Open AccessArticle

Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation

Zuriel Aquino-Santiago

¹,

J. O. Aguilar

Guillermo Becerra-Núñez

^2,3

and

O. A. Jaramillo

^4,*

Posgrado en Ingeniería en Energía, Universidad Nacional Autónoma de México, Priv. Xochicalco s/n, Temixco 62580, Morelos, Mexico

Division de Ciencias, Ingenieria y Tecnologia, Universidad Autónoma del Estado de Quintana Roo, Boulevar Bahía s/n, Chetumal 77019, Quintana Roo, Mexico

Investigadoras e Investigadores por Mexico, Consejo Nacional de Humanidades, Ciencias y Tecnologías, Av. Insurgentes Sur 1582, Alcaldía Benito Juárez 03940, Ciudad de México, Mexico

⁴

Instituto de Energías Renovables, Universidad Nacional Autónoma de México, Priv. Xochicalco s/n, Temixco 62580, Morelos, Mexico

Author to whom correspondence should be addressed.

Processes 2024, 12(12), 2653; https://doi.org/10.3390/pr12122653

Submission received: 1 October 2024 / Revised: 12 November 2024 / Accepted: 18 November 2024 / Published: 25 November 2024

(This article belongs to the Special Issue Advanced Heat Transfer Technologies for the Design, Operation and Optimization of Steam Power Systems)

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Abstract

Parabolic Trough Collectors (PTCs) are a well-established technology for efficiently generating hot water and low-enthalpy steam. For instance, PTCs can be used in steam power systems to drive small Organic Rankine Cycles (ORCs). This study evaluated the thermal efficiency of a PTC equipped with a receiver tube featuring a concentric annular cross-section. This receiver design consists of a tube with a concentric rod inside, forming an annular gap through which the working fluid flows. A thermodynamic model was developed to assess the PTC’s thermal efficiency in hot water and low-enthalpy steam applications. The evaluation considered the First and Second Laws of Thermodynamics, factoring in environmental losses. The model included a bare receiver tube with three-rod diameters—3/8, 1/2, and 3/4 inches—and a range of volumetric flow rates from 1 to 6 L per minute. The results showed improved heat transfer with the annular cross-section receiver compared to a conventional circular one, particularly at lower flow rates of 1 and 2 L per minute. The highest increase in thermal efficiency was observed with the 3/4-inch rod at a flow rate of 1 L per minute, where the maximum efficiency reached 40%.

Keywords:

concentrated solar energy; parabolic trough collector; Second Law analysis; low enthalpy processes; annular flow receiver

1. Introduction

Concentrated Solar Power (CSP) plants use Parabolic Trough Concentrators (PTCs) to produce renewable electricity by capturing thermal energy in a thermodynamic cycle. PTCs can operate at high temperatures between 300 and 570 °C for electricity generation, and at medium temperatures ranging from 85 to 250 °C for various industrial thermal processes such as evaporation, cleaning, pasteurization, distillation, sterilization, drying, cooking, and heating for domestic hot water and spaces, as well as power generation systems using Organic Rankine Cycle (ORC) systems.

Advancements in PTC technology and the integration of ORC systems are promising developments in the renewable energy sector. Continuous research and innovation are essential to unlock the full potential of CSP plants and address the challenges associated with efficiency, cost, and environmental impact. For instance, the integration of the ORC has emerged as a feasible option for electricity production in CSP plants. Recent improvements in CSP technology have made power plants that combine CSP and ORC technologies more economically appealing, offering a reliable and efficient solution with easy-to-operate ORC power block units that require low maintenance. However, the optimal capacity of the Rankine cycle in this context remains an open question that requires further exploration.

On the other hand, research has been conducted on various methods to enhance the thermal efficiency of PTCs. Emphasis has been placed on passive methods to improve heat transfer within PTC systems, aiming to optimize the thermal performance of PTCs further. A critical aspect of enhancing PTC efficiency lies in optimizing heat transfer between the solar energy incident on the absorber tube and the heat transfer fluid (HTF) to maximize the overall performance and effectiveness of the system.

In 2014, Natarajan et al. [1] investigated passive techniques to enhance heat transfer in absorber tubes, a critical area in solar thermal power engineering. Previous studies showed that increasing the fluid flow rate in a parabolic trough collector (PTC) plain absorber tube could improve heat transfer. Experiments demonstrated that an 85 kg/h flow rate resulted in a higher temperature difference under a solar flux of 850 W/m². In that study, computational fluid dynamics (CFD) simulations were conducted using different cross-sectional inserts (triangle, inverted triangle, and semi-circular) inside the absorber tube. The analysis used the SST

κ - ω

turbulence model and ANSYS CFX 12.1 software to calculate the heat transfer and pressure drop. The results indicated that triangular inserts provided optimized heat transfer and reduced thermal fatigue, though with a higher pressure drop than plain tubes without inserts. In the same year, Egbo et al. [2] simulated the effect of various annular gaps (0.010 m to 0.055 m) on the performance of a solar parabolic-trough collector, while maintaining constant reflector geometry, absorber-tube design, and meteorological data. Their results show that increasing the annular gap between the absorber tube and the surrounding glass cover leads to higher total heat loss and lower thermal efficiency. The research concludes that as the annular gap increases, the system’s thermal efficiency decreases, highlighting the importance of optimizing the gap size to improve the performance of solar parabolic-trough collectors.

In 2018, Chang et al. [3] proposed a study to enhance the reliability and thermal performance of a parabolic trough solar receiver by introducing concentric and eccentric rods as turbulators. The study analyzed the flow and convective heat transfer characteristics of molten salt in the receiver. They developed a three-dimensional model and validated it with experimental results and empirical equations. The findings indicate that both rod inserts significantly improve heat transfer performance, with the normalized Nusselt number increasing by 1.10 to 7.42 times compared to a plain receiver. They also showed that the integrated performance factor decreases as the Reynolds number increases, particularly when the rod diameter exceeds 0.8. Furthermore, the inserts help to make the temperature distribution uniform, significantly reducing thermal deflection and enhancing the receiver’s reliability.

In 2021, Peng et al. [4] studied the impact of a new insertion method, specifically a semi-annular and fin-shaped metal foam (SAFM) hybrid structure, on the thermohydraulic and thermodynamic performance of the absorber tube in a parabolic trough collector (PTC). They analyzed the effects of different SAFM fin shapes (rectangular, triangular, and trapezoidal) on turbulent forced convection heat transfer and flow under non-uniform heat flux using numerical methods. The study employed the local thermal non-equilibrium method to account for energy imbalances between the fluid and the metal foam structure, and the Monte Carlo ray-trace method to evaluate the non-uniform heat flux on the absorber tube wall. The findings indicate that SAFM significantly enhances the thermohydraulic and thermodynamic performance of the absorber tube, with Nusselt numbers increasing by 256.3% to 838.7%. Additionally, SAFM outperforms individual semi-annular or fin-shaped metal foam, particularly the triangular fin design, which yields superior performance compared to the rectangular and trapezoidal fins.

In 2024, Kumar et al. [5] conducted a study to assess the practical effectiveness of a locally developed parabolic trough collector system through experimental analysis. Their system consists of two parabolic trough collectors featuring one-ended evacuated tubes and inner concentric copper tubes for double-pass airflow, achieving average air temperatures between 130.22 °C and 166.72 °C, indicating its viability for industrial and domestic applications. Solar intensity, thermal performance, and pressure drop were measured after the first and second evacuated tubes. The experiments were conducted over six days with three airflow rates (0.0056 kg/s, 0.0038 kg/s, and 0.0021 kg/s), revealing a maximum air temperature of 210.5 °C. The highest thermal efficiency was recorded at 39.31%, with an overall average of 30.56%. Additionally, the maximum pressure drop reached 510 Pa at 0.0056 kg/s. In the same year, Abdala et al. [6] introduced a new design for absorber tubes in parabolic trough solar collectors. They focused on annular tubes with and without spiral tape as alternatives to traditional designs. Using numerical analysis with CFD code, they looked at different geometric ratios of the spiral tape—pitch (P/D), width (W/D), and height (H/D)—to see how they affected heat transfer performance, pressure drop, and entropy generation. The results showed that the annular absorber tube without spiral tape achieved a higher Nusselt number (Nu) than conventional tubes. Specifically, at a pitch ratio of 1.4, the Nusselt number increased, and the average Nu ratio was 24.5% greater at a height ratio of 0.3. Additionally, the spiral tape reduced total entropy generation compared to conventional tubes, while the lowest pitch ratio decreased wall temperature, which helped alleviate thermal stress. However, the friction ratio was 41.7 times higher with spiral tape than without.

