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This variable is the ratio of the cross sectional area of the fluid flowing through the shell to the total shell area. It is an indicator of how tightly packed the tubes are within the shell.To create the one through ten variable test problems that are used in this study, variables are added one Experimental DesignTable 1. Heat exchanger model response and independent variables(16)(17)(18)(19)Model ResponseResponse Symbol UnitsOverall heat transfer rate WModel VariablesVariable Symbol Units Main EffectCold stream inlet temperatureTC,inK Quasi-LinearHot stream inlet temperatureTH,inK Quasi-LinearTube thickness T m Quasi-LinearFlow Area Ratio Arn/a Quasi-LinearTube fouling heat resistanceRfm2K/W Quasi-LinearCold stream flow rate ṁC kg/s ExponentialHot stream flow rate ṁH kg/s ExponentialShell length L m ExponentialNumber of tubes N Integer ExponentialTube inner diameter D m Quasi-Quadratic sampling tend to fill the design space more evenly and provide more accurate results than Latin hypercube designs and orthogonal arrays. They also show that Hammersley sequencing is preferred to uniform designs when large sets of training data can be afforded. In contrast to an expensive computer simulation, all of the test functions used in this paper can be sampled rapidly and large sets of training data are available. Therefore, Hammersley sequence sampling is selected as the method for generating training and test points in this study.Performance AssessmentIn this study, the metamodeling techniques are evaluated based on the number of training points needed to achieve a predetermined error metric. The error metric used is the relative average absolute error (RAAE) and is given by (20):where yi is the actual value of the base model at the ith test points, ŷi is the predicted value from the metamodel, n is the number of sample points, and σ is the standard deviation of the response. To determine the requirednumber of training points, the quantity of training points is increased continuously until an RAAE value of 0.25 is achieved. The training points are generated with Hammersley sequence sampling. The RAAE is calculated using one hundred test points per variable (100D), which are also generated with a Hammersley sequence. One issue with this testing strategy is the possibility that some of the training points and test points overlap. For example, in the 1D case, all of the training points will overlap with test points if the number of test points is pisible by the number of test points. Overlapping is to be avoided because testing interpolating methods (kriging and RBF) at the trainingpoints results in zero error. To avoid overlapping in the 1D problem, 101 test points (a prime number) are used. In higher dimensions, test points and training points do not overlap in a Hammersley sequence provided that the number of training points does not equal the number test points.In addition to the number of sample points necessary to achieve the pre-specified error metric, the computational times required to build each model and to predict the 100D test points are also recorded. All experiments are performed on a 32-bit PC with an Intel Pentium® Dual-Core 2.50 GHz processor with 4.00 GB of RAM.Table 2 includes a summary of the tests to be performed. Metamodels are created in one through ten dimensions for the heat exchanger model and three kernel density estimation functions of varying modality. Performing this task with all three of the metamodeling methods results in a total of 120 tests.Graphical representations of the results of the study are provided in Figs. 2 to 5. In each figure, the abscissa indicates the dimensionality of each problem, while the ordinate represents the number of training points required to meet the error threshold of RAAE <0.25. The center of each circle represents the number of training points required for the specific dimension and metamodeling method. The size of the solid circles represents the relative training time for each metamodel and the translucent circles represent the relative prediction (20)Tests PerformedTest Test Function Scale Test PointsTermination Criteria1 2 mode kernel1-10D100*D(101 in 1D) RAAE < 0.252 D mode kernel32D mode kernel4Heat ExchangerTable 2. Complete experimental planResults and Discussion of dimensions. Also, the functions with higher modality require a higher number of training points. In most cases, radial basis function metamodels require the smallest number of training points of the three methods, followed by support vector regression. Kriging metamodels tend to need the highest number of training points of all. Th e ability of radial basis functions and support vector regression to model the base functions with few training points can be attributed partially to careful selection of user-defi ned tuning parameters. Recall from equations (6) and (10) that the Gaussian basis and kernel functions contain tuning parameters k and g for radial basis functions and support vector regression, respectively. Th e value of these parameters has a signifi cant eff ect on the quality of the resulting metamodel fi t. In this study, these parameters were selected by trial and error until optimal values were obtained. Kriging, on the other hand, has correlation parameters θi that are identifi ed automatically during the fi tting process. Th is automated identifi cation of the correlation parameters is eff ective but requires signifi cant computational expense.Th e results of the heat exchanger are slightly diff erent than those of the kernel function. Th e required number of training points to model the heat exchanger in most dimensions is very similar for kriging and radial basis functions, with support vector regression needing only slightly more in a few cases. Generally speaking, all methods are able to approximate the heat exchanger model accurately with very few training points when compared to the highly modal and nonlinear function produced using the kernel density estimation method.