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Uncertainty in indoor air quality and grey system method
ARTICLE IN PRESS
Building and Environment 42 (2007) 1711–1717 www.elsevier.com/locate/buildenv
Uncertainty in indoor air quality and grey system method Chihui Zhua,b, Nianping Lia,, Di Rea, Jun Guana a
College of Civil Engineering, Hunan University, Changsha 410081, China School of Civil Engineering, Guangzhou University, Guangzhou 510006, China
b
Received 4 May 2005; received in revised form 14 June 2005; accepted 24 January 2006
Abstract Based on analysis of uncertainty, this paper presents grey system theory to handle the ‘‘grey’’ characteristic of IAQ. Grey comprehensive analysis of indoor air quality reveals that we should pay more attention to the air purification and humidity control in the design and maintenance of HVAC. In order to represent grey characteristic of IAQ system, the educed grey IAQ models can identify the variation intervals of key IAQ model parameters that are lack of directly measurable messages in practical situations. Furthermore, grey assessment is an effective multifactor comprehensive assessment method that can express the integrative influence of contamination indexes on indoor air quality. We can determine the IAQ grade and make comparison according to the grey incidence matrix R. r 2006 Elsevier Ltd. All rights reserved. Keywords: Indoor air quality; Uncertainty; Grey system theory
1. Introduction There are several theories how to handle uncertain data: fuzzy logic, system identification, dimension analysis. A new one may be added: the grey system theory [1]. The systems with partially known and partially unknown information are named as grey systems. Grey system theory is especially useful when the complete set of factors involved in the system’s behavior is unknown or unclear, when the relationships of system factors to the system’s behavior and inter-relationships among factors are uncertain, when the system behavior is too complex to determine completely, or when only limited information on system behavior is available. To make it easier to the reader to understand the grey system an application to indoor air quality (IAQ) is given. Uncertainty exists in the influencing factors of IAQ, IAQ model, assessment of IAQ, etc. IAQ is a ‘‘grey’’ system that we are short of adequate messages to describe. How can we make use of these inadequate messages to extract or produce more useful information about the grey system? Corresponding author. Tel./fax: +86 731 8823115.
E-mail address: [email protected] (N. Li). 0360-1323/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2006.01.015
Based on analysis of the uncertainty contained in IAQ, this paper presents grey system analysis of IAQ, grey model of IAQ, grey assessment of IAQ to provide guides to design and maintenance of indoor environment.
2. Uncertainty in IAQ 2.1. Uncertainty in scope of IAQ In the research process, people modify their viewpoint of IAQ according to new discoveries. IAQ was seen as a series of indoor air contaminant indexes formerly. Recent study indicates that this viewpoint is incomprehensive and unable to express the meaning of IAQ. What we need are indoor air that is perceived as fresh, pleasant and stimulating, with no negative effects on health and a thermal environment perceived as comfortable by almost all occupants. ASHRAE Standard 62-2001 [2] provides the concept of acceptable IAQ that covers not only the objectivity but also subjectivity of IAQ. Thus IAQ is an interdisciplinary science. Along with the development of IAQ studies in depth and in extent, the scope involved in IAQ will vary and extend.
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2.2. Uncertainty in factors affecting IAQ The indoor air environment is a result of the interaction between outdoor air quality, building envelope, ventilation system, contaminant sources, and building occupants. Poor IAQ is aroused by numerous indoor air contaminants essentially. Moreover, only a few among the numerous contaminants are known by people (determinate or white). As to most contaminants, we do not know their characteristics and health effects exactly (uncertainty or grey). The effects of IAQ problems are often nonspecific symptoms rather than clearly defined illnesses. Based on industrial hygiene, we cannot identify the reasons for the complaint of occupants. This problem may be caused by the combined effects of multiple pollutants at low concentration. The influence of indoor air contaminants on human body is complex and because so many factors operate in the system, it is difficult to make certain the causality between them. Further we should reduce the randomicity and find out rule underlying during data processing, because the data measurement is affected by some uncontrollable factors. It is uncertainty contained in the previous studies that force us to look for suitable tools to deal with such questions. 2.3. Uncertainty in IAQ model In a narrow sense, IAQ model is the mathematic relation that describes spatia-temporal rule of contaminant transport and removal. According to the comprehension of internal rules, people present models such as uniformly mixed and distributed model, modified uniformly mixed and distributed model, double-zone model, etc. In the above-mentioned models, system parameters and system inputs are assumed to be deterministic. Although there are some causalities between the system inputs and the system outputs, system model parameters may vary in wide ranges due to practical situations and incidental factors. System structure and the relation among system parameters are alterable also. Moreover, uncertainty in system input exhibits in interval variation of system input parameters. The two types of uncertainties contribute to the grey characteristics of IAQ that can be dealt with by grey system theory. 2.4. Uncertainty in assessment of IAQ Which contaminant can denote the state of IAQ best? Assessment of IAQ relates to not only the health effect of indoor contaminants to occupant but also the subjective response of occupant to indoor air. Constituting new index according to several contaminants is an effective method. Nevertheless, evaluation index and grade criterion are empirical and have no physical significations. The degree of IAQ is a grey idea. It does not correspond to a series of determinate values of contaminants necessarily, but relates to concentration scopes. The assessment of IAQ is multi-
factor assessment; thus assessment method should express the integrative influence of contamination indexes on IAQ. 3. Grey system methods applied in IAQ Based on the above discussion, we know that the complete set of factors involved in IAQ is unknown or unclear, the relationships of influencing factors to IAQ and inter-relationships among factors are uncertain, the IAQ system behavior is too complex to determine completely and only limited information on IAQ system is available. In a word, IAQ is a system that we are short of adequate messages to describe. Prof. Deng Julong first originated grey system theory to deal with this type of system. Grey comprehensive analysis, grey modeling, grey forecast, etc. are effective methods for applying grey system theory. In the field of building environment, Wang [3] proposed a method based on grey prediction model to predict building energy consumption. Hu [4] studied grey model of direct solar radiation intensity on the horizontal plane for cooling loads calculation and building thermal process analysis with grey system method. Zeng [5] presented uncertainty and methodology in environmental science and engineering. Grey analysis is particularly applicable in instances with very limited data and in cases with little system knowledge or understanding. All these previous studies provide reasonable references for us to deal with the ‘‘grey’’ characteristics of IAQ. Thereinafter, we try to deal with grey IAQ system by using appropriate grey methods. 3.1. Grey comprehensive analysis of IAQ In the design of heating, ventilating and air conditioning (HVAC) system and management of IAQ, we should know which contaminant affects IAQ chiefly, which contaminant can arouse a certain symptom, which contaminant will deteriorate IAQ significantly with concentration increase. Furthermore, the measurement of contaminant concentration is a time-consuming and expensive process. As we know, some contaminants have the same sources and there is correlativity in their presence and intensity. Can we propose an integrative index or choose a certain contaminant to represent a group of contaminants? There are so many factors concerned in the previous problems that it is not an easy work to identify the causality specifically. Fortunately, the grey relation analysis provides an effective way to solve those problems. 3.1.1. Idea of grey relation analysis According to grey system theory, we can choose a parameter to represent the development (output) of a system; thus a time or spatial sequence can express the development of the system. The values for each input factor, like the output values, can be treated as a sequence. The input and output sequences can be compared to identify the system’s key factors. The fundamental idea of the grey relational analysis is that the geometrical
