1、 2016 Mathematical Contest in Modeling (MCM/ICM) Summary Sheet Summary A traditional bathtub cannot be reheated by itself, so users have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propose an optimal
2、 strategy for users to keep the temperature even and close to initial temperature and decrease water consumption. To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas. We establis
3、h two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simu
4、late the change of temperature field via CFD. As for the second sub-model, we define an iteration consisting of two process: heating and standby. According to energy conservation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time
5、 of adding hot water. We also use CFD to simulate the temperature field in second sub-model. In consideration of evaporation, we correct the results of sub-models referring to some scientists studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found
6、to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us. Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath ad
7、ditives. We focus on the influence of those factors to heat transfer and then conduct sensitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrease heat transfer and save water. Based on our model analysis and co
8、nclusions, we propose the optimal strategy for the user in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life. Key words: Heat transfer Thermodynamic system CFD Energy conservationFor office use only T1
9、T2 T3 T4 For office use only F1 F2 F3 F4 Team Control Number 44398 Problem Chosen A Team #44398 Page 2 of 51 Enjoy a Cozy and Green Bath Contents 1 Introduction 4 1.1 Background . 4 1.2 Literature Review . 4 1.3 Restatement of the Problem 4 2 Assumptions and Justification 6 3 Notations 7 4 Model Ove
10、rview . 7 5 Sub-model I : Adding Water Continuously 8 5.1 Model Establishment . 9 5.1.1 Control Equations and Boundary Conditions 9 5.1.2 Definition of the Mean Temperature 11 5.1.3 Determination of Heat Transfer Capacity . 11 5.2 Results 13 5.2.1 Determination of Parameters 13 5.2.2 Calculating Res
11、ults . 14 5.2.3 Simulating Results 15 6 Sub-model II: Adding Water Discontinuously 18 6.1 Heating Model 18 6.1.1 Control Equations and Boundary Conditions 18 6.1.2 Determination of Inflow Time and Amount . 19 6.2 Standby Model . 20 6.2.1 Process Analysis 20 6.2.2 Calculation of Parameters 20 6.3 Res
12、ults 21 6.3.1 Determination of Parameters 21 6.3.2 Calculating Results . 23 Team #44398 Page 3 of 51 6.3.3 Simulating Results 23 6.4 Conclusion 27 7 Correction and Contrast of Sub-Models . 27 7.1 Correction with Evaporation Heat Transfer 27 7.1.1 Correction Principle 27 7.1.2 Correction Results . 28
13、 7.2 Contrast of Two Sub-Models 30 7.2.1 Evaluation Criteria 30 7.2.2 Determination of Water Consumption 30 7.2.3 Conclusion . 31 8 Model Analysis and Sensitivity Analysis . 31 8.1 The Influence of Different Bathtubs 32 8.1.1 Different Volumes of Bathtubs 32 8.1.2 Different Shapes of Bathtubs . 34 8
14、.2 The Influence of Person in Bathtub . 36 8.2.1 When the Person Remains Static in a Bathtub 36 8.2.2 When the Person Moves in a Bathtub 37 8.2.3 Results Analysis and Sensitivity Analysis . 38 8.3 The Influence of Bubble Bath Additives . 42 8.4 The Influence of Radiation Heat Transfer 44 8.5 Conclus
15、ion 45 9 Further Discussion . 45 9.1 Different Distribution of Inflow Faucets 45 9.2 Model Application . 46 10 Strength and Weakness . 47 10.1 Strength . 47 10.2 Weakness . 47 Report . 49 Reference . 50 Team #44398 Page 4 of 51 1 Introduction 1.1 Background Bathing in a tub is a perfect choice for t
16、hose who have been worn out after a long days working. A traditional bathtub is a simply water containment vessel without a secondary heating system or circulating jets. Thus the temperature of water in bathtub declines noticeably as time goes by, which will influent the experience of bathing. As a
17、result, the bathing person needs to add a constant trickle of hot water from a faucet to reheat the bathing water. This way of bathing will result in waste of water because when the capacity of the bathtub is reached, excess water overflows the tub. An optimal bathing strategy is required for the pe
18、rson in a bathtub to get comfortable bathing experience while reducing the waste of water. 1.2 Literature Review Korean physicist Gi-Beum Kim analyzed heat loss through free surface of water contained in bathtub due to conduction and evaporation 1. He derived a relational equation based on the basic
