# Numerical Modeling and Experimental Validation of Heat Transfer Characteristics in Small PTCs with Nonevacuated Receivers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Collector

#### 2.2. Heat Transfer in Nonevacuated Parabolic Trough Receiver

- Direct normal irradiance (DNI) is concentrated using the parabolic reflecting mirrors. In this study, the slope, specular and tracking errors were considered by introducing an appropriate increase in the angular divergence of the solar beam.
- The concentrated solar radiation flux is transmitted through the glass envelope and reaches the absorber tube. During this step, a small part of the concentrated solar radiation energy is absorbed by the glass envelope. This amount of energy was considered in the present study.
- The absorber tube absorbs the concentrated solar flux through the selective coating deposited on the outer surface of the absorber. The angular distribution of the absorbed solar flux was considered in this study.
- The heat absorbed by the selective coating is conducted to the inner surface of the absorber tube and then transferred to the HTF through convection. At the same time, the selective coating exchanges energy with the inner surface of glass envelope through conduction, convection and radiative exchange in the annular air gap. All these phenomena were considered in this study.
- The outer surface of the glass envelope dissipates heat towards the environment through convective and radiative exchanges.

#### 2.3. Concentrated Solar Flux

^{2}for a DNI = 1000 W/m

^{2}. Table 2 summarizes the values of the main parameters used for the optical simulation.

- Method used: Monte Carlo ray-tracing (MCRT) simulation;
- Grid pattern: dithered rectangular in which for each cell of the rectangular grid, the starting point of each ray is chosen randomly within the cell with a uniform distribution;
- Spatial and angular distributions of the solar beam: the rays are distributed uniformly over the grid dimensions, while the direction of each ray is chosen randomly within a Gaussian angular distribution with a half-angle [31] equal to the assigned divergence of solar beam θ
_{sb}.

^{2}of the maximum value, and it is twice the standard deviation of the angular distribution [32].

- Sun shape error ${\sigma}_{sun}$ = 2.6 mrad;
- Slope error ${\sigma}_{slope}$ = 1.9 mrad;
- Specularity error ${\sigma}_{mirror}$ = 2 mrad;
- Tracking error ${\sigma}_{track}$ = 1 mrad.

- Factory settings of optical parameters;
- Reflection and refraction of the rays on the glass envelope;
- Partial reflection of the rays on the absorber surface.

#### 2.4. One-Dimensional Energy Balance Model

_{f}, DNI, T

_{a}), must satisfy the following differential equation:

_{p}is the average specific heat at constant pressure, T

_{f}is the mass-averaged inlet temperature of the HTF, T

_{a}is the ambient temperature, DNI is the direct normal irradiance and z is the axial coordinate of the receiver tube. With reference to Figure 2b, the above equation was applied to a portion of the receiver tube of length Δz

_{i}, where with T

_{f,i}and T

_{f,i}

_{+1}mass-averaged inlet and outlet temperatures are indicated, respectively.

_{f,}

_{0}and the expression of the function q′ (T

_{f,i}, DNI, T

_{a}), it was possible to calculate the temperatures in each portion of length Δz

_{i}of the receiver tube and, therefore, also the collector outlet’s mass average temperature, T

_{f,u}.

## 3. Model Equations

#### 3.1. Equations Applied to the Heat Transfer Fluid Domain

^{(l)}and k

^{(t)}are respectively the laminar and turbulent conductivity.

^{2}, known as the Richardson number [37], is much smaller than unity, varying from 7.82 × 10

^{−3}to 3.47 × 10

^{−2}in the temperature range considered.

^{(t)}appears. It, knowing the Prandtl turbulent number Pr

^{(t)}and the turbulent viscosity ${\mu}^{\left(t\right)}$, is deducible from the following equation [38,39]:

^{(t)}expression, the following Kays–Crawford correlation was used [38,39].

