experimental and theoretical case study on cross ventilation · the calculation of ventilation...

Nordic Journal of Building Physics Vol.2 1999, Schmidt, Maas, Hauser

Experimental and Theoretical Case Study on Cross Ventilation– designing a mathematical model

SUBMITTED: November 1998.REVISED: January 1999.PUBLISHED: June 1999.

Dietrich Schmidt, Researcher.Department of Construction PhysicsUniversity of KasselD-34109 Kassel, Germanye-mail: [email protected]

[email protected]

Anton Maas, Dr.-Ing.Department of Construction PhysicsUniversity of KasselD-34109 Kassel, Germanye-mail: [email protected]

Gerd Hauser, Univ.-Prof. Dr.-Ing.Department of Construction PhysicsUniversity of KasselD-34109 Kassel, Germanye-mail: [email protected]://www.bpy.uni-kassel.de/

KEYWORDS: Natural cross ventilation, Tracer gas measurement, Mathematical modelling.

SUMMARY: Natural ventilation through windows and other openings in the building envelope is still the mostcommon way to ensure the necessary fresh air supply in buildings. To date, there are few validated models forthe calculation of ventilation rates through large openings. In particular, there is a lack of information aboutnatural cross-ventilation. On the basis of an empirical study a mathematical model for the calculation ofcross- ventilation in buildings is presented in this paper.In order to investigate the different influences on the airflow through large openings in the building envelope,measurements of air change rates under natural conditions are carried out for certain types and configurationsof openings. By using the collected data, the influences of the buoyancy and wind driven airflow are analysed.The influences of the two phenomena on each other are determined in detail. Based on published numericalmodels for single-sided ventilation and knowledge about airflow patterns through windows, a new model forcross ventilation is derived. Again, this model is validated with the measured data. By using this new model,convection heat loss and thermal loads of buildings can be calculated more specifically.

1 IntroductionThe design of energy efficient buildings requires a balance between a good thermal performance (effectiveinsulation, appropriate selection of heating, cooling and lighting techniques) and an acceptable quality ofindoor climate (indoor air quality, ventilation rate and thermal comfort). Vast efforts have been made toimprove the thermal conditions of buildings in the last decades, for example by constructing nearly perfectbuilding envelopes and good thermal simulation tools. But knowledge of the real airflow in buildings ventilatedby natural means is still lacking. To achieve an acceptable air quality and to minimise the heat losses caused bythe necessary air exchange, estimation methods for the airflow are needed.

In most buildings the necessary air exchange is still provided by means of natural ventilation. To calculate theairflow under the condition of natural ventilation, some models for single sided ventilation have been published(Phaff 1980, Feustel 1990 Maas 1995 and Allard 1998). Some theoretical investigations have also beenconducted into the mechanism of natural air exchange through window openings (Daler 1984 andAllard 1998). The driving forces for the air exchange in buildings under the conditions of natural ventilation


are differences in wind pressure on the facades and differences between inside and ambient air temperature. Afew attempts to set up different mathematical models have been published on this basis (Phaff 1980 and Allard1998). Most of these investigations used tracer gas measurements to validate the airflow models.

This paper presents a new model calculating the airflow under cross ventilation conditions on the basis ofknown estimation methods for the airflow under single-sided ventilation circumstances. The model is based onthe analysis of the different effects on the air exchange rate using measurement data from tracer gasinvestigations in an experimental building.

2 Experimental investigations

ϑ ϕϕϑ ϑ

ϕϑ

ϕϑp

N2O N2SF6

weather station

Tracergas measurement system

testroom

ventilation system

ϕ ϑSensor location: temperaturehumidityp pressure

Fig. 1: Measurement setup inside the experimental building.

