- Email: [email protected]
Response surface models for CFD predictions of air diffusion performance index in a displacement ventilated office
Energy and Buildings 40 (2008) 774–781 www.elsevier.com/locate/enbuild
Response surface models for CFD predictions of air diffusion performance index in a displacement ventilated office K.C. Ng a,*, K. Kadirgama b,1, E.Y.K. Ng c,2 a
Department of Research & Applications, O.Y.L. R&D Center, Lot 4739, Jalan BRP 8/2, Taman Bukit Rahman Putra, 47000, Sungai Buloh, Selangor Darul Ehsan, Malaysia b Department of Mechanical Engineering, Universiti Tenaga Nasional, Km. 7, Jalan Kajang-Puchong, 43009 Kajang, Selangor Darul Ehsan, Malaysia c School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 29 January 2007; received in revised form 22 March 2007; accepted 13 April 2007
Abstract Based on the Response Surface Methodology (RSM), the development of first- and second-order models for predicting the Air Diffusion Performance Index (ADPI) in a displacement-ventilated office is presented. By adopting the technique of Computational Fluid Dynamics (CFD), the new ADPI models developed are used to investigate the effect of simultaneous variation of three design variables in a displacement ventilation case, i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on the comfort parameter ADPI. The RSM analyses are carried out with the aid of a statistical software package MINITAB. In the current study, the separate effect of individual design variable as well as the second-order interactions between these variables, are investigated. Based on the variance analyses of both the first- and second-order RSM models, the most influential design variable is the supply temperature. In addition, it is found that the interactions of supply temperature with other design variables are insignificant, as deduced from the second-order RSM model. The optimised ADPI value is subsequently obtained from the model equations. # 2007 Elsevier B.V. All rights reserved. Keywords: Response Surface Methodology (RSM); Computational Fluid Dynamics (CFD); Air Diffusion Performance Index (ADPI); Thermal comfort; Air ventilation
1. Introduction The cooling of occupied spaces, which is generally accomplished by mechanical ventilation, consumes a huge amount of non-renewable fossil energy in the world that leads to the pollution of atmospheric environment. Therefore, in order to minimise the energy usage while enabling good thermal comfort condition to be achieved, effective distribution of fresh air within an occupied space is of practical importance. For a long time, the heating, ventilating and air conditioning (HVAC) engineers and researchers have been realising that in order to optimise the comfort condition in an occupied space, efficient quantitative models that establish the relationship between a large group of independent parameters (design * Corresponding author. Tel.: +60 389286227; fax: +60 389212116. E-mail addresses: [email protected] (K.C. Ng), [email protected] (K. Kadirgama), [email protected] (E.Y.K. Ng). 1 Tel.: +60 389287255, fax: +60 389212116. 2 Tel.: +65 67904455, fax: +65 67911859. 0378-7788/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2007.04.024
variables) and output variable (response) are highly desirable. This can be accomplished by both the experimental and numerical approaches. In order to study the relationship between the response and independent design variables, a large number of experiments are undoubtedly required. This has reflected on the increased total cost of the study, which is particularly true in the case of employing physical experimentations. Therefore, numerical experiments such as those accomplished by CFD have been gaining immense popularity within the HVAC industry since the past few decades. Despite the fact that it is not totally free from errors, it serves as a practical design tool for building engineers nowadays. For example, by using pure numerical approach, Haghighat et al. [1] have investigated the relationship between the concentration level in a partitioned room and various positions of door, supply and exhaust. Lee and Awbi [2] have studied the effect of partition on ventilation effectiveness due to its location and gap underneath. Lim et al. [3] have determined the optimum position of an air-conditioned unit for achieving good thermal comfort condition (based on the
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Predicted Mean Vote) in a typical studio-type apartment. Very recently, by employing the commercial CFD code (FLUENT), Ooi et al. [4] have studied the temperature and velocity distributions in an air-conditioned room for various positions of the air conditioner blower. Based on the results of three blower positions simulated, the best position is then selected for maximum comfort of an occupant. In addition, some recommendations have been given by Bojic et al. [5], based on the pure CFD analyses (FLOVENT), the optimum placement of a window-type air conditioner in a residential bedroom in order to achieve minimum draft subjected to the calculation of Air Diffusion Performance Index (ADPI). It can be noted in general, however, most of the numerical studies focus on the one-factor-at-a-time design, without having any idea on the behaviour of response variable when two or more
775
design variables are varied at the same time. The current paper intends to consider this particular issue that involves making design decision based on several design variables, which is practically desirable. In order to demonstrate the method, the authors have considered the effect of simultaneous variations of three design variables in a displacement-ventilated office (refer to Fig. 1), i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on the behaviour of response variable (ADPI). The case considered here is taken from He et al. [6], in which detailed numerical and experimental studies have been performed to investigate the efficiency of contaminant removal for several ventilation systems. Here, the CFD model developed is firstly validated with the experimental data provided by He et al. [6], prior to
Fig. 1. Configuration of the mockup office equipped with a displacement ventilation system investigated by He et al. [6]. The measurement points are 1A, 2A, 3A and 4A. Exact dimensions and locations of the obstacles and measurement points can be found in He et al. [6]. The design variables (Ldd and Lex) are measured from the origin O, (a) isometric view and (b) top view.
776
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
running the response surface analyses. It has been recently noted by Abou-El-Hossein et al. [7] that RSM is one of the statistical techniques that saves cost and time in conducting experiments by reducing the total number of required tests. Furthermore, RSM helps to identify, with great accuracy, the effect of the interactions of different design variables on the response when they are varied simultaneously. In spite of this, within the community of building engineering, only a few research works based on RSM are reported, such as those by Klemm et al. [8] for multicriteria optimisation and Valencia et al. [9] for model development to predict asphalt pavement properties. In this study, based on RSM, the first- and second-order models for the comfort parameter ADPI are developed for a displacement ventilation case illustrated in Fig. 1. Based on the received RSM models from the CFD simulation, the most influential design variable is determined and the corresponding interactions between the design variables are subsequently identified. Also, based on the model equations obtained, one can easily identify the optimum design combination to achieve good thermal comfort condition based on the ADPI value, which is frequently used as a reference value for indoor airflow studies [10].
