Basic air infiltration

Buildrng and Environmenf, Vol. 32, No. 2, pp. 95-100. 1997 f$ 1997 Elsevier Saence Ltd. All rights reserved

Pergamon

PrInted10Great Britain 036&1323/97$17.@0+0.00

PII: SO360-1323(96)00048-O

Basic Air Infiltration M. D. LYBERG*

(Received 5 July 1996; accepted 28 August 1996) Some fundamental properties of building air injiltration are derived starting from propositions stressing the underlying assumptions. Possible structures of air infiltration models relating the flow to a leakage area are considered. Approximate expressions are derivedfor the value of the reference pressure necessary for mass conservation to hold. These results are applied to the case of enforced building pressurization. It is demonstrated that some currently usedmodels of air infiltration violate mass conservation. Other models are shown to lead to unnecessarily large errors when applied to pressurization. 0 1991 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

trusions and constrictions resulting in an uneven flow or even backflow at parts of the cross section [l]. The pressure drop across many such objects is approximately proportional to the square of the flow rate. Therefore, the flow function of a wall surface element is likely to be proportional to the square root of the pressure drop across it. Studies of the dependence of the flow rate on the pressure difference using a model where QcclApp18 mostly yield a value of j that is compatible with l/2, or sometimes somewhat larger. Many models where fl has been put equal to l/2 have been presented [24]. Models of this kind are easily cast in a non-dimensional form. Under natural conditions, the pressure difference across a building envelope is never the same for all portions of the envelope. The pressure difference may be due to stack effects, wind forces or mechanical ventilation. Each one of these effects will contribute to the pressure difference in a specific way.

SUBJECT a building with external area S to a pressurization (or depressurization) resulting in a pressure difference of absolute value lAppIbetween the indoor and outdoor air. This will cause an airflow Q across the building envelope. Consider a particular building and assume that the pressure difference is of the same order of magnitude as naturally occuring pressure differences. The airflow divided by the surface area, Q/S, is then in general a function of IApplonly, the flow function F(lAppl),and the characteristics of the building considered. To compare different buildings, a more general measure of the air leakiness needs to be defined. Imagine an external wall consisting of all the types of and holes present in a building envelope. Suppose that the divisions between different air leaks are so thin that the total volume of these walls is almost zero, or the porosity is equal to one. If a wall portion of area S is subjected to a pressure difference Ap, the ratio Q/s = F(IApl) is a function that cannot be exceeded by the flow function of any other wall. If an ordinary wall is subjected to the same pressure difference, one would expect the flow function to be A* lAppI,where A is some constant 0
*Department of Physics, University of S351 95 Sweden. E-mail: [email protected]

Wexio,

2. THE PRESSURE First consider density variations with height. Suppose the pressure at a height z,, is p,,. Assume that the temperature is constant. The pressure p at height z will then (as the pressure is proportional to the density for fixed temperature), according to the Boltzmann distribution, be equal to p = p,, exp( -pg(z-z,)/kT). Here, p is the average mass of the air molecules, k is the Stefan-Boltzmann constant and g is the constant of gravitation. For building applications, the exponent of the above expression will be very small, so approximately p = p,,( 1- pg(z - z,)/kT). For a 10 m building this gives a pressure drop from roof to basement of about 120Pa. However, it is only the pressure difference across the envelope that is interesting here. In fact, there may be two expressions of the kind above, one for the outdoor air and one for the indoor air, having different p0 and zO.To relate these two expressions, one must resort to the fact that for mass conservation to hold, the indoor and outdoor pressure must be the same at some height. Select z,, to be this height. The pressure difference will then be given by the expression Ap =p,pg(z-z,)AT/kT’~3410(z-z,)AT/T*, where AT

