# A Differentiable Model for Optimizing Hybridization of Industrial Process Heat Systems with Concentrating Solar Thermal Power

## Abstract

**:**

## 1. Introduction

#### Modeling, Simulation, and Optimization

## 2. Model Formulations

#### 2.1. Concentrating Solar Thermal Model

**Definition**

**1**

_{opt}[30])

**.**The optical efficiency of the PTC ${\eta}_{opt}$ is given by:

#### 2.2. Process System Model

- Heat losses to the environment from piping and heat exchangers is negligible.
- Heat losses to the environment from the TES is negligible.
- The process system requires low to medium temperature IPH (i.e., ≤250 ${}^{\circ}$C).
- Heat is always available at or above the minimum temperature as required by the IPH system.
- The IPH system energy demand is not dependent on the state or design decisions of the solar energy system.

- $({\dot{q}}_{s}^{i}<{\dot{q}}_{p}^{i})\wedge ({q}_{ts}^{i}<h({\dot{q}}_{s}^{i}-{\dot{q}}_{p}^{i}))\iff {\dot{q}}_{a}^{i}<0$The PTC is not providing enough energy to meet the demand of the process (TES moves into the discharging state) and there is not enough energy in the TES to meet the demand of the process over this time step. In this case, conventional energy must be supplied to the process to meet the full demand of the process.
- $({\dot{q}}_{s}^{i}>{\dot{q}}_{p}^{i})\wedge ({\dot{q}}_{p}^{\mathrm{peak}}{x}_{ts}-{q}_{ts}^{i}<h({\dot{q}}_{s}^{i}-{\dot{q}}_{p}^{i}))\iff {\dot{q}}_{a}^{i}>0$The PTC is providing enough energy to meet the demand of the process (TES moves into the charging state) and there is not enough available capacity to store all excess energy in the TES device. In this case, the excess energy must be rejected.
- $\left(\right)open="("\; close=")">({\dot{q}}_{s}^{i}{\dot{q}}_{p}^{i})\wedge ({q}_{ts}^{i}\ge h({\dot{q}}_{s}^{i}-{\dot{q}}_{p}^{i}))\iff {\dot{q}}_{a}^{i}=0$The TES is either in the discharging state with enough stored energy to meet the full demand, or it is in the charging state and has enough available capacity to store the solar energy in excess of the process demand.

**Theorem**

**1.**

**Proof.**

#### Smooth Approximation

**Theorem**

**2.**

**Proof.**

#### 2.3. Economic Model

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 2.4. Optimization

**Theorem**

**4.**

**Proof.**

## 3. Numerical Experiments and Results

#### 3.1. Constant IPH Demand, Commercial Fuel Rate

#### 3.2. Periodic IPH Demand, Commercial Fuel Rate

#### 3.3. Constant IPH Demand, Industrial Fuel Rate

#### 3.4. Periodic IPH Demand, Industrial Fuel Rate

#### 3.5. Computational Efficiency

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

B&B | Branch-and-Bound |

CSP | Concentrating Solar Power |

CST | Concentrating Solar Thermal |

IPH | Industrial Process Heat |

LCS | Lifecycle Savings |

NLP | Nonlinear Program |

NREL | National Renewable Energy Laboratory |

NSRDB | National Solar Radiation Database |

O&M | Operations and Maintenance |

PTC | Parabolic Trough Solar Concentrator |

TES | Thermal Energy Storage |

TMY | Typical Meteorological Year |

## Appendix A. Solar Position Model

**Definition**

**A1**

**.**Due to the eccentricity of the Earth’s orbit and tilt of its axis, the length of a day varies throughout the year. The variation is calculated by the solar time $ST$:

**Definition**

**A2**

**.**The solar declination angle δ is the angular distance of the sun’s rays north (or south) of the equator:

**Definition**

**A3**

**.**The hour angle γ is the angle between the meridian of an observer and the sun. It is negative before local solar noon and positive after local solar noon. At local solar noon, $\gamma ={0}^{\circ}$. The hour angle (in degrees) is defined as