assumed to be measurable, the response surface can be expressed mathematically as: y ¼ f ðx1 ; x2 ; . . . ; xk Þ
For practical design purpose, the goal is to optimise the response variable y, subjected to certain combination of design variables. In what next, the ADPI model in the form of Eq. (1) will be expressed. 3. Model of air diffusion performance index Draft is a frequent concern when designing indoor environments [11]. In order to account for the presence of draft, which is defined as any localised feeling of coolness or warmth of any position of the body due to both air movement and air temperature, the ADPI parameter is used in the current study. ADPI presents the percentage of locations where values are taken that meet specifications for effective draft temperature (1.7 K < u< 1.1 K) and air speed (WS < 0.35 m/s). If ADPI reaches its maximum value, i.e. 100%, the most desirable condition is thereby achieved [5]. The effective draft temperature is expressed as: u ¼ ðT x T c Þ aðWS bÞ
2. Introduction to response surface methodology Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building. By careful design of experiments, the objective is to optimise a response variable (output variable), which is influenced by several independent design variables (input variables). An experiment is a series of tests, called runs, in which changes are made in the input variables in order to identify the reasons for changes in the output response. Originally, RSM has been developed to model experimental responses and then migrated into the modelling of numerical experiments. The difference is in the type of error generated by the response. In physical experiments, inaccuracy can be due to measurement errors whereas in numerical experiments, errors may due to incomplete convergence of the iterative process, round-off errors and the discrete representation of continuous physical phenomena. In RSM, the errors are assumed to be random. RSM is a methodology of constructing approximations of the system behavior using results of the response analyses calculated at a series of points in the design variable space. Optimisation of RSM can be solved in the following three stages: Design of experiment. Building the response surface model. Solution of minimization/maximisation problem according to the criterion selected. The concept of a response surface involves a dependent variable y called the response variable and several independent design variables x1, x2, . . ., xk. If all of these variables are
(1)
(2)
where Tx is the local dry bulb temperature for air (8C), Tc the averaged room dry bulb temperature (8C) and WS is the air speed (m/s). The constants a and b are taken as 8 K s/m and 0.15 m/s, respectively. With reference to RSM, where the response variable is ADPI in the current study, the relationship between the investigated three design variables and the response variable can be represented by the linear Eq. (3): yð1Þ ¼ b0 x0 þ b1 x1 þ b2 x2 þ b3 x3
(3)
Here, y(1) is the first-order prediction model for ADPI and b is the model parameter. x0 is dummy variable (x0 = 1) and b0 is an arbitrary constant. The design variables such as x1, x2 and x3 are the location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex), respectively. In most of the practical cases, the response surface demonstrates some curvature effects in most ranges of the design variables. Therefore, it would be more useful for a designer to consider the second-order model. The practical importance of second-order model is to help one to understand the second-order effect of each design variable separately and the two-way interaction amongst these design variables. This second-order model y(2) can be represented by Eq. (4), in general, for three design variables: yð2Þ ¼ b0 x0 þ b1 x1 þ b2 x2 þ b3 x3 þ b11 x21 þ b22 x22 þ b33 x23 þ b12 x1 x2 þ b13 x1 x3 þ b23 x2 x3
(4)
Here the two indices (subscripts) of variable b represent the interaction between the corresponding variables. For example, b12 represents the significance of interaction between design variable 1 and 2.
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
4. Research methodology 4.1. CFD simulation The ADPI models, as discussed in the previous section, are determined numerically in the current work. The CFD package used has been constantly verified with the available experimental measurement and reference solution (see [12,13]). Here, prior to performing the sensitivity study based on RSM, the flow model is validated with the experimental data given by He et al. [6]. The ADPI values for various design combinations (obtained from the Box–Behnken method to be discussed later) are then determined by the validated CFD model. The flow solver is based on the finite-volume formulation on structured meshes using the cell-centered approach. It uses a non-staggered variable storage technique, which is more robust as compared to the traditional staggered arrangement [14]. Therefore, in order to avoid the pressure oscillations arisen due to the non-staggered arrangement, the pressure interpolation technique similar to the one proposed by Rhie and Chow [15] is adopted here. The issue of pressure–velocity decoupling associated with the current incompressible flow equations is resolved via the SIMPLE algorithm of Patankar [16]; more recent details of SIMPLE algorithm can be found in Jasak [17]. The Bi-Conjugate Gradient (Bi-CGSTAB) method proposed by Van der Vorst [18] has been used to solve the sparse matrix system arisen from the discretised flow equations. In the current study, the first-order upwind differencing scheme for convective discretisation is adopted for robustness purpose. This is acceptable in the current context due to the fact that trend analysis deduced from the simulation results of various designs is more important here. In order to model the flow turbulence, the RNG k–e equations are adopted. Buoyancy is modelled via the Boussinesq approximation. In order to promote numerical stability of the buoyant flow simulation, a transient approach has been used with a time step size of 0.1 s. The results are assumed to be
777
steady when the percentage difference of the successive change between the variables at current and previous time steps is less than 0.01%. The configuration of the displacement ventilation flow case has been illustrated in Fig. 1. The design variables in this particular design combination (based on that of [6]) are: Ldd = 1.5550 m, T = 15.9 8C and Lex = 2.33 m. Figs. 2 and 3 compare the predicted speed and temperature profiles with the available experimental data at four locations (see 1A, 2A, 3A and 4A in Fig. 1). In general, the predicted speed and temperature variations match the measurements and, by considering the coarseness of the mesh system employed (30 25 16), the agreements can be considered satisfactorily. The discrepancies between the predicted and measured speed profiles may due to, partly, the low speed values associated in most of the space in which the hot-sphere anemometers may fail to give accurate results [6]. For the temperature profiles, all the predictions follow the similar trends of those measured. Here, the predicted and measured temperature profiles have shown clear stratification associated with the displacement ventilation system. 4.2. Experimental design for RSM With the validation of the current CFD model, the ADPI models, i.e. Eqs. (3) and (4), are now readily to be determined. The model parameter b is calculated from the least square method, in which the calculation is performed by adopting the commercial statistical software, MINITAB. In order to reduce the total number of numerical tests and allow simultaneous variation of the three independent design variables, the numerical procedure has to be well designed. In the current study, the Box–Behnken design method, which is based on the combination of the factorial with incomplete block design, has been adopted. The attractive part of this method is that it does not require a large number of tests as it considers only three levels (lowest ‘‘1’’, middle ‘‘0’’ and
Fig. 2. Comparison of speed profiles on mesh 30 25 16 at four locations in a displacement ventilated room. H = 2.26 m. ^: Experiment [6], prediction.
