# A Thermal-Hydraulic Model for the Stagnation of Solar Thermal Systems with Flat-Plate Collector Arrays

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}flat-plate collectors. Stagnation was triggered by manually switching off the pump. From the results, they deduced that stagnation is manageable if the piping is routed in such a way that the steam forms a continuous volume and propagates monotonously downward. They provided rules for pipe routing resulting in the best possible emptying of the collector field. As another rule, they define the location of the check valve, which must be upstream from the connection point of the pressure maintenance. This allows displacement of liquid from the collector field in both the supply and return lines. Based on the measurement data from experiments on a real system with two 22 m

^{2}flat-plate collectors, Hausner and Fink [4] described stagnation as a sequence of five successive phases as (1) liquid expansion, (2) displacement of liquid by steam (3) emptying by boiling (phase with saturated steam), (4) emptying by boiling (phase with saturated and superheated steam) and (5) refilling of the collectors. From experimental data and qualitative considerations, they classified the hydraulic designs of collectors and their connections by external pipes according to their emptying behavior. This and the practical rules derived from it were published also by Hausner and Fink [5]. Streicher [6] addressed the danger of water hammer induced by rapid condensation of steam during stagnation. He explained, with examples, that those hydraulic designs in which the steam forms a continuous volume are to be preferred. Lustig [7] conducted experiments on systems with flat-plate collectors and vacuum-tube collectors focusing on the processes in the absorber. He recognized that the displacement of the liquid and the spreading of the vapor takes the form of a two-phase flow. For a collector with a meander type absorber tube, he identified the various flow patterns in all the tube sections. He used a plug-flow model to simulate the propagation of steam into a single pipe. The residual liquid was predefined as a fraction of the absorber volume. By adjusting the residual amount and the absorber volume, a fairly good agreement with the experiments could be achieved.

^{2}aperture area. Their experiments showed the strong influence of the emptying behavior on the steam range. Based on these experiments Scheuren [12] derived a more general empirical correlation for the steam range. He distinguished three classes of collector fields characterized by good, average and poor emptying behavior and provided a corresponding set of two parameters for the correlation. However, the class needs to be determined initially, based on qualitative considerations and practical experience.

## 2. Basic Models and Methods

#### 2.1. Experimental Facility

_{C}= 2.39 m. The location of the vapor front is detected by measuring the temperature of the pipe wall and the pressure at the collector outlet and at the connection of the membrane expansion vessel. Temperature sensors are attached to the outside of the pipe walls in regular intervals of 1.5 m. The temperature measured is compared to saturation temperature, which indicates the arrival of the vapor front.

#### 2.2. Collector Design

#### 2.3. Collector Models

#### 2.4. Heat Capacity of Different Absorber Regions

#### 2.5. Heat Loss Distribution

#### 2.6. Supply and Return Lines

^{®}HT tubes, whose properties are described in the datasheet [21]. Equivalent values are listed in Table A2. The inner diameter equals the outer diameter of the pipe, ${d}_{t,o}$. The outer diameter is denoted by, ${d}_{i}$. The heat capacity of the insulation is ignored.

_{t}, based on the Nusselt number for free convection from a vertical wall of the same height,

_{i}, of the insulating layer is calculated in a simplified way. The net heat flux is the sum of absorbed solar irradiation, the heat losses through the insulation layer and the heat flux from the surface to the environment by convection and radiation. In a stationary state, the net heat flux at the boundaries to the environment is equal to zero, as expressed by Equation (17).

_{k}, for the pipe section, k, related to the temperature difference between the pipe wall and the environment is calculated.

#### 2.7. Pressure Maintenance

^{®}used in the experiments. For this purpose, numerical routines for the vapor pressure of water, ${p}_{v}\left(T\right)$, as a function of temperature and the saturation temperature of TyfocorLS

^{®}, ${T}_{s,WG}\left(p\right)$, as a function of saturation pressure were implemented. The corresponding saturation pressure of steam is,

_{0}, at the reference temperature, T

_{0}, is calculated to be,

#### 2.8. Properties of the Liquid and Gaseous Phase within the Circuit

^{®}used in the experiments is a mixture of water, propylene glycol and anti-corrosion additives with a mass fraction of water, x

_{m}= 0.58. Figure 6 shows the phase diagram of a binary mixture of water and propylene glycol as a function of the molar fraction, x, calculated for a total pressure of three bar. Apparently, the vapor phase of the original mixture with a mass fraction of water, x

_{m}= 0.58 consists practically only of the vapor of water. Even at the dry stagnation temperature of the thermochromic collector, 167 °C, and the selective collector, 192 °C, the molar fraction of propylene glycol is quite small. It can be concluded from the diagram that the collector will never dry out entirely because the saturation temperature of pure propylene glycol will never be reached.