19、 theory of heat transfer to evaluate the performance of bath tubes. The major heat loss was found to be due to evaporation. Moreover, he found out that the speed of heat loss depends more on the humidity of the bathroom than the temperature of water contained in the bathtub. So, it is best to mainta
20、in the temperature of bathtub water to be between 41 to 45 and the humidity of bathroom to be 95%. When it comes to convective heat transfer in bathtub, many studies can be referred to. Newtons law of cooling states that the rate of heat loss of a body is proportional to the difference in temperatur
21、es between the body and its surroundings while under the effects of a breeze 2. Claude-Louis Navier and George Gabriel Stokes described the motion of viscous fluid substances with the NavierStokes equations. Those equations may be used to model the weather, ocean currents, water flow in a pipe and a
22、ir flow around a wing 3. In addition, some numerical simulation software are applied in solving and analyzing problems that involve fluid flows. For example, Computational Fluid Dynamics (CFD) is a common one used to perform the calculations required to simulate the interaction of liquids and gases
23、with surfaces defined by boundary conditions 4. 1.3 Restatement of the Problem We are required to establish a model to determine the change of water Team #44398 Page 5 of 51 temperature in space and time. Then we are expected to propose the best strategy for the person in the bathtub to keep the wat
24、er temperature close to initial temperature and even throughout the tub. Reduction of waste of water is also needed. In addition, we have to consider the impact of different conditions on our model, such as different shapes and volumes of the bathtub, etc. In order to solve those problems, we will p
25、roceed as follows: Stating assumptions. By stating our assumptions, we will narrow the focus of our approach towards the problems and provide some insight into bathtub water temperature issues. Making notations. We will give some notations which are important for us to clarify our models. Presenting
26、 our model. In order to investigate the problem deeper, we divide our model into two sub-models. One is a steady convection heat transfer sub-model in which hot water is added constantly. The other one is an unsteady convection heat transfer sub-model where hot water is added discontinuously. Defini
27、ng evaluation criteria and comparing sub-models. We define two main criteria to evaluate our model: the mean temperature of bath water and the amount of inflow water. Analysis of influencing factors. In term of the impact of different factors on our model, we take those into consideration: the shape
28、 and volume of the tub, the shape/volume/temperature of the person in the bathtub, the motions made by the person in the bathtub and adding a bubble bath additive initially. Model testing and sensitivity analysis. With the criteria defined before, we evaluate the reliability of our model and do the
29、sensitivity analysis. Further discussion. We discuss about different ways to arrange inflow faucets. Then we improve our model to apply them in reality. Evaluating the model. We discuss about the strengths and weaknesses of our model. Team #44398 Page 6 of 51 2 Assumptions and Justification To simpl
30、ify the problem and make it convenient for us to simulate real-life conditions, we make the following basic assumptions, each of which is properly justified. The bath water is incompressible Non-Newtonian fluid. The incompressible Non-Newtonian fluid is the basis of NavierStokes equations which are
31、introduced to simulate the flow of bath water. All the physical properties of bath water, bathtub and air are assumed to be stable. The change of those properties like specific heat, thermal conductivity and density is rather small according to some studies 5. It is complicated and unnecessary to co
32、nsider these little change so we ignore them. There is no internal heat source in the system consisting of bathtub, hot water and air. Before the person lies in the bathtub, no internal heat source exist except the system components. The circumstance where the person is in the bathtub will be invest
33、igated in our later discussion. We ignore radiative thermal exchange. According to Stefan-Boltzmanns law, the radiative thermal exchange can be ignored when the temperature is low. Refer to industrial standard 6, the temperature in bathroom is lower than 100 , so it is reasonable for us to make this
