Multi-Criteria Decision-Making Method for Simple and Fast Dimensioning and Selection of Glass Tube Collector Type Based on the Iterative Thermal Resistance Calculation Algorithm with Experimental Validation

Abstract

This paper presents an analytical method for the dimensioning and selection of the four glass tube collector types: single-glazed with an air layer, single-glazed with a vacuum layer, double-glazed with an air layer, and double-glazed with a vacuum layer. In the first part of the paper (dimensioning phase), the iterative thermal resistance calculation algorithms were developed for all glass tube collector types, whereby the iterative thermal resistance calculation algorithm of the single-glazed tube collector with an air layer was experimentally tested and validated. The second part of the paper (selection phase) uses a multi-criteria decision-making method to determine the optimal glass tube collector design. Unlike other papers, three indicator groups are taken into account in this case: geometric (mass, surface occupation, total surface occupation, volume occupation), economic (manufacturing and exploitation costs), and ecological (embodied energy and greenhouse gas emission). The proposed method is characterized by simple and fast calculations with satisfactory accuracy, which avoids high investment costs (experimental research), approximation and discretization of physical models (numerical research), and a large number of input parameters with boundary conditions (theoretical research). It should be noted that, with certain additions and changes, it can also be applied to other solar thermal collectors, so the authors believe such tools are handy for the global scientific public.

1. Introduction

Solar technology is constantly developing and is currently expanding. This scientific field includes a large number of different solar constructions, such as solar collectors (SCs), photovoltaic panels (PVs), and hybrid (photovoltaic–thermal) collectors (PVTCs).

Of all SCs, flat-plate collectors (FPCs) and evacuated tube collectors (ETCs) have the greatest practical and commercial applications [1,2] because they are distinguished by a relatively simple manufacturing process, reduced dimensions, satisfactory thermal performance, and relatively low price. In addition, ETCs also offer higher thermal efficiency [3] and working fluid temperature [4] than FPCs, so they can be found almost everywhere: in the residential sector [5], in the commercial (public) sector [6], in the industrial sector [7], etc. In the mentioned sectors, ETCs are used in heating systems [8], cooling systems [9], ventilation systems [10], air conditioning systems [11], hot water systems [12], etc. Some ETC constructions are modified by concentrators [13], reflectors [14], mirrors [15,16], and tracking systems [17,18,19] accompanied by a large number of experimental [20], numerical [21], and theoretical (analytical or mathematical) [22] investigations.

The hybrid configuration with ETCs is presented in [23]. The experimental investigation revealed that the solar thermal collector’s maximum outlet temperature, efficiency, and maximum cooling capacity were 87 °C, 56%, and 4.6 kW, respectively. In [24], a comparative study was carried out on the performance of an individual SC under laboratory conditions and field conditions (experimental study). A compound parabolic concentrator (CPC) with an ETC has been used to apply boiler feed-water preheating. The performance monitoring and evaluation of the actual installation were carried out over 10 months. The system was designed to supply an average of 448,500 kWh-thermal/year, saving the equivalent amount of fossil fuel. The ETC is developed in [25] to achieve high heating-medium temperature. They used 10 mol% of ethylene glycol in water as the heat transfer fluid. The results showed that the average temperature of the heat transfer fluid in a modified tube increased to 160.32 °C, with an efficiency increase of 34.96%.

Kaya et al. numerically investigated an ETC with a U-tube [26]. The thermal performance of the analyzed SC was simulated for different operating conditions. They used higher thermal conductivity working fluids: Ag, ZnO, and MgO nanoparticles in 30%:70% (by volume) ethylene glycol/pure water mixture and different nanoparticle volumetric concentrations.

For example, a new ETC construction with inner concentrating, which has 10% higher thermal efficiency than classic ETCs, was proposed, fabricated, and presented in [27]. Additionally, the mathematical model of the mentioned solar collector was developed. Ma et al. [28] mathematically considered a double-glazed ETC with a U-tube, where the absorber film is deposited on the outer surface of the absorber tube.

In the review paper presented in [29], Chopra et al. analyzed ETCs with direct flow and heat pipes, providing comprehensive theoretical analysis. The review paper [30] focused on air as a working fluid in ETCs and summarized design configurations and simulation works, using nanofluids and phase change materials to understand their influence on the thermal performance of these ETCs.

