A linear model of the e!ects of residence time distribution on mixing
pattern in a ventilated airspace
Chung-Min Liao
∗, Huang-Min Liang
Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, 10617, Taiwan, ROC Received 7 January 2002; accepted 10 January 2002
Abstract
The ability of a simple linear response model is evaluated to explain the mixing e0ciencies in an incomplete mixing ventilated airspace. Data interpretation and mean residence time calculation for a speci2ed output concentration pro2le can also be evaluated. The residence time distribution (RTD) functions take the form of the two-parameter gamma distribution and account for di!erent mixing types such as complete mixing, piston 4ow (no mixing), incomplete mixing, and various combinations of the above types. In these combinations, the di!erent mixing types simulated by di!erent RTDs conceptually represent air4ow regions in series. The mixing e0ciency was introduced to characterize the extent or degree of mixing in a ventilation system in that mixing e0ciency equals zero for piston 4ow (no mixing), unity for complete mixing, and a value in between these two extremes for incomplete mixing. An environmental chamber experiment was conducted to generate several output pro2les to evaluate the applicability of the model. Carbon dioxide was employed as the tracer gas. Our results show that an overall root-mean-squared error value of 8:64 ± 5:25 ppm is low, indicating that the combination mixing patterns are generally found to be minimally biased and give better 2tting than other simpler mixing patterns. Despite their neglect of molecular di!usion and possible temporal=spatial nonlinearities, these linear response models appear reasonably robust, making them at least as useful to building microenvironment designers in reconsidering the possibilities and consequences of various forms of incomplete mixing related to indoor air quality problems. ? 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Indoor air quality; Mixing; Residence time distribution; Ventilation
1. Introduction
Knowledge of the mixing patterns in a ventilated airspace is valuable at boththe design stage and during operation to determine the e!ective mixed proportion of the air vol-umes and existing contaminants and to assess the values of di!erent operational procedures as a means of optimizing process performance in indoor air quality control problems. Barber and Ogilvie [1] suggested that departure from com-plete mixing might be caused by the formation of multiple 4ow regions within the airspace or by short-circuiting of supply air to exhaust outlet. A work presented by Chen et al. [2] regarding the methods to measure dust production and deposition rates in buildings indicated that the assump-tion of complete mixing was not valid during tests. Their
∗Corresponding author. Tel.: +886-2-2363-4512; fax:
+886-2-2362-6433.
E-mail address: cmliao@ccms.ntu.edu.tw (C.-M. Liao).
experiments showed that tanks-in-series 4ow, i.e., di!erent 4ow regions behaving as a number of mixed tanks connected in series dominated the overall mixing process within the ventilated airspace. Their work also suggested that a more complicated multi-zone mixing model might be needed to account for incomplete mixing to better understand the be-havior of dust in local transport mechanisms.
Levenspiel [3] indicated that mixing in a continuous 4ow system could range from piston 4ow (plug 4ow or no mix-ing) as a lower limit, to complete mixing as an upper limit. Bothcomplete mixing and piston 4ow represent the limit-ing cases that seldom occur in ventilation systems. It is most probable that the type of mixing encountered in ventilation systems, in general, lies somewhere between these two ex-tremes. We call this type as incomplete mixing or partial mixing.
The most widely used analytical model for simulating the air4ow patterns and convection–di!usion distribution of a contaminant in building environment is the dispersive
0360-1323/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S0360-1323(02)00026-4
Nomenclature
Cin(t) time-dependent input concentration (mg m−3)
Cm;n measurement data point n in error analysis
Cs;n simulation data point n in error analysis
Cout(t) time-dependent output concentration (mg m−3)
Cout(T) output concentration as a function of T
(mg m−3)
C0 constant input concentration (mg m−3)
CB background gas concentration (mg m−3)
E(t) mean of the gamma distribution
f(t) residence time distribution function
g(t) two-parameter gamma residence time
distribu-tion funcdistribu-tion
N number of data points
Q volumetric air4ow rate through the system
(m3h−1)
sd(t) standard deviation of the gamma distribution
T mean residence time (h)
Tn normal mean residence time (h)
t time variable (h)
U(T) relative concentration as a function of T (dimen-sionless)
V air volume (m3)
VT total air volume in the system (m3)
Vc air volume occupied by complete mixing (m3)
Vi air volume occupied by incomplete mixing (m3)
Vp air volume occupied by piston 4ow (m3)
Greek symbols
; shape and scale parameters, respectively, in the gamma distribution
(•) gamma function
mixing volume factor I (VT=(Vc+ Vp))
mixing volume factor II (VT=(Vc))
air-exchange rate (h−1)
T relative residence time (dimensionless)
mixing e0ciency
residence time variable (h)
Abbreviations
pdf probability density function
RTD residence time distribution
RMSE root-mean-squared error model that utilizes the dispersivity concept in a distributed
scheme [4–8]. Reliable estimates of the dispersivity may be scale-dependent, however, are very di0cult to obtain. The dispersive model is also characterized by other limitations. Liu et al. [7] and Ho! and Bundy [9] pointed that the the-oretical foundation for the dispersive model in its multiple air4ow regions is not sound for systems other than air jet withturbulent or laminar 4ow.
