INCOMPLETE MIXING IN VENTILATED AIRSPACES, PART II.

SCALE MODEL STUDY

E. M. Barber and J. R. Ogilvie

Department of Agricultural Engineering, University of Saskatchewan, Saskatoon, Sask. S7N 0W0; and School ofEngineering, University of Guelph, Guelph, Ontario NIG 2WL

Received 29 August 1983, accepted 4 April 1984

Barber, E. M. and J. R. Ogilvie. 1984.study. Can. Agric. Eng. 26: 189-196.

Incomplete mixing in ventilated air spaces. Part II. Scale model

Rate-of-decay tracergas experiments conducted in a one-fifth scalephysicalmodelof a slot-ventilated airspacetestedthe validity of a two-parametermathematicalmixing model. The tracer gas was a density-balanced combination of C02andCH4. The mixingmodel, previously developed andvalidated for aqueous mixingsystems, accounted for incompletemixing in terms of short-circuiting and stagnant zones. Called the w,m-model, it was expressed as:

(Ce-Q/CCo-Q = net-wmv)

Extensive departure from complete mixing was shownto have occurredin the scale model airspace. Results from themodel study, using the tracerdata, invalidated the original assumptions by exceeding the bounds of the physical parameters n andm. The results suggested othermoresignificant reasons for departure fromcomplete mixing. The natureand extent of departure from complete mixing in a particular airspace was shown to be a function of the Archimedesnumber (corrected), the inlet jet momentum, the inlet/outlet configuration, and the geometry of internal obstacles toairflow.

INTRODUCTION

In a theoretical analysis of incompletemixing, Barber and Ogilvie (1982)showed that there were several models ofthe ventilated airspace in a livestock building. Experiments reported here tested certain hypotheses expanding on thesemodels.

Given a control volume with a definedinlet and outlet, two extreme ideal flowpatterns are theoretically possible. Thefirstof theseiszeromixing,orperfectplugflow. Withplug flow, incomingair wouldmove as an intact wave from the inlet to

ward the outlet. Given a change in someproperty of the air at the inlet (i.e. a stimulus), the response at the outlet would lookjust like the stimulus, but would be delayed by the length of time taken for theair to travel the distance between inlet andoutlet.

At the opposite extreme is the ideal flowregimeof perfect mixing. Air entering atthe inlet would be instantaneously anduniformly dispersed throughout the airspace.Theconcentration of a tracer at anypoint in the airspace, or at the outlet,wouldbe described by the mixing-dilutionequation,

non-ideal flow. In the first class are the

dispersed plug flow models, or simply dispersion models. The control volume contents are assumed to be displaced by plugflow, but dispersion occurs along themoving front due to backmixing. Dispersion models are most satisfactory whereflow conditions deviate only slightly fromplug flow.

The second class of models for non-

ideal flows consist of the mixed or "mix

ing" models. This class encompasses avariety of models wherein the control volume is envisaged as a network of flow regions connected in series or parallel. Flowoccurs within, between, and around theseinter-connected regions. Mixing modelsare used most satisfactorily for flow regimes which are closer to perfect mixing

than the extreme of plug flow. A comparison of the response of ideal and non-idealflow systems to a step change in the concentration of a tracer at the inlet is presented in Fig. 1.

Theoretically, a dispersed plug flow regime might be realized in livestock buildings under two conditions. One of these isthe case of summer ventilation in long airspaces where air enters through large inlets in one end wall and is exhausted

through fans at the opposite end of the airspace. The fitting of a dispersed plug flowmodel to this flow regime apparently hasnot been accomplished. The second mostlikely possibility for dispersed plug flowexists in the case of barns equipped withporous ceilings. Experience in Canada(Honey and Wrubleski 1977) and in Eng-

VdCldt = Q (C-O (1)

where symbols are as defined in Table I.Neitherplug flow (perfectnon-mixing)

nor perfect mixing are likely to be a realistic model of the actual flow regimewithin the ventilated airspace of a livestock building. Levenspiel (1972) definestwo general classes of models to describe

TABLE I. LIST OF SYMBOLS

a = vertical dimensions of inlet openingb = horizontal dimensions of inlet openingm = portionof the airspacethat is well mixed; remainder(1-m)is totally stagnantn = portion of the incoming air that enters the mixing zone; remainder (l-n) is

short-circuited to exhaust

t = time

A = value of slope of line of log CR versus /ARC = Archimedes numberB = length of building supplied by the inletC = concentration of tracer in the well mixed zoneCe = concentration of tracer in the exhaust airstreamQ = concentration of tracer in the incoming airstreamC0 = concentration of tracer in the mixing zone at time = 0CR = concentration ratio C/Ct or C-CJC0-CXH = ceiling height of buildingQ = air flow through the inletTs = surface temperature of the pigsT{ = outdoor air temperature or temperature of airentering through theinletV = volume of the airspace

CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984

Units

m

m

sec

sec-1

m

m3/m3

m3/m3

m3/m3

m3/m3

m

m3/sec

°C

°Cm3

189

PerfectMixing ^-

Mixed V">^'

\ STIMULUS

1 RESPONSE

'/^Dispersed' ^Plug flow

-•—Plug Flow

Time

Figure 1. Response of control volume fordifferent flow regimes.

land (Randall 1975) indicates that airflowin these buildings occurs in a series of rotary patterns which are more characteristicof mixed systems than plug flow systems.Under most conditions, then, mixingmodels appear to offer more promise thandispersed plug flow models for the studyof mechanically ventilated airspaces.