The state of the market, along with the available literature, shows that many ETC constructions never progress beyond the case-study phase to the commercial phase. The reasons include complex technological manufacturing procedures, costly experimental investigations, use of scarcely available materials, high manufacturing costs, high final product prices, and difficult transportation (including assembly and maintenance). Some solutions are much larger in terms of surface occupancy and volume occupancy [31] compared to the basic versions of solar collectors—indicators that authors often ignore in their works. Numerical and theoretical investigations intended to describe the functioning principles of such solar installations are further complicated by various problems: approximation and discretization of physical models, selection and description of physical and chemical phenomena, definition of properties of working fluids, determination of specific boundary conditions, and inclusion of a large number of influential parameters, among others. It has been noted that ecological indicators (embodied energy and greenhouse gas emissions) are also often neglected despite their importance for green technology, which is emphasized globally today [32,33,34].

Due to all of the above, this paper presents a simple and fast analytical method for the dimensioning and selection of glass tube collectors (GTCs) with direct water flow as the working fluid. The simplicity of the method is reflected in the development of a mathematical model for calculating thermal resistance in GTCs (based on an iterative calculation algorithm), regardless of the number and performance of both the glass tubes (GTs) and the interspace. The mathematical model was applied to the following types of GTCs: single-glazed tube collector (SGTC) with an air layer, SGTC with a vacuum layer, double-glazed tube collector (DGTC) with an air layer, and DGTC with a vacuum layer. The mathematical model for the SGTC with an air layer has been experimentally tested and verified using measurement data. Finally, the multi-criteria decision-making (MCDM) method was used to determine the optimal design of the GTC. This method is rarely used in such studies. For this purpose, four geometric criteria (mass, surface occupancy, total surface occupancy, volume occupancy), two economic criteria (manufacturing costs, exploitation costs), and two ecological criteria (embodied energy, greenhouse gas emissions) were used. The aim of this paper is to highlight the importance of evaluating GTC performance from several different angles. Only then can sustainable development be ensured, as well as energy, economic, and environmental security.

2. Research Area

This section delineates four configurations of glass tube collectors (GTCs) (Figure 1); they are:

• SGTC with air layer (collector type S1);
• SGTC with a vacuum layer (type S2 collector);
• DGTC with air layer (collector type S3);
• DGTC with a vacuum layer (collector type S4).

The basic structural and functional unit of all GTCs is the aluminum flat absorber plate (ABS) with an integrated circular cross-section flow channel (Figure 1). A selective SnAl₂O₃ coating is used to enhance the optical characteristics (absorptivity and emissivity [19]) of the ABS. The internal glass tube (IGT) is a common component for both SGTCs and DGTCs, while the external glass tube (EGT) is characteristic only of DGTCs. The space between the ABS and the IGT is filled with an air layer for the S1 collector type, whereas a vacuum layer is used for the S2 collector type. In the S3 collector type, two air layers are present: one between the ABS and the IGT and another between the IGT and the EGT. In the S4 collector type, these two air layers are replaced with two vacuum layers.

The geometric characteristics of the GTCs are shown in Figure 2. As illustrated, the width of the ABS is 100 mm, and its thickness is 2 mm. The diameter of the cross-section flow channel is Ø15 mm. The thickness of both the IGT and EGT is 3 mm. The distance between them (in the case of DGTCs) is 6 mm. The lengths of all components (ABS, IGT, and EGT) are 800 mm. The optical–thermal characteristics of the GTCs are presented in

Table 1. Geometric and thermal characteristics of the GTCs [19,35,36].

Component	Material	ρ	cp	k	ε	L
ABS	Aluminum	2700	900	203	0.25	0.8
IGT	Glass	2200	660	0.8	0.9
EGT

where ρ [kg/m³] is the density, c_p [J/(kgK)] is the specific heat, k [W/(mK)] is the thermal conductivity, ε [-] is the emissivity, and L [m] is the length.

.

3. Materials and Methods

To effectively analyze and optimize the performance of glass tube collectors (GTCs), it is crucial to understand the mechanisms of heat loss. Heat losses (Q_loss) in GTCs can be described using a thermal resistance model that accounts for various components of resistance along the heat transfer path. This model enables precise calculation of the amount of heat lost by the collector, which is essential for improving its energy efficiency.

3.1. General Thermal Resistance Model for GTCs

Heat losses Q_loss [W] in the GTCs can be described as Equation (1) [37]:

$$Q_{loss}={\displaystyle \frac{T_{abs}-T_o}{{\displaystyle \sum R_{loss}}}}={\displaystyle \frac{T_{abs}-T_o}{{\displaystyle \sum R_{loss,\perp }}+{\displaystyle \sum R_{loss,\angle }}}}\approx {\displaystyle \frac{T_{abs}-T_o}{{\displaystyle \sum R_{loss,\perp }}}}$$

(1)

where T_abs [K] is the absorber temperature, T_o [K] is the ambient temperature, ΣR_loss [K/W] is the total resistance to heat transfer [35], ΣR_loss,⊥ [K/W] is the total resistance to heat transfer in the transverse plane, and ΣR_loss,∠ [K/W] is the total resistance to heat transfer in the longitudinal (resistance to heat transfer in the longitudinal plane is negligibly small compared to resistance to heat transfer in the transverse plane [19]) plane.