The main parameter describing the temporal change in a continuous system is the mean residence time (or mean age) of air in the ventilation system [10]. For time-invariant systems, the mean age T is de2ned as [10] T =0∞tf(t) dt, where f(t) is the residence time distribution (RTD) function which describes the exit time distribution of 4uid element (air) entering the system at a given time t = 0. The RTD approachcan also be used to describe aerodynamic transport in a ventilated airspace. The methodology is general in the sense that the heterogeneity in transport properties may also be incorporated using various RTDs of speci2c interest.
Mathematically, f(t) is the probability density function, for the time t. Etheridge and Sandberg [11], Freeman et al. [12], Skarret and Mathisen [13], Grot and Lagus [14] and among others pointed out that f(t) could be obtained by a tracer gas (e.g., SF6) experiment or by measuring
the existing gaseous pollutant concentrations (e.g., CO2,
NH3) presented in the ventilated airspace. A parsimonious
representation of f(t) can be obtained by a simple uni-model and continuous mathematical function. A convenient and 4exible mathematical function is the gamma density
function. The gamma distribution is a distribution of Pear-son’s Type III in statistics. The unique feature of the gamma distribution is that one end of the distribution is bound to a 2xed value, whereas the other end is distributed over a large scale of the variate. The overall shape of the gamma distribution is not balanced as a normal distribution. The other reason to utilize the gamma distribution is that this ap-proach may reduce the mathematical terms in the analytical solution.
Three most common mixing patterns appearing in a con-tinuous 4ow system are the complete mixing, piston 4ow, and the complete-piston 4ow that simulates the combina-tion of complete mixing and piston 4ow [10]. The new mixing patterns discussed in this study are the complete– incomplete-piston 4ow, complete–incomplete mixing, incomplete–piston 4ow, and incomplete mixing patterns. It is obvious that incomplete mixing alone or in combination with the other mixing types is more likely to occur in a general ventilation system than piston 4ow alone, complete mixing alone, or the combination of complete mixing-piston 4ow.
A logical 2rst step in this assessment is an evaluation of ability of a simple linear response function model to account for di!erent types of mixing in general ventilation systems suchas piston 4ow (no mixing), incomplete mixing, com-plete mixing and their combinations which represent the si-multaneous existence of the three or any two of the mixing types in the system. The proposed linear response model is free from the limitations of the dispersive model (i.e., the
dispersivity problem) that makes them an e!ective tool for investigating indoor air quality problems. The linear re-sponse model is particularly useful for investigating sys-tems that lack detailed ventilation data, i.e., syssys-tems with unknown distribution of parameters. Considering that venti-lation systems in many commercial buildings and=or animal housing are characterized by little or no ventilation data, analysis of tracer data by the linear response model may be the only available approach for solving air-supply problems. The signi2cance of this work has two e!ects: (1) this study will alert designers of building environment to the pos-sibilities and consequences of various forms of incomplete mixing, and (2) the model can be used to quantify mixing characteristics to replace the complete mixing assumption in ventilation design calculation. Because of the inherent di0culties in 2eld measurements, an environmental cham-ber experiment was conducted to evaluate the ability of the model under controlled conditions.
2. Materials and methods
2.1. Environmental chamber experiments
The test facilities used in the experimental studies were composed of an environmental chamber, an air delivery sys-tem, a carbon dioxide gas generating system and a carbon dioxide gas sampling system. The ambient temperature dur-ing experiment was 31±0:6◦C. The relative humidity ranged
between 70% and 80%. We employed carbon dioxide as the tracer gas.
Fig. 1 gives the dimensions and general outline of the environmental chamber. The dimensions were 1:54×0:92× 0:95 m3for a volume of 1:35 m3. The dimensions were such
as to simulate one slot-ventilated enclosure witha negative pressure ventilation system. The chamber, fabricated from one Plexiglas and three sandwich-type boards.
