Simulation of a Solar Chimney Ventilation System for an Air Polluted Megacity

under the direction of Prof. Hamed Hamid Muhammed Department of Technology and Health Royal Institute of Technology

Research Academy for Young Scientists July 8, 2015 Abstract

Air pollution is a major issue in many megacities, since it is a threat to the general health of the population. In this study, attention is paid to the capital city of Iran, Tehran. Tehran suffers from heavy air pollution. Moreover, the problem is enhanced by the surrounding mountains, which obstruct airflow through the city and confine the pollutants. Using a computational simulation, solar chimneys have been investigated as a possible tool to ventilate the polluted air in the city. The simulation has shown that the ventilation efficiency is far too low to be beneficial. Even as many as 1000 chimneys are only capable of daily ventilating, on average, 1% of the city’s air. However, solar chimneys are still of interest for further research, because of their environmental and maintenance benefits. Contents

1 Introduction 1 1.1 The Issue of Air Pollution ...... 1 1.2 Solar Chimneys ...... 1 1.2.1 Concept ...... 1 1.2.2 Design ...... 2 1.2.3 Mathematical Modeling and Simulation ...... 4

2 Method 7 2.1 Topography Approximation ...... 7 2.2 Meteorological Data ...... 8 2.3 Simulation ...... 9

3 Results 12 3.1 Ambient Conditions ...... 12 3.2 Volume Flow Rate ...... 13 3.3 Air Changes per Hour ...... 15

4 Discussion 16

5 Acknowledgements 19

A Matlab script for linear approximation of daily variation of ambient conditions 21

B Matlab script for iteration loop 22 1 Introduction

1.1 The Issue of Air Pollution

Air pollution is a major health issue in many megacities, since exposure to toxic gases and particles can cause both acute and chronic affliction [1]. For exam- ple, cardiovascual and respiratory diseases [1]. One megacity with this problem is Tehran, the capital of Iran, where pollution is enhanced by the topography and wind conditions. The mountains surrounding the city obstruct airflow through the city, thus confining and concentrating the pollutants [1]. The aim of this study is to determine whether solar chimneys can prevent the accumulation of pollutants.

1.2 Solar Chimneys

1.2.1 Concept

The concept of solar chimneys was first proposed in 1903 by Cabanyes [2]. So- lar chimneys utilize solar radiation to induce upward-directed airflow. Ambient air enters the chimney’s collector system, which usually has a transparent upper side that lets solar radiation through to a bottom side of material with a high absorbance capacity [2]. The bottom absorber thus accumulates heat, which is conducted to the air [6]. When the heated air expands, it obtains a lower density in relation to colder ambient air and is affected by a boyancy force which drives it up through the chimney [6].

1 1.2.2 Design

The classical solar chimney constructions are power plants with a horizontally oriented collector, vertical chimney and an airﬂow driven, electricity producing turbine, as illustrated in Figure 1.

Figure 1: Model of vertical solar chimney power plant.

Another, less investigated type is the sloped solar chimney power plant, which is built with the chimney resting on a sloped surface. While the purpose of large-scale solar chimney power plants is producing electricity, small-scale solar chimneys are used as ventilation systems for rooms and buildings. The hypothesis proposed in this study is that it is possible to combine the concept of small-scale solar chimney ventilation with the design of large-scale

2 sloped solar chimney power plants. What is meant is a large-scale construction of sloped solar chimneys, located along the mountainsides surrounding Tehran, or other cities with similiar issues and environment, with the purpose to lead polluted air away and thus serve as a ventilation system for the city. Figure 2 displays a schematic model of the suggested ventilation system. The entire chimney is to function as a collector, with an upperside transparent to longwave solar radiation and an absorber backside to warm up the inside air and drive its movement upward.

Figure 2: Model of large scale solar chimney ventilation.