All measurements are carried out inside an experimental building at the University of Kassel / Germany. Thebuilding consists of a 58 m³ testroom and thermal buffer room on top of and under the testroom. The measuredairtightness of the testroom is n50=0,1 1/h. The total volume of the cross ventilated zone is 70 m3 . There is adouble casement tilting window in the south facade and a single casement tilting window in the north wall toallow measurements to be taken under cross ventilation conditions. The technical equipment is mainly installed


in the cellar under the testroom (see Fig. 1). A weather station beside the house provides the meteorologicaldata of interest, namely temperatures, windspeed and direction. The ambient temperature and pressure ismeasured close to the northern facade at a height of 2 m, the wind is measured 8 m above the roof level at thenorth-east corner of the experimental building. The sensors for indoor temperature and humidity are located inthe middle of the testroom with a distance of 0.3 m from the ceiling.

2.1 The experimental set-upTo estimate the rate of air exchange or the airflow through the testroom, the constant concentration method isused for tracer gas measurements. A control device provides a constant concentration of tracer gas inside thetestroom. The necessary tracer gas flow into the room is proportional to the airflow through the testroom. Thus,the air change rate can be easily calculated. This is true for long term measurements under natural conditionsas well. Presently, equipment for two different gases, N2O and SF6, is available. The four gas samples are takenfor one tracer gas at a distance of approximate 1 m from the eastern and western walls of the testroom at aheight of 0.25 m and 2.20 m respectively 1.20 m and 2.00 m (see Fig. 1).

To calibrate the tracer gas equipment a ventilation system is installed into the building. This system is able totransport an incoming and outgoing airflow of 50 up to 1200 m³/h separately, equivalent to an air change rateof 0,85 to 20 1/h. Furthermore, the mechanical ventilation system is used as reference. Detailed investigationsin order to optimise the tracer gas control system and to compare results of volumetric airflow ratemeasurements of different systems are done (see Fig. 2).

0

300

600

900

1200

0 200 400 600

Time

Propeller anemometer

Tracergas N2O

Tracergas SF6

[min]

[m³/h]

Fig. 2: Comparison of volumetric airflow measured with different techniques.

2.2 Measurement programmeA series of measurements was undertaken under natural conditions with the set-up described in section 2.1.During the project, valid data of approximately 1300 h were recorded. In Fig. 4 the recordings from the 18th ofFebruary 1998 are shown as an example.

From 00:00 to 08:00 hours only the northern tilting window was open in default position (see Table 1),representing single-sided ventilation conditions. After 8:00 o’clock the southern window was open in defaultposition as well, cross ventilation is recorded. The increasing airflow and dynamics of the airflow recording isremarkable. The characteristic airflow patterns for cross ventilation proved to be different from the single sidedone. It is obvious the airflow was more dependent on the windspeed than on the temperature differences


between inside and outside. To be able to investigate the dependency of the airflow, different cases of openingpositions of the windows were chosen.

Table 1: Variations of opening positions for the measurements.

s

α

a

Fig. 3: Definition of the opening position and –angle on the tilting window.

Table 2: Window sizes and description of the opening positions (see. Fig. 3).

north faced window south faced window

window breadth B 0,92 m 0,82 m

window height H 1,21 m 1,12 m

opening width a 15,5 cm 5 cm 13,5 cm 5 cm

clear opening width s 10,4 cm 2,5 cm 8,9 cm 2,6 cm

opening angle α 7,30° 2,37° 6,87° 2,56°

case opening width anorth-facing window

opening width asouth-facing window

default 15,5 cm 13,5 cm

single sided reduced 5 cm 13,5 cm

two sided reduced 5 cm 5 cm

H

B

α


0

300

600

900

[m³/h]

0

3

6

9[K]

0

1

2

3

4[m/s]

0

90

180

270

360

00:00 06:00 12:00 18:00

Daytime

[°]

[h] 24:00

Fig. 4: Recordings of the essential parameters influencing the air change rate at 02/18/1998.


3 Measurement resultsTo compare the validity of the collected data with the results from earlier published investigations, the airexchange induced through wind and buoyancy are considered separately.

3.1 Influence of temperature differencesGeneral considerations show the buoyancy-driven air exchange is proportional to the square root of thedifference of the inside and outside temperature [Phaff 1980]. Maas and Daler also used this model to fit theirresults for single sided ventilation at very low wind speeds [Maas 1995, Daler 1984]. To compare the results ofthe present study with those of Maas and Daler, their models have to be adapted to the same test conditions asdescribed in chapter 2.1. The calculated airflows through the two windows are simply added.