The main goal of this study is to analyze the thermal performance of a parabolic trough collector (PTC) equipped with concentric annular tube inserts for low enthalpy processes using the principles of thermodynamics. To estimate the behavior of the PTC with a concentric rod, we developed a thermodynamic model framework based on empirical correlations for calculating the heat transfer rate and pressure drop under fully developed conditions. This framework includes a Second Law analysis that provides valuable insights into the exergy efficiency of the PTC. The thermodynamic model framework is designed to study low-temperature parabolic trough concentrators similar in size to smaller-scale commercial solar collectors, operating within a temperature range of 70 to 110 °C. This framework can be used as the first stage in optimizing the performance of a PTC operating in low-enthalpy steam generation.

The paper is organized as follows: In Section 2, we present the relevant background to this work, describing previous PTCs developed by our group, which form the basis for this study. Section 3 details the thermodynamic model of the PTC. Thermal efficiency is calculated using the First Law of Thermodynamics. In Section 3.1, an energy balance in the receiver tube is carried out to establish the useful heat in steady-state conditions. Radiative and convective losses are considered. In Section 3.2, an analysis of the internal flow for a hollow and concentric tube is developed. This subsection presents formulations for the Reynolds, Nusselt, and Prandtl numbers for internal flow mass, as well as the pressure drop that considers the pumping power. In Section 4, the thermodynamic framework is reported. In Section 4.1, an energy balance is established, accounting for heat transfer and energy losses to determine the efficiency using the First Law. In Section 4.2, the efficiency using the Second Law is described to establish the reversible limit. In Section 4.2, the entropy generation number is presented to estimate the reduction in irreversibility based on the entropy generation. In Section 5, the main results and discussion on the results are reported. In Section 5.1.1 and Section 5.1.2, the thermal performances by the First and Second Laws are reported. In Section 5.1.3, the generation number results are included to compare the performance of the PTC when a hollow tube and a concentric annular tube are used. The main conclusions are reported in Section 5.2.

2. Background

Previous parabolic trough collector developments by the authors have given them experience in both design and experimental validation. Among the most important works are those carried out by Jaramillo et al. The authors of [7] developed a thermodynamic model to analyze the performance of a parabolic trough collector (PTC) using a twisted tape insert to enhance heat transfer. The model focused on determining the conditions under which the insert optimally improves efficiency and identified low twisted ratios and low Reynolds numbers as key factors. Supported by experimental data, the model predicted the thermal and exergy efficiency, revealing that the Nusselt number, removal factor, friction factor, and thermal efficiency increase as the twist ratios and Reynolds numbers decrease. However, these performance metrics do not improve with higher twist ratios. The results showed that twisted tape inserts are an effective passive method for enhancing heat transfer under specific conditions, particularly with a twist ratio near 1 and low Reynolds numbers corresponding to low flow rates (about 1 lpm). The swirling flow generated by the twisted tape enhances heat exchange but also causes a pressure drop. The Second Law analysis indicated that exergy efficiency improves only when the entropy generation number (NS, a) is less than one. The findings suggest the need for further investigation into passive heat transfer enhancement techniques, such as testing other types of inserts to evaluate their impact on PTC performance.

In 2016, Borunda et al. [8] analyzed a small concentrated solar power (CSP) plant coupled with an organic Rankine cycle (ORC) in a novel configuration. In this setup, the energy produced is directly used to power the block and charge the thermal storage. The system was applied to a medium-temperature cogeneration process in the textile industry. The results showed that this configuration reduces the required thermal storage size. Simulations conducted using TRNSYS modeled actual operating conditions, demonstrating promising potential for applications requiring both electricity and heat generation. The design and simulation results, including efficiency and production curves, suggest that the system is particularly suited for medium-temperature industrial processes, such as pasteurization. Compared to traditional configurations, the direct feed-storage configuration presented requires a smaller storage tank for the same storage duration. However, as the solar fraction increases, the energy and exergy efficiencies of both the thermal storage and the solar-ORC plant decrease. The overall system efficiency improves when waste heat is utilized as a heat source, achieving around 55% energy recovery. The study suggests that further exploration of other storage technologies, such as molten salts, concrete, or phase change materials, could enhance the system’s performance by balancing heat capacity and cost.

In 2017, Caldiño et al. [9] presented the design and performance analysis of a small organic Rankine cycle (ORC) system coupled to a parabolic trough solar collector (PTC). The system was analyzed using the First and Second Laws of Thermodynamics and was designed to generate 10 kWe of electricity for nearly 18 h a day, as well as provide energy for heating water. The system consists of three main components: parabolic trough solar concentrators (PTC), a thermal storage system, and an ORC system, which uses R245fa as the working fluid and a radial turbine as the expander. The system is designed to operate in Temixco, Mexico, utilizing local solar radiation data for performance estimation. Simulations, conducted using Matlab v2023a and CoolProp v6.5.0, calculated the ORC power, efficiency, mass flow, and thermal load stored in the PTC loop. The study provides insights into the potential efficiency and operation of solar-powered ORC systems in regions with suitable solar resources.

3. Heat Transfer for Evaluating the Thermal Performance of a PTC

Currently, there is no widely accepted international standard for evaluating the thermal performance of a PTC. However, one of the main references is the International Standard ASHRAE 93-1986 [10], which is used to evaluate solar thermal collectors and includes a section on methodologies applicable to solar concentrators.

The PTC’s thermal performance was modeled based on the ASHRAE 93-1986 (RA 91) standard. This document outlines test methods for evaluating the thermal performance of solar energy collectors that utilize single-phase fluids and do not have significant internal energy storage. In Section 8.2.1.1 of the ASHRAE 93-1986 (RA 91) standard, a test method for determining the thermal efficiency of a concentrating collector is detailed. This method is commonly employed to measure thermal efficiency and compare it with similar solar collectors.

In the thermal performance model of the PTC, two receiver configurations were used in the solar collector. The first configuration involves a hollow tube with a constant cross-section through which the thermal fluid flows and was used as a reference to determine the thermal efficiency of the solar collector with respect to an annular tube. The second configuration consists of a tube with a concentric rod inside, as shown in Figure 1. This setup allows for the creation of an annular flow around the internal tube and improving the thermal efficiency of the parabolic trough collector by means of the reduction of annular space. Three sizes of solid rod were used, with diameters of 3/8”, 1/2”, and 3/4”.

In Figure 1,

D_{o}

is the external diameter for both the hollow tube and the concentric rod,

D_{i}

refers to the inner diameter of the hollow tube,

d_{i}

denotes the annular rod, and

d_{o}

indicates the inner diameter of the concentric rod.

During the operation of a parabolic trough concentrator, a temperature difference arises between the receiver tube and the surrounding environment. This temperature gradient causes heat to transfer from the higher temperature receiver tube to the cooler environment, resulting in thermal losses. These losses occur through three mechanisms: conduction, convection, and radiation. In this model, thermal losses due to conduction are neglected because the receiver tube supports are assumed to be thermally insulated. Therefore, only convection and radiation are considered, as described below.

3.1. Energy Balance in the Receiver Tube

The calculation of the energy transferred to the working fluid, the useful heat

Q_{u}

, is carried out by means of an energy balance in the receiver tube under steady-state conditions, in which the energy absorbed by the receiver and the thermal losses to the environment are considered. The changes in the kinetic, potential, and work energies are also neglected.