Air entered the chamber through a long slot inlet mea-suring 92 × 4:5 cm2. A 30 cm diameter, 6-blade propeller
fan was used to exhaust air from the chamber. The exhaust location was 80 cm from 4oor level measured to the center of the outlet. The fan operated at constant speed by a volt-age indicator maintaining a constant air velocity. Fan out-put expressed as m3h−1 was measured near the end of a
discharged duct. Chamber ventilation rates were determined using the impeller method based on manufacturer speci2ca-tion. Ventilation rates used throughout the experiments were ranged from 850 to 1070 m3h−1to generate various forms
of carbon dioxide concentration output pro2les.
Gas generating system consisted of a high-pressure cylinder of carbon dioxide, a two-stage pressure regu-lator, plastic tubing, and three 4owmeters (Hsin Chuan M-type, No. HC-A02, Taiwan) (Fig. 1). Discharge from the cylinder of pure carbon dioxide was divided into three for the generating operation. The Hsin Chuan M-type 4owmeter had a millimeter scale that could be converted
to a volume 4ow scale for air withaid of calibration charts. Adjustment factors associated with the calibra-tion charts were used to calculate the gas 4ow rate for gases other than air. According to manufacture informa-tion, the adjustment factor is equal to the square root of the speci2c gravity of air divided by the speci2c gravity of carbon dioxide. The resulting value is 0.808. Carbon dioxide was used at a rate of 500 l h−1. In
or-der to investigate the variation of carbon dioxide con-centration in4uenced by the spatial change, the carbon dioxide generating points were deployed as shown in Fig. 1.
Two KRK Model 200-type carbon dioxide meters (Kasa-hara Chemical Instruments Corp., Japan) were used to mea-sure carbon dioxide concentration. Measuring principle for KRK Model 200-type carbon dioxide meter is based on the potentiometric methods in that carbon dioxide gas passed into the membrane of the electrode system in contacting with NaHCO3 solution results in a change of pH values.
Car-bon dioxide concentration could be measured through the potential di!erences derived from the proportional relations between the change of pH value and the partial pressure of carbon dioxide gas.
Before the chamber test, two carbon dioxide meters were calibrated withknown carbon dioxide concentrations of 0% and 10%, respectively. Sampling was carried out in two planes, 30.7 and 61:4 cm from a side wall along the y-direction of Fig. 1 and the sampling interval was 10 s. All sampling outputs were recorded on a computer-controlled data acquisition system (Fluke-2280A Data Logger, John Fluke MFG Co., Inc., U.S.A.). Manufacturer speci2ca-tion shows that the carbon dioxide reading to be accurate within 3% might be expected, indicating that the random error of ±3% was uniformly distributed to carbon dioxide concentration outputs.
2.2. Mathematical model
A simple linear response model is proposed to describe the tracer dynamics between output and input gas concentrations subjected to an appropriate RTD function,
Cout(t) =
t
0 Cin(t − )g() d + CB (1)
where Cout(t) is the output gas concentration (mg m−3),
Cin(t) is the input gas concentration (mg m−3), g(t) is th e
RTD function, also called the system response function or the weighting function, and CBis the background gas
con-centration (mg m−3). In the second term of Eq. (1), t is the
calendar time, and is the integration variable that repre-sents the residence time of air4ow.
In the case of the constant tracer material input with a constant air-exchange rate (h−1) in a ventilated airspace,
46 cm FLUKE-2280A DATA LOGGER KRK MODEL CO2 METERS % % FLOWMETER 95 cm 90 cm 80 cm 154 cm 30 cm SIX-BLADE EXHAUST FAN ENVIRONMENTAL CHAMBER Y Z Q X Q CO2 ELECTRODE PURE CO2 9 kg 56~57 kg/cm2 CO2 0.7 kg/cm2
Fig. 1. General outline of the dimensions of environmental chamber and experimental equipments.
total initial input concentration, the linear response model describing the output concentration remaining after t is Cout(t) =
t
0 Cin(t − ) e
−g() d + C
B: (2)
Our approachis to use the gamma distribution to de-scribe the RTD function in an incomplete mixing of ventilated airspace. Because of the complexity of air mixing, their RTD is not expected to follow a nor-mal distribution. The gamma distribution, thus, has a greater chance to describe the mixing behavior than does the normal distribution. The presented analytical solution demonstrates that it is possible to construct a systematic way of lumping parameters and the in-complete mixing behavior could be analyzed. Bu!ham and Gibilaro [15] also pointed out that the simplest
extension of the tanks-in-series model is the gamma model.
The two-parameter gamma probability density function (pdf) of t may be written as
g(t) ≡ f(t; ; ) =()1 e−t=t−1 t ¿ 0; (3)
where (the shape parameter) and (the scale parameter) are positive parameters, the normalization factor () = ∞
0 −1e−d, where is a dummy variable of integration,
is the gamma function of mathematical statistics. With the gamma pdf, the mean, E(t) and standard deviation, sd(t) are: E(t) = ; sd(t) = 1=2, respectively.
