Air and Heat Flow through Large Vertical Openings

 Rolf Jennings
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1 Air and Heat Flow through Large Vertical Openings J L M Hensen *, J van der Maas #, A Roos * * Eindhoven University of Technology # Ecole Polytechnique Federale de Lausanne After a short description of the physical phenomena involved, unified expressions are worked out descriing net airflow and net heat flow through large vertical openings etween stratified ones These formulae are ased on those of Cockroft for idirectional flow, ut are more general in the sense that they apply to situations of unidirectional flow as well The expressions are compatile with a pressure network description for multione modelling of airflow in uildings The technique has een incorporated in the flows solver of the ESPr uilding and plant energy simulation environment The relative importance of the governing variales (pressure difference, temperature difference and vertical air temperature gradients) is demonstrated y parametric analysis of energy performance in a typical uilding context It is concluded that vertical air temperature gradients have a major influence on the heat transferred through large openings in uildings and should e included in uilding energy simulation models Symols a i temperature profile coefficient for one i (K) i temperature gradient in one i (K/ m) C d discharge coefficient ( ) c p specific heat of air (J/ kgk) g acceleration of gravity (m/ s 2 ) h aperture height (m) M molecular mass of air (kg/ kgmole) m ij air mass flow from one i to one j (kg/ s) P pressure (Pa) q ij air volume flow from one i to one j (m / s) P ref reference pressure (Pa) R universal gas constant (J/ kgmole K) T temperature (K) u horiontal air velocity (m/ s) W aperture width (m) height coordinate (m) height of neutral level (m) 0 height of reference level (m) height of ottom of aperture (m) Φ ij heat flow from one i to one j (W) ρ air density (kg/ m ) ξ integration variale (m) α ( 0 )/h( ) C a P( +h)(pa) C P( )(Pa) hgk 1 T 1 ( 0 ) = C a C (Pa) K P ref M/ R (Pa kgk/ J) Z a Z INTRODUCTION C /2 a (m 2 Pa 1/2 ) C /2 (m 2 Pa 1/2 ) Airflows through doorways, windows and other large openings are important paths via which air (including moisture and pollutants) and thermal energy are transferred from one one of a uilding to another In case of large openings, the airflow at the top usually differs from the flow at the ottom of the opening Under certain conditions this may even result in idirectional flow through the opening In recent times there has een an increased interest in modelling airflow through large openings in uildings (eg Allard et al 1992) The current pulication seeks to e a asic contriution in this area y presenting and demonstrating a general * Eindhoven University of Technology # Ecole Polytechnique Federale de Lausanne Group FAGO, PO Box 51 La d Energie Solaire et de Physique du Batiment NL MB EINDHOVEN, Netherlands CH LAUSANNE, Switerland
2 approach for predicting airflow and heat flow through large vertical openings etween stratified ones Net Heat Flow when Zero Volume Flow For the mass and heat transfer through large vertical openings, Balcom et al (1984) and others like Boardman et al (1989) used the socalled isothermal one Bernoulli model According to Bernoulli, the maximum velocity u() in a large vertical opening etween two ones resulting from a static pressure difference (therey excluding any frictional losses) is given y: u() = 2 P ρ = 2 ρ ρ g( ) = = 2g T T( ) (m/s) [1] where indicates the height of the neutral level (ie the level at which the pressure difference P P 1 P 2 = 0 Pa), ρ ρ 1 ρ 2,and Tis the temperature difference etween one 1 and one 2ie T T 2 T 1 In this expression, it is implicitly assumed that T is independent of the height coordinate, ie that temperature gradients are equal and not too large When the toptoottom temperature difference over the opening is small compared to the asolute temperature, this approximation is highly accurate The heat flow Φ 21 from the warmer