{\dot{Q}}_{u} = {\dot{Q}}_{a b s} - {\dot{Q}}_{c o n v} - {\dot{Q}}_{r a d}

(1)

where

{\dot{Q}}_{a b s} = A_{a} G_{B} η_{o}

is the energy absorbed by the receiver tube and the remaining terms correspond to thermal losses. By substituting the expressions for each of the variables shown in Equation (1) and factoring the term

(T_{r} - T_{a})

, the balance energy equation is expressed as

{\dot{Q}}_{u} = A_{a} G_{B} η_{o} - A_{r} U_{L} (T_{r} - T_{a})

(2)

where

U_{L} = h_{c o n d} + h_{r a d}

is the heat loss coefficient.

The heat loss coefficient is calculated by evaluating the heat transfer coefficient

h_{r a d}

at the maximum receiver temperature

T_{r m}

. This temperature is obtained by an energy balance in the tube considering two criteria: neglect energy losses to the environment and set the same temperature for the inner and outer surfaces of the receiver tube, which is valid for

k ≫ l n (\frac{D_{o}}{D_{i}})

, where k is the thermal conductivity of the receiver tube. Thus, the maximum receiver temperature of the receiver is as follows:

T_{r m} = \frac{η_{o} C_{o} G_{B}}{h_{w}} + T_{f o m}

(3)

where

C_{o}

is the solar concentration ratio,

G_{B}

is the direct solar radiation and

h_{w}

is the fluid heat transfer coefficient. The maximum fluid outlet temperature

T_{f o m}

is obtained from:

T_{f o m} = \frac{η_{o} C_{o} G_{B} π D_{o} L}{\dot{m} C_{p}} + T_{f i}

(4)

where

C_{p} (\frac{J}{k g \cdot K})

is the fluid specific heat and

T_{f i}

is the inlet fluid temperature. It is important to note from (4) that increasing the mass flow causes a decrease in the fluid maximum outlet temperature, as well as an increase in the fluid’s convective coefficient, thus decreasing the maximum receiver temperature

T_{r m}

. The radiative heat transfer coefficient also decreases, causing a reduction in the overall loss coefficient

U_{L}

In practice, it is not possible to directly calculate the useful heat with Equation (2), because the receiver temperature depends on several design and atmospheric variables. For this reason, the useful heat expression will be redefined in terms of the fluid inlet temperature. For this purpose, the useful heat transferred through the tube will be expressed by means of its thermal resistances as

{\dot{Q}}_{u} = U_{o} A_{r} (T_{r} - T_{f})

(5)

where

T_{f}

is the local fluid temperature and

U_{o} (\frac{W}{m^{2} \cdot K})

is the global heat transfer coefficient based on the area of the receiver tube, which is expressed as follows:

U_{o} = {[\frac{D_{o}}{h_{w} D_{i}} + (\frac{D_{o}}{2 k_{c}} l n (\frac{D_{o}}{D_{i}}))]}^{- 1}

(6)

where

k_{c}

is the receiver’s tube thermal conductivity. For an annular concentric tube,

D_{i} = d_{o}

can be used. Finally, from (2) and (5), it is possible to eliminate

T_{r}

, so the useful transferred heat is expressed as

{\dot{Q}}_{u} = F^{'} A_{a} [η_{o} G_{b} - \frac{U_{L}}{C_{o}} (T_{f} - T_{a})]

(7)

From (7),

F^{'}

is the efficiency factor of the solar collector, and is expressed as

F^{'} = \frac{1}{U_{L}} [\frac{1}{U_{L}} + \frac{D_{o}}{h_{w} D_{i}} + (\frac{D_{o}}{2 k_{c}} l n \frac{D_{o}}{D_{i}})]

(8)

This efficiency factor represents the ratio of the actual useful energy to the calculated useful energy and is evaluated considering that the receiver temperature is the same as that of the local fluid, i.e.,

T_{r} = T_{f i}

3.1.1. Convective Losses

The model considers a forced convection between wind at the ambient temperature

T_{a}

and the receiver tube temperature

T_{r}

. Thus, the heat transfer

{\dot{Q}}_{c o n v}

is calculated according to [11]:

{\dot{Q}}_{c o n v} = A_{r} h_{c o n v} (T_{r} - T_{a})

(9)

where

A_{r}

(m^{2})

is the receiver area and

h_{c o n v}

is the convective heat transfer coefficient expressed as:

h_{c o n v} = N u_{v} \frac{k_{v}}{D_{o}}

(10)

where

k_{v}

is the wind thermal conductivity.

3.1.2. Radiative Losses

The amount of energy emitted

Q_{r a d}

by the receiver tube depends on the temperature difference between the receiver tube and the sky, i.e., the receiver area is considered a small object in a large cavity. Note that the sky temperature is taken as the ambient temperature for practical purposes. The radiative heat transfer is expressed as:

{\dot{Q}}_{r a d} = A_{r} h_{r a d} (T_{r} - T_{a})

(11)

where

h_{r a d}

({W / m}^{2} K)

is the radiative heat transfer coefficient. This coefficient is analogous to the convective coefficient and is expressed as

h_{r a d} = ε_{r} σ (T_{r} + T_{a}) (T_{r}^{2} + T_{a}^{2})

(12)

where

σ

({W / m}^{2} \cdot K^{2})

is the Stefan-Boltzmann constant,

ε_{r}

is the receiver’s surface emissivity, and

T_{r}

is the receiver’s temperature.

3.1.3. Heat Removal Factor

To define

{\dot{Q}}_{u}

as a function of the inlet temperature

T_{f i}

, the heat removal factor is introduced, which is defined as the ratio of the actual useful energy and the useful energy obtained when the receiver and inlet fluid temperatures are the same,

(T_{r} = T_{f i})

F_{R} = \frac{\dot{m} C_{p} (T_{f o} - T_{f i})}{A_{r} [η_{o} G_{B} C_{o} - U_{L} (T_{f i} - T_{a})]}

(13)

The fluid outlet temperature

T_{f o}

is obtained by means of an energy balance over a tube element, as shown in Figure 2. This balance is performed under steady-state conditions. Also, the useful energy incident per unit length of the tube

(\dot{q_{u}} = {\dot{Q}}_{u} / L)

on an infinitesimal length

δ_{y}

is considered. Then, the energy balance is described as

{\dot{q}}_{u} δ y + \dot{m} C_{p} T_{f} - \dot{m} C_{p} (T_{f} + \frac{d T_{f}}{d y} δ y) = 0

(14)

Dividing the Equation (14) by

δ y

, evaluating the limit when

δ y

\to 0

and substituting

{\dot{q}}_{u}

, returns

\dot{m} C_{p} \frac{d T_{f}}{d y} - F^{'} \frac{A_{a}}{L} [η_{o} G_{B} - \frac{U_{L}}{C_{o}} (T_{f} - T_{a})] = 0

(15)

Integrating Equation (15) by means of variable separation and considering

C_{p}

F^{'}

, and

U_{L}

as constants, gives

\int_{T_{f i}}^{T_{f o}} \frac{1}{T_{f} - T_{a} - η_{o} G_{B} C_{o} / U_{L}} d T_{f} = - \frac{A_{r} F^{'} U_{L}}{L \dot{m} C p} \int_{0}^{L} d y

(16)

Therefore, the outlet fluid temperature can be expressed as:

\frac{T_{f o} - T_{a} - η_{o} G_{B} C_{o} / U_{L}}{T_{f i} - T_{a} - η_{o} G_{B} C_{o} / U_{L}} = e x p (- \frac{A_{r} F^{'} U_{L}}{\dot{m} C_{p}})

(17)

Subtracting

T_{f o}

from (17) and substituting it into (13), returns

F_{R} = \frac{\dot{m} C_{p}}{A_{r} U_{L}} [1 - e x p (- \frac{A_{r} F^{'} U_{L}}{\dot{m} C_{p}})]

(18)

The Reynolds number (

R e

) for the fluid, in this case water without phase change, was determined. The definition of this parameter depends on the geometry over which the flow occurs. For a hollow tube,

R e_{t v}

is given by:

R e_{t v} = \frac{4 \dot{m}}{π D_{i} μ_{w}} .

(19)

For a concentric rod, the Reynolds number

R e_{c}

can be expressed as:

R e_{c} = \frac{4 \dot{m}}{π d_{h} μ_{w}} .

(20)

where

d_{h} = (d_{o} - d_{i})

is the hydraulic diameter of the concentric tube, and in both cases,

\dot{m}

is the mass flow rate and

μ_{w} (m^{2} / s)

is the dynamic fluid viscosity.

