Temperature Secret in Bathtub: A Model of Temperature Distribution of Bathtub Based on Heat Conduction Equation

We use the multidimensional heat conduction and heat transfer equations to model the temperature distribution of water in a bathtub by solving partial differential equations. We address optimal water addition and bathtub design. First, we establish a water surface cooling model using Newton's law of cooling to simulate heat exchange between air and water. Without new heat sources, the water temperature reaches a minimum in 40 minutes. We then simulate adding hot water with a one-dimensional heat conduction model, including air cooling effects. We determine that the optimal heat input is 80 Joules and the optimal water velocity is 0.042 m/s to maintain temperature and save water. The ideal bathtub dimensions are 1.5m length, 0.6m width, 0.42m depth, with rounded corners. Using finite difference methods and MATLAB's Pdetool, we solve the heat conduction equation and verify numerical stability, discussing the model's pros and cons and suggesting improvements.