3 1.2.3 Mathematical Modeling and Simulation

Nomenclature 2 Aab Area of absorber (m ) 2 Ac Area of cover (m ) 2 Ai Cross sectional area of chimney inlet (m ) 2 Ao Cross sectional area of chimney outlet (m ) Ar Ratio of Ao to Ai ACH Number of air changes per hour Cd Coefficient of discharge of air channel inlet −1 Cfl Specific heat of air (J kg K) −2 hab Conductive heat transfer coefficient for absorber (W m K) −2 hc Conductive heat transfer coefficient for cover (W m K) −2 hr,ab,c Conductive heat transfer coefficient between absorber and cover (W m K) Ls Stack height (m) m˙ Mass flow rate (kg s−1) −2 Sc Solar radiation heat flux absorbed by cover (W m ) −2 Sab Solar radiation heat flux absorbed by absorber (W m ) Ta Ambient temperature (K) Tab Mean temperature of absorber (K) Tc Mean temperature of cover (K) Tf Mean temperature of air in chimney (K) Tr Room temperature (K) −2 Ub Overall heat transfer coefficient between absorber and room (W m K) u Air velocity at outlet of chimney (m s−1) −2 Ut Overall heat transfer coefficient from top of cover (W m K) v Volume being ventilated (m3) V˙ Volume flow rate (m3 s−1) −3 ρf Density of air flow in chimney (kg m ) γ Constant for mean temperature approximation

For a room ventilation solar chimney with two openings and uniform ambient temperature, based on mathematical modeling developed by Bansal et. al. [4] and formulae derived by Andersen [5] for predicting natural ventilation by boyancy,

4 the mass ﬂow rate in kg s−1 through the chimney is expressed as

s ρf Ao 2gLs(Tf − Ta) m˙ = Cd √ . (1) 1 + Ar Ta

The coefficient of discharge of air channel inlet, Cd, is assumed to be 0.57, as suggested by Andersen [5]. The ambient air temperature is, consequently, also the temperature of the air going into the inlet. Equation (1) has been used in theoretical research on solar chimneys for room ventilation by Mathur et. al. [3], as well as Ong and Chow [7], whose works also involve experimental validations. Having a value for mass flow rate, the flow speed, volume flow rate and the number of air changes per hour can be determined using equations

m˙ u = , (2) ρf Ao

m˙ V˙ = (3) ρf and V˙ · 3600 ACH = . (4) v

Number of air changes per hour signifies how many times per hour the the air being ventilated is exchanged. In other words, the number of air changes per hour indicates the ventilation rate. Equation (1) proves that the airflow is dependent on the difference between its own temperature and the ambient temperature. The difference is in turn dependent on the amount of solar radiation absorbed by the solar chimney and amount of heat conducted to the airflow. To take this into account, there are radiation and heat transfer models. The model used in this

5 study is the same as Mathur’s [3] and consists of equations (5), (6) and (7). The energy balance equation for the glass cover can be described as [Incident solar radiation]+[Radiative heat gain by cover from absorber]=[Convective heat loss to airﬂow]+[Overall heat loss coeﬃcient from cover to ambient] and mathematically be expressed as

ScAc + hr,ab,cAab(Tab − Tc) = hcAc(Tc − Tf ) + UtAc(Tc − Ta). (5)

The energy balance equation for the absorber can be similiarly described as [Solar radiation]=[Convection to airﬂow]+[Long wave re-radiation to cover]+[Conduction to main room] and mathematically expressed as

SabAab = habAab(Tab − Tf ) + hr,ab,cAab(Tab − Tc) + UbAab(Tab − Tr). (6)

Lastly, the energy balance equation for the airﬂow, showing the relationship [Con- vection from absorber]=[Convection from cover]+[Useful heat gain by the air], can be written as

hcAcTc − (hcAc + habAab +mC ˙ fl/γ)Tf + habAabTab = −(mC ˙ fl/γ)Tr. (7)

The constant for mean temperature approximation, γ is assumed to be 0.74, as suggested by Ong and Chow [7]. When simulating large-scale sloped solar chimneys one must also consider the pressure diﬀerence between altitudes and the fact that absorbed solar radiation depends on the slope angle of the surface on which the chimney lies. This can be done with the sloped chimney divided into smaller vertical segments for which the

6 ambient conditions are uniform. The airﬂow through each segment can be calculated easily using the equations presented above. Then, by performing iterative calculations, the overall airﬂow in the entire solar chimney can be predicted [6].