The Reynolds number

R e_{o}

for flow over both the hollow and concentric tubes is given by:

R e_{c i l} = \frac{V D_{o}}{ν}

(21)

where V (m/s) is the constant flow velocity near the tube, and v (m²/s) is the kinematic viscosity. It is important to note that in both configurations, the outer cylinder

D_{o}

is exposed to the wind velocity.

The wind impacting the surface of the receiver tube can be approximated as a cross flow over a cylinder, meaning the flow is perpendicular to the cylinder’s axis. This flow develops without confinement restrictions, complicating the analytical calculation of heat transfer over the cylinder. The Nusselt number

(N u)

represents the enhancement of convective heat transfer compared to fluid conduction and is used as an empirical correlation to evaluate the average heat transfer coefficient. In this model, the average Nusselt number for forced convection was used, according to [12]:

N u_{v} = 0.446 R e_{c i l}^{0.5} P r_{v}^{0.35} + \frac{0.528 P r_{v}^{0.42}}{{({(6.5 e^{R e_{c i l} / 5000})}^{- 5} + {(0.031 R e_{c i l}^{0.8})}^{- 5})}^{1 / 5}}

(22)

From Equation (22)

R e

takes values from 2000

\leq R e_{c i l}

\leq 9 \times 10^{4}

and the Prandtl number

P r

takes values of

0.7

\leq P r_{v} \leq 176

3.2. Analysis of Internal Flow in Hollow and Concentric Rod

In the case where the working fluid circulates inside the hollow tube, the heat transfer rate depends on the regime associated with the Reynolds number. For a laminar flow (

R e_{t v} < 2300

), the Nusselt number value is a constant

N u_{l a m} = 4.364

[12]. On the other hand, if the flow is turbulent (

4000 < R e_{t v} \leq 10^{6}

) under conditions

0.1 \leq P r_{w} \leq 1000

and

D_{i} / L \leq 1

, the Nusselt number is defined as

{N u}_{t u r} = \frac{(f_{t v} / 8) (R e_{t v} - 1000) P r_{w}}{1 + 12.7 \sqrt{f_{t v} / 8} ({P r_{w}}^{2 / 3} - 1)} [1 + {(\frac{D_{i}}{L})}^{2 / 3}]

(23)

where L is the tube length and

f_{t v}

is the friction factor, defined as

f_{t v} = {(1.8 \log_{10} (R e_{t v}) - 1.5)}^{- 2}

(24)

If the fluid flow is in a transient regime (

2300 \leq R e_{t v} \leq 4000

) with

P r_{w} \geq 0.6

and

D_{i} / L \leq 1

, the Nusselt number is obtained by means of an interpolation of the laminar and turbulent Nusselt values, i.e., [13],

{N u}_{t r a} = 4.364 (1 - γ_{t v}) + γ_{t v} \frac{375 (0.04026) P r_{t v}}{1 + 12.7 \sqrt{0.04026 / 8} (P r_{t v}^{2 / 3} - 1)} [1 + {(\frac{D_{i}}{L})}^{2 / 3}]

(25)

where

γ_{t v}

is the intermittency factor at a given point in the flow field and is defined as the fraction of time the flow remains turbulent [14]. The value of the intermittency factor varies from 0 to 1 and was evaluated using the following equation:

γ_{t v} = \frac{R e_{t v} - 2300}{4000 - 2300}

(26)

On the other hand, for fluid flow in annular concentric tubes with an isolated rod and constant heat flow over the annular tube surface, the expression for the Nusselt number for the laminar regime (

R e_{t a} < 2300

) was obtained from an analytical solution developed by [15], which was calculated as a function of the diameter ratio

x = d_{i} / d_{o}

{N u}_{l a m} = - \frac{2 (- \frac{x^{2}}{2} + \frac{r_{m}^{2} x^{2}}{2} + \frac{x^{4}}{4} - r_{m}^{2} x^{2} \ln (x) + \frac{1}{4} - \frac{r_{m}^{2}}{2}) (1 - x)}{b_{1} + b_{2} + b_{3} + b_{4} + b_{5}}

(27)

where the terms

b_{1}, b_{2}, b_{3}, b_{4}, b_{5}

are as follows:

b_{1} = (\frac{1}{2} - \frac{1}{2} x^{2} - x^{2} {(\ln (x))}^{2}) r_{m}^{2} - (\frac{1}{2} x^{2} - \frac{1}{4} x^{4} - x^{2} r_{m}^{2}) \ln (x)

(28)

b_{2} = \frac{a (21 - 100 r_{m}^{2} + 108 r_{m}^{4}) - a x^{2} (- 36 + 108 x^{2} - 60 x^{4} + 9 x^{6})}{576 (1 + x^{2} - 2 r_{m}^{2})}

(29)

b_{3} = \frac{x^{2}}{4} - \frac{a x^{2} (72 (- 2 + x^{2}) \ln (x) - 4 r_{m}^{2} (- 27 + 81 x^{2} - 29 x^{4}))}{576 (1 + x^{2} - 2 r_{m}^{2})}

(30)

b_{4} = - \frac{x^{4}}{16} + \frac{4 a x^{2} r_{m}^{2} (6 (- 15 - 6 x^{2} + 5 x^{4}) \ln (x) + 72 {(\ln (x))}^{2})}{576 (1 + x^{2} - 2 r_{m}^{2})}

(31)

b_{5} = - \frac{3}{16} - \frac{36 a x^{2} r_{m}^{4} (3 x^{2} - 12 x^{2} \ln (x) + 8 (2 + x^{2}) {(\ln (x))}^{2})}{576 (1 + x^{2} - 2 r_{m}^{2})}

(32)

and the terms a and

r_{m}

are defined as:

r_{m} = {(\frac{1 - x^{2}}{2 \ln (\frac{1}{x})})}^{0.5}

(33)

and

a = \frac{\frac{2 r_{m}^{2}}{x} - \frac{1}{x} - x}{x - \frac{1}{2 x} - \frac{x^{3}}{2} + r_{m}^{2} (\frac{1}{x} - x + 2 x \ln (x))}

(34)

It is important to note that once the x value is established, the

N u_{l a m}

value remains constant and is, therefore, independent of the Reynolds and Prandtl numbers.

For a turbulent flow (

4000 < R e_{t a} \leq 10^{6}

) with

0.6 \leq P r_{w} \leq 1000

and

d_{h} / L \leq 1

, where

d_{h} = d_{o} - d_{i}

, the Nusselt number is calculated with the following expression [13]:

{N u}_{t u r} = \frac{(f_{t a} / 8) (R e_{t a} - 1000) P r_{w}}{1 + 12.7 \sqrt{f_{t a} / 8} ({P r_{w}}^{2 / 3} - 1)} [1 + {(\frac{d_{h}}{L})}^{2 / 3}] (0.9 - 0.15 x^{0.6})

(35)

where

f_{t a}

is the friction factor expressed as [13]

f_{t a} = {(1.8 \log_{10} (R e_{t a}^{*}) - 1.5)}^{- 2} .

(36)

In the above equation, the Reynolds number is redefined as

R e_{t a}^{*}

as follows:

R e_{t a}^{*} = R e_{t a} \frac{(1 + x^{2}) \ln (x) + 1 - x^{2}}{{(1 - x)}^{2} \ln (x)}

(37)

For a transient flow (

2300 \leq R e_{t a} \leq 4000

), the Nusselt number expression was obtained in a manner similar to that reported in [13], interpolating the laminar and turbulent regimens values of the annular tube:

N u_{t r a} = (1 - γ_{t a}) N u_{l a m} + γ_{t a} \frac{375 (0.9 - 0.15 x^{0.6}) f_{t a} P r_{w}}{1 + 12.7 \sqrt{f_{t a} / 8} ({P r_{w}}^{2 / 3} - 1)} [1 + {(\frac{d_{h}}{L})}^{2 / 3}]

(38)

The Nusselt number

N u_{l a m}

corresponds to Equation (27) and

f_{t a}

to Equation (36), evaluated using the following equation:

R e_{t a}^{*} = 4000 \frac{(1 + x^{2}) \ln (x) + 1 - x^{2}}{{(1 - x)}^{2} \ln (x)}

(39)

Finally, the parameter

γ_{t a}

is defined as

γ_{t a} = \frac{R e_{t a} - 2300}{4000 - 2300}

(40)

On the other hand, the pressure drop

(Δ P)

is an important factor in the calculation of the pumping power. In addition, its value has a direct relationship with entropy generation. The pressure drop of a fluid flow in a hollow tube is defined as:

Δ P_{t v} = \frac{8 f_{t v} {\dot{m}}^{2} L}{π^{2} ρ D_{i}^{5}}

(41)

where

ρ

({kg / m}^{3})

is the density.