Wexio,

95

is the temperature difference and I‘ix the geometric mean temperature. Assuming an outdoor indoor- temperature difference of IO K. the drop in pressure c)\‘er a building height of IO m will be roughly 4.5 Pa. The local wind pressure on a building en\elopc I\ 1” I)portional to the wind speed squared. The relevant hind speed is. in general. defined a\ the velocity of tht‘ free wind at such a height above the building that the ait stream is unobstructed by the presence of the building. The tocut wind pressure may then be defined ax (‘,,/J/ ’ 2. where C‘,, is the pressure coetficicnt that variez over the envelope. The wind speed discussed here iz not the meteorological wind speed which is. in general. measured at a height of IOm. For an isolated building. C‘,. may take ;I masimum value of about 0.6 on the windward side at a height about 1 3 of the building height. Locally. for example at corners. the pressure coefticient may be r‘ben largeI-. On the leeward side. there is in general a wake and the pre\sure coefficient may take a value of about -0. I down tcl - 0.3. For a building IO m tall and a wind speed of 5 m \. the maximal pressure on the windward side is then about IO Pa. and the pressure on the leeward side about ~~~ 3 Pa The numbers quoted here are very approximate: in reality. the building shape will be an important factor. In a built environment. the absolute values of the prcszurc coefficients are smaller than those quoted above. Data on the local wind speed. or pressure. are seldom available. One has to use data recorded at some other point. To establish a relation between the recorded prc\sure and the pressure at the building site is then a necebbity. A common assumption is the existence of ;I cot-relation such that the recorded pressure is equal to the wind speed squared at the building site. multiplied b\ some coefficient. or C,,~W‘2. The air flow rate produced by a mechanical tentilutlvn system may be assumed to be known or easy to record An important factor is the balancing of the presburc The pressures discussed above relate to pressure differences relative to various reference points. (‘onscqucntl>. for calculations one will habe to introduce a common reference pressure in order that the inRoN and outflox of air are equal. This reference pressure will depend on the temperature diff‘erence, the wind speed and the mechanical ventilation rate. However. in general. it will \‘ar! only slowly with these variables.

3. ANALYTICAL FORMULATION INFILTRATION

OF AIR

Using the arguments of the ver) short mtroductlon to the topic ofair infiltration given above. one may continue to define in a more precise form a set of assumptions relevant for the analytical treatment of air infiltration. Make the following propositions for the air flow, starting with the more fundamental ones. PI.

P2.

The local air Row depends only on the pressure difference across the leak path and the geometrical configuration of the leak path. The local flow direction is towards the tower pressure.

1’7. P3. 1’5.

rw

The local flow is zero if zero. The air HOM decreases pressure difference. The presence of exterior not influence the external The interior temperature

the pressure

difference

monotonically

ix

with the

leak path openings does pressure. is the same everywhere.

I-or constructing models. one wjitl also need some propositions uith a more detailed description of some pressure differences. as follows. I’:.

The pressure difference caused by the stack eftct i\ proportional to the indoor-~outdoor temperature difference and to the vertical distance of the exterior opening of a leak path from a reference height. f’X The pressure difference caused by wind is proportional to the wind velocity squared at some reference point. PC) Pressure drops across internal partitions are neglected; that is. we consider a unizone building interior. PI0 There is no interaction between the stack and the wind pressure difference apart from their influence on the reference pressure necessary for mass conservation to hold. (.‘onsider the flow dC, across a surface element dS of the envelope. The first five of the above propositions may be formulated as follows: dQ = ,4*F((Apl)*r(Ap)d.S.

(1)

I\ here dQ -= Hou rate [kg:s] Ip = pressure difference across the leak path [Pa] 4 = relative leakage area fi. = HO\V function, a monotonic function I;(O) = 0. [kg/(m’s)] ((_Y)= + I --I The next two propositions. may be summarized as

with

if.\->0 if\-CO. treating

the stack effect.

/J = air densit! 17 = Indoor outdoor temperature difference T = average of the indoor and the outdoor temperature !/ = earth gravitational constant : = height of the surface element considered _,/ - = height of reference level. independent of AT j J A 7’)is defined from equation (2). Similarly, the wind pressure (A/I),,,,,, = +C,-C’,,,

will be given by = c,(~‘)(C’~-C‘,,).

where

I = wind speed at a reference point C’,,= pressure coefficient at a point I‘,, = constant independent of the wind speed c,,,(r>)isdefined in equation (3).

