Modeling air leakage in buildings caused by the cyclic variation of the atmospheric pressure

Abstract

This article considers a new type of air infiltration through building envelopes caused by the barometric pressure variation. This process is independent from wind action or stack effect. A new building–atmosphere differential equation of air exchange is established. Based on the solution of the differential equation of air exchange, we propose the notion of pressure equilibration time $τ_{50}$ that characterizes the dynamic response of the building. Furthermore, experimental climatic data were processed using Fourier analysis methods in order to build an identification model based on the regular harmonics of external pressure oscillation. The barometric pressure reconstructed in its parameterized form was introduced in the differential equation of air exchange as a term that models the dynamics of the external action.

The analytic solution of the differential equation of air exchange demonstrates that the indoor–outdoor pressure difference is insignificant at less than 10⁻³Pa for any harmonic of the external pressure variation. At the same time, it is concluded that the airtightness of the envelope has little influence on the process, as the indoor–outdoor pressure equilibration is almost instantaneous in a continuous regime of variation.

The described mechanism of air infiltration explains the alternation of infiltration and exfiltration of air in buildings. For this, a mass balance of air exchange for the specific ranges of time is performed. We prove that the barometric pressure variation has an effect that accounts for 3.19% of the total quantity of air exchanged.

The advances provided by this paper constitute a useful instrument for further studies concerning the stack effect in thermal dynamic conditions.

Practical application: The paper proposes a novel methodology of determining the air exchange building–environment by considering a new component of infiltration and its cyclic variation: the barometric pressure. A new mechanism of natural air infiltration is determined and modeled and it should be added to the existing ones: wind action and stack effect. A complete methodology of analysis and extraction of the cyclic processes hidden in an envelope of stochastic variations is defined and applied with the support of signal processing techniques and spectral analysis. The refinement of the mathematical instrument was able to distinguish through a deterministic approach an influence of 3.19% of the new barometric component. Furthermore, with few adaptations, the methodology constructed in this study can be applied for investigation of the stack effect by considering the cyclic variation of the atmospheric temperature. Currently, the stack effect is analyzed only for average conditions of temperature, a simplistic approach that suggests a large potential for further improvement.

Keywords

Barometric pressure variation transient air infiltration signal processing air leakage air infiltration pressure equilibration time

Introduction

This paper proposes an analytic methodology for determining the natural air exchange of buildings caused by the cyclic variation of the barometric pressure.

This new discerned category of air infiltration through the building envelope comes with novelty and originality of approach as the current methods consider only the wind action or the stack effect (we exclude the effect of mechanical ventilation). These influences are determined by a permanent periodicity of the atmospheric pressure caused by the Earth’s rotation and the gravitational interaction between the Earth and the other celestial bodies of the solar system. The periodicity of barometric pressure variation generates successive stages of building–environment air exchange with impact on heat loss. This irreversible process is caused by the transport of warm air outside of the building in specific periods when the external pressure decreases.

Determining airflow rates is a process accompanied by several difficulties that involve a special methodology of normalization and parameterization of the variation embedded in random fluctuations. A positive point of this paper is the simplicity of the final solutions that will offer easy evaluations of the new considered process.

We point out that the barometric pressure variation should be treated as a process of air infiltration that is independent from wind actions or the stack effect. While the wind is characterized by sudden variations and is composed of turbulent movements of air masses, the barometric pressure has instead a smooth evolution. The wind interferes with the barometric pressure determining a combined action, but its effect (received by the barometer as a static pressure) is considered a random component equivalent to a noise type of external disturbance in terms of systems dynamics. The wind is sudden, non-uniformly distributed as direction and magnitude, while the barometric pressure variation determines a soft action and a uniformly distributed air leakage across the envelope. These differences are evident and a quantitative and qualitative comparison will recommend that the two influences should be treated separately. At the same time, comparing the barometric pressure variation with the stack effect, they have two representative common points: both are smooth and cyclic processes along the duration of the year. However, a limitation of the analysis of the stack effect that is carried out nowadays needs to be acknowledged because it is only performed for averaged conditions that ignore or replace the effect of cyclicity.

Currently, the state of the art inherently considers the barometric pressure influence as a subcomponent of wind action. We come up against this issue by treating this phenomenon separately and we demonstrate quantitatively and analytically that a share of the total air mass exchange for a building belongs to the dynamic variation of the barometric pressure itself.

In this paper, the air exchange is mathematically modeled, thus providing rigorous analytic solutions. Therefore, we quantify this influence through an explicit solution for which the designer/researcher will only need to know the main amplitudes and periods of the harmonics of pressure variation. The hourly weather data of the considered geographical area will be necessary for a duration of at least one year. Starting from the empirical data of weather, we will cover several stages of mathematical processing and finally we will quantify the effect through simple relations.

Today, the incongruence between the pressure test conditions and natural conditions of infiltration is bridged through empirical relations that in some cases include the weather particularities of the considered geographical location. While they offer sufficient information for rapid evaluation of the system, they lack the pure mathematical and physical deductions that the field of building physics usually asks for. Empirical relations are useful for practical cases in the real world, but they offer only a partial sufficiency for researchers that are preoccupied with the essence of the phenomena.

The most frequently used estimation of the annual air leakage considers the Kronvall–Persily^1,2 method ${ACH}_{nat} = {ACH}_{50} / 20$ that is used more as a rule of thumb.³ The other models of infiltration (Lawrence Berkeley Laboratory (LBL),^4,5 Variable Flow Exponent (VFE),⁶ Sherman’s model,¹ National Research Council (NRC),^7,8 Alberta Air Infiltration (AIM)-2,⁹ Canadian General Standards Board (CGSB)’s model,¹⁰ American Society for Testing and Materials (ASTM’s) model,¹¹ American Society of Heating Refrigerating, and Air-Conditioning Engineers (ASHRAE’s model)¹²) are more advanced, but have their own limitations. We evaluate that a deterministic approach that considers hourly time samples of the weather variation will improve the current models, just as determining several harmonics for the barometric pressure variation will change the perspective in the analysis of air leakage in buildings. This method rigorously identifies all the factors and all the variations that affect the interaction between the building and the environment. By providing an alternative analytical solution for each factor that influences the indoor–outdoor air exchange, an overall check of convergence between analytical and experimental solutions for the entire year is made possible for the final stages of analysis when it comes to buildings.

While today empiric evaluations account more and more for the progress of science, we still believe in the intrinsic value of the analytic instruments that are also addressing higher levels of concept development from processing measurement data (MD) in such a way that the generalization of results is possible, in order to benefit future and interdisciplinary research.

The main problem considered in this paper is providing an answer to the question of how and by how much does the registered regular variability of weather influences the air exchange of the building. Beyond the information regarding the level of influence of the pressure variation, the knowledge we have obtained of this phenomenon will become more refined and at a deeper, more profound level for this natural process.

Therefore, we propose the addition of a new term of barometric pressure variation effect $({\overset{\cdot}{V}}_{bar})$ to a well-known identity (a linear-additive model) that until now has only taken into account the infiltration air flow rates of the stack effect $({\overset{\cdot}{V}}_{stack})$ and of the wind action $({\overset{\cdot}{V}}_{wind})$ for computing the total air flow rate $({\overset{\cdot}{V}}_{nat})$ in a building in natural conditions (the effect of the mechanical ventilation it is not considered)

{\overset{\cdot}{V}}_{nat} = {\overset{\cdot}{V}}_{stack} + V_{bar} + {\overset{\cdot}{V}}_{wind} [m^{3} / h]

(1)

While stack effect is evaluated only in averaged weather conditions and the wind action is a stochastic process, the effect of barometric pressure variation will consider a time series of permanent and periodic variations.

In completing equation (1), Table 1 presents, in simplified form, how the models of infiltration will change when considering the new term. Such a diversity of models originates from the fact that while the components of air infiltration are independent, they nevertheless interact in time being guided by nonlinear and stochastic processes. This study focuses on the quantitative evaluation of the new term of air infiltration, thus we will only use the simplest infiltration model, the linear-additive one, thus avoiding other complications that may prevent the understanding of the principal objectives of the study: (a) the aeraulic influence of the barometric pressure and (b) the methodology itself that can be further applicable to other cyclic natural variations (atmospheric temperature, etc). Several models of air infiltration (Table 1) propose their specific superposition principles and overall they showcase an intense level of debates focused on how to combine the different actions on the building’s envelope.

Table 1.

Representative models of air infiltration in natural conditions (completed with the new barometric term).

Model	Basic equation
Linear additive	${\overset{\cdot}{V}}_{nat} = {\overset{\cdot}{V}}_{stack} + {\overset{\cdot}{V}}_{bar} + {\overset{\cdot}{V}}_{wind} [m^{3} / h]$ (1)
Linear weighted	${\overset{\cdot}{V}}_{nat} = C_{1} {\overset{\cdot}{V}}_{stack} + C_{2} {\overset{\cdot}{V}}_{bar} + C_{3} {\overset{\cdot}{V}}_{wind} [m^{3} / h]$ (2)
Simple quadrature (LBL model), 1980	${\overset{\cdot}{V}}_{nat} = ({\overset{\cdot}{V}}_{stack}^{2} + {\overset{\cdot}{V}}_{bar}^{2} + {\overset{\cdot}{V}}_{wind}^{2})^{1 / 2} [m^{3} / h]$ (3)
General quadrature (LBL model), 1980	${\overset{\cdot}{V}}_{nat} = ({\overset{\cdot}{V}}_{stack}^{2} + {\overset{\cdot}{V}}_{bar}^{2} + {\overset{\cdot}{V}}_{wind}^{2} + \sum_{i, j, k = \bar{1, 3}} C_{i} {\overset{\cdot}{V}}_{j} {\overset{\cdot}{V}}_{k})^{1 / 2} [m^{3} / h]$ (4)
VFE model, 1983	${\overset{\cdot}{V}}_{nat} = ({\overset{\cdot}{V}}_{stack}^{1 / n} + {\overset{\cdot}{V}}_{bar}^{1 / n} + {\overset{\cdot}{V}}_{wind}^{1 / n})^{n} [m^{3} / h]$ (5)
Kronvall–Persily, 1985	${\overset{\cdot}{V}}_{nat} = {\overset{\cdot}{V}}_{50} / 20 [m^{3} / h]$ (6)
Sherman’s model, 1987	${\overset{\cdot}{V}}_{nat} = {\overset{\cdot}{V}}_{50} / N [m^{3} / h]$ (7)
NRC model, 1989	${\overset{\cdot}{V}}_{nat} = F ({\overset{\cdot}{V}}_{stack}^{1 / n} + {\overset{\cdot}{V}}_{bar}^{1 / n} + {\overset{\cdot}{V}}_{wind}^{1 / n})^{n} [m^{3} / h]$ (8)
AIM-2 model, 1990	${\overset{\cdot}{V}}_{nat} = ({\overset{\cdot}{V}}_{stack}^{1 / n} + {\overset{\cdot}{V}}_{bar}^{1 / n} + {\overset{\cdot}{V}}_{wind}^{1 / n} + \sum_{i, j, k = \bar{1, 3}} C_{i} {\overset{\cdot}{V}}_{j}^{1 / 2 n} {\overset{\cdot}{V}}_{k}^{1 / 2 n})^{n} [m^{3} / h]$ (9)

$C_{1}, C_{2}, C_{3}, F$ : interaction coefficients; n: power law air infiltration index (from the pressure test); j, k: permutation numbers, N: coefficient of the environment influence.