The Moody or Darcy factors depend on the flow regime. For laminar flow, these can be determined using the Hagen–Poiseuille equation:

f_{t v} = \frac{64}{R e_{t v}}

(42)

For turbulent flow (

R e_{t v} > 4000

), the friction factor is defined by Equation (24). In the transient regime, (

2300 \leq R e_{t v} \leq 4000

), the friction factor is obtained by means of interpolation between the laminar and turbulent regime values:

f_{t v} = (1 - γ_{t v}) \frac{16}{575} + (0.04026) γ_{t v}

(43)

where

γ_{t v}

is defined by Equation (26).

In this way, the equation to calculate the fluid drop pressure in an annular concentric rod is as follows:

Δ P_{t a} = \frac{8 f_{t a} {\dot{m}}^{2} L}{π^{2} ρ {(d_{o} + d_{i})}^{2} {(d_{o} - d_{i})}^{3}}

(44)

where the friction factor

f_{t a}

is determined using Equation [13]:

f_{t a} = \frac{64}{{R e}_{t a}^{*}}

(45)

and the parameter

R e_{t a}^{*}

is calculated from Equation (37). For a turbulent flow, the friction factor is obtained from Equation (36). For transient flow, (

2300 \leq R e_{t a} \leq 4000

), the friction factor is obtained by means of interpolation between the laminar and turbulent regime values, using the following equation:

f_{t a} = (1 - γ_{t a}) \frac{64}{{R e}_{t a}^{*}} + γ_{t a} {(1.8 \log_{10} (R e_{t a}^{*}) - 1.5)}^{- 2}

(46)

The

R e_{t a}^{*}

and

γ_{t a}

values were calculated by means of Equations (36) and (26), respectively.

4. Thermodynamic Framework for the Performance of the PTC

In practice, many problems involve mass flow across system boundaries. As a result, the First Law of Thermodynamics can be applied to open systems (or control volumes), where the mass entering and exiting the control volume carries energy. The First Law establishes the energy balance, accounting for heat transfer and work as forms of energy interaction, while ensuring total energy conservation. The Second Law of Thermodynamics, on the other hand, addresses irreversibilities caused by factors such as friction from fluid flow within the duct and heat transfer due to finite temperature differences.

4.1. Thermal Efficiency by the First Law of Thermodynamics

Once the heat removal factor has been defined, see Equation (18), the useful heat can be expressed as a function of the inlet fluid temperature as:

{\dot{Q}}_{u} = F_{R} [G_{B} η_{o} A_{a} - A_{r} U_{L} (T_{f i} - T_{a})]

(47)

Useful heat

{\dot{Q}}_{u}

plays an important role because it allows evaluating the thermal efficiency of the solar collector using the following equation:

η_{I} = \frac{{\dot{Q}}_{u}}{{\dot{Q}}_{*}}

(48)

where

{\dot{Q}}_{*} = A_{a} G_{B}

is the incident solar radiation on the collector’s aperture area. Therefore, the efficiency can be expressed as a lineal equation:

η_{I} = F_{R} η_{o} - \frac{F_{R} U_{L}}{C_{o}} (\frac{T_{f i} - T_{a}}{G_{B}})

(49)

where

F_{R} η_{o}

can be considered as the ordinate to the origin and

F_{R} U_{L} / C_{o}

can be considered as the equation slope.

Since two configurations will be used in the receiver tube (see Figure 1), it is necessary to define the thermal efficiency improvement factor as follows:

Δ η_{I} = \frac{η_{I, t a}}{η_{I, t v}}

(50)

where

η_{I, t v}

is the efficiency of the concentrator with flow in the hollow tube (without rod inside) and

η_{I, t a}

is the efficiency of the collector with an annular concentric receiver. This factor has the value of 1 (

0 %

percent) when the use of an annular flow does not represent an improvement with respect to the flow in a hollow tube, and in the case of an improvement, the improved efficiency factor is

Δ η_{I} > 1

, which means that there would be an increase in efficiency using concentric annular tubes instead of hollow tubes.

4.2. Thermal Efficiency by the Second Law of Thermodynamics

Calculating the thermal efficiency by means of the Second Law of Thermodynamics evaluates the deviation or approximation of a system with respect to the reversible limit, and is defined as

η_{I I} = \frac{{\dot{X}}_{i n c} - {\dot{X}}_{d e s}}{{\dot{X}}_{i n c}}

(51)

where the

{\dot{X}}_{i n c}

parameter is the exergy incident on the system and

{\dot{X}}_{d e s}

is the energy destruction parameter. The efficiency

η_{I I}

is 1 when the system is reversible and 0 when all the exergy is destroyed.

In a parabolic trough solar collector, the incident exergy is associated with the solar radiation on its surface, as follows:

{\dot{X}}_{i n c} = {\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}})

(52)

The parameter

T_{*}

is the apparent temperature of the sun and its value is established as 4500 K. On the other hand, exergy destruction,

{\dot{X}}_{d e s}

, is determined through the entropy generation resulting from heat transfer and fluid friction and can be calculated from entropy generation by means of the Gouy–Stodola theorem. However, since the exergy incident on the collector is directly dependent on solar radiation, any variations in solar radiation will linearly affect both the available exergy and the calculated exergy destruction. Thus, the entropy generation rate of the collector is defined as follows: [12]

{\dot{S}}_{g e n} = (\frac{{\dot{Q}}_{l o s s}}{T_{a}} + \frac{{\dot{Q}}_{u}}{T_{f i}} - \frac{{\dot{Q}}_{*}}{T_{*}}) + (\frac{\dot{m}}{ρ} \frac{Δ P}{T_{a}})

(53)

where

{\dot{Q}}_{l o s s}

{\dot{Q}}_{u}

, and

{\dot{Q}}_{*}

are the heat losses, the useful heat, and the solar beam radiation collected by the PTC, respectively.

Δ P

is the pressure drop on the hollow or annular tube. The irreversibilities are due to the heat transfer ratio (first parenthesis) and the fluid friction (second parenthesis).

Substituting the heat loss expression

{\dot{Q}}_{l o s s} = {\dot{Q}}_{*} - {\dot{Q}}_{u}

in Equation (53), the exergy destruction of the system is obtained by means of the following equation:

{\dot{X}}_{d e s} = {\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}}) - {\dot{Q}}_{u} (1 - \frac{T_{a}}{T_{f i}}) + \frac{\dot{m} Δ P}{ρ}

(54)

Finally, by substituting Equations (53) and (54) in Equation (51), the thermal efficiency is obtained, written as a linear equation:

η_{I I} = b - p (\frac{T_{f i} - T_{a}}{G_{B}})

(55)

where b can be considered as the ordinate to the origin:

b = F_{R} η_{o} (1 - \frac{T_{a}}{T_{f i}}) {(1 - \frac{T_{a}}{T_{*}})}^{- 1} - \frac{\dot{m} Δ P}{ρ} {[A_{a} G_{B} (1 - \frac{T_{a}}{T_{*}})]}^{- 1}

(56)

and p can be considered the slope defined as follows:

p = \frac{F_{R} U_{L}}{C_{o}} (1 - \frac{T_{a}}{T_{f i}}) {(1 - \frac{T_{a}}{T_{*}})}^{- 1}

(57)

The improvement factor to compare the efficiencies obtained by means of the First and Second Laws of Thermodynamics is expressed as

Δ η_{I I} = \frac{η_{I I, t a}}{η_{I I, t v}}

(58)

where

η_{I I, t v}

is the thermal efficiency of a solar collector with a hollow tube and

η_{I I, t a}

is the thermal efficiency of a solar collector with an annular concentric rod.

