Particle transport in low-energy ventilation systems.

Part 2: Transients and experiments

Introduction

Buildings and occupied spaces are transient by theirnature as people move, equipment is turned on andoff, external conditions change, etc. While making asteady-state assumption is often useful in the initialstudy of contaminant distribution (Musser andPersily, 2002) it does not provide all the informationas it neglects the temporal nature of contaminantsources and building ventilation rates (e.g. Bolsterand Caulfield, 2008).In general, an efficient manner of air quality

control is to remove the contaminant before it hastime to mix throughout the space, which meansextracting it from near the source, where theconcentration will be the highest. Therefore, it is

important to understand how the contaminant willdisperse in the room. In most indoor environmentsthe transport of the pollutant is advective, as thediffusion rate is relatively small. An approximatediffusion timescale is 5 days for a typical space, yetfrom experience we know that the actual dispersionrate is more on the order of minutes or hoursdepending on the space in question. As such under-standing the temporal nature of this dispersion canprovide some useful insight into effective removal ofindoor air contaminants.Much previous work has been carried out in the field

of contaminant modeling in buildings. Most of it isbased on reduced order zonal models (Feustel, 1999;Schneider et al., 1999; Zhao et al., 2004). In zonalmodels, buildings are typically broken into several

Abstract Providing adequate indoor air quality while reducing energy con-sumption is a must for efficient ventilation system design. In this work, we studythe transport of particulate contaminants in a displacement-ventilated space,using the idealized �emptying filling box� model (P.F. Linden, G.F. Lane-serffand D.A. Smeed (1990) Emptying filling boxes: the fluid mechanics of naturalventilation, J. fluid Mech., 212, 309–335.). In this paper, we focused on transientcontaminant transport by modeling three transient contamination scenarios,namely the so called �step-up�, �step-down�, and point source cases. Using ana-lytical integral models and numerical models we studied the transient behavior ofeach of these three cases. We found that, on average, traditional and low-energysystems can be similar in overall pollutant removal efficiency, although quitedifferent vertical gradients can exist. This plays an important role in estimatingoccupant exposure to contaminant. A series of laboratory experiments wereconducted to validate the developed models.

D. T. Bolster, P. F. Linden

Department of Mechanical and Aerospace Engineering,University of California, San Diego, La Jolla, CA, USA

Key words: Low-energy ventilation; Contaminant;Particles.

Diogo T. BolsterDepartment of Geo-EngineeringTechnical University of Catalunya (UPC)Campus Road08034 BarcelonaSpainTel.: +34 655 039 065Fax: +34 944 016 890e-mail: diogobolster@gmail.com, dbolster@ucsd.edu

Received for review 21 January 2008. Accepted forpublication 4 August 2008.� Indoor Air (2008)

Practical ImplicationsThe results presented here illustrate that the source location plays a very important role in the distribution ofcontaminant concentration for spaces ventilated by low energy displacement-ventilation systems. With these resultsand the knowledge of typical contaminant sources for a given type of space practitioners can design or select moreeffective systems for the purpose at hand.

Indoor Air 2008www.blackwellpublishing.com/inaPrinted in Singapore. All rights reserved

� 2008 The AuthorsJournal compilation � Blackwell Munksgaard 2008

INDOOR AIRdoi:10.1111/j.1600-0668.2008.00569.x

1

zones, which interact with one another. Each of thezones is assumed to be well mixed at all times.This approximation of uniform and instantaneous

mixing is very convenient and can often be justified forcases with strong internal air motion. However, thisassumption may not always be appropriate, particu-larly in low-energy ventilation systems where there is adeliberate creation of non-uniformity in temperature.In such cases, exposure to the contaminants maydepend significantly on a person�s location with aspace. Ozkaynak et al. (1982) found that pollutantlevels in a kitchen with the oven on depended heavilyon the sampling location. Rodes et al. (1991) discov-ered that personal sampling almost always reveals ahigher exposure to contaminants that would bepredicted from indoor air monitoring that assumesperfect mixing. Lambert et al. (1993) showed that thelevels of suspended particles in a non-smoking sectionof a restaurant were 40–65% less than that in thesmoking section.Various computational fluid dynamics (CFD) studies

have been conducted looking at particulate transportin ventilated spaces (e.g. Holmberg and Li, 1998;Zhang and Chen, 2006; Zhao et al., 2004). However,even with current computer speeds, full-scale simula-tions can be prohibitively expensive, particularly forlarge buildings with multiply connected spaces. Accu-rate CFD simulations of such flows can be verydifficult, because of uncertain boundary conditions(Cook et al., 2003), the difficulties involved in accu-rately modeling thermal plumes (Yan, 2007) and theappropriate selection of a wide choice of availableturbulence models (Ji et al., 2007), which can all affectthe flows involved in a non-negligible manner.Reduced analytical models (such as the one we

present in this paper) can be useful, as they can beintegrated into zonal computer models (e.g. Energy-Plus; Department of Energy, Berkeley, CA, USA)where connected spaces are treated as nodes thatcommunicate with one another via conservation equa-tions (e.g. temperature, contaminant concentration,etc.). Further, such analytical models provide impor-tant benchmarks, necessary for the validation of CFDmodels. Also, while useful for specific cases, CFDstudies do not always provide practical informationabout the general physics and behavior of contami-nants in low-energy ventilation systems.In this work, we compare contaminant transport in a

traditional �mixing� system model with two low-energy�displacement� ventilation models using reduced ana-lytical integral models to study the average concentra-tions as they vary with time. We then use a numericalmodel (Germeles, 1975) to study the detailed verticaldistribution of contaminant. These models are thencompared with the results of laboratory experiments tovalidate the information obtained with the mathemat-ical models.

