2016美赛论文A

For office use only

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43945

Problem Chosen

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F2________________

F3________________

F4________________

2016

MCM/ICM

Summary Sheet

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In order to stop the hot water becoming cold, we develop a model to determine optimal shape of bathtub to keep the temperature throughout the bathtub and as close as possible to the initial temperature without wasting too much water.

We found that there are only two ways to lose heat: heat loss from the water surface and heat transfer loss from inner surface bathtub. Then we develop a Thermal Energy Transfer Model to calculate the total heat sunk. Q=AF+BS(A is heat transfer coefficients from the water surface, B is the heat transfer coefficients from inner surface bathtub)

Concerning different shapes of bathtub will influence the area of water surface, the contact area of between the water and https://www.360docs.net/doc/ed17673739.html,pared with circle, oval, sector, square, the circle has greater area than of the other shape, if they have the same perimeter. So we choose the cylinder.Then we use a linear programming model to determine the radius and the height.

Based on the total heat loss equals to the total heat resource,and the water inflow is a constant trickle of hot water from the faucet, so the optimal strategy is the water flow velocity that satisfied the equation.

Concerning the influence of additive, the specific Heat Capacity of the water can be changed and the bubble will cover the water surface, which will attributed to can reduction of the evaporation from the water surface.Taking the two factors into consideration, the temperature will lower after we fill the additive.

Based on the Thermal Energy Transfer Model, if some variable changes, such as the constant temperature, the volume of human body, the area of the water surface, the result of our strategy can be different.

According to our market research and human body structure, we simplify our model. Neglecting the Comfort of the bathtub is the weakness.

The improvement of the model is to put a cover on the top of the bathtub. Reduce the area of evaporation area. The velocity of water inflow is depend on the temperature fluctuation.

Key words:Thermal Energy Transfer evaporation convection thermal conductivity

linear programming

contents

1.Introduction (2)

2. Asumption (2)

3. Notation (3)

4. Thermal Energy Transfer Model (4)

4.1 Introduction (4)

4.2 Analysis (4)

5. Determine shape (7)

5.1 Bottom shape (7)

5.1.1 Notation (7)

5.1.2 Analysis (8)

5.2 Determine r and h (9)

5.2.1 Introdution (9)

5.2.2 Analysis (9)

6. Water Strategy (11)

6.1 Analysis (11)

6.3 Model test (11)

7. The influence of bubble bath (13)

8. Sensitivity Analysis (14)

8.1 sensitivity analysis of F (14)

8.2 sensitivity analysis of temperature (15)

8.2 sensitivity analysis of volume of human body (16)

9. Evaluating Model (16)

9.1 Strength: (16)

9.2 Weakness (17)

10. Improvement of the model (17)

11.A letter to users (19)

12.Reference (20)

In our daily life, taking a bath is a comfortable way to clean and relax ourselves. After we fill bathtub in hot water, it will soon get noticeably cooler. Concerning the bathtub is a containment vessel without a secondary heating system and circulating jets, we have to add a constant trickle of hot water from the faucet to reheat the bathing water, at the same time let the excess water escapes through an overflow drain.

Task 1 We must develop a model of the temperature of the bathtub water in space and time to determine a optima shape to keep the temperature throughout the bathtub and as close as possible to the initial temperature without wasting too much water;

Task 2 If we take a bubble bath, initially fill the bathtub with additive, this will influence the specific Heat Capacity of the water as time goes by, we must calculate the influence of our model;

Task 3 Based on the Thermal Energy Transfer Model, if some variable changes, such as the constant temperature, the washing time, the area of the water surface.