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resemblance between the curves of the sequences represents the relation between the factors. In the IAQ study, uncomfortable sensation and sick symptoms can be seen as system outputs; indoor air contaminants can be treated as system inputs. Through field investigation, we can obtain the dissatisfactory rates of indoor air, sick symptom rates and concentrations of several recognized important indoor contaminants in several buildings. As discussed above, these values constitute corresponding sequences that can be used to identify key factor and inter-relation. We use the grey relational grade to quantificationally weigh the relation between those sequences. The higher grey relational grade value implies that the relation between factors is closer. Thus, grey relation analysis can reveal the relationships of related factors to IAQ and interrelationships among these factors. The relationships can then be used to suggest appropriate instructions in design and maintenance of indoor environment. 3.1.2. Calculation of grey relational grade Because sequences include different parameters that have different dimensions and the numerical values of these parameters may have great ranges, we should make these sequences equivalent through pretreatment in order to remove the influence of dimensions. The ordinary pretreatment is to calculate the initial images or average images of these sequences. After pretreatment, the output sequence that constituted by the dissatisfactory rates of indoor air or sick symptoms rates in l buildings is expressed as X ¼ fxð1Þ; xð2Þ; . . . ; xðkÞ; . . . ; xðlÞg.
(1)
In the same way, the n input sequences can be expressed as Y j ¼ fyj ð1Þ; yj ð2Þ; . . . ; yj ðkÞ; . . . ; yj ðlÞg, j 2 N ¼ f1; 2; . . . ; ng,
ð2Þ
where n denotes that there are n affecting factors (formaldehyde, CO, CO2, NO2, SO2, respirable particle, pathogen, etc.). The grey relational grade rj between system output sequence X and one affecting factor sequence Yj can be calculated by Eq. (3) and the grey relational grade rj shows the correlation between a specific ill symptom and a specific contaminant: rj ¼
l 1X x ðkÞ, l k¼1 j
(3)
xj ðkÞ ¼
minj mink jxðkÞ yj ðkÞj þ lmaxj maxk jxðkÞ yj ðkÞj , jxðkÞ yj ðkÞj þ lmaxj maxk jxðkÞ yj ðkÞj ð4Þ
where l ðl 2 ½0; 1; commonly l ¼ 0:5Þ is distinguishing coefficient which is used to regulate difference significance of grey relational grade.
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In the comprehensive grey analysis what we concern mostly is the sort order of grey relational grades rather than the real numerical values. The highest value implies that the corresponding contaminant is most closely correlated with the acceptability of IAQ. Lower values are associated with contaminants which are less important to the IAQ. Grey relation analysis is an improved method of identifying and prioritizing key factors and is useful for variable independence analysis. 3.1.3. Grey analysis of IAQ In modern office buildings, the concentrations of indoor air parameters are low and steady commonly, but the requirement of occupant is relatively high. This type building is object suiting for IAQ grey system analysis. Through field investigation, Shen [6,7] obtained the dissatisfactory rates of indoor air, sick symptoms rates and concentrations of several recognized important indoor contaminants in several office buildings. Using the method discussed above, we have calculated the grey relational grades between the dissatisfactory rates of indoor air (or sick symptoms rates) and indoor air contamination levels. According to the sort order of these grey relational grades, we can draw the following results [8]: 1. The grey relational grade between respirable particle and the dissatisfactory rate of indoor air has the highest value, which illuminates that respirable particle is the key reason for poor air quality. This may be due to that respirable particle to which gaseous air contaminants and biologic particles adhere can enter respiratory tract and stimulate mucous membrane. 2. The second highest value is the grey relational grade between pathogen and the dissatisfactory rate of indoor air. It is possible that when indoor space is close, ventilating effect is poor and moisture accumulates, pathogen will grow quickly and exert an important effluence on the acceptability of IAQ. 3. The grey relational grades between indoor contaminants and most sick symptoms have the similar sort orders. The high grey relational grade value indicates IAQ will deteriorate rapidly with the concentration raise. In other words, reducing the concentration of corresponding contaminants will improve IAQ remarkably. Consequently, we should pay more attention to the air purification and humidity control in the design and maintenance of HVAC. The grey relational grade between the formaldehyde and the dissatisfactory rate of indoor air (or sick symptom) has the relatively low value, which shows that the dilution of VOCs cannot improve IAQ effectively. On the other hand, because the main sources of VOCs are furnishings, wall coverings and occupancy activities, the effective method to improve IAQ is to select the materials and arrange occupancy activity reasonably.
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In the same way, grey analysis is applied to determine the correlation among diverse contaminants. High relational grades between contaminants may indicate contaminants interdependence, suggesting there is correlativity in their presence and intensity. Based on the full matrix constituted by grey relational grades, we may divide contaminants into different types according to critical value of grey relational grades. Thus, we can select a certain contaminant representing a group of contaminants in order to simplify the measurement of IAQ parameters. This technique is mostly suitable for occupancy and owner (not expert and researcher). 3.2. Grey model of IAQ It can be seen from above that we should control key contaminants to obtain better IAQ. Based on comprehension of IAQ system, people develop different IAQ models to describe spatia-temporal rules of contaminant transport and removal. Under certain initial and boundary conditions, these IAQ models can be expressed in the following form: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ,
(5)
where y1 ; y2 ; . . . ; yn are n model parameters, X is the input vector of the model, Y is one air quality index (for example CO2) and f denotes a certain function form. Model parameters in Eq. (5) are generally assumed to be fully known and constant. Although there is determinate causality between the input and output in above models, uncertainty lies in the system development and variation. Except statistical randomness, uncertain indicates epistemic uncertainty and system information inadequacy. 3.2.1. Uncertainty in IAQ Because of the influences of indoor contaminant sources distribution, plan of supply air outlets and return inlets and numerous incidental factors, model parameters y1 ; y2 ; . . . ; yn may vary in wide intervals. Furthermore, because the complexity of the natural world greatly exceeds our ability to model it, we may use a small number of variables to represent a complex phenomenon, choose incorrect functional forms, and set inappropriate boundaries. Therefore, another uncertainty is the lack of correspondence between the model and reality. In the view of grey system theory, this type of uncertainty can be referred to as uncertainty in system structure and be expressed in grey structure and grey parameters marked by ‘‘’’. Under such case, function form is grey structure f; model parameters are grey parameters yi ði ¼ 1; 2; . . . ; nÞ. The usual form of grey parameter is grey interval number that can be expressed as following: yi ¼ ½y^ i di;1 ; y^ i þ di;2 ði ¼ 1; 2; . . . ; nÞ,
(6)
where y^ i is the whitening value of the ith parameter, di;1 and di;2 indicate the variational scope of grey parameter.