Mathematical models

To understand the fate of particles in a ventilated spaceit is necessary to understand the flow within the space.We consider three models, which are shown schemat-ically in Figure 1. We analyze two low-energy ventila-tion models [(b) and (c)] and one traditional mixingmodel (a). In the low-energy models we consider thespace either mechanically or naturally ventilated withfresh air entering through a low-level vent and hotbuoyant air leaving via a vent at high level. Heatsources in the space are represented by ideal plumes.

Contamination scenarios considered

For each of the models depicted in Figure 1, weconsider three types of contamination situations:

1. Step-down (natural attenuation): This is the situa-tion where a space is initially uniformly filled with acontaminant and fresh, uncontaminated air isintroduced into the space.

2. Step-up (external contaminant): Here we consider aspace that is initially uncontaminated. Then a contam-inant is introduced in through the ventilation system.Atsteady state this scenario is equivalent to the externalcontaminant source considered in Part 1 of this work.

3. Isolated source in plume: Here we consider a spacethat is initially uncontaminated. Then a contami-nant is introduced as a point source located withinthe plume. We choose this location because peopleare often the source of heat as well as the source ofcontaminants in buildings. In addition, it is also theonly type of point source that we can currentlyadequately describe with our model. At steady state,this scenario is equivalent to the internal contami-nant source considered in Part 1 of this work.

Model (a) – entirely well-mixed space

In this model, we treat the entire room as well mixed(Figure 1a). The concentration in a well-mixed space,Kwm, satisfies the conservation equation

dKwmdt¼ �Qin þQfall

SHKwm þ

QinSH

Kin þQinKsSH

; ð1Þ

where Qin is the flow rate into the space, Qfall = vfall Sis defined as the settling flow rate, vfall is the settlingvelocity, assumed here to be the Stokes settlingvelocity, Kin is the concentration of contaminant inthe incoming air, Ks is the concentration associatedwith a point source, S is the room cross sectional area,and H is the height of the room. Qfall quantifies theamount of deposition that will take place. We neglectdeposition of particles to the ceiling and sidewalls andassume that particles settle out of the lower and upper

Bolster & Linden

2

layers at this settling velocity. While this is a strongassumption, in Part 1 of this work we illustrated thatthis assumption, while providing quantitatively differ-ent results, still allows for detailed qualitative analysis.

Model (b) – well-mixed two-layer model

In this section, we consider model (b) from Figure 1.We take an approach similar to that of Hunt and Kaye(2006) and assume that the upper and lower layers arealways well mixed. Thus, the governing equations forconservation of contaminant in each of the layers are

dKldt¼ � Qp þQfall

Sh

� �Kl þ

QfallSh

Ku þQinSh

Kin

dKudt¼ QpSðH�hÞKl�

QfallþQpSðH�hÞ

� �Kuþ

QpKsSðH�hÞ ð2Þ

where K1 and Ku are the concentrations of contaminantin the lower and upper layers respectively, h is theheight of the lower layer, and Qp is the plume flow rateacross the interface and at steady state Qp = Qin.

Model (c) – mixed lower layer, unmixed upper layer

Here we consider model (c) from Figure 1. For a well-mixed lower layer and unmixed upper layer theconservation equations for the upper and lower layerare slightly modified from the previous section. Insteadof removing fluid of the average concentration ofcontaminant, we must now account for the fact thatthe upper layer can have concentration gradients in it,thus leading to the following conservation equations

dKldt¼ � Qp þQfall

Sh

� �Kl þ

QfallSh

Kuðz ¼ hÞ þQinSh

Kin;

d �Kudt¼ Qp

SðH� hÞ ðKl � Kuðz ¼ HÞÞ

� QfallSðH� hÞ

� �Kuðz ¼ hÞ þ

QpKsSðH� hÞ : ð3Þ

To understand the dynamics of the upper layer it isimportant to understand the flow within the space,which is determined by coupling the plume flow withthe environment outside the plume. Exact details ofthis are given in Bolster and Linden (2007).