**Definition**

**A4**

**.**The solar zenith angle ϕ (in degrees) is the angle between the sun’s rays and the vertical plane:

**Definition**

**A5**

**.**The solar elevation or altitude angle α (in degrees) is the angle between the sun’s rays and the horizontal plane:

**Definition**

**A6**

**.**The solar azimuth angle ζ is the angle between the sun’s rays and due South for the Northern Hemisphere. If the sun rises and/or sets south of the East–West line (i.e., $cos\gamma >tan\delta /tan\left[{\mathit{Latitude}}^{\circ}\right]$), then we have:

**Definition**

**A7**

**.**The surface tilt angle β of a horizontal surface with a N-S axis orientation and E-W tracking is given by:

**Definition**

**A8**

**.**The surface incidence angle θ (in degrees) is the angle between the sun’s rays and the normal of the horizontal surface. For fixed N-S axis with E-W tracking, the incidence angle can be calculated as:

## Appendix B

${\u03f5}_{1}$, Shadowing Factor | 0.98 |

${\u03f5}_{2}$, Tracking Error | 0.994 |

${\u03f5}_{3}$, Geometry Error | 0.98 |

${\u03f5}_{4}$, Mirror Dirt Factor | $0.88/{r}_{m}\approx 0.941$ |

${\u03f5}_{5}$, Glass Receiver Envelope Dirt Factor | $(1+{\u03f5}_{4})/2\approx 0.971$ |

${\u03f5}_{6}$, Unaccounted Losses | 0.96 |

${r}_{m}$, Mirror Reflectance | 0.935 |

${a}_{m}$, Receiver Tube Absorptance | 0.94 |

$\tau $, Glass Receiver Envelope Transmittance | 0.963 |

${C}_{PTC,0}$, specific installed cost of PTC ($/m${}^{1.84}$) | 425.0 |

${C}_{TES,0}$, specific installed cost of TES ($/kWh${}_{\mathrm{th}}^{0.91}$) | 45.14 |

${C}_{th,0}$, commercial initial specific fuel cost ($/kWh${}_{\mathrm{th}}$) | 7.232/293.1≈ 0.02467 |

industrial initial specific fuel cost ($/kWh${}_{\mathrm{th}}$) | 3.420/293.1≈ 0.01167 |

r, discount rate (-) | 10% |

${r}_{c}$, capital financing APR (-) | 6.5% |

${r}_{th}$, fuel cost annual inflation rate (-) | 1% |

${t}_{debt}$, loan term (years) | 10 |

${t}_{life}$, lifetime of project (years) | 30 |

${\mathbf{x}}^{L}=({x}_{ts}^{L},{x}_{a}^{L})$, variable lower bound (h, m${}^{2}$) | $(0.001,0.01)$ |

${\mathbf{x}}^{U}=({x}_{ts}^{U},{x}_{a}^{U})$, variable upper bound (commercial fuel case) (h, m${}^{2}$) | $(16.0,6\times {10}^{4})$ |

variable upper bound (industrial fuel case) (h, m${}^{2}$) | $(14.0,5\times {10}^{4})$ |

$\xi $, minimum solar fraction (-) | 0 |

**Table A3.**The annual average DNI (kWh/m${}^{2}$/day) for each location is resolved by month. The data are calculated from the hourly TMY values obtained from [41].

Month | Weston | Aurora | Firebaugh |
---|---|---|---|

Janary | 3.73 | 4.68 | 3.21 |

Febrary | 4.41 | 5.25 | 4.24 |

March | 4.61 | 6.36 | 6.07 |

April | 5.31 | 6.42 | 7.03 |

May | 5.33 | 6.72 | 8.68 |

June | 5.32 | 7.98 | 10.16 |

July | 5.60 | 7.14 | 9.54 |

August | 5.37 | 6.82 | 9.30 |

Sepember | 5.24 | 7.21 | 7.70 |

Octorber | 4.13 | 6.33 | 6.54 |

November | 3.72 | 5.44 | 4.50 |

December | 3.39 | 4.08 | 3.36 |

Ann. Avg. | 4.68 | 6.21 | 6.71 |

## Appendix C. Spatial Branch-and-Bound (B&B) Global Optimization

**Definition**

**A9**

**.**Consider an interval ${X}^{i}\subset X$. The upper-bounding problem is given by:

**Definition**

**A10**

**.**Consider an interval ${X}^{i}\subset X$. The lower-bounding problem is given by:

**Figure A1.**(

**Left**) The decision space X (outer box) is shown overlaying the feasible set $\mathcal{F}$ and is partitioned into subintervals using a bisection strategy. Note that ${X}^{7}\cap \mathcal{F}=\varnothing $ and therefore the corresponding upper- and lower-bounding problems are infeasible. (