- The efficiency of the absorber region is defined by the saturation temperature of the residual liquid, T
_{s,r}, which itself is a function of the water content. - The glycol content of the vapor is neglected. Henceforth, the gas phase is referred to as steam.
- Contribution of the overheated regions of the absorber to the energy balance is neglected.
- The pressure and the fluid properties of the original mixture define the saturation temperature of steam within the pipes.

_{m}

_{0},

_{s,r}, is iteratively calculated by applying Raoult’s law,

#### 2.9. Time Evolution of Circuit Temperatures

#### 2.10. Two-Phase Mixture Model

#### 2.11. Drift-Flux Correlation

_{gj}, is defined as the difference between the local velocity of the gas phase, w

_{g}, and the sum of the superficial velocities of the gas and liquid phases, j

_{g}and j

_{l}.

_{0}, and a corrected average drift velocity, U

_{gj}.

## 3. Derivation of the Model

#### 3.1. Residual Liquid

- The meander and header tubes are completely wetted. All parts of the absorber contribute to vaporization according to their efficiency.
- Only steam leaves the wetted and steam-filled parts of the absorber. The residual liquid is considered stationary, which is expressed by a liquid superficial velocity of ${j}_{l}=0$ everywhere in the absorber.
- The influence of two-phase pressure losses on the saturation temperature is neglected. Consequently, each absorber within the collector array undergoes the same process and the saturation temperature is the same everywhere.
- The steam flow distributions in the upper and lower parts of the absorber are symmetrical.

_{m}

_{0}, and summing up over all the meander tube sections,

^{th}upper header tube results from the sum of steam flows in the upper half of one to k meanders and the steam flows in one to k header tubes,

^{th}bottom header tube is calculated analogously.

#### 3.1.1. Corrections for the Residual Mass of Water

#### 3.2. Temperature Rise

_{s}. The temperature rise during the heating-up period is described by Equation (52), where T is the temperature of the absorber at an arbitrary location along the flow path.

_{s}, within a finite time interval.

#### 3.3. Liquid Displacement

#### 3.4. Steam Generation

#### 3.5. Steam Propagation into the Circuit

_{k}, depends on the specific heat capacities of the pipe and the steam.

_{v}, results in a differential equation for the location, x, of the steam front within the control volume defined by the pipe section, k.

_{C}collectors and regions $X\in \left\{Ht,M,Hb\right\}$ as follows:

#### 3.6. Numerical Procedure

_{S}, for the effective stagnation temperature in Equation (2), was chosen. The distribution parameter, δ, was varied until the measured and simulated steam ranges matched. The corresponding efficiency of the meander region, η

_{M}, was stored for curve fitting. The time evolutions of the measured and simulated steam ranges were compared. If the gradients of the simulated steam range, which depend on the efficiency, were too small, the value of the parameter, K

_{S}, was increased and vice versa. This procedure was repeated until the steam ranges and the time evolutions matched. Finally, the parameters of the distribution function, Equation (49), were determined, based on the final values δ(η

_{M}).

## 4. Results and Discussion

#### 4.1. Comparison of Experimental Data and Simulations

_{SC}= 300 s, and τ

_{TC}= 600 s.

- For the system with standard selective absorbers, the simulated values for the total steam ranges lie well within the confidence limits. No significant difference between active and open check valves can be determined. It follows that the effect of the check valves on displacement and evaporation processes within the collector array is negligible.
- For the system with thermochromic absorbers, the steam ranges with open check valves are within the confidence range, whereas the model tends to overestimate steam ranges in the cases with active check valves.

#### 4.1.1. Conventional Selective Absorbers, Deactivated Check Valves

#### 4.1.2. Conventional Selective Absorbers, Check Valves Active

#### 4.1.3. Thermochromic Absorbers—Deactivated Check Valves

#### 4.1.4. Thermochromic Absorbers—Active Check Valves

#### 4.2. Uncertainties

_{φ}= 0.25 m to the uncertainty in the steam range.

