Mathematical modeling on the air indoor exchange.docx

1、Mathematical modeling on the air indoor exchangeMathematical modeling on the air indoor exchangeLinke XiaoYongkang RenZhongyu RenChongqing unniversity of science and technologyUnniversity town, Chongqing CityAbstractIts cold in winter in the northern China, and the temperature indoor and outdoor dif

2、fers much. To keep both the air inside fresh and the house warm, we need to contemplate how long and how often we open windows and doors. We take the climate of Jinan City as a sample and design a mathematical model to investigate the issue. Under the given condition that the amount of the people in

3、door is invariable, we set an open-windows and doors standard, which is, whether the density of carbon dioxide and the temperature indoor may pose the people sick. Combining physiological data of human, seams of windows and doors and flow velocity of air, we devise mathematical models to investigate

4、 how the density of carbon dioxide indoor changes during the daytime and night. Then considering heat changes when opening windows and doors, we devise another model to investigate heat changes over time, utilizing Matlab to describe it. Finally, we compromise the factors above and obtain an optimal

5、 configuration.Keywords: air indoor exchange, heat exchange, mathematical modelingIntroductionQuestion:In the winter of the northern China, the temperature indoor and outdoor differs much because of the chill weather. And only the temperature indoor kept above 18 can people feel suave. Hence people

6、there used to close windows and doors to keep warm. Meanwhile air exchange decreases with windows and doors close, which leads to worse air condition and increases the risk people contract respiratory disease.“Often open windows and doors, let fresh air in” This concept has aroused great concern amo

7、ng people in northern China. Therefore, with a given house scale and known temperatures indoor and outdoor, we need to research how often and how long should people open windows and doors. Assume that the house is 2.7 meters high, 3 people live indoor. Please design mathematical models to tackle the

8、 issues below.1. Discuss the conditions that house scale is respectively controlled in 20m2, 40m2, 60 m2 and 80 m2.2. The house scale needed at least with windows and doors close the whole winter.3. Any other aspects you should take into consideration while discussing the air condition indoor.And to

9、 tackle these issues, we need to make clear the unknowns below.1. How the concentration of carbon dioxide indoor changes over time. 2. How heat changes over time with windows and doors open.3. The frequency and the length of the time people open windows and doors. Analysis of issue Because of the fr

10、eezing weather in winter in northern China, people there used to close windows and doors to keep warm. Thus it declines the air exchange, posing a terrible impact on peoples health. This problem, which is called indoor air pollution, is aroused by architecture materials, commodities, chemicals and h

11、uman activities. Normally, endogenous chemical contaminants (such as carbon dioxide, ammonia and hydrogen sulfide) released from human breath, sweat and urine, exogenous pollutants (such as benzene, methylbenzene, styrene, methyl alcohol) released from human breath, and virus and bacteria from cough

12、 and sneeze constitute the major source of the pollution. Although indoor air pollution consists of several factors, we used to estimate air condition by one of them. Carbon dioxide indoor mostly originates from humans breath. While an adult is resting, the carbon dioxide he breathes out is about 20

13、L. And while he is laboring, the carbon dioxide he breathes out is 0.51 times more. With the labor load increasing, the rest of his body initiates to discharge the pollution air (such as decomposition product from sweat.) And the concentration of oxygen is decreasing at the time the concentration of

14、 carbon dioxide is increasing. When the concentration of carbon dioxide goes above 0.07%, people start to feel sick. So we can utilize the concentration of carbon dioxide as an indoor air pollution standard. The concentration of carbon dioxide ought to be restricted under 0.07%. Although seams of wi

15、ndows and doors can also complete air exchange, it doesnt affect significantly when the degree of the carbon dioxide from breath is high and when the house is enormous. Thus, we need to take human factor into account while devise an air exchange model. To investigate the issue, we should reconcile b

16、oth the concentration of carbon dioxide and the temperature indoor. We stipulate that the concentration of carbon dioxide indoor shouldnt runs above 0.07%, or people will open windows and doors. With windows and doors open, heat initiates exchanging, cooling down the indoor temperature by means of h

17、eat convection. Hence we consider the appropriateness of the temperature indoor at the same time. If the house scale is large enough, there will be enough seams to control the density of carbon dioxide within 0.07%. Therefore people there dont need to open windows and doors to exchange air.Assumptio