**Right**) An illustration of the B&B tree corresponding to the decision space partitioning.

Algorithm 1: Branch & Bound Maximization |

## References

- US Energy Information Administration. Annual Energy Outlook 2018 with Projections to 2050; Technical Report for US Department of Energy; US Energy Information Administration: Washington, DC, USA, 2018.
- US Department of Energy. Available online: https://www.energy.gov/eere/amo/dynamic-manufacturing-energy-sankey-tool-2010-units-trillion-btu (accessed on 15 May 2018).
- Kurup, P.; Turchi, C. Initial Investigation into the Potential of CSP Industrial Process Heat for the Southwest United States; Technical Report of NREL; National Renewable Energy Laboratory: Golden, Colorado, 2015.
- Farjana, S.H.; Huda, N.; Mahmud, M.P.; Saidur, R. Solar process heat in industrial systems–A global review. Renew. Sustain. Energy Rev.
**2018**, 82, 2270–2286. [Google Scholar] [CrossRef] - Kalogirou, S. The potential of solar industrial process heat applications. Appl. Energy
**2003**, 76, 337–361. [Google Scholar] [CrossRef] - Lauterbach, C.; Schmitt, B.; Jordan, U.; Vajen, K. The potential of solar heat for industrial processes in Germany. Renew. Sustain. Energy Rev.
**2012**, 16, 5121–5130. [Google Scholar] [CrossRef] - Mekhilef, S.; Saidur, R.; Safari, A. A review on solar energy use in industries. Renew. Sustain. Energy Rev.
**2011**, 15, 1777–1790. [Google Scholar] [CrossRef] - Martinez, V.; Pujol, R.; Moia, A. Assessment of Medium Temperature Collectors for Process Heat. Energy Proc.
**2012**, 30, 745–754. [Google Scholar] [CrossRef] - Sharma, A.K.; Sharma, C.; Mullick, S.C.; Kandpal, T.C. Solar industrial process heating: A review. Renew. Sustain. Energy Rev.
**2017**, 78, 124–137. [Google Scholar] [CrossRef] - Panwar, N.L.; Kaushik, S.C.; Kothari, S. Role of renewable energy sources in environmental protection: A review. Renew. Sustain. Energy Rev.
**2011**, 15, 1513–1524. [Google Scholar] [CrossRef] - Ramos, C.; Ramirez, R.; Beltran, J. Potential Assessment in Mexico for Solar Process Heat Applications in Food and Textile Industries. Energy Proc.
**2014**, 49, 1879–1884. [Google Scholar] [CrossRef] - Murray, J. Aluminum Production Using High-Temperature Solar Process Heat. Sol. Energy
**1999**, 66, 133–142. [Google Scholar] [CrossRef] - Hockaday, S.; Dinter, F.; Harms, T. Opportunities for concentrated solar thermal heat in the minerals processing industry. In Proceedings of the 4th Southern African Solar Energy Conference (SASEC) 2016, Stellenbosch, South Africa, 31 October–2 November 2016. [Google Scholar]
- Mekhilef, S.; Faramarzi, S.Z.; Saidur, R.; Salam, Z. The application of solar technologies for sustainable development of agricultural sector. Renew. Sustain. Energy Rev.
**2013**, 18, 583–594. [Google Scholar] [CrossRef] - Hussain, M.I.; Lee, G.H. Utilization of Solar Energy in Agricultural Machinery Engineering: A Review. J. Biosyst. Eng.
**2015**, 40, 186–192. [Google Scholar] [CrossRef] [Green Version] - Beesing, M.E. Textile Drying Using Solar Process Steam. In Proceedings of the Energy Conference ASME 1979 International Gas Turbine Conference and Exhibit and Solar, San Diego, CA, USA, 12–15 March 1979. [Google Scholar] [CrossRef]
- Sharma, A.K.; Sharma, C.; Mullick, S.C.; Kandpal, T.C. GHG mitigation potential of solar industrial process heating in producing cotton based textiles in India. J. Clean. Prod.