## 5. Conclusions

^{2}collector area [17,36]. The actual size is limited mainly by the validity of the assumption that pressure losses during stagnation are negligible. Tools of this kind will become increasingly important as the key to designing both cost effective and operationally safe solar systems.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Quantity | Unit | Standard | Thermochromic | |
---|---|---|---|---|

Critical temperature Tc | °C | - | 68 | |

Temperature range | T ≤ T_{c} | T > T_{c} | ||

zero-loss efficiency factor | - | η_{0} = 0.785 | η_{0} = 0.757 | η_{0,TC} = 0.83 |

Heat loss coefficient | W/Km^{2} | ${a}_{1}$= 4.19 | ${a}_{1}$ = 4.27 | ${a}_{1,TC}$ = 6.17 |

Heat loss coefficient | W/K^{2}m^{2} | ${a}_{2}$ = 0.0135 | ${a}_{2}$ = 0.0065 | ${a}_{2,TC}$ = 0.0103 |

Stagnation temperature | °C | 192.1 | 167 | |

Dry meander region heat capacity | J/K | 2793 | ||

Meander volume | l | 0.924 | ||

Dry header region heat capacity | J/K | 566 | ||

Header volume | l | 0.660 | ||

Aperture area | m^{2} | 2.33 | ||

Hydraulic elements | Length m | d_{i} m | Number of nodes - | |

Absorber tube | 21.88 | 0.0082 | 88 | |

Header tube | 1.1 | 0.02 | 1 |

Heat Conductivity | W/Km | W/°C^{2}m | W/°C^{3}m |
---|---|---|---|

0.03764 | 7.7 × 10^{−05} | 8 × 10^{−07} | |

Absorption coefficient - | Emissivity - | Inner diameter mm | Outer diameter mm |

0.9 | 0.95 | 22 | 60 |

Inner Diameter | Outer Diameter | Density | Specific Heat | Conductivity |
---|---|---|---|---|

mm | mm | kg/m^{3} | J/kgK | W/Km |

20 | 22 | 8600 | 382 | 300 |

#### Properties of the Heat Carrier

^{®}yielded a mass fraction, ${x}_{m0}=0.579$, for the system with conventional selective absorbers and, x

_{m}0 = 0.557, for the system with thermochromic absorbers. The saturation temperature of the mixture is calculated based on the simplifying assumption of an ideal two-component mixture of water and propylene glycol using properties listed in Table A4. Within the temperature range considered here, the vapor pressure is sufficiently well described by a power function,

^{®}[39].

Molar Mass | Coefficients for Vapor Pressure in Pa | ||
---|---|---|---|

Component | kg/kmol | a | b |

Water | 18.02 | 0.7517·10^{−3} | 4.047 |

Propylene-glycol | 76.1 | 1.8745·10^{−8} | 5.601 |

TyfocorLS | - | 0.7517·10^{−3} | 4.225 |

^{®}[39]. The surface tension of TyfocorLS

^{®}is modelled based on experimental data presented by Chang, et al. [40] for a binary mixture of propylene glycol with a molar fraction x = 0.85 of water.

## Appendix B

Symbol | Unit | Symbol | Unit | ||
---|---|---|---|---|---|

A | m^{2} | Area | V | m^{3} | Volume |

${a}_{1}$ | W/Km^{2} | Heat loss coefficient | $\dot{W}$ | W | Work rate |

${a}_{2}$ | W/K^{2}m^{2} | Heat loss coefficient | w | m/s | Velocity |

c | J/kgK | Specific heat capacity | w_{gj} | m/s | Drift velocity |

C | J/K | Heat capacity | x | m | Location of steam front |

C_{0} | - | Distr. parameter | $\dot{x}$ | m/s | Velocity of steam front |

d | m | Diameter | x | - | Molar fraction |

D | m | Outer diameter | x_{m} | - | Mass fraction |

G | W/m^{2} | Solar irradiation | Greek symbols | ||

h_{C} | m | Height difference | α | - | Steam power exponent |

h | J/kg | Specific enthalpy | α | - | Absorption coefficient |

h_{v} | J/kg | Spec. evap. enthalpy | α_{c} | W/Km^{2} | Conv. heat transfer coeff. |