Cholette and Cloutier (1959), in theirwork with chemical reaction vessels, developed a mixing model which takes intoaccount an effective volume of mixing,possible short circuiting and stagnantzones. In the simplest of their models, thecontrol volume, V, (Fig. 2) is visualizedas two distinct zones: a fraction, m, of thesystem volume is considered to be completely mixed and the remaining fraction,1-m, is considered to be stagnant. Furthermore, the incoming airstream is visualized in two parts: a fraction, n, of theincoming flow enters the completelymixed zone while the fraction, 1-n, isshort-circuited to the outlet.

The two parameters, n and m, may beestimated in rate-of-decay tracer experiments. A mass balance on the tracer gasin the completely mixed zone provides theequation:

mVdCldt = -nQ(C-Cd (2)

Assuming an equilibrium concentration,C0, at t = 0, and given a non-zero inlet

QCj

Figure 2. Volume-apportionment model ofincompletely-mixed airspace.

190

tracer concentration, C\, during the decayperiod following a step change in the inlettracer concentration, the solution to Eq. 2may be written:

(C-Q/(C0-Q = *-**v> (3)

A mass balance on the entire system yieldsthe relationship:

(1 - n)gC, + nQC = QCe

which simplifies to:

C = (C.-C, + nO/n (4)

Substituting Eq. 4 for C in Eq. 3,

((Ce-Ci + nQM-Q/^o-Q = e<-n<*"*v> (5)

Resolving to

(C.-CJ/(Co-Q = n*-"*^ (6)

Equation 6 is the mathematical expressionof the volume-apportionment model and iscalled the n,m-model. The parameters canbe obtained by expressing the relationshipin linear form (Y = A + BX) if Eq. 6 istransformed as;

In (C. - Q/(C0 - Cd = In n + {-nlm QtIV)(7)

The plot of the log of the concentrationratio, ln(CR), versus the air exchangenumber, QtIV, is a straight line, where Inn is the intercept on the logarithmic axis(at t = 0), and the slope of the line is— nlm.

Cloutier and Cholette (1968) successfully validated the two-parameter modelof Eq. 6 for small-scale mixed tanks. Mil-bury et al. (1964) likewise used physicalscale models to validate the n,m-model forpredicting flow in aeration vessels.

The traditional mixing model employedin ventilation research has been the singleparameter model, derived from Eq. 1. Ina rate-of-decay tracer gas experiment, theplot of ln((C-Ci)/(C0-Q) versus t will bea straight line with slope equal -A, whereA has units of time _1 and represents theeffective air exchange rate. If Q and Vbothare known, the log concentration ratio, In(CR)9 may be plotted against the dimen-sionless air exchange number, QIVt, andthe slope of the line will be -K, where Kis the ratio between the effective air ex

change rate, A, and the calculated air exchange rate, QIV. The equation of thestraight line may be written as

In (C - Q/(C0 -Cd= ~ K QtIV (8)

A comparison of Eqs. 8 and 3 will showthat the single parameter model (hereinafter referred to as the ^T-model) is simplya special case of the more general n,m-model where K = nlm.

The jfif-model accounts for two possiblesources of departure from complete mix

ing:

1. short-circuiting. If some of the incoming air travels directly to the outletwithout mixing with room air, the numberof effective air exchanges within the roomwill be less than the calculated theoretical

number of air changes and K will be lessthan 1.

2. stagnant zones. If a portion of theairspace is occupied by dead space (i.e.,air that essentially does not move or exchange with the main airflow region), theeffective volume of the ventilated portionof the room will be reduced; hence thenumber of effective air exchanges withinthe well-mixed region will appear to begreater than the theoretical number of airchanges and K will be greater than one.

The ^f-model provides no basis for distinguishing between the counteracting effects of stagnant zones and short-circuiting where both occur simultaneously. Thetwo-parameter n,m-model could theoretically overcome this deficiency.

OBJECTIVES AND HYPOTHESESExperiments were designed to test two

hypotheses related to the w,m-model:I. The ventilated airspace behaves as an

incompletely mixed fluid system whereina portion of the incoming air is short-circuited to the outlet and a portion of theairspace volume is stagnant. The extent ofthese causes of departure from completemixing may be defined by two coefficientsin the n,m-model and may be experimentally assessed for a particular airspace bya stimulus-response tracer gas technique.

