4.2 Design of experiments (DOE)

With the aim of obtaining regression equations for Nu number, the design of experiments is applied. A second-order regression model for each of the variables is used. The DOE used was the Box-Behnken response surface design. The advantages of this method are as follows: it is a second-order model, and the experimental


### Table 3.

Parameters used in the solver of the CFD simulations.

Figure 7. Validation of nu number on the hot side (G) for the finned-tube geometry.

Figure 8. Validation of nu number on the cold side (EG) for the finned-tube geometry.

Exhaust Gas Heat Recovery for an ORC: A Case Study DOI: http://dx.doi.org/10.5772/intechopen.86075

Figure 9. Validation of nu number on the hot side (G) for the helical-tube geometry (G).

Figure 10. Validation of nu number on the cold side (EG) for the helical-tube geometry.

Figure 11. Local Nusselt number for ethylene glycol side. Helical heat exchanger.

points are within the experimental space. Other methods like CCD have experimental points outside the experimental space which present diver ence in the CFD simulations. Therefore, the Box-Behnken method was used in the computer simulations. The results are:


The equations obtained are

Figure 12. Local Nusselt number for gas side. Helical heat exchanger.

Figure 13. Local Nusselt number for ethylene glycol side. Finned heat exchanger.

Figure 14. Local Nusselt number for gas side. Finned heat exchanger.

Exhaust Gas Heat Recovery for an ORC: A Case Study DOI: http://dx.doi.org/10.5772/intechopen.86075

QG<sup>00</sup> ¼ � <sup>19483</sup> � <sup>63</sup>ID � <sup>92</sup>ED <sup>þ</sup> <sup>38412</sup>MASS <sup>þ</sup> <sup>86</sup>:9TEMP <sup>þ</sup> <sup>3</sup>:354ID<sup>2</sup> <sup>þ</sup> <sup>2</sup>:66ED<sup>2</sup> � <sup>3</sup>:20ID <sup>∗</sup> ED � <sup>0</sup>:164ID <sup>∗</sup> TEMP � <sup>658</sup>ED <sup>∗</sup> MASS (10) � 0:471ED ∗ TEMP þ 225MASS ∗ TEMP QEG<sup>00</sup> ¼ � <sup>16212</sup> � <sup>111</sup>ED <sup>þ</sup> <sup>29547</sup>MASS <sup>þ</sup> <sup>73</sup>TEMP <sup>þ</sup> <sup>2</sup>:9ID<sup>2</sup> <sup>þ</sup> <sup>2</sup>:59ED<sup>2</sup> � <sup>3</sup>:22ID <sup>∗</sup> ED � <sup>0</sup>:131ID <sup>∗</sup> TEMP � <sup>585</sup>ED <sup>∗</sup> MASS (11) � 0:403ED ∗ TEMP þ 212:9MASS ∗ TEMP PEG <sup>¼</sup> <sup>84</sup> <sup>þ</sup> <sup>12</sup>:31ID � <sup>9</sup>:4ED <sup>þ</sup> <sup>186</sup>MASS <sup>þ</sup> <sup>0</sup>:0657ID<sup>2</sup> <sup>þ</sup> <sup>0</sup>:0955ED<sup>2</sup> � <sup>0</sup>:000053TEMP<sup>2</sup> � <sup>0</sup>:1834ID <sup>∗</sup> ED <sup>þ</sup> <sup>4</sup>:39ID <sup>∗</sup> MASS � <sup>0</sup>:00292ID ∗ TEMP � 3:49ED ∗ MASS þ 0:00238ED ∗ TEMP � 0:093MASS ∗ TEMP (12) PG ¼ 5:52 � 0:1187ID þ 0:058ED þ 26:86MASS þ 0:00319TEMP <sup>þ</sup> <sup>0</sup>:000386ID<sup>2</sup> � <sup>0</sup>:000328ED<sup>2</sup> <sup>þ</sup> <sup>0</sup>:000157ID <sup>∗</sup> ED � <sup>0</sup>:000014ID ∗ TEMP þ 0:00058MASS ∗ TEMP (13)

$$\begin{aligned} \mathbf{AG} &= 227 + 4.84 \mathbf{ID} - 2.05 \mathbf{ED} - 647 \mathbf{M} \mathbf{ASS} - 0.688 \mathbf{TEMP} + 0.02127 \mathbf{ID}^2 \\ &+ 0.0092 \mathbf{ED}^2 + 588.4 \mathbf{M} \mathbf{ASS}^2 + 0.000430 \mathbf{TEMP}^2 - 0.03658 \mathbf{ID} \\ &\ast \mathbf{ED} - 4.608 \mathbf{ID} \ast \mathbf{M} \mathbf{ASS} - 0.004547 \mathbf{ID} \ast \mathbf{TEMP} + 3.26 \mathbf{ED} \ast \mathbf{M} \mathbf{ASS} \\ &+ 0.00300 \mathbf{ED} \ast \mathbf{TEMP} + 0.6390 \mathbf{M} \mathbf{ASS} \ast \mathbf{TEMP} \end{aligned}$$

(14)


Table 4.

Standard deviation and mean square error for the case of the helical-tube geometry.


Table 5.