(3)

Basic Air Injiltration Summing up different contributions, difference across an envelope section is

the pressure

AP = - c,(AT)(z - zo)+ c,(u) CC,- C,) -~,er(n,AT) = A$-pren

(4)

which also defines the pressure difference Ap’. To obtain the net air flow across the envelope, Qnet, one has to sum up all contributions from the building exterior surfaces and add the contribution from mechanical ventilation, Q,,,“. In the limit that the surface elements dS approach zero, the sum will go over into an integral over the surface, and one arrives at the following:

Qnet= ~~~*~W4,*444d~+

Qmv

(5)

Obviously, for sections of the envelope where there are no leak paths, A will be zero and there will be no contribution to the integral. Mass conservation requires that Qnet = 0. This is achieved by the reference pressure taking an appropriate value. The two constants z0 and C,, must be assigned numerical values. Assume that the air flow due to mechanical ventilation of equation (5) has been fixed. It is then often an advantage to choose z0 such that the reference pressure is zero when there is no wind, and C, such that the reference pressure is zero when there is no stack effect. The reference pressure may now be determined from equation (5) if the analytical form of the flow function F is known. So far, two factors have not been considered. It is assumed that there are no openings large enough to allow an inflow as well as an outflow through the same opening (for a review of effects of large openings, see [5]). It will also be assumed that rapid fluctuations of external pressures do not affect the time-averaged air exchange. Turbulence may have an effect of this kind, but so far rather few measurements have been carried out. For a review of this problem, see [6].

ment. If the flow function is a power of the pressure difference with an exponent between l/2 and 1, it may not be possible to use more than the first two terms in the series development of equation (6). The reason is that the integral in equation (5) may not be convergent for the third term in equation (6). Instead, higher order estimates of the reference pressure have to be obtained by an iterative procedure. One has to put pref = p$+p$‘,, where the first term on the right hand side is equal to the estimate of the interior pressure from the first two terms of equation (6). One may then expand equation (5) around Ap = 0 to obtain the second order correction, and so on.

5. PRESSURIZATION Pressurization, or depressurization, means that a building is subjected to an internal over- or underpressure, in general by using a powerful fan. The purpose may be to find leak paths or to determine the leakiness of the building. The pressure applied may be selected so large (or so small) that wind and stack effects may be neglected. There will then be either an inflow or an outflow of air through all leak paths. For simplicity, assume that the mechanical ventilation rate is zero, albeit this is not necessary. The over- or underpressure should be measured relative to the value of the reference pressure when there is no pressurization. Denote the enforced pressure by II (which may be positive or negative), and assume that everywhere on the building envelope ]Apl 5 III\. Also, assume that the flow function is a power of the pressure difference with an exponent p that may have a fixed value, or be a model parameter, that is, F(lT+ Ap) = F,, (Il+Ap)B. Expanding the flow function in inverse powers of II and inserting the result into equation (5) one obtains Q.et = III(Bc(Il)F,$I

dS+ IIIlB-‘fiF&4Ap dS

-+~I-I~a-2@I)/?(~- l)F&4(App)*dS+

4. DETERMINATION OF THE REFERENCE PRESSURE

It has been demonstrated that a model may be selected in such a manner that in a u*-AT plane the reference pressure is zero along both coordinate axes (see Fig. 1). One may expand the flow function around the reference pressure equal to zero as F(lA~p0 = F(IAP’I) -P~~+(AP’)F’(IAP’I) +P?Z’(IAP’~)/~+

97

.

(7)

The higher order terms will have coefficients that are integrals over powers of the pressure difference when there is no pressurization. Note that the t sign term will only appear in every second term. Keeping the first two terms of the expansion, equation (7) may be written as

with ,

(6)

where F’ is the derivative of F with respect to the argu-

AT

Fig. 1. Areas in the &AT plane where the reference pressure is approximately zero.

A, = F&4 dS; BL = /?F&-tAp dS.

(9)

The coefficient of the first term contains the integral of the leakage area A over the building envelope and is obviously a measure of the average leakiness of the building. It may be referred to as the leak area, AL. Assume that one wants to determine the leakage area from a set of pressurization data. Also, assume that a two-parameter model is to be used. There are then two obvious choices for a model. One possibility is to use the first two terms of equation (7) with a fixed value of /I, for example equal to l/2. Another possibility is to use as a model just the first term of equation (7), but let the exponent be a model parameter. The value given to the normalization factor F. deter-

mines the scale of the flow function, but it will not affect the value of the relative leakage area. For the model used in equations (8) and (9). a natural choice for p = I /2 is to put F, = ,,z. which gives the flow the right dimension without any further dimensional correction factors.

6. MODELS

OF

VENTILATION-INFILTRATION INTERACTION

From equation (3) it follows that the infiltration rate. Qm,. is given by the largest one of the absolute values of the influx or outflux of air, or Q,., = sup]JJ‘AF(]Ap])(i’( i Ap) dSi.