System parameter is not a constant, but a grey interval number. Besides the uncertainty in system structure, there is uncertainty in system input also. For example, emit rate of indoor air contaminant is not only dependent on the characteristics of contaminant sources but also affected by environment factors (temperature, humidity, air velocity etc.). Air infiltration rate is related to building envelope, distribution of indoor and outdoor air pressure, climate etc. Biologic contamination of human being varies greatly with age, sex, activity level, and body condition. IAQ system input has no single value and will vary in large interval. In other words, system input is grey parameter X. 3.2.2. Grey IAQ model As discussed above, Eq. (5) does not take numerous unknown and incidental factors into consideration. It can be called as determinate model that is whiting description of grey IAQ system. Based on Eq. (5), we introduce grey structure f, grey parameters yi , grey input X to represent grey characteristics of IAQ system, therefore obtain the corresponding grey IAQ model: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ.
(7)
When we have appropriately described the essence of IAQ system, system structure is whiting structure and grey IAQ model can be simplified as the following generally adopted equation: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ.
(8)
Grey IAQ model describes IAQ comprehensively and reasonably by combining definite physical model with grey parameters. It makes best use of the existing knowledge and measurable messages about the IAQ and employs grey variables, grey series, and grey arrays to describe the important parameters that are lack of directly measurable messages. (1) Optimal grey parameters: Based on grey system theory, the optimal grey parameters of IAQ model can be defined as the optimal grey intervals which include all unknown and accidental factors’ set effects. The whiting values y^ 1 ; y^ 2 ; . . . ; y^ n of grey parametersyi ði ¼ 1; 2; . . . ; nÞ reflect the average influence of numerous factors on system; the two boundary values ðdi;1 and di;2 Þ of the optimal interval relate to the upper limitation and lower limitation of the system’s dynamic development. System essence is embedded in the measured values concerned with system input and output. Through field measurement, we can gain system input X and a discrete distribution sequence Y ¼ ½Y 1 ; Y 2 ; . . . ; Y m constituted by values of a certain IAQ index. Further, from Y we can educe two sequences Y¯ ¼ ½Y¯ 1 ; Y¯ 2 ; . . . ; Y¯ m and Y ¼ ½Y 1 ; Y 2 ; ; Y m2 ðm1 þ m2 ¼ mÞ which reflect the upper and lower limitations of the unknown and accidental factors’ set effects on the IAQ index. The optimal grey parameters and whiting values can be identified from X, Y¯ ,
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Y , Y and corresponding grey IAQ model by using optimization criteria and optimization method. The detailed process is referred to Ref. [9]. (2) Operations of grey interval numbers: When there is uncertainty in the system input, the system input presents in the grey interval parameter X. Traditional arithmetic operation principles are substituted by algorithms of grey interval numbers. The result should also be an interval grey number. We can calculate the grey IAQ model according to the algorithms among grey interval numbers. 3.2.3. The application of grey IAQ model Eq. (8) is deduced from a certain physical model. Various grey IAQ models can be obtained by changing the parameters in the corresponding determinate models into grey ones. (1) Grey double zone IAQ model: Ev is introduced to express the ratio of the effective supply air into the working zone to the total supply air in the double zone IAQ model. Ev represents the ventilating effect which relates to ventilating mode, ventilating rate, supply air quality, characteristic and plan of indoor air contamination sources etc. As we know, it is not easy to show the relation between Ev and affecting factors. The rest parameters can be measured easily. Consequently, we can educe the corresponding grey double zone IAQ model by replacing Ev with grey parameter E v . It is convenient to estimate the optimal grey parameter E v , in different ventilating strategies from the system input and corresponding measured output data through the method of least squares. More messages we know about the studied system, more it is convenient for parameter estimation. In order to reduce the complexity of system analysis, we analyze every element of grey IAQ model to make certain which element is measured and controlled easily and the sources of uncertainty and their rough variation scopes. On the other hand, the integrative effect of the known and accidental factors on system behaviors should be included in the grey IAQ model. Consequently, we need to select appropriate inputs to stimulate the grey system in order to obtain the date that contain system characteristics as much as possible. Combined with field measurement, grey IAQ model can identify the variation intervals of key IAQ model parameters that are lack of directly measurable messages in practical situations, especially when you feel dissatisfied with IAQ. (2) Grey uniformly mixed and distributed IAQ model: The grey method can be applied to the general used uniformly mixed and distributed IAQ model also. Since system inputs C0 (concentration of supply air), Q (supply air rate), and N (contaminant emanation rate) vary in intervals, they can be substituted by the corresponding grey parameters to get the grey model. According to existing knowledge, the grey parameters C 0 ,Q, N can be recommended for given condition. In this case, we can attain the upper limitation and lower limitation of contaminant index through grey
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numbers operations and can make the comparison between calculated values and measured values [10]. When we know more information of ventilation system (room structure, occupant constitution, operating mode, etc.), the varying scope can be reduced and the system characteristic is known more specifically. 3.3. Grey assessment of IAQ The assessment of IAQ is a multifactor comprehensive assessment that should express the integrative influence of these parameters on IAQ. Generally, people pay more attention to these indoor pollutions: formaldehyde, carbon monoxide (CO), carbon dioxide (CO2), nitrogen oxides (NO2), sulfur dioxide (SO2), respirable particles, pathogen, and radon. Every kind of substance mentioned above provides the information of a group of residential pollutions that have the similar properties, or sources and reflects the present status of IAQ. Because radon is not the general indoor pollution in modern buildings, we select the following substances as the parameters measured: formaldehyde, CO, CO2, NO2, SO2, respirable particles, pathogen. Thus, it is an important problem how to assess IAQ according to contaminant indexes. 