^{(t)}and ${\mu}^{\left(t\right)}$ are not thermophysical properties of the fluid but only of the motion and vary strongly as the position varies.

#### 3.2. Equations Applied to the Air Gap Fluid Domain

^{3}to 1.25 × 10

^{4}in the temperature range considered, while the critical value between laminar motion and transition zone was 10

^{8}. The equations related to laminar motion are the following:

#### 3.3. Equations Applied to Solid Domains

#### 3.4. Radiation Heat Transfer

_{i}, the temperatures of which are assumed uniform. The adopted balance equations are the following [40]:

_{i}is the net power lost by surface A

_{i}, ${\epsilon}_{i}$ is the emissivity of surface A

_{i}and F

_{i}

_{−k}are the view factors between i-th and k-th surface.

_{i}is the radiosity that represents the total radiation leaving the surface A

_{i}per unit time and per unit area. Its expression is given by [40]:

_{i}is the total radiation incident on surface A

_{i}per unit time and per unit area and ${\rho}_{i}$ is the reflectance of surface A

_{i}.

#### 3.5. Boundary Conditions

- Diathermic oil velocity at the inlet of the receiver [41]:

- Turbulence intensity entering the receiver, defined through the following equation [19]:

_{T}

_{0}, we can calculate the turbulent kinetic energy k

_{0}at the inlet;

- Characteristic scale of the turbulence at the inlet, L
_{T}_{0}= 0.07D [30], where D is the inner diameter of the absorber tube; - Turbulent kinetic energy dissipation in the input, defined by the following expression depending on k
_{0}and L_{T}_{0}:

- Gradients of k and ε in the outlet section in the z direction equal to 0, namely:

- Conductive flux gradients in the z direction, in the outlet and inlet sections, equal to 0;
- Temperature of the diathermic oil entering the absorber tube, variable from 100 to 250 °C;
- Concentrated solar flux absorbed by the absorber tube, ${q}_{b}\left(\theta \right)$, where θ is the angular; coordinate. This flux was deduced from the ray-tracing analysis described in Section 2.3;
- Absolute pressure in the air gap varying from 1.19 to 1.42 bar depending on the temperature of the diathermic oil entering the receiver tube (as described in Section 3.7);
- For fluid domains, the inlet flow rates and the outlet pressures are specified;
- Thermal loss from the glass tube to the ambient air given by the following formula:

#### 3.6. Mesh and Solver Implemented

#### 3.7. Pressure Variation in the Air Gap

_{f}, can be deduced from the following formula:

## 4. Results and Discussion

#### 4.1. Three-Dimensional Thermo-Fluid Dynamic Analysis

- Fluid domains are modeled through conjugate heat transfer and nonisothermal flow, which incorporate basic computational fluid dynamic (CFD) equations;
- Laminar and turbulent flow are both supported and can be modeled with natural and forced convection;
- Turbulence can be modeled using Reynolds-averaged Navier–Stokes equations trough “k-ε” or “low Reynolds k-ε” models;
- Pressure work and viscous dissipation can be considered to evaluate the effects on the temperature distribution.

^{2}with steps of 250 W/m

^{2}.

^{2}represent the thermal losses of the receiver tube towards the environment and, therefore, are reported with a minus sign.

_{a}) was determined. It represents the thermal power transferred to the HTF per unit length (W/m) as a function of the average mass temperature of the diathermic oil, the DNI and the ambient temperature. The regression function used was the following:

**M**is the design matrix of the fitting problem, whose diagonal elements are the variances (squared uncertainties) of the fitting parameters.

#### 4.2. Experimental Set-Up and Results Obtained

^{2}. Finally, the ambient temperature was measured with a thermo-hygrometer (model VAISALA HMP63), suitably ventilated and shielded from solar radiation, with an accuracy greater than 0.5 K.

- Temperature: u = 0.08 °C;
- Flow rate: u = 0.1%;
- Direct solar irradiance: u = 6 W/m
^{2}; - Ambient temperature: u = 0.3 °C.

