# A Machine Learning Approach to Correlation Development Applied to Fin-Tube Bundle Heat Exchangers

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## Abstract

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**Dataset License:**CC-BY-SA

## 1. Introduction

- application of error estimation and adaptive sampling
- direct inclusion of predictive CFD model data in model regression
- extended validity range of geometric parameters towards the weight optimum indicated by earlier work

## 2. Method

#### 2.1. Initial Database

- Data points outside the ranges defined in Table 2 were omitted. The upper limit on ${U}_{{F}_{\mathrm{min}}}$ was relatively restrictive since kinematic viscosity for air is about three times higher at 500 ${}^{\circ}\mathrm{C}$ compared to usual test conditions (∼ 100 ${}^{\circ}\mathrm{C}$). Hence, many experimental data points were excluded, but the resulting Reynolds number range (cf. Table 2). was considered representative of the possible operating conditions of a WHRU
- Geometries with only heat transfer or only pressure drop data were removed. A power law function was fitted to the heat transfer data of each remaining geometry and interpolated to the Reynolds numbers at which the (adiabatic) pressure drop was measured. This is necessary because the chosen model building method requires both outputs to be defined at each data point.
- A tube bank array of 30 ${}^{\circ}$ was considered in this work, as it is the most compact arrangement. Tube banks with array angles in the range 25${}^{\circ}$–35${}^{\circ}$ were corrected using Equation (1), derived from the Escoa correlation [18], to obtain data corresponding to $\beta =30{}^{\circ}$. The maximum applied corrections were 5% for the heat transfer data and 9% for the pressure drop data, respectively. Other tube bank data were discarded.
- The number of streamwise tube rows was not considered as a parameter in this work. Data point duplicates were removed such that only data for the largest number of tube rows were retained.$$\begin{array}{c}\hfill \mathrm{Nu}{\mathrm{Pr}}_{{30}^{\circ}}=\mathrm{Nu}\mathrm{Pr}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{1+{e}^{-1/(2tan\beta )}}{1+{e}^{-1/(2tan30{}^{\circ})}},\\ \hfill {\mathrm{Eu}}_{{30}^{\circ}}=\mathrm{Eu}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{1.1+1.8{e}^{-1/tan\beta}}{1.1+1.8{e}^{-1/tan30{}^{\circ}}}\end{array}$$

#### 2.2. Correlation Development (ALAMO)

#### 2.3. Numerical Model

- Fully periodic computational domain (Figure 2) was discretized primarily with hexahedral cells. A graded boundary layer grid was used in the wall normal direction in the space between the fins (${y}^{+}<1$).
- Density and thermophysical properties were considered constant, properties for air were used for the external fluid and the fin thermal conductivity was set corresponding to carbon steel ($48.5$ $\mathrm{W}$/$\mathrm{m}$/$\mathrm{K}$).
- The steady-state Reynolds Averaged Navier–Stokes (RANS) equations were solved together with the energy equation and the Spalart–Allmaras turbulence model equation [23] using the open source CFD toolbox OpenFOAM v4.1.
- The Spalart–Allmaras turbulence model was selected due to its simplicity, robustness and suitability for simulating boundary layers under adverse pressure gradient conditions. It also yields similar results as other eddy viscosity turbulence models when applied to finned tube bundles [24]. Model constants were kept at their default value, including the turbulent Prandtl number.
- Second order upwind discretization was used for all convective terms.
- The conjugate heat transfer between the fin and the external fluid was modeled explicitly, resolving the temperature field in the fin. The tube wall thermal resistance was neglected—a uniform temperature was applied at the fin root and on the tube surface. The fluid bulk temperature was specified at the leftmost periodic boundary, avoiding source terms in the energy equation.
- Fin efficiency was evaluated by solving the energy equation a second time, subsequent to RANS model convergence, assuming a frozen flow field and a uniform temperature boundary condition on one fin-tube row. The resulting heat flux was used to compute the fin efficiency in the first simulation having finite thermal conductivity in the fin.
- The computed heat flux, bulk temperature and total pressure drop were normalized into Nusselt and Euler numbers according to standard practice (see, e.g., [17]).

