Immersive Translate

Restatement of the problem

1.1 Background of the problem

With the development of modern science and technology, people are becoming more and more dissatisfied with the current life, people not only want to "eat enough", but also "eat well"; Due to the fact that human activities in modern society have had a great impact on our dear natural world, extreme weather occurs from time to time, seriously affecting our daily life, and the most widespread and troublesome atmospheric problems have become a factor that many people value.

Based on the above situation, a device that combines the functions of air conditioner, humidifier and air purifier is designed according to the requirements. He can replenish moisture while adjusting the indoor temperature, increase the humidity of the room, and the air purifier part can also absorb impurities and dust particles in the air, purify the indoor air, and make the indoor circulating air more fresh. This 3-in-1 product not only reduces the space it takes, but also greatly improves safety.

The ensuing question is how to use the limited indoor space to design a three-in-one air conditioner that can make the most full use of the indoor space and provide people with a comfortable life, which is an interesting and closely related to life.

1.2 Question Requirements

In the case of giving the indoor size and various performance parameters of the air conditioner (maximum air intake, maximum air outlet speed, rated power, etc.), an air conditioner that meets the requirements is designed. The specific requirements are as follows:

Question 1:

In a room of size to $5m×8m×3m$ be renovated, analyze the location, size, location and number of air inlets and outlets of the air conditioner, and the influence on the temperature control effect of the air conditioner.

Under the limited conditions, the different indoor temperature conditions in winter and summer are discussed, and the optimization model of the shape of the air conditioner is established, and the possible optimal shape and size of the air conditioner are designed.

Question 2:

Considering the purification effect of the three-in-one air conditioner on the air, according to the different shapes of the three-in-one air conditioner, a mathematical optimization model of the influence of the air purifier on the air purification effect was established, and according to the mathematical optimization model, the optimal shape and size of the size and shape of the three-in-one air conditioner in the room to be renovated were calculated.

Question 3:

Similarly, considering the air humidification effect of the three-in-one air conditioner, a mathematical optimization model is established according to the influence of different shapes and sizes of the three-in-one air conditioner on the humidification effect, and the optimal size and shape of the three-in-one air conditioner in the space are given.

Question 4:

Based on the size and shape optimization model of the air conditioner established in questions 1-3, considering a more comprehensive situation, on the basis of combining the appearance design of the three-in-one air conditioner, humidifier and air purifier, the appearance of the three-in-one device is designed by considering the lower energy consumption, giving people a more comfortable experience, and the better purification effect and humidification effect, and giving the shape and size parameters of the three-in-one device.

Problem Analysis:

This problem is an optimization problem of the shape and size design of the air conditioner, which needs to take into account the constraints, the actual situation, the mathematical and physical equations behind each function, and the requirements of the computer to solve the problem, and comprehensively consider the design parameters of the shape and size of the air conditioner. For optimization problems, we can usually use heuristic algorithms (such as genetic algorithm, ant algorithm, simulated annealing algorithm, etc.) to obtain a local optimal solution.

Question 1:

Air conditioning efficiency is affected by a variety of factors, including the location of the air conditioner, the layout of air intakes and outlets, wind direction and angle, wind speed and volume, etc. Together, these factors determine the performance of the air conditioning system when it comes to regulating the indoor temperature.

Air conditioning location: The location of the air conditioner affects the uniformity of the distribution of hot and cold air, which in turn affects the efficiency of temperature regulation throughout the space.

Inlet and outlet layout: The number, location, and direction of the intake and outlet outlets determine the path and efficiency of air flow.

Wind direction and angle: The wind direction and angle determine the direction of air flow, affecting the uniformity and directness of air distribution.

Wind speed and air volume: Wind speed and air volume directly affect the cooling and heating capacity of the air conditioning system, as well as the speed at which air circulates.

Seasons: Different seasons directly affect the indoor temperature, which in turn affects the heat spread throughout the space.

Regarding the relationship between gas velocity and temperature in space, we can use the thermodynamic equation of space with heat source and the NS equation of non-compressible fluid to quantitatively describe the heat distribution in space.

$\frac{∂T}{∂t} = a^{} ∇^{} T+ \frac{q}{C_{} ρ} and ρ (\frac{∂ v}{∂ t} + (v ∙ ∇) v) = − ∇ p + ε ∇^{} v +F$

（（1）

Due to the low velocity of the gas blown out of the air conditioner, it can be approximated that the fluid blown out of the air conditioner flows steadily. For the influence of multiple factors, they essentially affect the temperature distribution and velocity distribution of the space, so just know their shape and size, you can get the relationship between the temperature, the number of degrees, and the pressure of the space, and then calculate the heat distribution of the space.

Question 2:

Air purifiers are an important device used in modern homes to improve indoor air quality. Its design directly affects the purification efficiency, including the path of air flow, the efficiency of contaminant capture, etc. Therefore, it is of great significance to analyze the influence of air purifier shape on purification effect for the design of efficient and energy-saving air purifiers. On the whole, the factors that affect the effect of air purification are as follows:

Air flow path: The shape of the air purifier determines the path of air flow, which in turn affects the purification efficiency.

Diffusion coefficient: Different pollutants in the air and the same pollutant at different temperatures have different diffusion coefficients, which will change the diffusion equation of the gas, thus affecting the purification effect.

Types of pollutants: Different pollutants have different diffusion coefficients in the same space, so the final purification effect is also different.