0

50

100

150

200

250

0 5 10 15 20

Temperature difference

Daler

Maas

Correlation

Measurement

[K]

[m³/h]

Fig. 5: Volumetric airflow rate vs temperature difference. Default case, 1h mean values, wind speed u<0,3 m/s.

Fig. 5 shows the measured data and model results for the default case as an example. The results calculatedaccording to Daler differ by about +14.8%, the results according to Maas differ by about -1.7% from the presentcorrelation. Maas has undertaken his investigations in the experimental building in Kassel as well. This mayexplain the good agreement. The scatter of the test results could be explained by small fluctuating wind-drivenairflows. The effects of very low windspeeds under the conditions of cross ventilation are remarkable.


0

50

100

150

200

0 5 10 15 20

Temperature difference

Correlation default

Default according to Maas

Correlation one-sided reduced

One sided reduced according to Maas

[K]

[m³/h]

Fig. 6: Volumetric airflow vs temperature difference. Different cases. 1h mean values, wind speed u<0,3m/s.

The comparison of the adaptations of the measurement from Fig. 5 to the measurement case single sidedreduced is shown in Fig. 6. The fits for both cases are close to the results according to Maas.

3.2 Influence of wind speedFor a more detailed investigation of wind-induced air exchange, the characteristics of the wind has to bestudied closely. The experimental building is sited in an urban environment and thus surrounded by otherbuildings with the same height at a distance of some hundred meters, see [Maas 1995]. For furtherinvestigations, the wind direction is split up in four main directions. In addition to the main directions, thesouth-west direction is considered, too.

North 315° - 45°East 45° - 135°South 135°- 225°West 225°- 315°South-West (marked) 180° - 270°

(Wind North 360° => Northerly Wind, from 360°)


0 2 4 6

0°30°

60°

90°

120°

150°180°

210°

240°

270°

300°

330°

[m/s]

(North)

Fig. 7: Wind speed vs wind direction at the measurement location. Measurement case default. 1h mean values,temperature difference 2K<∆ϑ<4K.

For the diagrams shown in Fig. 8 and Fig. 9, data for temperature differences between 2K and 4K are selecteddue to availability. The airflow is not significantly influenced by such small temperature differences as 2K to4K.

0

400

800

1200

0 1 2 3 4

Wind velocity

Correlation

Measurement

[m/s]

[m³/h]

Wind direction: South

Fig. 8: Volumetric airflow rate vs wind speed for wind from the south. Default case, 1h mean values,temperature difference 2K<∆ϑ<4K.

The figures show the subordinate effect of the wind direction on the airflow. The wind speed is the dominantparameter.


The airflow characteristics for main wind direction south-west are nearly identical to the wind from the south.

0

400

800

1200

0 1 2 3 4

Wind velocity

Correlation

Measurement

[m/s]

[m³/h]

Wind direction: West

Fig. 9: Volumetric airflow rate vs wind speed for wind from the west. Default case, 1h mean values,temperature difference 2K<∆ϑ<4K.

3.3 Simultaneous temperature and wind influence

0

400

800

1200

0 1 2 3 4

Wind velocity

∆ϑ=0− 2Κ

∆ϑ=15− 18Κ

[m/s]

[m³/h]

Fig. 10: Volumetric air flow rate vs wind speed for southern wind. Measurement case default, 1h mean values.

The airflow dependency on both wind and temperature difference is shown in Fig. 10. Daler [Daler 1984] andMaas [Maas 1995] as well as Warren [Warren 1977] confirm that these effects cannot be superposed forreliable results. No better solution to estimating the resulting airflow has yet been found. For cross-ventilation,


the temperature difference has a small effect on the airflow rate. Even for particularly large differences such as15 to 18K, the effect is very low.

The measurement results shown in Fig. 10 indicate that the temperature difference even has a contrary effect onthe airflow. An increased temperature difference leads to a decreased airflow.

4 Theoretical investigationsIn the following, a model is set up for single sided ventilation and is then extended for the special case of crossventilation.