**2017**, 145, 74–84. [Google Scholar] [CrossRef] - Sharma, A.K.; Sharma, C.; Mullick, S.C.; Kandpal, T.C. Incentives for promotion of solar industrial process heating in India: a case of cotton-based textile industry. Clean Technol. Environ. Policy
**2018**, 20, 813–823. [Google Scholar] [CrossRef] - Frey, P.; Fischer, S.; Drück, H.; Jakob, K. Monitoring Results of a Solar Process Heat System Installed at a Textile Company in Southern Germany. Energy Proc
**2015**, 70, 615–620. [Google Scholar] [CrossRef] - Sharma, A.K.; Sharma, C.; Mullick, S.C.; Kandpal, T.C. Potential of Solar Energy Utilization for Process Heating in Paper Industry in India: A Preliminary Assessment. Energy Proc.
**2015**, 79, 284–289. [Google Scholar] [CrossRef] - Sharma, A.K.; Sharma, C.; Mullick, S.C.; Kandpal, T.C. Carbon mitigation potential of solar industrial process heating: paper industry in India. J. Clean. Prod.
**2016**, 112, 1683–1691. [Google Scholar] [CrossRef] - Meier, A.; Bonaldi, E.; Cella, G.M.; Lipinski, W.; Wuillemin, D. Solar chemical reactor technology for industrial production of lime. Sol. Energy
**2006**, 80, 1355–1362. [Google Scholar] [CrossRef] - Romero, M.; Steinfeld, A. Concentrating solar thermal power and thermochemical fuels. Energy Environ. Sci.
**2012**, 5, 9234–9245. [Google Scholar] [CrossRef] - Kalogirou, S.; Lloyd, S.; Ward, J. Modelling, optimisation and performance evaluation of a parabolic trough solar collector steam generation system. Sol. Energy
**1997**, 60, 49–59. [Google Scholar] [CrossRef] - Duffie, J.A.; Beckman, W.A. Solar Engineering of Thermal Processes; John Wiley & Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Kalogirou, S.A. A detailed thermal model of a parabolic trough collector receiver. Energy
**2012**, 48, 298–306. [Google Scholar] [CrossRef] - Kalogirou, S.A. Solar Energy Engineering Processes and Systems, 2nd ed.; Academic Press: Oxford, UK, 2014. [Google Scholar]
- Ghobeity, A.; Mitsos, A. Optimal design and operation of a solar energy receiver and storage. J. Sol. Energy Eng.
**2012**, 134. [Google Scholar] [CrossRef] - Ghasemi, H.; Sheu, E.; Tizzanini, A.; Paci, M.; Mitsos, A. Hybrid solar-geothermal power generation: Optimal retrofitting. Appl. Energy
**2014**, 131, 158–170. [Google Scholar] [CrossRef] - Gunasekaran, S.; Mancini, N.; El-Khaja, R.; Sheu, E.; Mitsos, A. Solar-geothermal hybridization of advanced zero emissions power cycle. Energy
**2014**, 65, 152–165. [Google Scholar] [CrossRef] - Silva, R.; Berenguel, M.; Perez, M.; Fernandez-Garcia, A. Thermo-Economic Design Optimization of Parabolic Trough Solar Plants for Industrial Process Heat Applications with Memetic Algorithms. Appl. Energy
**2014**, 113, 603–614. [Google Scholar] [CrossRef] - Stuber, M.D. Optimal Design of Fossil-Solar Hybrid Thermal Desalination for Saline Agricultural Drainage Water Reuse. Renew. Energy
**2016**, 89, 552–563. [Google Scholar] [CrossRef] - Yan, C.; Wang, S.; Ma, Z.; Shi, W. A simplified method for optimal design of solar water heating systems based on life-cycle energy analysis. Renew. Energy
**2015**, 74, 271–278. [Google Scholar] [CrossRef] - Baniassadi, A.; Momen, M.; Amidpour, M. A new method for optimization of Solar Heat Integration and solar fraction targeting in low temperature process industries. Energy
**2015**, 90, 1674–1681. [Google Scholar] [CrossRef] - Frasquet, M. SHIPcal: Solar heat for industrial processes online calculator. Energy Proc.