Non-dimensionalization

We non-dimensionalize as follows:

t ¼ s SHQp

; K ¼ Krefj; h ¼ Hf; ð4Þ

WM

(a)

(b)

(c)

Qin

Qin

Qin

Qout

Qout

Qout

H

WM

WM

Qp

H

h

WM WM

Qp

H

h

Fig. 1 Models of displacement ventilation systems. (a) Singlewell-mixed layer, (b) two well-mixed layers, and (c) two-layermodel with well-mixed lower layer and unmixed upper layer.WM stands for well mixed

Particle transport in low-energy ventilation systems

3

where Kref is a reference concentration, which will bedifferent for each of the three situations considered.Here s, j , and f are dimensionless time, dimensionlessconcentration, and dimensionless interface height,respectively. For the step-down method, it is the initialconcentration of contaminant in the space (Kref = K0).For the step-up system, it is the concentration ofcontaminant entering the spaces (Kref = Kin) and forthe point source case, it is the concentration of thesource (Kref = KS). This results in the followingdimensionless equations:(a)

djds¼ �ð1þ aÞjþ jin þ js; ð5Þ

(b)

djlds¼ �ð1þ aÞjl þ aju þ jin

f;

djuds¼ jl � ð1þ aÞju þ js

1� f ; ð6Þ

(c)

d �jlds¼ a

fðjuðf ¼ fhÞ � �jlÞ þ

1

fðjin � �jlÞ;

d�juds¼ �a1�fðjuðf¼fhÞÞþ

1

1�fð�jl�juðf¼1ÞþjsÞ: ð7Þ

where a¼Qfall=Qp, which is a dimensionless represen-tation of the particle settling velocity.

Results

All of the above equations are linear and can be solvedanalytically. In this section, we present the generalsolutions to the governing equations without consid-ering the specific contamination conditions. The con-tamination scenarios mentioned previously will bediscussed later in this section.

General results of model equations

Model (a). The governing equation (5), can be inte-grated to give the average concentration throughoutthe room. It is described by a simple exponentialequation

jðaÞ ¼ joe�ð1þaÞs þjin þ js1þ a ð1� e

�ð1þaÞsÞ; ð8Þ

where jo is the initial contaminant concentration in theroom, and jin is the concentration of contaminantentering the room.

Model (b). The system of equations (6), can be solvedby Laplace transforms or as an eigenvalue problem.The solution is

where

A1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2aþa2þ4f2þ4f2aþ4f2a2�4af�4f�4fa2

q:

ð11Þ

Model (c). To close this system of equations (7), wemust find analytical expressions for jðcÞu ðf¼fh;tÞ andjðcÞu ðf¼1;tÞ. It is necessary to examine the full temporalevolution of the contaminant transport to determinethese quantities.

jðbÞl ¼ð1þ aÞjin þ ajs

a2 þ aþ 1 þ e��ð1þaÞ2fð1�fÞ s

�ðjo �

ð1þ aÞjin þ ajsa2 þ aþ 1 Þ coshð

A1s2fð1� fÞÞ

þ joða� 1þ 2fÞða2 þ aþ 1Þ þ jinða2 þ 1� 2fða2 þ aþ 1ÞÞ � jsða2 þ aÞ

Aða2 þ aþ 1Þ sinhðA1s

2fð1� fÞÞ� ; ð9Þ

jðbÞu ¼jin þ ð1þ aÞjs

a2 þ aþ 1 þ e��ð1þaÞ2fð1�fÞ s

�ðjo �

jin þ ð1þ aÞjsa2 þ aþ 1 Þ coshð

A1s2fð1� fÞÞ

þ joðaþ 1� 2faÞða2 þ aþ 1Þ þ jinð�1� aÞ � jsða2 þ 2aþ 1� 2fða2 þ aþ 1ÞÞ

Aða2 þ aþ 1Þ sinhðA1s

2fð1� fÞÞ� ; ð10Þ

Bolster & Linden

4

In the limit of a fi 0, jin = 0, and in the currentnon-dimensionalization this solution collapses to thatof Hunt and Kaye (2006). Also, at steady state, allthe results presented in Part 1 of this work arerecovered.Because diffusion is being neglected due to high

Peclet numbers, the method of characteristics can beused to determine the concentration of contaminant inthe upper layer at the interface height. This concen-tration will be that at the top of the room a period sdearlier, where sd is the time taken for a front ofdescending particles to travel from the top of the roomto the interface. It can be shown that

sd ¼3

5fh ln

1� ð1� 35aÞfhf� ð1� 35aÞfh

!: ð12Þ

Therefore, to close this system of equations, the onlyquantity required is an estimate of the contaminantconcentration at the top of the room. Assuming thatthe plume spreads instantaneously across the entireplan area of the ceiling, and that a perfectly mixed layerforms, we can equate the background concentration tothat of the plume at the top of the room (seeAppendix A).

jðcÞu ðf¼1;sÞ¼5ð1�fhÞ�juðsÞþ3ð�jlðsÞþjsÞfh

5�2fh1

1þ 3af5�2f

!

jðcÞu ðfh; sÞ ¼ jðcÞu ðf ¼ 1; s� sdÞ: ð13Þ

An analytical solution to this system of equations hasbeen found and can be computed. However, it is verylaborious and involves many summations which mustbe computed and gives very little insight into thephysical behavior of the system. Instead, the equationsare solved numerically using a Runge–Kutta scheme,which is more efficient. The solutions of the numericaland analytical systems are identical.

