Comparison of probabilistic approaches to the mould growth problem

Transcription

1 Comparison of probabilistic approaches to the mould growth problem Zoltán Sadovský & Oľga Koronthályová & Peter Matiašovský Institute of Construction and Architecture, Slovak Academy of Sciences, Bratislava, Slovakia

2 Mould growth conditions NECESSARY CONDITIONS: Sufficient duration of temperature (T) and relative humidity (RH) levels Dependent on: - fungus species - substrate Obtained by exploring hourly time series of T and RH STEADY STATE CONDITIONS: Sedlbauer s isopleth system for spore germination, valid for all fungi, substrate category I and II 2

3 TOW mould growth criteria TIME-OF-WETNESS CONCEPT TOW (fluctuating states) by Adan 1994, Adan & Samson 2011 DURATIONS OF GROWTH CONDITIONS BY TOW ON CYCLES Continuous time periods, which besides of hours with RH exceeding the critical values include hours enabling continuation of mould growth by absorbed moisture despite of RH less than RH crit (T) TOW = cyclic wet period / cyclic period (wet + dry) TOW on cycle is defined by: TOW constant (0.5) TIME SERIES resulting from the TOW concept continuous durations of favourable mould growth conditions TOW on cycles and dry periods TOW off cycles 3

4 TOW on/off process EXAMPLE: 2010 November 4 to TOW off TO OW on (hours) Time (days) 4

5 Probabilistic approaches 1. TIME INVARIANT APPROACH A single random variable output is studied, e.g. : a) the totals of hours with RH(T) RH crit (T) in a reference period b) the mould growth potential RH(T) / RH crit (T) in a reference period Statistical characteristics and cdf are estimated. 2. TIME DEPENDENT APPROACH new model - J Build Physics online A stochastic process representing TOW on cycles is developed and studied for estimating the first-passage probability of exceeding a considered duration of TOW on cycle in a reference period. 3. TIME DEPENDENT APPROACH block maxima method The stochastic process of TOW on cycles is represented by maxima of TOW on cycles in the reference periods. The probability of exceeding a considered maximum of TOW on cycle is estimated. 5

6 Example studied BUILDING STRUCTURE part of a residential house two external walls of a room of m3 treated as a 1- zone space external walls: mineral plaster 15 mm; AAC 300 mm; min. plast. 15 mm RANDOM VARIABLE MATERIAL PROPERTIES 3 parameters of heat and moisture transfer of AAC: thermal conductivity, water vapour resistance factor and moisture diffusivity at 100% RH TIME DEPENDENT INPUT time series climatic conditions: 13 years hourly measurements in Bratislava heating and ventilation system & user behaviour: natural ventilation and ideal heating, activities of 4-member family (scenarios) REFERENCE PERIOD the first heating season from as-built state 6

7 Deterministic calculations INDOOR AIR RELATIVE HUMIDITY simulation tool NPI, based on the solution of the water vapour mass balance equation in the single zone space RH & T ON INTERNAL SURFACE OF THE EXTERNAL WALL as hourly time series by 1-D simulation tool NEV3 (Koronthályová & Matiašovský 1998) based on the solution of two coupled equations for heat and moisture transfer FAVOURABLE MOULD GROWTH CONDITIONS Because of minor temperature fluctuations around 19 C RH RH crit = 80 % (point-in-time hourly criterion) 7

8 1. Time invariant approach a) Totals of hours with RH RH crit in the reference period (13 realisations) emp pirical F Gumbel max Seasonal hours with RH-RHcrit >= 0 Sample cdf is approximated Non-linear regression: N = 13 exp(-exp(-1.283/b*(x/a-1)-0.577))#gumbel 0 <= x <= 500 Least square minimized a = ; b = Esimated using 1/n: Variance of residuae = Correlation coefficient = PROBABILITY OF EXCEEDING GIVEN d of the totals P(D > d) = 1 F (d) Structure of durations?! 8

9 2. Time dependent approach / 1 MODEL FOR THE TOW ON STOCHASTIC PROCESS 2010 November 4 to November 4 to 29 TO OW off TOW on (hours) Time (days) TOW on spik kes (hours) Random occurences & durations of TOW cycle Time (days) Practically important durations are rare events 9

10 2. Time dependent approach / 2 STUDY OF THE TOW ON PROCESS For applying the well developed theory of extreme values the independence or at least near independence of the data is required. Then the rare events can be studied as exceedances over high threshold applying the Poisson spike process. CLUSTERS AND DECLUSTERING OF TOW ON EVENTS Clusters indicate dependence of data. Declustering can be done by reducing the sample of exceedances to that of the cluster maxima THE UPPER TAIL APPROXIMATIONS OF TOW ON DURATIONS of the sample cdf of ordered durations d i are required i ( d ) = The Generalised Pareto cdf (GP) is employed F i 814 occurrences of TOW on 10

11 2. Time dependent approach / GP F em mpirical and GP cdf Upper TOW on cycles [hours] Non-linear regression: N = 11 1-(1+c*(x-a)/b)^(-1/c) #GP 15 <= x <= 530 Least square minimized a = b = c = Estimated using 1/n: Variance of residuae = E-007 Correlation coefficient = PROBABILITY OF THE FIRST UP-CROSSING OF d P(D > d) = 1 exp[ ] where + ν + d is the up-crossing rate and ν d = F( d)[1 F( d)] ν ν = 814/13 is the mean rate occurrence of TOW on cycles in a season 11

12 l and GEV cfd Block maxima method ONLY SEASONAL MAXIMA OF TOW are studied avoiding dependencies empirical GEV F Seasonal max TOW on [hours] Non-linear regression: N = 13 exp(-(1+c*(x-a)/b)^(-1/c)) #GEV 5 <= x <= 530 Least square minimized a = 50.34; b = c = Estimated using 1/n: Variance of residuae = Correlation coefficient = GEV stands for Generalised extreme value cdf PROBABILITY OF EXCEEDING GIVEN d P(D > d) = 1 F (d) 12

13 Probabilities by time-dependent stochastic & block maxima methods 0.7 Days - TOW on Probability of level crossing P( D> a) - max TOW on P( D> a) - TOW on process Hours - TOW on Seasonal maxima provide somewhat more conservative probabilities than the stochastic process approach including all rare events. 13

14 Comparison of probabilistic approaches total of hours with RH >= RHcrit 500 max TOW on in hours 500 Total of hours with RH >= RHcrit threshold of 48 hours max TOW on in hours 1998_ _ _ _ _ _ _05 1) Sum of hours with RH RHcrit blue symbol line 2) Seasonal max TOW on cycles brown symbol dashed line 3) Exceedances > 2 days include also TOW on`s < max TOW on full symbols 2005_ _07 Heating season 2007_ _ _ _11 14

15 Conclusions The time invariant probabilistic approaches to mould growth problem provide information on totals of point-in-time (hourly) favourable mould growth conditions. With respect to practically important durations this information may be misleading. The probabilities provided by the stochastic analysis of rare events and by the block maxima method are in overall agreement. The stochastic approach is more general including possibly more important data and e.g. allowing estimation of probability of double occurrence of an event. The block maxima method is simpler avoiding to care about the data dependencies. The time dependent probabilistic approaches can properly take into account the randomness of input time series resulting mainly from the variability of climatic conditions. 15

16 Thank you for attention