3.3.1. Standard sequences for assessment We grade the objects according to contaminant concentrations usually. Allowing for the fact that the concentrations of indoor contaminants are usually low, we grade IAQ to four grades: cleanness, noncontamination, low contamination and high contamination. When the indoor air is considered as cleanness, the concentrations of indoor contaminants are background values and this indoor air environment is suit to human being. When indoor air is considered as noncontamination, the major of the occupants feel well and the indoor air environment is not harmful to human being. When indoor air contaminant concentrations are exposure limits, the occupants can feel IAQ unacceptable obviously and indoor air is considered as low contamination. When indoor air is heavily polluted, the health of human is affected markedly. In such multifactor comprehensive assessment, we obtain the values of j representative contaminants of l buildings through field measurement and these values constitute measured sequences P01 ; P02 ; ; P0l . According to the achievement in the research of IAQ, the standard values of m grades can be proposed and these values also constitute standard sequences Q01 ; Q02 ; . . . ; Q0m . We can assess IAQ according to the relations between measured sequences and standard sequences. 3.3.2. Pretreatment of original sequences The elements of each sequence have different dimensions and can vary in large numerical ranges. We know that the response is proportional to the logarithm of the stimulus from the Weber–Fechner Law: R ¼ k log S [11]. In order to reflect the perceived intensity and make comparison
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between different contaminants, we can formulate the following equation to treat contaminant concentrations: n L ¼ k log , (9) n0 where n is the measured value, n0 is the background concentration that is suit for human being and k is the constant for different contaminant. k can be obtained by assuming that L is equal to 2 when n is exposure limit. After pretreatment, the concentration values have been converted to dimensionless values that give a much better approximation to human perception of IAQ. The l measured sequences and m standard sequences that have been pretreated can be expressed as following: Pl ¼ fpl ð1Þ; pl ð2Þ; . . . ; pl ðkÞ; . . . pl ðjÞg, Qm ¼ qm ð1Þ; qm ð2Þ; . . . ; qm ðkÞ; . . . ; qm ðjÞ .
(10) (11)
3.3.3. Mathematical model of grey assessment As we know, the basic idea of the grey relational analysis in grey system theory is that the geometrical resemblance between the curves of these sequences represents the relation between them. Thus, we can determine the IAQ grade and make comparison between different objects by the grey relational grades that quantificationally weigh the relations between the measured sequences and the standard sequences. The improved grey relational grade rl,m can be calculated according to the following equation: rl;m ¼
j 1X
j
xl;m ðkÞ ¼
xl;m ðkÞ,
(12)
k¼1
minl minm mink jpl ðkÞ qm ðkÞj þ lmaxl maxm maxk jpl ðkÞ qm ðkÞj . jpl ðkÞ qm ðkÞj þ lmaxl maxm maxk jpl ðkÞ qm ðkÞj
(13) All the grey relational grade rl,m values constitute a l m grey incidence matrix: 1 0 r1;1 r1;2 r1;m C Br B 2;1 r2;2 r2;m C C B (14) R¼B . .. C. .. .. B .. . C . . A @ rl;1
rl;2
rl;m
The elements of the grey incidence matrix provide abundant information for assessment and comparison of IAQ: (1) Elements in every row of the matrix are the grey relational grades between a certain object and the corresponding grades. The grade that the maximal value relates to is the assessment of a certain object. The sort order of the grey relation grades indicates the trend of IAQ also.
(2) In the other hand, elements in every column of the matrix are the grey relational grades between a certain grade and different objects. So we can obtain the IAQ comparison between two objects by comparing the corresponding elements of two columns of the grey incidence matrix. The higher grey relational grade in excellent IAQ grades implies IAQ is better. In contrast, the higher value in poor IAQ grades implies IAQ is worse. If the grey relational grades that imply Building A is better than Building B are dominant, we can make the conclusion that IAQ of Building A is better than IAQ of Building B. As an example, we calculate the grey incidence matrix using the indoor contaminant concentrations of four buildings [12]. Based on the fundamental principles of grey system, we can determine the IAQ grades and make comparison according to the grey incidence matrix R. IAQ has much to do with the influence of contaminants on human comfort and health. The subject response to contaminations influences the assessment of IAQ greatly also. In order to make grey assessment more significative, we should find out the contaminations that influence the subject response greatly according to subject investigation of IAQ and regulate the standard sequences. Grey assessment provides an additional effective method to evaluate the status of IAQ.
4. Conclusions Grey comprehensive analysis of IAQ indicates that we can select a certain contaminant representing a group of contaminants to simplify the measurement of IAQ parameters and we should pay more attention to the air purification and humid control in the design and maintenance of HVAC. Various grey IAQ models can be obtained by changing the parameters in the corresponding determinate models into grey ones. We propose grey assessment method to determine the IAQ degree and make comparison. Grey assessment is an effective multifactor comprehensive assessment method that can express the integrative influence of contamination indexes on IAQ. Because investigation on uncertainty in IAQ using grey system theory is a bran-new mission, there are still many problems which need to be solved. Moreover, it is a meaningful work to combine grey system theory with stochastic analysis and fuzzy mathematics in the study of IAQ.
Acknowledgments The authors are very grateful to the National Natural Science Foundation of China and Guangzhou University Research Foundation for the support.
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References [1] Deng JL. Grey control systems. 2nd ed. Wuhan: Huazhong University of Science and Technology Press; 1992. [2] ASHRAE. ASHRAE Standard 62-2001, Ventilation for acceptable indoor air quality. Atlanta: American Society of Heating, Refrigerating and Air-condition Engineers, Inc. [3] Wang X, Chen Z, Yang C, Chen Y. Grey predicting theory and application of energy consumption of building heat–moisture system. Building and Environment 1999;34(4):417–20. [4] Hu WB, Hua B, Yang CZ. Building thermal process analysis with grey system method. Building and Environment 2002;37(6):599–605. [5] Zeng GM, Zuo L, Yang CP. The application of grey system and method in environmental systems. Beijing: The Science and Technology Press of China; 1994. [6] Shen JM, Mao JC, Cai W. The subject evaluation of indoor air quality for office buildings in Shanghai. Ventilation and Dust control 1996;15(2):14–7.