one (2) to the colder one (1) is carried y air flowing from 2 to 1 aove the neutral level The heat flow Φ 12 from the colder one (1) to the warmer one (2) takes place elow the neutral level These contriutions are given y: Φ 21 = c p C d W Φ 12 = c p C d W +h ρ 2()u()T 2 ()d (W) [2a] ρ 1()u()T 1 ()d (W) [2] Balcom s expression for the net heat flow through the aperture is otained y inserting the expression for u() into the expressions for Φ 21 and Φ 12, therey assuming that the temperature profiles in oth ones are linear, ie T i () = a i + i, and assuming that the net volume flow is ero, ie that the neutral level is located in the middle of the aperture The expression reads: Φ 12 +Φ 21 = C d ρ c p W g T h/2 T 1/2 * * T + 0 h( ) (W) [] and is good approximation when the thermal gradients in oth ones are equal, and not too large T 1 P ρ 1 1 positive flow direction h u() reference level o neutral level n ero level Figure 1 Definition of various parameters + h T 2 P ρ 2 2 = 0 From equation [] it is seen that y including the temperature gradients 1 and 2 the heat flow is increased y the factor h( ) In T practice 1 and 2 are not well known, ut the importance of this correction factor for small T shows the need to include the effect of stratification in uilding energy simulation environments like ESPr Mass Flow etween Stratified Zones In the following, an expression for the mass flow through a large opening separating two ones of different temperature and pressure is derived The general case is considered, ie there is a static pressure difference at reference level 0 etween one 1 and 2, and different vertical temperature profiles occur in the two ones These temperature profiles are assumed linear, though the temperature gradients can e different The situation is depicted in Figure 1 First we assume that conditions are such that the neutral level, ie the level at which the pressure is equal in ones 1 and 2, is located in the opening, so that idirectional air flow occurs The stack pressure difference etween a point at height and a point at reference height 0 is calculated from:
3 P() P( 0 )= 0 g ρ 2(ξ) ρ 1 (ξ) dξ (Pa) [4] gk 1 T 1 ( 0 ) ( 0) (Pa) [7] Although density variations due to pressure variations are negligily small, those resulting from temperature differences should e taken into account, especially when temperature gradients are large The air density in one i is inversely proportional to the temperature, namely: ρ i = P ref M (kg/ m ) RT i where P ref is some reference pressure, eg the atmospheric pressure, M is the molecular weight of air, and R is the universal gas constant To a very good approximation one may write: ρ i = K T i (kg/ m ) with K constant The expression for P() now reads: P() P( 0 )= = gk 1 0 T 2 (ξ ) 1 dξ (Pa) [5] T 1 (ξ ) Assuming a linear temperature profile T i () = a i + i one otains: P() P( 0 )= =gk 1 0 a ξ 1 dξ a ξ = =gk 1 ln T 2() 2 T 2 ( 0 ) 1 ln T 1() 1 T 1 ( 0 ) (Pa) [6] If the temperature gradient in oth ones is not too large, we have to a very good approximation: ln T i() T i ( 0 ) T i() T i ( 0 ) ( ), T i ( 0 ) ie the first order approximation is highly accurate Inserting the linear temperature profile gives: ln T i() T i ( 0 ) i 0 T i ( 0 ) so that in first order approximation: P() P( 0 ) ( ) This means that P() changes linearly with the height coordinate when temperatures in oth ones differ at the reference height 0 Note that the first order approximation results in equation [7] which is independent of the temperature gradients in oth ones, 1 and 2 Inserting the second order approximation, ie ln T i() T i ( 0 ) T i() T i ( 0 ) 1 T i ( 0 ) 2 T i () T i ( 0 ) T i ( 0 ) in equation [6] gives: P() P( 0 ) gk 1 T 1 ( 0 ) ( 0) gk T2 2( 0) 1 T1 2( 0) ( 0) 2 (Pa), showing that the temperature gradients give only a second order contriution to P() The relative error made y assuming that ln T i() T i ( 0 ) T i() T i ( 0 ) T i ( 0 ) ( ) is of the order of (T i () T i ( 0 )) / 2 T i ( 0 ) Even for a ceilingtofloor temperature difference of 6 K, this relative error will e 