3.1.1. Thermal Resistance Model for SGTCs

In this case (for SGTCs), the total thermal resistance to heat transfer (in the transverse plane) can be calculated as Equation (2) [38]:

$${\displaystyle \sum R_{loss}}\approx {\displaystyle \sum R_{loss,\perp }}\approx R_{abs\text{-}igt}+R_{igt}+R_{igt\text{-}o}$$

(2)

where R_abs-igt [K/W] is the resistance to heat transfer between the absorber and the internal glass tube, R_igt [K/W] is the resistance to heat transfer through the internal glass tube, and R_igt-o [K/W] is the resistance to heat transfer between the internal glass tube and the ambient air.

The next figure (Figure 3) illustrates the heat transfer and resistance network for the two analyzed SGTC types: SGTC with an air layer (Figure 3a) and SGTC with a vacuum layer (Figure 3b). For the SGTC with an air layer, thermal energy (heat losses) between the ABS and the IGT is exchanged through radiation and convection. In contrast, for the SGTC with a vacuum layer, the convective component is eliminated.

All values presented in Equation (2) are calculated using the simple and double iterative calculation algorithm (calculation method developed for this study). Figure 4 shows an iterative calculation algorithm for SGTC with an air layer.

$$h_{rad(abs\text{-}igt)}={\displaystyle \frac{\left(T_{abs}^2+T_{igt}^2\right) \left(T_{abs}+T_{igt}\right)}{\frac{1}{\sigma {\epsilon }_{abs}}+\frac{A_{abs,e,tot}}{A_{igt,i,tot}}\left(\frac{1}{\sigma {\epsilon }_{igt}}-\frac{1}{\sigma }\right)}}$$

(3)

$$h_{conv(abs\text{-}igt)}={\displaystyle \frac{N{u}_{ia}{k}_{ia}}{D_{igt,i,eq}}}$$

(4)

$$h_{rad(igt\text{-}o)}=\sigma {\epsilon }_{igt}\left(T_{igt}^2+T_o^2\right) \left(T_{igt}+T_o\right)$$

(5)

$$h_{conv(igt\text{-}o)}=2.8+3W$$

(6)

where [39,40] σ [W/(m²K⁴)] is the Stefan–Boltzmann constant, A_igt,i,tot [m²] is the total internal area of the internal glass tube, Nu_ia [-] is the Nusselt number of the internal air layer [41,42,43,44], k_ia [W/(mK)] is the thermal conductivity of the internal air layer, D_igt,i,eq [m] is the equivalent diameter of the internal air layer, and W [m/s] is the wind speed.

Before entering the calculation algorithm, two data groups must be adopted: SGTC geometric, thermal, and optical characteristics (Chapter 2)—first group, and meteorological parameters (values T_o and W)—second group. After that, the first stage (step I) of calculation is approached—adopting initial values of T_abs and T_igt. In step II, Equations (3)–(6) are used to determine all components in Equation (2) and then the total heat losses from Equation (1). In step III, the control q_abs-igt and q_igt-o equations are used to check the agreement of the adopted T_abs and T_igt values with the obtained results q_abs-igt ≈ q_igt-o (step IV) [45]. If the control heat flux results were approximately equal (q_abs-igt ≈ q_igt-o, i.e., deviation ≤ 0.1 W/m²), the iterative procedures would be completed, and the results would be finally printed (step V).

3.1.2. Thermal Resistance Model for DGTCs

The total thermal resistance to heat transfer for DGTCs is given by Equation (7) [38]:

$${\displaystyle \sum R_{loss}}\approx {\displaystyle \sum R_{loss,\perp }}\approx R_{abs\text{-}igt}+R_{igt}+R_{igt\text{-}egt}+R_{egt}+R_{egt\text{-}o}$$

(7)

where R_igt-egt [K/W] is the resistance to heat transfer between the internal and external glass tubes, R_egt [K/W] is the resistance to heat transfer through the external glass tube, and R_egt-o [K/W] is the resistance to heat transfer between the external glass tube and the ambient air.

On the one side, values R_abs-igt and R_igt-egt could be a function of the rad(abs-igt) and conv(abs-igt), that is, in the function of the rad(igt-egt) and conv(igt-egt)—for air layers (Figure 5a). On the other side, for vacuum layers (Figure 5b), the same values are only in function of the rad(abs-igt) and rad(igt-egt), respectively.