Standard deviation and mean square error for the case of the finned-tube geometry.

$$\begin{aligned} \textbf{AEG} &= -2184 - 5.78 \textbf{ID} + 15.3 \textbf{ED} + 259 \textbf{MASS} + 4.686 \textbf{TEMP} - 0.0198 \textbf{ID}^2 \\ &- 0.0837 \textbf{ED}^2 - 1329 \textbf{MASS}^2 - 0.003373 \textbf{TEMP}^2 + 0.0611 \textbf{ID} \ast \textbf{ED} \\ &+ 0.00095 \textbf{ID} \ast \textbf{TEMP} - 1.37 \textbf{ED} \ast \textbf{MASS} - 0.00121 \textbf{ED} \ast \textbf{TEMP} \\ &+ 0.573 \textbf{MASS} \ast \textbf{TEMP} \end{aligned}$$

Tables 4 and 5 show the parameters that have a significant effect as well as the standard deviation(s) and mean square error (R) for the helical and finned geometries.

From the regression parameters, it is evident that the heat flux on the gas side (QG") and the heat flux on the ethylene glycol side (QEG") for both types of heat exchangers have a high standard deviation. However, the regressed expressions for these parameters seem to adjust very well with the results of CFD simulations as shown in Figures 15–18. The legend RS fitting stands for response surface fitting.

The curves for the gas side and ethylene glycol side for the case of helical heat exchanger show a similar behavior since the distance separating both gas side and

Figure 15. Plot of QG" vs. exchanger configuration for the helical-tube geometry.

Figure 16. Plot of QEG" vs. exchanger configuration for the helical-tube geometry.

Exhaust Gas Heat Recovery for an ORC: A Case Study DOI: http://dx.doi.org/10.5772/intechopen.86075

ethylene glycol side is small. This length corresponds to the inner tube thickness. So, the heat transfer area for the gas side and the ethylene glycol is similar. However, a difference in value exists between QG" and QEG", and that difference can be observed in Figures 15 and 16.

In the case of the finned heat exchanger, the surface area of the gas side differs from that of ethylene glycol side. In this case the QG" and QEG" plots show a different behavior. This is shown in Figures 17 and 18.

In order to generate correlations for local Nusselt numbers exclusively for the bank of heat exchangers simulated, each one of the heat exchangers was divided in sections using the software Fluent [14]. These sections represent dimensionless distance from 0.05 to 0.95. In this way local Nusselt number can be obtained. The helical type of each exchanger was divided in 13 sections, and the finned type of each exchanger was divided in 19 sections. These divisions were done on each of the 41 configurations of the helical heat exchangers and 25 configurations of the finned heat exchangers. Figure 19 shows the section on each heat exchanger.

Figure 17. Plot of QG" vs. exchanger configuration for the finned-tube geometry.

Figure 18. Plot of QEG" vs. exchanger configuration for the finned-tube geometry.

Figure 19. Section for the determination of the local nu number. (a) Helical heat exchanger and (b) finned heat exchanger.

Next, correlations for each exchanger geometry to fit the CFD results are proposed. For the finned-tube heat exchanger, the ethylene glycol side exhibits a laminar flow regime, while the gas side exhibits a turbulent regime. The correlations for the hot side local Nu number at a dimensionless distance of 0.5 for all configurations are presented in Figure 20. Figure 21 presents the local cold side Nusselt number for a dimensionless distance of 0.45 for all configurations. The correlations have the form.

$$\text{For ethylene glycol (EG): } \text{Nu}\_{\text{x}} = mRe^{\imath}Pr^{L} \tag{16}$$

$$\text{For gas (G): } Nu\_x = mRe^n[(A\_f/A\_t)Pr]^L \tag{17}$$

In the case of the helical-tube heat exchanger, a similar correlation for the ethylene glycol was proposed. On the gas side, a factor was proposed by dividing internal diameter of the gas side over the difference of the outside diameter of the annular side minus the internal diameter of the gas side. In the same way, a

parameter (δ) was used. This parameter is the ratio between the helix diameter and the internal diameter of the gas side. The correlations for u number are presented in Figure 22 for the hot side and Figure 23 for the cold side for dimensionless distance of 0.7 and 0.1, respectively. The correlations have the form.

Figure 20. Correlation for gas side (G) in the finned-tube geometry.

Figure 21. Correlation for cold side (EG) in the finned-tube geometry.

Figure 22. Correlation for the hot side (G) of the helical-tube geometry.

Heat and Mass Transfer - Advances in Science and Technology Applications

$$Nu\_{\infty} = mRe^{\imath}Pr^{L} \tag{18}$$

$$Nu\_{\mathfrak{x}} = mRe^{n}[(D\_{i}/(D\_{o} - D\_{i}))(\mathbf{1}/(\delta)]^{L} \tag{19}$$

### 4.3 Neural network

An artificial neural network approach is proposed to fit the response variables of the DOE; these are the inputs given by the DOE (experiments) and CFD simulations. The object is to train the neural network using the input and the corresponding output data derived from the experimental measurements. This process is known as single training cycle or iteration. The cycle is repeated sequentially using a back-propagation algorithm so that training proceeds iteratively until the mean square error between the predicted outputs and corresponding measured values is reduced to an acceptable level. So, the results were introduced in the neural network, and the outputs of the network match very well with some results obtained from CFD. It is observed that the neural network can do a good fitting for the Nu number and heat flux for both types of heat exchangers. Figures 24–27

Figure 23. Correlation for the cold side (EG) of the helical-tube geometry.

Figure 24. Neural network fitting for heat flux on the hot side (gas).

Exhaust Gas Heat Recovery for an ORC: A Case Study DOI: http://dx.doi.org/10.5772/intechopen.86075

Figure 25. Neural network fitting for the Nu number on the hot side of the finned tube exchanger.

Figure 26. Neural network fitting for the heat flux on the hot side (gas).

Figure 27. Neural network fitting for the Nu number on the hot side of the helical exchanger.

show the results of the fitting for heat flux and Nu number. NN fitting stands for neural network fitting.