Dividing the process of air exchange in its constituents, analyzing them separately and finally reintegrating them in a unique, global term of air infiltration is a divide et impera classical approach that follows after a deterministic quantification of all effects. Separating and subtracting the term of periodic external pressure variation leads to a better assessment of the other surveyed effects: wind action and stack effect.

This new term is a latent one and manifests less obviously, but its quantification leads to a better representation of the phenomenon of aeraulic interaction. In terms of magnitude or impact, it will not radically change the ways in which the problems of building engineering are viewed or defined today. Instead, it is a result of a focalized analysis toward improving the evaluation of energy efficiency of buildings where every percent of influence is important.

In its essence, air infiltration requires a multivalent analysis, as its nonlinearity and degree of randomness require the quantifiable inclusion of any active factor, in order to keep the overall solution under control. The existing difficulties in the process of air exchange analysis have so far constrained the aeraulics of the building to be rather based on empiric equations centered mostly on statistical compliance, like for instance in the case of the power law (PL). The problem is that the established PL for a building offers only one piece of information that is available, which is rather for static conditions than for dynamic or cyclic ones. This study will use a linearized form of the PL that uses a linear extrapolation at low-pressure differences specific for natural conditions.

If our solutions for the extra factor discovered in the expression of infiltration will prove to be practical for the aeraulic design of the buildings, then it will be marked as a stepping stone in the advancement of the rigorous evaluation of the entire exchange process.

The influence of the periodicity of pressure variation will not be on par as magnitude with the influence of the wind or of the stack effect. However, this technique of dealing with variable weather conditions can prove useful in future attempts of more balanced analyses of other factors, for instance of stack effect. While in this paper we treat the variability of atmospheric pressure, the stack effect analysis would instead be concerned with the variability of external temperature benefiting from the progress of this study.

The proposed instrument will prove to be valuable when dealing with variable conditions in the environment. Its novelty brings forth methods of deterministic analysis for each of the three components of equation (1), methods that are easier to validate in simulated environments, easier to use in design of buildings, and ultimately easier to integrate in known methodologies of energy-efficiency estimation.

Previous approaches to air exchange in transient conditions

Various studies dedicated to the measurement of airtightness for various platforms of research^13–20 have demonstrated the impact of air exchange from an energy point of view. Climatic conditions, cultural distinctiveness, and the level of development also create differences from country to country^21,22 including the assessment of airtightness of the buildings.

Deterministic studies of variable air infiltration are showing increasing interest,^4,23–31 while the classical approaches (such as the case of the blower door test) for fixed conditions of test under averaged conditions of weather have their limitations. The pressure test presents the effects of static conditions of air exchange, while the results are to be used for prediction of the aeraulic interaction in dynamic conditions specific to the natural processes. There is a kind of disparity inherent to this approach, but at the same time, progress in the fields of stochastic analysis using averaging and correlations makes it possible to pass over this issue of using static descriptions to estimate dynamic effects. In the long run, however, the premise is flawed, because a lot of important information is lost, specifically information dealing with transient properties, with time response of systems, without which the connection to fields like systems theory, control, automation, or systems engineering is not possible. Thus, in addition to static analysis, this study also integrates modern numerical instruments of discovering regular variations hidden by stochastic processes (like Fourier analysis and other specific tools in the field of signal processing).

The dynamic action of the wind^32–40 often determines the largest share of air exchange (depending on the specifics of climate) of the building and subsequently fuels preoccupations for stochastic, harmonic, and time-series analysis.^41–46 The time-based measurements of air leakage provide an opportune and reliable source of inspiration concerning its mathematical apparatus as they deliver the best evaluation at low-pressure variations by revealing the induced harmonics of the pressure signal through Fourier analysis. In this regard, notable works are those of Sherman et al.^47–51 where the principles of processing MD of the induced fluctuating pressure are underlined. Further developments have centered around the diversification of the measuring equipment^52–59 and the implementation of Fourier analysis on this kind of signals.^60,61 Several measurement procedures are detailed in Cooper et al.⁵⁶ and Cooper and Etheridge.⁶² This study selects some mathematical procedures from these techniques as we ascertained that there are closed similarities between the natural oscillation of the atmospheric pressure and the induced harmonics of pressure. The only difference is that the harmonics of pressure of the unsteady techniques has a much shorter time period and lower amplitude but mathematically both cases are treated through the same methodology of Fourier analysis.

The barometric pressure variation of the atmosphere and of the external conditions in general has received its share of attention.^63–73 Nevertheless, it did not raise the same interest as for the temperature, solar radiation, and wind action. An explanation comes from the actions of wind that interact directly with this phenomenon masking it almost completely.

While several applications like ASCOS (1981), AIRNET (1989), COMIS (1989) were issued in the 1980s, an increasing interest in their use and evaluations can be seen in the 1990s when the theory of air infiltration in buildings gained a well-defined form. Multizone problems of air infiltration created, through their complexity, the necessary impulse to mainstream the use of computing techniques for faster and more accurate evaluations. Programming languages like Fortran and later C, or environments like Matlab, gave substantial facilities for modular formulation of the problems at numerical level along with the facilities brought by considering an associated network for the model of a building. DOMVENT3D, a model of infiltration and exfiltration built for Matlab considers the walls as porous media with a linear model of pressure distribution.^74–76 The stack and wind pressure diagrams are considered additive creating the possibility for the integration of the overall equation of infiltration along of the wall. Despite their actual complexities and different numerical and programming particularities, the comparison of the results of analysis platforms like CONTAM and DOMVENT3D provides differences in results that are less than 1%.^74–76

While the vast majority of the studies concerning airflow in buildings consider the dynamic analysis dedicated for wind action, they ignore the variations of the barometric pressure, instead only taking into account the averaged conditions of static pressure. Such premises are ideal for complex analysis, but they lead to a loss of information if their contribution is not quantified.

We propose a model focused on the variation of the barometric pressure extracted from MD with a sampling time of 1 h, read for the duration of one year (the measurements are performed for multiple years but the balance is performed for one year). Thus, sensible aspects linked to the periodic variation around a mean value are included in the analysis and will provide notable results. In this paper, all the final relations are backed up by mathematical deductions and thermodynamic transformations that characterize this special process of air infiltration.

Comparison of the new type of air leakage with the existing ones

We give a synthetic classification (Table 2) of six cases of air infiltration in order to depict their distinctiveness, causes, and conditions. The new term of air infiltration will be better identified.

Table 2.

A qualitative comparison of the components of air infiltration rates in a building.

	Variable	Driving force	Spatial distribution	Time distribution
Natural conditions	${\overset{\cdot}{V}}_{stack}$	Stack effect (thermal effect)	Lower zones vs. upper zones (opposed direction of infiltration)	The process is permanent. i.e. there is a permanent indoor–outdoor temperature difference. The process is periodical. Randomness is present. The actual balance can only be defined for static and averaged conditions.
	${\overset{\cdot}{V}}_{bar}$	Barometric pressure oscillation	Uniformly distributed on the entire envelope: “air in” when $p_{e} ↗$ and “air out” when $p_{e} ↘$	The process is permanent with successive periods of air infiltration and exfiltration. The process is periodical, as is the external pressure. Randomness is present. The mass balance involves the solution of the differential equation of air exchange driven by pressure variation.
	${\overset{\cdot}{V}}_{wind}$	Wind effect	Upwind zones (air in) vs. downwind zones (air out)	The effect is cvasi-periodical and governed by stochastic processes. It manifests with randomness of amplitude and direction. High and sudden variations of the amplitude and direction.
Forced conditions	${\overset{\cdot}{V}}_{ven}$	Mechanical ventilation effect	Upstream zones of ventilation vs. downstream zones of ventilation.	The effect is regular, active by context and programmable/controllable. It is permanent if the equipment is maintained active.
Forced conditions	${\overset{\cdot}{V}}_{press . test}$	Pressure difference created mechanically.	Uniform spatial distribution along the entire envelope	The external conditions (the wind) for this process are checked to be stable. The pressure test is performed at constant external conditions. Used for qualification or certification of the building’s energy efficiency.

There are several empirical well-established relations that provide an evaluation of the aeraulic interaction of the building, but are less appropriate for generalization. The main scope of this paper is the analytic investigation of the effect of the periodic variation of external pressure on the energy balance of the building.

The exchange of thermal energy that, as a phenomenon, is characterized by the enthalpy variation of the air that transits the envelope, also has a correspondent in the process of air mass transfer. The overall heat balance of the infiltrated air is

\begin{matrix} Q = \int_{0}^{T} \overset{\cdot}{q} d τ = \int_{0}^{T} {\overset{\cdot}{m}}_{out \to in} c_{p} (T_{i} - T_{e}) d τ \\ = - \int_{0}^{T} {\overset{\cdot}{m}}_{in \to out} c_{p} (T_{e} - T_{i}) d τ \\ = \int_{0}^{T} \frac{| \overset{\cdot}{m} |}{2} c_{p} (T_{i} - T_{e}) d τ \neq 0 [kW] \end{matrix}

(10)

where

{\overset{\cdot}{m}}_{out \to in} = [\overset{\cdot}{m}]_{+}

is the positive part of the air flow rate

\overset{\cdot}{m}

that has a similar distribution in both directions because of its sinusoidal variation.

The corresponding overall air mass balance determined by the difference $p_{i} - p_{e}$ of the building along a complete cycle of pressure variation is

\begin{matrix} \int_{0}^{T} \overset{\cdot}{m} d τ = \int_{- T / 2}^{T / 2} {\overset{\cdot}{m}}_{out \to in} d τ | \begin{matrix} \frac{{dp}_{e}}{d τ} > 0 \end{matrix} \end{matrix}

(11)

\begin{matrix} + \int_{T / 2}^{3 T / 2} {\overset{\cdot}{m}}_{in \to out} d τ | \begin{matrix} \frac{{dp}_{e}}{d τ} < 0 \end{matrix} = 0 [kg] \end{matrix}

Equations (10) and (11) are coupled for the characterization of the airflow process that consists in a permanent indoor–outdoor equilibration. Each event characterized by a decrease in the environment’s pressure ( $p_{e} ↘$ and $p_{i} - p_{e} > 0$ ) determines the evacuation of a mass of heated air from inside. The consequence is an irreversible heat loss associated to the airflow rate, which needs to be determined. The exhausted warm air is replaced in a successive cycle by outdoor external air that will require a heating load $\overset{\cdot}{Q} = {\overset{\cdot}{m}}_{out \to in} c_{p} (T_{e} - T_{i})$ in order to meet the permanent indoor air temperature T_i.

Along the whole cycle of pressure variation, from a hydraulic point of view, the air mass balance is null, while by thermodynamic balance is not null.

A discussion is necessary following the context of equations (10) and (11). The air mass balance of equation (11) has an increased importance when it is coupled with equation (10) concerned with thermal energy balance. The most powerful reason of performing the air mass balance is concern with the warm air that is lost by the building that translates in exergy loss and heating costs. One of the results of this study will be the representation of the heat rates following the finding of the air mass flow rates caused by the barometric pressure variation.