2 Method

2.1 Topography Approximation

The most densely built-up and traﬃcated area of Tehran, seen in Figure 3, is chosen for ventilation. Thus, the polluted air is estimated to cover an area of 498.6 km2 [10] and from the lower part of the city to reach a height of 500 m. As a result, the goal is to ventilate 282.7 km3 of polluted air. It is assumed that the chimneys should be built with the bottom opening at an elevation of 100 m above the city center and from there reach a height of 1000 m.

Figure 3: Satellite image of Tehran with surroundings [10]. Marked area selected for ventilation.

7 2.2 Meteorological Data

Meteorological data is needed in order to estimate the ambient conditions and amount of polluted air that needs to be ventilated. Monthly average minimum and miximum air temperatures, presented in Table 1, and number of sunlight hours, presented in Table 2, were obtained from BBC Weather [12]. Mean values for averaged monthly solar downward radiative ﬂux years 2000-2005, presented in Table 3 were obtained from databases by NASA [11]. From monthly mean values, seasonal mean values were calculated. The average maximum value for solar downward radiative ﬂux, occuring in the middle of the day, was assumed to be the mean value times three.

Table 1: Averaged Ambient Temperature Conditions [12].

Season Mean minimum temperature [K] Mean maximum temperature [K] Summer 294.15 308.82 Autumn 285.15 297.48 Winter 272.48 282.48 Spring 279.48 294.82

Table 2: Average number of sunlight hours [12].

Season Average number of sunlight hours Summer 11.3 Autumn 8.3 Winter 6.3 Spring 7.7

8 Table 3: Averaged Downward Radiative Flux years 2000-2005 [11].

Season Mean ﬂux [W m−2] Approximated maximum ﬂux [W m−2] Summer 330 990 Autumn 290 870 Winter 250 750 Spring 290 870

2.3 Simulation

The simulation was performed using an iterative method in Matlab. The solar chimney was divided into smaller vertical segments with a stack height of 20 m, for which the following assumptions were made:

1. The ambient conditions, such as temperatures and atmospheric pressure, were considered to be uniform throughout the chimney.

2. The airﬂow was considered to be laminar.

3. The transparent cover was considered to be opaque to infrared radiation.

4. The air inside the chimney was considered to be non-radiation absorbing.

5. Thermo physical properties were evaluated at average temperatures.

Using the meteorological data presented in Tables 1 and 3, linear approximations of the daily ambient condition changes for each season were made, see appendix A for the script. In this approximation, the time interval between two points where the incident solar radiation, H, is equal to 50 W m−2, was obtained. This was done because the simulation can’t produce output when H < 50. Another limitation with the simulation was that it couldn’t produce output for all possible chimney

9 dimensions either. The opening of the chimney was limited to 1 m and the altitude difference to 1000 m. Therefore, only one combination of dimensions was tested. The reason for the limitations lies in the mathematical model, which is applicable only when the airflow is warmer than the ambient air. It is seen in equation (1) that Tf < Ta will result in complex numbers. For each segment, temperature changes and flow is predicted by a function, EISCRV, which simulates a vertical solar chimney for room ventilation based on the equations for energy balance and airflow properties. EISCRV was written by Meshram and can be obtained online [9]. The function’s input and output values are presented in Table 4.

Table 4: Input and output values for EISCRV

Input Nomenclature Unit Incident solar radiation H W m−2 Initial temperature of cover Tc,i K Initial temperature of inside airflow Tf,i K Initial temperature of absorber Tab,i K Ambient temperature Ta K Stack height Ls m Chimney width w m Gap between cover and absorber d m Opening width z m 3 Room volume Vr m Maximum number of iterations maxiter Maximum error to be incorporated maxerr Output Nomenclature Unit Mean temperature of inside airflow Tf K Mean temperature of cover Tc K Mean temperature of absorber Tab K Mass flow rate M kg s−1 Volume flow rate V m3 s−1 Air changes per hour ACH

In this case, the width of the chimney, as well as the opening and gap between cover

10 and absorber, were the same. The same applies for ambient and room temperature. The room volume was set to the estimated volume of polluted air in the city. An iteration loop was added to EISCRV, see appendix B for the script, in order to simulate airﬂow in 50 segments at diﬀerent seasonal ambient conditions. The process is described below:

1. Retrieve input values for H, Tc,i, Tab,i, Tf,i, Ta for given season, time of

day and height. Use constant input values for Ls = 20 m, w, d, z = 1 m,

11 3 V = 2.827 · 10 m , maxiter = 1000 and maxerr = 0.0001.