Task 4 We must send a non-technical explanation for users of the bathtub that describes our strategy while explaining why it is so difficult to get an evenly maintained temperature throughout the bath water;

2. Asumption

1.The water inflow equals to the water outflow, and the velocity of two flow remains

constant, from the beginning, the water is full of the bathtub;

2.We do not take into account the volume of water loss during evaporation;

3.We assumed that the initial temperature of the bathtub is the same as the air, the

initial temperature of the water is the appropriate temperature and the

temperature of the water inflow is 100℃;

4.Once the water flow into the bathtub ,it can mixed up quickly;

5.The body exposed outside the water do not exchange heat with the air, since the

surface of the head is very small compared with other factors;

6.The temperature of the water equals to bathtub surface inside;

7.Under normal circumstance, the room is like containment vessel, so the wind

speed is a certain small number, and we assumed that the volume of the room is

10m3;

8.The heat loss from external surface of the bathtub make negligible influence on the

total heat sunk and temperature of the air;

9.The surface of the bathtub and the platform is the same;

3. Notation

temperature gradient between initial temperature of

room and water in bath

The temperature of water inflow minus the

4. Thermal Energy Transfer Model

4.1 Introduction

We develop the total heat loss speed model, which is sum of heat loss from water surface and bathtub inner surface.

Q=A·F+B·S

B=q=?k

A=Pa

0.623L [22.0+12.5w2+2.0(t?AFT

?t0)]

0.5

?[C p(t?AFT

?t0)+0.623L

(p′?p)]

4.2 Analysis

Thermal Energy Transfer is a process that contains two parts, the first part is the transfer between the water and the bathtub, the second part is between the water and the air. We have already define the area of the water surface exposed in the air as F, and the area of inner surface is S. We suppose A is the heat transfer coefficients from the water surface exposed in the air. A is a comprehensive influence coefficient affected by three factors: evaporation, thermal convection, thermal radiation. We define B is the heat transfer coefficients from inner surface bathtub. The heat sunk from the first part is A multiply F, the heat sunk from the second part is B multiply S. So we can get the total heat loss.

Q=A·F+B·S

Concerning the first part is thermal conduction that spontaneously flows from a hotter water to a colder bathtub, B is affected by the thermal conductivity of the material. According to the previous study, we found that the law of heat conduction,[3] also known as Fourier's law, states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area, at right angles to that gradient, through which the heat flows. q, is equal to the product of thermal conductivity; , and the negative local temperature gradient,. The heat flux density is the amount of energy that flows through a unit area per unit time. Then we can get the formula:

B=q=?k

Total heat sunk from the bathtub inner surface:

B·S=q·S

In this way, we can get two circumstances. If the thermal conductivity of the air is less than the bathtub, the heat can be lost faster through the bathtub, so the thinner the material the slower the heat can be lost. On the country, the material is so good that its thermal conductivity is less than the air, since the warmed surface is getting wider and wider, the material lost more and more heat. So the best choice is also make the bathtub thinner.We could easily come to the conclusion that the surface must be the bathtub inside, but not outside of it.

Since there are three dominant form of heat transfer in liquids and gases, the second part contains three process: The fundamental modes of heat transfer are evaporation, thermal convection, thermal radiation. So the total lost heat from the surface is sum of the three factors. Thermal energy can be transferred by radiation, but radiation have less influence on the transfer compared with the conduction and convection. In addition from the formula, which is d Q c=εσ(t+273)4dF [1] When d Q c is a very small number, when σ,Stefan-Boltzmann constant equals to 5.6*10?8W/m2·K4, we can get the same conclusion. Because this phenomenon accident significantly in a vacuum. So the major factor is conduction and convection. So total heat sunk is sum of the lost heat from convection and heat of Vaporization.

Thermal energy can be transferred by convection: [1]

d Q a=α(t?θ)dF

Thermal energy can be transferred by evaporation: [1]

d Q b=β(p′?p)dF

So the total heat sunk from the water surface is as follows:

d Q a+dQ b=[ d Q a+ d Q b]dF

=[α(t?θ)+β(p′?p)]dF

According to the study of Bowen LS [2 ], the ratio of heat losses by conduction and by evaporation from any water surface, we can simplify the formula above:

αβ=b=

P a C p

0.623L

d Q a+dQ b= [P a C p

0.623L

β(t?θ)+β(p′?p)]dF

β is an evaporation coefficient can be defined as

β=[22.0+12.5w2+2(t?θ)]0.5

,it is a variable affected by the wind speed and temperature of the air and water.

p′is saturated steam pressure of water surface, which can be changed with temperature of the water, it can be calculated by the formula follows:

ln p′=12.062?