Step-down

Here we consider the removal of a uniformly distrib-uted contaminant from the space by introducinguncontaminated air in through the ventilation system.In this case, jin = 0, jo = 1, and js = 0 which canbe substituted into the results presented in the previ-ous section �General results of model equations�.Figure 2 displays a sample of solutions for the average

0 50

0.5

1

τ

κ

0 5τ

κ

0 0.5τ

κ

0 50

0.5

1

τ

κ

0 5τ

κ

0 0.5τ

κ

0 50

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

0

0.5

1

τ

κ

0 5τ

κ

0 0.5τ

κ

Fig. 2 Step-down case – average concentration for three model with various a and f. One-layer well mixed ()), two-layer well mixed(o), two-layer unmixed (x). The top row corresponds to f = 0.25, the middle row to f = 0.5, and the bottom row to f = 0.75

Particle transport in low-energy ventilation systems

5

contaminant concentration for each of the threemodels at three values of interface height, f = 0.25,0.5 and 0.75, and three values of a = 0.1, 1 and 10.Except for the case of a = 10 and f = 0.25 theaverage amounts of contaminant are similar for thethree cases, suggesting that each system exhibitscomparable efficiency vis-a-vis overall contaminantremoval. This conclusion applies over a wide range ofparameter space corresponding to typical physicalsituations. Although the average concentrations areapproximately the same, this does not imply that thevertical concentration profiles are also the same, merelythat the concentration being extracted is similar. In thenext section of the paper, we study these verticalconcentration profiles.

Quantity of particulates deposited. A quantity that maybe useful to know is the percentage of particulatecontaminants that escape the room through the uppervent and the percentage that is deposited on the floor.The percentage of contaminant deposited, Gdep, can befound as the integral of the product concentration andsettling flow rate over time

Cdep ¼Z s!10

jlðs0Þafds0: ð14Þ

The quantity Gdep is a measure of the percentage ofcontaminant that a person occupying the lower layerwill be exposed to, as all deposition occurs in the lowerlayer.Figure 3 displays Gdep for all three models over a

wide range of parameter space. The quantitative valuesof Gdep are roughly similar for the three cases;although, there are interesting qualitative differences.For the two-layer models, (b) and (c), there is amonotonic increase in Gdep as f increases. For theunmixed upper layer, the quantity of deposited con-taminant is always smaller than for the well-mixed two-layer system. As shown in Bolster and Linden (2007),where we compare the effective draining rates of theupper and lower layers of both these systems, it can beshown that the unmixed upper layer case is lesseffective at removing contaminant from the upperlayer. The mechanism of contaminant removal is thesame here and, in addition, there is transport backdown to the lower layer from the upper layer becauseof settling. This leads to the larger quantities ofdeposition observed in model (c). The largest differ-ences in Gdep between models (b) and (c), equal to 0.11,occurs for f = 0.35 and a = 2. A contour plot of thesedifferences is shown in Figure 4.Figure 5 compares Gdep for the single-layer well-

mixed space to the two-layer models. The line wherethe predicted amount of deposition is equal for bothmodels is shown on each plot. For smaller particles (i.e.a small) and lower interfaces (i.e. f small), the two-

layer systems predict lower values of Gdep, while forlarger values of these parameters the single well-mixedspace predicts lower values. As Gdep is effectively ameasure of the quantity of contaminant that occupantsin the lower layer have been exposed to, this plot showsthat for certain locations of interface and particle sizes,the low-energy systems reduce exposure. However, for

(a)

(b)

(c)

Fig. 3 Isocontours of the fraction of total initial particles thatare deposited on the floor during a step-down study for variousvalues of a and f for each of the three models, (a–c), consideredhere

α

ζ

0.11

0.09

0.07

0.05

0.03

10–1 100 101

0.2

0.4

0.6

0.8

0.01

Fig. 4 Isocontours for a range of a and f of the difference in thefraction of deposited particles during a step-down study betweenmodels (c) and (b) Ccdep � Cbdep

Bolster & Linden

6

larger particles and higher interfaces, the traditionalmixing ventilation system outperforms the low-energysystems.

Step-up

Now we consider the opposite situation, where a roomis initially completely uncontaminated and a contam-inant is introduced via the ventilation system. This cancorrespond to a number of scenarios, such as a leak ina ventilation system, a malicious release, or an externalcontaminant entering the building though naturalventilation.Here jo = 0, jin = 1, and js = 0, which should be

substituted into the results presented in section �Generalresults of model equations�. Unlike, the step-downscenario considered previously, the step-up case hassome interesting steady-state characteristics as dis-cussed in Bolster and Linden (2008). For passivecontaminants this steady state corresponds to auniformly distributed concentration of contaminantequal to that of the source. However, the influence ofgravitational settling leads to non-trivial steady-statedistributions.Here we present the transient evolution of average

concentration for the step-up case for the same valuesof a and f as we did for the step-down case. The resultsare plotted in Figure 6. In this case, as we predicted inthe discussion on steady states, the evolution ofconcentrations can be very different. Notice particu-larly, that for larger particles, a = 10, the differencecan be quite significant, although the time scales to

reach steady state is comparable. This figure is slightlydeceptive in that it makes the low-energy systemsappear more efficient at removing large particles, whichthey are, provided only average concentrations areconsidered. However, recall that the lower layerconcentration is always higher than the well-mixedcase, thus exposing occupants to higher levels ofcontaminant.

