A simplified mathematical approach for modelling stack ventilation

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Building and Environment 71 (2014) 121e130
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Building and Environment
journal homepage: www.elsevier .com/locate/bui ldenv
A simplified mathematical approach for modelling stack ventilation in
multi-compartment buildings
Andrew Acred a, Gary R. Hunt b,*
aDepartment of Civil and Environmental Engineering, Imperial College, London SW7 2AZ, UK
bDepartment of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
a r t i c l e i n f o
Article history:
Received 23 May 2013
Received in revised form
15 September 2013
Accepted 19 September 2013
Keywords:
Natural ventilation
Multi-storey buildings
Multizone
Atrium
Solar chimney
Stack effect
* Corresponding author. Tel.: þ44 1223768449.
E-mail address: gary.hunt@eng.cam.ac.uk (G.R. Hu
0360-1323/$ e see front matter � 2013 Elsevier Ltd.
http://dx.doi.org/10.1016/j.buildenv.2013.09.004
a b s t r a c t
A simple mathematical model of stack ventilation flows in multi-compartment buildings is developed
with a view to providing an intuitive understanding of the physical processes governing the movement
of air and heat through naturally ventilated buildings. Rules of thumb for preliminary design can be
ascertained from a qualitative examination of the governing equations of flow, which elucidate the
relationships between ‘core’ variables e flow rates, air temperatures, heat inputs and building ge-
ometry. The model is applied to an example three-storey office building with an inlet plenum and
atrium. An examination of the governing equations of flow is used to predict the behaviour of steady
flows and to provide a number of preliminary design suggestions. It is shown that control of venti-
lation flows must be shared between all ventilation openings within the building in order to minimise
the disparity in flow rates between storeys, and ensure adequate fresh air supply rates for all
occupants.
� 2013 Elsevier Ltd. All rights reserved.
1. Introduction
Passive stack ventilation is a popular energy-saving design
feature in modern architecture. Significant architectural features
such as atria, solar chimneys and double façades are often incor-
porated into large building designs with a view to assisting stack
ventilation and thereby delivering a comfortable internal envi-
ronment. Such multi-compartment buildings pose a particular
design challenge due to the interaction between heat and air flows
through different building zones.
Computational tools such as multizone building simulation
software [1e4] and computational fluid dynamics (CFD) [5e7] are
routinely used to tackle these complex design problems, and are
capable of detailed, multi-variable analysis of heat and air flows.
Due to their flexibility, however, effective use of these tools requires
specialist knowledge to ensure reliability of results e which can
vary significantly based on choice of code, grid, domain and user
input [8,9] e and carries associated costs in time, labour and
computing power.
nt).
All rights reserved.
Simplified mathematical models e which can readily be solved
by hand, or numerically with a small computational overhead e
therefore still form a crucial part of the design process. Whilst not
able to capture the same level of detail as computational tools,
these simple models elucidate some of the key relationships be-
tween design parameters and thereby provide rapid and intuitive
guidance at the preliminary design stage. Indeed, industry stan-
dards and design guidance (e.g. Refs. [10,11]) are underpinned by
these simple models.
Much existing guidance for stack ventilation focusses on the
simple case of a single room, the initial mathematical model for
which was developed and validated in small-scale laboratory ex-
periments by Linden et al. [12]. Numerous experimental studies
have since extended this work to multi-compartment buildings.
Holford and Hunt [13], for example, developed and experimentally
validated a simplified mathematical model of stack ventilation
flows in a room attached to an atrium; this model was further
validated by Ji et al. using CFD [5]. Livermore and Woods [14]
experimentally validated a similar model of flows in a two-storey
building with a ventilation stack. Chenvidyakarn and Woods [15]
also showed that a simple model can capture the behaviour of
multiple flow regimes in two interconnected heated spaces. Based
on the robustness of these models, the behaviour of stack ventila-
tion flows inmulti-compartment buildings and the implications for
design have also been investigated in theoretical studies [16e18].