If is held constant, then as approaches in2nity the gamma distribution function approaches the Gaussian dis-tribution, the variance of which decreases as increases.
The gamma distribution also satis2es the physical require-ment that residence time must be non-negative. The scale parameter serves to determine boththe mean and standard deviation of the t’s. The mean and variance of each gamma distribution governs the characteristic properties of the lin-ear response model.
The e!ect of and has the following characteristics. For 0 ¡ ¡ 1, muchof the air corresponds to very small values of residence time t near 0; for = 1, an exponential distribution is obtained, for which t=0 is still the most likely value. As increases above 1 the density function becomes increasingly symmetric withsmall values of t occurring less frequently. Firstly, we are taking the complete-piston 4ow pattern as an example to get insight into the concept of and .
The occupied air volume relationship among the total vol-ume (VT), complete mixing volume (Vc), and piston 4ow
volume (Vp) in the complete-piston 4ow pattern is VT=Vc+
Vp. For = 1, RTD function in Eq. (3) reduces to
g(t) = 1e−t=; t ¿ 0: (4)
Based on Eq. (4) and air volume relations, the complete-piston 4ow mixing pattern can be mathematically expressed as g(t) = Te−t=T t ¿ 0; 0; t ¡ 0; (5) where = T; =VVT c; T = VT Q: (6)
The complete-piston 4ow pattern simulates a completely mixed tanks-in-series witha region where air4ow moves in piston 4ow. Therefore, the complete-piston 4ow is a special case of the two-parameter gamma pdf when is de2ned by Eq. (6) and = 1 and results in the mean and standard deviation of the complete-piston 4ow are E(t)=sd(t)=T=. The complete-piston 4ow combines the complete mixing withthe piston 4ow. For = 1, the complete-piston 4ow gives the complete mixing, and for → ∞, it reduces to the piston 4ow. Substituting = 1 or letting → ∞ in Eq. (6) yields = T for the complete mixing, and = 0 for the piston 4ow. The mean is T in bothcomplete mixing and piston 4ow, and variance is T2 for complete mixing and 0
for piston 4ow.
These values can also be obtained from the mean and variance of the two-parameter gamma pdf. In other words, complete mixing, piston 4ow, and their combinations (which represent air4ow regions in series of the di!erent mixing types) are all accounted for by the two-parameter gamma pdf. For this reason, we propose the general mixing pat-tern, which takes the functional form of the two-parameter gamma pdf, for simulating mean residence times in ven-tilation systems, and consequently the additional 2tting
parameters will give some information about the structure of the system.
The air volume relations of a general mixing pattern of a ventilated airspace in a time-invariant system is
VT= Vc+ Vp+ Vi; (7)
where Vi is the air volume of incomplete mixing region.
In a time-varying system, Eq. (7) has the form as VT(t) =
Vc(t) + Vp(t) + Vi(t). Based on Eqs. (6) and (7), after some
mathematical manipulation, the shape parameter and scale parameter can be obtained as
= VT Vp+ Vc+ Vi = 1 1= + (1 − 1=); ¿ 0:5; VT Vp+ Vc+ (1 − )Vi = 1 1= + (1 − )(1 − 1=); 6 0:5; (8) =(V T T=(Vc+ Vi))= T 1 + − (9) in which =V VT c+ Vp (10) and = VT Vc; (11)
where is the mixing e0ciency describing the portion of ventilation air4ow that has mixed with room air, and and may be referred to as the mixing volume factor I and II, respectively. The detailed deviation of and is given in Appendix A.
Substituting Eq. (3) into Eq. (2) for a constant Cin= C0
gives an output and input relations as a function of T as, U(T) ≡ Cout(T) C0 = 1 ( + 1) + CB C0; (12)
where U(T) is de2ned as a relative concentration. Substitut-ing the parameters of and of any of the mixSubstitut-ing patterns into Eq. (12) gives the U(T) of that particular mixing type. In case of the combination mixing patterns of complete– incomplete-piston 4ow,
U(T) =[(T=)((1=) + − =) + 1]1 +CCB
0: (13)
Table 1 summarizes the expressions of the RTD functions and the U(T)s for the several forms of mixing patterns. The 2rst three cases in Table 1 correspond to the mixing patterns of complete-piston 4ow, complete mixing, and pis-ton 4ow, respectively. The mixing patterns corresponding to cases 4, 5, and 6 are designated as the incomplete-piston 4ow, complete–incomplete mixing, and incomplete mixing, respectively. Case 7 represents the combination mixing pat-terns of complete–incomplete-piston 4ow. These seven mix-ing patterns can give other patterns, as special cases, by changing the types of mixing or mixing e0ciency.
