Water temperature distribution of bathtub

Abstract

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,