#### 4.3. Comparison between Theoretical and Experimental Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | section of the tube (m^{2}) |

A_{a} | aperture area of the concentrator (m^{2}) |

a_{1}, a_{2} | regression parameters |

b_{0}, b_{1}, b_{2} | regression parameters |

c_{p,m} | specific heat at constant pressure (J/(kg K)) |

C_{µ}, C_{c1}, C_{c2} | parameters of the k-ε model |

D | inner diameter of the receiver tube (m) |

D_{abs} | outer diameter of absorber tube (m) |

D_{g} | outer diameter of glass envelope (m) |

DNI | direct normal irradiance (W/m^{2}) |

f | friction factor |

f_{c}, f_{µ} | damping functions of Low Reynolds k-ε model |

g | gravity acceleration (m/s^{2}) |

G | variable that allows the determination of the wall distance (m^{−1}) |

Gr | Grashof number |

k | turbulent kinetic energy (J/kg) |

k | thermal conductivity (W/(m^{2} K)) |

k^{(l)} | laminar conductivity (W/(m K)) |

k^{(t)} | turbulent conductivity (W/(m K)) |

h_{f} | convective heat coefficient from absorber tube to diathermic oil (W/(m^{2} K)) |

h_{w} | convective heat coefficient from glass tube to ambient air (W/(m^{2} K)) |

I | identity matrix |

l | tube length (m) |

l_{ref} | distance beyond which the objects are described more thoroughly (m) |

I_{w} | distance from the receiver wall (wall distance) that satisfies the Eikonal equation (m) |

$\dot{m}$ | flow rate (kg/s) |

n | unit vector normal to the tube surface |

Nu | Nusselt number |

Pr | Prandtl number |

p_{A} | absolute pressure (Pa) |

$\overline{p}$ | Reynolds-averaged pressure (Pa) |

q | concentrated solar flux on the receiver tube (W/m^{2}) |

q_{b}(θ) | concentrated solar flux on the outer surface of the absorber tube in function of the angular coordinate θ (W/m^{2}) |

q′ | thermal power per unit length from the absorber tube to the thermal fluid (W/m) |

q_{z}′ | thermal power per unit length transferred to the thermal fluid (W/m) |

${q}_{g}^{\u2033}$ | heat flux from the glass envelope to the ambient air (W/m^{2}) |

${q}_{f}^{\u2033}$ | heat flux from the absorber tube to the diathermal oil (W/m^{2}) |