Experimental measurements from the passive house Politehnica

The passive house “Politehnica” (Figure 1) is built on the campus of University Politehnica of Bucharest as a result of the cooperation between several organizations and is the subject of representative studies.^20,77–92 Several results obtained from this research platform have been gathered into a rich experimental measurements database that is now used in this study.

Figure 1.

Representation of real and theoretical model of the building^82–85: (a) Passive house Politehnica and (b) virtual model of the passive house. Image taken with permission from Elsevier.⁸³

The building is a cluster of two houses: “East House” and “West House.” The heating system of “East House” is based on an air-ground heat exchanger (EAHX), while for “West House” it is based on a geothermal heat pump (GHP) (Figure 2). The building’s characteristics are provided in Table 3.

Figure 2.

Thermal instalations of the “Politehnica” building^82–85: (a) East House and (b) West House. a) 1 – Solar Colector, 2 – Cold Water Inlet, 3 – Hot Water Tank, 4 -Domestic Hot Water Outlet, 5 Electric Resistance Heater, 6 – Heat Recovery Unit (HRU), 7 – Water-Air Heat Exchanger, 8 – Pumps Station, 9 – Geothermal HEat Exchanger, 10 – Passive Cooling Heat Exchanger, 11 – Hydronic Radiant Panel b) 1 – Solar Colector, 2 – Cold Water Inlet, 3 – Hot Water Tank, 4 - Domestic Hot Water Outlet, 5 Electric Resistance Heater, 6 – Heat Recovery Unit (HRU), 7 – EAHX By-Pass, 8 – Condensate Drain Well, 9 – Earth To Air Heat Exchanger (EAHX), 10 – Electric Radiant Panel. Image taken with permission from Elsevier.⁸³

Table 3.

Geometry and energy performance of the building.^82–85

	Quantity			Requirement/ recommendation
Parameter	“East House”	“West house”	Overall	Requirement/ recommendation
Positioning/orientation	Lat. 44.438 N, Long. 26.047 E, Alt. 76.6 m//South			South orientation
Overall dimensions L × B × H [m]	16 × 11.8 × 7.5			–
Treated floor area [m²]	140	140	280	–
Envelope area [m²]	481.6	481.6	797.2	–
Indoor air volume [m³]	435.3	435.3	870.6	–
Compactness C = V/S [m³/m²]	1.628	1.628	1.967	Cubic = ideal
Pressure test—air changes per hour [h⁻¹]	=	0.586	≤0.586	≤0.6
Specific space heat demand, [kWh/m^2/y]	13	13	12.5	≤15
Specific space DHW demand, [kWh/m^2/y]	18	18	18	–
Specific space electric energy consum., [kWh/m²/y]	45.8	30.5	38.1	–
Total specific primary energy demand, [kWh/m²/y]	119	79	99	≤120

DHW: domestic hot water.

In the process of qualification of the building as a passive house, tests of depressurization and pressurization were performed²⁰ (Figure 3), based on which the PL of air leakage was determined.

Figure 3.

Measurement data of the pressure test performed for the passive house Politehnica. Image taken with permission from Elsevier²⁰

We retain the relationship of depressurization performed for “East House”²⁰ and we extrapolated it for the entire building

\overset{\cdot}{V} = 51.4 Δ p^{0.587} [m^{3} / h]

(12)

The extrapolation of the measurements of a patterned partition of the building for the entire building by virtue of symmetry is a conservative criterion that allows this case study to double the values of the airflow rates in a limited margin.⁸⁸ These assumptions consider, with approximation, a proportionality that allows further developments, and come as a consequence of several limitations of the available MD because of the availability of a single test equipment instead of two. Multizone analysis can insulate the influence of a partitioning wall^93–95 through strategies specific to the classical (physical) principle of superposition (of the overlay effects) along with the substantial support brought by the modeling through thermo-aeraulic analogy of the associated networks on the building. Jones and Lowe follow the path of deterministic approach by considering a vertical pressure gradient on the porous walls^96–98 in order to make an integration of the infiltration effect on the entire envelope area. Such analytical approach may help in dedicated analysis with discovering the influence of the internal partitioning walls compared with the alternative hypothesis of considering them external walls of the envelope.

We resorted to a reductionist strategy that makes the mathematical model able to use the test results as a starting point. The model sufficiently reduces the complexity of the problem due to symmetry, but with a slight reduction in accuracy. The test performed for the reference partition house (East House) was performed by keeping the entrance door open for the other half of the house. The concrete structure of the partitioning wall has similar geometry with the external walls, thus almost the same infiltration characteristics.

We appreciate that the extrapolation of the infiltration properties to the entire building (generally speaking, from a patterned part to the whole), due to the condition of symmetry, does indeed offer an acceptable condition to advance further in the proposed topic that is unrolled in the following sections.

For the pressure tests, a Mineapolis Blower Door DG-700 (Figure 4) as used, manufactured by The Energy Conservatory.^99–103

Figure 4.

The equipment of pressure test^99–103: (a) Mineapolis blower door equipment and (b) functioning scheme. Image taken with permission from Elsevier.⁸⁸

In what follows, a reference of comparison for the results will be the equation proposed by Sherman¹ obtained through tracer gas techniques and available for air infiltration in natural conditions

{ACH}_{nat} = {ACH}_{50} / 20 [h^{- 1}]

(13)

A discussion on equation (13) is necessary. Although it has been chosen in this study because of its simplicity, there are questions regarding its accuracy. The results of the air mass balance of the action of the barometric pressure will be related to equation (13) that estimates the total air mass exchanged by the building.

Sherman¹ proposed a correction of the rule-of-20 given to leakage infiltration ratio (LIR) and with this correction, equation (13) becomes

\begin{matrix} {ACH}_{nat} = \frac{{ACH}_{50}}{\bar{N}} = \frac{{ACH}_{50}}{N_{0} \cdot {CF}_{1} \cdot {CF}_{2} \cdot {CF}_{3}} \\ = \frac{{ACH}_{50}}{(15 \dots 30) \cdot (0.7 \dots 1.0) \cdot (0.9 \dots 1.2) \cdot (0.7 \dots 1.4)} \\ = \frac{{ACH}_{50}}{(6.615 \dots 50.400)} [h^{- 1}] \end{matrix}

(14)

where

\bar{N}

is the annual average LIR, N₀ is the basic LIR, and

{CF}_{1, 2, 3}

are correction factors. It is obvious that the function of the specific conditions of the location, the range (6.615\ldots50.400) can offer quite significant differences from the value

LIR = \bar{N} = 20

of the rule-of-20. Nevertheless, we will assume in our study the value

LIR = \bar{N} = 20

as the theory of variation of LIR has significant potential to be improved in the future. Whether they are updated or not, the current methods of evaluation for each existing model have their own deficiencies. Therefore, a consensus toward a common, more inclusive model, is necessary in the near future.

We used hourly air pressure data (from years 2014 and 2015) from the international weather station Bucuresti–Baneasa, which belongs to Meteo Romania (National Meteorological Administration). Long-term trend analysis shows no statistically significant changes in air pressure since 1961.¹⁰⁴ At the same time, we have used as an independent source the archived data of the Wunderground website¹⁰⁵ containing daily MD (2006–2015). Figure 4 offers a time-based representation of pressure variation.

The cyclicity of the external pressure (Figure 5) is evidently visible and justifies the translation through signal processing computing techniques of such natural variations in a sum of regular and representative harmonics that can be processed further through formal techniques. A cyclic behavior can be observed for the external temperature as well. Various studies have already considered this property especially from an energy analysis perspective. We can say that the mathematical instrument validated in this study for the barometric pressure variation can be further used for the analysis of the stack effect in cyclic thermal conditions.

Figure 5.

Representation of the atmospheric conditions: (a) daily atmospheric pressure, 2006–2015.¹⁰⁵; (b) hourly atmospheric pressure, 2014–2015¹⁰⁴; and (c) hourly atmospheric temperature, 2014–2015.¹⁰⁴

The parameterization of the external pressure by using signal processing techniques

This section proposes the modeling of the atmospheric pressure as a sum of periodic functions. One challenge comes from the random components of variation that are aimed to be disposed of in order to obtain the pure signal. A parameterized form of the pressure variation model obtained in this section gives the possibility for analytic evaluation of the air mass transfer and for the balance of the thermodynamic process of the indoor air.

The weather data are a discrete time-series that require spectral analysis combined with knowledge about the orbital periods and phases of the Earth and the Moon. There is a connection between the oceanic tides and the barometric pressure variations known as atmospheric tides that are caused by the Earth, Sun, and Moon’s gravity and even by the planets of the Solar System at an insignificant margin.

The technical literature^63–73 mentions five main harmonics of variation for the external pressure (bi-diurnal, diurnal, bimonthly, monthly, and annual) justified astronomically. These harmonics along with the others that will be identified in this section will be extracted through spectral analysis considering the MD as a time series of the atmospheric pressure signal.

Many heuristic actions that negotiate with the nonlinearity and the randomness of the signal (see Figure 5(a,b)) are considered. Absolute and local peaks, patterns of the signal, periodicities, trends, AC/DC components are explored. The reconstruction of the signal considers a finite number of relevant harmonics, while the noise contribution is removed. The harmonics are additive and their composition is governed by the principle of superposition (we make the observation that the superposition of the waves should be considered different than the superposition of the components of air infiltration in natural condition).

Thus, the atmospheric pressure can be modeled in the following general form

p_{e} = TE + \sum_{k_{H} = 1}^{n_{H}} H_{i} + \sum_{k_{H} = 1}^{n_{H}} ξ_{i} [kPa]

(15)

where TE is the trend of evolution (usually a constant, DC component or a nonperiodic analytical function), H is a cyclic component, while ξ is the noise specific for each harmonic (including all noise sources, from measurement errors to exogen contributions that are not relevant to the core signal). As a truncated Fourier series, the atmospheric pressure can be modeled with the following explicit form

p_{e} (τ) = \frac{A_{0}}{2} + \sum_{h = 1}^{n_{h}} A_{h} sin (ω_{h} τ + ϕ_{h}) + \sum_{h = 1}^{n_{h}} ξ_{h} [kPa]

(16)

where n_h is the number of the main harmonics composing the signal.

The Fourier series of the signal is characterized by the formalism presented in Appendix 1 and requires significant computing resources for processing the discrete Fourier transform (DFT). As an alternative to DFT, the Fast Fourier Transform (FFT) is preferred, which accelerates the computation (from a numerical standpoint). Using FFT, we have extracted the permanent signal that is buried within a measured signal filled of noise. Figure 6 shows the frequency–domain amplitude characteristics (on a logarithmic scale) where local peaks amplitude are identified after performing some preliminary filtering.

Figure 6.

Magnitude spectrogram of the barometric pressure: (a) hourly atmospheric pressure, years 2014–2015 and (b) daily average atmospheric pressure, years 2006–2015.

The peaks of the time periods of 12 hours, 1 day, 14 days, 30 days, and 1 year (H1, H5, H8, H16, H17) can be noted among the main harmonics, confirming the scientific literature.^63–73 They deserve a special attention due to their astronomical background. Alongside these main harmonics, there are a few others that are also permanent and have a comparable effect. Their phenomenological origin still needs to be surveyed further in dedicated studies, but their existence is certified by the results revealed in the spectrograms of Figure 6.

