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In this paper, we do the best to extend the time of bathing without wasting too much water. In the Lumped Parameter model (1) , we predict the change of temperature by the use of Runge-Kutta method. We also consider the heat dissipation between the water and the bathtub wall as well as the air. And we discuss the change of temperature under different height of water surface. In our opinion, when a person is sitting in the tub, he just gets heat energy from water. We add a new equation of conservation of energy. Compared with the heat dissipation on the water, the body has less influence on the temperature of water. A person is in motion instead of stationary when bathing. In other to correct the evaporative coefficient of water, we introduce a correction coefficient (we call it α). Under the depth of 0.43 meters, we can determine to reach the value of the correction coefficient (α) of 38 within 15 minutes, where α=1.15. After researching the ℃ relationship between different rate of drainage and temperature, we use α to predict the time arriving at 38 for the second time. Compared with the longest suitable bathing time, it is longer ℃ than 30 minutes in general. It can be proved that our strategy is able to keep the temperature throughout the bathtub and as close as possible to the initial temperature without wasting too much water. A bathtub, bath or tub (informal),a water pipe equipment for the use of bathing or showering, is usually equipped in the bathroom .Namely, it is a large capacity for holding water in which a person may bathe. A number of modern bathtubs are made of fiberglass or acrylic, but alternatives are available in enamel on steel or cast iron. Recently, because of the growing popularity of acrylic bathtub with heating system, most different shapes are becoming available. The analysis of the comfort of bathing has been a universal concern. Moreover, there have been a lot of researches into the studies of the standard and shape of the bathtub. The problem requires us to do several jobs to determine the best strategy the person in the bathtub can adopt to keep the temperature even throughout the bathtub. We need to research the change trend of water temperature with a person in the water, who has motions. In this problem, we need to consider the heat dissipation between water and air as well as that between water and the surface tub. Further, we consider the influence from the parameter with some motions in the bathtub.

Keyword: Goal programming Heat transfer theory Newton cooling formula Abstract. In this paper, we build the model to determine the best strategy that the person in the bathtub can keep the temperature. Firstly, we assume that the bathtub is a fixed size water containment vessel. In case of no external heat source, we set up a water temperature model of the bathtub by using the knowledge of heat transfer, get the figure of temperature change with time. Secondly, adding a single faucet which can inject hot water continuously keeps the temperature of the water in the bathtub. Let the heat of the hot water equal to the loss of heat,obtaining the relationship between the temperature and the flow rate of the hot water. By listing the target of water-saving amount, constrained conditions of temperature, constrained conditions of water consumption, and set up the goal programming model. Using MATLAB to solve the model, we obtain the best strategy that the minimum water consumption is 2016L,temperature of hot water injected is 54.5℃, and the flow rate is 0.28L/s .

we use the governing equation and continuity equation to build a two-dimensional model of water temperature on micro element theory. Because the partial differential equation is difficult to obtain its analytical solution, we use finite difference method to work out numerical solution of the model. Introduction In the bathtub, water temperature is determined by heat exchange between the interior and exterior parts of water. The heat exchange between the water and exterior environment is determined by the temperature brought by inflowing hot water and that taken away by overflow.The heat exchange in the interior of water including:The water vertical flowing caused heat exchange and the heat exchange caused by water density convection. Problem analysis There are many other factors influence water temperature in the bathtub, mainly including: The liquidity of water: including the heat conduction and diffusion, the geometry of a water, and the location of overflow drain, etc. The heat exchange at the water surface:  Convection: Water and air heat transfer between two different media mainly by convection  Heat conduction: If the air temperature and water temperature is different, energy exchange can occur through heat conduction.  Evaporation: On the border of water and air, when some water molecules have enough energy, they will escape to the air. Water and energy exchange in the bathtub: the heat exchange of bathtub with water is by solid heat conduction. Assumption  The initial air temperature and humidity are fixed value in the bathroom.  The bathroom is an independent system.  Per unit time, the addition of hot water is constant.  The temperature of the water flowing into bathtub is 45°C.  Initial water temperature in the bathtub is 39°C. Water Temperature Distribution of Bathtub We consider the distribution of temperature in the water and we use the finite element method to analyze the temperature distribution in the water in this model. 2nd Workshop on Advanced Research and Technology in Industry Applications (WARTIA 2016) © 2016. The authors Published by Atlantis Press 603 Fig.1 The infinitesimal Assuming that bathtub isothermal surface is horizontal upward, the water temperature structure in bathtub composed of different temperature of countless isothermal surfaces . From figure 1, the infinitesimal of hierarchical model is presented. We take out an infinitesimal body along the constant-height surface. According to the heat conduction differential equations

It is quite difficult to get the temperature field of flow liquid from the three-dimensional perspective by hand calculation. But we figure out an unusual method to obtain the temperature, and we also use 3D software to simulate and test. In order to facilitate the calculation, we use cuboid bathtub as our model to study. In our model, we analyze the temperature field in an innovative perspective. We take part of the bathtub water around the faucet as a heat source based on Boundary Layer Theory, which simplifies the influence of the inlet water. We separate the temperature field into three directions and make analysis one by one to modify the difficulty of calculating it.As a result, we obtain the concrete distribution of the temperature field in three dimensions. We use the division of heat source and cooler area to get the temperature field around the faucet. Considering the similarity of the heat-dissipating pattern between bathtubs and fins, we optimize the calculation of the temperature field. Finally, we test our model with the simulation software ANSYS and the results support our theoretical calculation, which demonstrates that our model is reasonable and practical.