Entropy Generation Number

A direct application of entropy generation is in the area of heat transfer enhancement. In this application, the goal is to decrease the irreversibilities of a given system by redesigning the flow configuration. For this purpose, the entropy generation number is defined as [12]:

N_{S, a} = \frac{{\dot{S}}_{g e n, a}}{{\dot{S}}_{g e n, o}}

(59)

where

{\dot{S}}_{g e n, o}

is the irreversibility of the device’s initial configuration (no improvements) and

{\dot{S}}_{g e n, a}

represents the device’s irreversibility when the flow configuration is redesigned. The redesigned flow configuration shows improvements over the initial configuration when

N_{s, a} < 1

In this model, the simplified entropy generation number is expressed as follows:

N_{S, t a} = \frac{{\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}}) - {\dot{Q}}_{u, t a} (1 - \frac{T_{a}}{T_{f i}}) + \frac{\dot{m}}{ρ} Δ P_{t a}}{{\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}}) - {\dot{Q}}_{u, t v} (1 - \frac{T_{a}}{T_{f i}}) + \frac{\dot{m}}{ρ} Δ P_{t v}}

(60)

where the irreversibilities are due to the heat transfer rate

{\dot{S}}_{Δ T}

and the fluid friction

{\dot{S}}_{Δ P}

. Thus, Equation (60) can be rewritten with the irreversibilities distribution rate

ϕ (W / K)

to analyze the contribution of each irreversibility as follows:

N_{S, t a} = \frac{{\dot{S}}_{Δ T, t a} (1 + ϕ_{t a})}{{\dot{S}}_{Δ T, t v} (1 + ϕ_{t v})}

(61)

where

ϕ_{t a}

corresponds to the annular flow:

ϕ_{t a} = \frac{{\dot{S}}_{Δ P, t a}}{{\dot{S}}_{Δ T, t a}} = \frac{\frac{\dot{m}}{ρ} Δ P_{t a}}{{\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}}) - {\dot{Q}}_{u, t a} (1 - \frac{T_{a}}{T_{f i}})}

(62)

and

ϕ_{t v}

to flow through a hollow tube,

ϕ_{t v} = \frac{{\dot{S}}_{Δ P, t v}}{{\dot{S}}_{Δ T, t v}} = \frac{\frac{\dot{m}}{ρ} Δ P_{t v}}{{\dot{Q}}_{*} (1 - \frac{T_{a}}{T_{*}}) - {\dot{Q}}_{u, t v} (1 - \frac{T_{a}}{T_{f i}})}

(63)

In the following section, the thermal efficiencies of the PTC are modeled for both the hollow tube and concentric annular tube configurations based on the First and Second Laws of Thermodynamics. Additionally, the entropy generation numbers for each configuration are presented and analyzed.

5. Results and Discussion

The calculations were based on the parabolic trough collector parameters reported in reference [7]. The physical parameters of the PTC are shown in Table 1. The PTC from which the design parameters were taken is shown in Figure 3, as a reference image.

The thermophysical properties of water at a constant pressure (1.21 atm) and an average temperature range (20–90) °C are presented in Table 2. Additionally, the thermophysical properties of air at a constant pressure (1 atm) are included. The above properties of air and water are implicit (e.g., mass flow and convective heat transfer coefficient) in the theoretical development presented in Section 3 and Section 4.

5.1. Thermal Efficiency

5.1.1. First Law Thermal Efficiency

The thermal efficiency value

η_{I}

from Equation (49) is directly proportional to the heat removal factor

F_{r}

, (Equation (18)). Therefore, calculating

η_{I}

as a function of the mass flow

\dot{m}

will enable the analysis of thermal efficiency trends. Figure 4 illustrates the heat removal factor for both types of receiver configurations: hollow and annular concentric flow, within a range of 0–6 L per minute (

l p m

). For the concentric rod, three inner diameters were considered:

d_{i}

9.525 mm (3/8”), 12.7 mm (1/2”), and 19.05 cm (3/4”) for the concentric solid rod. Figure 4 shows that each curve exhibits a distinct point (or inflection point) where the rate of change of

F_{r}

with respect to the mass flow suddenly increases. These points correspond to mass flow rates of 1.8, 2.7, 2.9, and 3.3

l p m

, and are indicated by arrows on the x-axis; each represents the transition from laminar to turbulent flow within the receiver pipe.

Figure 4 demonstrates that for mass flow rates exceeding 3.5

l p m

, the heat removal factor

F_{r}

does not vary significantly between the 1/2” and 3/8” diameter annular tubes and the hollow tubes. The 3/4” diameter annular tube consistently exhibits the highest

F_{r}

value across all mass flow rates, while the hollow tube shows higher

F_{r}

values than the 1/2” and 3/8” diameter annular tubes within the range of 2.5 to 4

l p m

Figure 5 shows the thermal efficiency of the PTC at a constant mass flow rate of 1

l p m

. The dotted curve presents the lowest efficiency and corresponds to the concentrator with a hollow receiver tube. The thermal efficiency of PTCs with an annular concentric receiver was obtained as well for 1/2” and 3/8” diameter; these are shown in Figure 5 for three different diameters (

d_{i} = 3 / 8 ”, d_{i} = 1 / 2 ”

and

d_{i} = 3 / 4 ”

), where it can be observed that efficiency increases as the internal diameter of the annular tube increases. The vertical displacement order of the curves is similar to those of the removal factor at a constant flow rate of 1

l p m

(Figure 4).

In general, as the mass flow increases, the difference in thermal efficiency between the concentric receiver tubes decreases. However, the largest diameter (3/4”) always has the highest efficiency value compared to the other diameters, which can be seen in Table 3. At mass flow rates of 1 and 2

l p m

, the hollow receiver tube has a higher thermal efficiency than the 3/8” and 1/2” annular receiver tubes (see Figure 6), but at 3

l p m

, the thermal efficiency of the hollow tube is higher than the 3/8” and 1/2” annular receiver tubes (see Figure 7). This behavior is related to the heat removal factor. Figure 4 shows that at a mass flow of 3 lpm, the heat removal factor is higher in the hollow tube than in the 3/8” and 1/2” diameter annular tubes. When the mass flow reaches a value of 4

l p m

, the thermal efficiency increases for each receiver tube used (see Figure 8). In particular, the 3/4” diameter annular tube has an improvement factor of 2%. The 1/2” annular tube has an improvement value of 1%. The other tubes show no improvement in thermal efficiency.

Finally, for mass flows of 5 and 6

l p m

(see Figure 9), there is an increase in the thermal efficiency for all four receiver tubes compared to lower mass flows (1 to 4

l p m

However, the overall efficiency increase rate is lower as the mass flow rate increases. The 3/4” annular tube has a thermal efficiency improvement factor of 2% for both mass flow rates (5 and 6

l p m

), the 1/2” annular tube has an improvement factor of 1%, and the other tubes (3/8” annular and hollow), a value of less than 1%. The thermal efficiency to 6

l p m

shows similar behavior to 5

l p m

; therefore, that curve is omitted.

Table 3 presents the values of the loss coefficient, heat removal factor, and thermal efficiency factor for mass flow rates of 1 and 2

l p m

5.1.2. Second Law Thermal Efficiency

In this section, the thermal efficiency of PTCs with four types of receiver tubes was calculated by means of the Second Law of Thermodynamics. For the analysis, constant values of the mass flow rate from 1 to 6

l p m

were used. The thermal efficiency was established as a function of

Δ T / G_{B}

, where the inlet temperature

T_{f i}

was defined in the range of 20–90 °C.

Figure 10 shows the thermal efficiency curves at a constant mass flow rate of 1

l p m

. The curve with the lowest efficiency corresponds to the PTC with the hollow receiver tube. The remaining curves were obtained from a PTC with a concentric annular rod, and it is observed that the efficiencies of these curves increase as the internal diameter of the concentrator tube increases. Note that the order of the vertical position of the curves is very similar to the thermal efficiencies obtained by means of the First Law of Thermodynamics (see Figure 5). However, the values on the ordinate axis differ by one order of magnitude.