4.1 Modelling approachPhaff presents a model estimating the airflow for natural ventilation as a function of the wind velocity and thedifference of inside and ambient air temperature [Phaff 1980]. This more general airflow model, the de Gidsand Phaff method, integrates the most important parameters in an empirical correlation [Allard 1998]. Eventhe airflow occurring at open windows, when wind or buoyancy effects are absent, could be predicted. Again,this model is modified by Maas [Maas 1995] and based on:

flowIn uAV ⋅⋅ζ=& ( 1)

With a flow resistance ζ , a clear opening area A and a velocity uflow the model can be formulated such that,

sidedone,Ex322

1IIn VCHCuCA21

V && =+ϑ∆⋅⋅+⋅⋅⋅⋅Θ= ( 2)

In this formula the flow resistance is represented by window airflow ratio Θ , describing the opening position ofthe window, as defined below. The clear opening area is the total window opening AI. The whole square rootterm presents a velocity. The local windspeed u is used to take the wind introduced air exchange into account,together with a parameter C1. The height of the window H and the difference between room and ambienttemperature ∆ϑ are used to count for the buoyancy driven airflow, together with the parameter C2. The lastfactor C3 is equivalent to an effective pressure that provides ventilation in the absence of stack effect or steadywind. Experimental results have shown that fluctuating effects are responsible for that airflow. Fluctuatingflows are attributed to the turbulence characteristics of the incoming wind and / or to turbulence induced by thebuilding itself. This turbulence in the airflow causes simultaneous positive and negative pressure fluctuations ofthe inside air [Allard 1998]. These parameters Cx are used to adapt the calculation to the measurement results.The different coefficients are calculated by a four-dimensional Chi-square-optimisation according to theLevenberg-Marquardt-algorithm, described in [Press 1992].

A definition of a window airflow ratio is

( )

( )°=αα

=Θ90A

A

eq

eq ( 3) where ( )2

S2

I

eq

A1

A1

1A

+=α ( 4)

The figures below show the geometry of a tilting window:


Al

α

H

B

Fig. 11: Definitions of the geometry opening dimensions on a tilting window.

HBAI ⋅= ( 5)

+

α⋅⋅

α⋅⋅= B

2cosH

2sinH2A S ( 6)

Unlike the single sided ventilation, a part of the whole airflow rate is caused by the airflow through thebuilding. The other part is caused by air exchange processes, like the ones at single sided ventilation. Theexchanged airflow at every window has to be added to the airflow through the entire building [Daler 1984].With the mass flow of air

ThExcross mmm &&& += ( 7)

Using the simplification that the air density is nearly constant for small temperature differences .)const( ≈ρ , itfollows that

ThExcross,In VVV &&& += ( 8)The approach presented in equation 2 above is extended and adapted to the cross ventilation. The volumetricairflow through each window is added to give

2,Ex1,ExEx VVV &&& += ( 9)

32,122

12I21I1Ex CHCuCA21

A21

V +ϑ∆⋅⋅+⋅⋅

⋅⋅Θ+⋅⋅Θ=& ( 10)

For the airflow passing both windows, an approach for two openings behind each other in flow direction ischosen [Daler 1984]:

ρ∆⋅

⋅Θ+

⋅Θ

= tot

2

2I2

2

1I1

Thp2

A1

A1

1V& ( 11)

The assumption that the total pressure difference over the entire building is caused by wind only leads to

24

tot uCp2 ⋅=

ρ∆⋅

( 12)

The whole model for cross ventilation with two windows in opposite walls of a building turns out to be

2

2I2

2

1I1

24

32,122

12I21I1cross,In

A1

A1

uCCHCuCA

21

A21

V

⋅Θ+

⋅Θ

⋅++ϑ∆⋅⋅+⋅⋅

⋅⋅Θ+⋅⋅Θ=& ( 13)