#### 2.4. Accuracy Evaluation

#### 2.5. Case Study and Verification of Optimal Point

## 3. Results and Discussion

#### 3.1. Correlation Development

#### 3.2. Case Study

#### 3.3. Sensitivity Analysis

## 4. Conclusions

- The choice of correlation is decisive for the outcome of tube bundle weight optimization, at least for the boiler section considered in the case study. The developed correlations suggest a radically different design compared to the Escoa correlations.
- The trends of the developed correlations generally match well with data from the CFD model. The sensitivity to the design variables close to the optimal point for the case study is, however, exaggerated for some variables.
- The PFR correlation for the Euler number is the most robust reference correlation with regards to the trends in the design variables, indicating that the hydraulic diameter can be an appropriate length scale for pressure drop modeling.
- The Nusselt number is relatively insensitive to all design parameters other than the flow velocity and the tube diameter (i.e., the Reynolds number) around the studied design points.
- In general, the Nusselt number appears more difficult to model accurately, compared to the Euler number. A possible explanation, given the preceding bullet point, is that particular geometries cause complex flow redistribution that only a highly nonlinear model can represent.
- Quantitative accuracy on the case study is good for the developed heat transfer correlation, but disappointing for the pressure drop correlation. The accuracy of the Escoa correlations is also poor at the case study optimum. More data are most likely needed in the range of compact designs with low tube diameter, if further accuracy improvements are to be achieved.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ALAMO | Automated Learning of Algebraic Models for Optimization |

ANN | Artificial Neural Network |

BIC | Bayesian Information Criterion |

CFD | Computational Fluid Dynamics |

RANS | Reynolds Average Navier–Stokes |

RBNN | Radial Basis function Neural Network |

RMSE | Root Mean Square Error |

SVR | Support Vector Regression |

WHRU | Waste Heat Recovery Unit |

## Nomenclature

Roman symbols | |

${A}_{f}$ | fin heat transfer area [${\mathrm{m}}^{2}$] |

${A}_{t}$ | tube heat transfer area [${\mathrm{m}}^{2}$] |

${c}_{f}$ | fin tip-to-tip clearance [$\mathrm{m}$] |

${d}_{o}$ | outer tube diameter [$\mathrm{m}$] |

${h}_{f}$ | total fin height [$\mathrm{m}$] |

${h}_{s}$ | segmented height [$\mathrm{m}$] |

${N}_{r}$ | number of streamwise tube rows [-] |

${N}_{t}$ | number of transverse tube rows [-] |

p | total pressure [$\mathrm{Pa}$] |

${P}_{t}$ | transverse tube pitch [$\mathrm{m}$] |

${P}_{l}$ | longitudinal tube pitch [$\mathrm{m}$] |

${s}_{f}$ | fin pitch [$\mathrm{m}$] |

${\widehat{s}}_{f}$ | fin aperture (=${s}_{f}-{t}_{f}$) [$\mathrm{m}$] |

${t}_{f}$ | fin thickness [$\mathrm{m}$] |

${t}_{w}$ | tube wall thickness [$\mathrm{m}$] |

${u}_{{F}_{min}}$ | mean velocity in minimum free flow area [$\mathrm{m}/\mathrm{s}$] |

${w}_{s}$ | segment width [$\mathrm{m}$] |

Greek symbols | |

${\alpha}_{o}$ | outer heat transfer coefficient [$\mathrm{W}/{\mathrm{m}}^{-2}/\mathrm{K}$] |