˙ 2. Use EISCRV to calculate output values for Tc, Tab, Tf , m˙ , V and ACH and store these in array. This step in itself involves iteration, as the Gauss Seidel method is used to solve the system of linear equations for calculation of temperatures [9].

3. Repeat step 1-2 for next segment, using output Tf from previous segment

as input Tf,i. For the next value of Ta, substract 0.13 from previous value in order to take the temperature diﬀerence due to altitude into account [8]. This step is represented in Figure 4.

4. Repeat step 1-3 until the end of the time interval.

5. Retrieve values for V˙ and ACH for each segment. Plot these as a function of time in given interval. For the same interval, calculate average V˙ and ACH.

11 Figure 4: Segment division for iterative calculation.

3 Results

3.1 Ambient Conditions

Figure 5 shows the linear approximation of the variation of ambient conditions during sunlight hours. The solar radiations is most intense in the middle of the day, while temperature reaches its maximum after two thirds of the day has passed. Note that the scaling on the axes may vary, as it is set so that the graphs ﬁt in the diagram. The time on the x-axes indicates the number of hours passed since the moment when solar radiation exceeded 0 W m−2.

12 1000 Solar 310 1000 Solar radiation (W/m2) 290 radiation Temperature (K) (W/m2 ) Temperature (K) ) ) 2 2

500 300 500 280 Temperature (K) Temperature Temperature (K) Temperature Solar radiation (W/m Solar radiation (W/m

0 2 4 6 8 10 1 2 3 4 5 6 Time (h) Time (h)

(a) Summer conditions (b) Winter conditions

1000 300 1000 Solar radiation (W/m2) 300 Solar radiation (W/m2) Temperature (K) Temperature (K) ) ) 2 2

500 290 500 290 Temperature (K) Temperature Temperature (K) Temperature Solar radiation (W/m Solar radiation (W/m

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Time (h) Time (h)

Figure 5: Linear approximation of the variation of ambient conditions during sunlight hours.

3.2 Volume Flow Rate

Results for volume flow rate show that airflow is present during sunlight hours all year round. Daily, the airflow is most substantial in the middle of the day and annually during summer. This is demonstrated in Figure 6. The graphs actually consist of several parallel lines that represent the flow in each separate segment. Time on the x-axis indicates the number of passed hours since the moment when solar radiation reaches 50 W m−2 - the starting point of all simulations.

13 1.6 Summer conditions Autumn conditions 1.4 Winter conditions Spring conditions

1.2 /s) 3 1 V (m

0.8

0.6

0 2 4 6 8 10 12 Time (h)

Figure 6: Daily variation of volume ﬂow rate under seasonal ambient conditions.

Table 5a presents the seasonal mean volume ﬂow rate in one solar chimney. Ta- ble 5b presents how much air volume one solar chimney is capable of transporting under diﬀerent seasonal ambient conditions.

Table 5: Mean volume ﬂow rate and total volume transportation (a) Mean volume ﬂow rate under seasonal ambi- (b) Daily total volume transportation. ent conditions.

Season V [m3 · 104] Season V¯ [m3 s−1] TOT Summer 4.1963 Summer 1.0863 Autumn 2.9301 Autumn 1.0408 Winter 0.9947 Winter 2.1056 Spring 1.0429 Spring 2.7257 Annual mean 2.9894 Annual mean 1.0412

14 3.3 Air Changes per Hour

Figure 7 demonstrates how the number of air changes per hour, namely the ventilation rate, conducted by one solar chimney is changing throughout the day. The number of air changes per hour is proportional to the volume ﬂow rate, which implies that it follows the same pattern of variation; Daily it is most substantial in the middle of the day and annually during summer. Note that the values on the y-axis are multiplied by 10−6. The graphs actually consist of several parallel lines that represent the ﬂow in each separate segment. Time on the x-axis indicates the number of passed hours since the moment when solar radiation reached 50 W m−2 - the starting point of all simulations.