: current
778
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Fig. 3. Comparison of temperature profiles on mesh 30 26 16 at four locations in a displacement ventilated room. H = 2.26 m. T* = (T Ts)/(Te Ts). Ts is the supply temperature (15.9 8C), Te is the exhaust temperature (24.8 8C). ^: Experiment [6], : current prediction. Table 1 Levels of design variables Design variable
Coding of levels 1 (lowest)
Location of the displacement diffuser, Ldd [m] Supply temperature, T [8C] Exhaust position, Lex [m]
0.700 13 0.00
0 (middle) 1.905 16 2.36
1 (highest) 3.110 19 4.72
second-order models. By using MINITAB, the simulation conditions of 15 tests are generated, as shown in Table 2. Based on these testing conditions, the comfort parameter (response variable), ADPI is then computed from the in-house CFD package as described earlier. The CFD-predicted ADPI values are plotted in Fig. 4 for different test numbers, on top of those predictions based on RSM, which will be discussed in the next section. 5. Results and discussions
highest ‘‘1’’) of each design variable. The maximum and minimum levels (constraints) of each design variable are normally determined based on the recommendations given by the manufacturer as well as users’ preferences. The levels of the three design variables are given in Table 1. The Box–Behnken design is normally used for non-sequential experimentation, where a test is conducted only once, which in turn allows efficient evaluation of the model parameters in the first- and
5.1. Development of first-order ADPI model After performing the 15 numerical tests using CFD, the ADPI simulated is used to find the model parameters appearing in the postulated first-order model (see Eq. (3)). In order to perform the calculation of these parameters, the least square method is used with the aid of MINITAB. The first-order linear
Table 2 CFD simulation conditions according to Box–Behnken design and the predicted ADPI models based on CFD and RSM Test number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Location of the displacement diffuser, Ldd [m]
Supply temperature, T [8C]
Exhaust position, Lex [m]
ADPI (%) CFD
1st-order RSM
2nd-order RSM
3.110 3.110 1.905 0.700 1.905 1.905 3.110 0.700 3.110 0.700 0.700 1.905 1.905 1.905 1.905
16 16 13 19 19 13 13 16 19 16 13 16 16 16 19
4.720 0.000 4.720 2.360 4.720 0.000 2.360 4.720 2.360 0.000 2.360 2.360 2.360 2.360 0.000
36.23 35.70 22.87 40.96 42.21 22.77 21.92 35.02 42.59 33.97 26.08 35.04 35.04 35.04 41.72
34.27 33.72 24.99 43.13 43.45 24.44 24.76 34.17 43.23 33.62 24.66 33.95 33.95 33.95 42.90
35.42 35.14 23.33 40.62 42.00 22.99 22.26 35.58 43.62 34.78 25.06 35.71 35.71 35.71 41.25
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
779
locations of the displacement diffuser (Ldd) and the exhaust (Lex) do not contribute much to the variation of the ADPI. In general, the increase of all the design variables will cause the ADPI to become larger, which is desirable from the design point of view. Chung and Lee [10] have performed a similar trend analysis on ADPI based on different values of inlet air temperature. It is worth to mention here that, as illustrated in Fig. 5, the current predicted model agree qualitatively well with that of Chung and Lee [10], in which the ADPI values increase as the supply temperature increases. As seen from Fig. 4, the predicted ADPI values obtained from the first-order model agree well with the CFD values. The adequacy of the first-order model is verified by using the analysis of variance (ANOVA). At a level of confidence of 95%, the model is checked for its adequacy. As shown in Table 3, the P-value of 0.236 (> 0.05) is not significant with the lack-of fit and F-ratio is 3.61. This implies that the model can fit and it is adequate [19].
Fig. 4. Comparison of ADPI models against CFD predictions.
5.2. Development of second-order ADPI model Here, the second-order model is formulated to describe the effect of the three design variables investigated on the ADPI, given by MINITAB: ADPIð2Þ ¼ 82:2641 6:2873Ldd þ 12:3392T þ 0:3862Lex þ 0:0074L2dd 0:3143T 2 0:0875L2ex þ 0:4004Ldd T 0:0452Ldd Lex þ 0:0143TLex
Fig. 5. Comparison of ADPI model based on supply temperatures. For RSM, Ldd is 3.11 m and Lex is 4.72 m.
model for predicting the ADPI can be expressed as: ADPIð1Þ ¼ 15:6377 þ 0:0241Ldd þ 3:0770T þ 0:1153Lex (5) From this linear expression, by examining the values of the coefficients, one can easily deduce that the response variable ADPI is significantly affected by the supply temperature (T). Also, it is interesting to note that the
(6)
Similar to the first-order model, by examining the coefficients of the first-order terms, the supply temperature (T) has the most dominant effect on the ADPI. The contribution of exhaust location (Lex) is the least significant here. Also, owing to the P-value of interaction is 0.248 (>0.05), one can easily deduce that the interactions of distinct design variables are not significant here. In other words, the most dominant design variable T has minimum interaction with others in the current context. As seen from Fig. 4, the predicted ADPI using the secondorder RSM model is able to produce values close to those computed using CFD, and, as it should be the case, it exhibits better agreement as compared to those from the first-order RSM model. The ANOVA shown in Table 4 indicates that the model is adequate as the P-value of the lack-of-fit is not significant (>0.05).
Table 3 Analysis of variance (ANOVA) for first-order equation (from MINITAB) Source of variation
Degree of freedom (d.f.)
Sum of squares
Mean squares
F-ratio
P-value
Regression Linear Residual error Lack-of-fit Pure error Total
3 3 11 9 2 14
682.312 682.312 45.991 43.324 2.667 728.303
227.440 227.440 4.181 4.814 1.333
54.400 54.400
0.000 0.000
3.610
0.236
780
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Table 4 Analysis of variance (ANOVA) for second-order equation (from MINITAB) Source of variation
Degree of freedom (d.f.)