II. The nature and extent of departurefrom complete mixing in a particular airspace is a function of the corrected Archimedes number, the inlet jet momentum, the inlet/outlet configuration, and thegeometry of internal obstacles to airflow.

Acceptance of this latter hypothesis willsupport the selection of the Archimedesnumber as the most appropriate dynamicsimilarity criterion for the design of non-isothermal scale-model ventilation stud

ies. Furthermore, if the contribution to explanation of variance is sufficient, the resulting empirical model may haveapplication as a tool for prediction of incomplete mixing.

Experimental FacilityThe experiments were performed in a

one-fifth scale model, shown in Fig. 3, ofa single, self-contained room in the SwineResearch Centre, Arkell, Ont. The insidedimensions of the model were 2 m x 2.4m x 0.64 m. A removable plywood bafflecreated a floor-level obstruction 0.25 m

high and perpendicular to the primary air-

CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984

®5

® Thermocouple& Gas Probe

1,2 3,4® L

Figure 3. Scale model building.

Figure 4. Schematic of model airspace showing tracer gas and temperaturemonitoring locations.

lyzer was recorded on a digital data logger(Fluke Model 2240B), then stored onmagnetic tape prior to analysis on a microcomputer (Tektronic 4051).

Rate-of-decay tracer gas experimentswere performed using a density-balanced

matograph (Hewlett-Packard Model5830A - Chromosorb 102 80/100 col

umn). The controlled flow rate througheach pump circuit was monitored by inline flowmeters.

The analog signal from the infrared ana-

M

IS

InfraredCOoAnalyzer

V

{.GasChromatograph

^I rVv

Hj-

D Q ffi>

v

0

• Sample frommodel airspaceat location i

Variable

sample linelength

Foam-rubber

filter

3-way solenoidvalve in

de-energizedposition

Rotameter

Manual flow-

control valve

Diaphragmpump

Q) Peristalticpump

Exhaust to

atmosphere

Intake from

atmospherefor flow-ratebalancing

flow. Electric heater cable, to which variable power input up to 1200 W could beapplied, was fixed to the floor of themodel. The scale model was placed insidea controlled environment chamber andventilated with chilled air from within thechamber.

At the top of one 2-m-long wall, aplexiglass baffle created a continuous slotinlet of variable opening height. A 75-mm-diameter exhaust port was located inthe center of each 2-m-long wall, one onthe wall beneath the inlet and one on theopposite side of the airspace from the inlet. In any one trial, one of the exhaustports was sealed shut. Air from the openport was exhausted from the environmental chamber andaway from the laboratorybuilding.

Air samples were withdrawn from sample locations 1,2,3, and 4 (shown in Fig.4). These samples, all drawn at the sameflow rate, were blended into a compositesample prior to analysis. This compositesample will be referred to as the innerchambersample. An outlet air sample waswithdrawn from sample location 5 or 6 depending on whicn exhaust port was active.

The gas sampling system was designedto enable sequential, repeated samplingsfrom both the inner chamber and the outletlocationduring a single trial. The system,outlined in Fig. 5, continuously drew samples through two separate sampling lines.Three-way solenoid valves were used toswitchthe sample line being purged. Oneline exhaustsdirectly to atmospherewhilethe sample drawn by the other pumppassed through an infrared C02 gas analyzer (Beckman Model 864) before discharge to atmosphere or to a gas chro- Figure 5. Schematic of air sampling andanalysis system.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984191

composite tracer gas consisting of C02 andCH4. Introduction of the gas occurred in amixingchamber just upstream from the inlet baffle and was controlled by a solenoidvalve.

In a typical trial, the ventilation rate wasset, the temperature of the supply air andof the airspace were adjusted and the system was allowed to come to equilibrium.Tracer gas then was introduced into thesupply air at a rate that would establish anequilibriumC02 concentrationof between1.5 and 2.5% within the model airspace.The supply of tracer gas to the mixingchamber was then abruptly halted and theC02 concentration was monitored in theexhaust air and, in most trials, at the innerchamber sample locations.

The exact time the tracer gas supply wasshut off, relative to the time of samplewithdrawal from the airspace, was corrected to account for transport delay timesand a zero time (t0) was established foreach sample location. The transport delaytime in each sample line was determinedby introducing a burst of C02 at the openend of the line and measuring the time until a response was noted at the analyzer.Accurate estimation of the intercept term,In n, in Eq. 7 depends on precise measurement of the zero time.