Delta:1_given name
Delta:1_surname
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Nomenclature
Symbol
A vent area, m2
A0 effective vent area, m2
A* combined effective vent area, m2
B buoyancy flux, m4 s�3
b thermal expansion coefficient, K�1
cd discharge coefficient, e
cp specific heat capacity (of ambient air), J kg�1 K�1
Cp wind pressure coefficient, e
Eg geopotential energy, J
Ek kinetic energy, J
g gravitational acceleration, m s�2
g0 reduced gravity, m s�2
H zone height, m
DH atrium height above top storey, m
L number of flow loops, e
N number of building zones, e
Dpcrack pressure drop across a crack, N m�2
Dpenergy energetic inertia pressure, N m�2
Dpstack stack pressure, N m�2
Dpturn pressure drop due to stack turning, N m�2
Dpvent pressure drop across a vent, N m�2
Dpwind wind pressure, N m�2
Q ventilation flow rate, m3 s�1bQ dimensionless ventilation flow rate, e
RA� ratio of vent areas, e
Re Reynolds number, e
r density of air, kg m�3
S stack cross-sectional area, m2
t time, s
T temperature, K
u wind speed, m s�1
V number of vents, e
V zone volume, m3
W heating rate, W
z vertical coordinate, m
Subscript
a atrium
c ceiling-level
e external environment
eff effective
f floor-level
i storey number
in inlet vent
l flow loop index
m, n zone index
out outlet vent
p plenum
pa combined value for plenum and atrium
s storey
tot total
v vent index
1 For example, when T ¼ 290 K and Te ¼ 280 K, ðre � rÞ=re ¼ 0.033 [21].
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130122
However, few studies explicitly tackle a general approach to
stack ventilation in multi-compartment buildings. Etheridge [19],
for example, extends the ‘explicit method’ e a simplified mathe-
matical approach to preliminary design presented in CIBSE guid-
ance [11] e to a number of example multi-compartment buildings.
Axley [20] also presents a general ‘loop equation’ method for
forming the equations governing ventilation flows in multi-
compartment buildings, focussing on applications in multizone
software.
In this paper we adapt and build upon this existing work to
develop a generalised method for modelling stack ventilation
flows in multi-compartment buildings, focussing on the intuitive
value of the method for use in preliminary design. In particular, we
examine the qualitative relationships between ‘core’ ventilation
variables e flow rates, temperatures, heat inputs and building
geometry e in order to inform the sizing of ventilation openings.
The mathematical model used is deliberately simple; where rele-
vant, we have highlighted how additional detail may be included.
This work is intended, firstly, to provide one possible approach for
extending existing preliminary design guidance for stack ventila-
tion to multi-compartment buildings; and, secondly, to provide a
‘sense check’ for software modellers with a view to reducing the
computational overhead and costs associated with the design
process.
In Section 2 we outline the assumptions and approximations
used in developing our model; in Section 3 we develop a model for
stack ventilation in a general multi-compartment building; and in
Section4 we apply this general model to an example three-storey
building with an atrium and inlet plenum, testing some of the
qualitative predictions of the model by numerically solving for flow
rates in a number of scenarios.
2. Assumptions and approximations
2.1. Variations in density and temperature
Stack ventilation flows are driven by differences in density be-
tween internal and external air. These differences in density are
typically small such that (re � r)/re � 1, where r is the density of
internal air and re is the density of external air.1 Air may then be
regarded as incompressible to leading order (Boussinesq approxi-
mation [22]) and variations in density and temperature are ignored,
except where they appear in driving ‘reduced gravity’ terms,
g0 ¼ g
re � r
re
¼ gbðT � TeÞ; (1)
where T and Te are the internal and external air temperatures,
respectively, bz 1/Te is the thermal expansion coefficient of air and
g is gravitational acceleration. Since the reduced gravity is a mea-
sure of the internal temperature excess, T � Te, we interchangeably
use the terminology ‘reduced gravity’ and ‘temperature excess’
throughout.
2.2. Heat transfer
Variations in air density and temperature within a building are
generated by heat inputs from occupants, office equipment, solar
gains, and so on. This heat is then removed by ventilation, or by
transfer into or through the building fabric. For simplicity, we
Fig. 1. ‘Well-mixed’ building zone with air at uniform temperature, T, and corre-
sponding temperature excess g0 . Air enters through a low-level vent at height zin and
exits through a high-level vent at height zout.
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130 123
assign a net heat input to each building zone, and do not explicitly
model heat sources or heat transfer through the building fabric.