The discovery of the harmonics H1\ldotsH17 is not based simply on the graphic representation in Figure 6. In order to identify these harmonics, several levels of data filtration (through averaging for different bins of time) were performed (rigorously in Matlab), thus providing with a robust and sustainable mechanism for this identification.

The two spectrograms (Figure 6) expressed through two different data series have an obvious similitude and reveal roughly the same peaks of magnitude. Through the local peaks, we can discover 17 principal harmonics that are grouped (Table 4) in two categories, one corresponding to those of periods less than a month (H5\ldotsH17), and the other for harmonics with period of over one month (H1\ldotsH4). Harmonics periods under seven days were better underlined by Figure 6(a) (corresponding to the data of Figure 5(a)), while those with periods over seven days by Figure 6(b) (corresponding to the data of Figure 5(b)). We considered that only the harmonics of periods less than a month are sensibly influencing the air exchange of the building and only these will be used in the balance equations of the following sections. The pressure noise is excluded from consideration for aeraulic balances as it cannot be quantified in a deterministic approach. It has a significant effect (because originates mainly from the wind action), but in this study only the permanent variations are considered (which is the normal practice for deterministic continuous models; further, specialized studies are required in order to include the noise component into the model, which is stochastic in nature). Table 4 details the information of Figure 6 and creates the possibility of explicit use of the modeled pressure in balance equations. The marked columns (H1, H5, H8, H16, H17) of the table represent the five permanent harmonics attested by the literature. The share of influence of each harmonic (Table 4) related to the combined action of all harmonics will be discussed after the explanation of the air mass balances.

Table 4.

Time periods, amplitudes, and phases of the main harmonics of atmospheric pressure variation.

Harmonics with the time period greater than a month
	H0	H1	H2	H3	H4	H5	H6	H7	H8	H9	H10
Period [days]	–	365.2	152.2	93.6	48.7	30	20.99	17.56	14.61	11.86	9.84
Amplitude [Pa]	100649	366.6	113.1	108.3	103.6	78.2	85.6	87.9	60.2	53.7	56.9
Phase (2014) [rad]	–	1.7976	1.2553	1.9684	3.7723	1.9869	2.5802	−0.6240	3.1426	0.8892	3.9650
Share of influence [%]		0.482	0.288	0.678	1.582	2.358	2.120	2.446	3.333	3.207	2.319
Harmonics with the time period less than a month (the most representative for air infiltration)
	H11	H12	H13	H14	H15	H16	H17	–	–	–	–
Period [days]	6.82	5.18	3.82	3.11	2.53	1	0.5	–	–	–	–
Amplitude [Pa]	110	87.2	54.4	41.64	24.4	37.1	47.7	–	–	–	–
Phase [rad]	3.9106	3.5430	1.5901	4.2986	5.0379	6.0233	2.8167	–	–	–	–
Share of influence [%]	6.455	6.745	5.706	5.370	3.866	14.844	38.200

Thus, the determined harmonics can be associated to equation (16) resulting the following form

\begin{matrix} p_{e} (τ) = p_{m} + p_{A_{1}} sin (ω_{1} τ + ϕ_{1}) + \dots \\ + p_{A_{17}} sin (ω_{17} τ + ϕ_{17}) + \sum_{h = 1}^{n_{h} = 17} ξ_{h} [kPa] \end{matrix}

(17)

Figure 7 showcases the reconstructed signal based on equation (17) that uses the harmonics (H1\ldotsH17) of Figure 6 and Table 4. For the parametric form of the pressure model, a series of tests can be performed to determine the fitting of the experimental data against the model by using the following statistical indicators: Pearson correlation coefficient (PCC); root mean squared error (RMSE); root mean squared relative error (RMSRE); mean absolute error (MAE), and mean absolute relative error (MARE) (Table 5).

Figure 7.

Representation of the reconstructed atmospheric pressure signal with noise removed.

Table 5.

Statistical indicators of degree of fitting of the parameterized pressure to the empirical data.

PCC	RMSE	RMSRE	MAE	MARE
$\frac{\sum_{i}^{n} (x_{i} - \bar{x}) (y_{i} - \bar{y})}{\sqrt{\sum_{i}^{n} (x_{i} - \bar{x})^{2}} \sqrt{\sum_{i}^{n} (y_{i} - \bar{y})^{2}}} (18)$	$\sqrt{\frac{1}{N} \sum_{i}^{} {(x_{i} - y_{i})}^{2}} (19)$	$\sqrt{\frac{1}{N} \sum_{i}^{} {(\frac{x_{i} - y_{i}}{y_{i}})}^{2}} (20)$	$\frac{1}{N} \sum_{i}^{} {\| x_{i} - y_{i} \|}^{} (21)$	$\frac{1}{N} \sum_{i}^{} {\| \frac{x_{i} - y_{i}}{y_{i}} \|}^{} (22)$
0.42	652.38 Pa	0.0065	525.97 Pa	0.0052

PCC: Pearson correlation coefficient; RMSE: root mean squared error; RMSRE: root mean squared relative error; MAE: mean absolute error; MARE: mean absolute relative error.

These statistical indicators show a high level of fitting with the measured data and validate the parameterized form that in more advanced studies may be confronted with other models. Figure 7 shows a good replication of the real signal provided by the obtained model (without noise). At this stage, the basis for further analytic investigation of the air exchange of the building is established.

The remaining noise is considered to be caused by the wind action that is a subject for different techniques of analysis involving stochastic processes.

The free aeraulic response of the building and the pressure equilibration time

This section prepares the ground for a later determination of the building’s aeraulic response to external variations of pressure. The aeraulic properties of the surveyed building are determined, revealing the reaction of the system in the process of pressure equilibration after an imposed initial standard pressure difference of 50 Pa.

The problem formulates the evolution toward a pressure equilibrium after an initial pressure difference of 50 Pa is applied as disturbance to the system. Particularly, we are interested in the pressure equilibration time $(τ_{50})$ , which is the time necessary for the the system to traverse freely the pressure range from 50 Pa to 4 Pa. Furthermore, the system evolves asymptotically toward null pressure difference (in steady state). A tighter building will take a longer time to reach the equilibrium with the outdoor, while a leakier one will respond in a much shorter period.

There are considered two scenarios of air infiltration corresponding to two distinct real cases:

The case of pressure test ruled by PL of flow, where the pressure difference is significantly greater than zero value

\overset{\cdot}{V} = C Δ p^{a} = 51.4 Δ p^{0.587} [m^{3} / h]

(23)

The hypothetical case of laminar law of flow (LLF) (see Appendices 2 and 3)

\begin{matrix} \overset{\cdot}{V} = \overset{\cdot}{V} | \begin{matrix} Δ p ≅ 0 \end{matrix} = \frac{C Δ p_{\inf}^{a}}{Δ p_{\inf}} Δ p = \underset{C^{*}}{\underset{︸}{C Δ p_{\inf}^{a - 1}}} Δ p \\ = C^{*} Δ p = 28.994 Δ p [m^{3} / h] \end{matrix}

(24)

Case (b) is specific to the infiltration determined by the barometric component as it comes with soft variations in time.

While in pressure tests, the flow regime is preponderantly turbulent, the air infiltration due to the atmospheric pressure variation is instead laminar. Two criteria justify this description: (a) The process is slow due to the slow variation of the barometric pressure allowing to the internal pressure to keep a closed gap value to it due to the effect of pressure uniformization that translates into a small pressure difference between the two sides of the wall; (b) the small pressure difference will be demonstrated in the next section that will describe the solution of the differential equation of the pressure difference indoor–outdoor.

Equation (24) will permit further the elaboration of the corresponding differential equation of the air infiltration due to the barometric pressure cyclicity. After the integration, the obtained general solution will appear as a sum of harmonics that will be considered to have their independent aeraulic action each. The determined air mass flow rate will be then related to the empirical formula Kronvall–Persily in order to have a perspective of how much influence has the new barometric term.

We generalize the algorithm provided by Mattsson and Claesson^26,27 and we prepare the problem to be solved further in the following section when the periodical action of the atmosphere is also taken into account. The disturbance will not only be an initial difference but a periodic and continuous variation. The differential equation of air exchange (DEAE) is

\begin{matrix} \frac{dm}{d τ} + \sum_{i} {\overset{\cdot}{m}}_{i} = \frac{dm}{d τ} + ρ \overset{\cdot}{V} = \frac{d}{d τ} (\frac{pV}{RT}) + ρ \overset{\cdot}{V} \\ = \frac{V}{RT} \frac{dp}{d τ} + C Δ p^{a} = \frac{V}{RT} \frac{d (Δ p)}{d τ} + ρ C Δ p^{a} = 0 \end{matrix}

(25)

We make the notation for the time constant $τ_{c} = \frac{V}{C ρ RT}$ and the final form of the differential equation (25) corresponding to turbulent flow and the hypothetical laminar flow, respectively is

\frac{d (Δ p)}{d τ} + \frac{1}{τ_{c}} Δ p^{a} = 0 (turbulentinfiltration)

(26)

\frac{d (Δ p)}{d τ} + \frac{1}{τ_{c}^{*}} Δ p = 0 (laminarinfiltration)

(27)

The solutions of the differential equations (26) and (27) are presented in Table 6 and the representation of the solution of equation (26) is illustrated in Figure 8 (see also Appendix 2).

Table 6.

Representation of the solutions of the differential equation of pressure equilibration.

Case Variable	Power law infiltration $\overset{\cdot}{V} = C Δ_{p} a$	Laminar infiltration $\overset{\cdot}{V} = C^{*} Δ_{p}$
Time constant	$τ_{c} = \frac{V}{C ρ RT} = 0.601 s$ (28)	$τ_{c}^{} = \frac{V}{C^{} ρ RT} = τ_{c} \frac{C}{C^{*}} = 1.773 s$ (29)
Pressure diff., $Δ p (\underline{τ})$	$Δ p^{=} Δ p_{0} [1 - \frac{(1 - a)}{τ_{c} Δ p_{0}^{1 - a}} τ]^{\frac{1}{1 - a}}$ (30)	$Δ p^{=} Δ p_{0}^{- \frac{τ}{τ_{c}}}$ (31)
Flow rate, $\overset{\cdot}{V} (\underline{τ})$	$\overset{\cdot}{V} = C Δ p_{0}^{a} [1 - \frac{(1 - a)}{τ_{c} Δ p_{0}^{1 - a}} τ]^{\frac{a}{1 - a}}$ (32)	$\overset{\cdot}{V} = C^{*} Δ p_{0}^{- \frac{τ}{τ_{c}}}$ (33)

Figure 8.

Characteristics of aeraulic evolution of the building in transient conditions: (a) pressure difference $Δ p$ and (b) volume airflow rate $\overset{\cdot}{V}$ .

The pressure equilibration time applicable for the case of the PL of infiltration (where it is considered a significant pressure difference) is derived from equation (30) (in Table 6) and has the form

\begin{matrix} τ_{50} = τ | \begin{matrix} Δ p_{\inf} = 4 \\ Δ p_{0} = 50 \end{matrix} = \frac{τ_{c} Δ p_{0}^{1 - a}}{1 - a} [1 - (\frac{Δ p_{\inf}}{Δ p_{0}})^{1 - a}] \\ = 4.7 [s], forPLofflow \end{matrix}

(34)

The primary dynamic properties of the aeraulic system are characterized by the time constant $τ_{c}$ and by pressure equilibration time $τ_{50}$ that explains how the system tends to return to equilibrium when an initial excitation of pressure difference is applied. At this point, we have established the basis for analysis of the system in forced conditions determined by the external pressure that acts with periodicity. The next problem is concerned with how the system reacts under a periodic excitation, being met with a higher degree of complexity for solving it.