In this paper, the change of water temperature in the bathtub is mainly studied. Through the simulation of the water temperature distribution in time and space and the construction of the thermal conduction model, we discuss the water injection scheme. According to the Fourier's law and Newton cooling law, we establish a one-dimensional heat conduction differential model with water temperature, bath depth and time as variables. Then the model is classified into two parts because of the heat source's existence. By the evaporation model we can conclude the relationship between the water temperature and time, then solve the partial differential equation in the usage of implicit format of finite difference method. Equation of the water speed and temperature and the optimal scheme can be determined through the Improved Genetic Algorithm on the base of law of conservation of energy. The judging equation can be obtained after the analysis of the various influence factors of the strategy plan. Finally, we draw a conclusion that: the shape and volume of bathtub infect the strategy most.

Several mathematical models are proposed to determine the shapes of bathtub and manipulative time, which are not only related to the heat source, the maximum capacity and volume of water, but also the motions of user and ingredients of water.

Firstly we build a set of models to describe the temperature of the bathtub water in time. In this section, we analyze the variation process of water temperature under the condition of Natural cooling and injecting hot water respectively through a macro perspective. The Newton’s cooling law and the energy conservation law are used in these two models orderly. And we draw the conclusion that the temperature has a linear correlation with time under the condition of small change in temperature. Secondly, we improve our model by taking natural cooling factor into consideration while injecting hot water. Moreover, on the basis of the analysis of the last step, we simplify our model. Then we use the optimal analysis method to find the best strategy on the foundation of our improved model. And the best strategy is to keep the best rate of injected water constantly. In this way, we can keep the temperature of water as close as possible to the comfortable temperature without wasting too much water only by adjusting the appropriate rate of the injected water. Introduction Imagine you drag your tired body home after a busy day, have a hot bath and then read a book on the sofa with clean and relax. Isn’t it an exciting thing in a day? However, the problem is that the bathwater will get cooler after a while, and you have to make your enjoy time interrupted to add hot water into the bathtub. But meanwhile you don’t want to waste too much water. In this case, we should propose a solution that makes sure you have a comfortable bath without paying excessive costs for water. Fundamental Assumptions ●No heat loss and no thermal power conversion in the system. And the system keep a dynamic balance in the heat exchange process. ●The bathroom is a closed system without ventilation so that we don’t take the impact of the wind in the cooling process into consideration. ●The evaporation of water is not included in the calculating of water. ●The temperature in the bathroom is constant, maintained at 25°C; And the temperature of the injected water is constant. ●The location of the faucet in the bathtub is fixed. To reduce water consumption, the faucet is fixed at the bottom of the bathtub. ● After entering the bathtub, the quality of the water in the bathtub stays the same 2nd Workshop on Advanced Research and Technology in Industry Applications (WARTIA 2016) © 2016. The authors Published by Atlantis Press 761 The traditional models about thermodynamics Table.1Parameter description Parameter Description h Convective heat transfer coefficient 1 T The water temperature of any time r T Indoor temperature A Cross section area c Specific heat 0 m The quality of the water in the bathtub 0 T The initial temperature e T The temperature of water when it starts to cool 1 q The flow rate of the injected water,the quality of injection water per second s T The target temperature q ∆ The speed difference between the inflow and outflow water T The temperature of the injection water 1 m The quality of injection water 2 m The quality of outflow water The temperature of the bathtub water in space and time is solved by using three traditional mathematical models, a model of nature cooling, a model of injecting hot water, and a model of the water temperature distribution in the bathtub. The first two models describe the temperature in the bathtub in time from a macro perspective. The temperature distribution of the water in bathtub shows the temperature in the bathtub in space from a micro perspective. Newton gave a description on the decrease rate of the object ‘s temperature with the law of cooling. And it reveals that when the object’s temperature is higher than the surrounding media’s, it will transfer heat to others and cool down following the law. When there is temperature difference between the surface and the surrounding, the heat loss from the unit area per unit time is proportional to the difference, and the proportional coefficient is called heat transfer coefficient[1]. It can be described as follows: ( ) 1 r dQ h T T dSdt = − Make integration on the cross section of heat conduction, ( ) 1 r dQ Ah T T dt = − According to the formula of heat calculation, when the temperature changed from 0 T to 1 T ,the heat emitted ( ) ( ) 1 0 0 1 0 1 r dT Q cm T T cm Ah T T dt = − − = − After derivation calculus to t , we can get Thus 1 kt s r T e T + = + , Where 0 Ah k cm = − , s is a constant. We assume the temperature when it starts to cool is e T , and we can determine the value of based on s initial condition , that is ( ) ln e r s T T = − Thus, the temperature of the water at any time when it naturally cool can be described as follows: ( ) 1 kt e r r T e T T T = − + Based on the formula of heat calculation, and the law of conservation of energy, we can obtain the equation of the heat exchange on a macroscopic level: ( ) ( )( ) 1 2 0 2 2 1 cm T T c m q t m T T − = − + ∆ − −  [2]. We approximately simplifies the equation, and it turns to: ( ) 0 1 1 1 1 / m q t T t q T q T + ∆ ∆ ∆ + =  . Assume t ∆  tends to infinity and we can get the following