It is convenient to express heat inputs, W in Watts, in terms of
buoyancy inputs, B in m4 s�3, given by
B ¼ gb
recp
W ; (2)
where cp is the specific heat capacity of ambient air. We inter-
changeably use the terminology ‘buoyancy input’ and ‘heat input’
throughout.
Similarly, the rate of heat transfer between building zones due
to a flow rate Q of air with temperature excess g0 can be expressed
in terms of a buoyancy flux, in m4 s�3, given by g0Q.
2.3. Driving stack pressure
Buoyant air (i.e. with g0 s 0) contained within a building pro-
vides the driving stack pressure for natural ventilation flows. In
general, the stack pressure depends upon the vertical distribution
of buoyant air within the building and is linked with the density
stratification within each building zone. However, for simplicity,
and to ensure compatibility with existing multizone models, we
make the approximation that the air within each building zone is at
uniform temperature, or ‘well-mixed’.
Fig. 1 shows a simple building zone in which the air is ‘well-
mixed’, with temperature excess g0. The zone is ventilated through
a low-level inlet vent at height zin, and a high-level outlet vent at
height zout. Assuming a hydrostatic pressure distributionwithin the
zone, the driving stack pressure, Dpstack, between the inlet and
outlet vents is given by
Fig. 2. Example three-storey building with inlet plenum, which doubles as an entrance ha
intended flow pattern. From left to right: cool, ambient air flows into the plenum and is supp
through ceiling-level vents (yellow arrows) and exits the atrium through a high level vent
reader is referred to the web version of this article.)
Dpstack ¼ reg
0H; (3)
where H¼ zin � zout is the vertical distance between the inflow and
outflow vents in the zone (note that Hflow loops corresponds
to the number of loop equations needed to fully describe flows
through the building.
Flow loops can be identified using graph theory. Fig. 4 shows a
graphical representation of the example building from Figs. 2 and 3,
inwhich each node represents a building zone, with one additional
zone for the external environment, and each edge represents a
ventilation opening connecting two adjacent zones. For a building
with N zones (excluding the external environment), and V ventila-
tion openings, the number of independentflow loops, L, through the
building is given by L¼V�N [32,33]. In this case, V¼ 8 andN¼ 5, so
there are 3flow loops through the building, one through each storey.
Note, however, that the loops shown in Fig. 4 do not necessarily
represent all possible flows through the example building. Flows
could travel in the opposite direction to that shown (e.g. for a
summer ventilation scheme in which air is drawn in via the atrium
and cooled before being supplied to the storeys); recirculation of
flows between storeys via the connecting spaces is also possible.
However, these additional flows can be described by a linear
combination of the flow loops shown; the equations describing
flows around these loops therefore form a complete description of
the possible flows within the building.
3.2. Flow equation for a given flow loop
Passing around a given flow loop, l, and applying Equation (4) at
each ventilation opening, the total pressure drop across vents on
the flow loop is given by
Dpvent;l ¼
X
v
reQ
2
v;l
A02
v;l
; (6)
where the subscript v, l denotes quantities at vent v on flow loop l
(see LHS of Equation (11) for this applied to the example building).
Similarly, applying Equation (3) to each zone along a flow loop, the
total stack pressure driving flows around the loop is
Dpstack;l ¼
X
n
reg
0
n;lHn;l; (7)
where the subscript n, l denotes quantities within zone n on flow
loop l (see RHS of Equation (11) for this applied to the example
building). Since each flow loop is closed, the total pressure loss at
ventilation openings is balanced by the stack pressure, i.e.
Dpvent,l ¼ Dpstack,l. By combining Equations (6) and (7), the loop
equation for flow loop l is then given by
X
v
Q2
v;l
A02
v;l
¼
X
n
g0n;lHn;l: (8)
For a building with L flow loops, applying Equation (8) to each
flow loop generates L coupled equations.
3.3. Conservation of volume
As we are concerned with steady flows, we must also conserve
mass within each zone, which for Boussinesq flows reduces to
conservation of volume, i.e.