1% at most As in the mass flow calculation the square root of P() is integrated over the height of the opening, the resulting error will even e smaller In the following, this error will therefore e neglected If the opening through which the air flows extends from to + h, the pressure difference etween the two ones at ottom level will e equal to: P( ) = P( 0 )+ [7a] + gk 1 T 1 ( 0 ) ( 0 ) and at the top of the opening: P( + h) = P( 0 )+ gk 1 T 1 ( 0 ) ( + h 0 ) (Pa) (Pa) 2 [7] Now, if EITHER P( )>0 and P( + h) <0 OR P( )<0 and P( + h) >0 then the neutral level is located inside the opening and idirectional airflow occurs If P( ) and P( + h) have the same sign, or if one of them is ( )
4 ero, only unidirectional flow takes place According to Bernoulli s Law, a pressure difference P() results in a local air velocity u() proportional to the square root of P() Therefore, an infinitesimal volume flow d q through an element of height d in the opening can e written as: d q = Wu()d (m / s) [8] If we consider the case where T 2 > T 1 and where the pressures at reference level 0 in oth ones are such that the neutral level is located inside the opening (so idirectional airflow will occur), then the mass flow from 2 to 1 is equal to: +h m 21 = ρ 2d q = +h = C d W 2ρ 2 P()1/2 d (kg/ s) [8a] and the mass flow from 1 to 2 is equal to: m 12 = ρ 1d q = = C d W 2ρ 1 P()1/2 d (kg/ s) [8] where C d is an empirical constant In these expressions, the error made y placing 2ρ i in front of the integral sign is negligile ecause density variations are very small over the integration interval when compared to variations in P() Inserting the linear expression for P() into the integrals gives for m 21 and ṁ 12 the following expressions: m 21 = 2 C h dw 2ρ 2 C /2 a (kg/m ) [9a] m 12 = 2 C h dw 2ρ 1 ( C /2 C ) (kg/m ) [9] t where: hgk 1 T 1 ( 0 ) = C a C C a P( +h) (Pa) C P( ) (Pa) (Pa) Note that in the situation in figure 1, the pressure difference at the top level of the opening, C a P( +h) is negative, so that C /2 a is an imaginary numer To keep the value of m 21 real, the asolute value of C a should e taken It is convenient, however, to write the net mass flow of air through the opening as a complex quantity, ie: m net = ṁ 21 + ṁ 12 = 2 2 C dw h * * ρ 2 C /2 a ρ 1 C /2 (kg/ s) [10] This expression was first derived y Cockroft (1979) The net mass flow is a complex numer, of which the real part gives the flow from 1 to 2 and the imaginary part gives the flow from 2 to 1 It must e emphasied that the Cockroft formula for m net in the form given aove only holds for the special case depicted in Figure 1! There are two reasons why it is necessary to modify the expression: 1 st If one 1 on the left were the warmer one instead of the cooler one, m 12 would take place aove the neutral level, and ṁ 21 elow it The integration interval for oth contriutions would e interchanged, so that in Cockroft s expression, the term containing C a is now ṁ 12 and the term containing C is now ṁ 21 The formula now reads: m net = ṁ 12 + ṁ 21 = 2 2 C dw h * * ρ 1 C /2 a ρ 2 C /2 (kg/ s) [10a] However, the real part still gives the flow from 1 to 2 and the imaginary part still gives the flow from 2 to 1 2 nd If the external pressures in oth ones differ consideraly, the neutral level will shift to a height elow or aove (ie outside) the opening, so that the airflow ecomes unidirectional In this situation, one of the flow terms results from an integration over the entire opening, ie from to + h, while the other term is canceled In the situation of unidirectional flow, the pressure differences at the ottom and top of the opening, C and C a, have the same sign (unless one of them vanishes), so that C /2 a C /2 is either a real or a pure imaginary numer