The response of the building under cyclic dynamic action of the external pressure

In this section, the internal pressure variation is modeled under dynamic external action. An analytic survey for a harmonic “h” of pressure variation is performed. The solution for this component is further adapted for the rest of harmonics of the spectrum.

The differential equation of the transient process of air exchange has the parametric form

\begin{matrix} \frac{V}{RT} \frac{dp}{d τ} + C Δ p^{a} = \frac{V}{RT} \frac{d (p - p_{e})}{d τ} + C Δ p^{a} + \frac{V}{RT} \frac{{dp}_{e}}{d τ} \\ = \frac{V}{RT} \frac{d (Δ p)}{d τ} + C Δ p^{a} + \frac{p_{h} V}{RT} ω_{h} cos (ω_{h} τ + ϕ_{h}) = 0 \end{matrix}

(35)

The standard differential equation has the form

\begin{matrix} C_{τ} \frac{d (Δ p)}{d τ} + \frac{1}{R_{τ}} Δ p^{a} = C_{τ} \frac{d}{d τ} [p_{h} cos (ω_{h} τ + ϕ_{h} - \frac{π}{2})] \\ = C_{τ} ω_{h} p_{h} sin (ω_{h} τ + ϕ_{h} - \frac{π}{2}) [kPa] \end{matrix}

(36)

The DEAE in transient state is

\underset{\begin{matrix} damping \\ response \end{matrix}}{\underset{︸}{\frac{d (Δ p)}{d τ}}} + \underset{\begin{matrix} elastic \\ response \end{matrix}}{\underset{︸}{\frac{1}{τ_{c}} Δ p^{a}}} = \underset{\begin{matrix} driving \\ pressure \end{matrix}}{\underset{︸}{ω_{h} p_{h} sin (ω_{h} τ + ϕ_{h} - \frac{π}{2})}} [kPa]

(37)

where

τ_{c} = \frac{V}{C ρ RT} = R_{τ} C_{τ}

is time constant (aeraulic) of the building,

C_{τ} = \frac{V}{ρ RT}

is the aeraulic capacitance, and

R_{τ} = \frac{1}{C}

is the infiltration resistance. The oscillation of the system determined by the driving pressure will follow its periodicity, but with a reduced amplitude (due to the damping effect of the envelope) and subjected to a dampening effect.

By virtue of superposition, the general effect of the process will be the sum of the effects of all harmonics

\tilde{Δ p} = \sum_{h} {\tilde{Δ p}}_{h} = \sum_{h = 1}^{n_{h}} | {\bar{Δ p}}_{h} | sin (ω_{h} τ + ϕ_{h}) [kPa]

(38)

According to the spectral representation theorem, if an evolution is a homogeneous mean-square continuous real-time process, then it can be expressed as a sum of independent sinusoidal processes.¹⁰⁶

Figure 9 expresses, in terms of network analogy, the aeraulic interaction between the building and the environment. For this study, it is also of interest to describe equation (37) in terms of general form of the Helmholtz Oscillator Equation (HOE)^107–114

\begin{matrix} a \overset{··}{Δ p} + (b_{1} + b_{2} Δ p^{2} + b_{3} | \overset{\cdot}{Δ p} |) \overset{\cdot}{Δ p} + c Δ p \\ + d Δ p^{2} + e Δ p^{3} = A \cdot sin (ω τ + ϕ) \end{matrix}

(39)

Figure 9.

Network analogy representation of the dynamic interaction building–environment.

Comparing equation (37) with (39) of HOE that it is very suitable for wind-induced pressure or other dynamic interactions in its general form, we can see that a particular form of HOE (considering $a = b_{2} = b_{3} = d = e = 0$ ) covers also the action of the barometric pressure (that is a soft action)

b_{1} \overset{\cdot}{Δ p} + c Δ p = A \cdot sin (ω τ + ϕ)

(40)

At natural air infiltration corresponding to low-pressure differences, the time constant takes the new adapted form $τ_{c}^{*} = \frac{V}{C^{*} ρ RT} = τ_{c} \frac{C}{C^{*}}$ as per equation (29) that corresponds to the laminar air infiltration. Subsequently, equation (37) has the new form

\frac{d (Δ p)}{d τ} + \frac{1}{τ_{c}^{*}} Δ p = ω_{h} p_{h} sin (ω_{h} τ + ϕ_{h} - \frac{π}{2}) [kPa]

(41)

Following several stages of transformation and solving the differential equation, we find that

\begin{matrix} \begin{matrix} Δ p = {Δ p_{0} - \frac{τ_{c}^{* 2} ω_{h} p_{h}}{1 + τ_{c}^{* 2} ω_{h}^{2}} [\frac{1}{τ_{c, l}} s i n (φ_{h} - \frac{π}{2}) \\ - ω_{h} c o s (ω_{h} τ + φ_{h} - \frac{π}{2})}] e^{- \frac{t}{τ_{c}^{*}}} \end{matrix} \\ \begin{matrix} + \frac{τ_{c}^{* 3} ω_{h} p_{h} s i n (ω_{h} τ + φ_{h} - \frac{π}{2})}{1 + τ_{c}^{* 2} ω_{d}^{2}} \\ - \frac{τ_{c}^{* 2} ω_{h}^{2} p_{h} c o s (ω_{h} τ + φ_{h} - \frac{π}{2})}{1 + τ_{c}^{* 2} ω_{h}^{2}} [k P a] \end{matrix} \end{matrix}

(42)

Removing the term containing $e^{- \frac{t}{τ_{c}^{*}}}$ that rapidly converges to zero and applying the superposition of two coherent waves (for the sine and cosine terms of the equation) that interfere and form a resultant wave (see Appendix 4), we obtain

\begin{matrix} Δ p = \underset{p_{i, h}}{\underset{︸}{(\frac{τ_{c}^{* 2} ω_{h}}{1 + τ_{c}^{* 2} ω_{h}^{2}} \sqrt{τ_{c}^{* 2} + ω_{h}^{2}}) p_{h}}} sin [ω_{h} τ + a tan (\frac{τ_{c}^{*} sin (ϕ_{h} - π / 2) + ω_{h} sin (ω_{h} τ + ϕ_{h} - π)}{τ_{c}^{*} cos (ϕ_{h} - π / 2) + ω_{h} cos (ϕ_{h} - π)})] [kPa] \end{matrix}

(43)

p_{i, h} = p_{h} \underset{≅ 1}{\underset{︸}{\sqrt{1 + \frac{τ_{c}^{* 4} ω_{h}^{2} (τ_{c, l}^{2} + ω_{h}^{2})}{(1 + τ_{c}^{* 2} ω_{h}^{2})^{2}} + 2 \frac{τ_{c}^{* 2} ω_{h}}{1 + τ_{c}^{* 2} ω_{h}^{2}} \sqrt{τ_{c}^{* 2} + ω_{h}^{2}} cos [ϕ_{h} - atan (\frac{τ_{c}^{*} sin (ϕ_{h} - \frac{π}{2}) + ω_{h} sin (ω_{h} τ + ϕ_{h} - π)}{τ_{c}^{*} cos (ϕ_{h} - \frac{π}{2}) + ω_{h} cos (ϕ_{h} - π)}) - \frac{π}{2}]}}} ≅ p_{h}

(44)

The internal pressure thus has the expression $p_{i} = p_{e} + Δ p = p_{i, h} sin (ω_{i, h} τ + ϕ_{i, h})$ for the hypothesis of daily harmonic variation and subsequently the daily component of the internal pressure has the amplitude and the phase as follows

\begin{matrix} ϕ_{i, h} = ϕ_{h} + \underset{≅ 0}{\underset{︸}{atan (\frac{τ_{c}^{*} sin (ϕ_{h} - π / 2) + ω_{h} sin (ω_{h} τ + ϕ_{h} - π)}{τ_{c}^{*} cos (ϕ_{h} - π / 2) + ω_{h} cos (ϕ_{h} - π)})}} ≅ ϕ_{h} \end{matrix}

(45)

We have performed a test for all harmonics of the external pressure variation and we found that the indoor pressure converges to the external pressure within a margin of 10⁻³Pa. It has a very small decay and amplitude difference from the external pressure as the airtightness of the envelope has a small effect of preventing the pressure uniformization of the two sides. This result was intuitively foreseen before the analysis, but now it is mathematically demonstrated. Subsequently, we stated that the airtightness of the building influences only marginally the difference between the internal and external pressure as the external pressure variation is sufficiently slow to favor a pressure uniformization indoor–outdoor and a small decay phase. We point out that the statement is only valid for the action of the barometric pressure variation (it is not valid for the wind influence or pressure tests). The small pressure difference indoor–outdoor determines the regime of airflow in the process of infiltration, which will be laminar.

At this point, we can assume in a safe margin that the indoor pressure is the same with the external one and this aspect will significantly simplify the problem.

The equation of state $p_{i} = ρ_{i} {RT}_{i}$ applies for the space of the room. The standard air mass of the building is $m_{0} = \frac{p_{m} V}{{RT}_{i}}$ . Results that $ρ_{i} = \frac{p_{e}}{{RT}_{i}}$ and $dm = d \tilde{m} = V d ρ_{i} = \frac{V}{{RT}_{i}} {dp}_{e} = \frac{V ω_{h} p_{h}}{{RT}_{i}} cos (ω_{h} τ + ϕ_{h}) d τ$ and in this way the flow rate for the harmonic “h” of the pressure is obtained

{\overset{\cdot}{m}}_{h} = {\tilde{\overset{\cdot}{m}}}_{h} = \frac{V ω_{h} p_{h}}{{RT}_{h}} sin (ω_{h} τ + ϕ_{h} + \frac{π}{2})

(46)

According to the principle of superposition for linear systems, the response caused by two or more stimuli is the sum of the responses caused by each stimulus separately. Thus, the system assimilates all the components of the external pressure variation in the same way as each one acts separately. The total air mass flow rate (Figure 10) is

\overset{\cdot}{m} = \tilde{\overset{\cdot}{m}} = \sum_{h}^{n_{h}} {\overset{\cdot}{m}}_{h} = \sum_{h}^{n_{h}} {\tilde{\overset{\cdot}{m}}}_{h}

(47)

Figure 10.

The air mass flow rates through the envelope: (a) air mass flow rate along of the years 2014–2015 and (b) air mass flow rate along of the month of January, 2015.

By combining the processsed information represented in Figure 5(c) (that represents the temperature variation) with equation (47), from which we have retained only the positive air flow rates, we can now obtain the heating rate

Q = \sum_{h}^{n_{h}} [{\overset{\cdot}{m}}_{h}]_{+} \cdot c_{p} \cdot (T_{i} - T_{e})

(48)

where

[{\overset{\cdot}{m}}_{h}]_{+}

is the positive part of the air mass flow rate considered to be from indoor to oudoor. The equation is determined only for the cold season (15 October–15 April) that is roughly the heating period at the northern hemisphere with the reference comfort temperature of the indoor air

T_{i} = 20^{\circ} C

. This result can also be extended for the warm season if a cooling system is considered in order to ensure an internal comfort temperature

T_{i} = 26^{\circ} C

Equation (48) represents the heating rates (Figure 11) generated by the air exchange building–atmosphere in the cold seasons and it registers among the other existing heat losses of the building (convection–conduction, radiation, airflow through large openings, etc).