Experimental DesignRandall and Battams (1979), concluded

that there is no significant relationship between the direction of airflow in a confined airspace and the Reynolds number.Theydemonstrated instead, that the initialpathof air entering and mixingwithintheairspace is completely described by geometrical relationships and a corrected Ar

chimedes number defined explicitly as:

Atc (5.89 (ab)(BH)(B+H)(T-Td)l(Q2(546+ J,+ 7,))

The Archimedes number represents theratio between buoyancy and inertial forcesacting on the inlet jet. For isothermal conditions, buoyancy forces will not be a factor and it might be expected that the behavior of the air jet would be related to theReynolds number. For non-isothermalconditions, however, buoyancy forcescannot be neglected and intuitively the jetbehavior should be described better by theArchimedes number than by the Reynoldsnumber. Following the recommendationsof Randall and Battams (1979), the experiments reported herein were designedon the basis of Archimedes number simi

larity.Six parameters were selected for study:

(1) ventilation rate, Q (L/sec), (QNOM);(2) mean inlet jet velocity, ux (m/sec),(VNOM); (3) inlet slot height, a (m),(SLOT); (4) inlet-outlet separation,(IOSEP); (5) presence or absence of a bar

rier to airflow, (BAFFLE); and (6) temperature difference between inside and incoming air T (°C) (DTNOM).

Although Q, ux and a are not linearlyindependent, all three parameters are retained for convenience in interpretation ofthe results.

The experimental design layout (TableII) shows how the various parameters wereintroduced. QNOM, VNOM, SLOT andDTNOM represented nominal or designvalues for the ventilation rate (L/sec), inletjet velocity (m/sec), inlet slot height (mm)and airspace to supply air temperature difference (°C), respectively. The number ineach box represents the case numbershown in Table III with the second digitbeing the building configuration (i.e., 10-3 is Case 10, building type 3).

The model ventilation rates of 4, 12,and 36 L/sec represent prototype ventilation rates of 2.1, 6.3, and 18.9 air exchanges per hour; such rates are typical ofcold to cool weather ventilation rates in

commercial swine barns (Turnbull andBird 1979). The inlet jet velocities are rep-

TABLE II. EXPERIMENTAL DESIGN

Building

DTNOM °C

Typel Type 2 Type 3 Type 4

5 15 5 15 5 15QNOML/sec

SLOT

mm

VNOM

m/sec 5 15

4

4

12

12

36

36

1.0

1.73

1.73

5.20

5.20

8.00

2.5

1.5

4.3

1.5

4.3

2.5

8-lt10-1

4-1

7-1

1-1

2-1

11-1

12-1

6-1

9-1

3-1

5-1

8-2

10-2

4-2

7-2

1-2

2-2

11-2

12-2

6-2

9-2

3-2

5-2

8-3

10-3

4-3

7-3

1-3

2-3

11-3

12-3

6-3

9-3

3-3

5-3

8-4 11-4

10-4 12-4

4-4 6-4

7-4 9-4

1-4 3-4

2-4 5-4

tCase no. - Building type.

TABLE III. MEAN TRACER STATISTICS SUMMARIZED BY TREATMENTS WITHIN MAIN DESIGN

Case 1 2 3 4 5 6 7 8 9 10 11 12

Ar 3 4 6 7 10 17 20 30 50 55 80 130

Ji(x 103) 62 36 62 21 36 21 7 4 7 2 4 2

QNOM 36 36 36 12 36 12 12 4 12 4 4 4

VNOM 4.5 2.5 4.5 4.5 2.5 4.5 1.5 2.5 1.5 1.5 2.5 1.5

DTNOM 5 5 15 5 15 15 5 5 15 5 15 15

Param. Bldg1 0.93 0.90 0.94 1.04 0.99 1.01 1.05 1.02 1.01 0.97 0.92 0.92

K 2 1.08 1.09 1.00 1.02 1.03 0.94 0.93 0.86 0.84 0.85 0.84 0.83

3 0.98 0.99 1.06 1.02 1.04 1.02 1.09 1.04 1.11 1.05 0.95 0.95

4 0.97 0.99 0.98 0.96 0.96 0.91 0.84 0.85 0.80 0.82 0.80 0.81

1 1.00 _ _ 0.99 0.98 0.97 1.00 0.97 0.95 0.98 1.02 1.01

Al 2 _ _ 0.97 1.06 1.01 0.98 0.99 0.93 1.02 1.01 1.00

3 0.99 _ _ 0.97 0.96 0.97 1.00 0.97 0.95 1.01 1.00 1.00

4 1.03 0.95 - 0.97 0.99 0.99 0.96 0.98 0.95 1.01 1.01 1.00

1 0.93 0.91 0.94 1.01 1.00 0.99 1.03 1.06 1.02 1.01 1.04 1.01

n 2 1.05 1.09 1.02 1.00 1.04 0.97 0.91 0.93 0.87 0.92 0.95 0.94

3 1.00 0.99 1.11 1.01 1.08 1.02 1.07 1.09 1.12 1.13 1.06 1.07

4 0.99 1.02 0.99 0.95 1.00 0.94 0.84 0.89 0.83 0.88 0.92 0.90

1 1.00 1.01 1.00 0.97 1.01 0.98 0.98 1.04 1.01 1.04 1.13 1.10

m 2 0.97 1.00 1.02 0.98 1.01 1.03 0.98 1.08 1.04 1.09 1.13 1.14

3 1.02 1.00 1.04 0.99 1.04 1.00 0.98 1.05 1.01 1.07 1.12 1.13

4 1.02 1.03 1.01 0.99 1.04 1.04 0.97 1.04 1.03 1.07 1.15 1.11

192CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984

resentative of prototype jet velocitiesranging between 3.3 and 9.6 m/sec,bracketing the 5-m/sec level commonlyrecommended for good air distribution inslot-ventilated airspaces (Turnbull andBird 1979).