Q | useful thermal power extracted from the collector (W) |

r | radial coordinate of the receiver (m) |

Re | Reynolds number |

Ri | Richardson number |

r_{t} | inner radius of the receiver tube (m) |

t_{abs} | thickness of the absorber tube (m) |

t_{glass} | thickness of the glass tube (m) |

$\overline{T}$ | Reynolds-averaged temperature (°C) |

T_{abs} | absorber temperature (°C) |

T_{a} | ambient air temperature (°C) |

T_{g} | glass envelope temperature (°C) |

T_{in} | heat transfer fluid temperature at the inlet of the collector (°C) |

T_{out} | heat transfer fluid temperature at the outlet of the collector (°C) |

T_{m} | mean temperature of the heat transfer fluid (°C) |

T_{f} | average mass temperature of the heat transfer fluid (°C) |

T_{st} | internal temperatures of the steel tube (°C) |

T_{sky} | apparent sky temperature (°C) |

U | velocity vector (m/s) |

$\overline{\mathit{u}}$ | Reynolds-averaged velocity vector (m/s) |

u | standard uncertainty |

u_{z,in} | diathermic oil velocity at the inlet (m/s) |

$\u2329{u}_{z,in}\u232a$ | average diathermic oil velocity at the inlet (m/s) |

v_{w} | wind velocity (m/s) |

z | axial receiver coordinate (m) |

Greek symbols | |

α | absorber solar absorbance |

τ | glass solar transmittance |

ρ | mirror solar reflectance |

ρ | density of the heat transfer fluid (kg/m^{3}) |

ε | turbulent kinetic energy dissipation (J/(kg s)) |

ε | absorber emissivity |

ε_{g} | glass emissivity |

θ | angular coordinate (rad) |

θ_{sb} | angular divergence of solar beam (rad) |

µ | viscosity (Pa s) |

${\mu}^{\left(t\right)}$ | turbulent viscosity (Pa s) |

µ_{wall} | wall viscosity (Pa s) |

σ | Boltzmann constant (W/(m^{2} K^{4})) |

σ_{k}, σ_{ε}, σ_{w} | low Reynolds k-ε model parameters |

Subscripts | |

abs | absorber |

a | ambient air |

b | beam |

calc | calculated |

cond | conductive |

conv | convective |

meas | measured |

rad | radiative |

f | fluid |

g | glass |

in | inlet |

out | outlet |

sb | solar beam |

st | steel tube |

w | wind |

Abbreviations | |

2D | two dimensional |

3D | three dimensional |

CFD | computation fluid dynamic |

DAQ | data acquisition system |

DCS | distributed control system |

DNI | direct normal irradiance |

FEM | finite element method |

FVM | finite volume method |

HTF | heat transfer fluid |

IAM | incident angle modifier |

MCRT | Monte Carlo ray-tracing |

PTC | parabolic trough collector |

RMSE | root mean square error |

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**Figure 1.**View of a portion of the parabolic trough concentrator (