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130 125
X
m
�
Qm;n � Qn;m
� ¼ 0; (9)
whereQm,n denotes the flow rate from zonem into zone n (i.e. flows
into zone n) and Qn,m denotes the flow rate from zone n into zonem
(i.e. flows out of zone n). For example, if air flows from zone 1 to
zone 2 at a rate Q, we have Q1,2 ¼ Q and Q2,1 ¼0. For a building with
N zones, applying Equation (9) to each zone generates N coupled
equations.
3.4. Steady heat balance
Equations (8) and (9) form a general airflow model, allowing
ventilation flow rates to be calculated when the temperature dis-
(11)
tribution within the building is known. The temperature distribu-
tion, in turn, depends upon the flow rates through the building and
the heat inputs within each zone, and can be calculated by
considering the heat balance for each building zone. For the general
case of time-varying flows, the airflow and heat balance models
must be solved separately [34]. However, for simplicity, we focus on
steady flows only such that the airflowand heat balancemodels can
be solved simultaneously.
For a zone, n, with heat input Bn, the steady heat balance isX
m
�
g0nQn;m � g0mQm;n
� ¼ Bn; (10)
where g0mQm;n and g0nQn;m correspond to heat transfer rates due to
ventilation flows into and out of zone n, respectively. For a building
with N zones, applying Equation (10) to each zone generates N
coupled equations.
Equations (8)e(10) form a complete model of the air and heat
flows through a general building e comprising L þ 2N ¼ V þ N
coupled equations e and can be solved simultaneously for Q and g0
when all values of A0 and B are known.
4. Application to an example building
In order to illustrate how the general model presented above
may be applied in practice, we consider again the example three-
storey building shown in Figs. 2 and 3.
The key features of the building are labelled in Fig. 3. Each
storey has a floor-to-ceiling height H; the atrium extends a
height DH above the top floor. The plenum vent has effective
area A0
p (cf. Equation (5)), the atrium vent has effective area A0
a,
and the storeys have floor- and ceiling-level vents with effective
areas A0
fi and A0
ci respectively (where i ¼ 1, 2, 3 denotes the storey
number). The temperatures in the plenum, storeys and atrium
are g0p, g0si and g0a, respectively. The heat inputs within the
plenum, storeys and atrium are Bp, Bsi and Ba, respectively. Note
that, because this analysis focusses on steady flows, we do not
need to specify plan areas or zone volumes for the example
building.
We assume that the air inside the building is warmer than the
external air, i.e. with g0 > 0, and that there is no wind such that air
enters the building via the low level plenum inlet, and exits the
building via the high level atrium outlet, following the flow loops
shown in Fig. 4. The flow rates through the plenum vent, storeys
and atrium vent are Qp, Qi and Qa, respectively.
4.1. Flow loop equations
The flow loops around the building are shown in Fig. 4. Applying
Equation (8) to each flow loop gives three governing equations:-
where, conserving volume in the plenum and atrium,
Qtot ¼ Qp ¼ Qa ¼ Q1 þ Q2 þ Q3 is the total flow rate through the
building, and
1
A�2
i
¼ 1
A02
fi
þ 1
A02
ci
and
1
A�2
pa
¼ 1
A02
p
þ 1
A02
a
(12)
are combined effective vent areas for the storeys (‘inner’ vents)
and the plenum and atrium (‘outer’ vents), respectively. Note that,
in defining Q1, Q2 and Q3, we have implicitly conserved volume
within each storey such that the flow rates into and out of each
storey are equal. Equation (9) has therefore been applied once to
each building zone, in accordance with the general model in Sec-
tion 3.
4.2. Qualitative interpretation of model
The key quantities in Equation (11) have been highlighted to
allow us to gain a qualitative understanding of how the flows
through this type of building behave.
4.2.1. Driving buoyancy terms
The terms on the RHS of Equation (11) describe the buoyancy
distribution within the building which provides the driving pres-
sure for stack ventilation.
In a typical building of this type, ambient air is supplied to the
inlet plenum and is heated as it passes through the building so that
g0p A�
2 > A�
1. Furthermore,
Fig. 5. Variation of dimensionless flow rates, bQ i ¼ Qi=ðBtotA�2
paDHÞ
1
3 , with vent area
ratio, RA� ¼ A�
s=A
�
pa , for a three-storey building with DH/H ¼ 1, Ba ¼ 0 and equal
heating rates and vent areas on each storey. The dashed line shows the flow rate
through an equivalent isolated storey, i.e. a room with the same heat input and vent
areas, but not attached to an atrium or plenum. For RA�(0:1, control of flow rates is
dominated by the size of the ‘inner’ vents, A�
s ; for RA�a1:0, control is dominated by the
size of the ‘outer’ vents, A�
pa .