Figure 11.

The heat rate through the envelope: (a) heat rate along of the years 2014–2015 and (b) heat rate along of the month of January, 2015.

At this stage, an analytic construction is necessary that will relate the accounted air mass exchanged to the total quantity of exchanged air given by equation (13).

Air mass balance from the barometric pressure variation

The successive pulses of air exchange present interest because the temperature difference indoor–outdoor gives an energy sense to the cycles of air exchange. We start by studying the effect of one harmonic “h” that causes “+” paths of warm air exhaust and “−” paths of cold air admission.

Figure 12 presents a set of increases and decreases in pressure (denomination paths/tracks in what follows). The zones with red arrows are for warm air evacuation while the portions with arrows in blue are the inlet of cold air from outside. A complete cycle of pressure variation contains the following

– For the path $\frac{{dp}_{e}}{d τ} < 0$ (or $p_{e} ↘$ and $p_{i} - p_{e} > 0$ ), warm air is evacuated from inside by virtue of internal pressure adaption to the external pressure;

– For the path $\frac{{dp}_{e}}{d τ} > 0$ (or $p_{e} ↗$ and $p_{i} - p_{e} < 0$ ), cold air is admitted from outside that will determine an indoor pressure increase. A heating load will be required in order to maintain the temperature T_i of the indoor air.

Figure 12.

Cycles of air exchange indoor–outdoor through the building’s envelope.

Each descendent track of the pressure involves a mass of warm air ${\tilde{m}}_{h, warm}$ corresponding to the harmonic “h”

\begin{matrix} {\tilde{m}}_{h, warm} = {\tilde{m}}_{h} | \begin{matrix} \frac{{dp}_{e}}{d τ} < 0 \end{matrix} = \int_{\frac{(2 k + 1) π / 2 - ϕ_{h}}{ω_{h}}}^{\frac{(2 k + 3) π / 2 - ϕ_{h}}{ω_{h}}} | {\tilde{\overset{\cdot}{m}}}_{h} | d τ \end{matrix}

(49)

\begin{matrix} = \frac{2 V p_{h}}{{RT}_{i}} \int_{\frac{(2 k + 1) π / 2 - ϕ_{h}}{ω_{h}}}^{\frac{(2 k + 2) π / 2 - ϕ_{h}}{ω_{h}}} sin (ω_{h} τ + ϕ_{h}) d τ = \frac{2 {Vp}_{h}}{{RT}_{i}} \end{matrix}

The density is $ρ_{i} = \frac{p_{e}}{{RT}_{i}} = \frac{p_{0}}{{RT}_{i}} + \frac{p_{h}}{{RT}_{i}} sin (ω_{h} τ + ϕ_{h}) = ρ_{m} + \tilde{ρ}$ such that for the variable component $\tilde{ρ} = \frac{p_{h}}{{RT}_{i}} sin (ω_{h} τ + ϕ_{h})$ we perform a normalization inside of the range $[\frac{(2 k + 1) π / 2 - ϕ_{h}}{ω_{h}}, \frac{(2 k + 3) π / 2 - ϕ_{h}}{ω_{h}}]$ and the variable component of the density becomes

\begin{matrix} \bar{{\tilde{ρ}}_{h, warm}} = \frac{2}{T_{h}} \int_{\frac{(2 k + 1) π / 2 - ϕ_{h}}{ω_{h}}}^{\frac{(2 k + 3) π / 2 - ϕ_{h}}{ω_{h}}} | \tilde{ρ} | d τ \end{matrix}

(50)

\begin{matrix} = \frac{4}{T_{h}} \frac{p_{h}}{{RT}_{i}} \int_{\frac{(2 k + 1) π / 2 - ϕ_{h}}{ω_{h}}}^{\frac{(2 k + 2) π / 2 - ϕ_{h}}{ω_{h}}} sin (ω_{h} τ + ϕ_{h}) d τ = 0 \end{matrix}

In this way, for the range $[\frac{(2 k + 1) π / 2 - ϕ_{0 h}}{ω_{h}}, \frac{(2 k + 3) π / 2 - ϕ_{0 h}}{ω_{h}}]$ , the density is $ρ_{i} = ρ_{m} = \frac{p_{0}}{{RT}_{i}}$ . Subsequently, the volume of warm air lost in the period $[\frac{(2 k + 1) π / 2 - ϕ_{0 h}}{ω_{h}}, \frac{(2 k + 3) π / 2 - ϕ_{0 h}}{ω_{h}}]$ corresponding to the harmonic “h” is

{\tilde{V}}_{h, warm} = \frac{{\tilde{m}}_{h, warm}}{ρ_{m}} = \frac{2 π V p_{h}}{p_{0}}

(51)

The total airflow rate derived from the overlaid periodicities of the external pressure is

\bar{\overset{\cdot}{V}} = \sum_{i}^{n} \bar{{\overset{\cdot}{V}}_{i}} = \frac{2 π V}{p_{m}} \sum_{i = 5}^{n_{h} = 17} \frac{p_{i}}{τ_{i}}

(52)

The total mean air exchange per hour is the sum of the effects of all considered harmonics

\begin{matrix} \bar{ACH} = 3600 \frac{\bar{\overset{\cdot}{V}}}{V} = \sum_{i}^{n} {\bar{ACH}}_{i} \end{matrix}

(53)

\begin{matrix} = \frac{2 \cdot 3600 \cdot π}{p_{m}} \sum_{i}^{n} \frac{p_{i}}{τ_{i}} = 0.00059529 [h^{- 1}] \end{matrix}

We consider, at this point, the measurements of the passive house Politehnica with the rate of infiltration of ${ACH}_{50} = 0.586 [h^{- 1}]$ . The air exchange per hour in natural conditions along of the year is

{ACH}_{nat} = {ACH}_{50} / 20 = 0.02930 [h^{- 1}]

(54)

For the standard case that considers the permanent harmonics identified as local peaks in the spectrogram, the ratio of the infiltrated air due to the pressure variation will be

f = \frac{\bar{ACH}}{{ACH}_{nat}} \cdot 100 = 0.02032 = 2.03 %

(55)

The share of influence of each harmonic (Table 4) related to the combined action of all harmonics of the barometric pressure reveals that the bi-daily (38.200% influence) and the daily (14.844% influence) harmonics have by far the most important influences because of the frequency of the cycles combined with the magnitude of the amplitudes.

If we complete the standard case by considering all the harmonics of Figure 5, non-filtered and filtered at the conservative level of 50%, the ratio f is 31.24% and 3.19%, respectively. However, the value provided by equation (55) is considered a minimum derived from the most restrictive/conservative modeling.

Considering the information provided by Table 7, we appreciate that Case 1 that proposes the ratio of 2.03 % is the most sustainable as a concept as it is based on local peaks of the spectrogram. It represents the lowest possible threshold of aeraulic influence of the atmosphere. Case 2 considers all harmonics of the spectrogram and serves only as reference regarding the maximum possible impact that can be caused by the atmospheric pressure oscillation. A large portion of its impact contains the wind’s influence that explains the high value of 31.24%. Case 3 applies a common technique of filtering of the pressure signal at a specific level of 50% with the consequence of selection of 46 harmonics from a total of 960 items.

Table 7.

Evaluation of the infiltration air as function of selected harmonics of pressure.

Criterion	Case 1 (standard case)	Case 2 (nonfiltered harmonics)	Case 3 (50% filtered harmonics)
Number of harmonics	17	960	46
Ratio, f [%]	2.03%	31.24%	3.19%

The level of aeraulic influence of 3.19% appears to be the closest to the real impact of the phenomenon and represents only a slight correction to the standard case (Case 1) such that we kept it as the most representative value. This final result suggests that the barometric pressure variation has an effect that should be considered in all aeraulic balances of buildings.

A problem that should be pointed out is the variance of the overall air infiltration rate in natural conditions ( ${\overset{\cdot}{V}}_{nat}$ ), which is discussed in detail in Sherman¹ and Jones et al.² The value ${\overset{\cdot}{V}}_{nat} = {\overset{\cdot}{V}}_{50} / 20$ may not be the most accurate and may offer an overestimation of the total air infiltration rate, but we choose it arbitrary due to its simplicity.

The value of 3.19% may not be impressive, but it is permanent and determined with rigorous methodology that involves an interdisciplinary effort. We consider that our initiative is a step forward toward a more accurate and rigorous evaluation of the air exchange in buildings. Moreover, we have constructed a methodology that can serve for the analysis of other phenomena of air mass exchange, such as the stack effect. Stack effect is currently analyzed for average atmospheric conditions, but in reality the atmospheric temperature has a periodical variation. Therefore, we consider that the methodology we developed could be extended to the study of air exchange as a result of the stack effect.

Conclusion

In this study, a new component of air leakage in natural conditions is identified, which is generated by the periodic variation of the barometric pressure.

The DEAE is introduced for the dynamic interaction building–atmosphere. Based on its solution, the notion of pressure equilibration time $τ_{50}$ is introduced. This is the time necessary for the building to transit freely from the standard pressure difference of 50 Pa to the minimum prescribed difference of 4 Pa for the pressure tests. Unlike the parameter ${ACH}_{50}$ that characterizes the airtightness in steady state conditions of the pressure difference, $τ_{50}$ characterizes the airtightness of the building in dynamic conditions of pressure equilibration and is more connected to the processes in real conditions that are cyclic and transitory. For the passive house Politehnica, we found the dynamic responsiveness with the time $τ_{50} = 4.7 s$ that is a relatively long time and is specific for tight buildings.

Considering the weather conditions from two independent sources for the range of years 2006–2015 (daily pressure data) and of years 2014–2015 (hourly pressure data), we used techniques from the field of signal processing (Fourier transforms and statistical analysis) in order to reconstruct a parameterized model of the external pressure cyclic variation. Analyzing the spectrograms of the two data sets, we confirmed the five main periodic components of variation that also have an astronomical background: a bi-diurnal component, a diurnal, a bimonthly, a monthly, and an annual component. These five harmonics are included among the 17 main harmonics (derived of local amplitude peaks on the spectrogram) that we considered in our analysis of the balance equations. The aeraulic balances have only considered the harmonics with time periods that are less than a month (H5 \ldots H17). The cyclicity of each component of the barometric pressure affects the dynamics and quantity of air mass exchange between the building and environment. All the harmonics are adding up in conformance with the principle of superposition of the waves.

Considering that the barometric pressure variation conducts the evolution of the internal pressure, we described the aeraulic interaction building–atmosphere by devising the DEAE. The analytic solution of the differential equation has permitted us to model the airflow rates traversing the envelope. We have mathematically proven that in free conditions, the difference between the external and internal pressure is almost insignificant inside a margin of 10⁻³Pa and this result explains that the airtightness provided by the envelope affects only marginally the air exchange building–environment under the action of the barometric pressure. However, for the case of wind action (which is more sudden), the influence of the airtightness of the envelope is much more important as our statement is provided only for slow variations.