Point source

Here we also consider a situation, where a room isinitially completely uncontaminated. This time thoughthe contaminant is introduced from a point source. Forthis case jo = 0, jin = 0, and js = 1, which can besubstituted into the results of section �General results ofmodel equations�. Once again we track the averageconcentrations predicted by the three models forvarious values of settling velocity and interface height.The results are shown in Figure 7.As with the step-up case we see that both the two-

layer models predict similar values. However, thesevalues can differ significantly from those predicted bythe well-mixed model (a). For small a as the interfaceheight, f, increases the average concentration for thetwo-layer systems decreases, while the one-layer systemremains unchanged, as the average concentration forthe two-layer case is javg = fjl + (1)f)ju. Now, fromBolster and Linden (2008) we know that for a pointsource with small a, the vast majority of the contam-inant remains in the upper layer and that the steadystate concentration for the upper layer is independentof f. Therefore, as the interface height rises, there is lesscontaminant in the space. As a increases the differencebetween the upper and lower layer concentrationsdecreases, because now more particles can fall throughto the lower layer. Therefore, the average concentra-tion of the system is closer to that of a well-mixedspace. Again, it is worth pointing out that the lowerlayer concentration is always less than this average andalso lower than the well-mixed value.

Numerical method – Germeles algorithm

While the solutions to model (c) above are interestingfor comparison purposes, it provides only sufficientinformation to calculate the average amount ofcontaminant and provides no description of the con-taminant distribution within the upper layer. Thismodel displays stratification of contaminant in theupper layer in a manner that models (a) and (b) do not.The detailed structure of the upper layer is not resolvedand, therefore, to determine the vertical concentrationprofile we can employ a modification of a numericalscheme originally developed by Germeles (1975).In this scheme, the background ambient fluid is

discretized into a finite number of layers, n, and it is

α

ζ

10−1 100 101

0.2

0.4

0.6

0.8 (a)

(b)

Γ dep (a) < Γ

dep (c)

α

ζ

10−1 100 101

0.2

0.4

0.6

0.8

Γ dep (a) < Γ

dep (b)

Fig. 5 Comparison of Gdep for the single-layer well-mixed spaceto the two-layer models. (a) To the right of the line Cadep

assumed that the plume evolves on a faster timescalethan the ambient. Therefore, for any given time step,the equations associated with the plume are solvedassuming that the background does not vary. Theequations can be solved through the entire height of theroom using a Runge–Kutta scheme.Once the plume equations have been solved the

background layers, whose concentration and densityremain unchanged during a particular time step, areadvected with the background velocity. This processcaptures the entrainment of fluid from each layer bythe rising plume as the advected layers reduce inthickness at each time step. When the plume reachesthe ceiling a new layer is added, the thickness of whichis determined by the flow rate of the plume at the top ofthe room and size of the chosen time step. Thecontaminant concentration assigned to this new layeris the same as the concentration of contaminant in theplume at the top of the room. For details of thisnumerical method and the plume equations see Bolsterand Linden (2007) and Germeles (1975).

Case 1 – step-down

A sample set of solutions for three values of a atvarious times at an interface of f = 0.5 are shown inFigure 8. As expected, when using the Germeles

algorithm, the upper layer is concentration stratifiedand the well-mixed assumption is questionable. As withpassive contaminants (Bolster and Linden, 2007) theinterface location plays an important role, alwayscorresponding to the maximum concentration in theupper layer. The biggest difference between this caseand passive contaminants is that the concentration atthe interface decreases with time. In the passive casethere is a front that falls from the ceiling towards theinterface, below which the concentration remains thatwhich is initially in the room. For the passive case thedescent time of this front to the interface can be shownto be infinite. However, for particulate contaminants,this front will fall more quickly due to gravitationalsettling and actually reach the interface in finite time.Physically, this can be interpreted as the fact thatparticles can fall through the interface, whereas passivecontaminants cannot. The descent time of this firstfront, sd can be calculated from (18) and decreases withincreasing particle size. It approaches infinity asa fi 0.In addition, the interface always corresponds to the

region of maximum concentration, because, if theconcentrations in the upper layer fall below those ofthe lower layer (which does not happen for passivecontaminants), the lower layer has the highest concen-tration levels and thus so does the lower side of

0 50

0.5

1

τ

κ

0 50

0.2

0.4

0.6

τ

κ

0 0.50

0.05

0.1

τκ

0 50

0.5

1

τ

κ

0 50

0.2

0.4

0.6

τ

κ0 0.5

0

0.05

0.1

τ

κ

0 50

0.5

1

τ

κ

0 50

0.2

0.4

0.6

τ

κ

0 0.50

0.05

0.1

τ

κ

Fig. 6 Step-up case – average concentration for three model with various a and f. One-layer well mixed ()), two-layer well mixed (o),two-layer unmixed (x). The top row corresponds to f = 0.25, the middle row to f = 0.5, and the bottom row to f = 0.75

Bolster & Linden

8

the interface. From an occupant�s perspective thisincreased exposure in the lower layer and interfacialregion is a concern.Once again, as with passive contaminants, the reason

this low-energy ventilation system is not more efficientat contaminant removal than a traditional system canbe explained by the plots in Figure 8. Low-energysystems are efficient from a heat removal perspectivebecause the warmest fluid being extracted from the topof the room. However, the location of the warmestfluid does not coincide with the region of maximumcontamination. Therefore, this energy-efficient mecha-nism does not translate across to contaminant removalefficiency.