×10 -6 2 Summer conditions Autumn conditions 1.8 Winter conditions Spring conditions

1.6

1.4 ACH 1.2

0.8

0.6 0 2 4 6 8 10 12 Time (h)

Figure 7: Daily variation of number of air changes per hour under seasonal ambient conditions.

15 Table 6a presents the seasonal mean of the number of air changes per hour conducted by one solar chimney. Table 6b presents the daily percentage of air that one solar chimney is capable of ventilating.

Table 6: Mean air changes per hour and daily total air change (a) Mean value for air changes per hour under (b) Daily total air change (ACH ) seasonal ambient conditions. TOT

Season ACH¯ Season ACHTOT (%) Summer 0.0015 Summer 1.3834 · 10−6 Autumn 0.0010 Autumn 1.3254 · 10−6 Winter 1.2666 · 10−6 Winter 0.0007 Spring 1.3281 · 10−6 Spring 0.0009 Annual mean 1.3259 · 10−6 Annual mean 0.0010

4 Discussion

The simulation shows that an airflow in the solar chimney is present during sunlight hours all year round. It can be concluded that the airflow is more substantial at higher ambient temperatures and during more intense solar radiation. However, the ventilation effect is not great enough to continuously ventilate the polluted air in the city. According to the simulation, one chimney ventilates 0.0007-0.0015% of all the air per day. 100 chimneys would then on average ventilate 0.1% per day. Even if there were enough resources to build as much as 1000 chimneys, on average, only 1% of all air would be ventilated. The solar chimneys would therefore not provide any significant improvement of air quality. It is possible that solar chimneys with other dimensions could be more efficient. The problem is that

16 the simulation can’t be used to test all dimensions. In order to determine the relationship between the chimney’s dimensions and airflow, another simulation method is needed. Still, the low efficiency does not necessarily imply that the concept of solar chimney ventilation outside rooms and buildings is completely useless. It may not be powerful enough to ventilate the air of an entire megacity such as Tehran, but perhaps it could be useful on a more local scale. In this study, the simulation does not produce values that are entirely correct, because it neglects several influencing factors. First of all, the simulation is limited to a certain time interval, namely sunlight hours when the solar radiation exceeds 50 W m−2. What happens during night hours is unknown. Secondly, the simulation is based on mean ambient conditions. In reality, the weather fluctuates. Cold and cloudy periods do occur. Lower ambient temperature and weaker solar radiation result in weaker airflow and ventilation effect. Thirdly, the simulation is based on simplified mathematical modeling with the premade assumption that the airflow is laminar. Hence, if the airflow at some point becomes turbulent, the simulation is no longer realistic. The simulation’s accuracy is not only affected by its limitations, but also by the accuracy of the meteorological data used as input. The only information available was not entirely up to date. Data for average temperature was 4 years old and downward longwave radiative flux 10 years old. It is most likely that the current ambient conditions are different. Regarding solar radiation, the average downward longwave radiative flux is not the best estimation of radiation absorbed by the chimney. It would be more accurate to gather information about radiation on the horizontal surface and use a radiation model to predict radiation absorbed by the sloped surface of the chimney.

17 Further research is needed, particularly on the following subjects: First of all, gathering of more accurate and up to date meteorological data with more information about variations and fluctuations. With that, taking into account more ambient air properties, such as relative humidity. Solar radiation absorbed can be estimated more exact using radiation models with consideration of the surface’s slope angle, reflectance and diffused radiation. Secondly, development of a simulation that can test a wider range of possible chimney dimensions, such as greater width and opening, in order to find the ideal design that results in maximum efficiency. Also, development of a simulation that can test a wider range of ambient conditions, such as solar radiation weaker than 50 W m−2, in order to evaluate how the efficiency may vary. Thirdly, investigation of the occurence and effects of turbulunce. This could be done using Computational Fluid Dynamics (CFD) with a suitable turbulence model. Accordingly, the airflow’s viscosity, its interac- tion with the chimney’s walls and the thermal boundary layer will be taken into account. Lastly, investigation of whether solar chimney ventilation can be used locally. That is, if it is capable of efficiently ventilating a smaller city area. The best case scenario would be theoretical studies combined with the building of a prototype for experimental validation. In conclusion, large-scale solar chimneys have been evaluated to be a poor ventilation system for air polluted megacities. Nevertheless, solar chimneys are still an important subject for research. With pollution and environmental problems, the solar chimneys’ environmental friendliness and ease of maintenance make them favorable for society. Therefore, development of research methods and further investigations of the possibilities are meaningful.