(10)

where the supremum is to be taken by varymg the sign of the argument of the H-function. and 1 ifu>O KY) = O ifs
Q,(An = SSAF(l(A~)st.i,kl)H((A~p),,.,,k) a.

(II)

where the pressure difference is the one given by equation (2) and

Qw(l,) = 1’SAF(l(A~r,,~,.dl)u((A~l,,,,.,,,dS‘.

(12)

where the pressure difference is the one given by equation (3). According to [lo], a relation describing the interaction between pressures of different origin should read as follows:

Pm = ~/(QE+Q~-~Q,QU,+?rlQm,.

(13)

where c(and ;’ are model parameters. Obviously, equation (13) is trivially satisfied when two of the three flows Q\. Q,, and Pm, are zero. One can demonstrate that as soon as more than one of them is different from zero. there will appear an error that grows with the magnitude of the flows, leading to a violation of the law of mass conservation Assume that the wind induced flow is not small, the mechanical ventilation rate is zero. and the stack induced flow is non-zero but small. It will be treated as a perturbation. Define a small stack induced pressure as (ApLuck= 6c,(AT)*(= - -,,)r

(14)

where the perturbation coefficient 6c, is as defined in equation (2) and such that almost everywhere on the building envelope the stack pressure is very small.

I(A~hx~l << l(~~Lp)w,A

except where the wind induced pressure is itself close to zero. However. at these points the total pressure will be nearly zero, and there will be almost no flow leaving a contribution when integrating over the building envelope. ,Also. from Section 3, the interior pressure is zero when there is just a wind driven pressure, so the interior pressure will remain small even if the stack pressure is nonzero, but still small. Consequently, one may expand the infiltration fow of equation (10) for a small stack effect &, and a small interior pressure dp, as

Q,,,, = Qm+~~~~‘(l(A~P)u,,~l)O((A~)~,“~) x (&,(z - 2,))- bp,) dS. For the circumstances will read

described

above, equation

Q;“, = Q:(&,(: - :J) + Q;-aQ,(Gc,(z

( 15) (14)

- z,,))*Qw. (16)

Taking the square of equation (15), identifying the infiltration flow of equation (15) with that of equation ( 16), cancelling identical terms on the right hand and left hand sides. and neglecting second order terms, one arrives at

= - c$4F(l&,(z

- zJ)O(&,(z - z”)) dS.

( 17)

The left hand side of equation (17), stemming from equation (15), does explicitly depend on the magnitude of the wind pressure. while the right hand side, originating from equation (17) does not. Thus, there is a mass flow that is not accounted for in equation (I 3); that is. equation (13) violates the law of mass conservation. This conclusion is independent of assumptions regarding the leak distribution or the analytical form of the flow function. The only exception is when the derivative of the flow function, F’. appearing in the left hand side of equation (17). is constant. The flow function is then proportional to the pressure difference, that is, the flow IS laminar. Similar conlusions may be shown to be valid for other choices of the dominant factor causing a pressure difference. The same conclusions will of course also be valid when none of the pressures is small. One may then summarize as follows. The calculation scheme of equation (13) violates mass conservation for all configurations of the leak area distribution, for all environmental conditions, and for all analytical forms of the flow function with the exception of when the flow is laminar.

7.

PRESSURIZATION

AND MODELS

FLOW

FUNCTION

At the end of Section 4 different two-parameter models possible to determine the leak area from a set of pressurization data were briefly discussed. One model is Qnc, = A,JH]‘%(H)+B,lfI~’ where the model parameters model is

‘.

(18)

are A, and B,, and the other

Basic Air Injiltration

99

0 MODEL 1: EXPONENT FIXEI TO >.AVERAGE ERROR 9.7%.

25

q MODEL

2: EXPONENT FREE PARAMETER. AVERAGE ERROI; 36.7 %

2

6

10

14

16

22

26

30

34

36

42

46

50

54

56

62

66

70

REPRODUCIBILITY ERROR [%] Fig. 2. Frequency distribution of the reproducibility error for two models, The reproducibility of the twoparameter models of equation (18) (Model 1) and equation (19) (Model 2) has been tested on a set of pressure measurements from 115 objects. The pressurization measurements have been carried out twice on each object. The model of equation (18), where the flow rate is to first order proportional to the square root of the pressure difference, gives an error of about 10%. For the model where the flow rate is an arbitrary power of the pressure difference, the error is almost four times as large.