Table 4 shows the thermal efficiency enhancement values for different annular concentric receiver tubes. It should be noted that these values are similar to those shown in Table 3, which indicates that viscous effects are negligible in the calculation of thermal efficiency by means of the Second Law of Thermodynamics.

The efficiencies obtained at a constant flow rate of 2

l p m

are shown in Figure 11. The figure shows an increase in the thermal efficiencies compared to those obtained at 1

l p m

(Figure 10). However, the enhancement factor decreases due to a higher rate of change of the hollow tube with respect to the annular receiver tube.

Figure 12 shows the efficiencies obtained at a constant mass flow rate of 3

l p m

. Note that the vertical order of position of the curves is similar to the thermal efficiency curves presented in Figure 7. However, the enhancement factor for the

3 / 4 ”

tube decreased by one unit, i.e., viscous effects contribute to thermal efficiency. The

1 / 2 ”

and

3 / 8 ”

diameter annular concentric tubes have a null enhancement factor, indicating that the heat removal factor has a directly proportional relationship with the thermal efficiency for both configurations.

Figure 13 shows thermal efficiencies obtained at a constant flow mass rate of 4

l p m

. The enhancement factors are

2 %, 1 %

, and null to

3 / 4 ”

1 / 2 ”

, and

3 / 8 ”

diameter annular tubes, respectively.

Finally, Figure 14 shows the thermal efficiencies at a mass flow of 5

l p m

. The thermal efficiency has an enhancement of

1 %

for the

3 / 4 ”

and

1 / 2 ”

annular receiver tubes and less than

1 %

for the

3 / 8 ”

annular tube. On the other hand, at a mass flow of 6

l p m

, all configurations have a thermal efficiency factor less than

1 %

, and are omitted for this reason.

5.1.3. Entropy Generation Number Results

The entropy generation numbers

N_{S, t a}

for the mass flow-annular diameter pairs are presented below. The mass flows used were from 1 to 6

l p m

. Each inequality shown in Table 5 is valid for

Δ T / G_{B} = (T_{f i} - T_{a}) / G_{B}

, where

T_{f i}

take values from 20 to 90 °C; for example,

N_{S, t a}

< 1 for the pairs 1

l p m

—annular

3 / 4 ”

diameter tubes is valid in the respective

Δ T / G_{B}

range. The pairs

N_{S, t a}

< 1 are advantageous because the irreversibilities decrease with the implementation of annular flow. The

3 / 4 ”

diameter annular receiver tube presents the better enhancement value, regardless of the mass flow used. On the other hand, the

1 / 2 ”

diameter annular receiver tube is subject to the same condition (

N_{S, t a}

< 1), except for 3

l p m

mass flow, in which case, there is no improvement to thermal efficiency. The

3 / 8 ”

diameter annular receiver tube presents values of

N_{S, t a}

< 1 only for mass flow values of 1, 2, 5, and 6

l p m

The distribution ratio of the irreversibilities for an annular tube and hollow tube are

ϕ_{t a} \approx ϕ_{t v} \approx 0

, independently of

\dot{m}

and

Δ T / G_{B}

, i.e., the heat transfer irreversibilities are dominant in both flow configurations. As a consequence, the

N_{S, t a}

is approximately equal to the entropy generation ratio for both configurations, which is caused by heat transfer.

5.2. Conclusions

A thermodynamic model to simulate the heat transfer of a parabolic trough collector with a concentric cylindrical rod into the receiver tube (annular tube) was developed. A constant mass flow on the receiver tube, no phase change and an operation range to use in low enthalpy processes were considered. Thermal and exergy efficiencies and entropy generation number were obtained from the model.

The thermal efficiencies obtained at 1

l p m

and 2

l p m

show a remarkable increase in heat transfer in the annular tube in comparison to the hollow tube. The maximum efficiency improvement factor values,

Δ η_{I}

(35–40%), correspond to the

3 / 4 ”

annular tube at 1

l p m

(see Figure 5). In addition, the value of

Δ η_{I}

decreased as the diameter of the inserted rod decreased.

The results from the Second Law of Thermodynamics show that enhancement factors

Δ η_{I I}

obtained at 1

l p m

are equal to those obtained from the First Law of Thermodynamics,

Δ η_{I}

, with the same mass flow. The maximum values obtained of

Δ η_{I}

(35–40%) correspond to a

3 / 4 ”

diameter annular receiver tube|. This indicates the drop in pressure after inserting the rod into the receiver tube is negligible. However, the

Δ η_{I}

of the

3 / 4 ”

diameter of an annular receiver tube decreases slightly (by 1%) for mass flows greater than 1

l p m

Additionally, the entropy generation number,

N_{S, t a}

, with values lower than 1, showed a decrease of PTC irreversibilities after implementation of the concentric rod, i.e., the technique presents thermodynamic advantages. This fact is in agreement with the obtained thermal and exergy efficiencies. Additionally, it was shown that the heat transfer irreversibilities are dominant in both flow configurations and the pressure drop is almost negligible.

Author Contributions

Conceptualization, O.A.J.; Methodology, Z.A.-S., G.B.-N. and O.A.J.; Validation, J.O.A. and G.B.-N.; Formal analysis, Z.A.-S., J.O.A. and O.A.J.; Investigation, Z.A.-S., J.O.A. and O.A.J.; Resources, G.B.-N.; Data curation, G.B.-N.; Writing—original draft, O.A.J. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

Symbols
$A$	Area $[m^{2}]$
$C_{o}$	Concentration ratio $[-]$
$C_{p}$	Specific heat at constant pressure $[kJ / kg K]$
$D_{i}$	Inner diameter $[m]$
$D_{o}$	Outer diameter $[m]$
$d_{i}$	Inner diameter annular section $[m]$
$d_{o}$	Outer diameter annular section $[m]$
$F^{'}$	Efficiency factor $[-]$
$F_{R}$	Heat removal factor $[-]$
$f$	Friction factor $[-]$
$G_{B}$	Direct solar radiation $[W / m^{2}]$
$h$	Heat transfer coefficient $[W / m^{2} K]$
$L$	Length $[m]$
$\dot{m}$	Mass flow rate $[kg / s]$
$N_{s}$	Entropy generation number $[-]$
$N u$	Nusselt number $[-]$
$P$	Pressure $[kg m / s^{2}]$
$P r$	Prandtl number $[-]$
$\overset{\cdot}{Q}$	Heat per unit time $[W]$
${\overset{\cdot}{Q}}_{l o s s}$	Heat loss $[W]$
${\overset{\cdot}{Q}}_{u}$	Useful heat $[W]$
${\overset{\cdot}{Q}}_{*}$	Solar beam radiation collected by the PTC $[W]$
$R e$	Reynolds number $[-]$
${\overset{\cdot}{S}}_{g e n}$	Entropy generation rate $[W / K]$
$T$	Temperature $[K]$
$T_{a}$	Ambient temperature $[K]$
$T_{i n}$	Temperature at the input of the receiver tube $[K]$
$T_{o u t}$	Temperature at the output of the receiver tube $[K]$
$T_{r}$	Temperature of the receptor $[K]$
$U$	Loss coefficient $[W / m^{2} K]$
V	Velocity $[m / s]$
$\dot{X}$	Exergy per unit time $[W]$
Greek letters
$α$	Absorptivity $[-]$
$α = k / ρ C_{p}$	Thermal diffusivity $[m^{2} / s]$
$Δ η_{I}$	Enhancement factor by First Law $[-]$
$Δ η_{I I}$	Enhancement factor by Second Law $[-]$
$Δ P$	The drop pressure $[kg m / s^{2}]$
$ε$	Emissivity $[-]$
$η_{o}$	Optical efficiency $[-]$
$k$	Thermal conductivity $[W / m K]$
$η_{I}$	First Law efficiency $[-]$
$η_{I I}$	Second Law efficiency $[-]$
$μ$	Dynamic viscosity $[\frac{kg}{m s}]$
$ν$	Kinematic viscosity $[m^{2} / s]$
$ρ$	Density $[kg / m^{3}]$ , Reflectivity $[-]$
$σ$	Stefan-Boltzmann constant $[5.67051 \times 10^{- 8} W m^{- 2} K^{- 4}]$
$ϕ$	Irreversibilities distribution rate $[W / K]$

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Figure 1. Types of receiver tubes used; (a) Hollow tube and (b) concentric annular tube.