18 5 Acknowledgements

I would like to thank my colleague Alberte Kofoed Lauridsen for her cooperation and dedication to this project. I am grateful to our supervisor, Professor Hamed Hamid Muhammed of the department of Technology and Health at the Royal Institute of Technology, for his helpful guidance and support. Special thanks to Research Academy for Young Scientists, its organizer group, especially my coun- selor Anna Broms and the project manager Philip Frick for making this project possible.

19 References

[1] Naddaﬁ K., Hassanvand M.S., Yunesian M., Momeniha F., Nabizadeh R., Faridi S., Gholampour A. Health impact assessment of air pollution in megacity of Tehran, Iran, Iranian J Environ Health Sci Eng. Published online 2012-12-17. Retrieved 2015-07-04, Available from: http://tinyurl.com/nwxrmly.

[2] Cao F., Zhao L., Guo L. Simulation of a sloped solar chimney power plant in Lanzhou, Energy Conservation and Management, Volume 52, Issue 6, P. 2360–2366, (2011).

[3] Mathur J., Bansal N.K., Mathur S., Jain M., Anupma Experimental investigations on solar chimney for room ventilation, Solar Energy, Volume 80, Issue 8, P. 927–93, (2006).

[4] Bansal N.K., Mathur R., Bhandari M.S. Solar chimney for enhanced stack ventilation, Building and Environment, Volume 28, Issue 3, P. 373-377, (1993).

[5] Andersen K.T. Theoretical considerations on natural ventilation by thermal boyancy, ASHRAE Transactions, Volume 101, Issue Pt 5, P. 1103-1117, (1995).

[6] Bilgen E., Rheault J. Solar chimney power plants for high latitudes, Solar Energy, Volume 79, Issue 5, P. 449-458, (2005).

[7] Ong K.S., Chow C.C. Performance of a solar chimney, Solar chimney, Volume 74, Issue 1, P. 1-17, (2003).

[8] Portland State Aerospace Society A Quick Derivation relating altitude to air pressure, updated 2004-12-22. Retrieved 2015-07-03. Available from: http://tinyurl. com/pnnqdmt.

[9] Meshram A. Experimental Investigation on Solar Chimney for room ventilation, updated 2013-05-14. Retrieved 2015-06-25. Available from: http://tinyurl.com/nnrahj7.

[10] Google Earth "Tehran", 2015-06-23.

[11] Atmospheric Science Data Center, NASA, NASA Surface meteorology and Solar Energy: Interannual Variability, (2000-2005). Retrieved 2015-07-02. Available from: http://tinyurl.com/ooqsj26.

[12] BBC Weather, Tehran Average Conditions, (2011) Available from: http://www. bbc.com/weather/112931.

20 A Matlab script for linear approximation of daily

variation of ambient conditions

Ambient conditions approximated are of downward radiatiove ﬂux and temperature variation during solar hours. Also, the time interval between two moments when H = 50 is found. Input values hold here are valid for summer conditions.

1 Hsu1= linspace(0,990,20340);% increase radiation

2 Hsu2= linspace(990,0,20340);% decrease radiation

3 Hsu=[Hsu1 Hsu2];% radiation change vector

5 Tsu1= linspace(294.15,308.82,2 ∗ l e n g t h(Hsu)/3);% increase temperature

6 Tsu2= linspace(308.82,294.15,length(Hsu)/3);% decrease temperature

7 Tsu=[Tsu1 Tsu2];%temperature change vector

9 tsu= linspace(0,40680,length(Hsu));% time vector

11 indsu=find(Hsu>=50);% start index

12 tsu(indsu(1));% start time

13 tsu(indsu(end));% end time

14 t i n t s u= tsu(indsu(end)) −tsu(indsu(1))% time interval

15 Tsu(indsu(1))% start temperature

16 Tsu(indsu(end))% end temperature

21 B Matlab script for iteration loop Volume ﬂow rate and air changes per hour are calculated and plotted. Input values hold here are valid for summer conditions.