**a**) and details of the receiver tube (

**b**).

**Figure 2.**(

**a**) Cross-section of the receiver tube with inlet and outlet thermal heat flows; (

**b**) longitudinal section of a portion of the receiver tube.

**Figure 4.**Meshed computational domain of 3D thermal model applied to a portion of the receiver tube 1 m long.

**Figure 5.**Thermal field in the receiver tube (

**a**); 3D simulation with inlet fluid temperature at 100 °C (

**b**); 3D simulation with inlet fluid temperature at 250 °C.

**Figure 7.**Photos of the experimental set-up with the small-PTC under test (

**a**) and details of the plant’s components (

**b**).

**Figure 8.**(

**a**) Difference between calculated and measured outlet temperatures; (

**b**) calculated power output vs. measured ones.

Parameter | Symbol | Value |
---|---|---|

Aperture area | A_{a} | 13.6 m^{2} |

Aperture of optical system | W | 2.37 m |

Focal length | f_{c} | 0.82 m |

Rim Angle | ϕ_{r} | 72.68° |

Mirrors length | L | 6 m |

Outer diameter of absorber tube | D_{abs} | 42.4 mm |

Thickness of absorber tube | t_{abs} | 2 mm |

Outer diameter of glass tube | D_{glass} | 70 mm |

Thickness of glass tube | t_{glass} | 2.2 mm |

Description | Symbol | Value |
---|---|---|

Direct normal irradiance on the aperture area | DNI | 1000 W/m^{2} |

Mirrors solar reflectance | ρ | 0.94 |

Absorber tube solar absorbance | α | 0.93 |

Glass solar transmittance | τ | 0.92 |

Glass solar absorptance | α_{g} | 0.04 |

Direction of solar rays | - | On-axis |

Angular distribution of solar flux | - | Gaussian |

Assigned angular divergence of solar beam | θ_{sb} | 10 mrad |

Parameter | Mesh 1 | Mesh 2 | Mesh 3 |
---|---|---|---|

Type of mesh | Coarse | Normal | Fine |

Domain elements per unit length | 18,432 | 33,840 | 54,782 |

Boundary elements per unit length | 8256 | 14,294 | 19,894 |

Edge elements per unit length | 1412 | 2064 | 2578 |

Thermal power per unit length q’ (W/m) | 1388.2 | 1376.5 | 1375.3 |

Relative difference (%) | - | 0.8% | 0.09% |

**Table 4.**Trend of the pressure in the air gap as a function of the temperature of the diathermic oil.

Parameter | Values | ||||||
---|---|---|---|---|---|---|---|

T_{f} (°C) | 100 | 125 | 150 | 175 | 200 | 225 | 250 |

p_{air gap} (bar) | 1.19 | 1.21 | 1.25 | 1.27 | 1.34 | 1.36 | 1.42 |

Parameter | Symbol | Value |
---|---|---|

Ambient air temperature | T_{a} | 20 °C |

Apparent sky temperature | T_{sky} | 10 °C |

Wind speed | v_{w} | 3 m/s |

Mass flow rate of diathermic oil | $\dot{m}$ | 0.441 kg/s |

Absorber thermal conductivity | k_{abs} | 14.8 + 0.0153 (T_{abs} + 275.15) W/(m K) |

Absorber hemispherical emittance | ε_{abs} | 0.05 + 0.001 T_{abs} |

Glass thermal conductivity | k_{g} | 1.38 W/(m K) |

Glass hemispherical emittance | ε_{g} | 0.89 |

T (°C) | DNI (W/m^{2}) | ||||
---|---|---|---|---|---|

0 | 250 | 500 | 750 | 1000 | |

100 | −59.59 | 354.8 | 766.5 | 1176 | 1582 |

125 | −87.27 | 330.2 | 745.4 | 1158 | 1569 |

150 | −119.5 | 299.7 | 716.9 | 1132 | 1545 |

175 | −155.9 | 265.0 | 684.2 | 1102 | 1518 |

200 | −199.3 | 222.2 | 642.4 | 1061 | 1478 |

225 | −246.3 | 175.1 | 595.3 | 1014 | 1432 |

250 | −302.3 | 119.1 | 539.4 | 958.5 | 1377 |

Parameter | Unit | Value | Standard Error | T-Ratio |
---|---|---|---|---|

a_{1} | W m^{−1} K^{−1} | −0.38 | 0.01 | 25.6 |

a_{2} | W m^{−1} K^{−2} | −0.00401 | 0.00008 | 50.9 |

b_{0} | m | 1.571 | 0.005 | 291.1 |

b_{1} | m K^{−1} | 0.00109 | 0.00008 | 14.1 |

b_{2} | m K^{−2} | −0.0000027 | 0.0000003 | 10.4 |

Parameter | Value |
---|---|

Site | ENEA Research Centre Trisaia |

Latitude | 40°09′ N |

Longitude | 16°38′ E |

Inclination and azimuth | Single-axis tracking oriented in E–W direction |

Heat transfer fluid | Diathermic oil—Therminol^{®} 66 [45] |

Average flow rate | 0.441 kg/s |

Mean DNI | 910 W/m^{2} |

DNI (W/m ^{2}) | T_{a}(°C) | T_{in}(°C) | T_{out}(°C) | $\dot{\mathit{m}}$ (kg/s) | T_{m}(°C) | c_{p,m}(J/(kg K)) | Q_{meas}(W) | |
---|---|---|---|---|---|---|---|---|

