2016 Mathematical Contest in Modeling (MCM/ICM) Summary Sheet Summary
A traditional bathtub cannot be reheated by itlf, so urs have to add hot water from time to time. Our goal is to establish a model of the temperature of bath water in space and time. Then we are expected to propo an optimal strategy for urs to keep the temperature even and clo to initial temperature and decrea water consumption.
To simplify modeling process, we firstly assume there is no person in the bathtub. We regard the whole bathtub as a thermodynamic system and introduce heat transfer formulas.
We establish two sub-models: adding water constantly and discontinuously. As for the former sub-model, we define the mean temperature of bath water. Introducing Newton cooling formula, we determine the heat transfer capacity. After deriving the value of parameters, we deduce formulas to derive results and simulate the change of temperature field via CFD. As for the cond sub-model, we define an iteration consisting of two process: heating and standby. According to energy conrvation law, we obtain the relationship of time and total heat dissipating capacity. Then we determine the mass flow and the time of adding hot water. We also u CFD to simulate the temperature field in cond sub-model.
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In consideration of evaporation, we correct the results of sub-models referring to some scientists’ studies. We define two evaluation criteria and compare the two sub-models. Adding water constantly is found to keep the temperature of bath water even and avoid wasting too much water, so it is recommended by us.
Then we determine the influence of some factors: radiation heat transfer, the shape and volume of the tub, the shape/volume/temperature/motions of the person, the bubbles made from bubble bath additives. We focus on the influence of tho factors to heat transfer and then conduct nsitivity analysis. The results indicate smaller bathtub with less surface area, lighter personal mass, less motions and more bubbles will decrea heat transfer and save water.
Bad on our model analysis and conclusions, we propo the optimal strategy for the ur in a bathtub and explain the reason of uneven temperature throughout the bathtub. In addition, we make improvement for applying our model in real life.
Key words: Heat transfer Thermodynamic system CFD Energy conrvation For office u only
T1 T2 T3 T4 For office u only F1 F2
西部省
F3 F4 Team Control Number 44398 Problem Chon A
3至6岁Enjoy a Cozy and Green Bath
Contents
1 Introduction (4)
1.1 Background (4)
1.2Literature Review (4)
1.3Restatement of the Problem (4)
2Assumptions and Justification (6)
3Notations (7)
4Model Overview (7)
5 Sub-model I : Adding Water Continuously (8)
5.1 Model Establishment (9)我也是生活的主角
5.1.1 Control Equations and Boundary Conditions (9)
5.1.2 Definition of the Mean Temperature (11)
5.1.3 Determination of Heat Transfer Capacity (11)
5.2 Results (13)
嫣然一笑请上勾
5.2.1 Determination of Parameters (13)
5.2.2 Calculating Results (14)
5.2.3 Simulating Results (15)
6 Sub-model II: Adding Water Discontinuously (18)
6.1 Heating Model (18)
6.1.1 Control Equations and Boundary Conditions (18)
6.1.2 Determination of Inflow Time and Amount (19)
6.2 Standby Model (20)
6.2.1 Process Analysis (20)
6.2.2 Calculation of Parameters (20)
6.3 Results (21)
6.3.1 Determination of Parameters (21)
6.3.2 Calculating Results (23)第三次长沙会战
6.3.3 Simulating Results (23)
6.4 Conclusion (27)
7 Correction and Contrast of Sub-Models (27)
7.1 Correction with Evaporation Heat Transfer (27)
7.1.1 Correction Principle (27)
7.1.2 Correction Results (28)
7.2 Contrast of Two Sub-Models (30)
7.2.1 Evaluation Criteria (30)
7.2.2 Determination of Water Consumption (30)
7.2.3 Conclusion (31)
8 Model Analysis and Sensitivity Analysis (31)
8.1 The Influence of Different Bathtubs (32)
8.1.1 Different Volumes of Bathtubs (32)
8.1.2 Different Shapes of Bathtubs (34)
8.2 The Influence of Person in Bathtub (36)
8.2.1 When the Person Remains Static in a Bathtub (36)
8.2.2 When the Person Moves in a Bathtub (37)
8.2.3 Results Analysis and Sensitivity Analysis (38)
8.3 The Influence of Bubble Bath Additives (42)
8.4 The Influence of Radiation Heat Transfer (44)
8.5 Conclusion (45)
9 Further Discussion (45)
9.1 Different Distribution of Inflow Faucets (45)
9.2 Model Application (46)
10 Strength and Weakness (47)
10.1 Strength (47)
10.2 Weakness (47)
Report (49)
7年级上册数学书
竹菜板Reference (50)