RESULTS AND DISCUSSIONA total of 136 trials were conducted

within the experimental plan of Table II.At least two trials were carried out for eachtreatment. The order in which trials were

conducted was randomized. The equilibrium C02 concentrations ranged between1.5 and 2.3%. The density of the composite tracer was nearly equal to that ofnormal air, varying between 0.5% heavierthan air to 0.4% lighter than air. Withinthis range, the results of the tests were independent of tracer gas density (Barber1981).

Actualventilationrates achievedduringthe trials varied from 4.1 to 4.5 L/sec, 12.1to 13.0 L/sec, and 36.7 to 37.6 L/sec fortreatments designated 4, 12 and 36 L/sec,respectively. The temperature differenceachieved between the inlet and the bulkairspace varied from 4.6 to 7.6 °C fortreatments designated as DTNOM = 5,and from 15.0 to 18.5°C for treatmentsdesignated as DTNOM = 15.

For each trial conducted at either 4 or12 L/sec, plots of CR versus (t -10) weregenerated for the outlet and inner-chambersample locations. In trials at 36 L/sec, therate of decay of tracer concentration wastoo rapid to sample from two locations;hence, for most trials, only the outlet location was monitored. Figure 6 shows theresults from separate trials.

In accordance with the w,m-model, foreach sample location plots were preparedof In (CR) versus the dimensionless airexchange number, QtIV. A linear regressionwas performed for each plot and theslope and intercept were recorded. The results of this analysis are shown in TableHI. The parameter, Al, is the interceptterm in the best-fit linear equation for theresponse of the inner chamber sample location. The inner chamber was not sampled in all trials at 36 L/sec hence somevalues for Al are missing from Table III.

Testing the n,m-modelAccording to the concepts embodied in

both the AT-model and the n,m-model, theregression plots for the inner chamber andoutlet sample locations should have identical slopes. The hypothesis, that there isno difference between the slope of theregression plots for the outlet and innerchamber sample locations, was tested forthe 88 trials where both locations were

<-iIda

128

188

88

68

48

28

8

-28

128

+*♦♦♦* ^-^

TRIAL NO. 810

>♦♦♦+++++

-18 -5 8 5 18 15 28 25 38

TIME Cln minutes from ZERO tla«)

• 1i ' • « • 1

J • 8MVUM6 OIKS)

i*! TRIAL NO. 682

1 *\ *

_ _ J

48

28

-28-18 -5 8 5 18 15 28 25 38

TIME On mlnuUs from ZERO tim)

-5 8 5 18 15 28

TINE Cln ninuUs fro. ZERO tlM>

Figure 6. Typical decay curves.

monitored in the same trial. A high correlation coefficient (R2 = 0.992) and a t-test indicate that the means of the twoslope estimates were not significantly different(P > 0.95). Thenullhypothesis wasaccepted on the basis that the mean difference between the two sample meanswas less than 0.5% which was very smallcompared to potential experimental error(Barber 1981). The slopes of the regressionplotsare the samefor the twosamplelocations.

The two coefficients in the n,m-modelare estimated from the slopes and the intercepts of regression curves from rate-of-decay tracer data sampled from the outletlocation. Figure 7 shows the regressioncurves for both sample locations in one

trial. The results presented in Table III indicate, however, that certain theoreticalpremises underlying the w,m-modelare violated with the airspace modeled in theseexperiments.

The /*,m-model presumes that the airspace behaves as a single mixed zone withan attached stagnant zone. Rate-of-decaydata collected from within the mixed zoneshould, when plotted as in In (CR)versusQtIV, produce a straight line with an intercept of 1.0, if, in fact the zone is wellmixed. As shown in Table III, several intercepts different from 1.0 are indicatedfrom the current trials.

Inner-chamber intercepts deviatingfrom 1.0 could arise as a consequence oferrors in estimation of zero time, the inlet

CANADIANAGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984193

Qt/V

0.0 0.5 1.0 1.5 2.0 2.5

Qt/V

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0 1 r-

8106

o

0.0

-0.5

-1.0

i i i i

— -0.5 ^« X x^1

6"io

Trial No. Trial No. 810

O -1.0

6"

O

i-1.5

O -1.5

c

o-i -2.0

-2.5

K = 0.940. Al = 1.01

0)

O

c

o>o-J

-2.0

-2.5

-3.0

. K =n =

m

=0.944

=0.946= 0.998

Figure 7. Regression plots from infrared analyzer data.