5 By carefully comparing the expressions for m net which can e estalished for the different cases of unidirectional and idirectional flow, ie y "tuning" the temperature difference and the pressure difference etween one 1 (left) and one 2 (right), the following very convenient formula for m net which holds in all cases can e otained: m net = ṁ 12 + ṁ 21 (kg/ s) [11] m 12 = ρ 1 Re(Z a Z ) 0 (kg/ s) [11a] m 21 = ρ 2 Im(Z a Z ) 0 (kg/ s) [11] where: Z a 2 2 Z 2 2 C /2 a (m 2 Pa 1/2 ) C /2 (m 2 Pa 1/2 ) As the direction 1 2 is, y definition, the positive direction, the contriution ṁ 21 should e nonpositive, which explains the minus sign appearing in it The artificial complex quantites Z a and Z are introduced for convenience and have no physical meaning In the complex plane, Z a Z is located either on the positive real axis (when there is a unidirectional flow 1 2), on the positive imaginary axis (when there is a unidirectional flow 2 1), or in the first quadrant of the complex plane (when the flow is idirectional) When for a given temperature difference etween one 1 and 2 the external pressure difference P( 0 ) is continuously increased from highly negative to highly positive, Z a Z descries a smooth continuous curve Heat Flow etween Stratified Zones Just as for the mass flow, a convenient expression for the idirectional heat flow through a large opening etween stratified ones can e derived, giving Φ 12 and Φ 21 as real and imaginary parts of complex quantities Whereas mass flows are calculated y evaluating integrals of the type: ρ i()d q (kg/s) heat flows are calculated y evaluating integrals of the type: c pt i ()ρ i ()d q = = c p C dw 2ρ i ()T i () P()d in an analogous way (W) To e ale to evaluate these integrals analytically for linear temperature profiles T i () = T i ( 0 ) + i ( 0 ), the integrand P() aove, should e of the form 2ρ i ()T i () [polynomial] P(), which means that 2ρ i ()T i () should e approximated y its "est linear fit", which is (as can e checked easily): 2ρ i ( 0 ) [T i ( 0 ) + 1/2 i ( 0 )] ( kg/m K) Evaluation of the integrals is a rather laorious task, which will not e documented here due to space constraints (we will gladly provide the full derivation to readers wishing to otain this material) However, when these integrals are worked out in the same way as was done for the mass flows, we otain convenient expressions for the heat flows Φ 12 and Φ 21, namely: Φ net =Φ 12 +Φ 21 (W) [12] Φ 12 = c p ρ 1 * [12a] *Re( T 1a ( 0 )Z a T 1 ( 0 )Z ) 0 (W) Φ 21 = c p ρ 2 * [12] *Im( T 2a ( 0 )Z a T 2 ( 0 )Z ) 0 (W) where: T ia ( 0 ) T i ( 0 ) i h C a + α (K) for i = 1, 2 T i ( 0 ) T i ( 0 ) i h C + α 5 2 (K) for i = 1, 2 in which α is a dimensionless reference height (α ( 0 )/h), and the densities ρ 1, respectively ρ 2, are evaluated at the reference level 0 APPLICATION Equations [11] and [12] have een incorporated into the large vertical openings component of the flows solver (Hensen 1991) of the ESPr uilding and plant energy simulation environment (Aasem et al 199) This particular solver is ased on a nodal network mass alance approach, and can e used  amongst others  for multione modelling of airflow in uildings In the following some calculation results are given, which demonstrate the relative importance of the flow governing variales y means of parametric analysis For this we started from a asecase involving two
6 uilding ones connected y a door opening with width W = 10 m, heigth h = 20 m, and reference height α = 05 The discharge coefficient C d was assumed to e 050 Various cominations of pressure difference, temperature difference, and vertical air temperature gradients were considered m net [kg/s] T1  T2 = 10 K T1  T2 = 5 K T1  T2 = 0 K m [kg/s] P1  P2 [Pa] 10 m 12 m 21 m net P1  P2 [Pa] Figure Net mass flow rate ṁ net (kg/ s) vs pressure difference P 1 P 2 (Pa) as a function of asolute temperature difference T 1 ( 0 ) T 2 ( 0 ) (K) Figure 2 Mass flow rate ṁ (kg/ s) vs pressure difference P 1 P 2 (Pa) for T 1 ( 0 ) T 2 ( 0 ) =10K direction of 2 1 when P changes from positive to negative Figure 2 shows the mass flow results as a function of the pressure difference etween one 1 and one 2, for an asolute temperature difference of 10 K From equation [11] follows that the temperature gradients do not influence the mass flows Flow m 12 will e aove the neutral level when one 1 is the