We found that in the process of air exchange of the building, there is an intermittent evacuation of the warm internal air at specific ranges of time. For each interval of time, we performed an averaging of the mass flow rate that leaves the building and translates in irreversible heat losses in the cold season. It should be noted that these quantities of exchanged air are detectable in calculations only by studying the system in dynamic conditions; while otherwise in averaged conditions of the barometric pressure would had been impossible for any result to be properly quantified.

We noticed a magnitude of impact of 3.19% caused by the barometric pressure variation from the total air exchanged in natural conditions. The level of air exchange is computed based on the permanent cycles of the atmosphere while in reality random variations (derived from the main pressure waves and independent of the wind action) will generate additional exchanges of air masses. We appreciate that this margin of air infiltration it is at a level that should not be ignored as it is determined through exact methods and it happens permanently with quasi-regular harmonics. We point out that the share of influence of the barometric component was related to the Kronvall–Persily estimation of the average air infiltration in natural conditions along a year. It was chosen because of its simplicity although its accuracy raised a series of debates. If more exact evaluations of the air infiltration in natural conditions can be provided in the future, then the actual share of the barometric pressure influence may be updated and it is possible to discover a larger influence than the actual one revealed in this study.

This paper offers a methodology of treating the variable conditions of the weather. It will serve as an instrument to further investigate the stack effect based on the variability of atmospheric temperature. Currently, most studies consider the stack effect only in averaged conditions, a practice that comes with a loss of information on how the system evolves in time. While today the stack effect is evaluated by considering a static diagram of pressure, our deterministic instrument will offer the possibility of analysis of air mass transfer in transient conditions as well.

Footnotes

Acknowledgments

We would like to express special thanks to Mrs. Despina Baracu for her support for the elaboration of this research, and also to professors Claude-Alain Roulet, Max H. Sherman, Mattheos Santamouris, and Helmut E. Feustel for the answer and the expertise provided regarding the content of the article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Appendix

References

Sherman

. Estimation of infiltration from leakage and climate indicators. Energy Build 1987; 10: 81–86.

Jones B, Persily A, Sherman M H. The Origin and Application of Leakage-Infiltration Ratios, 2016 Conference Proceeding by ASHRAE and AIVC IAQ, https://www.techstreet.com/standards/the-origin-and-application-of-leakage-infiltration-ratios?product_id=1928359#jumps (2016, accessed 19 December 2017).

Berge A. Analysis of methods to calculate air infiltration for use in energy calculations, 2011 Master of Science Thesis, Chalmers University Of Technology, Göteborg, Sweden 2011.

Sherman

Grimsrud

. Infiltration-pressurization correlation: simplified physical modeling. Lawrence Berkeley Laboratory Report, LBL-10163. ASHRAE Trans 1980; 86: 778–807.

Sherman MH. Air infiltration in buildings. PhD Thesis, University of California, Berkeley, 1980. Lawrence Berkeley Laboratory Report, LBL-10712.

Wilson

Pittman

. Correlating measured infiltration for wind from a single direction. ASHRAE Trans 1983; 89: 211–227.

Shaw

. A correlation between air infiltration and airtightness for houses in a developed residential area. ASHRAE Eunsactwns 1981; 87: Pt 11.

Reardon J, Air infiltration modelling study, Report. No. CR5446.3, National Research Council of Canada (NRC), Ottawa , 1989.

Walker IS and Wilson DJ, AIM-2: The alberta air infiltration model, University of Alberta (Department of Mechanical Engineering Report 71), Edmonton, Alberta, 1990.

10.

CGSB. Determination of the overall envelope air tightness of buildings by the fan pressurization method using the building’s air handling systems. Gatineau, Canada: Canadian General Standards Board, 1999.

11.

ASTM E779-2003: Standard Test Method for Determining Air Leakage Rate by Fan Pressurization. Conshohocken, PA: American Society for Testing and Materials, 2003.

12.

ASHRAE. Fundamentals handbook (SI units), Atlanta: American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 2013.

13.

Kim

M-H

J-H

Jeong

J-W-

. Feasibility of building envelope air leakage measurement using combination of air-handler and blower door. Energy Buildings 2013; 62: 436–441.

14.

ASTM E779-2003: Standard Test Method for Determining Air Leakage Rate by Fan Pressurization. Conshohocken, PA: American Society for Testing and Materials, 2003.

15.

Iordache

Nastase

Damian

et al.

Average permeability measurements for an individual dwelling in Romania. Build Environ 2011; 46: 1115–1124.

16.

Jeong

J-W

Firrantello

Bahnfleth

et al.

Case studies of building envelope leakage measurement using an air-handler fan pressurisation approach. Build Serv Eng Res Technol 2008; 29: 137–155.

17.

Cooper E, Zheng X, Wood Ch, et al. Field trialling of a new airtightness tester in a range of UK homes. In: 36th AIVC conference, At Madrid, Spain, September 2015.

18.

Damian A, Iordache V, Nastase I- Systems of measurement of the air permeability of the buildings (Sistem de masura a permeabilitatii la aer a cladirilor), The XVth Conference on Comfort, Efficiency, Energy Conservation and Environment Protection, 26-27 Nov. 2008.

19.

Iordache V, Damian A. Influence of the building envelope permeability upon the annual heating load. Romanian Journal of Civil Engineering (Revista Română de Inginerie Civilă) 2013(2); 4: 98–107.

20.

Iordache

Teodosiu

et al.

Permeability measurements of a passive house during two construction stages. Energy Procedia 2016; 85: 279–287.

21.

Sfakianaki

Pavlou

Santamouris

et al.

Air tightness measurements of residential houses in Athens, Greece. Build Environ 2008; 43: 398–405.

22.

Kalamees

. Air tightness and air leakages of new lightweight single-family detached houses in Estonia. Build Environ 2007; 42: 2369–2377.

23.

Sherman MH, Grimsrud DT, Sonderegger RC. The Low Pressure Leakage Function of a building,1979, LBL-9162C.2, Berkeley, CA: Lawrence Berkeley Laboratory, University of California, 1979.

24.

Nishioka

Chen

Arai

et al.

Unsteady pressurization method to measure the airtightness of the building envelope. In: Proceedings of the Second International Conference on Building Physics, 14-18 September 2003, Leuven, Belgium, 2003.

25.

Fürbringer

J-M

Maas

. Suitable algorithms for calculating air renewal rate by pulsating air flow through a single large opening. Build Environ 1995; 30: 493–503.

26.

Mattsson B and Claesson J-A. Transient pressurisation method for air infiltration measurements of building envelope components. In: The 7th symposium on building physics in the Nordic countries, Reykjavik, 13–15 June 2005.

27.

Mattsson

Claesson

J-A

. Transient pressurization method for measurements of airtightness. J Build Phys 2007; 31: 35–53.

28.

Han

H-T

Yom

C-M

. Modeling of turbulent ventilation through an opening due to outdoor pressure fluctuations. Kor J Air-Condition Refrigerat Eng 2008; 20: 121–127.

29.

Han

Kim

Han

C-H

. Effect of outdoor pressure fluctuations on natural ventilation in a room with two openings. Int J Air-Condition Refrigerat 2012; 20: 1250003–1250003.

30.

Walker

Wilson

. Evaluating models for superposition of wind and stack effect in air infiltration. Build Environ 1993; 28: 201–210.

31.

Warren PR and Webb BC. The relationship between tracer gas and pressurization techniques in dwellings. In: 1st AIVC Conference “Instrumentation and measuring techniques”, Windsor 6-8 October 1980, pp.245–276.

32.

Kronvall J- Air flow in building components, 1980 Division Of Building Technology, Lund Institute Of Technology, Report TVBH-1002, Lund, Sweeden, 1980.

33.

Feustel HE and COMIS. An international multizone air-flow and contaminant transport model. LBNL-42182, 1998.

34.

Walker IS and Wilson DJ. Field validation of algebraic equations for stack and wind driven air infiltration calculations, LBNL 42361, 1998.

35.

Pietrzyk K. Probabilistic modelling of air infiltration and heat loss in low-rise buildings. PhD Thesis, Chalmers University of Technology, Göteborg, Sweden, 2000.

36.

Deru M and Burns P. Infiltration and natural ventilation model for whole-building energy simulation of residential buildings. Report NREL/CP-550-33698, 2003.

37.

Kraniotis D, Aurlien T and Thiis TK. Wind direction and leakage distribution in buildings. A CFD transient analysis of their impact on air exchange rates under unsteady wind conditions. In: 6th European-African conference on wind engineering (EACWE 2013), Cambridge, 7–11 July 2013.

38.

Kraniotis D, Thiis TK and Aurlien T-A. Numerical study on the impact of wind gust frequency on air exchanges in buildings with variable. In: 33rd AIVC conference—2nd tightvent conference, Copenhagen, 10–11 October 2012.

39.

Haghighat

Brohus

Rao

. Modelling air infiltration due to wind fluctuations—a review. Build Environ 2000; 35: 377–385.

40.

Sharma RN. The fundamentals of building internal pressure dynamics induced through a dominant opening.

41.

Haghighat

Rao

Fazio

. The influence of turbulent wind on air change rates a modelling approach. Build Environ 1991; 26: 95–109.

42.

Holmes JD. Mean and fluctuating internal pressures induced by wind. Townsville: Department of Civil and Systems Engineering, James Cook University of North Queensland, Townsville, Australia, 1978.

43.

Harris

. The propagation of internal pressures in buildings. J Wind Eng Industr Aerodyn 1990; 34: 169–184.

44.

Carey

Etheridge

. Direct wind tunnel modelling of natural ventilation for design purposes. Build Serv Eng Res Technol 1999; 20: 131–142.

45.

Kraniotis D. Dynamic characteristics of wind-driven air infiltration in buildings—the impact of wind gusts under unsteady wind conditions. PhD Thesis, Norwegian University of Life Sciences, 2014. DOI: 10.13140/RG.2.1.3607.9444, 2004.

46.

Etheridge

. Unsteady flow effects due to fluctuating wind pressures in natural ventilation design-instantaneous flow rates. Build Environ 2000; 35: 321–337.

47.

Sherman, DT Grimsrud, Sonderegger RC. The Low Pressure Leakage Function of a Building, 1979 Lawrence Berkeley Laboratory, LBL-9162; EEB-ENV-79-10, University of California, Berkeley, USA, 1979.

48.

Modera MP and Sherman MH. AC pressurization a technique for measuring leakage area in residential buildings. Berkeley: Lawrence Berkeley Laboratory LBL-18395, 1986.

49.

Sherman MH and Modera M. Low-frequency measurement of leakage in enclosures. Berkeley: Lawrence Berkeley Laboratory LBL-20121, 1986.

50.

Sherman MH and Modera MP. Signal attenuation due to cavity leakage. J Acoust Soc Am 1988; 84: 0001-4966/88/122163-07$00.80.

51.

Sherman MH. Air change rate and airtightness in buildings. American Society for Testing and Materials, ASTM, Reoort STP1067, Philadelphia, PA, 1990.

52.

Card WH, Sallman A, Graham RW, et al. Infrasonic measurement of building air leakage: a progress report, American Society for Testing and Materials ASTM STP719, Philadelphia, PA, 1980.

53.

Carey

Etheridge

. Leakage measurements using unsteady techniques with particular reference to large buildings. Build Serv Eng Res Technol 2001; 22: 69–82.

54.