Sum of squares
Mean squares
F-ratio
P-value
Regression Linear Square Interaction Residual error Lack-of-fit Pure error Total
9 3 3 3 5 3 2 14
720.851 682.312 30.050 8.489 7.452 4.785 2.667 728.303
80.095 227.44 10.017 2.830 1.490 1.595 1.333
53.740 152.610 6.720 1.900
0.000 0.000 0.033 0.248
1.200
0.485
5.3. Design optimisation With the ADPI models obtained, the optimised response variable (ADPI) can then be determined. Here, the goal is to maximise the ADPI from the correct combination of the design variables. For optimisation purpose, the response variable is transformed using a specific desirability function shown in Fig. 6. The weight defines the shape of the desirability function for the response, which can be selected from 0.1 to 10.0 to emphasise or de-emphasize the target value (set to 100%). A weight can be Less than one (minimum is 0.1) places less emphasis on the target, or Equal to one places equal importance on the target and the bounds, or Greater than one (maximum is 10.0) places more emphasis on the target, which is the main concern of the current work. Therefore, the weight is set to 10.0. From the second-order ADPI model, the optimized ADPI value is 43.41% (calculated from MINITAB), subjected to the following combination of the design variables: Ldd ¼ 3:11 m; T ¼ 19 C; Lex ¼ 4:6487 m:
(7)
In order to verify the optimised ADPI value predicted from RSM, the CFD simulation is performed again, by adopting the combination of design variables shown in Eq. (7). The ADPI computed is 42.68% (%difference = 1.71%), and it is worth to mention here that it is indeed the highest ADPI value as compared to those from the previous 15 CFD tests (see Table 2). Apparently, all the design variables are approaching their maximum values (level = 1) in the case of maximum ADPI value is desired. This condition holds true even for the first-
order model, in which the linear model has recommended the ceiling values of those design variables in order to achieve the most desirable comfort condition, by maximising the ADPI value in the current context. 6. Conclusion CFD studies have been applied extensively to the simulation of indoor/outdoor airflow. However, most of the numerical tests are based on a one-factor-at-a-time design, without having any idea about the behaviour of an output parameter (response) when two or more design variables are varied simultaneously. The current study focuses on the effect of simultaneous variations of three design variables in a displacement-ventilated office, i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on behaviour of the response variable (air diffusion performance index). In the current work, the response surface methodology has been proven to be a successful technique to perform the trend analysis of air diffusion performance index with respect to various combinations of three design variables. By using the least square method, the first- and second-order models have been developed based on the test conditions in accordance with the Box–Behnken design method. The models have been found to accurately representing the ADPI values with respect to those simulated using CFD. The equations have been checked for their adequacy with a confidence interval of 95%. Both RSM models reveal that the supply temperature is the most significant design variable in determining the ADPI response as compared to the others. In general, within the working range of the supply temperatures considered here, ADPI increases as the supply temperature increases. Based on the second-order RSM model, the supply temperature does not interact much with the remaining design variables. Therefore, one may exclude both locations of displacement diffuser and exhaust for indoor comfort design purpose (based on ADPI) in the current design case. With the model equations obtained, a designer can subsequently select the best combination of design variables for achieving optimum comfort condition. Acknowledgements
Fig. 6. Desirability function for maximising the response.
The first author would like to express his sincere appreciation to his former colleague, Dr. T.K. Lim (now in AMD, Singapore) for his recommendation. We also acknowl-
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
edge Mr. W.M. Chin (OYL R&D, Malaysia) for showing consistent interest and support on the current work. The software facilities provided by Universiti Tenaga Nasional (UNITEN) are greatly appreciated. Also, special thanks to Mr. Anuar (UNITEN), for developing the Graphical User Interface of the current CFD package. Reference [1] F. Haghighat, Z. Jiang, J. Wang, A CFD analysis of ventilation effectiveness in a partitioned room, Indoor Air 4 (1991) 606–615. [2] H. Lee, H.B. Awbi, Effect of partition location on the air and contaminant movement in a room, in: Proceedings of Indoor Air ‘99, vol. 1, Edinburgh, August 8–13, (1999), pp. 349–354. [3] T.K. Lim, Y.L. Ong, M. Hamdi, ‘Locating the Optimum Position to Place an Indoor Air Conditioner in Studio-Type Apartment to Achieve Good Thermal Comfort’, FLOVENT Technical Paper, Paper V45, total number of pages: 7, 2005. [4] Y. Ooi, I.A. Bahruddin, P.A.A. Narayana, Airflow analysis in an air conditioning room, Building and Environment 42 (3) (2007) 1531– 1537. [5] M. Bojic, F. Yik, T.Y. Lo, Locating air-conditioners and furniture inside residential flats to obtain good thermal comfort, Energy and Buildings 34 (2002) 745–751. [6] G. He, Z. Yang, J. Srebric, Removal of contaminants released from room surfaces by displacement and mixing ventilation: modelling and validation, Indoor Air 15 (2005) 367–380. [7] K.A. Abou-El-Hossein, K. Kadirgama, M. Hamdi, K.Y. Benyounis, Prediction of cutting force in end-milling operation of modified AISI P20 tool steel, Journal of Materials Processing Technology 182 (2007) 241–247.
781
[8] K. Klemm, W. Marks, A.J. Klemm, Multicriteria optimisation of the building arrangement with application of numerical simulation, Building and Environment 35 (2000) 537–544. [9] L.E.C. Valencia, A.M. Ramirez, G.L. Barcenas, E.A. Guzman, Modelling of the performance of asphalt pavement using response surface method, Building and Environment 40 (8) (2005) 1140–1149. [10] K.C. Chung, C.Y. Lee, Predicting air flow and thermal comfort in an indoor environment under different air diffusion models, Building and Environment 31 (1) (1996) 21–26. [11] J. Toftum, Air movement-good or bad? Indoor Air 14 (Suppl. 7) (2004) 40–45. [12] K.C. Ng, ‘Multigrid solution using high-resolution NVF differencing schemes for solution-adaptive unstructured meshes’, Ph.D. Thesis, Department of Mechanical Engineering, Universiti Tenaga Nasional, Malaysia, 2006. [13] K.C. Ng, M.Z. Yusoff, E.Y.K. Ng, Higher-order bounded differencing schemes for compressible and incompressible flows, International Journal for Numerical Methods in Fluids 53 (1) (2007) 57–80. [14] J. Zhu, On the higher-order bounded discretisation schemes for finite volume computations of incompressible flows, Computer Methods in Applied Mechanics and Engineering 98 (1992) 345–360. [15] C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA Journal 21 (1983) 1525–1532. [16] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1981. [17] H. Jasak, ‘Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flow’, Ph.D. Thesis, Imperial College, University of London, UK, 1996. [18] H.A. Van der Vorst, BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM Journal of Scientific and Statistical Computing 13 (2) (1992) 631–644. [19] C.R. Hicks, Fundamental Concepts in the Design of Experiments, fourth ed., Oxford University Press, USA, 1993.