1 %[T,M,V,ACH]= EISCRV(H,T, Ta,Ls,W,z,d,Vr,maxiter,maxerr)

3 c l o s e all

4 c l e a r all

6 h1 = 1400;% lowest altitude

7 h2 = 2400;% highest altitude

8 % in practise only difference(h2 −h1) of importance

10 Tmax = 308.82;% max temperature of day

11 Tstart = 294.71;% start temperature of day

12 Tend = 295.26;% end temperature of day

14 t= linspace(0,10.73);% time vector

16 TT1= linspace(Tstart, Tmax,round(2 ∗ l e n g t h(t)/3));% temperature increase vector

17 TT2= linspace(Tmax,Tend,round(length(t)/3));% tempereature decrease vector

18 TT=[ TT1 TT2];% temperature variation vector

20 H1= linspace(50,990,length(t)/2);% solar radiation increase vector

21 H2= linspace(990,50,length(t)/2);% solar radiation decreaase vector

22 H=[ H1 H2];% solar radiation variation vector

24 Hlen= length(H);

25 ndx= 1;

26 Ttub= TT(ndx)+2;% initial warm up of air in tube by two degrees

27 antal_steg= round((h2 −h1)/20)

28 antal_temp= length(TT)

29 kk= 0;

31 % iterative segmential calculations

32 f o r ii =1:antal_temp

33 k= 0;

34 i f ii == 1

22 35 T= [(TT(ndx)+Ttub)/2; Ttub; TT(ndx)];

36 e l s e

37 T= [(TT(ndx)+mtris((ndx −2)∗50+1,2))/2; mtris((ndx −2)∗50+1 ,2) +2; TT(ndx)];

38 end

39 i i

40 f o ri=1:antal_steg

41 [T,M,V,ACH] = EISCRV(H(ndx),T,TT(ndx) −k ∗ 0.13, 20, 1, 1, 1, 2827000000, 1000, 0 . 0 0 0 1 ) ;

42 k=k+1;

43 kk= kk +1;

44 mtris(kk,:) = [T(1)T(2)T(3)MV ACH];

45 T= [((TT(ndx) −k ∗ 0 . 1 3 )+T(2))/2;T(2); TT(ndx) −k ∗ 0 . 1 3 ] ;

46 end

48 i f ndx == length(TT)

49 ndx= 1;

50 e l s e

51 ndx= ndx+1;

52 end

53 end

55 f i g u r e% plotting ambient conditions variation

56 [ hAx,hLine1,hLine2] = plotyy(t,H,t,TT)

57 legend({’Solar radiation(W/m^{2})’,’Temperature(K)’})

58 x l a b e l(’Time(h)’)

59 y l a b e l(hAx(1) ,’Solar radiation(W/m^{2})’)% lefty −a x i s

60 y l a b e l(hAx(2) ,’Temperature(K)’)% righty −a x i s

62 meanV= mean(mtris(:,5),’omitnan’)% calculate mean volume flow rate

63 meanACH= mean(mtris(:,6),’omitnan’)% calculate mean air changes per hour

65 f i g u r e% plotting number of air changes per hour

66 y1= reshape(mtris(:,6) ,[ antal_steg, antal_temp]);

67 p l o t(t,y1’,’ −−y’,’LineWidth’ ,2)

68 y l a b e l(’ACH’)

69 x l a b e l(’Time(h)’)

23 71 f i g u r e% plotting volume flow rate

72 y2= reshape(mtris(:,5) ,[ antal_steg, antal_temp]);

73 p l o t(t,y2’,’r’,’LineWidth’ ,2)

74 y l a b e l(’V(m^{3}/s)’)

75 x l a b e l(’Time(h)’)