Day 1 | 884 | 23.2 | 105.3 | 115.3 | 0.437 | 110.3 | 1872 | 8204 |

887 | 23.4 | 105.1 | 115.1 | 0.437 | 110.1 | 1871 | 8199 | |

885 | 23.5 | 105.1 | 115.1 | 0.436 | 110.1 | 1872 | 8158 | |

886 | 23.4 | 105.3 | 115.2 | 0.436 | 110.3 | 1872 | 8109 | |

Day 2 | 955 | 23.3 | 151.8 | 161.6 | 0.436 | 156.7 | 2037 | 8652 |

957 | 23.4 | 152.3 | 161.9 | 0.446 | 157.1 | 2039 | 8717 | |

959 | 23.6 | 153.2 | 162.7 | 0.446 | 158.0 | 2042 | 8631 | |

947 | 23.1 | 151.9 | 161.5 | 0.436 | 156.7 | 2037 | 8527 | |

Day 3 | 916 | 21.0 | 201.4 | 209.2 | 0.448 | 205.3 | 2214 | 7688 |

904 | 21.4 | 199.8 | 207.8 | 0.448 | 203.8 | 2208 | 7876 | |

918 | 21.3 | 198.2 | 206.2 | 0.448 | 202.2 | 2203 | 7862 | |

921 | 21.2 | 196.7 | 204.6 | 0.449 | 200.7 | 2197 | 7769 | |

Day 4 | 874 | 23.7 | 251.5 | 258.2 | 0.437 | 254.8 | 2399 | 7040 |

871 | 23.5 | 250.5 | 257.0 | 0.436 | 253.7 | 2394 | 6777 | |

892 | 23.4 | 250.1 | 256.8 | 0.437 | 253.4 | 2393 | 7004 | |

895 | 24.5 | 251.3 | 257.8 | 0.435 | 254.6 | 2398 | 6800 |

T_{out, meas}(°C) | T_{out, calc}(°C) | Difference % | Q_{meas}(W) | Q_{calc}(W) | Difference % | |
---|---|---|---|---|---|---|

Day 1 | 115.3 | 115.49 | 0.16% | 8204 | 8342 | 1.85% |

115.1 | 115.33 | 0.20% | 8199 | 8373 | 2.26% | |

115.1 | 115.33 | 0.20% | 8158 | 8353 | 2.26% | |

115.2 | 115.53 | 0.20% | 8109 | 8362 | 2.33% | |

Day 2 | 161.6 | 161.69 | 0.05% | 8652 | 8821 | 0.88% |

161.9 | 161.98 | 0.11% | 8717 | 8839 | 1.87% | |

162.7 | 162.88 | 0.11% | 8631 | 8855 | 1.88% | |

161.5 | 161.70 | 0.18% | 8527 | 8740 | 3.11% | |

Day 3 | 209.2 | 209.41 | 0.05% | 7688 | 7989 | 1.35% |

207.8 | 207.73 | 0.06% | 7876 | 7890 | 1.65% | |

206.2 | 206.31 | 0.05% | 7862 | 8045 | 1.32% | |

204.6 | 204.85 | 0.03% | 7769 | 8090 | 0.64% | |

Day 4 | 258.2 | 258.13 | 0.09% | 7040 | 6983 | 3.58% |

257.0 | 257.14 | 0.05% | 6777 | 6964 | 2.09% | |

256.8 | 256.93 | 0.05% | 7004 | 7178 | 1.91% | |

257.8 | 258.16 | 0.06% | 6800 | 7193 | 2.39% |

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## Share and Cite

**MDPI and ACS Style**

Ebolese, A.; Marano, D.; Copeta, C.; Bruno, A.; Sabatelli, V.
Numerical Modeling and Experimental Validation of Heat Transfer Characteristics in Small PTCs with Nonevacuated Receivers. *Solar* **2023**, *3*, 544-565.
https://doi.org/10.3390/solar3040030

**AMA Style**

Ebolese A, Marano D, Copeta C, Bruno A, Sabatelli V.
Numerical Modeling and Experimental Validation of Heat Transfer Characteristics in Small PTCs with Nonevacuated Receivers. *Solar*. 2023; 3(4):544-565.
https://doi.org/10.3390/solar3040030

**Chicago/Turabian Style**

Ebolese, Amedeo, Domenico Marano, Carlo Copeta, Agatino Bruno, and Vincenzo Sabatelli.
2023. "Numerical Modeling and Experimental Validation of Heat Transfer Characteristics in Small PTCs with Nonevacuated Receivers" *Solar* 3, no. 4: 544-565.
https://doi.org/10.3390/solar3040030