nomenon. The intercept, n, would beoverestimated by about 6% if C0 wasunderestimatedby 0.1 (i.e., if C0 were incorrectly measured as 1.7% instead of thecorrect value of 1.8%). Errors of this magnitude seem unlikely. Errors in estimationof the zero time are possible but wouldhave the least effect at low flow rateswhere the largestvalues forn appear. Bothof the above types of error would applyequally to values for Al and n. Thus, if all

120

values for n were reduced by 10% to correct for supposed errors, then Al valueswould be reduced in many cases to numbers much less than 1 and could not be

explained by the w,m-model. Errors in estimation of Cx could cause incorrect estimates for n; however, errors in C{ estimation areeasily detected since they showup as a curvature in the plots of log concentration ratio versus QtIV, curvature thatwas not apparent in these trials.

As in the case of values for n, valuesgreater than 1 for m (Fig. 10) also arephysically impossible. Error in measurement of the ventilation rate could lead tosuch errors in m. The overestimation ofvalues for m also could be due to an over-

estimation of n since m is derived from the

slope divided by n.Assuming that the above-noted incon

sistencies between the data for m and nand the theory of the n,m-model could beaccounted for by experimental error, therestill are other reasons to doubt the validityof the H,m-model as a reasonable model ofthe experimental airspace. For example:

(1) If all n values are assumed to be 10%too high, then a significant amount ofshort-circuiting would be indicated forbuilding configurations 1 and 3, even at

tracer concentration, and the equilibriumtracer concentration (Barber 1981). Errorsin t0, however, shouldbe consistent for alltrials for a particular flow rate and hencewill not contribute significantly to within-treatment variations of Al. The combinederrors due to the remaining causes seemlikelyto account for atmost a variation of±0.025 in the estimation of Al (Barber1981). The intercept values in Table IIIsuggest that the linear regression plot forthe inner chamber sample location doesnot, for all treatments, necessarily passthrough the origin.

The second inconsistency of the datarelative to the n,m-model is the occurrenceof intercepts greater than 1.0 for the outletsampleregressions (Fig. 8). Decaycurvesfor the two sample locations within twotrials differing only in outlet location arecompared inFig. 9. For trial 501 thedecaycurve for the outlet sample locationis seento lie entirely beneath the decay curve forthe inner chamber sample location, whiletheopposite is true for trial 612. Whiletheformereffect might be explained in termsof short-circuiting, the n,m-model doesnot offer anexplanation forthe latter phe

zo

iiorHZIdOZoo

orId

%or

100- o«oO«o^0

80

60

40

20

0

"ini

Figure 8. Values for n versus corrected Archimedes number (Arc).

194

UJ>

UJor

-20,

o

• o

° 0 TRIAL NO. 501* • o

• o• o

• o

•**-^

"**>.**w

-ICH -5 0 5 10 15 20 25 30

TIME (lN MINUTES FROM ZERO TIME)

zo

<or

120

100

80

60

^•^•o^**cd

i— 1 1 1 »

• L° i ° r

o •

o •

o • To •

o •

°°V*<*

RIAL NO. 612oo

orUJo 40<or

UJ>

UJor

20

0

-20,-IO -5 0 5 10 15 20 25 30

TIME (in minutes from ZERO time)

Figure 9. Comparison of decay curves attwo sample locations.

CANADIAN AGRICULTURALENGINEERING, VOL. 26, NO. 2, WINTER 1984

high Arc (Fig. 8). However, at high Arc,the supply air jet would be expected to descend to the floor on entry and short-circuiting should be minimal for the configuration of the outlet on a wall oppositefrom the inlet.

(2) Insertion of the baffle appears tohave decreased the extent of short-circuiting for building configuration 1 and tohave increased the extent of short-circuiting for building configuration 2 (Fig. 8).There does not appear to be a physical explanation for this phenomenon that is consistentwith the H,m-model of the airspace.

(3) The data for m indicate that insertionof the baffle decreased the extent of thestagnant region for building configuration1 (Fig. 10). There does not appear to beany physicalbasis for believing that majorobstructions to air flow at floor level willdecrease stagnant regions and improvemixing. Such a finding would be contraryto the results of the many flow visualization studies of Randall (1975).

(4) Assuming that m has been consistentlyoverestimated, Fig. 10 indicates thatthe extent of stagnant zones decreasedwith decreasing ventilation rate. This curious result is unlikely to be true, the morelikelyexplanation being that the variationsin the parameter m are due to some effectother than the existence of stagnant zones.

The data do not support Hypothesis I.It would appear that reasons must existother than short-circuiting and stagnantzones to account for the response of theairspace to a step change in tracer concentration. Barber and Ogilvie (1982) havesuggested a multicompartment model forthe ventilated airspace as an alternative tothe simpler K- and w,m-models. In thismodel, the airspace is visualized as a network of inter-connected flow regimeswhere air is continuously inter-mixed between primary and secondary zones. Themore complicated model can account formany of the apparent inconsistencies betweenthe data and the simpler models, andis shown in Barber and Ogilvie (1984).