Case 2 – step-up

A sample set of solutions for the step-up case is shownin Figure 9. Again, notice that during the initialtransient stage there can be significant gradients is theconcentration field in the upper layer. This timethough, the concentration in the upper layer increaseswith height, because through re-entrainment of con-taminant from the background into the plume, theconcentration at the top of the room is increasing.Also note that the upper layer concentration is

always less than that of the lower layer, as found at

steady state in Part 1 of this project (Bolster andLinden, 2008). The steady-state values shown inFigure 9 correspond to the previously predicted values.Therefore, if the source of contamination is the inlet ofthe ventilation system occupants in the lower layer willbe exposed to the highest levels of contaminant in theroom.Again, for this case these low-energy ventilation

systems do not exploit the same mechanism as they dowith heat removal, thus reducing their potentialcontaminant removal efficiency.

Case 3 – point source

Figure 10 shows the vertical concentration profiles,as calculated with the Germeles algorithm for a pointsource located in a plume, for various values of aand an interface at half the height of the room.There are significant gradients in the concentrationfield in the upper layer, particularly at early times.As with the step-up case the concentration in theupper layer increases with height, because viare-entrainment of contaminant from the backgroundinto the plume, the concentration at the top of theroom is increasing. Therefore, the concentration atthe top of the room is the maximum concentration inthe upper layer.

0 50

0.5

1

τ

κ

0 50

0.2

0.4

0.6

τ

κ

0 0.50

0.05

0.1

τκ

0 50

0.5

1

τ

κ

0 50

0.2

0.4

0.6

τκ

0 0.50

0.05

0.1

τ

κ

0 50

0.5

1

τ

κ0 5

0

0.2

0.4

0.6

τ

κ

0 0.50

0.05

0.1

τ

κ

Fig. 7 Point source – average concentration for three model with various a and f. One-layer well mixed ()), two-layer well mixed (o),two-layer unmixed (x). The top row corresponds to f = 0.25, the middle row to f = 0.5, and the bottom row to f = 0.75

Particle transport in low-energy ventilation systems

9

The concentration in the upper layer is also alwaysgreater than the concentration in the lower layer. Thismakes this scenario very different from either of the

previous two. Now, the concentration being extractedfrom the top of the space is always the maximum con-centration in the space. Therefore, the concentration

0 0.5 1 0

0.5

1 τ = 0.05

0 0.5 1 10

0.5

1 τ = 0.1

0 0.5 0

0.5

1 τ = 0.2

0 0.5 1 0

0.5

1 τ = 0.25

0 0.5 1 0

0.5

1 τ = 0.75

0 0.5 1 0

0.5

1 τ = 2.25

0 0.5 1 0

0.5

1 τ = 5

0 0.5 1 0

0.5

1 τ = 3

0 0.5 1 0

0.5

1 τ = 1.5

a =

0.1

a =

1a

= 1

0

Fig. 8 Step-down Germeles vertical concentrations for f = 0.5 over several a at different times

0 0.5 1 0

0.5

1 τ = 0.5

α =

0.1

α

= 1

α

= 1

0

0 0.5 1 0

0.5

1 τ = 1

0 0.5 1 0

0.5

1 τ = 2

0 0.5 1 0

0.5

1 τ = 0.2

0 0.5 1 0

0.5

1 τ = 0.5

0 0.5 1 0

0.5

1 τ = 1

0 0.05 0.1 0

0.5

1 τ = 0.03

0 0.05 0.1 0

0.5

1 τ = 0.06

0 0.05 0.1 0

0.5

1 τ = 0.2

Fig. 9 Step-up Germeles vertical concentrations for f = 0.5 over several a at different times

Bolster & Linden

10

extraction process is exploiting the same mechanismas the temperature extraction mechanism that makesthese low energy systems so appealing. This translatesinto optimum efficiency of contaminant removal,which can clearly be seen in the previous discussion

on averages and transients in this scenario. Anotherfeature shown in Figure 10 is that ratio of theconcentration of upper to lower layers approachesunity as a increases, thus approaching an entirelywell-mixed space.