To do this, since the diffusion of gas molecules is involved, we can use the gas diffusion equation in conjunction with the continuous equation:

$\frac{∂ρ}{∂ t} + ∇∙ (ρ ν)=0 and \frac{∂ C}{∂ t} + ∇∙ (v C)= ∇∙(D∇C)$

（（2）

The initial velocity field can be calculated according to the shape and size of different air purifiers, and the pollutant concentration distribution in the space can be calculated. $v = μi+νj+ϖk$ For different kinds of pollutants, only benzene, an air pollutant that is harmful to people and common in decoration, is considered here, and the rest will not be discussed.

Question 3:

A humidifier is a device used to regulate indoor humidity and improve living comfort. Its design has a direct impact on its humidification effectiveness, including the uniformity of moisture distribution and humidification efficiency. Therefore, it is of great significance to analyze the influence of humidifier shape on humidification effect for the design of efficient humidifiers. According to the topic, it can be known that the main factors affecting humidifiers are:

Initial distribution of water vapor in a room: The initial distribution of water vapor has a significant impact on the effectiveness of humidification.

Water vapor velocity and direction: Different humidification directions and humidification degrees will affect the spatial humidification distribution of water vapor, so it also has a very important impact on the humidification effect.

The shape of the humidifier: The humidifier itself takes up a certain amount of space, which also affects the flow of water vapor in the air.

Then, since the air humidity can be measured by relative humidity (%), or absolute humidity ( $kg/ m^{})$ , here for the same unit, and consistent with the second question, absolute humidity (the concentration of water vapor) is used, so that an optimization model can be used twice!

Question 4:

The 3-in-1 device integrates the functions of an air conditioner, humidifier and air purifier, and its design needs to take into account energy efficiency, human comfort, purification and humidification. This comprehensive optimization design is essential to achieve the best performance and user experience of the device, and a good design is usually pleasant~

First of all, let's take a look at the goal of this question: according to the shape and size of the air conditioner, purifier and humidifier optimized by the first three questions, and comprehensively consider the energy consumption, human comfort, purification effect and humidification effect, to design the best appearance and size of the three-in-one air conditioner. This is equivalent to two variables: size and shape. The goal has changed from the original single objective to multiple objectives (four), and you only need to change the objective function, and you are OK.

Model assumptions:

In order to facilitate our modeling and solution in the future, we put forward the following reasonable assumptions:

It is assumed that the gas emitted by the air conditioner flows at a steady rate;

Suppose that the indoor gas is a non-pressurable, non-viscous fluid (because the gas flows at a low velocity, it is approximately assumed);

It is assumed that the gas is not affected by gravity during short-distance movement, so the influence of gravity is ignored.

Assuming a comfortable human body temperature of 26°C;

Suppose that the shape of an air conditioner can only take cubes and cylinders, and there is no other way (and most of them do);

It is assumed that there is only one air outlet and air inlet of the air conditioner (this is enough to cover the vast majority of air conditioners in the market);

It is assumed that the chamber is adiabatic from the outside world, there is no heat exchange, and the four walls of the chamber are the second type of adiabatic boundary conditions.

It is assumed that the room temperature in questions 2 and 3 remains constant, i.e., the gas diffusion coefficient remains constant;

Description of the symbol

symbol	illustrate
$dx,dy,dz$	Differential fractions of $x,y,z$ space in the direction respectively
$x,y,z$	The length, width, and height of the 3-in-1 air conditioner

Model building and solving

By observing these four questions, it can be clear that this is an optimization problem, in which the decision variables are the shape, size, location of the space, airflow velocity, angle inlet and outlet location of the air conditioner and other factors; The objective function is slightly different depending on the problem, the first question is temperature regulation, the second question is air purification, the third question is air humidification, and the fourth question is the first three plus air conditioning energy consumption; The constraints are:

The maximum air intake and maximum air outlet are both $600 m^{} /ℎ$

The maximum outlet speed is $8m/s$

The air conditioner is $1800w$ rated at

The maximum volume of an air conditioner is $0.1 m^{}$

From there, mathematical models can be built on a case-by-case basis, which can then be solved using optimization algorithms such as genetic algorithms.

5.1 Establishment and solution of problem 1 model

5.1.1 Question 1: Preparation for model establishment

First, create a spatial Cartesian coordinate system and put this room in:

In order to be able to solve the problem, we also need to mesh the space in which the room is located, and subdivide them into $nx,ny,nz$ parts along $x,y,z$ the axis, so that we can get their spatial coordinates:

At this time, there are:

${\begin{matrix} dx= \frac{5}{nx} \\ dy= \frac{8}{ny} \\ dz= \frac{3}{nz} \end{matrix}$

（3）

At the same time, we need to simplify the hypotheses of the model:

The first is its shape

Here we assume that it only has two types: a cube and a cylinder.

And then there's its size

Let's assume that its volume is the maximum volume $0.1 m^{}$ , and when it is a cube, let $长 =x ，宽 =y ，高=z，则 \frac{x}{y} = a ， \frac{y}{z} = b$ , and then from it being a cube, $a=b=1$ start stretching to both sides, $a,b$ and take it

$a$	$\frac{1}{4}$	$\frac{1}{3}$	$\frac{1}{2}$	$1$	2	3	4
$b$	$\frac{1}{4}$	$\frac{1}{3}$	$\frac{1}{2}$	$1$	2	3	4

Table 5-1 a,b are the values

Therefore, we get:

${\begin{matrix} x= \sqrt[]{0.1 a^{} b} \\ y= \sqrt[]{0.1 \frac{b}{a}} \\ z= \sqrt[]{\frac{0.1}{a b^{}}} \end{matrix}$

（4）

Easy to obtain, $在此处键入公式。$

Team Number :	apmcm24201031
Problem Chosen :	B