Fig. 6. Variation of dimensionless flow rates, bQ i ¼ Qi=ðBtotA�2
paDHÞ
1
3 , with vent area
ratio, RA� ¼ A�
s=A
�
pa , for a three-storey buildingwith equal heating rates and vent areas
on each storey for three cases: (a) DH/H ¼ 1, Ba/Btot ¼ 0.75, (b) DH/H ¼ 4, Ba/Btot ¼ 0.75,
(c) DH/H ¼ 4, Ba/Btot ¼ 0. The dashed line in each figure shows the dimensionless flow
rates through an equivalent isolated storey.
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130126
the greater the disparity between temperatures in the plenum,
storeys and atrium, the greater the required difference in vent
areas; this is significant for buildings where the atrium is heated,
for example.
Conversely, for a building with equal vent areas on all storeys,
i.e. A�
3 ¼ A�
2 ¼ A�
1, we expect that there will be a disparity be-
tween flow rates through the occupied storeys, i.e. Q3Fig. 6(a) and (b), we note that increasing DH has the
desirable effect of reducing the difference in flow rates between the
top and bottom storeys, as expected from Section 4.2.2. Conversely,
comparing Fig. 6(b) and (c), we note that increasing the heat input
in the atrium increases the disparity between the top and bottom
storeys.
For RA�(0:1, increasing either Ba or DH increases the assisting
effect of the atrium, increasing the flow rate through all storeys
relative to the case of an isolated storey.
4.3.3. Implications for design
The first conclusion to draw from this analysis is that a naturally
ventilated multi-storey atrium building should not be designed
with equal vent areas on all storeys, for the case of equal heat
inputs on all storeys; this will always result in a reduction in flow
rates on ascending the building, with the potential for inadequate
ventilation and overheating on the top storeys. Achieving a
balanced design with equal flow rates on all storeys therefore re-
quires that the storey vent sizes, A�
si, increase on ascending the
building.
The second key implication for design is that ventilation control
must be shared between the ‘inner’ and ‘outer’ vents. If the ‘inner’
vents are too small (small RA� ), exchange flows may develop at the
atrium outlet; conversely, if the ‘outer’ vents are too small (large
RA� ), flows become choked at the atrium outlet, rendering the ‘in-
ner’ (storey) vents ineffective for ventilation control. From Figs. 5
and 6, a value of RA�w0:1 provides a satisfactory balance of con-
trol between ‘inner’ and ‘outer’ vents for the example building in
this study.
The conclusions drawn in this example study are intended to
provide qualitative ‘rules-of-thumb’ only. The appropriate sizing of
vents for similar multi-storey buildings has been considered in
more detail by Hunt and Holford [39] and Acred and Hunt [40].
5. Conclusions
We have developed a general simplified mathematical approach
to modelling stack ventilation in multi-compartment buildings.
Following the loop equation method, a pressure balance around
closed flow loops is used to form the governing equations of flow. A
qualitative examination of the equations of flow can be used to gain
an intuitive understanding of how flow rates, air temperatures and
building geometry interact and what, therefore, might constitute
an effective design.
The simplifications used in developing the model are high-
lighted. The effects of density stratification in all building zones
have been neglected. In addition, the only pressure terms consid-
ered are the driving stack pressure and pressure losses across
ventilation openings; potential errors introduced by neglecting
other pressure terms are highlighted.
The model is applied to an example three-storey building with
an inlet plenum, open-plan occupied spaces and an atrium or solar
chimney. A number of rules-of-thumb for preliminary design are
suggested from a qualitative examination of the governing equa-
tions of flow. The equations are solved numerically for the case of
equal heat inputs on all storeys, to further elucidate the implica-
tions for design. It is shown that choosing equal vent areas on all
storeys always results in a disparity between flow rates through the
storeys; lower flow rates and higher temperatures are observed on
the upper storeys. This disparity is amplified by heating the atrium
2 For brevity, we drop the subscripts l, n and v for the pressure terms listed in the
Appendices.