$\beta $ | tube bundle layout angle [${}^{\circ}$] |

${\eta}_{f}$ | fin efficiency [-] |

$\lambda $ | thermal conductivity [$\mathrm{W}/\mathrm{m}/\mathrm{K}$] |

$\nu $ | kinematic viscosity [${\mathrm{m}}^{2}/\mathrm{s}$] |

$\tilde{\nu}$ | modified turbulent viscosity [${\mathrm{m}}^{2}/\mathrm{s}$] |

$\rho $ | density [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}$] |

${\sigma}_{y}$ | yield stress [-] |

Dimensionless numbers | |

$\mathrm{Re}={u}_{{F}_{\mathrm{min}}}{d}_{o}/\nu $ | Reynolds number |

$\mathrm{Eu}=\Delta p/\left(\right)open="("\; close=")">{N}_{r}\frac{1}{2}\rho {u}_{{F}_{\mathrm{min}}}^{2}$ | Euler number |

$\mathrm{Nu}={\alpha}_{o}{d}_{o}/\lambda $ | Nusselt number |

$\mathrm{Pr}=\nu \rho {c}_{p}/\lambda $ | Prandtl number |

## Appendix A

## References

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**Figure 1.**Process for correlation development, testing and benchmarking. Green boxes indicate where CFD simulations are employed.

**Figure 3.**Predictive accuracy on regression and validation data (208 data points): correlations developed in this work (Supplementary Materials) and correlations from Holfeld [17], Escoa [18] and PFR [26].

**Figure 4.**Optimized boiler geometry using the Escoa correlations (

**left**) and correlations developed in this work (

**right**), without explicit diameter constraints.

**Figure 5.**Trends in correlations and CFD simulator around the midpoint in the design space (marked by a circle). CFD simulations were independently sampled (not part of the dataset used for model development). Each parameter was varied independently, with remaining parameters held constant.

**Figure 6.**Trends in correlations and CFD simulator around the midpoint in the design space (marked by a circle). CFD simulations were independently sampled (not part of the dataset used for model development). Each parameter was varied independently, with remaining parameters held constant.

**Figure 7.**Trends in correlations around the case study optimum found using correlations developed in this work (marked by a circle). CFD simulations were independently sampled (not part of the dataset used for model development). Each parameter was varied independently, with remaining parameters held constant at the optimum.

**Figure 8.**Trends in correlations around the case study optimum found using correlations developed in this work (marked by a circle). CFD simulations were independently sampled (not part of the dataset used for model development). Each parameter was varied independently, with remaining parameters held constant at the optimum.

**Table 1.**Published work on thermal-hydraulic heat exchanger modeling using machine learning methods.

Data Source | Experimental | Correlation or CFD |
---|---|---|

Fully connected ANN | [5,6,7,8,9,10] | [11] |

SVR, RBNN, Kriging | [9] | [12,13] |

**Table 2.**Considered design space for compact fin-tube bundles. Ancillary variables are adjusted to achieve a reasonable number of segments per fin revolution and a representative fin efficiency. Geometric parameters are shown in Figure 2.

