PERFORMANCE ASSESSMENT OF A NATURALLY VENTILATED MULTIZONE BUILDING 1 Henrik Brohus, Ph.D. ABSTRACT Accurate performance assessment is a prerequisite of an optimum building design. Some buildings require special attention due to significant influence of fluctuating forcing functions on the building model response - like naturally ventilated multizone buildings. Multizone modelling is a way to determine the air flows in a complex ventilated building subject to internal and external loads. The purpose of this work is to consider and quantify the influence of randomness in the load parameters, which is accomplished by means of a stochastic multizone model, as part of the performance assessment. In the first place a deterministic multizone model is applied. The model is capable of predicting air flow and pressure distribution within a building divided into an arbitrary number of zones and flow paths. The air flow is driven by pressure differences due to wind and stack effect. Next, the multizone model is coupled with a Monte Carlo Simulation module generating multiple realisations of input according to the statistical distribution of data. Statistical treatment of output data results in output distributions and statistics of the resulting air flows. The method is illustrated by a case study indicating that a stochastic approach gives substantial and detailed information on the building performance in which case a deterministic approach would have been quite inadequate. INTRODUCTION In building design it is crucial to be able to predict the building performance with a satisfactory accuracy, especially, if the aim is to perform an optimisation of the building. Key performance parameters may be the energy consumption, environmental load (e.g. CO2 emission) and the indoor climate (e.g. thermal comfort and indoor air quality). In terms of performance assessment prediction of air infiltration and ventilation plays an important role in the design of energy efficient buildings as well as healthy buildings. Traditionally, building performance is based on a deterministic approach, which implies that the spread of input parameters is zero. The deterministic approach is valid if the effect of fluctuations in the model input (external air temperature, wind speed and direction, solar heat gain, occupants´ behaviour, internal heat gain, etc.) is negligible when compared with the mean value. However, when random fluctuations of the forcing functions are significant, the influence on the output (energy consumption, indoor climate, etc.) can no longer be disregarded, for example in case of naturally ventilated buildings, where the entire ventilation capacity is supplied by natural driving forces. Most parameters influencing the ventilation capacity of a naturally ventilated building are stochastic by nature. For instance wind induced pressure is a complex process that is highly influenced by the turbulent nature of wind. This paper addresses the performance assessment of a naturally ventilated multizone building in terms of air flow rate. The influence of the stochastic nature of external air temperature and wind speed and direction are considered. 1 Henrik Brohus is associate professor, Department of Building Technology and Structural Engineering, Aalborg University, Denmark. Multiple compartments in ventilated buildings often give rise to complex air flow patterns. This is especially the case for naturally ventilated buildings characterised by a relatively close interaction with external weather conditions and significant mutual flows between the compartments. The multiple external and internal air flows in this study are found by means of multizone modelling. In a multizone model, a nonlinear air flow equation is formulated for each zone and the equations are solved for the unknown zone pressures after which the air flows between the zones can be calculated (Feustel and Dieris 1992). The external loads on a naturally ventilated building are stochastic processes, fluctuating both in time and place. In this work, stochastic models for the external air temperature, the wind direction and speed are adapted from the Danish Design Reference Year (Brohus et al. 2002; Jensen and Lund 1995). The Monte Carlo Simulation (MCS) approach is used to estimate the variability of the resulting air flows in the building, given as mean values, standard deviations and distribution functions. Multiple realisations of the input quantities are generated according to the input distribution functions. The statistics of the output are then obtained by statistical sampling among the simulated data. The approach will be illustrated by means of a test case with a twelve zone naturally ventilated building. METHODS Multizone model The building is divided into n zones representing rooms or part of rooms, see Figure 1. FIGURE 1. Multizone building model showing location of pressure nodes and air flow paths as well as general model topology. Pressure differences between the zones induce air flow through cracks, openings, duct work, etc. The main task of a multizone model is to determine these air flows in order to design the HVAC system, natural airing devices, etc. Defining air flow out of the zone to be positive, the balance equation for Zone i can be written as, see Figure 2. n skl n n m g i (P) = ∑∑∑ g ic,kl + ∑∑ g i0,kl = 0 k =1 l =1 c =1 l >k k =1 l =1 ( ) i = 1, … , n (1) with g c i ,kl Qklc ∆Pklc (Pk , Pl ) = − Qklc ∆Pklc (Pk , Pl ) 0 ( ( Q 0 ∆Pkl0 (Pk ) g i0,kl = kl 0 ) i =k i =l if if ) (2) else i =k if (3) else where Qklc is the air flow component c between internal zone k and l (m3/s), Qkl0 is the air flow between internal zone k and external air node l (m3/s), ∆Pklc is the pressure difference corresponding to air flow component Qklc (Pa), ∆Pkl0 is the pressure difference corresponding to air flow component Qkl0 (Pa), Pk is the reference pressure in zone k (Pa), n is the number of zones, m is the number of external pressure nodes, and skl is the number of air flow components between zone k and l. FIGURE 2. Example of air flow patterns and corresponding indexing in the multizone model. The air flow through cracks and openings are modelled by (Feustel and Dieris 1992) D c ( ∆Pklc ) nkl Q ( ∆P ) = klc c − Dkl ( − ∆Pklc ) nkl c c kl c kl ∆Pklc ≥ 0 if (4) else where Dklc is an empirical coefficient (m3/(s·Pa)) and nklc is another empirical coefficient (n.d.) By considering the stack effect, the pressure difference, ∆Pklc, over an internal opening is given by ( ) ( ∆Pklc (Pk , Pl ) = Pk − Pl + ρ k g hk − hklc − ρ l g hl − hklc ) (5) where ρk is the zone air density (kg/m3), g is the gravitational acceleration, hk is the reference height for zone k (m), and hklc is the height corresponding to air flow component Qklc (m). The zone air density is obtained from the following expression (Feustel and Raynor-Hoosen 1990) ρk = Patm (1 + W k ) 461.518(θ k + 273.15 )(W k + 0.62198 ) (kg/m3) (6) where θk is the zone air temperature (°C) and Wk is the zone water humidity ratio (kg/kg). Similarly, the pressure difference, ∆Pkl0, over an external opening is given by ( ) ∆Pkl0 (Pk ) = Pk + ρ k g hk − hl0 − Pl 0 0 (7) 0 where Pl is the pressure in external node l (Pa) and hl is the height corresponding to external pressure node l (m). 0 The external pressures, Pl , are depending on the stack effect and on the wind pressures on the building envelope Pl 0 = Patm − ρ 0 ghl0 + 1 ρ 0C Pl v 2 2 (8) where Patm is the atmospheric pressure at ground (zero) level (Pa), ρ0 is the density of the external air l 3 (kg/m ), CP is the pressure coefficient, related to geometry and wind speed (n.d.) and v is the wind speed corresponding to building height (m/s). The wind speed, v, is modelled by the following wind profile H v = v ext z0 α (9) where vext is the characteristic wind speed corresponding to reference height (m/s), z0 is the reference height above the ground level (m), α is a coefficient depending on terrain roughness (n.d.) and ρk is the zone air density (kg/m3). Equation (1) is solved for the unknown pressure vector, P, by Newton’s method, extended by a Steffensen iteration scheme (Walton 1995). When the unknown pressure vector has been obtained, the various flow components, Qklc and Qkl0, are calculated. Monte Carlo Simulation N values, xk, of the random input parameter vector, X, are simulated in the Monte Carlo Simulation approach using appropriate input probability distributions. Then N corresponding values, yk, of the random output parameter vector, Y, are obtained using the multizone model. Finally, the statistics of the output quantities and the output distribution functions are estimated by statistical sampling. First, N independent standard normal variables, uk, are generated using a standard random generator. Using the so-called Nataf transformation (Sørensen 1995), realisations of the input are obtained by x i ,k = FX−i1 (Φ( z i ,k )) (10) z k = Tu k (11) TTT = ρ e (ρ) (12) where Fxi is the distribution function for Xi (n.d.), Φ is the standard normal distribution function (n.d.), zk is a realisation of a dependent standard normal vector (n.d.), T is a lower triangular matrix (n.d.), ρe is an equivalent correlation coefficient matrix (n.d.) and ρ is the correlation coefficient matrix (n.d.). Equation (12) is solved using Cholesky decomposition. The correlation coefficient matrix takes into account the dependence between the different input quantities, the equivalent correlation coefficient matrix, ρe, takes into account the distribution types of the different variables (Sørensen 1995). RESULTS AND DISCUSSION A twelve-zone building test case is analysed. The zones, air flow paths and external pressure nodes are shown in Figure 1 and the opening heights and types are defined in Figure 3. Three different opening types are considered, with parameters given in Table 1. Opening type A is considered to be a large opening like a stairwell, opening type B may simulate windows or another purpose provided opening, whereas type C may represent component cracks. It is assumed that a vertical temperature gradient prevails ranging from 20°C at the first floor to 23°C at the fourth floor. The external climate data comprise both deterministic and stochastic data. FIGURE 3. Geometry of 12-zone building test case. Three types of openings are applied corresponding to Table 1. The deterministic internal and external conditions are summarised in Tables 2 – 3. The zone temperature and water humidity are given in Table 2. Deterministic data for the external climate are given in Table 3. The pressure coefficients, Cpk, are obtained for the openings as functions of the wind direction, based on data from Orme et al. (1998), and are shown in Figure 4. Opening types A B C TABLE 1. Opening types cf. Figure 3 D (m3/(s·Pa)) n (n.d.) 0.04 0.6 0.02 0.6 0.005 0.6 θ (Deg) Zones 1,2,3 4,5,6 7,8,9 