0 0.5 1 0

0.5

1 τ = 0.5

α =

0.1

α

= 1

α

= 1

0 0 0.5 1

0

0.5

1 τ = 2

0 0.5 1 0

0.5

1 τ = 6

0 0.5 1 0

0.5

1 τ = 0.25

0 0.5 1 0

0.5

1 τ = 1

0 0.5 1 0

0.5

1 τ = 2.5

0 0.05 0.1 0

0.5

1 τ = 0.05

0 0.05 0.1 0

0.5

1 τ = 0.1

0 0.05 0.1 0

0.5

1 τ = 0.4

Fig. 10 Point source Germeles vertical concentrations for f = 0.5 over several a at different times

0 1 20

0.5

1

τ

κ

ζ = 0.95

0 1 20

0.5

1

τ

κ

ζ = 0.8

0 1 20

0.5

1

τ

κ

ζ = 0.6

0 1 20

0.1

0.2

0.3

τ

κ

ζ = 0.3/0.1

Fig. 11 Experimental results for a = 0.1. The solid line represents the theoretically predicted concentration using the Germeles model.The error bars represent the experimental measurements

Particle transport in low-energy ventilation systems

11

Experiments

To validate the models presented here and gain furtherinsight into the dynamics of particles in low-energyventilation systems, a series of full scale laboratoryexperiments were conducted. A chamber of cross

sectional area 1.3 · 2.6 m and height 1.8 m wasventilated by a displacement system. Although theexperimental chamber is not as tall as a typical interiorspace, it is large enough to ensure that the mainphysical processes are represented accurately. A heatsource of 65 W is placed in the center of the room. The

0 1 2 0

0.1

0.2

0.3

τ

κ

ζ = 0.95

0 1 20

0.1

0.2

0.3

τ

κ

ζ = 0.8

0 1 2 0

0.1

0.2

0.3

τ

κ

ζ = 0.6

0 1 20

0.1

0.2

0.3

τ

κ

ζ = 0.1/0.3

Fig. 13 Experimental results for a = 2.5. The solid line represents the theoretically predicted concentration using the Germeles model.The error bars represent the experimental measurements

0 1 2 0

0.5

1

τ

κ

ζ = 0.95

0 1 2 0

0.5

1

τ

κ

ζ = 0.8

0 1 2 0

0.5

1

τ

κ

ζ = 0.6

0 1 2 0

0.1

0.2

0.3

τ κ

ζ = 0.3/0.1

Fig. 12 Experimental results for a = 0.625. The solid line represents the theoretically predicted concentration using the Germelesmodel. The error bars represent the experimental measurements

Bolster & Linden

12

temperature in the background was measured atvarious heights (30, 60, 90, 120, and 150 cm) usingtype K thermocouples. The heat source consisted of alight bulb, which is encased in a specially constructedwooden enclosure (0.2 · 0.2 · 0.22 m) to minimizeradiative losses. Wrapping the box in materials oflow emissivity, such as aluminium foil, had negligibleeffect on the background temperature and so radiativelosses were considered small. Because of the difficultiesinvolved in generating a step-up or step-down situationwith particles, the only situation which we considerhere is the point source in the plume, which stilldemonstrates the interesting dynamics associated withthis flow.The interface dividing the upper and lower layers is

designed to sit at a height corresponding to half theheight of the room (i.e. f = 0.5). Three values ofa = 0.1, 0.625, and 2.5 are considered. Unfortunatelyvalues of a > 2.5 were inaccessible with our currentequipment. Once the temperature in the room hasreached steady state, monodisperse particles of knownsize are injected vertically in coflow into the plume witha medical nebulizer. The particles used are polystyrene�microbead NIST traceable particle size standard�manufactured by Polysciences Inc. (Warrington, PA,USA) and are manufactured to within a ±2.5% sizestandard. The flow rate of injection is very smallcompared with the flow rate in the plume and so itseffect is considered to be negligible. Running thenebulizer does not affect the temperature field and sothis seems a reasonable conclusion.The contaminant source was turned on for a certain

amount of time son ¼ 56 s, where s is the flushing time(i.e. Volume=Qin) and then switched off. A model237A/B Met One Particle Counter (Hach Ultra, GrantsPass, OR, USA) was used to detect and count particles.It is placed at various heights within the room andparticle concentrations were measured periodicallyevery 30 s. The particle concentrations are tracked untilthey return to the background noise levels, so that boththe increase and drop in concentrations were captured.The results of these experiments are shown in

Figures 11–13. The error bars on the experimentaldata are associated with the uncertainty in themeasurement of particle concentration and the rate ofrelease by the nebulizer, which corresponds to roughly±20%. In all three cases the qualitative agreementbetween experiment and models is good. The quanti-tative agreement is generally within the ±20% errorbars. However, there are some regions of largerdisagreement and occasional measurements that arequite far off the predicted value. These can stem fromexternal infiltration or some other contaminant sourcethat we could not control in our experiment.An important feature to note in the experimental

data, which is particularly obvious in Figures 11 and12, is that the upper layer is clearly not well mixed as

there is a measurable time delay in the concentrationfield at f = 0.6 compared to f = 0.95 associated withthe advection of particles from the top of the room tothat height. Experimentally, this time delay is not soobvious for the larger particles (i.e. a = 2.5) because,as a result of faster settling, the time for particles todescend becomes comparable to the measurement timeand so the concentration fields for the three uppervalues of f become indistinguishable.In addition, for the largest particles, there seems tobe a

delay in the experimental data for the initial rise ofconcentration in the upper layer compared to thetheoretical prediction. The peak-predicted value iscomparable, but the time taken to get there is longer.This is probably, because we neglect the effect of settlingwithin the plume. The present models predict an instantincrease in concentration once the source is turned on.This comes from the assumption that the plume timescale ismuch faster than that of the background,which istrue. However, there may be a lag in this because ofsettling effects. This theoretical over-prediction of theupper layer concentration also leads to an over-predic-tion of the lower layer concentration.

