall in intensity. This happened because the sugar crystals immediately got in the way of laser light. As sugar dissolved, the intensity increased and at around t = 32s it was stable enough to consider the tea and sugar to be mixed. The intensities of both laser lights aren t completely equal because the lasers aren t perfectly identical and neither are the photoresistors. We made multiple measurements, analyzed the results and obtained the following dissolution times: Honey (21 ± 1)s Sugar (32 ± 2)s The errors in the results are estimated. These results serve as a comparison to our 3D printed cup. Fig. 8. Intensity over time for sugar for an ordinary cup, given in arbitrary units. B. The 3D printed cup For this cup we carried out several measurements as well. The dimensions of the printed cup and the household cup are practically the same so we find it reasonable to directly compare them. The same motor was used in both measurements, as well as the same propeller. The experimental set up with the 3D printed cup can be seen in Figure 9. The motor was connected to a 1.5 Volt battery and light from the lasers was passed through the cup apertures. The photoresistors and Arduino were used to measure the intensity of light. Sample measurement graphs can be seen in Fig. 10 and Fig. 11. On the graph for honey the motor was started at t = 2.5s, as noted by a sharp fall in both curves, much unlike the case for the household cup. We believe the reason for this is that the honey was not blocking either of the lasers when it was settled on the bottom. The intensity curves stabilized very quickly. The graph for sugar is very similar to the one for the household cup. It takes longer for the sugar to dissolve than it does for honey. That is because honey is already sugar dissolved in water but in a greater concentration. Nevertheless, in both cases the dissolution was much faster than in the household cup. We made several measurements, performed a statistical analysis and obtained the following dissolution times:

5 5 Fig. 9. The experimental set up Honey (9 ± 1)s Sugar (19 ± 2)s The results show that both honey and sugar are dissolved considerably faster in our cup than in the average household one. Fig. 11. Intensity over time for sugar for the 3D printed cup, Fig. 10. Intensity over time for honey for the 3D printed cup, C. Adding a lemon slice After conducting the previous measurements we repeated all of them with a lemon slice. The graphs of the intensities were identical as before and the lemon had no effect on the dissolving of honey or sugar. While conducting the measurements without the lemon, the propeller of the motor produced a small vortex in the cup. Adding the lemon slice additionally decreased the size of the vortex. It could be expected that with a smaller vortex the rate of stirring also decreased but we haven t seen that happen. III. CONCLUSION The aim of this task was to construct a device that will mix the honey in the tea in the shortest time. It is not possible to construct a device and claim that it is the fastest since one would somehow have to compare it to all possible devices or use the mathematics of the behaviour of fluids inside the cup which is complex to say the least. However, using some simple physical reasoning we managed to drastically lower the time needed to stir the honey/sugar, compared to the average cup. We managed to improve the time that it takes for honey to dissolve by 57% and the time it takes for the sugar to dissolve by 41%. We consider that our cup, with some additional modification, would be usable in everyday life. The entire cup can be powered by a single 1.5V battery.