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130128
(as in the case of a solar chimney), and reduced by increasing the
extension of the atrium above the top storey. The relative sizing of
the atrium, plenum and storey ventse quantified in the ratio RA� ¼
A�
s=A
�
pa e is also significant for control. It is shown that control must
be shared between all vents within the building, with RA�w0:1, in
order to provide the appropriate degree of control at the storey
level, and to avoid undesirable exchange flows developing at the
atrium outlet.
This work is intended to form a reference point for future
theoretical and experimental studies on stack ventilation in large
buildings.
Acknowledgements
A. P. Acred and G. R. Hunt gratefully acknowledge the financial
support of the EPSRC for this research.
Appendix
Using an effective zone height to cater for stratification
Wehave assumed that the air in each building zone is at uniform
temperature or ‘well-mixed’. However, a broad range of stratifica-
tion patterns are possible in practice and are closely linked with the
geometry and location of heat sources within the space ewhich, in
general, may be localised [12,41,42], horizontally distributed
[43,44] or vertically distributed [45e47]. The location and geome-
try of ventilation openings can also affect stratification and flow
patterns within the space [48,49].
It is possible to account for the effect of stratification on driving
stack pressures using what we refer to as an effective zone height
parameter. Consider the building zone shown in Fig. 7, which is
stratified such that g0 ¼ g0(z). The zone has low- and high-level
ventilation openings at heights zin and zout, respectively. The flow
rate through the zone is Q. Air entering the zone is at temperature
g0in; air exiting the zone is at temperature g0out. The heat input within
the zone is B such that, from Equation (10), g0out � g0in ¼ B=Q .
We define an effective zone height Heff as follows
Heff ¼
Zzout
zin
g0ðzÞ
g0out
dz: (16)
The driving stack pressure due to buoyant air within the zone is
then given by
Dpstack ¼ reg
0
outHeff : (17)
This formulation is convenient since g0out is equivalent to the
uniform zone temperature used in Section 3 and can be readily
calculated using Equation (10). In order to cater for stratification,
therefore, the only required alteration to the flow loop equations in
Equation (8) is to replace the zone heights, Hn,l, with effective zone
heights, Heff,n,l.
Note that this method of accounting for stratification is only
valid for zoneswith outlets at a single height. For zoneswith outlets
at multiple levels, the temperature of air exiting the zone will vary
depending upon the stratification.
General pressure balance
The pressure balance model in Equation (8) only takes into ac-
count stack pressure and losses in pressure across ventilation
openings. In practice, numerous additional factors can enter into
the pressure balance, a number of which we consider here. A more
detailed pressure balance for a given flow loop through a building is
as follows2:
Dpvent þ Dpcrack þ Dpturn ¼ Dpstack þ Dpwind þ Dpenergy; (18)
where Dpcrack is the drop in pressure across adventitious open-
ings, Dpturn is the drop in pressure required to change the di-
rection of flows entering stacks or other vertical spaces, Dpwind
is the wind pressure assisting flows and Dpenergy is the pressure
arising from unsteady changes in internal energy within the
building. Each of these terms is discussed in more detail below.
Additional pressure terms: wind
Wind can contribute significantly to assisting [50] or, in some
cases, opposing ventilation flows [51]. Along a given flow loop
which passes through the external environment, the wind pressure
assisting flows is given by
Dpwind ¼ 1
2
reu
2�Cp;in � Cp;out
�
; (19)
where u is wind speed and Cp,in and Cp,out are the wind pressure
coefficients at the vents where flows enter and exit the building,
respectively. When Dpwinddiscussed in detail and tabulated for various
openings in Ref. [53]. Flows through cracks can also be modelled by
changing the effective value of cd in Equation (4)e see Refs. [53,54].
Additional pressure terms: turning and frictional losses
Flows along long, narrow ventilation stacks or ducts are subject
to pressure losses due to friction and changes in the direction of
flow. As for flows through adventitious openings, these losses can
be catered for by modifying the effective value of cd in Equation (4).
An example calculation for a long duct is presented in Ref. [11].