4039.558

We can substituted into the equation above:

0.623L

[22.0+12.5w2+2.0(t?θ)]0.5[C p(t?θ)+

0.623L

(p′?p)]

∫ (dQ a+dQ b)dF =A?F

∫(dQ a+dQ b)dT=∫AFdT=F?∫AdT

The heat sunk from the water surface will make the air get warmer and warmer as time goes by. The process will produce interaction effects .The speed of losing heat through water surface will become slower as the temperature of the air become higher. So we can develop a formula to calculate the relationship between the two factors. θis the temperature of the room, this variable can be determined by the sum of the initial room temperature and the increment. The real-time temperature can be calculate by θ=AFT

c air m air

+t0, so we

use AFT

c air m air

+t0to replaceθin the formula above

A?F=Pa

0.623L [22.0+12.5w2+2.0(t?θ)]0.5[C p(t?θ)+0.623L

(p′?p)]

=Pa 0.623L [22.0+12.5w2+2.0(t?A?F?T

?t0)]

0.5

?[C p(t?A?F?T

?t0)+0.623L

(p′?p)]

Figure1 the relationship between A and time

5. Determine shape

5.1 Bottom shape

Since the smaller the surface water exposed in the air, the less heat will loss, in addition, we assume that the platform of the bathtub is the same as the top of it. In this model, we choose way to preliminarily decide which shape of platform have the greater area of the same perimeter. Compared with circle, oval, sector, square, the circle has greater area than of the other shape, if they have the same perimeter.

5.1.1 Notation

5.1.2 Analysis

Based on the market research, we find different kinds of shapes of bathtub. The platform of the bathtub can be classified into four shapes: circle, sector, oval and rectangle. In order to make the problems easier, we can assume that the bathtub is a pillar build on the four platform. So the first thing we must determine the shape of the bottom. The area of the shape above can be defined as S1, S2, S3, S4respectively.

s1=2πr12

C1=2√πS1

C1=πr12

]√S

?C=C1?C2= [2√π?(2+θ)√2

S2=2r2+θr2

C2= (2+θ)√2S2

C2=1

θr22

When θ∈[0,2π], +√θ≥2√2

?C=2√π-(

+√θ) √2≥2√π-4

√θ

?C<0

So C2

We can get the conclusion that perimeter of the circle is less than the sector, when they both have the same area.

S 4=a 2

C 4=4√S 4 C 4=4a ?C =C 1?C 4=4√4√1√π)√S ≥0

s 1=2πr 12

C 1=2√πS 1 C 1=πr 12

So C 1

5.2 Determine r and h

5.2.1 Introdution

Since we have already know the optimal platform of bathtub is circle (see model one), in addition, we consume the bathtub is a cylinder, volume is a certain number, then we can know the restricted condition is The Radius of the circle and the height.

In this part, we develop a linear programming model. The objective function is minimum Q. Radius and height are variables. The range of parameters is based on our market research, and human body structure.

5.2.2 Analysis

Because Q=A ?F +B ?S, the value of S and F can be determined by minimum Q. If we make partial derivative of the function equals to zero. Then we can get a optimal value of the F, which is not include S. In this way, we can easily get the ratio of S/F. dQ

=d(A ?F +B ?S)/dF = (A+B) +√π?V ?B ?(F)

When F=(√π?V?B

A+B

3, Q is the minimum value

When

F=(√π?V?B

A+B

)23, Then

S F =√π?V

F√F

+1=2?A B+3

We can derived the ratio of S

F =2A

+3, which is a variable about A and B. B is a constant

number if we choose a certain kinds of material (see model two, B is only changed with different kind of thermal conductivity. K).A is a variable changed with time, since the temperature of air will influence A, as time goes by. We define A0is a certain number, when T is a fixed time.

S F =2?A0

?F+2A0?F+3B?F

Q=A?F+B?S

The equation reveals that the only if A0is the minimum, the value of Q is the minimum. Because A0>0, under ideal circumstances, the minimum A0can equal to zero.