j | m/s | Superficial velocity | α_{r} | W/Km^{2} | Rad. heat transfer coeff. |

K | - | Correction factor | β | 1/K | Expansion coefficient |

L | m | Char. length | β | - | Steam-filled fraction |

l | m | Length | γ | - | View factor |

m | kg | Mass | δ | - | Distribution parameter |

$\dot{m}$ | kg/s | Mass flow | ε | - | Emissivity |

n | - | Number of nodes | η | - | Efficiency |

n_{C} | - | Number of collectors | λ | W/Km | Heat conductivity |

ν | m^{2}/s | Kinemat. viscosity | |||

P_{v} | W | Steam power | $\rho $ | Kg/m^{3} | Density |

p | Pa | Pressure | σ | Pa | Surface tension |

$\dot{Q}$ | W | Enthalpy change, gain | τ | s | Interval |

R_{D} | - | Displacement ratio | φ | rad | Inclination angle |

S | m^{2} | Surface | Dimensionless numbers | ||

T | K, °C | Temperature | Nu | - | Nusselt number |

U_{L} | W/Km^{2} | Heat loss coefficient | Pr | - | Prandtl number |

U_{gj} | m/s | Average drift velocity | Re | - | Reynolds number |

u | J/kg | Specif. inner energy | Ra | - | Rayleigh number |

Subscripts | |||||

a | Ambient | i,k,q | Indices | ||

C | Collector | ||||

F | Field, array | S | Stagnation | ||

G | Glycol | s | Saturation | ||

g,l | Gas, liquid phase | t | Tube, pipe | ||

H,b | Bottom header | v | Vapor | ||

H,t | Top header | W | Water | ||

i | Insulation | WG | Water-Glycol mixture | ||

M | Meander | α | Inlet | ||

m | Mean value | ω | Outlet | ||

Constants | |||||

g | Acceleration of gravity | 9.81 | m/s^{2} | ||

σ | Stefan-Boltzmann constant | 5.67037·10^{−8} | W/K^{4}m^{2} |

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**Figure 1.**Array of four flat-plate collectors with (

**a**) thermochromic absorbers and (

**b**) standard selective absorbers.

**Figure 2.**Hydraulic diagram of the circuit, components and temperature sensor positions, valid for both systems.

**Figure 3.**Solar collector array with meander-shaped absorber tubes and integrated header tubes connected in parallel.

**Figure 4.**Collector efficiency curves for conventional selective coating (black lines) and thermochromic coating (red lines), valid for G = 1000 W/m

^{2}and T

_{a}= 30 °C.

**Figure 5.**Normalized temperature distribution during dry stagnation, as a function of the distance from the lower header axis. The black markers indicate values measured.

**Figure 10.**Comparison of total steam ranges from experimental data and simulations for the system with thermochromic absorbers (red) and conventional selective absorbers (black). Markers with white crosses correspond to the dataset 08−16. The marker with a red star corresponds to the dataset 07−25.

**Figure 12.**Experimental and simulation results for the system with standard selective absorbers. Opened check valves. Case 08−16. (

**a**) Steam range and level height, (

**b**) Steam power, heat losses and irradiation, (

**c**) Change rates of heat, (

**d**) Overpressure and steam volume, (

**e**) Saturation temperature and gas temperature inside MEV.

**Figure 13.**Experimental and simulation results for the system with standard selective absorbers. Active check valves. Dataset 07−26. (

**a**) Steam range and level height, (

**b**) Steam power, heat losses and irradiation, (

**c**) Change rates of heat, (

**d**) Overpressure and steam volume, (

**e**) Saturation temperature and gas temperature inside MEV.

**Figure 14.**Experimental and simulation results for the system with thermochromic absorbers. Inactive check valves. Dataset 08−16. (

**a**) Steam range and level height, (

**b**) Steam power, heat losses and irradiation, (

**c**) Change rates of heat, (

**d**) Overpressure and steam volume, (

**e**) Saturation temperature and gas temperature inside MEV.

**Figure 15.**Steam range from experiment and simulation for the system with thermochromic absorbers. Active check valves. Dataset 07−25.