Response surface models for CFD predictions of air diffusion performance index in a displacement ventilated office K.C. Ng a,*, K. Kadirgama b,1, E.Y.K. Ng c,2 a
Department of Research & Applications, O.Y.L. R&D Center, Lot 4739, Jalan BRP 8/2, Taman Bukit Rahman Putra, 47000, Sungai Buloh, Selangor Darul Ehsan, Malaysia b Department of Mechanical Engineering, Universiti Tenaga Nasional, Km. 7, Jalan Kajang-Puchong, 43009 Kajang, Selangor Darul Ehsan, Malaysia c School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore Received 29 January 2007; received in revised form 22 March 2007; accepted 13 April 2007
Abstract Based on the Response Surface Methodology (RSM), the development of first- and second-order models for predicting the Air Diffusion Performance Index (ADPI) in a displacement-ventilated office is presented. By adopting the technique of Computational Fluid Dynamics (CFD), the new ADPI models developed are used to investigate the effect of simultaneous variation of three design variables in a displacement ventilation case, i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on the comfort parameter ADPI. The RSM analyses are carried out with the aid of a statistical software package MINITAB. In the current study, the separate effect of individual design variable as well as the second-order interactions between these variables, are investigated. Based on the variance analyses of both the first- and second-order RSM models, the most influential design variable is the supply temperature. In addition, it is found that the interactions of supply temperature with other design variables are insignificant, as deduced from the second-order RSM model. The optimised ADPI value is subsequently obtained from the model equations. # 2007 Elsevier B.V. All rights reserved. Keywords: Response Surface Methodology (RSM); Computational Fluid Dynamics (CFD); Air Diffusion Performance Index (ADPI); Thermal comfort; Air ventilation
1. Introduction The cooling of occupied spaces, which is generally accomplished by mechanical ventilation, consumes a huge amount of non-renewable fossil energy in the world that leads to the pollution of atmospheric environment. Therefore, in order to minimise the energy usage while enabling good thermal comfort condition to be achieved, effective distribution of fresh air within an occupied space is of practical importance. For a long time, the heating, ventilating and air conditioning (HVAC) engineers and researchers have been realising that in order to optimise the comfort condition in an occupied space, efficient quantitative models that establish the relationship between a large group of independent parameters (design * Corresponding author. Tel.: +60 389286227; fax: +60 389212116. E-mail addresses: [email protected] (K.C. Ng), [email protected] (K. Kadirgama), [email protected] (E.Y.K. Ng). 1 Tel.: +60 389287255, fax: +60 389212116. 2 Tel.: +65 67904455, fax: +65 67911859. 0378-7788/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2007.04.024
variables) and output variable (response) are highly desirable. This can be accomplished by both the experimental and numerical approaches. In order to study the relationship between the response and independent design variables, a large number of experiments are undoubtedly required. This has reflected on the increased total cost of the study, which is particularly true in the case of employing physical experimentations. Therefore, numerical experiments such as those accomplished by CFD have been gaining immense popularity within the HVAC industry since the past few decades. Despite the fact that it is not totally free from errors, it serves as a practical design tool for building engineers nowadays. For example, by using pure numerical approach, Haghighat et al. [1] have investigated the relationship between the concentration level in a partitioned room and various positions of door, supply and exhaust. Lee and Awbi [2] have studied the effect of partition on ventilation effectiveness due to its location and gap underneath. Lim et al. [3] have determined the optimum position of an air-conditioned unit for achieving good thermal comfort condition (based on the
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Predicted Mean Vote) in a typical studio-type apartment. Very recently, by employing the commercial CFD code (FLUENT), Ooi et al. [4] have studied the temperature and velocity distributions in an air-conditioned room for various positions of the air conditioner blower. Based on the results of three blower positions simulated, the best position is then selected for maximum comfort of an occupant. In addition, some recommendations have been given by Bojic et al. [5], based on the pure CFD analyses (FLOVENT), the optimum placement of a window-type air conditioner in a residential bedroom in order to achieve minimum draft subjected to the calculation of Air Diffusion Performance Index (ADPI). It can be noted in general, however, most of the numerical studies focus on the one-factor-at-a-time design, without having any idea on the behaviour of response variable when two or more
775
design variables are varied at the same time. The current paper intends to consider this particular issue that involves making design decision based on several design variables, which is practically desirable. In order to demonstrate the method, the authors have considered the effect of simultaneous variations of three design variables in a displacement-ventilated office (refer to Fig. 1), i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on the behaviour of response variable (ADPI). The case considered here is taken from He et al. [6], in which detailed numerical and experimental studies have been performed to investigate the efficiency of contaminant removal for several ventilation systems. Here, the CFD model developed is firstly validated with the experimental data provided by He et al. [6], prior to
Fig. 1. Configuration of the mockup office equipped with a displacement ventilation system investigated by He et al. [6]. The measurement points are 1A, 2A, 3A and 4A. Exact dimensions and locations of the obstacles and measurement points can be found in He et al. [6]. The design variables (Ldd and Lex) are measured from the origin O, (a) isometric view and (b) top view.
776
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
running the response surface analyses. It has been recently noted by Abou-El-Hossein et al. [7] that RSM is one of the statistical techniques that saves cost and time in conducting experiments by reducing the total number of required tests. Furthermore, RSM helps to identify, with great accuracy, the effect of the interactions of different design variables on the response when they are varied simultaneously. In spite of this, within the community of building engineering, only a few research works based on RSM are reported, such as those by Klemm et al. [8] for multicriteria optimisation and Valencia et al. [9] for model development to predict asphalt pavement properties. In this study, based on RSM, the first- and second-order models for the comfort parameter ADPI are developed for a displacement ventilation case illustrated in Fig. 1. Based on the received RSM models from the CFD simulation, the most influential design variable is determined and the corresponding interactions between the design variables are subsequently identified. Also, based on the model equations obtained, one can easily identify the optimum design combination to achieve good thermal comfort condition based on the ADPI value, which is frequently used as a reference value for indoor airflow studies [10].