Window-opening area

Clear window-opening area

AS


4.2 Adaptation of the mathematical modelThe estimation of a window airflow ratio Θ , which presents the real relation between opening position andresulting airflow, is quite unreliable. Section 4.1 gives a calculation method for these ratios on the basis of thegeometrical dimensions of the windows. But these ratios Θ cal do not give a good estimation of the predictedairflow, only present a way to achieve the necessary values direct from the geometric dimensions. So anempirical method is introduced to estimate window airflow ratios Θ by the help of measurements. Because themeasured window airflow ratios Θ give an even better model fit than the calculated ratios, the measured dataare used for the model correlation. To evaluate the different ratios, tracer gas measurements using theconcentration decay method are carried out [Maas 1995], and the ratios are calculated as

( )( )°=α

α=Θ90V

V

In

In&

&( 14)

For the presented project, a correlation of the measured ratios is used.

0

0,1

0,2

0,3

0 2 4 6 8

Opening angle

Measurement

Correlation

Calculation

[°]

[-]

Fig. 12: Airflow ratios Θ of a tilting window vs opening angle α [Maas 1995].

Using these measured window airflow ratios Θ , the coefficients Cx could be calculated using the Levenberg-Marquardt-algorithm on equation 13.

The results of these optimisations are given in Table 3:

Table 3: Calculated coefficients of the modelling equations for different cases.


case C1 C2 C3 C4

./. m/s²K m²/s² ./.

default 0,01965 1,896.10-3 0,01706 0,01946

single-sided reduced 2,612.10-5 1,331.10-4 8,768.10-4 0,01061

two-sided reduced 5,558.10-3 1,737.10-4 2,949.10-3 3,734.10-3

0

250

500

750

1000

0 250 500 750 1000

Measured airflow[m³/h]

[m³/h]

Fig. 13: Comparison of the measured and calculated airflow for all cases. 1h mean values.


0

1

2

3

0 250 500 750 1000

Measured airflow

Rel. error of Points

Polynomial trendline

[m³/h]

[-]

Fig. 14: Absolute value of relative error of the measured airflow for all cases. 1h mean values.

5 ExampleTo give an example of a calculation, values are taken from the example day Feb. 18th 1998. These are 6 minmean values from 10:38 to 10:44 o’clock. At this time both windows, the southern indicated by 1, the northernone by a 2, are in opening position default (see Table 1).

Temperature inside the test room: C8.11TI °=

Ambient temperature: C00.6TA °= K8.5TTT AI =−=∆

Wind velocity: sm46,2u =

Measured airflow: h³m

meas,In 6,523V =&

The geometric dimensions of the windows and window openings are taken from Table 1 and

Table 2. The resulting measured window opening ratios according to Fig. 12 are

( ) south215.030.7meas,1 =°Θ ( ) north226.078.6meas,2 =°Θ

The window opening areas are, according to Fig. 11 and equation 5

21I m918.0m12.1m82.0A =⋅= 2

2I m11.1m21.1m92.0A =⋅=which gives the following airflows:

The exchanged airflow at window 1, in the south façade:

h³m2.137C

2HH

CuCA21V 3

212

211I11Ex =+ϑ∆⋅+⋅+⋅⋅

⋅⋅Θ=&

The exchanged airflow at window 2, in the north façade:

h³m5.174C

2HH

CuCA21V 3

212

212I22Ex =+ϑ∆⋅+⋅+⋅⋅

⋅⋅Θ=&


The flow through the building and both windows

h³m0.192

A1

A1

uCV

2

2I2

2

1I1

24

Th =

⋅Θ+

⋅Θ

⋅=&

The total airflow is

h³m7.503VVVV Th2Ex1Excal,In =++= &&&&

In this case the relative error becomes

( ) %80.3V

VVf

meas

meascalmeasured −=−=Θ &

&&

Using calculated window opening ratios (equation 3 and 4), instead of the measured ones, the resulting airflowis

( ) south29.030.7cal,1 =°Θ ( ) north301.078.6cal,2 =°Θ

h³m5.674VVVV Th2Ex1Excal,In =++= &&&&

For this different case the relative error becomes

( ) %82.28V

VVf

meas

meascalcalculated +=−=Θ &

&&

6 ConclusionsThe theoretical investigations and measurements presented in this paper lead to the following conclusions:

• As long as buoyancy-forced air exchange can be provided, the results of investigations of single sidedventilation are valid for cross-ventilation with openings at the same height as well. The various airflowsthrough the windows are simply superposed.