The A'-model was introduced previously and shown to be a special case ofthe /i,m-model. The single parameter, K,in this simplest of the mixing models, isestimated as the negative slope of eitherthe inner chamber or the outlet regressionplots. For this experimental airspace,therefore, predictions for the value of Kvary between 0.8 and 1.1 (Fig. 11). Bythe usual interpretation, short-circuitingamounting to at least 20% of the total ventilation rate is predicted by the lowestvalue of K, while stagnantzones equal toat least 10% of the total airspace volumeare predicted by the highest value of K.

1.2

1.1

1.0

m

... ,1 i 1 . 1 1

1 r

o •2l

3 i r

3 a

41 1

0.9

0.8

2 5 10 20 50 100 200

Arc

Figure 10. Values of m versus corrected Archimedes number (Arc).

K

1.2-

1.1-

1.0-

0.9-

0.8-

.II1 It II1' f I I

1 ii ii

1 r

^ •2l

3 i r

3 a

4l I

2 5 10 20 50 100 200

Arc

Figure 11. Values of K versus corrected Archimedes number (Arc).

This interpretation, however, seemshighly unlikely in view of the discrepancies already noted for the n,m-model.There is no apparent physical explanationfor greater stagnation regions at the 12 L/sec flow rate compared to the 4 L/sec flowrate as predicted by K values in Table III.Furthermore, the possibility of short-circuiting in building configuration 3 for thelowest flow rate is highly unlikely.

Prediction of Incomplete MixingThe second hypothesis tested in this

study concerns the prediction of incomplete mixing and the validity of the Arc asthe correct dynamic similarity criterion.Whereas the slope and intercept data presentedin Table III cannotbe explainedbythe w,m-model, nor by the ^f-model, presumably the magnitude of their departurefrom unity does represent some measureof the differences in completeness of mixing within the ventilated airspace.

Barber et al. (1982) discussed the concepts of Archimedesnumber and a jet momentum function, Ji, as controlling parameters describing the extent of stablemixing patterns in ventilated airspaces.Thejet momentum functionrepresentstheamount of energy per unit volume of the

airspace that can be transferred from theinlet jet to the room air.

A multiple linear regression analysiswas performed to test Hypothesis II, thatthe slope, K, (the departure from completemixing) of the linear plot of In CRversusQtIVis a significant function ofArC9 Ji andthe building configuration.

For purposes of the linear model, thebuilding configuration was considered intwo parts, one part defining the presenceor absence of the flow barrier (BAFFLE)and the other part defining the location ofthe outlet relative to the inlet (IOSEP).These two non-parametric variables weretreated as dummy variables, and assigneda value of 0 or 1 as follows:

IOSEP = 0 if outlet on same wall asinlet;IOSEP = 1 if outlet on wall oppositethe inlet;BAFFLE = 0 if no baffle; andBAFFLE = 1 if baffle in placeThe best linear model is given by Table

IV. That the model explains a significantportion of the variation in K is indicatedby an R2 value of 0.90. Furthermore, theF value of 62.85 greatly exceeds the significant F(13,96,0.99) of 2.4. Therefore,Hypothesis II is accepted.

CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984 195

TABLE IV. BEST LINEAR MODEL FOR PREDICTION OF OUTLET REGRESSION PLOT SLOPE tilation rate were a function of the inlet-

Variables in the equation

Xi Partial-F

ISOEP -0.13346

BAFFLE 0.78929x10-»

IOSEP*BAFFLE -0.10635

4 - 0.55700 X101

Arc 0.31784X10-2

l 0.61004X102

Ar* -0.18720X10"4

J>*Arc 0.15919

JflOSEP 0.13384x102

7i2*IOSEP - 0.13329 xlO3

Arc2*ISOP 0.96196 xlO"5

7i*Arc*IOSEP -0.36890

7i*Arc*BAFFLE -0.11736

Constant -0.97920

F(13,96) = 62.850 R2 = 0.895

51.26

24.18

87.54

22.04

46.77

16.22

47.42

9.10

121.95

57.18

40.77

43.19

5.10

Presuming that the slope, K, is indicative of mixing conditions in the airspace,the above analyses confirm dependence ofairflow on the corrected Archimedes num

ber and the inlet jet momentum, and support the Archimedes number as the correctdynamic similarity criterion. The significant effect of the inlet jet momentum function suggests that non-isothermal scale-model studies should be designed forequivalence between the temperature differences in the model and the prototype.Otherwise, the Archimedes number andthe jet momentum number become opposing design criteria and distortion of themodel results will be inevitable.

SUMMARY AND CONCLUSIONS

The design of livestock building ventilation systems is based on the premise ofcomplete mixing. This assumption involves equating the thermodynamic properties of the exhaust air to the averagethermodynamic properties of the bulk airspace so as to enable the calculation ofsteady-state heat and moisture balances.The major objective of this project was totest the validity of the complete mixing assumption for slot-ventilated livestockbuildings.