assumed to be measurable, the response surface can be expressed mathematically as: y ¼ f ðx1 ; x2 ; . . . ; xk Þ
For practical design purpose, the goal is to optimise the response variable y, subjected to certain combination of design variables. In what next, the ADPI model in the form of Eq. (1) will be expressed. 3. Model of air diffusion performance index Draft is a frequent concern when designing indoor environments [11]. In order to account for the presence of draft, which is defined as any localised feeling of coolness or warmth of any position of the body due to both air movement and air temperature, the ADPI parameter is used in the current study. ADPI presents the percentage of locations where values are taken that meet specifications for effective draft temperature (1.7 K < u< 1.1 K) and air speed (WS < 0.35 m/s). If ADPI reaches its maximum value, i.e. 100%, the most desirable condition is thereby achieved [5]. The effective draft temperature is expressed as: u ¼ ðT x T c Þ aðWS bÞ
2. Introduction to response surface methodology Response Surface Methodology (RSM) is a collection of mathematical and statistical techniques for empirical model building. By careful design of experiments, the objective is to optimise a response variable (output variable), which is influenced by several independent design variables (input variables). An experiment is a series of tests, called runs, in which changes are made in the input variables in order to identify the reasons for changes in the output response. Originally, RSM has been developed to model experimental responses and then migrated into the modelling of numerical experiments. The difference is in the type of error generated by the response. In physical experiments, inaccuracy can be due to measurement errors whereas in numerical experiments, errors may due to incomplete convergence of the iterative process, round-off errors and the discrete representation of continuous physical phenomena. In RSM, the errors are assumed to be random. RSM is a methodology of constructing approximations of the system behavior using results of the response analyses calculated at a series of points in the design variable space. Optimisation of RSM can be solved in the following three stages: Design of experiment. Building the response surface model. Solution of minimization/maximisation problem according to the criterion selected. The concept of a response surface involves a dependent variable y called the response variable and several independent design variables x1, x2, . . ., xk. If all of these variables are
(1)
(2)
where Tx is the local dry bulb temperature for air (8C), Tc the averaged room dry bulb temperature (8C) and WS is the air speed (m/s). The constants a and b are taken as 8 K s/m and 0.15 m/s, respectively. With reference to RSM, where the response variable is ADPI in the current study, the relationship between the investigated three design variables and the response variable can be represented by the linear Eq. (3): yð1Þ ¼ b0 x0 þ b1 x1 þ b2 x2 þ b3 x3
(3)
Here, y(1) is the first-order prediction model for ADPI and b is the model parameter. x0 is dummy variable (x0 = 1) and b0 is an arbitrary constant. The design variables such as x1, x2 and x3 are the location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex), respectively. In most of the practical cases, the response surface demonstrates some curvature effects in most ranges of the design variables. Therefore, it would be more useful for a designer to consider the second-order model. The practical importance of second-order model is to help one to understand the second-order effect of each design variable separately and the two-way interaction amongst these design variables. This second-order model y(2) can be represented by Eq. (4), in general, for three design variables: yð2Þ ¼ b0 x0 þ b1 x1 þ b2 x2 þ b3 x3 þ b11 x21 þ b22 x22 þ b33 x23 þ b12 x1 x2 þ b13 x1 x3 þ b23 x2 x3
(4)
Here the two indices (subscripts) of variable b represent the interaction between the corresponding variables. For example, b12 represents the significance of interaction between design variable 1 and 2.
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
4. Research methodology 4.1. CFD simulation The ADPI models, as discussed in the previous section, are determined numerically in the current work. The CFD package used has been constantly verified with the available experimental measurement and reference solution (see [12,13]). Here, prior to performing the sensitivity study based on RSM, the flow model is validated with the experimental data given by He et al. [6]. The ADPI values for various design combinations (obtained from the Box–Behnken method to be discussed later) are then determined by the validated CFD model. The flow solver is based on the finite-volume formulation on structured meshes using the cell-centered approach. It uses a non-staggered variable storage technique, which is more robust as compared to the traditional staggered arrangement [14]. Therefore, in order to avoid the pressure oscillations arisen due to the non-staggered arrangement, the pressure interpolation technique similar to the one proposed by Rhie and Chow [15] is adopted here. The issue of pressure–velocity decoupling associated with the current incompressible flow equations is resolved via the SIMPLE algorithm of Patankar [16]; more recent details of SIMPLE algorithm can be found in Jasak [17]. The Bi-Conjugate Gradient (Bi-CGSTAB) method proposed by Van der Vorst [18] has been used to solve the sparse matrix system arisen from the discretised flow equations. In the current study, the first-order upwind differencing scheme for convective discretisation is adopted for robustness purpose. This is acceptable in the current context due to the fact that trend analysis deduced from the simulation results of various designs is more important here. In order to model the flow turbulence, the RNG k–e equations are adopted. Buoyancy is modelled via the Boussinesq approximation. In order to promote numerical stability of the buoyant flow simulation, a transient approach has been used with a time step size of 0.1 s. The results are assumed to be
777
steady when the percentage difference of the successive change between the variables at current and previous time steps is less than 0.01%. The configuration of the displacement ventilation flow case has been illustrated in Fig. 1. The design variables in this particular design combination (based on that of [6]) are: Ldd = 1.5550 m, T = 15.9 8C and Lex = 2.33 m. Figs. 2 and 3 compare the predicted speed and temperature profiles with the available experimental data at four locations (see 1A, 2A, 3A and 4A in Fig. 1). In general, the predicted speed and temperature variations match the measurements and, by considering the coarseness of the mesh system employed (30 25 16), the agreements can be considered satisfactorily. The discrepancies between the predicted and measured speed profiles may due to, partly, the low speed values associated in most of the space in which the hot-sphere anemometers may fail to give accurate results [6]. For the temperature profiles, all the predictions follow the similar trends of those measured. Here, the predicted and measured temperature profiles have shown clear stratification associated with the displacement ventilation system. 4.2. Experimental design for RSM With the validation of the current CFD model, the ADPI models, i.e. Eqs. (3) and (4), are now readily to be determined. The model parameter b is calculated from the least square method, in which the calculation is performed by adopting the commercial statistical software, MINITAB. In order to reduce the total number of numerical tests and allow simultaneous variation of the three independent design variables, the numerical procedure has to be well designed. In the current study, the Box–Behnken design method, which is based on the combination of the factorial with incomplete block design, has been adopted. The attractive part of this method is that it does not require a large number of tests as it considers only three levels (lowest ‘‘1’’, middle ‘‘0’’ and
Fig. 2. Comparison of speed profiles on mesh 30 25 16 at four locations in a displacement ventilated room. H = 2.26 m. ^: Experiment [6], prediction.