For buildings similar in form to the example building in Fig. 2, a
typical feature is a flow in which air flows horizontally into a stack
or other vertical space before turning through 90� and flowing
vertically upwards. Livermore and Woods [14] showed that the
drop in pressure required to change the direction of flow is typically
an order of magnitude greater than frictional losses in the stack and
is given by
Dpturnw
reQ
2
S2
; (21)
where S is the cross-sectional area of the stack or other vertical
space. Comparing Equation (21) with (6) we find that Dpturn/
Dpvent w (A0/S)2. For narrow stacks for which S w A0 , Dpturn must,
A. Acred, G.R. Hunt / Building and Environment 71 (2014) 121e130 129
therefore, be taken into account. However, we assume that both the
plenum and atrium in the example building considered in this
paper are ‘wide’, with S [ A0 , such that Dpturn can be neglected
without introducing significant error.
Additional pressure terms: effect of unsteady changes in internal
energy
The application of Bernoulli’s theorem along a streamline
through a building zone is a special case of conservation of me-
chanical energy within the zone. Axley [55] notes that, when
considering the general, time-varying mechanical energy balance
within a zone, the accumulation or dissipation of internal energy
within a zone must be taken into account. This gives rise to an
additional ‘energetic inertia’ pressure term, Dpenergy, which can
either oppose or assist flows through the zone. For a given zone,
Dpenergy ¼ �1
Q
d
dt
�
Eg þ Ek
�
; (22)
where Q is the flow rate through the zone, t denotes time, and Eg
and Ek are the geopotential energy and kinetic energy of air within
the zone, respectively.
For the high Richardson number flows synonymous with the
turbulent convection typically observed in stack ventilation flows,
we expect that Eg [ Ek. For a zone of height H, at uniform tem-
perature g0, the geopotential energy relative to a datum at floor
level is given approximately by Egwreðg � g0ÞHV, where V is the
volume of the zone. The energetic inertia pressure is then given by
Dpenergyw
reHV
Q
dg0
dt
: (23)
We note from Equation (23) that when the air within a zone is
cooling (dg0/dt 0), Dpenergy > 0 and acts to assist flows
through the zone.
To assess the significance of Dpenergy, consider an unsteady flow
inwhich the temperature varies linearly by an amount Dg0 in a time
Dt, such that dg0/dt ¼ Dg0/Dt. Comparing Equations (3) and (23), we
find,
Dpenergy
Dpstack
w
Dg0
g0
V
QDt
: (24)
For temperature changes occurring over a period of one hour
(i.e. Dt¼ 1 h) in a typical office building inwhich Q=V ¼ 1 ACH (air
change per hour), we then have Dpenergy/Dpstack w Dg0/g0. A 20%
change in temperature over this time period then introduces an
error of w20% in applying the simplified loop equation model in
Equation (8), which does not include Dpenergy. The energetic inertia
may, therefore, play a significant role in unsteady flows, particularly
during early transients when the temperature within a zone is
changing rapidly and the stack pressure is small.
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	A simplified mathematical approach for modelling stack ventilation in multi-compartment buildings
	1 Introduction
	2 Assumptions and approximations
	2.1 Variations in density and temperature
	2.2 Heat transfer
	2.3 Driving stack pressure
	2.4 Pressure drop across ventilation openings
	2.5 Additional pressure terms
	3 Multizone buoyancy-driven ventilation model
	3.1 Identifying flow loops
	3.2 Flow equation for a given flow loop
	3.3 Conservation of volume
	3.4 Steady heat balance
	4 Application to an example building
	4.1 Flow loop equations
	4.2 Qualitative interpretation of model
	4.2.1 Driving buoyancy terms
	4.2.2 Interaction of flow rate and vent areas (Q-A interaction)
	4.3 Steady state case study – equal storey vent areas, equal heat inputs
	4.3.1 Effect of varying RA*=As*/Apa*
	4.3.2 Effect of varying ΔH and Ba
	4.3.3 Implications for design
	5 Conclusions
	Acknowledgements
	Appendix
	Using an effective zone height to cater for stratification
	General pressure balance
	Additional pressure terms: wind
	Additional pressure terms: flow through adventitious openings
	Additional pressure terms: turning and frictional losses
	Additional pressure terms: effect of unsteady changes in internal energy
	References