Table 1
The residence time distribution (RTD) functions and relative concentration U(T) for di!erent mixing patterns as background gas concentration is negligible
Mixing pattern RTD function: g(t) U(T) = Cout=C0
1. Complete-piston Texp(−t=T) T
+ 1 −1
2. Complete mixing 1
Texp(−t=T) [T + 1]−1
3. Piston 4ow (t − T) exp(−T)
4. Incomplete-piston t−1 T − ()exp T −t − T −+ 1− 5. Complete–incomplete t−1 T 1 +− ()exp T −t 1 +− T 1 + − + 1− 6. Incomplete mixing t−1 T ()exp −t T T + 1 − 7. Complete–incomplete-piston t−1 T 1 +− ()exp T −t 1 +− T 1 + − + 1−
Cholette and Cloutier [16] and Jennings and Armstrong [17] pointed out that incomplete mixing describes the type of mixing between piston 4ow and complete mixing in that the mixing e0ciency is introduced and de2ned. The ap-proaches presented by Cholette and Cloutier [16] and Jen-nings and Armstrong [17] for characterizing the overall de-cay rate when mixing is not complete is to 2t an expres-sion of the type in terms of measurement data as C(t) ≈ C(0) exp(−Q=Vt), in which C(t) is time-dependent tracer concentration, C(0) is the initial condition of tracer concen-tration, Q=V = T−1is the inverse mean residence time of
ventilation system; Q is the ventilation rate, and V is the air volume.
Incomplete mixing can range from near complete mixing to near piston 4ow, but never reaches these extremes. We in-troduce the concept of mixing e0ciency to specify the exact location of incomplete mixing in the range between piston and complete mixing. In general ventilation systems, mix-ing is due to mechanical dispersion and molecular di!usion. The proposed linear response model is applicable to systems where molecular di!usion is negligible. Accordingly, we de-2ne the mixing e0ciency to describe the extent of mixing resulting from factors other than molecular di!usion: =
1 for complete mixing; 0 for piston 4ow
and incomplete mixing is characterized by 0 ¡ ¡ 1. There are two premises which must be met before the linear response model can be applied. The capable model accounting for the e!ects of molecular di!usion is only the dispersive model. The proposed model assumes that molec-ular di!usion is negligible, i.e., the RTD function of the tracer or existing gaseous contaminant and air are identical
and bothare related to the distribution of air velocities. Fur-thermore, the application of the model requires that an ideal tracer be injected into and measured in the 4ux concentra-tion. The proposed linear response model thus assumes that injected and measured in the 4ux has the same response function as the tracer material. These conditions, however, are met which are restrictive for two reasons: (1) molecular di!usion is negligible compared withmechanical dispersion in most 2eld situations, especially in large-scale systems, and (2) 4ux concentration samples are easy to obtain from typical occupied airspace.
The general ventilation systems are very complex. The complexities arise from 2eld-scale heterogeneity that create signi2cant variability in the air velocity. The complexities make characterization of mixing in ventilation system a very di0cult task. The task becomes even more di0cult when the systems are viewed as distributed-parameter systems. An alternative approachfor considering spatially distributed parameter or processes is to quantify the lumped response of mixing in the system. The concept of mixing e0ciency, therefore, is adopted for the purpose of this study.
The major di!erence between linear response model and dispersive model, therefore, is that the linear response model employs the concept of mixing e0ciency, whereas the dispersive model utilizes the dispersivity=di!usivity that may be scale dependent. Therefore, the proposed model simulates incomplete free from the dispersivity=di!usivity limitation.
As an example of changing the mixing e0ciency, con-sider now the complete–incomplete mixing pattern. If is changed from 0 ¡ ¡ 1 to 0, then the complete–incomplete mixing will give the complete-piston 4ow; if is changed to 1, then the resulting mixing pattern will be the complete
mixing. As an example of changing the mixing type, again consider the complete–incomplete mixing type. Suppose the mixing type is changed from complete–incomplete mixing to just complete mixing, the complete–incomplete mixing will give the complete mixing pattern. Cases 4 and 5, that are shown in Table 1, are more likely to occur in actual ventilation systems than piston 4ow, complete mixing, or the combination of complete-piston 4ow; and case 6 can be modeled di!erently by the dispersive model.
Analytical determination of the mean age in case of con-stant tracer input requires no additional information if the selected model is an one-parameter gamma pdf, suchas pis-ton 4ow or complete mixing patterns. For two-parameter gamma pdf, a priori knowledge of the parameters other than the mean age, the unknown, is necessary. Practically, how-ever, a priori knowledge of these parameters is very di0cult to obtain.
