Part II ........................................................................................................... 16 VII. Strengths and Weaknesses....................................................................... 17 VIII. Explanation for users .............................................................................. 18 IX. References......................................................................................................19
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Team Control Number
46731
Problem Chosen
A
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this page.
Summary
Our experience of everyday life tells us that, bathing quality is relevant to the temperature of the water. Hence the optimality of the water-adding strategy in bathing process has become an issue.
Part II Finding the best strategy ...................................................................12 VI. Results ........................................................................................................ 12
Key words: bathtub; heat-conduction; temperature variation and distribution;
water strategy; finite volume method;
Team # 46731
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Contents
Summary ................................................................................................................ 1 I. Introduction......................................................................................................4 II. Analysis ........................................................................................................4 III. Assumptions ................................................................................................. 6 IV. Notation ........................................................................................................ 7 V. Models .......................................................................................................... 8
Part I.............................................................................................................12 Results of Air’s Heat Radiation Model ............................................12 Results of Bathtub Wall’s Heat Radiation........................................13 Results of “Person In”......................................................................13 Results of “Hot Water In” ................................................................ 13 Results of bubble existed model ...................................................... 15 Results of Water Temperature Variation and Distribution Model....15
2016
MCM/ICM Summary Sheet (Your team's summary should be included as the first page of your electronic submission.) Type a summary of your results on this page. Do not include the name of your school, advisor, or team members on
In this article, we establish two models. One is water temperature variation and distribution model and the other one is finding best strategy model. We put forward some acceptable hypothesis to simplify the model. What’s more, we clear the meaning of the word “noticeably”.
The second part aims to finding the best water strategy. Different strategies are considered for adding hot water and the amount of the hot water. The key of strategies is how to choose the moment of opening the faucet and the lasting time. By developing an optimization model of the strategy, we manage to determine a strategy for different users and extend the model with a finite volume method. Under the condition of the same bathing time, we study what’s the ideal time to add the hot water to make it to the initial temperature.
The first part has five sections: air’s heat radiation, bathtub wall’s heat radiation, person in, hot water in, bubble existed. We discuss some factors that affect water temperature, such as the shape and the volume of the bathtub and person, especially the motions made by the person in the bathtub because the temperature in the bathtub has a great connection with person. Finally, we get the water temperature variation and distribution model.
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We transform the weight p. With the restrict condition of the temperature in the bath and the amount of the hot water, the question can be transformed to a nonlinear optimization. We get the conclusion and optimal results through the PDE (Partial Differential Equations) toolbox of MATLAB. And the result shows a clear superiority for choosing a suitable strategy for different users.