34、 assumption. The temperature of the adding hot water from the faucet is stable. This hypothesis can be easily achieved in reality and will simplify our process of solving the problem. Team #44398 Page 7 of 51 3 Notations Table 1 Notations Symbols Definition Unit h Convection heat transfer coefficien
35、t 2W/ m K k Thermal conductivity W/ m K pc Specific heat J/ kg K Density 2kg/m Thickness m t Temperature K 、 Time s min h、 、 mq Mass flow kg/s Heat transfer power W T A period of time s min h、 、 V Volume 3mL、 Mm、 Mass kg A Area 2m a b c、 、 The size of a bathtub 3m where we define the main parameters
36、 while specific value of those parameters will be given later. 4 Model Overview In our basic model, we aim at three goals: keeping the temperature as even as possible, making it close to the initial temperature and decreasing the water consumption. We start with the simple sub-model where hot water
37、is added constantly. At first we introduce convection heat transfer control equations in rectangular coordinate system. Then we define the mean temperature of bath water. Afterwards, we introduce Newton cooling formula to determine heat transfer capacity. After deriving the value of parameters, we g
38、et calculating results via formula deduction and simulating results via CFD. Secondly, we present the complicated sub-model in which hot water is added discontinuously. We define an iteration consisting of two process: heating and standby. As for heating process, we derive control equations and boun
39、dary conditions. As for standby process, considering energy conservation law, we deduce the relationship of total heat dissipating capacity and time. Team #44398 Page 8 of 51 Then we determine the time and amount of added hot water. After deriving the value of parameters, we get calculating results
40、via formula deduction and simulating results via CFD. At last, we define two criteria to evaluate those two ways of adding hot water. Then we propose optimal strategy for the user in a bathtub. The whole modeling process can be shown as follows. Fig.1 Modeling process 5 Sub-model I : Adding Water Co
41、ntinuously We first establish the sub-model based on the condition that a person add water continuously to reheat the bathing water. Then we use Computational Fluid Dynamics (CFD) to simulate the change of water temperature in the bathtub. At last, we evaluate the model with the criteria which have
42、been defined before. Team #44398 Page 9 of 51 5.1 Model Establishment Since we try to keep the temperature of the hot water in bathtub to be even, we have to derive the amount of inflow water and the energy dissipated by the hot water into the air. We derive the basic convection heat transfer contro
43、l equations based on the former scientists achievement. Then, we define the mean temperature of bath water. Afterwards, we determine two types of heat transfer: the boundary heat transfer and the evaporation heat transfer. Combining thermodynamic formulas, we derive calculating results. Via Fluent s
44、oftware, we get simulation results. 5.1.1 Control Equations and Boundary Conditions According to thermodynamics knowledge, we recall on basic convection heat transfer control equations in rectangular coordinate system. Those equations show the relationship of the temperature of the bathtub water in
45、space. We assume the hot water in the bathtub as a cube. Then we put it into a rectangular coordinate system. The length, width, and height of it is a , b and c . Fig.2 The water cube in the rectangular coordinate system In the basis of this, we introduce the following equations 5: Continuity equati
46、on: +0u v wx y z （ 1） where the first component is the change of fluid mass along the X-ray. The second component is the change of fluid mass along the Y-ray. And the third component is the change of fluid mass along the Z-ray. The sum of the change in mass along those three directions is zero. Team
47、 #44398 Page 10 of 51 Moment differential equation (N-S equations): 2222 2 2( ) ( )u u u p u u uu v wx y z x x y z （ 2） 2 2 22 2 2( ) ( )v v v p v v vu v wx y z y x y z （ 3） 2222 2 2( ) ( )w w w p w w wu v w gx y z z x y z （ 4） Energy differential equation: 2 2 22 2 2( + ) ( )p t t t t t tc u v wx y
48、 z x y z （ 5） where the left three components are convection terms while the right three components are conduction terms. Having derived those equations, we give the boundary conditions listed as follows: In the inflow water area intt （ 6） On the top surface of bath water, it transfer heat directly into air without heat conduction, so we have 1t h t