Conclusion

In this paper, we have considered the transient trans-port of particulate contaminants in a displacement-ventilated space. We compared three models, onerepresenting a traditional ventilation system and theother two representing the displacement-ventilatedspace. We considered three types of contaminationscenarios, namely a step-down, a step-up and a pointsource contaminant. Several important differencesbetween the traditional and low-energy systems werenoted.While for passive contaminants it is often argued

that studying one of the step-up, step-down and pointsource contamination scenarios is equivalent to study-ing them all, this is not the case for particulatecontaminants and the low-energy displacement systemdescribed here. The influence of gravitational settling isto introduce an �irreversibility� or �preferential direc-tion� to the flow, which destroys the symmetry of thestep-up, step-down and point source methods. There-fore, it is important to understand the differencesbetween these three scenarios as each one givesimportant information about real contamination sce-narios. The governing equations presented in thispaper are all linear and, therefore, each scenario canbe studied separately, and superposition of solutionscan be used to study more complex situations.It is widely believed that low-energy displacement

ventilation systems can be better than traditionalmixing systems at removing contaminants from aspace. This is because there is a belief that thesesystems will use the same mechanism for contaminant

Particle transport in low-energy ventilation systems

13

removal as they do for heat removal, where they areclearly more efficient. The heat extraction problemexploits the natural stratification that develops,extracting the warmest air that naturally sits at thetop of the room. However, there is no physicaljustification as to why this location should correspondto the location of maximum contaminant concentra-tion too. In fact many times it does not (Bolster andLinden, 2007).For the step-down case we considered the quantity of

particles deposited to the floor as an indicator of thefraction of contaminant that an occupant has beenexposed to. For smaller particles and low interfaces(i.e. small a and f) the low-energy displacement systemperforms better than the traditional mixing system.However, for larger particles this index indicates thatthe low-energy systems cause higher levels of exposure.Allowing for gradients in the upper layer [i.e. model (c)]leads to higher exposure than the well-mixed upperlayer, because the concentration at the interface ishigher than at the top of the room (like the passivecontaminant case in Bolster and Linden, 2007), thusallowing higher concentrations to settle back down tothe lower layer.Similarly for the step-up case, we showed that, at

steady state, the concentration in the lower layer isgreater than that of the upper layer. This is because thelevel of maximum concentration is in the lower layer,but the level of air extraction is in the upper layer, thusnot exploiting the �energy� benefit displacement venti-lation offers where the warmest air is extracted at thetop of the room.On the other hand, when considering the point

source scenario, we predict a higher steady-stateconcentration in the upper layer compared with thelower layer. The lower layer concentration will alwaysbe less than that in an equivalent traditional system,

thus reducing occupants� exposure to contaminants.Here, unlike the step-up and step-down cases, thesituation exploits the same mechanism as the heatremoval problem, always extracting the most contam-inated air from the top of the room.As mentioned in Part 1 of this work, the information

obtained in this work clearly has an impact on thedesign of an effective ventilation system and the specificdesign requirements. In some cases a displacementsystem may provide better air quality than a traditionalmixing ventilation system. On the other hand, if weconsider a naturally ventilated space, where externalsources can play an important part in contamination,the step-up scenario may be relevant and a mixingsystem could be preferred.Finally, one point that should hopefully be clear

from both Part 1 and Part 2 of this work is that it isnot sufficient to know the average concentration ofcontaminant in a space, both from an exposure andfrom a modeling perspective as such limited knowl-edge can lead to over- and under-predictions ofcontaminant concentrations, which in turn can leadto over- and under-designed ventilation systems,which can be wasteful of energy or not adequate interms of indoor air quality. Integrating models suchas those presented here can aid in better design. Inaddition, there is no reason to believe that themodels presented here could not be developed toinclude further transport mechanisms associated withimportant contaminants (e.g. chemical reactions,VOC transport, etc.).

Acknowledgement

The authors would like to thank the CaliforniaEnergy Commission for their financial support of thisproject.

References

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Appendix A. Concentration at top of room

When a new layer forms at the top of the room it has[Qp(z = H)]Dt fluid being supplied to it by the plumeand (Qout)Dt being extracted by the vent. The size ofthe new layer, because of gravitational settling, is[Qp(z = H) + Qf)Qout]Dt. Therefore, assuming per-fect mixing in this new layer the concentration of thisnew layer will be

C ¼ QpQp þQf

Pðz ¼ HÞ; ðA:1Þ

which can readily be shown to be

C ¼ 11þ 3af5�2f

Pðz ¼ HÞ: ðA:2Þ

Particle transport in low-energy ventilation systems

15