3. Results and discussion 3.1. Numerical experiments
Knowledge of the relative concentration (U) and the air exchange rate (), and the other parameters in Eq. (13), al-lows the determination of mean age of air T. The charac-teristics of the mixing behavior can be illustrated in Fig. 2, where U is plotted against relative mean residence time T for di!erent mixing types for a constant input concentration and constant and withdi!erent of 0.25 and 0.75, re-spectively. Fig. 2A shows that, for instance, for U =0:8 and =0:25 the mean age is T =0:4= for complete–incomplete mixing pattern, whereas in incomplete mixing 4ow, mean age is T = 1=. The real age lies somewhere between the extreme values of Fig. 2. Its better estimate depends on the choice of an adequate mixing pattern. The relationship for the incomplete mixing reduces to those of the complete mix-ing for = 1 and = 1, and the piston 4ow for = 0 and = 1.
The RTD functions can assume any shape, depending on the values of the shape parameter . It is obvious that the 4exibility of the RTD functions is very important if a realistic mixing type is derived to describe the complex mixing behavior of the general ventilation systems.
The minimum number of 2tting parameters requires modeling the combinations of complete–incomplete or incomplete-piston 4ow is three: the mean age of air (T), the mixing e0ciency (), and mixing volume factor I () in th e systems. On the other hand, the minimum number of pa-rameters required to model the complete–incomplete-piston 4ow type is four: T; ; , and mixing volume factor II, .
Figs. 3A and B illustrate the RTD function of di!erent shapes constructed for two of the mixing patterns devel-oped in the study: the complete–incomplete mixing and the incomplete-piston 4ow types. Fig. 3A shows that the graph with Vc= 0:9VT(90% of the system is completely mixed),
0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1
Relative residence time (λT )
Relative residence time (λT )
Relative concentration ( U ) complete-piston complete ( =1) piston ( =0) incomplete-piston complete-incomplete incomplete (a) =0.25 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 Relative concentration ( U ) complete-piston complete(=1) piston (=0) incomplete-piston complete-incomplete incomplete (b) =0.75
Fig. 2. Relative concentration as a function of the relative residence time for a constant input concentration and for di!erent mixing models: (A) = 0:25 and (B) = 0:75.
closely approximates that of the complete mixing. Fig. 3B shows that as the volume occupied by piston 4ow increases, the variances of the graph decrease and their peaks increase, i.e., the graphs approach piston 4ow resulting from increas-ing Vp and decreasing imply piston 4ow. The
numeri-cal example reveals that the development of the complete– incomplete-piston 4ow patterns withone set of parameters to account for the three mixing types (complete, incomplete, and piston 4ow) and their combinations.
3.2. Fitting linear response model to chamber experiment Systems that tend to be highly mixed can be modeled withthe incomplete mixing type withhighmixing e0ciency (i.e., the modeler has to specify a high value for ), or the complete–incomplete mixing pattern with high and=or high Vc, or the complete–incomplete-piston 4ow with high
complete-incomplete mixing
0 0.4 0.8 1.2 1.6 0 0.2 0.4 0.6 0.8 1 RTD ( g (t )) (a) 0 5 10 15 20 -0.2 0 0.2 0.4 0.6 0.8 1 RTD ( g (t )) (b) Vc/VT=1.0 Vc/VT=0.9 Vc/VT=0.75 Vc/VT=0.5 Vc/VT=0.25 Vc/VT=0.1 Vp/VT=0.9 Vp/VT=0.75 Vp/VT=0.5 Vp/VT=0.25 Vp/VT=0.1 Time (t)incomplete-piston mixing
Time (t)Fig. 3. The RTD function of di!erent shapes constructed for two of mixing models: (A) complete–incomplete mixing and (B) incomplete-piston 4ow. (VT= total air volume, VC= air volume occupied by complete mixing, VP= air volume occupied by piston 4ow).
Systems characterized by a low degree of mixing, can be simulated by the incomplete mixing pattern with a low , or the incomplete-piston 4ow with low and=or high Vpwith
low and=or high Vp, or the complete–incomplete-piston
4ow withlow and=or high Vp. In the case of the complete–
incomplete-piston 4ow, the incomplete mixing volume (Vi)
will provide a realistic transitional mixing zone between the highly or complete mixing volume (Vc) and the piston 4ow
volume (Vp).