**2016**, 91, 611–619. [Google Scholar] [CrossRef] - Ayub, M.; Mitsos, A.; Ghasemi, H. Thermo-economic analysis of a hybrid solar-binary geothermal power plant. Energy
**2015**, 87, 326–335. [Google Scholar] [CrossRef] - Al-Aboosi, F.Y.; El-Halwagi, M.M. An Integrated Approach to Water-Energy Nexus in Shale-Gas Production. Processes
**2018**, 6, 52. [Google Scholar] [CrossRef] - Stuber, M.D.; Sullivan, C.; Kirk, S.A.; Farrand, J.A.; Schillaci, P.V.; Fojtasek, B.D.; Mandell, A.H. Pilot demonstration of concentrated solar-powered desalination of subsurface agricultural drainage water and other brackish groundwater sources. Desalination
**2015**, 355, 186–196. [Google Scholar] [CrossRef] - Dudley, V.E.; Kolb, G.J.; Mahoney, A.R.; Mancini, T.R.; Matthews, C.W.; Sloan, M.; Kearney, D. Test Results: SEGS LS-2 Solar Collector; Technical Report of Sandia National Labs.; National Technical Information Service: Albuquerque, NM, USA, 1994.
- Forristall, R. Heat Transfer Analysis and Modeling of a Parabolic Trough Solar Receiver Implemented in Engineering Equation Solver; Technical Report of NREL; National Renewable Energy Laboratory: Golden, CO, USA, 2003.
- National Renewable Energy Laboratory. NSRDB: National Solar Radiation Database. Available online: https://nsrdb.nrel.gov/nsrdb-viewer (accessed on 21 May 2018).
- International Renewable Energy Agency. Renewable Energy Technologies: Cost Analysis Seris—Concentrating Solar Power; Technical Report of IRENA; International Renewable Energy Agency: Abu Dhabi, UAE, 2012. [Google Scholar]
- Falk, J.E.; Soland, R.M. An Algorithm for Separable Nonconvex Programming Problems. Manag. Sci.
**1969**, 15, 550–569. [Google Scholar] [CrossRef] - Bezanson, J.; Edelman, A.; Karpinski, S.; Shah, V.B. Julia: A Fresh Approach to Numerical Computing. SIAM Rev.
**2017**, 59, 65–98. [Google Scholar] [CrossRef] [Green Version] - US Energy Information Administration. Available online: https://www.eia.gov/dnav/ng/NG_PRI_SUM_DCU_NUS_M.htm (accessed on 16 April 2018).
- Wächter, A.; Biegler, L.T. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear Programming. Math. Program.
**2006**, 106, 25–57. [Google Scholar] [CrossRef] - Dunning, I.; Huchette, J.; Lubin, M. JuMP: A modeling language for mathematical optimization. SIAM Rev.
**2017**, 59, 295–320. [Google Scholar] [CrossRef] - Wilhelm, M.; Stuber, M.D. Easy Advanced Global Optimization (EAGO): An Open-Source Platform for Robust and Global Optimization in Julia. In Proceedings of the AIChE Annual Meeting 2017 Minneapolis, Minneapolis, MN, USA, 31 October 2017. [Google Scholar]
- Wilhelm, M.; Stuber, M.D. EAGO: Easy Advanced Global Optimization Julia Package. Available online: https://github.com/PSORLab/EAGO.jl (accessed on 1 May 2018).
- Kreith, F.; Kreider, J.F. Principles of Solar Engineering; Hemisphere Publishing Corporation: Washington, DC, USA, 1978. [Google Scholar]