where AL and b are the model parameters These two models have been employed to analyse a set of pressurization-depressurization measurement data from 115 objects, most of them row houses. For every object two different pressurization measurements have been carried

out; in most cases there was a period of about one year between the two pressurizations. Nominally, the objects have not been refurbished or altered in any other way between the two measurements, although small changes may have gone unnoticed in some cases. For all objects, the models have been used to predict the leak area, that is, the parameter Ai_. The model parameters have been determined by linear regression, for equation (19) after taking the logarithm of both sides. The difference in the predicted leak area between the two measurements is a measure of the reproducibility of the model for the object. The reproducibility of the two models has been determined for the 115 objects. The results are presented in Fig. 2 in terms of the two measured leak areas of an object, ALI and AL2

that is, in terms of the percentage relative deviation. The average deviation of the model of equation (20) is 37%, while that of the model of equation (19) where the exponent is fixed, is almost four times smaller, or less than 10%. Thus, the model where the exponent is a model

parameter has an unnecessarily large reproducibility error compared to at least one other model. A test of the model stability has been carried out by increasing the number of model parameters to three. This has been achieved by introducing a fixed correction term, z’, to the measured pressure by the substitution fl -+ fl + n’. For the model of equation (19) the parameter 71’in most cases takes a value not exceeding a few Pascal and only marginally reduces the reproducibility error. This indicates that the model is very stable and can be only a little improved by adding more model parameters. For the model of equation (20) the improvement is much larger; the average error is reduced from 37 to 25%. However, this is achieved by 7~’taking a value close to the smallest absolute value of the measured pressures, in many cases meaning a pressure correction of the order of 20 or 30 Pa. It is very unlikely that any of the measured data sets would have pressure measurement errors of this magnitude. Therefore, this model improvement must be rejected as spurious and rather indicates an instability of this model.

8. CONCLUSIONS A set of simplified physical principles upon which to base models of air infiltration has been formulated. These

M. D. Lyber
100

principles have been applied to give a general analytic form for air infiltration models. In particular, approximations based on series expansions of the general form have been given for the case that the flow is assumed to perform as a power of the pressure difference. A special case treated is that of pressurization. An examination of models currently used in air infiltration studies does not lead to a positive picture. One kind of model that is claimed to describe the interaction between different sources of air flows, such as stack induced, wind induced and flows by forced ventilation. turns out to violate the law of mass conservation. One model of the analytical structure of the flow function

gives a reproducibility error that is almost four times larger than that of another model with the same number of parameters. Furthermore, this second model is theoretically more palatable and from a non-dimensional point of view more tasteful. It is of course often a matter of taste which models to choose. However. using models that do not conform with basic physical principles or that have an inherent tendency to result in large errors is not likely to enhance the status of building science. Any model must mainly be judged according to how well it conforms with general principles, and only partially according to how well it may fit some data sets.

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3.

B . Air flow> through cracks. Building Swr~rs Enginrering. 1974. 42, 1233129. Etheridge, D. W., Crack flow equations and scale e&t. Building and Enrwonmen/, 1977, 12, 181. Baker, P. H.. Sharples, S. and Ward. I. C.. Air flow through cracks. Building and Enrironment, 1987,

Hopkins. L. P. and Hansford.

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Etheridge,

D. W. and Sandberg,

M.. A simple parametric

study of ventilation.

Building

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van der Maas. J.. Air flow through large openings in buildings. Technical Report from Annex 20. International Energy Agency. Swiss Federal Institute of Technology, 1992. Haghighat, F., Rao, J. and Fazio. P.. The Influence of turbulent wind on air change rates. a modellmg approach. Building and Erwironmenr, 1991. 26, 95- 105. Modera, M. and Peterson, F.. Simplified methods for combining mechamcal ventilation and natural infiltration. Lawrence Berkeley Laboratory Report 18955. 1985. Reardon, J.. Air infiltration modeling study. National Research Council of Canada, Report No. CR5446.3. 1989. Walker, I. S. and Wilson, D. J., AIM-Z: the Alberta air mliltration model. Department of Mechanical Engineering Report 71, University of Alberta. Alberta. 1990. Sherman. M. H.. Superposition in infiltration modelhng. Indoor Air. 1992. 2, 101-l 14.