Four selected scenarios designated as Systems I, II, III, and IV that obtained from environmental chamber experi-ments were given to evaluate the applicability of the pro-posed linear response model. In our analysis, we eliminate the background carbon dioxide concentration (CB) to
re-4ect the characteristics of the output carbon dioxide dynam-ics. To select the proper RTDs for interpreting the transient times, we use the knowledge of the carbon dioxide output
0 50 100 150 200 250 300 0 4 8 12 16 20 24 Time (h) CO 2 concentration (ppm) Measurement Simulation
System I
Fig. 4. Fitting of linear response model to System I as calculated from the complete–incomplete-piston mixing pattern with = 0:34 ± 0:18 and T = 6:34 ± 3:45 h(background CO2 concentration = 315 ppm).
concentrations as well as the nature of ventilation systems. Table 2 summarizes the simulation results of 2tting model to the chamber experiments.
The 2tting procedure in this work is preformed by guess-work and no objective criteria are applied. A low number of experimental data as well as a limited signi2cance of the residence time of the investigated systems did not justify a great e!ort needed for a more rigorous reinterpretation. Whenever the 2tting is performed by trial and error, the term “quantitative” is rather an arbitrary one.
Based on the above information, we chose the incomplete mixing pattern withrelatively short T and moderate for the interpretation of System I. Using this mixing pattern, the best 2t is obtained for T =6:34±3:45 h, =0:34±0:18. The model performance was evaluated by the root-mean-squared error (RMSE), computed from
RMSE =
N
n=1[Cm;n− Cs;n]2
N ; (14)
where N denotes the number of measurements, Cm;n is the
measurement data, and Cs;n is the simulation result
corre-sponding to data point n. Table 2 lists the RMSE values for the model performances, indicating that RMSE values ranged from 2.10 to 15:70 ppm. The overall RMSE value of 8:64 ± 5:25 ppm is low, considering that the average stan-dard error of measurements is already 11:07 ppm.
The predicted and observed output carbon dioxide con-centration pro2les for System I are shown in Fig. 4 in that the steady state is never obtained. This nonattainment of equilibrium leads to a large over estimate of the ventila-tion rate occurring in the test chamber. On comparison with the model, the RMSE value in the estimates as de2ned by Eq. (14) was 15:74 ppm, indicating that the numerics of the model were performing unfavorably well in the 13:00– 18:00 h shown in Fig. 4. The relatively large scatter of the
Table 2
Simulation results of 2tting the linear response model to measurements obtained from environmental chamber experiment
System I II III IV
Mean residence time: T 6:43 ± 3:45 2:44 ± 0:37 2:80 ± 0:22 3:50 ± 0:20
(mean ± sd) (h)
Mixing e0ciency: 0:34 ± 0:18 0:53 ± 0:02 0:48 ± 0:05 0:56 ± 0:01
(mean ± sd)
RMSE (ppm) 15.70 2.10 5.43 11.34
Mixing pattern Complete–incomplete-piston Complete–incomplete Complete-piston Incomplete
0 20 40 60 80 100 120 0 2 4 6 8 10 12 Measurement Simulation
System II
CO 2 concentration (ppm) Time (h)Fig. 5. Fitting of linear response model to System II as calculated from the complete–incomplete mixing pattern with = 0:53 ± 0:02 and T = 2:44 ± 0:37 h(background CO2 concentration = 315 ppm).
measurement data in 13:00–18:00 hwas probably caused by the variable air4ow rates. Therefore, more data would be necessary to interpret properly the mean residence time of this case. The interpretation gives an approximation of the mean residence time and mixing e0ciency.
For System II, the complete–incomplete 4ow pattern withrelatively low T and moderate was selected. The best 2t is obtained for T = 2:44 ± 0:37 h, = 0:53 ± 0:02, and it is given in Fig. 5 in that the RMSE value is 2:10 ppm, indicating that the model were performing well. The inter-pretation of the data in this case conforms the use of more re2ned model gives more con2dence and additional de-tails concerning the physical parameters of the investigated systems.
For System III, the mean transient time of air in the system withthe complete-piston mixing type was selected owing to the general tendency of such system to be highly mixed. Fig. 6 shows the optimal 2t obtained T = 2:80 ± 0:22 h, = 0:48 ± 0:05. In this case, the obtained 2t is found to be favorably good at the RMSE value 5:43 ppm.
For System IV, the mean transient time of air in the sys-tem withthe incomplete mixing pattern was selected owning to the general tendency of such system to be highly mixed. Fig. 7 shows that the best 2t obtained T = 3:50 ± 0:2 h, and = 0:56 ± 0:2. The RMSE value is found to be 11:34 ppm,
0 50 100 150 200 250 300 0 2 4 6 8 10 12 Measurement Simulation
System III
Time (h) CO 2 concentration (ppm)Fig. 6. Fitting of linear response model to System III as calculated from the complete-piston 4ow mixing with =0:48±0:05 and T =2:80±0:05 h (background CO2 concentration = 315 ppm). 0 50 100 150 200 250 300 350 0 2 4 6 8 10 12 Measurement Simulation
System IV
CO 2 concentration (ppm) Time (h)Fig. 7. Fitting of linear response model to System IV as calculated from the incomplete mixing pattern with = 0:56 ± 0:01 and T = 3:50 ± 0:20 h (background CO2 concentration = 315 ppm).
indicating that the deviation is well within the normal ex-perimental accuracy.