**Figure 1.**(

**left**) An illustration of a parabolic trough solar concentrator (PTC) with the defined array length ${L}_{m}$, aperture length ${L}_{a}$ and the aperture plane with incident solar radiation. (

**right**) The large-aperture PTC (Skyfuel, Lakewood, CO, USA) powering a solar thermal desalination pilot by Stuber et al. [38].

**Figure 2.**The block-flow diagram representation of the hybrid solar industrial process heat system being modeled.

**Figure 3.**The thermal performance of the global optimal hybridization strategies for constant IPH demand and the commercial initial specific fuel cost.

**Figure 4.**The objective function surfaces ${f}_{LCS}^{fix}$ (

**left column**) and the solar fraction surfaces $S{F}_{s}$ (

**right column**) for constant IPH demand for commercial fuel rate show similar qualitative trends between each region (rows).

**Figure 5.**The objective function surfaces ${f}_{LCS}^{fix}$ (

**left column**) and ${f}_{LCS}^{disc}$ (

**right column**) for constant IPH demand for industrial fuel rates are plotted for each geographic location. The discount-pricing model exhibits nonconvexity in all cases and suboptimal local minima for Aurora and Firebaugh.

**Table 1.**The optimal solar-IPH hybridization strategies for the three geographic locations for constant IPH demand with ${\overline{q}}_{p}={10}^{5}$ kW${}_{\mathrm{th}}$ for the commercial fuel rate. The global optimal results for the concave capital cost model in Equation (14) are included for comparison against the linear capital cost model in Equation (15).

City, State | Lat., Long. | Time Zone | Capital Model | ${\mathit{x}}_{\mathit{ts}}^{*}$ (h) | ${\mathit{x}}_{\mathit{a}}^{*}$(m${}^{2}$) | ${\mathit{SF}}_{\mathit{s}}^{*}$ | ${\mathit{f}}_{\mathit{k}}^{*}$ |
---|---|---|---|---|---|---|---|

Weston, MA | 42.37,−71.30 | −5 | $k=fix$ (Linear), Equation (15) | 11.24 | 46,436.4 | 0.516 | $2.976 M |

$k=disc$ (Concave), Equation (14) | 11.72 | 48,229.1 | 0.523 | $2.795 M | |||

Aurora, CO | 39.73,−104.66 | −7 | $k=fix$ (Linear), Equation (15) | 13.85 | 50,618.1 | 0.721 | $6.487 M |

$k=disc$ (Concave), Equation (14) | 14.58 | 53,238.5 | 0.731 | $6.377 M | |||

Firebaugh, CA | 36.85,−120.46 | −8 | $k=fix$ (Linear), Equation (15) | 11.42 | 42,601.7 | 0.695 | $7.533 M |

$k=disc$ (Concave), Equation (14) | 11.72 | 43,615.2 | 0.698 | $7.320 M |

**Table 2.**The optimal solar-IPH hybridization strategies for the three geographic locations for periodic IPH demand with ${\overline{q}}_{p}={10}^{5}$ kW and ${\sigma}_{p}=0.1$ for the commercial fuel rate. The global optimal results for the concave capital cost model in Equation (14) are included for comparison against the linear capital cost model in Equation (15).

City, State | Lat., Long. | Time Zone | Capital Model | ${\mathit{x}}_{\mathit{ts}}^{*}$ (h) | ${\mathit{x}}_{\mathit{a}}^{*}$ (m${}^{2}$) | ${\mathit{SF}}_{\mathit{s}}^{*}$ | ${\mathit{f}}_{\mathit{k}}^{*}$ |
---|---|---|---|---|---|---|---|

Weston, MA | 42.37,−71.30 | −5 | $k=fix$ (Linear), Equation (15) | 9.537 | 46,248.8 | 0.507 | $3.079 M |

$k=disc$ (Concave), Equation (14) | 9.966 | 47,945.8 | 0.521 | $2.881M | |||

Aurora, CO | 39.73,−104.66 | −7 | $k=fix$ (Linear), Equation (15) | 11.92 | 50,006.8 | 0.706 | $6.593 M |

$k=disc$ (Concave), Equation (14) | 12.59 | 53,238.7 | 0.731 | $6.460 M | |||

Firebaugh, CA | 36.85,−120.46 | −8 | $k=fix$ (Linear), Equation (15) | 9.732 | 42,688.4 | 0.691 | $7.641 M |

$k=disc$ (Concave), Equation (14) | 9.966 | 43,615.2 | 0.698 | $7.411 M |

**Table 3.**The optimal solar-IPH hybridization strategies for the three geographic locations for constant IPH demand with ${\overline{q}}_{p}={10}^{5}$ kW for the industrial fuel rate. The global optimal results for the concave capital cost model in Equation (14) are included for comparison against the linear capital cost model in Equation (15).