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[7] Shen JM, Mao JC, Sun GQ. The objective evaluation of indoor air quality for office buildings in Shanghai. Ventilation and Dust control 1995;14(2):14–7. [8] Zhu CH, Li NP, Wei W. Grey incidence analysis on subjective assessment of indoor air quality. Journal of Guangzhou University (Natural Science Edition) 2003;2(3):273–6. [9] Zeng GM, Ye SZ. A grey model for river water quality and its grey parameters’ optimal estimation. The Journal of Grey System 1989;1: 53–64. [10] Zhu CH. Assessment method of indoor air quality and its application based on grey system theory. Master degree thesis. Hunan University, Changsha, People’s Republic of China, 2002. [11] Jokl MV. Evaluation of indoor air quality using the decible concept based on carbon dioxide and TVOC. Building and Environment 2000;35(8):677–97. [12] Li NP, Zhu CH, Wen W. Grey assessment of indoor air quality. Journal of Hunan University (Natural Sciences Edition) 2002;29(4): 85–91.
Building and Environment 42 (2007) 1711–1717 www.elsevier.com/locate/buildenv
Uncertainty in indoor air quality and grey system method Chihui Zhua,b, Nianping Lia,, Di Rea, Jun Guana a
College of Civil Engineering, Hunan University, Changsha 410081, China School of Civil Engineering, Guangzhou University, Guangzhou 510006, China
b
Received 4 May 2005; received in revised form 14 June 2005; accepted 24 January 2006
Abstract Based on analysis of uncertainty, this paper presents grey system theory to handle the ‘‘grey’’ characteristic of IAQ. Grey comprehensive analysis of indoor air quality reveals that we should pay more attention to the air purification and humidity control in the design and maintenance of HVAC. In order to represent grey characteristic of IAQ system, the educed grey IAQ models can identify the variation intervals of key IAQ model parameters that are lack of directly measurable messages in practical situations. Furthermore, grey assessment is an effective multifactor comprehensive assessment method that can express the integrative influence of contamination indexes on indoor air quality. We can determine the IAQ grade and make comparison according to the grey incidence matrix R. r 2006 Elsevier Ltd. All rights reserved. Keywords: Indoor air quality; Uncertainty; Grey system theory
1. Introduction There are several theories how to handle uncertain data: fuzzy logic, system identification, dimension analysis. A new one may be added: the grey system theory [1]. The systems with partially known and partially unknown information are named as grey systems. Grey system theory is especially useful when the complete set of factors involved in the system’s behavior is unknown or unclear, when the relationships of system factors to the system’s behavior and inter-relationships among factors are uncertain, when the system behavior is too complex to determine completely, or when only limited information on system behavior is available. To make it easier to the reader to understand the grey system an application to indoor air quality (IAQ) is given. Uncertainty exists in the influencing factors of IAQ, IAQ model, assessment of IAQ, etc. IAQ is a ‘‘grey’’ system that we are short of adequate messages to describe. How can we make use of these inadequate messages to extract or produce more useful information about the grey system? Corresponding author. Tel./fax: +86 731 8823115.
E-mail address: [email protected] (N. Li). 0360-1323/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2006.01.015
Based on analysis of the uncertainty contained in IAQ, this paper presents grey system analysis of IAQ, grey model of IAQ, grey assessment of IAQ to provide guides to design and maintenance of indoor environment.
2. Uncertainty in IAQ 2.1. Uncertainty in scope of IAQ In the research process, people modify their viewpoint of IAQ according to new discoveries. IAQ was seen as a series of indoor air contaminant indexes formerly. Recent study indicates that this viewpoint is incomprehensive and unable to express the meaning of IAQ. What we need are indoor air that is perceived as fresh, pleasant and stimulating, with no negative effects on health and a thermal environment perceived as comfortable by almost all occupants. ASHRAE Standard 62-2001 [2] provides the concept of acceptable IAQ that covers not only the objectivity but also subjectivity of IAQ. Thus IAQ is an interdisciplinary science. Along with the development of IAQ studies in depth and in extent, the scope involved in IAQ will vary and extend.
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2.2. Uncertainty in factors affecting IAQ The indoor air environment is a result of the interaction between outdoor air quality, building envelope, ventilation system, contaminant sources, and building occupants. Poor IAQ is aroused by numerous indoor air contaminants essentially. Moreover, only a few among the numerous contaminants are known by people (determinate or white). As to most contaminants, we do not know their characteristics and health effects exactly (uncertainty or grey). The effects of IAQ problems are often nonspecific symptoms rather than clearly defined illnesses. Based on industrial hygiene, we cannot identify the reasons for the complaint of occupants. This problem may be caused by the combined effects of multiple pollutants at low concentration. The influence of indoor air contaminants on human body is complex and because so many factors operate in the system, it is difficult to make certain the causality between them. Further we should reduce the randomicity and find out rule underlying during data processing, because the data measurement is affected by some uncontrollable factors. It is uncertainty contained in the previous studies that force us to look for suitable tools to deal with such questions. 2.3. Uncertainty in IAQ model In a narrow sense, IAQ model is the mathematic relation that describes spatia-temporal rule of contaminant transport and removal. According to the comprehension of internal rules, people present models such as uniformly mixed and distributed model, modified uniformly mixed and distributed model, double-zone model, etc. In the above-mentioned models, system parameters and system inputs are assumed to be deterministic. Although there are some causalities between the system inputs and the system outputs, system model parameters may vary in wide ranges due to practical situations and incidental factors. System structure and the relation among system parameters are alterable also. Moreover, uncertainty in system input exhibits in interval variation of system input parameters. The two types of uncertainties contribute to the grey characteristics of IAQ that can be dealt with by grey system theory. 2.4. Uncertainty in assessment of IAQ Which contaminant can denote the state of IAQ best? Assessment of IAQ relates to not only the health effect of indoor contaminants to occupant but also the subjective response of occupant to indoor air. Constituting new index according to several contaminants is an effective method. Nevertheless, evaluation index and grade criterion are empirical and have no physical significations. The degree of IAQ is a grey idea. It does not correspond to a series of determinate values of contaminants necessarily, but relates to concentration scopes. The assessment of IAQ is multi-