warmer one, otherwise it will e elow neutral level From the results it is clear that there is only a small and in P for which idirectional flow occurs It should e noted however that the corresponding airflows are quite large; eg for P = 0 25 Pa m 12 is 0 75 kg/ s or 2250 m / h This implies that there will also e a large heat flow associated with that If we would make graphs for the heat flows Φ (and assuming that there are no vertical temperature gradients), then the shapes would e quite similar to the ones in Figure 2 Oviously the yaxis values will e different and would range from 600 kw to 600 kw for the range of pressure and temperature differences in Figure 2 Figure shows the net mass flow results for various asolute temperature differences At very low or ero temperature difference there will only e unidirectional flow and the airflow will e similar to the flow through a large orifice Figure indicates that an increase in temperature difference "smooths" the transition from flow in the direction of 1 2 to the net heat flow [W] T1 = 0 C T2 = 20 C 1 = 4 K/m 1 = 2 K/m 1 = 0 K/m 1 = 2 K/m 1 = 4 K/m 1 = [K/m] Figure 4 Net heat flow Φ net (W) vs vertical temperature gradient 2 (K/ m) for one 2 as a function of gradient 1 (K/ m) for one 1; P( 0 ) = 0 (Pa) As indicated aove, the mass flows are not influenced y the vertical temperature gradients This is clearly not the case for the heat flows as can e seen in Figure 4 This figure shows the net heat flow (ie Φ net =Φ 12 +Φ 21 ) etween one 1 and one 2, assuming that the reference temperature in
7 one 1 is 0 C and is 20 C in one 2 For this case there is no pressure difference at reference height; ie P( 0 ) = 0 Pa From Figure 4 follows that net heat flow for this case would e 1600 W when the temperature gradients would not e taken into account (ie 1 = 2 = 0 K/ m) If there would only e a gradient in one of the ones (eg 1 = 0 and 2 0) then (for this particular case) the change in net heat flow is aout 100 W for each unit change in vertical temperature gradient If oth gradients are nonero then the changes can even e igger as can e seen in Figure 4 For instance for a common case where there are vertical temperature gradients of 1 K/ m in each one, then the net heat flow would e 1800 W instead of 1600 W, which is a difference of 12% system, Doctoral dissertation Eindhoven University of Technology (FAGO) (ISBN X) EO Aasem, JA Clarke, JW Hand, JLM Hensen, CEE Pernot, and P Strachan 199 ESPr A program for Building Energy Simulation; Version 8 Series, Energy Simulation Research Unit, ESRU Manual U9/1, University of Strathclyde, Glasgow CONCLUSION A general solution is presented for predicting (net) airflow and (net) heat flow through large vertical openings etween stratified uilding ones The solution proved to e compatile with a nodal network description of leakages for multione modelling of airflow in uildings By parametric analyses, the relative importance of the flow governing variales is demonstrated From the results it is clear that  apart from the other governing variales like pressure and temperature difference  vertical air temperature gradients have a major influence on the heat exchange y interonal airflows References F Allard, D Bienfait, F Haghighat, G Lieecq, J van der Maas (Ed), R Pelletret, L Vandaele, and R Walker 1992 Air flow through large openings in uildings, IEA Annex 20 sutask 2 technical report, LESOPB, EPFL, Lausanne JD Balcom, GF Jones, and J Yamaguchi 1984 Natural convection airflow measurement and theory, in Proc 9th National Passive Solar Conference, pp , Columus, OH CR Boardman, A Kirkpatrick, and R Anderson 1989 Influence of aperture height and width on interonal natural convection in a full scale, airfilled enclosure, in Proc National Heat Transfer Conf, Philadelphia Sumitted to ASME J Solar Energy Eng JP Cockroft 1979 Heat transfer and air flow in uildings, PhD thesis University of Glasgow JLM Hensen 1991 On the thermal interaction of uilding structure and heating and ventilating
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