Cooper EW and Etheridge DW. Measurement of building leakage by unsteady pressurisation. In: 25th AIVC conference “ventilation and retrofitting.” Prague, Czech Republic, 15–17 September 2004.

55.

Cooper

Etheridge

. Determining the adventitious leakage of buildings at low pressure. Part 2—pulse technique. Build Serv Eng Res Technol 2007; 28: 81–96.

56.

Cooper E, Zheng X, Gillot M, et al. A nozzle pulse pressurisation technique for measurement of building leakage at low pressure. In: 35th AIVC conference “Ventilation and airtightness in transforming the building stock to high performance,” Poznań, Poland, 24–25 September 2014.

57.

Watanabe Y, Kobayashi H and Utsumi Y. Development of validation of ac-pressurization measuring of leakage area of houses. In: Proceedings Buildings Simulation ’99, Sixth International IBPSA Conference, Kyoto, Japan, 13-15 September 1999, pp. 807–814.

58.

Lee

Kim

Ryou

. Air tightness measurement with transient methods using sudden expansion from a compressed chamber. Build Environ 2011; 46: 1937–1945.

59.

Bae

Moon

Lee

et al.

Improvement in the applicability of the air tightness measurement using a sudden expansion of compressed air. Build Environ 2013; 61: 133–139.

60.

Dewsbury

. AC pressurisation analysis by non-linear optimisation. Build Serv Eng Res Technol 1996; 17: 65–71.

61.

Dewsbury

. AC pressurisation Fourier analysis and the effect of compressibility of air. Build Serv Eng Res Technol 1996; 17: 73–77.

62.

Cooper E and Etheridge DW. An innovative technique for measurement of building leakage at low pressures. In: The 6th international conference on indoor air quality, ventilation & energy conservation in buildings IAQVEC 2007, Sendai, Japan, 28–31 October 2007.

63.

Roden GI. On atmospheric pressure oscillations along the pacific coast of North America, 1873–1963. J Atmosph Sci 1965; 22.

64.

Gupta BRD and Sing B. A power spectrum analysis of the mean daily pressure over the Mediterranean and neighborhood during November 1967 to April 1968. 1977; 29: 382–384.

65.

Chapman

Lindzen

. Atmospheric tides—thermal and gravitational, Dordrecht, Holland: D. Reidel Publishing Company, 1970.

66.

Lindzen RS. Dynamics in atmospheric physics, Berkeley: 2005.

67.

Mahongo SB and Francis J. Monthly variations in sea level at the Island of Zanzibar, J. Mar. Sci. 2010; 9(1): 1–16.

68.

Whiteman CD and Bian X. Solar semidiurnal tides in the troposphere detection by radar profilers, Bulletin of the American Meteorological Society 1996: 77(3).

69.

Randall DA. The Laplace Tidal Equations and Atmospheric Tides, Fort Collins, CO: Colorado State University, 2007.

70.

Nanthakrishnan R, Maliekal JA and Aralikatti SS. Atmospheric tidal oscillations part 1. Historical development, Current Science 1984; 53(18).

71.

Lu N. Time-series analysis for determining vertical air permeability in unsaturated zones, Journal of Geotechnical and Geoenvironmental Engineering 1999; 125(1).

72.

Carlson GC jr and Hastenrath S. Diurnal variation of wind, pressure, and temperature, Monthly Weather Review 1970; 98(5).

73.

Bohm S, et al. High-frequency signals of oceans and atmosphere in Earth rotation atmosphere. Viena University of Technology, TU Wien, EGU GA 2012, Vienna, Session G5.1.

74.

Lowe

. Ventilation strategy, energy use and CO₂ emissions in dwellings—a theoretical approach. Build Serv Eng Res Technol 2000; 21: 179–185.

75.

Jones

Lowe

Davies

et al.

Modelling uniformly porous facades to predict dwelling infiltration rates. Build Serv Eng Res Technol 2014; 35: 408–416.

76.

Jones

Das

Chalabi

et al.

Assessing uncertainty in housing stock infiltration rates and associated heat loss: English and UK case studies. Build Environ 2015; 92: 644–656.

77.

Baracu T. Contributii la modelarea numerica a cladirilor (Contributions to the numerical modeling of the buildings). PhD Thesis, University Politehnica of Bucharest, Romania, 2013.

78.

Baracu

Croitoru

A-M

Badea

. Theoretical investigations of the solar radiation at the site of passive house “Politehnica” from Bucharest. UPB Scient Bull Series C: Electr Eng 2014; 76: 255–266.

79.

Baracu T, Teodosiu C, Birsan M-V, et al. Theoretical evaluations of the natural infiltration of the air in buildings. In: 6th International conference on energy and environment, CIEM 2013, Bucharest–Romania, 2013.

80.

Baracu T, Muresan M and Badea A. A comparative analisys of the controllers on-off and PI for the heating system of a building. In: Proceedings of francophone multidisciplinary colloquium on materials, environment and electronics, PLUMEE 2013, Bacau, Romania, 22–25 May 2013. ISSN 2343–9092.

81.

Ionescu

Baracu

Vlad

G-B

et al.

The historical evolution of the energy efficient buildings. Renew Sustain Energy Rev 2015; 49: 243–253. ISSN 1364-0321.

82.

Badea A. Plus energy houses—Politehnica plus energy building. In: Proceedings of 5th international conference on energy and environment—CIEM, Bucharest, 2011.

83.

Baracu

Tanasiev

Mamut

et al.

A transient thermal anal-ysis by thermal networks of the Passive House ‘POLITEHNICA’ from Bucharest. Int J Sustain Build Technol Urban Develop 2013; 4: 146–159.

84.

Georgescu MS, Ochinciuc CV and Andone R. Politehnica University’s Passive Houses—architectural and structural details. In: 5th Trade fair and conferences for renewable energy and energy efficiency. Passive Houses, Renexpo® South-East Europe, Bucharest, 2011.

85.

Badea

Baracu

Dinca

et al.

A life-cycle cost analysis of the passive house “Politehnica” from Bucharest. Energy Build 2014; 80: 542–555.

86.

Baracu

Costinas

Ghiaus

et al.

New analytical methodologies for radiative heat transfer in enclosures based on matrix formalism and network analogy. Applied Thermal Engineering 2016; 107: 1269–1286.

87.

Baracu T and Grigoras-Benescu S. Computational analysis of the flow around a cylinder and of the drag force. In: The 2nd Conference of the Young Researchers from TUCEB, Technical University of Civil Engineering of Bucharest, Bucharest, Romania, 17–18 November 2011.

88.

Baracu

Badescu

Teodosiu

et al.

Consideration of a new extended power law of air infiltration through the building’s envelope providing estimations of the leakage area. Energy Build 2017; 149: 400–423.

89.

Attia

Eleftheriou

Xeni

et al.

Overview and future challenges of nearly zero energy buildings (nZEB) design in Southern Europe. Energy Build 2017; 155: 439–458.

90.

Attia S, Eleftheriou P, Xeni F, et al. Overview of challenges of residential nearly Zero Energy Buildings (nZEB) in Southern Europe. Liege, Belgium: Sustainable Buildings Design Lab, Technical Report, ISBN 9782930909059, 2016.

91.

Badescu V, Baracu T, Avram R, et al. On the design and optimization of constructal networks of heat exchangers by considering entropy generation minimization and thermoeconomics. In: Constructal law & second law conference, 15–16 May 2017, Bucharest, Romania.

92.

Muresan

Attia

. Energy efficiency in the Romanian residential building stock: a literature review. Renew Sustain Energy Rev 2017; 74: 349–363.

93.

Stymne

Boman

Kronval

. Measuring ventilation rates in the Swedish housing stock. Build Environ 1994; 29: 373–379.

94.

Hult

Sherman

. Estimates of uncertainty in multi-zone air leakage measurements. Int J Ventilat 2014; 12: 359–368.

95.

Guyot

Ferlay

Gonze

et al.

Multizone air leakage measurements and interactions with ventilation flows in low-energy homes. Build Environ 2016; 107: 52–63.

96.

Jones

Das

Chalabi

et al.

The effect of party wall permeability on estimations of infiltration from air leakage. Int J Ventil 2013; 12: 17–30.

97.

Jones B, Lowe R. Predicting the Optimum Air Permeability of a Stock of Detached English Dwellings. In: 35th Air Infiltration and Ventilation Centre (AIVC) Conference, Poznan, Poland, 2014.

98.

Jones B, Das P, Chalabi Z, et al. The relationship between permeability and infiltration in conjoined dwellings. In: 33rd AIVC and 2nd TightVent Conference: Optimising Ventilative Cooling and Airtightness for [Nearly] Zero-Energy Buildings, IAQ and Comfort, Copenhagen, Denmark, 2012.

99.

The Energy Conservatory—Minneapolis blower door—operation manual for model 3 and model 4 systems. The Energy Conservatory, www.energyconservatory.com, Mineapolis, 2010 (accessed 19 December 2011).

100.

The Energy Conservatory- Operating Instructions for the DG-700 Pressure and Flow Gauge, Mineapolis, 2010 The Energy Conservatory, www.energyconservatory.com, Mineapolis (2010, accessed 19 December 2017).

101.

Damian A, Iordache V and Nastase I. System of measurement of the permeability of the buildings (Sistem de masura a permeabilitatii la aer a cladirilor), Technical University of Civil Engineering from Bucharest (TUCEB), 2008.

102.

Teodosiu C. Determination of air permeability. Technical Report of UTCB, 2010.

103.

Teodosiu C. Presentation of the stand of permeability measurements (Ch. 10). Technical Report of UTCB, 2016.

104.

Cheval

Birsan

Dumitrescu

. Climate variability in the Carpathian Mountains Region over 1961–2010. Global Planet Change 2014; 118: 85–96. DOI: 10.1016/j.gloplacha.2014.04.005.

105.

Wunderground. Meteorological data for Bucharest, 2016, wunderground.com (15 November 2016).

106.

Varpasuo

. Seismic design and analysis of nuclear power plant structures. Eng Sci 2013; 11: 10–19.

107.

Feng Z and Meng Q. First integrals for the damped Helmholtz oscillator. International Journal of Computer Mathematics 2010; 87; 2798–2810.

108.

Innes D and Grighn DG. On a non-linear differential equation modelling Helmholtz resonator response. Journal of Sound and Vibration 1989; 131(2); 323–330.

109.

Almendral JA and Sanjuan MAF. Integrability and symmetries for the Helmholtz oscillator with friction, Journal of Physics A: Mathematical and General; 36: 3.

110.

Ortiz A, Seoane JM, Yang JH, et al. Dynamics of the Helmholtz oscillator with fractional order damping, http://www.redalyc.org/html/416/41629563003 (2012, accessed 19 December 2017).

111.

Askari H, Saadatnia Z, Younesian D, et al. Approximate periodic solutions for the Helmholtz–Duffing equation. Computers and Mathematics with Applications 2011; 62: 10.

112.

Hersb AS and Walker B. Fluid mechanical model of the Helmholtz resonator, Washington, D.C: National Aeronautics and Space Administration, NASA Cr-2904, 1977.

113.

Maas LRM. On an oscillator equation for tides in almost enciosed basins of non-uniform depth, 1998.

114.

Salas AH. Exact solutions to Helmholtz oscillator equation for an electrical circuit with quadratic nonlinearity, Chaos, Solitons & Fractals; 81: 68–77.