10,11,12 20 21 22 23 Patm (Pa) 101325 W0 (kg/kg) 0.006 TABLE 2. Deterministic zone parameters W (kg/kg) ρ = ρ (θ , W, Patm) (kg/m3) 0.008 1.1983 0.008 1.1942 0.008 1.1902 0.008 1.1862 TABLE 3. Deterministic external climate parameters g (m/s2) z0 (m) H (m) α (n.d.) 10 12 0.28 9.82 0.6 CP (n.d.) 0.3 0.0 -0.3 -0.6 -0.9 0 45 90 135 180 225 270 315 360 Opening 1 Opening 2 Opening 3 Opening 4 Opening 5 Opening 6 Opening 7 Opening 8 Opening 9 Opening 10 Opening 11 Wind Direction (°) FIGURE 4. Pressure coefficients for the multizone building based on measurement data (Orme et al. 1998) The stochastic data for the external climate: external air temperature, reference wind speed and direction (discretised into eight wind sectors) are considered to be the random forcing functions in the test case. Thus, the random input variable vector is (where D is the wind sector) X = [θ ext v ext D]T (13) A probabilistic load model based on the Danish Design Reference Year is applied to model the external forcing functions (Brohus et al. 2002). Time varying distribution functions for the external air temperature, the wind speed and the wind sector are given. The wind direction is divided into eight sectors, for each a distribution function for the wind speed is given. This model is used in the present application to generate realisations of the input vector. In the present application only one point in time is considered - January 1, at 12.00 – to show the idea. The mean value and standard deviation for each of the air flows are obtained by simulating 10,000 input vectors and using the multizone model to calculate the corresponding air flows, see Table 4. Flow Component Q1,2 Q2,3 Q4,5 Q5,6 Q7,8 Q8,9 Q10,11 Q11,12 Q2,5 Q5,8 Q8,11 Q1,4 Q3,6 Q4,7 TABLE 4. Mean value and standard deviation of air flows, see Figure 1 Flow µQ (m3/s) σQ (m3/s) µQ (m3/s) σQ Component 0.034 0.044 0.008 Q6,9 -0.012 0.046 Q7,10 0.010 0.022 0.049 0.010 Q9,12 0.002 0.047 -0.040 Q1,10 0.019 0.056 -0.019 Q3,50 0.007 0.049 -0.024 Q4,20 0.011 0.054 0.001 Q6,60 0.014 0.046 Q7,30 -0.022 0.046 0.015 0.005 Q9,70 0.066 0.013 -0.021 Q10,40 0.078 0.022 0.007 Q12,80 0.006 0.003 0.021 Q10,90 0.007 0.002 0.074 Q11,100 0.008 0.002 Q12,110 0.017 (m3/s) 0.002 0.003 0.003 0.044 0.046 0.050 0.047 0.058 0.050 0.066 0.052 0.016 0.047 0.011 It can be observed that the mean values are rather low compared with the standard deviations. This is caused by the fact that the wind direction is included in the model and that the sign of each air flow component changes according to the wind directions. Apparently this is close to zero in mean, in this case. F Q (n.d.) 1.00 0.75 Q01,1 Q03,5 0.50 Q07,3 Q09,7 0.25 Q012,8 Q011,10 0.00 -0.2 -0.1 0 Q (m3/s) 0.1 0.2 0.3 FIGURE 5. Selected distribution functions for the air flow rate in the naturally ventilated multizone building Selected distribution functions for the airflow are presented in Figure 5. It can be observed that the air flows are strongly non-Gaussian distributed and that the air flows in general prevail in both directions in most cases (alternating sign, see sign convention in Figure 1). If the mean values are applied without considering the spread of data erroneous results may occur in terms of flow capacity, flow direction, distribution in time, etc. This is due to the fact that the building in this example is solely naturally ventilated without any kind of control. Adapting appropriate control and maybe hybrid ventilation, i.e. dampers and fans, the building is more likely to provide unidirectional air flow and requested ventilation capacity. The study clearly shows that a deterministic approach using mean values to describe the behaviour of the input variables would lead to a significant – and unknown – level of uncertainty. The distribution functions in Figure 5 provide information on direction of flow, size of air flow rate, and probabilities of a certain result like flow from outside into a room, a certain air flow rate, etc. Those results are very useful when building performance is assessed and a solution is to be evaluated and optimised. CONCLUSION The performance assessment of a naturally ventilated multizone building is investigated. A stochastic multizone model is presented and applied in the investigation. The model is capable of predicting the variability of air flow and pressure distributions within a building that may be divided into an arbitrary number of zones and air flow paths. The Monte Carlo Simulation approach is used to generate realisations of external air temperature, wind speed and wind direction, from which the corresponding air flows and pressure distributions are calculated. By statistical sampling among output, it is possible to estimate mean values, standard deviations and distribution functions. It is found that disregarding the spread of input data may lead to highly inaccurate performance assessment and erroneous conclusions regarding the function and capacity of the building ventilation. REFERENCES Brohus, H., Frier, C. and Heiselberg, P., 2002. Stochastic Single and Multizone Models of a Hybrid Ventilated Building – A Monte Carlo Simulation Approach, Technical Report Annex 35, ISSN 13957953 R0219, Aalborg University, Denmark. Feustel, H.E. and Dieris, J. 1992. A Survey of Air flow Models for Multizone Structures, Energy and Buildings, 18, 79-100. Feustel, H.E. and Raynor-Hoosen, A. (ed.). 1990. 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