factor assessment; thus assessment method should express the integrative influence of contamination indexes on IAQ. 3. Grey system methods applied in IAQ Based on the above discussion, we know that the complete set of factors involved in IAQ is unknown or unclear, the relationships of influencing factors to IAQ and inter-relationships among factors are uncertain, the IAQ system behavior is too complex to determine completely and only limited information on IAQ system is available. In a word, IAQ is a system that we are short of adequate messages to describe. Prof. Deng Julong first originated grey system theory to deal with this type of system. Grey comprehensive analysis, grey modeling, grey forecast, etc. are effective methods for applying grey system theory. In the field of building environment, Wang [3] proposed a method based on grey prediction model to predict building energy consumption. Hu [4] studied grey model of direct solar radiation intensity on the horizontal plane for cooling loads calculation and building thermal process analysis with grey system method. Zeng [5] presented uncertainty and methodology in environmental science and engineering. Grey analysis is particularly applicable in instances with very limited data and in cases with little system knowledge or understanding. All these previous studies provide reasonable references for us to deal with the ‘‘grey’’ characteristics of IAQ. Thereinafter, we try to deal with grey IAQ system by using appropriate grey methods. 3.1. Grey comprehensive analysis of IAQ In the design of heating, ventilating and air conditioning (HVAC) system and management of IAQ, we should know which contaminant affects IAQ chiefly, which contaminant can arouse a certain symptom, which contaminant will deteriorate IAQ significantly with concentration increase. Furthermore, the measurement of contaminant concentration is a time-consuming and expensive process. As we know, some contaminants have the same sources and there is correlativity in their presence and intensity. Can we propose an integrative index or choose a certain contaminant to represent a group of contaminants? There are so many factors concerned in the previous problems that it is not an easy work to identify the causality specifically. Fortunately, the grey relation analysis provides an effective way to solve those problems. 3.1.1. Idea of grey relation analysis According to grey system theory, we can choose a parameter to represent the development (output) of a system; thus a time or spatial sequence can express the development of the system. The values for each input factor, like the output values, can be treated as a sequence. The input and output sequences can be compared to identify the system’s key factors. The fundamental idea of the grey relational analysis is that the geometrical
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resemblance between the curves of the sequences represents the relation between the factors. In the IAQ study, uncomfortable sensation and sick symptoms can be seen as system outputs; indoor air contaminants can be treated as system inputs. Through field investigation, we can obtain the dissatisfactory rates of indoor air, sick symptom rates and concentrations of several recognized important indoor contaminants in several buildings. As discussed above, these values constitute corresponding sequences that can be used to identify key factor and inter-relation. We use the grey relational grade to quantificationally weigh the relation between those sequences. The higher grey relational grade value implies that the relation between factors is closer. Thus, grey relation analysis can reveal the relationships of related factors to IAQ and interrelationships among these factors. The relationships can then be used to suggest appropriate instructions in design and maintenance of indoor environment. 3.1.2. Calculation of grey relational grade Because sequences include different parameters that have different dimensions and the numerical values of these parameters may have great ranges, we should make these sequences equivalent through pretreatment in order to remove the influence of dimensions. The ordinary pretreatment is to calculate the initial images or average images of these sequences. After pretreatment, the output sequence that constituted by the dissatisfactory rates of indoor air or sick symptoms rates in l buildings is expressed as X ¼ fxð1Þ; xð2Þ; . . . ; xðkÞ; . . . ; xðlÞg.
(1)
In the same way, the n input sequences can be expressed as Y j ¼ fyj ð1Þ; yj ð2Þ; . . . ; yj ðkÞ; . . . ; yj ðlÞg, j 2 N ¼ f1; 2; . . . ; ng,
ð2Þ
where n denotes that there are n affecting factors (formaldehyde, CO, CO2, NO2, SO2, respirable particle, pathogen, etc.). The grey relational grade rj between system output sequence X and one affecting factor sequence Yj can be calculated by Eq. (3) and the grey relational grade rj shows the correlation between a specific ill symptom and a specific contaminant: rj ¼
l 1X x ðkÞ, l k¼1 j
(3)
xj ðkÞ ¼
minj mink jxðkÞ yj ðkÞj þ lmaxj maxk jxðkÞ yj ðkÞj , jxðkÞ yj ðkÞj þ lmaxj maxk jxðkÞ yj ðkÞj ð4Þ
where l ðl 2 ½0; 1; commonly l ¼ 0:5Þ is distinguishing coefficient which is used to regulate difference significance of grey relational grade.
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In the comprehensive grey analysis what we concern mostly is the sort order of grey relational grades rather than the real numerical values. The highest value implies that the corresponding contaminant is most closely correlated with the acceptability of IAQ. Lower values are associated with contaminants which are less important to the IAQ. Grey relation analysis is an improved method of identifying and prioritizing key factors and is useful for variable independence analysis. 3.1.3. Grey analysis of IAQ In modern office buildings, the concentrations of indoor air parameters are low and steady commonly, but the requirement of occupant is relatively high. This type building is object suiting for IAQ grey system analysis. Through field investigation, Shen [6,7] obtained the dissatisfactory rates of indoor air, sick symptoms rates and concentrations of several recognized important indoor contaminants in several office buildings. Using the method discussed above, we have calculated the grey relational grades between the dissatisfactory rates of indoor air (or sick symptoms rates) and indoor air contamination levels. According to the sort order of these grey relational grades, we can draw the following results [8]: 1. The grey relational grade between respirable particle and the dissatisfactory rate of indoor air has the highest value, which illuminates that respirable particle is the key reason for poor air quality. This may be due to that respirable particle to which gaseous air contaminants and biologic particles adhere can enter respiratory tract and stimulate mucous membrane. 2. The second highest value is the grey relational grade between pathogen and the dissatisfactory rate of indoor air. It is possible that when indoor space is close, ventilating effect is poor and moisture accumulates, pathogen will grow quickly and exert an important effluence on the acceptability of IAQ. 3. The grey relational grades between indoor contaminants and most sick symptoms have the similar sort orders. The high grey relational grade value indicates IAQ will deteriorate rapidly with the concentration raise. In other words, reducing the concentration of corresponding contaminants will improve IAQ remarkably. Consequently, we should pay more attention to the air purification and humidity control in the design and maintenance of HVAC. The grey relational grade between the formaldehyde and the dissatisfactory rate of indoor air (or sick symptom) has the relatively low value, which shows that the dilution of VOCs cannot improve IAQ effectively. On the other hand, because the main sources of VOCs are furnishings, wall coverings and occupancy activities, the effective method to improve IAQ is to select the materials and arrange occupancy activity reasonably.