A mathematical model of the airspacecurrently used in ventilation research is theA'-model. This model accounts for incom

plete mixing in terms of a single parameter, K, usually defined as the ratio of theeffective ventilation rate to the calculated

rate. Values of K greater than 1 are interpreted to indicate stagnant zones whilevalues of K less than 1 are attributed toshort-circuiting of air to the exhaust outlet.

The tf-model provides no basis for distinguishing between the counteracting effects of stagnant zones and short-circuiting when both occur simultaneously. A

more complicated model, termed the n,m-model, is presented in this paper and includes two parameters to measure theseeffects. This two-parameter model hadbeen successfully validated for aqueousmixing systems in chemical reactors.

Experiments were designed to estimatethe two parameters, n and m, in the modelfor the special case of the slot-ventilatedlivestock building. Rate-of-decay tracertechniques were used to measure both theslope and the intercept through a plot ofthe tracer concentration ratio versus the airexchange number.

A one-fifth scale physical model wasdesigned and built on Archimedes numbersimilarity criterion. The building wasbased on a prototype swine grower-finisher building operating under winter conditions.

The main experimental design involved48 treatments in a partial factorial arrangement, with three ventilation rates, threeinlet jet velocities, two inlet-outlet temperature differences, two inlet-outlet arrangementsand two internalbuildinggeometries.

The following conclusions were madeas a result of the analysis of the results of136 trials.

(1) The mathematical n,m-model formixing did not successfully explain variations in the tracer data, and was rejected.Short-circuiting and stagnant zones couldnot, therefore, be the only causes of incomplete mixing in the experimental airspace.

(2) The effective ventilation rate in theexperimental airspace varied from 80 to110% of the calculated rate, according tothe AT-model, indicating extensive departure from complete mixing. Ventilationrates, in full scale buildings, designed onthe assumption of complete mixing couldbe in error by the same amount.

(3) That variations in the effective ven-

outlet arrangement, the presence or absence of a barrier to air flow, the correctedArchimedes number, and the inlet jet momentum was accepted. A second-orderlinear model involving only the above variables explained 90% of the variation inthe data.

ACKNOWLEDGMENTSThis research was funded by Agriculture

Canada Grant no. 9032, "Airflow patterns inswine buildings' * to J. R. Ogilvie, and by aNatural Sciences and Engineering ResearchCouncil of Canada Scholarship to E. M. Barber. The contributionof physical facilities bythe Department of Animal and Poultry Science, University of Guelph is also acknowledged.

REFERENCES

BARBER, E. M. 1981. Scale model study ofincomplete mixing in a ventilated airspace.Unpublished PhD dissertation. University ofGuelph, Guelph, Ontario.

BARBER, E. M. and J. R. OGILVIE. 1982.Incomplete mixing in ventilated airspaces.Part 1. Theoretical considerations. Can.

Agric. Eng. 24: 25-29.BARBER, E. M., S. SOKHANSANJ, W. P.

LAMPMAN, and J. R. OGILVIE. 1982.Stability of airflow patterns in ventilated airspaces. Paper no. 82-4551, presented atASAE Winter meeting, Chicago, 111. ASAE.St. Joseph, Mich.

BARBER, E. M. and J. R. OGILVIE. 1984.Interpretation of tracer gas experiments inventilation research. J. Agric. Eng. Res. 30:57-63.

CHOLETTE, A. and L. CLOUTIER. 1959.Mixing efficiency determination for continuous flow systems. Can. J. Chem. Eng. 37:105-112.

CLOUTIER, L. and A. CHOLETTE. 1968.Effect of various parameters on the level ofmixing in continuous flow systems. Can. J.Chem. Eng. 46: 82-88.

HONEY, L. and M. WRUBLESKI. 1977. Anevaluation of a porous ceiling in a free stallbarn. Paper No. 77-404. Presented at CSAEAnnual meeting, Guelph, Ont. July.

LEVENSPJEL, O. 1972. Chemical reactionengineering. 2nd ed., John Wiley and Sons,New York.

MILBURY, W. F., W. O. PIPES, and R. B.GRIEVES. 1964. A laboratory study ofmixing conditions in small aeration vessels.Proc. 19th Ind. Waste Conf., Eng. Ext. Ser.No. 117, Purdue Univ. Lafayette, Ind. pp.927-939.

RANDALL, J. M. 1975. The prediction of airflow patterns in livestock buildings. J.Agric. Eng. Res. 20: 199-215.

RANDALL, J. M. and V. A. BATTAMS.1979. Stability criteria for airflow patternsin livestock buildings. J. Agric. Eng. Res.24: 362-374.

TURNBULL, J. E. and N. BIRD. 1979. Confinement swine housing, Publ. 1451, Agriculture Canada, Ottawa, Ont.

196 CANADIAN AGRICULTURAL ENGINEERING, VOL. 26, NO. 2, WINTER 1984