=3 S=3F

2F=2π?r?h2π?r2=2π?r?h r=h

S=F+2π?r?h

Based on the structure of human body, the height of the bathtub is between 0.45m and 0.65m so as to cover the human body. The Radius of the circle is between 0.35m and

0.75m.Only satisfying the condition, a person in the bathtub can feel comfortable.

We define the ratio of S/F equals to ρ, so S=ρ?F

Min Q=AF+BS

=A?F+F?ρ?B

=F?(A+ρ?B)

=2?π?r2(A+ρ?B)

We have already know that A, B, ρis certification, so we define fixed number e to replace A+ρ?B.

Min Q =2?e?π?r2

s. t. h=r

0.35≤ r ≤0.75

0.45≤h≤0.65

As r is declining, Q is declining too. We should choose minimal radius of the circle. The optimal value is r=h=0.45

F=π?r2=0.636m2

S=2π?r?h=1.909m2

6. Water Strategy

6.1 Analysis

Based on the total heat loss equals to the total heat resource, the velocity of water can be measured by the model as follows.

Q T=∫(A?F+B?S)?dT

m=∫(A?F+B?S)dT

c??t v=m

Q T=c?m??t

Because the water inflow is a constant trickle of hot water from the faucet, so the strategy is make the water flow velocity equal to v.

6.3 Model test

Under normal circumstances, we can make an assumption that T equals to one hour (3600 seconds). Initial Temperature of the air equals to 23℃, constant temperature of water is 43℃,=43℃-23℃=20℃. The material of bathtub pottery, and its thermal conductivity; equals to 1.2W/(m·K).

According our market research, the average thickness of the bathtub is 0.02 meters.AT is an integral value, we can calculate by MATLAB.

F=π?r2=0.636m2

S=2π?r ?h =1.909m 2 ∫AdT =48.022

The cross-section of cash inflow Q T =∫(A ?F +B ?S)?dT

= (0.636×48.022×3600+1.909×20×1.2×0.02×3600)?T =1.135×105

Q T =c ?m ??t=1.135×105J

m=Q T

=0.4731kg

v=m

T =4.18cm/s

If we keep the water inflow velocity equal to 4.18cm 3/s, from the chart below, we can know that the temperature is around 43℃. From start to about 30 minutes, the temperature is declining, from 30 minutes to 60 seconds, the temperature is increasing. The greatest fluctuation is less than 1℃. The Strategy can satisfy the task, which is keeping the temperature throughout the bathtub and as close as possible to the initial temperature without wasting too much water.

We can use MATLAB plot the point, then we can get a curve about the relationship between temperature of the water and time.

T =c w ?m ??t ?t =

∫Q T dT

C?V?ρ

t=43℃+?t ?

∫Q T dT

C?V?ρ

Figure 1 the relationship between total heat loss and time

Figure2 : the relationship between temperature and

Figure3 : the total heat resource minus total heat loss as time goes by

The result shows that this water strategy is credible and the parameters tested are accordant with the practical situation, when v equals to 4.18cm/s.

7. The influence of bubble bath

If we want have a bubble bath, the additive will have influence on our model in two ways: The bubble covers on the water surface can influence the evaporation coefficient,β.Because the bubble can reduce the evaporation from the water surface. So the heat transfer coefficients from the water surface A, will become smaller.

According to Thermal Energy Transfer Model,

A=α(t?θ)+β(p′?p)

Q T=∫(A?F+B?S)?dT

If the bubble covers the surface, the β will decline toβ′, A is certain to become small. Q T will become small too.

The additive will make the Specific Heat Capacity of the water become smaller. The change of concentrations will makes the specific heat capacity of the water becomes smaller. After we fill the bathtub with additive, we define the new specific heat capacity of the water is c x.

Q T=c w?m??t=c x?m x??t′

Q T is the total heat resource from the water inflow. If we use the water strategy in the model above, we can know?t′≤?t. We can get the conclusion that the temperature will become lower after we put additive.

Taking the two factors into consideration, the temperature will lower after we fill the additive.