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In the same way, grey analysis is applied to determine the correlation among diverse contaminants. High relational grades between contaminants may indicate contaminants interdependence, suggesting there is correlativity in their presence and intensity. Based on the full matrix constituted by grey relational grades, we may divide contaminants into different types according to critical value of grey relational grades. Thus, we can select a certain contaminant representing a group of contaminants in order to simplify the measurement of IAQ parameters. This technique is mostly suitable for occupancy and owner (not expert and researcher). 3.2. Grey model of IAQ It can be seen from above that we should control key contaminants to obtain better IAQ. Based on comprehension of IAQ system, people develop different IAQ models to describe spatia-temporal rules of contaminant transport and removal. Under certain initial and boundary conditions, these IAQ models can be expressed in the following form: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ,
(5)
where y1 ; y2 ; . . . ; yn are n model parameters, X is the input vector of the model, Y is one air quality index (for example CO2) and f denotes a certain function form. Model parameters in Eq. (5) are generally assumed to be fully known and constant. Although there is determinate causality between the input and output in above models, uncertainty lies in the system development and variation. Except statistical randomness, uncertain indicates epistemic uncertainty and system information inadequacy. 3.2.1. Uncertainty in IAQ Because of the influences of indoor contaminant sources distribution, plan of supply air outlets and return inlets and numerous incidental factors, model parameters y1 ; y2 ; . . . ; yn may vary in wide intervals. Furthermore, because the complexity of the natural world greatly exceeds our ability to model it, we may use a small number of variables to represent a complex phenomenon, choose incorrect functional forms, and set inappropriate boundaries. Therefore, another uncertainty is the lack of correspondence between the model and reality. In the view of grey system theory, this type of uncertainty can be referred to as uncertainty in system structure and be expressed in grey structure and grey parameters marked by ‘‘’’. Under such case, function form is grey structure f; model parameters are grey parameters yi ði ¼ 1; 2; . . . ; nÞ. The usual form of grey parameter is grey interval number that can be expressed as following: yi ¼ ½y^ i di;1 ; y^ i þ di;2 ði ¼ 1; 2; . . . ; nÞ,
(6)
where y^ i is the whitening value of the ith parameter, di;1 and di;2 indicate the variational scope of grey parameter.
System parameter is not a constant, but a grey interval number. Besides the uncertainty in system structure, there is uncertainty in system input also. For example, emit rate of indoor air contaminant is not only dependent on the characteristics of contaminant sources but also affected by environment factors (temperature, humidity, air velocity etc.). Air infiltration rate is related to building envelope, distribution of indoor and outdoor air pressure, climate etc. Biologic contamination of human being varies greatly with age, sex, activity level, and body condition. IAQ system input has no single value and will vary in large interval. In other words, system input is grey parameter X. 3.2.2. Grey IAQ model As discussed above, Eq. (5) does not take numerous unknown and incidental factors into consideration. It can be called as determinate model that is whiting description of grey IAQ system. Based on Eq. (5), we introduce grey structure f, grey parameters yi , grey input X to represent grey characteristics of IAQ system, therefore obtain the corresponding grey IAQ model: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ.
(7)
When we have appropriately described the essence of IAQ system, system structure is whiting structure and grey IAQ model can be simplified as the following generally adopted equation: Y ¼ f ðy1 ; y2 ; . . . ; yn ; X Þ.
(8)
Grey IAQ model describes IAQ comprehensively and reasonably by combining definite physical model with grey parameters. It makes best use of the existing knowledge and measurable messages about the IAQ and employs grey variables, grey series, and grey arrays to describe the important parameters that are lack of directly measurable messages. (1) Optimal grey parameters: Based on grey system theory, the optimal grey parameters of IAQ model can be defined as the optimal grey intervals which include all unknown and accidental factors’ set effects. The whiting values y^ 1 ; y^ 2 ; . . . ; y^ n of grey parametersyi ði ¼ 1; 2; . . . ; nÞ reflect the average influence of numerous factors on system; the two boundary values ðdi;1 and di;2 Þ of the optimal interval relate to the upper limitation and lower limitation of the system’s dynamic development. System essence is embedded in the measured values concerned with system input and output. Through field measurement, we can gain system input X and a discrete distribution sequence Y ¼ ½Y 1 ; Y 2 ; . . . ; Y m constituted by values of a certain IAQ index. Further, from Y we can educe two sequences Y¯ ¼ ½Y¯ 1 ; Y¯ 2 ; . . . ; Y¯ m and Y ¼ ½Y 1 ; Y 2 ; ; Y m2 ðm1 þ m2 ¼ mÞ which reflect the upper and lower limitations of the unknown and accidental factors’ set effects on the IAQ index. The optimal grey parameters and whiting values can be identified from X, Y¯ ,
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Y , Y and corresponding grey IAQ model by using optimization criteria and optimization method. The detailed process is referred to Ref. [9]. (2) Operations of grey interval numbers: When there is uncertainty in the system input, the system input presents in the grey interval parameter X. Traditional arithmetic operation principles are substituted by algorithms of grey interval numbers. The result should also be an interval grey number. We can calculate the grey IAQ model according to the algorithms among grey interval numbers. 3.2.3. The application of grey IAQ model Eq. (8) is deduced from a certain physical model. Various grey IAQ models can be obtained by changing the parameters in the corresponding determinate models into grey ones. (1) Grey double zone IAQ model: Ev is introduced to express the ratio of the effective supply air into the working zone to the total supply air in the double zone IAQ model. Ev represents the ventilating effect which relates to ventilating mode, ventilating rate, supply air quality, characteristic and plan of indoor air contamination sources etc. As we know, it is not easy to show the relation between Ev and affecting factors. The rest parameters can be measured easily. Consequently, we can educe the corresponding grey double zone IAQ model by replacing Ev with grey parameter E v . It is convenient to estimate the optimal grey parameter E v , in different ventilating strategies from the system input and corresponding measured output data through the method of least squares. More messages we know about the studied system, more it is convenient for parameter estimation. In order to reduce the complexity of system analysis, we analyze every element of grey IAQ model to make certain which element is measured and controlled easily and the sources of uncertainty and their rough variation scopes. On the other hand, the integrative effect of the known and accidental factors on system behaviors should be included in the grey IAQ model. Consequently, we need to select appropriate inputs to stimulate the grey system in order to obtain the date that contain system characteristics as much as possible. Combined with field measurement, grey IAQ model can identify the variation intervals of key IAQ model parameters that are lack of directly measurable messages in practical situations, especially when you feel dissatisfied with IAQ. (2) Grey uniformly mixed and distributed IAQ model: The grey method can be applied to the general used uniformly mixed and distributed IAQ model also. Since system inputs C0 (concentration of supply air), Q (supply air rate), and N (contaminant emanation rate) vary in intervals, they can be substituted by the corresponding grey parameters to get the grey model. According to existing knowledge, the grey parameters C 0 ,Q, N can be recommended for given condition. In this case, we can attain the upper limitation and lower limitation of contaminant index through grey