**Table 1.**Propagation parameters and pipe direction with respect to the inlet and outlet of the collector field.

Supply Line | Return Line | ${\mathit{\delta}}_{\mathit{k}}$ | ${\mathit{\delta}}_{\mathit{q}}$ |
---|---|---|---|

horizontal | horizontal | 1 | 1 |

horizontal | downward | 1 | 0 |

downward | horizontal | 0 | 1 |

downward | downward | 1 | 1 |

**Table 2.**Average irradiation on the system with the standard selective (SC) and thermochromic (TC) absorber coating. Experimental and simulation results of the total steam range and simulation results of the residual mass. Active check valves (CV) are indicated by “✓”.

Dataset | CV | p_{0} | <G> | Residual Mass | Total Steam Range | |||||
---|---|---|---|---|---|---|---|---|---|---|

mm−dd | - | Bar | W/m^{2} | W/m^{2} | kg | kg | m | m | ||

SC | TC | SC | TC | SC | TC | SC | TC | |||

Exp. | Exp. | Sim. | Sim. | Exp. | Exp. | Sim. | Sim. | |||

07−24 | ✓ | 1 | 960 | 960 | 1.61 | 2.24 | 19.0 | 4.5 | 17.5 | 6.0 |

07−25 | ✓ | 1 | 1014 | 1014 | 1.52 | 1.74 | 20.5 | 9.0 | 20.2 | 12.4 |

07−26 | ✓ | 1 | 960 | 956 | 1.60 | 2.16 | 17.5 | 2.3 | 18.0 | 6.3 |

07−27 | ✓ | 1 | 902 | 903 | 1.63 | 2.17 | 20.5 | 6.5 | 19.6 | 7.5 |

08−02 | ✓ | 2 | 960 | 960 | 2.03 | 6.59 | 13.5 | 0.8 | 11.1 | 0.9 |

08−03 | ✓ | 2 | 988 | 988 | 1.78 | 3.72 | 15.8 | 0.8 | 15.0 | 1.6 |

08−06 | ✓ | 2 | 997 | 997 | 1.89 | 5.15 | 13.5 | 0.8 | 12.5 | 0.9 |

08−16 | − | 1 | 986 | 990 | 1.59 | 2.40 | 18.0 | 4.5 | 18.9 | 5.2 |

09−04 | − | 1 | 892 | 890 | 1.82 | 6.90 | 16.5 | 4.5 | 14.9 | 3.4 |

09−05 | − | 1 | 948 | 950 | 1.61 | 2.70 | 16.5 | 3.0 | 19.1 | 4.0 |

09−18 | − | 2 | 948 | 948 | 2.39 | 6.83 | 7.5 | 0.0 | 5.9 | 0.0 |

**Table 3.**Uncertainties of material properties and boundary conditions and their effect on the simulated steam range, based on data set 08−16.

Quantity | Uncertainty | Effect on Steam Range M | |
---|---|---|---|

Standard | Thermochromic | ||

Solar irradiance | ±2% | +1.06/−1.01 | +0.87/−0.90 |

Collector efficiency | ±4% | +0.49/−0.49 | +0.17/−0.28 |

Preset gauge pressure 1bar | ±0.1 bar | +0.46/−0.51 | +0.22/−0.34 |

Heat conductivity of pipe insulation | ±10% | −0.8/+0.96 | −0.3/+0.32 |

Absorber Type | Steam Range | Average Uncertainty | Relative Uncertainty |
---|---|---|---|

m | m | - | |

Conventional selective | 18 | 4.0 | 0.22 |

Thermochromic | 4.5 | 2.9 | 0.63 |

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## Share and Cite

**MDPI and ACS Style**

Eismann, R.; Hummel, S.; Giovannetti, F. A Thermal-Hydraulic Model for the Stagnation of Solar Thermal Systems with Flat-Plate Collector Arrays. *Energies* **2021**, *14*, 733.
https://doi.org/10.3390/en14030733

**AMA Style**

Eismann R, Hummel S, Giovannetti F. A Thermal-Hydraulic Model for the Stagnation of Solar Thermal Systems with Flat-Plate Collector Arrays. *Energies*. 2021; 14(3):733.
https://doi.org/10.3390/en14030733

**Chicago/Turabian Style**

Eismann, Ralph, Sebastian Hummel, and Federico Giovannetti. 2021. "A Thermal-Hydraulic Model for the Stagnation of Solar Thermal Systems with Flat-Plate Collector Arrays" *Energies* 14, no. 3: 733.
https://doi.org/10.3390/en14030733