徐振 徐正華 彭健 譚凱

【Abstract】This article describes a simple bath temperature model design. First， a list of mathematical physics equations and continuous N-S equation and momentum equation are consisting of convection-diffusion equation to solve the bath fluid （water molecules） convection-diffusion problems. Then calculated by MATLAB simulation software is used to analyze simulation results and verify the rationality of the design， so it provides the basis for the water temperature.

【Keywords】Convection；The Linear Regression Equation；Water Temperature Model

Water temperature optimization model based on matlab

Xu Zhen1，Xu Zhenghua1*，Peng Jian2，Tan Kai3

1.Institution of Mechanical Engineering， University of the South of China，421000， Hengyang，China

1.Institution of Mathematics， University of the South of China，421000， Hengyang，China

2.Institution of Computer Science and Technology，University of the South of China，421000， Hengyang，China

3.Institution of Nuclear Science and Technology，University of the South of China，421000， Hengyang，China

Introduction

In this article， bath tub， not with auxiliary heating system insulation， but it is just a simple tank. When a person is in the bath， in order to maintain the temperature of the bath and to make people feel a degree of comfort， faucet will continue to increase a certain amount of hot water bath. When the water bath reaches its capacity， the excess water will overflow through the gutter. In order to maintain a constant temperature， a certain amount of hot water is added continuously. Supposing the bath already filled with hot water and convection dominating diffusion at this time， a bath water temperature model performance can be showed by a temperature and time curve variation proves by convection diffusion equation and the method of finite element method[1].

1、Convection analysis of bath water

1.1 Convective Heat Transfer Conditions Hypothesis

a-the fluid is a continuous medium；b-the fluid is incompressible Newtonian one； c-all object parameters are constant （ρ ， Cp， k， μ）； d-the variables involved are all different； e- bath walls are adiabatic， namely， the heat will be diffused only by the direct contact with the surrounding environment of the surface；f- cross-sectional size of the bath in the direction of vertical flow maintains the same.

1.2 Symbol Description

Ρ- density of water； C- the specific heat capacity of water； k- thermal properties of water； μ-dynamic viscosity of water； v- infinitesimal fluid velocity along the x-axis direction； w-infinitesimal fluid velocity along the z-axis direction； t-time； P- aquarium water pressure； T-temperature；T1-aquarium water temperature initial time； T2-environmental temperature；Qcovn- The flow of faucet； S- The cross-sectional area of bath crock.

2、Model preparation

According to the basic knowledge of heat transfer， Convection can be divided into natural convection and forced convection[2]. The temperature gradient in the fluid can cause the change of density gradient. If the low density fluid is lower， the high density fluid will be natural convection under the action of gravity.

3、Model Establishment and simulation

In order to facilitate the discussion and understanding of the model， we will first study a simple rectangular bath with its length， breadth and height were a， b and c respectively. And we select the ordinate origin in the center of the rectangular side.

Figure1.the ordinate origin in the center of the rectangular side

In the process of lateral convection， we only analyze its cross-section， that is， only consider the two-dimensional variation of the water temperature in the bath. Thus， three-dimensional convective heat conduction function becomes two-dimensional[3] as follows：

Using MATLAB software[4] and computer programming， we get the temperature model inthermal diffusion coefficient of convectiondiffusionequation. Within the interval [0.023， 0.101]， time T = 2 h = 0.06， bath water temperature in the cross section temperature field distribution of 2 d and 3 d diagram.As you can see from figure 2 and figure 3， as the distance is more and more distant from the injection port， field distribution of bath water temperature is a gradient descent， and it can reflectthe actual distribution.

Figure2.Pr = 0.06， get T = 2 （hour） moment in the cross section temperature field distribution of 2 d diagram

Figure3.Pr = 0.101， get T = 2 （hour） moment in the cross section temperature field distribution of 3 d diagram

4、Conclusion

By solving the equation of temperature to determine the spatial and temporal distribution of water bath in convection heat transfer， we can determine a certain velocity and flow faucet using MATLAB software simulation. When considering the bathtub section temperature changes， the actual bathtub is just a three-dimensional volume. According to the experimental data， simulation result is consistent with the actual phenomenon， and it also verifies the reasonableness of the temperature model design.

References

[1] Cao Bo， Wei Yong Tao， Yu Jianhua.Transient heat conduction equation of Space Finite Element Method [J] Southwest University for Nationalities （Natural Science）， 2001，04： 429-432.

[2] Xia Ming. Fluid-Africa center symmetry point source Attenuation of shock wavefront testing and propagation characteristics[D]. Nanjing University of Science and Technology， 2003.3.

[3] Liu Ciqun.Analytical solution uniform flow in the two-dimensional diffusion equation [J] Hydrodynamics Research and Development， 1987.02： 81-84.

[4] Li Ming. MATLAB Solution of Partial Differential Equations [J] Hunan Agricultural Machinery， 2010.05： 89-91.

作者簡介

徐振（1989.06-），男，漢族，山東曲阜人，南華大學機械工程學院研究生，研究方向：機電測試及控制。