8. Sensitivity Analysis

8.1 sensitivity analysis of F

A?F=Pa

0.623L [22.0+12.5w2+2.0(t?A?F?T

?t0)]

0.5

?[C p(t?A?F?T

?t0)+0.623L

(p′?p)]

Giving the certain value of the area of the water surface, the formula, which describe the relationship between A?F and A?F?T, will reveal the relationship between A and T.

In order to prove the correctness of the formula, for each T and AT, we use MATLAB to plot a point .Then we can get fitting curve describe the relationship between A and T. Taking an example, we assume that the wind speed is 0.3m/s. The standard pressure of air equals to 0.1×104Pa, the initial room temperature,t0is 23℃, the water temperature t is 43℃,the room is a 10m3Confined space, evaporation heat of liquids L equals to 1.005

KJ/mol·℃, C p equals to 1.005 KJ/kg·℃.

Cause F is a variable, so we can assume F=1, 2, 3, 4 m2

Then we can get a formula about the relationship between AF and T.

A?F=0.1×104

1557.5[23.125+2?(20?AFT

0.832

)0.5[1.005(20?AFT

0.832

)+0.068471]

Figure4: the relationship between A and time

As the chart above shows that the larger area of the water surface, the greater of A, which means losing more heat per unit time. The result shows that this model is credible and the parameters tested are accordant with the practical situation.

8.2 sensitivity analysis of temperature

The model we developed, we assume the temperature of water is 43℃. But the most comfortable temperature is changed by personal preference and different season.

If the range of t is between 40℃and 50℃, the optimal water strategy can be described by the chart below. As the temperature of the water increase, the demand of the water is grow too.

Figure5: the demand of water changes with the temperature

8.2 sensitivity analysis of volume of human body

In the model we develop, we do not consider the human body. If we add the condition, the human body take up the volume of the water, so the volume of water become decline. We do not consider the head exposed on the air, because it will have little influence on the total heat loss. The human body submerge under the water do not absorb heat, since the temperature of human body do not grow. And the total heat loss from the human body can be a small certain number.

Q T=c w?m??t,

When m becomes smaller, Q T will become bigger. So we can get the conclusion that when a human body takes account, the heat loss speed is faster.

9. Evaluating Model

9.1 Strength:

1.We do the Qualitative Analysis and Quantitative Analysis when we are determine the

shape of bathtub;

2.We make full use of our market research, and based on the structure of human body,

we make some reasonable hypothesis to simplify our model;

3.We consider that the temperature of the room will have interact influence on the

temperature of water.

9.2 Weakness

1. The model is under ideal circumstances. The fitted value of the real temperature fluctuate greater than the real condition;

2. We do not consider the comfort when people use the bathtub.

10. Improvement of the model

10.1 give a cover on the top

Since the A is bigger than B, the water surface can loss more heat than the material of the bathtub, we should make the area surface water exposed in the air as small as possible. So we can cover the bathtub with a cap, which leaves a small hall to let the head extend to outside. Because the water surface trends to zero, so the only way to lose heat is through the bathtub. So Thermal Energy Transfer Model can be changed into the equation follows:

Q=B(S+?S)

10.2 water strategy

From the chart above, we can know the temperature of the water initially declining, after a while, it increase as time goes by.

If the water inflow is not constant, we need more hot water from the time being and gradually reduce the water. In this way, we can keep the temperature as close as the constant temperature.

A letter to users

Dear users,

I am glad you choose the bathtub we design. The shape of our material can keep the temperature throughout the bathtub and as close as possible to the initial temperature without wasting too much water. You need add a constant trickle of hot water from the faucet to reheat the bathing water.

Under the experimental condition, the area of the bath room is 10m2, the comfortable temperature is 43℃,the temperature of bathroom is 23℃, a trickle of hot water can satisfy you.

I am so sorry to tell you the temperature can fluctuate greatly, since it is really hard to control the water inflow velocity. From the time being, the temperature is declining, after a while, it will grow. You had better to have a cover on the top of the bathtub, it will reduce the evaporation.

Thank you very much!