: current
778
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Fig. 3. Comparison of temperature profiles on mesh 30 26 16 at four locations in a displacement ventilated room. H = 2.26 m. T* = (T Ts)/(Te Ts). Ts is the supply temperature (15.9 8C), Te is the exhaust temperature (24.8 8C). ^: Experiment [6], : current prediction. Table 1 Levels of design variables Design variable
Coding of levels 1 (lowest)
Location of the displacement diffuser, Ldd [m] Supply temperature, T [8C] Exhaust position, Lex [m]
0.700 13 0.00
0 (middle) 1.905 16 2.36
1 (highest) 3.110 19 4.72
second-order models. By using MINITAB, the simulation conditions of 15 tests are generated, as shown in Table 2. Based on these testing conditions, the comfort parameter (response variable), ADPI is then computed from the in-house CFD package as described earlier. The CFD-predicted ADPI values are plotted in Fig. 4 for different test numbers, on top of those predictions based on RSM, which will be discussed in the next section. 5. Results and discussions
highest ‘‘1’’) of each design variable. The maximum and minimum levels (constraints) of each design variable are normally determined based on the recommendations given by the manufacturer as well as users’ preferences. The levels of the three design variables are given in Table 1. The Box–Behnken design is normally used for non-sequential experimentation, where a test is conducted only once, which in turn allows efficient evaluation of the model parameters in the first- and
5.1. Development of first-order ADPI model After performing the 15 numerical tests using CFD, the ADPI simulated is used to find the model parameters appearing in the postulated first-order model (see Eq. (3)). In order to perform the calculation of these parameters, the least square method is used with the aid of MINITAB. The first-order linear
Table 2 CFD simulation conditions according to Box–Behnken design and the predicted ADPI models based on CFD and RSM Test number
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Location of the displacement diffuser, Ldd [m]
Supply temperature, T [8C]
Exhaust position, Lex [m]
ADPI (%) CFD
1st-order RSM
2nd-order RSM
3.110 3.110 1.905 0.700 1.905 1.905 3.110 0.700 3.110 0.700 0.700 1.905 1.905 1.905 1.905
16 16 13 19 19 13 13 16 19 16 13 16 16 16 19
4.720 0.000 4.720 2.360 4.720 0.000 2.360 4.720 2.360 0.000 2.360 2.360 2.360 2.360 0.000
36.23 35.70 22.87 40.96 42.21 22.77 21.92 35.02 42.59 33.97 26.08 35.04 35.04 35.04 41.72
34.27 33.72 24.99 43.13 43.45 24.44 24.76 34.17 43.23 33.62 24.66 33.95 33.95 33.95 42.90
35.42 35.14 23.33 40.62 42.00 22.99 22.26 35.58 43.62 34.78 25.06 35.71 35.71 35.71 41.25
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
779
locations of the displacement diffuser (Ldd) and the exhaust (Lex) do not contribute much to the variation of the ADPI. In general, the increase of all the design variables will cause the ADPI to become larger, which is desirable from the design point of view. Chung and Lee [10] have performed a similar trend analysis on ADPI based on different values of inlet air temperature. It is worth to mention here that, as illustrated in Fig. 5, the current predicted model agree qualitatively well with that of Chung and Lee [10], in which the ADPI values increase as the supply temperature increases. As seen from Fig. 4, the predicted ADPI values obtained from the first-order model agree well with the CFD values. The adequacy of the first-order model is verified by using the analysis of variance (ANOVA). At a level of confidence of 95%, the model is checked for its adequacy. As shown in Table 3, the P-value of 0.236 (> 0.05) is not significant with the lack-of fit and F-ratio is 3.61. This implies that the model can fit and it is adequate [19].
Fig. 4. Comparison of ADPI models against CFD predictions.
5.2. Development of second-order ADPI model Here, the second-order model is formulated to describe the effect of the three design variables investigated on the ADPI, given by MINITAB: ADPIð2Þ ¼ 82:2641 6:2873Ldd þ 12:3392T þ 0:3862Lex þ 0:0074L2dd 0:3143T 2 0:0875L2ex þ 0:4004Ldd T 0:0452Ldd Lex þ 0:0143TLex
Fig. 5. Comparison of ADPI model based on supply temperatures. For RSM, Ldd is 3.11 m and Lex is 4.72 m.
model for predicting the ADPI can be expressed as: ADPIð1Þ ¼ 15:6377 þ 0:0241Ldd þ 3:0770T þ 0:1153Lex (5) From this linear expression, by examining the values of the coefficients, one can easily deduce that the response variable ADPI is significantly affected by the supply temperature (T). Also, it is interesting to note that the
(6)
Similar to the first-order model, by examining the coefficients of the first-order terms, the supply temperature (T) has the most dominant effect on the ADPI. The contribution of exhaust location (Lex) is the least significant here. Also, owing to the P-value of interaction is 0.248 (>0.05), one can easily deduce that the interactions of distinct design variables are not significant here. In other words, the most dominant design variable T has minimum interaction with others in the current context. As seen from Fig. 4, the predicted ADPI using the secondorder RSM model is able to produce values close to those computed using CFD, and, as it should be the case, it exhibits better agreement as compared to those from the first-order RSM model. The ANOVA shown in Table 4 indicates that the model is adequate as the P-value of the lack-of-fit is not significant (>0.05).
Table 3 Analysis of variance (ANOVA) for first-order equation (from MINITAB) Source of variation
Degree of freedom (d.f.)
Sum of squares
Mean squares
F-ratio
P-value
Regression Linear Residual error Lack-of-fit Pure error Total
3 3 11 9 2 14
682.312 682.312 45.991 43.324 2.667 728.303
227.440 227.440 4.181 4.814 1.333
54.400 54.400
0.000 0.000
3.610
0.236
780
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
Table 4 Analysis of variance (ANOVA) for second-order equation (from MINITAB) Source of variation
Degree of freedom (d.f.)