The results of the present study illustrate a high degree of modeling success associated withthe linear
approximations, based exclusively on the measurements from the environmental chamber experiments. Simulation results also con2rm that the two-parameter gamma RTD function gives a good 2t and would be considered as more reliable in those cases. The results presented here should be interpreted as providing evidence that a linear approxima-tion, invoking the principle of superposiapproxima-tion, may be rea-sonable if the system parameters do not result in a dramatic shift in ventilation functions.
The linear response model supplies additional physical information on the ventilation system of T or . The linear response model does not require any physical knowledge prior to the data interpretation, but the obtained system re-sponse function supply physical information.
The results of the present study are generally supportive of the linear systems approach; a more rigorous test of the methodology would include application of dispersive model to an even broader consideration. Parsimony, however, is a legitimate modeling objective and additional model com-plexity (often in the form of nonlinearity) should only be advocated if it can be shown to signi2cantly reduce uncer-tainties normally attributed to random system or measure-ment errors.
4. Conclusions
A simple linear response function model that is capable of interpreting output concentration pro2les and simulating a variety of mixing regimes was presented. The gamma dis-tribution was chosen because it is characterized by only two parameters. The mixing types of piston 4ow (no mixing), incomplete mixing, and complete mixing can be obtained from various forms of RTD functions. Incomplete mixing can be modeled separately or in combination withcomplete mixing or piston 4ow resulted in complete–incomplete and the incomplete-piston 4ow type, respectively.
Heterogeneous mixing is simulating free from the dis-persivity limitations (scale dependence). The shape 4ex-ibility of the system response functions presented in this paper is comparable with those of the dispersive model. The linear response model has the potential for estimating mean residence time in ventilation systems characterized by di!erent mixing regimes. In the case of the linear re-sponse model it is hoped that additional experimental data withwell-documented test information can be used to con2rm the overall applicability of a comparable linear approachto modeling indoor air quality responses to input disturbance.
Appendix A. The deviation of and in Eqs. (8) and (9) We consider a general case where the air volume relations in a general mixing pattern of a ventilated airspace in a
time-invariant system is given in Eq. (7) as VT=Vc+Vp+Vi.
Apply this concept to Eq. (6) with respect to
= T
; (A.1)
where = VT=((Vc+ Vp+ Vi)).
The scalar parameter thus can be derived based on Eq. (A.1) and knowledge that for Vc: = 1 and for Vp: = 0,
= T = T [VT=(Vc+ Vp+ Vi)]= T [VT=(Vc+ Vi)] =T Vc+ Vi VT =T Vc+ Vi+ Vp+ Vc− Vc− Vp VT =T Vc+ (Vi+ Vp+ Vc) − (Vc+ Vp) VT =T Vc VT + − Vc+ Vp VT : (A.2)
In view of Eq. (A.2), we de2ne
= VT
Vc+ Vp (A.3)
and make an appropriate transformation of VT=Vcto
follow-ing expression as VT Vc = VT Vc+ Vp Vc+ Vp Vc = ; (A.4)
where = (Vc+ Vp)=Vcis already given in Eq. (6).
Substituting Eqs. (A.3) and (A.4) into Eq. (A.2), the 2nal expression of scalar parameter can be obtained as
=(V T T=(Vc+ Vi))= T 1 + − : (A.5)
The shape parameter can be derived as follows:
= VT Vc+ Vp+ Vi = VT=VT [(Vi+ Vp+ Vi)=VT] = 1 [(Vi+ Vp+ Vi)=VT] =[(V 1 c+ Vp)=VT+ Vi=VT] =[(V 1 c+ Vp)=VT+ (Vi+ Vp+ Vc− Vc− Vp)=VT] =[(V 1 c+ Vp)=VT+ (Vi+ Vp+ Vc) − (Vc+ Vp)=VT] =[(V 1 c+ Vp)=VT+ − ((Vc+ Vp)=VT)]: (A.6)
Substituting Eq. (A.3) and de2nition of given in Eq. (6) into Eq. (A.6), we can obtain the 2nal expression of shape parameter as
=1= + (1 − 1=)1 ; (A.7)
when ¿ 0:5. References
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