City, State | Lat., Long. | Time Zone | Capital Model | ${\mathit{x}}_{\mathit{ts}}^{*}$ (h) | ${\mathit{x}}_{\mathit{a}}^{*}$(m${}^{2}$) | ${\mathit{SF}}_{\mathit{s}}^{*}$ | ${\mathit{f}}_{\mathit{k}}^{*}$ |
---|---|---|---|---|---|---|---|

Weston, MA | 42.37,−71.30 | −5 | $k=fix$ (Linear), Equation (15) | 0.001 | 0.01 | 0.000 | $920.2 |

$k=disc$ (Concave), Equation (14) | 0.001 | 0.01 | 0.000 | $920.2 | |||

Aurora, CO | 39.73,−104.66 | −7 | $k=fix$ (Linear), Equation (15) | 0.001 | 15,720.7 | 0.250 | $323.8 k |

$k=disc$ (Concave), Equation (14) | 0.001 | 15,648.7 | 0.248 | $95.59 k | |||

Firebaugh, CA | 36.85,−120.46 | −8 | $k=fix$ (Linear), Equation (15) | 0.046 | 16,581.5 | 0.298 | $698.1 k |

$k=disc$ (Concave), Equation (14) | 0.001 | 16,424.3 | 0.295 | $468.3 k |

**Table 4.**The optimal solar-IPH hybridization strategies for the three geographic locations for periodic IPH demand with ${\overline{q}}_{p}={10}^{5}$ kW for the industrial fuel rate. The global optimal results for the concave capital cost model in Equation (14) are included for comparison against the linear capital cost model in Equation (15).

City, State | Lat., Long. | Time Zone | Capital Model | ${\mathit{x}}_{\mathit{ts}}^{*}$ (h) | ${\mathit{x}}_{\mathit{a}}^{*}$(m${}^{2}$) | ${\mathit{SF}}_{\mathit{s}}^{*}$ | ${\mathit{f}}_{\mathit{k}}^{*}$ |
---|---|---|---|---|---|---|---|

Weston, MA | 42.37,−71.30 | −5 | $k=fix$ (Linear), Equation (15) | 0.001 | 0.01 | 0.000 | $773.4 |

$k=disc$ (Concave), Equation (14) | 0.001 | 0.01 | 0.000 | $773.4 | |||

Aurora, CO | 39.73,−104.66 | −7 | $k=fix$ (Linear), Equation (15) | 0.001 | 16,894.0 | 0.267 | $350.2 k |

$k=disc$ (Concave), Equation (14) | 0.001 | 16,879.1 | 0.267 | $120.8 k | |||

Firebaugh, CA | 36.85,−120.46 | −8 | $k=fix$ (Linear), Equation (15) | 0.033 | 17,972.1 | 0.320 | $759.4 k |

$k=disc$ (Concave), Equation (14) | 0.001 | 17,839.1 | 0.318 | $529.4 k |

City, State | Section 3.1 | Section 3.2 | Section 3.3 | Section 3.4 | ||||
---|---|---|---|---|---|---|---|---|

$\mathit{k}=\mathit{fix}$ | $\mathit{k}=\mathit{disc}$ | $\mathit{k}=\mathit{fix}$ | $\mathit{k}=\mathit{disc}$ | $\mathit{k}=\mathit{fix}$ | $\mathit{k}=\mathit{disc}$ | $\mathit{k}=\mathit{fix}$ | $\mathit{k}=\mathit{disc}$ | |

Weston, MA | 1.09 | 10.4 | 0.632 | 13.3 | 0.103 | 0.42 | 0.103 | 0.39 |

Aurora, CO | 0.70 | 6.69 | 0.495 | 6.40 | 0.187 | 15.5 | 0.208 | 19.4 |

Firebaugh, CA | 0.70 | 10.1 | 1.06 | 9.92 | 0.604 | 12.77 | 0.517 | 12.67 |

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**MDPI and ACS Style**

Stuber, M.D.
A Differentiable Model for Optimizing Hybridization of Industrial Process Heat Systems with Concentrating Solar Thermal Power. *Processes* **2018**, *6*, 76.
https://doi.org/10.3390/pr6070076

**AMA Style**

Stuber MD.
A Differentiable Model for Optimizing Hybridization of Industrial Process Heat Systems with Concentrating Solar Thermal Power. *Processes*. 2018; 6(7):76.
https://doi.org/10.3390/pr6070076

**Chicago/Turabian Style**

Stuber, Matthew D.
2018. "A Differentiable Model for Optimizing Hybridization of Industrial Process Heat Systems with Concentrating Solar Thermal Power" *Processes* 6, no. 7: 76.
https://doi.org/10.3390/pr6070076