Sum of squares
Mean squares
F-ratio
P-value
Regression Linear Square Interaction Residual error Lack-of-fit Pure error Total
9 3 3 3 5 3 2 14
720.851 682.312 30.050 8.489 7.452 4.785 2.667 728.303
80.095 227.44 10.017 2.830 1.490 1.595 1.333
53.740 152.610 6.720 1.900
0.000 0.000 0.033 0.248
1.200
0.485
5.3. Design optimisation With the ADPI models obtained, the optimised response variable (ADPI) can then be determined. Here, the goal is to maximise the ADPI from the correct combination of the design variables. For optimisation purpose, the response variable is transformed using a specific desirability function shown in Fig. 6. The weight defines the shape of the desirability function for the response, which can be selected from 0.1 to 10.0 to emphasise or de-emphasize the target value (set to 100%). A weight can be Less than one (minimum is 0.1) places less emphasis on the target, or Equal to one places equal importance on the target and the bounds, or Greater than one (maximum is 10.0) places more emphasis on the target, which is the main concern of the current work. Therefore, the weight is set to 10.0. From the second-order ADPI model, the optimized ADPI value is 43.41% (calculated from MINITAB), subjected to the following combination of the design variables: Ldd ¼ 3:11 m; T ¼ 19 C; Lex ¼ 4:6487 m:
(7)
In order to verify the optimised ADPI value predicted from RSM, the CFD simulation is performed again, by adopting the combination of design variables shown in Eq. (7). The ADPI computed is 42.68% (%difference = 1.71%), and it is worth to mention here that it is indeed the highest ADPI value as compared to those from the previous 15 CFD tests (see Table 2). Apparently, all the design variables are approaching their maximum values (level = 1) in the case of maximum ADPI value is desired. This condition holds true even for the first-
order model, in which the linear model has recommended the ceiling values of those design variables in order to achieve the most desirable comfort condition, by maximising the ADPI value in the current context. 6. Conclusion CFD studies have been applied extensively to the simulation of indoor/outdoor airflow. However, most of the numerical tests are based on a one-factor-at-a-time design, without having any idea about the behaviour of an output parameter (response) when two or more design variables are varied simultaneously. The current study focuses on the effect of simultaneous variations of three design variables in a displacement-ventilated office, i.e. location of the displacement diffuser (Ldd), supply temperature (T) and exhaust position (Lex) on behaviour of the response variable (air diffusion performance index). In the current work, the response surface methodology has been proven to be a successful technique to perform the trend analysis of air diffusion performance index with respect to various combinations of three design variables. By using the least square method, the first- and second-order models have been developed based on the test conditions in accordance with the Box–Behnken design method. The models have been found to accurately representing the ADPI values with respect to those simulated using CFD. The equations have been checked for their adequacy with a confidence interval of 95%. Both RSM models reveal that the supply temperature is the most significant design variable in determining the ADPI response as compared to the others. In general, within the working range of the supply temperatures considered here, ADPI increases as the supply temperature increases. Based on the second-order RSM model, the supply temperature does not interact much with the remaining design variables. Therefore, one may exclude both locations of displacement diffuser and exhaust for indoor comfort design purpose (based on ADPI) in the current design case. With the model equations obtained, a designer can subsequently select the best combination of design variables for achieving optimum comfort condition. Acknowledgements
Fig. 6. Desirability function for maximising the response.
The first author would like to express his sincere appreciation to his former colleague, Dr. T.K. Lim (now in AMD, Singapore) for his recommendation. We also acknowl-
K.C. Ng et al. / Energy and Buildings 40 (2008) 774–781
edge Mr. W.M. Chin (OYL R&D, Malaysia) for showing consistent interest and support on the current work. The software facilities provided by Universiti Tenaga Nasional (UNITEN) are greatly appreciated. Also, special thanks to Mr. Anuar (UNITEN), for developing the Graphical User Interface of the current CFD package. Reference [1] F. Haghighat, Z. Jiang, J. Wang, A CFD analysis of ventilation effectiveness in a partitioned room, Indoor Air 4 (1991) 606–615. [2] H. Lee, H.B. Awbi, Effect of partition location on the air and contaminant movement in a room, in: Proceedings of Indoor Air ‘99, vol. 1, Edinburgh, August 8–13, (1999), pp. 349–354. [3] T.K. Lim, Y.L. Ong, M. Hamdi, ‘Locating the Optimum Position to Place an Indoor Air Conditioner in Studio-Type Apartment to Achieve Good Thermal Comfort’, FLOVENT Technical Paper, Paper V45, total number of pages: 7, 2005. [4] Y. Ooi, I.A. Bahruddin, P.A.A. Narayana, Airflow analysis in an air conditioning room, Building and Environment 42 (3) (2007) 1531– 1537. [5] M. Bojic, F. Yik, T.Y. Lo, Locating air-conditioners and furniture inside residential flats to obtain good thermal comfort, Energy and Buildings 34 (2002) 745–751. [6] G. He, Z. Yang, J. Srebric, Removal of contaminants released from room surfaces by displacement and mixing ventilation: modelling and validation, Indoor Air 15 (2005) 367–380. [7] K.A. Abou-El-Hossein, K. Kadirgama, M. Hamdi, K.Y. Benyounis, Prediction of cutting force in end-milling operation of modified AISI P20 tool steel, Journal of Materials Processing Technology 182 (2007) 241–247.
781
[8] K. Klemm, W. Marks, A.J. Klemm, Multicriteria optimisation of the building arrangement with application of numerical simulation, Building and Environment 35 (2000) 537–544. [9] L.E.C. Valencia, A.M. Ramirez, G.L. Barcenas, E.A. Guzman, Modelling of the performance of asphalt pavement using response surface method, Building and Environment 40 (8) (2005) 1140–1149. [10] K.C. Chung, C.Y. Lee, Predicting air flow and thermal comfort in an indoor environment under different air diffusion models, Building and Environment 31 (1) (1996) 21–26. [11] J. Toftum, Air movement-good or bad? Indoor Air 14 (Suppl. 7) (2004) 40–45. [12] K.C. Ng, ‘Multigrid solution using high-resolution NVF differencing schemes for solution-adaptive unstructured meshes’, Ph.D. Thesis, Department of Mechanical Engineering, Universiti Tenaga Nasional, Malaysia, 2006. [13] K.C. Ng, M.Z. Yusoff, E.Y.K. Ng, Higher-order bounded differencing schemes for compressible and incompressible flows, International Journal for Numerical Methods in Fluids 53 (1) (2007) 57–80. [14] J. Zhu, On the higher-order bounded discretisation schemes for finite volume computations of incompressible flows, Computer Methods in Applied Mechanics and Engineering 98 (1992) 345–360. [15] C.M. Rhie, W.L. Chow, Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA Journal 21 (1983) 1525–1532. [16] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1981. [17] H. Jasak, ‘Error Analysis and Estimation for the Finite Volume Method with Applications to Fluid Flow’, Ph.D. Thesis, Imperial College, University of London, UK, 1996. [18] H.A. Van der Vorst, BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems, SIAM Journal of Scientific and Statistical Computing 13 (2) (1992) 631–644. [19] C.R. Hicks, Fundamental Concepts in the Design of Experiments, fourth ed., Oxford University Press, USA, 1993.