• The dependency of wind-forced air exchange with wind direction is small. In the studied case the airflow ismainly dependent on wind speed. Other studies showed a dependence of the wind direction or pressurecoefficients cp at the openings on the airflow, but prediction methods like the ASHRAE and the Aynsleymethod do not take the wind direction into account [Allard 1998].

• Both temperature difference and wind have a significant influence on the airflow. But results obtained showthat the influence of temperature difference is subordinate. Measurements indicate, in some cases, that anincrease in temperature difference leads to a decreased airflow.

• A mathematical model to calculate the airflow is determined with measured data. This model describes thequantity of air exchanged as a function of: local wind velocity, the difference of the inside and the ambienttemperature and a constant term of turbulence effects / diffusion in the window openings. Due to the scatterof the measured results, there are still variations between measured and calculated airflows. For wideopened windows and small airflows, the relative error of calculated data is between -62.1% and +347.7%.

• The model is only valid in the case of cross ventilation with tilting windows at the same height.


7 AcknowledgementThis report presents a part of the research project ”Meßtechnische und theoretische Untersuchungen zumLuftaustausch in Gebäuden”, which has been carried out at Kassel University / Germany from September 1997to August 1998. The authors thank the Deutsche Forschungsgemeinschaft (DFG) for financing the project.

8 List of symbolsnotation dimension Meaning

A m² Area

a m window opening width

B m breath

C1 - wind speed coefficient 1 of modelling approach

C2 m/s²K temperature coefficient 2 of modelling approach

C3 m²/s² turbulence coefficient 3 of modelling approach

C4 - wind speed coefficient 4 of modelling approach

H m height

p Pa pressure

s m clear opening width

u m/s wind speed

V& m³/h airflow

α ° window opening angle

∆ϑ K temperature difference

ζ - flow resistance coefficient

Θ - window airflow ratio

Indicescal calculated

cross cross ventilation

eq equivalent

Ex exchanged

I inner

In incoming

meas measured

S at the side

Th through the building

tot total


9 ReferencesAllard F. (ed.)(1998). Natural Ventilation in Buildings. A design Handbook. James & James. Ltd. London.

Daler R., Hirsch E., Haberd, F., Knöbel U. and Krüger W. (1984). Bestandsaufnahme von Einrichtungen zurfreien Lüftung im Wohnungsbau, Bundesministerium für Forschung und Technologie, Researchreport T 84-028.

Feustel H. E. and Raynor-Hoosen A. (1990). Fundamentals of the multizone air flow model - COMIS,Technical Note AIVC 29, Air Infiltration and Ventilation Centre.

Maas A. (1995). Experimentelle Quantifizierung des Luftwechsels bei Fensterlüftung. Dissertation, Universityof Kassel, Department of architecture.

Phaff J. C., de Gids W. F., Ton J. A., van der Ree D. V. and Schijndel L. L. M. (1980). Ventilatie vonGebouwen. Onderzoek naar de Gevolgen van Het Openen van een Raam op Het Binnenklimaat von een Kamer.Rapport C 448, Instituut voor Milieuhygiene en Gezondheidstechniek, Delft, Netherlands.

Press W. H., Teukolsky S. A., Vetterling W. T. and Flannery B. P. (1992). Numerial recipes in FORTRAN:The art of scientific computing. Cambridge University Press.

Van der Maas J. (ed.)(1992). Air flow through large openings in buildings. Annex 20: Air Flow Patternswithin Buildings. Subtask 2: Air Flows between Zones. International Energy Agency, Energy Conservation inBuildings and Community Systems Programme.

Warren P. R. (1977). Ventilation through openings on one wall only. Proc. of the International Conference onHeat and Mass Transfer in Buildings, Dubrovnik.

Nordic Journal of Building Physics 1999.Available at http://www.ce.kth.se/bim/bphys

experimental and theoretical case study on cross ventilation · the calculation of ventilation...

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