Design Variables | Min | Max |
---|---|---|

${u}_{{F}_{\mathrm{min}}}$ [m/s@500 ${}^{\circ}$C] | 2.53 | 30.4 |

${d}_{o}$ [mm] | 9.65 | 50.8 |

${h}_{f}$ [mm] | 1.4 | 25.4 |

${h}_{s}/{h}_{f}$ [-] | 0.0 | 1.0 |

${\widehat{s}}_{f}$ [mm] | 0.49 | 4.9 |

${c}_{f}$ [mm] | 0.39 | 8.0 |

Ancillary and Derived Variables | ||

Re [-] | 310 | 19,000 |

${t}_{f}$ [mm] | 0.3 | 0.75 |

${w}_{s}$ [mm] | 2.0 | 4.0 |

$\beta {\phantom{\rule{3.33333pt}{0ex}}}^{1}$ [deg] | 30.0 | 30.0 |

Exhaust mass flow [kg/s] | 75.0 |

Exhaust pressure drop [Pa] | 1500 |

Steam pressure [Pa] | 25 × 10${}^{5}$ |

Steam/water temperature [${}^{\circ}$C] | 224 |

Cold end temperature difference [${}^{\circ}$C] | 20 |

Transferred heat [W] | ≥15× 10${}^{6}$ |

Narrow gap flow velocity [m/s] | ≤30 |

Frontal cross-section | square |

**Table 4.**Accuracy on validation dataset, computed from Latin hypercube sample of design space (30 CFD simulations).

Model | Eu | NuPr${}^{-1/3}$ | |||
---|---|---|---|---|---|

${\mathit{R}}^{2}$ | RMSE | ${\mathit{R}}^{2}$ | RMSE | ||

[-] | [%] | [-] | [%] | ||

This work | |||||

database only | 0.75 | 28 | 0.70 | 33 | |

after sampling | 0.94 | 15 | 0.76 | 25 | |

Holfeld [17] | −0.08 | 58 | 0.75 | 35 | |

Escoa [18] | −0.05 | 43 | 0.42 | 34 | |

PFR [26] | 0.79 | 72 | 0.59 | 33 |

**Table 5.**Case study: Geometry optima for correlations from this work and Escoa, without (left column) and with explicit diameter constraint of 25.4 mm (right column). Eu and NuPr${}^{-1/3}$ data are computed at the closest geometry possible to simulate with CFD.

Correlations: | This Work | Escoa | ||
---|---|---|---|---|

Geometry | ||||

Re [-] | 3600 | 9400 | 3600 | 9400 |

${d}_{o}$ [mm] | 9.65 | 25.4 | 9.65 | 25.4 |

${h}_{f}$ [mm] | 10.9 | 12.7 | 6.28 | 15.8 |

${h}_{s}/{h}_{f}$ [mm] | 1.0 | 1.0 | 0.96 | 1.0 |

${t}_{f}$ [mm] | 0.50 | 0.50 | 0.50 | 0.50 |

${\widehat{s}}_{f}$ [mm] | 0.49 | 0.49 | 4.90 | 4.90 |

${w}_{s}$ [mm] | ||||

${c}_{f}$ [mm] | 0.39 | 0.39 | 8.00 | 8.00 |

${N}_{r}$ [-] | 2.2 | 1.2 | 21.8 | 17.1 |

${N}_{t}$ [-] | 140 | 102 | 105 | 52 |

Normalized objective function | 1.00 | 1.49 | 0.87 | 2.33 |

Eu [-] | ||||

This work | 5.64 | 10.8 | ||

Escoa | 0.594 | 0.76 | ||

CFD | 3.81 | 6.28 | 0.52 | 0.75 |

Deviation [%] | +48 | +72 | +13 | −2.2 |

NuPr${}^{-1/3}$ [-] | ||||

This work | 66.6 | 125.0 | ||

Escoa | 56.8 | 96.5 | ||

CFD | 67.7 | 136 | 38.9 | 74.4 |

Deviation [%] | −1.7 | −8.0 | −46 | +30 |

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## Share and Cite

**MDPI and ACS Style**

Lindqvist, K.; Wilson, Z.T.; Næss, E.; Sahinidis, N.V.
A Machine Learning Approach to Correlation Development Applied to Fin-Tube Bundle Heat Exchangers. *Energies* **2018**, *11*, 3450.
https://doi.org/10.3390/en11123450

**AMA Style**

Lindqvist K, Wilson ZT, Næss E, Sahinidis NV.
A Machine Learning Approach to Correlation Development Applied to Fin-Tube Bundle Heat Exchangers. *Energies*. 2018; 11(12):3450.
https://doi.org/10.3390/en11123450

**Chicago/Turabian Style**

Lindqvist, Karl, Zachary T. Wilson, Erling Næss, and Nikolaos V. Sahinidis.
2018. "A Machine Learning Approach to Correlation Development Applied to Fin-Tube Bundle Heat Exchangers" *Energies* 11, no. 12: 3450.
https://doi.org/10.3390/en11123450