Figure 2. Energy balance in a δ_y tube element.

Figure 3. Parabolic trough collector.

Figure 4. Heat removal factor, (

F_{r}

) vs. mass flow (

\dot{m}

Figure 4. Heat removal factor, (

F_{r}

) vs. mass flow (

\dot{m}

Figure 5. Thermal efficiency at constant mass flow of 1

l p m

Figure 5. Thermal efficiency at constant mass flow of 1

l p m

Figure 6. Thermal efficiency at constant mass flow of 2 lpm.

Figure 7. Thermal efficiency at constant mass flow of 3 lpm.

Figure 8. Thermal efficiency at constant mass flow of 4 lpm.

Figure 9. Thermal efficiency at constant mass flow of 5

l p m

Figure 9. Thermal efficiency at constant mass flow of 5

l p m

Figure 10. Second Law thermal efficiency at constant mass flow of 1

l p m

Figure 10. Second Law thermal efficiency at constant mass flow of 1

l p m

Figure 11. Second Law thermal efficiency at constant mass flow of 2

l p m

Figure 11. Second Law thermal efficiency at constant mass flow of 2

l p m

Figure 12. Second Law thermal efficiency at constant mass flow of 3

l p m

Figure 12. Second Law thermal efficiency at constant mass flow of 3

l p m

Figure 13. Second Law thermal efficiency at constant mass flow of 4

l p m

Figure 13. Second Law thermal efficiency at constant mass flow of 4

l p m

Figure 14. Second Law thermal efficiency at constant mass flow of 5 lpm.

Table 1. Physical parameters of parabolic trough collector.

Optical efficiency, $η_{o}$	$0.7 [-]$
Aperture area, $A_{a}$	$5.187 [m^{2}]$
Receiver area, $A_{r}$	$0.438 [m^{2}]$
Concentration ratio, $C_{o}$	$11.84 [-]$
Length	$4.88 [m]$
Outer diameter (Hollow tube $D_{o}$ )	$0.0285 [m]$
Inner diameter (Hollow tube $D_{i}$ )	$0.0252 [m]$
Outer diameter (Annular tube $d_{o}$ )	$0.0252 [m]$
Inner diameter (Annular tube $d_{o}$ )	$0.0127 [m]$ $0.0158 [m]$ $0.0222 [m]$
Tube material	$C o p p e r$
Emissivity, $ε_{r}$	$0.96 [-]$

Table 2. Relevant values used in the modeling process.

Property	Value	Property	Value
Average properties at operating range of 20–90 °C
Water density, $ρ$	$992.23 \frac{kg}{m^{3}}$	Copper thermal conductivity, $κ$	$380.5 \frac{W}{m \cdot K}$
Water thermal diffusivity, $α_{w}$	$17.65 \times 10^{- 6} \frac{m^{2}}{s}$	Air thermal diffusivity, $α_{v}$	$1.515 \times 10^{- 7} \frac{m^{2}}{s}$
Water thermal conductivity, $κ_{w}$	$0.628 \frac{W}{m \cdot K}$	Air thermal conductivity, $κ_{v}$	$0.0259 \frac{W}{m \cdot K}$
Water kinematic viscosity, $ν_{w}$	$0.658 \times 10^{- 6} \frac{m^{2}}{s}$	Air kinematic viscosity, $ν_{v}$	$12.49 \times 10^{- 6} \frac{m^{2}}{s}$
Water heat capacity, $C_{p, w}$	$4.1794 \frac{kJ}{kg \cdot K}$	Wind velocity, V	$2.3 \frac{m}{s}$
Water dynamic viscosity, $μ_{w}$	$0.653 \times 10^{- 3} \frac{kg}{m s}$
Temperature and solar radiation parameters
Irradiance, $G_{B}$	$865 \frac{W}{m^{2}}$	Ambient temperature, $T_{a}$	$293.15 K$
Solar apparent temperature, $T_{*}$	$4500 K$

Table 3. Thermal efficiency at different mass flows.

	$U_{L} [\frac{W}{m^{2} K}]$	$F_{R} [-]$	$Δ η_{I} [-]$
Receiver tube type
(1 $l p m$ )
Hollow Tube	43.96	0.6243	- - -
Annular Tube (3/8”)	0.5269–2.6889	0.5592–2.7926	0.6024–2.962
Annular Tube (1/2”)	0.5483–2.778	0.5826–2.8900	0.6013–2.9575
Annular Tube (3/4”)	0.5936–2.9669	0.6324–3.0975	0.6456–3.1422
Receiver tube type
(2 $l p m$ )
Hollow Tube	41.71	0.7641	- - -
Annular Tube (3/8”)	0.6446–3.1327	0.6592–3.1906	0.6659–3.2164
Annular Tube (1/2”)	0.6475–3.1449	0.6228–3.2055	0.6692–3.2302
Annular Tube (3/4”)	0.6634–3.2113	0.6737–3.2512	0.679–3.2712

Table 4. Thermal efficiency enhancement to different annular receiver tubes.

Receiver Tube Type	1 $lpm$	2 $lpm$
Annular Tube (3/8”)	20–23%	4–5%
Annular Tube (1/2”)	25–29%	8–9%
Annular Tube (3/4”)	35–40%	18–19%

Table 5. Entropy generation numbers for a mass flow ranging from 1 to 6

l p m

Table 5. Entropy generation numbers for a mass flow ranging from 1 to 6

l p m

Mass Flow $(lpm)$	Annular $3 / 8 ”$	Annular $1 / 2 ”$	Annular $3 / 4 ”$
1	$N_{S, t a} < 1$	$N_{S, t a} < 1$	$N_{S, t a} < 1$
2	$N_{S, t a} < 1$	$N_{S, t a} < 1$	$N_{S, t a} < 1$
3	$N_{S, t a} > 1$	$N_{S, t a} > 1$	$N_{S, t a} < 1$
4	$N_{S, t a} > 1$	$N_{S, t a} < 1$	$N_{S, t a} < 1$
5	$N_{S, t a} < 1$	$N_{S, t a} < 1$	$N_{S, t a} < 1$
6	$N_{S, t a} < 1$	$N_{S, t a} < 1$	$N_{S, t a} < 1$

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Aquino-Santiago, Z.; Aguilar, J.O.; Becerra-Núñez, G.; Jaramillo, O.A. Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation. Processes 2024, 12, 2653. https://doi.org/10.3390/pr12122653

AMA Style

Aquino-Santiago Z, Aguilar JO, Becerra-Núñez G, Jaramillo OA. Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation. Processes. 2024; 12(12):2653. https://doi.org/10.3390/pr12122653

Chicago/Turabian Style

Aquino-Santiago, Zuriel, J. O. Aguilar, Guillermo Becerra-Núñez, and O. A. Jaramillo. 2024. "Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation" Processes 12, no. 12: 2653. https://doi.org/10.3390/pr12122653

APA Style

Aquino-Santiago, Z., Aguilar, J. O., Becerra-Núñez, G., & Jaramillo, O. A. (2024). Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation. Processes, 12(12), 2653. https://doi.org/10.3390/pr12122653

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Enhancing the Thermal Efficiency of Parabolic Trough Collectors by Using Annular Receivers for Low-Enthalpy Steam Generation

Abstract

1. Introduction

2. Background

3. Heat Transfer for Evaluating the Thermal Performance of a PTC

3.1. Energy Balance in the Receiver Tube

3.1.1. Convective Losses

3.1.2. Radiative Losses

3.1.3. Heat Removal Factor

3.2. Analysis of Internal Flow in Hollow and Concentric Rod

4. Thermodynamic Framework for the Performance of the PTC

4.1. Thermal Efficiency by the First Law of Thermodynamics

4.2. Thermal Efficiency by the Second Law of Thermodynamics

Entropy Generation Number

5. Results and Discussion

5.1. Thermal Efficiency

5.1.1. First Law Thermal Efficiency

5.1.2. Second Law Thermal Efficiency

5.1.3. Entropy Generation Number Results

5.2. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Nomenclature

References

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