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numbers operations and can make the comparison between calculated values and measured values [10]. When we know more information of ventilation system (room structure, occupant constitution, operating mode, etc.), the varying scope can be reduced and the system characteristic is known more specifically. 3.3. Grey assessment of IAQ The assessment of IAQ is a multifactor comprehensive assessment that should express the integrative influence of these parameters on IAQ. Generally, people pay more attention to these indoor pollutions: formaldehyde, carbon monoxide (CO), carbon dioxide (CO2), nitrogen oxides (NO2), sulfur dioxide (SO2), respirable particles, pathogen, and radon. Every kind of substance mentioned above provides the information of a group of residential pollutions that have the similar properties, or sources and reflects the present status of IAQ. Because radon is not the general indoor pollution in modern buildings, we select the following substances as the parameters measured: formaldehyde, CO, CO2, NO2, SO2, respirable particles, pathogen. Thus, it is an important problem how to assess IAQ according to contaminant indexes. 3.3.1. Standard sequences for assessment We grade the objects according to contaminant concentrations usually. Allowing for the fact that the concentrations of indoor contaminants are usually low, we grade IAQ to four grades: cleanness, noncontamination, low contamination and high contamination. When the indoor air is considered as cleanness, the concentrations of indoor contaminants are background values and this indoor air environment is suit to human being. When indoor air is considered as noncontamination, the major of the occupants feel well and the indoor air environment is not harmful to human being. When indoor air contaminant concentrations are exposure limits, the occupants can feel IAQ unacceptable obviously and indoor air is considered as low contamination. When indoor air is heavily polluted, the health of human is affected markedly. In such multifactor comprehensive assessment, we obtain the values of j representative contaminants of l buildings through field measurement and these values constitute measured sequences P01 ; P02 ; ; P0l . According to the achievement in the research of IAQ, the standard values of m grades can be proposed and these values also constitute standard sequences Q01 ; Q02 ; . . . ; Q0m . We can assess IAQ according to the relations between measured sequences and standard sequences. 3.3.2. Pretreatment of original sequences The elements of each sequence have different dimensions and can vary in large numerical ranges. We know that the response is proportional to the logarithm of the stimulus from the Weber–Fechner Law: R ¼ k log S [11]. In order to reflect the perceived intensity and make comparison
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between different contaminants, we can formulate the following equation to treat contaminant concentrations: n L ¼ k log , (9) n0 where n is the measured value, n0 is the background concentration that is suit for human being and k is the constant for different contaminant. k can be obtained by assuming that L is equal to 2 when n is exposure limit. After pretreatment, the concentration values have been converted to dimensionless values that give a much better approximation to human perception of IAQ. The l measured sequences and m standard sequences that have been pretreated can be expressed as following: Pl ¼ fpl ð1Þ; pl ð2Þ; . . . ; pl ðkÞ; . . . pl ðjÞg, Qm ¼ qm ð1Þ; qm ð2Þ; . . . ; qm ðkÞ; . . . ; qm ðjÞ .
(10) (11)
3.3.3. Mathematical model of grey assessment As we know, the basic idea of the grey relational analysis in grey system theory is that the geometrical resemblance between the curves of these sequences represents the relation between them. Thus, we can determine the IAQ grade and make comparison between different objects by the grey relational grades that quantificationally weigh the relations between the measured sequences and the standard sequences. The improved grey relational grade rl,m can be calculated according to the following equation: rl;m ¼
j 1X
j
xl;m ðkÞ ¼
xl;m ðkÞ,
(12)
k¼1
minl minm mink jpl ðkÞ qm ðkÞj þ lmaxl maxm maxk jpl ðkÞ qm ðkÞj . jpl ðkÞ qm ðkÞj þ lmaxl maxm maxk jpl ðkÞ qm ðkÞj
(13) All the grey relational grade rl,m values constitute a l m grey incidence matrix: 1 0 r1;1 r1;2 r1;m C Br B 2;1 r2;2 r2;m C C B (14) R¼B . .. C. .. .. B .. . C . . A @ rl;1
rl;2
rl;m
The elements of the grey incidence matrix provide abundant information for assessment and comparison of IAQ: (1) Elements in every row of the matrix are the grey relational grades between a certain object and the corresponding grades. The grade that the maximal value relates to is the assessment of a certain object. The sort order of the grey relation grades indicates the trend of IAQ also.
(2) In the other hand, elements in every column of the matrix are the grey relational grades between a certain grade and different objects. So we can obtain the IAQ comparison between two objects by comparing the corresponding elements of two columns of the grey incidence matrix. The higher grey relational grade in excellent IAQ grades implies IAQ is better. In contrast, the higher value in poor IAQ grades implies IAQ is worse. If the grey relational grades that imply Building A is better than Building B are dominant, we can make the conclusion that IAQ of Building A is better than IAQ of Building B. As an example, we calculate the grey incidence matrix using the indoor contaminant concentrations of four buildings [12]. Based on the fundamental principles of grey system, we can determine the IAQ grades and make comparison according to the grey incidence matrix R. IAQ has much to do with the influence of contaminants on human comfort and health. The subject response to contaminations influences the assessment of IAQ greatly also. In order to make grey assessment more significative, we should find out the contaminations that influence the subject response greatly according to subject investigation of IAQ and regulate the standard sequences. Grey assessment provides an additional effective method to evaluate the status of IAQ.
4. Conclusions Grey comprehensive analysis of IAQ indicates that we can select a certain contaminant representing a group of contaminants to simplify the measurement of IAQ parameters and we should pay more attention to the air purification and humid control in the design and maintenance of HVAC. Various grey IAQ models can be obtained by changing the parameters in the corresponding determinate models into grey ones. We propose grey assessment method to determine the IAQ degree and make comparison. Grey assessment is an effective multifactor comprehensive assessment method that can express the integrative influence of contamination indexes on IAQ. Because investigation on uncertainty in IAQ using grey system theory is a bran-new mission, there are still many problems which need to be solved. Moreover, it is a meaningful work to combine grey system theory with stochastic analysis and fuzzy mathematics in the study of IAQ.
Acknowledgments The authors are very grateful to the National Natural Science Foundation of China and Guangzhou University Research Foundation for the support.
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