18、ns Note: the models we use are based on the climate of Jinan City.Here are the parameters:NameValueAverage wind velocity outdoor3.2m/sAverage wind velocity of major direction4.3m/sMajor directionENEIncidence of major direction15%Average altitude51.6mTemperature outdoor-7Density of CO2 outdoor0.588g/

19、 m3 NameValueConcentration of CO2 outdoor0.039%Limit of the concentration of CO20.07%Quantity of flow on CO2 from breath40g/hQuantity of flow on CO2 while sleeping25g/hDensity of air outdoor (18)1.213kg/ m3Density of air outdoor (-7)1.327kg/ m3Specific heat capacity outdoor1.009kJ/(kg.)Specific heat

20、 capacity indoor1.013kJ/(kg.)Assumptions on the house structure:Monolayer iron door, size (width*height): 1.5*2m, inevitable seam: 9mMonolayer glass window, size (width*height): 0.5*1m, inevitable seam: 8mFlow quantity of air osmosis of door seam (each meter): 2.86m3/(m*h)Flow quantity of air osmosi

21、s of window seam (each meter): 1.98m3/(m*h)Proof cannot preserve heat.Here are the illustrations of window and door:Other assumptions:1. With windows and doors open, we dim the air flow is one-dimensional steady. 2. The temperature indoor is controlled at 18 with windows and doors close.3. We assume

22、 one more window every 20 m2 gained of the house scale. 4. We assume only carbon dioxide affects the air condition.5. We assume the circumstance conditions outside (such as temperature outside, frequency of wind, flow direction) are steady.6. The air has been mingled instantly when exchanged.7. Igno

23、re seams with windows and doors closed.8. Ignore resistance when air is flowing. 9. The air is uniform distributed and air flows only around exits and entrances.10. The tree people are a family, which contains two adults and a kid. And the gas a kid breath is half that of an adult.Symbol definitionS

24、ymbolDefinition1Air density outdoor2Initiate air density indoorL1Flow quantity of air osmosis of door seamL2Flow quantity of air osmosis of window seamV1House volumeVqFlow quantitySHouse scaleSymbolDefinitionv1Flow velocity1Door seam scale2Window seam scalec1specific heat capacity outdoorc2specific

25、heat capacity indoorT1Initiate temperature indoorT2Temperature outdoorT3Limit temperature indoorSmDoor scaleScWindow scaleABreath quantity ASleeping breath quantityCco2Concentration of carbon dioxide outdoorCco2Concentration of carbon dioxide of sick pointCco2Limit concentration of carbon dioxideMod

26、el1We initiate to investigate the change of carbon dioxide over time, regardless of temperature limit. We detect that people should open windows and doors in such circumstance: (1) (2)In the inequation above, denotes the time charged to decline the concentration of carbon dioxide. At the left side i

27、n parentheses, the first term represents the density of carbon dioxide indoor when people breathe. The second term represents the density change of the osmosis quantity from seams. Those two terms difference multiplied by the flow quantity represents the sum quantity of carbon dioxide indoor. When t

28、he value goes above the limit that makes people sick, people should open windows and doors.Model2Based on model1, we initiate to consider the temperature factor. Considering that the flow-in quantity values from about eighty-percent to ninety-percent of the flow-out quantity in reality, we assume th

29、at the proportion is eighty-five percent. And here is the mass of carbon dioxide when open windows and doors.M1=M1+-(M1+)/V1* v1*Sc*dt+1.372*0.039*0.01* v1*Sc*dt*0.85.M1=M1+-(M1+)/V1*1.6dt+1.372*0.039*0.01*1.6dt*0.85.(3) Heat keeps transferring through different systems until the temperature reaches

30、 equality. The whole process obeys the energy conservation law and the energy transfer law. Energy transfers from the high-temperature substance to the low-temperature substance. Virtually its a transfer of internal energy, that is, the absorption of the internal energy from the low-temperature subs

31、tance is equal to the release of the internal energy from the high-temperature substance. The thermal balance equation (Q1=Q2) only works in the process in adiabatic system, that is, the whole system with no energy loss. And moreover, the initial state and the final state should be equilibrium state

32、. Usually, the release of calorie results from incline of temperature, solidification, liquidation and burning. And on the contrary, the absorption of calorie results from increase of temperature, dissolution and evaporation. According to the thermal balance law, we come up with the equation below: (4)Where Vq is equal to v1 multiplied S, and i