Due to the larger number of boundary (glass) surfaces in the construction of DGTCs, the iterative calculation algorithm should satisfy one more criterion—based on a triple check (calculation method developed for this study). For example, for triple-glazed tube collectors (TGTCs), the iterative calculation algorithm would be based on a quadruple check. In Figure 6, the triple iterative calculation algorithm for a double-glazed tube collector (DGTC) with an air layer is shown. This algorithm incorporates various parameters and coefficients, including heat transfer coefficients and areas, which are essential for accurate thermal analysis. Each coefficient and parameter depicted in the figure plays a crucial role in determining the overall thermal performance of the DGTC.

The figure illustrates the calculation process involving various heat transfer coefficients and areas. Specifically, h_rad(igt-egt) [W/(m²K)] is the radiation heat transfer coefficient between the internal and the external glass tubes (Equation (8)), h_{conv(igt-egt)} [W/(m²K)] is the convection heat transfer coefficient between the internal and external glass tubes (Equation (9)), A_egt,avg,tot [m²] is the total average area of the external glass tube, D_egt,e [m] is the external diameter of the external glass tube, D_egt,i [m] is the internal diameter of the external glass tube, A_egt,e,tot [m²] is the total external area of the external glass tube, h_rad(egt-o) [W/(m²K)] is the radiation heat transfer coefficient between the external glass tube and the ambient air (Equation (10)), h_conv(egt-o) [W/(m²K)] is the convection heat transfer coefficient between the external glass tube and the ambient air (Equation (11)), q_igt-etg [W/m²] is the specific heat flux exchanged between the internal and external glass tubes, and q_etg-o [W/m²] is the specific heat flux exchanged between the external glass tube and the ambient air.

$$h_{rad(igt\text{-}egt)}={\displaystyle \frac{\left(T_{igt}^2+T_{egt}^2\right) \left(T_{igt}+T_{egt}\right)}{\frac{1}{\sigma {\epsilon }_{igt}}+\frac{A_{igt,e,tot}}{A_{egt,i,tot}}\left(\frac{1}{\sigma {\epsilon }_{egt}}-\frac{1}{\sigma }\right)}}$$

(8)

$$h_{conv(igt\text{-}egt)}={\displaystyle \frac{N{u}_{ea}{k}_{ea}}{D_{egt,i,eq}}}$$

(9)

$$h_{rad(egt\text{-}o)}=\sigma {\epsilon }_{egt}\left(T_{egt}^2+T_o^2\right) \left(T_{egt}+T_o\right)$$

(10)

$$h_{conv(egt\text{-}o)}=2.8+3W$$

(11)

where [39,40] Nu_ea [-] is the Nusselt number of the external air layer [41,42,43,44], k_ea [W/(mK)] is the thermal conductivity of the external air layer, and D_igt,i,eq [m] is the equivalent diameter of the external air layer.

In step I of the triple iterative calculation algorithm (Figure 6), the initial values are T_abs, T_igt, and T_egt. All Equation (7) values are determined in step II using Equations (3), (4), and (8)–(11). The control equations (step III) now are q_abs-igt, q_igt-egt, and q_egt-o [45]. When the control heat flux results were approximately equal (q_abs-igt ≈ q_igt-egt ≈ q_egt-o, with the same deviation, step IV), the calculation would be finished (step V).

3.2. Experimental Model

Experimental validation of the analytical (mathematical) model was performed on the example of the SGTC with an air layer (S1 collector type, Chapter 2). The isometric view of the mentioned SC is shown in Figure 7a, while the complete experimental installation with the measuring chain is shown in Figure 7b.

The experimental installation with a measuring chain (Figure 7b) is formed by two subsystems: solar construction (Figure 7a) and a meteorological station. The experimental installation operation principle with open circulation circuits (without circulation pumps) is based on hydrostatic pressure—the driving force for water flow through the SC. Accompanying measuring equipment for collecting appropriate parameters (solar irradiance intensity on a horizontal surface, ambient air temperature, wind speed, water inlet temperature, water outlet temperature, and water mass flow rate) required for thermal analysis consisted of a Kipp & Zonen SMP3 pyranometer (accuracy < 5%), a Kipp & Zonen data logger METEON (accuracy < 0.1%), WZP-035 Ø5 × 50 mm Pt-100 temperature probes (accuracy ± 0.2 °C), and a HERZ STROMAX manual control valve [19].

Experimental Q_loss value was measured indirectly by Equation (12):

$$Q_{loss}=Q_{sun}-Q_{heat}=A_{abs(up)}{I}_{TOT}-{\dot{m}}_w{c}_p\left(T_{w,out}-T_{w,in}\right)$$

(12)

where Q_sun [W] is the solar heat power, Q_heat [W] is the heat power of the SGTC with an air layer, A_abs(up) [m²] is the upside area of the absorber, I_TOT [W/m²] is the total incident solar irradiance (for the tilted surfaces, the value I_TOT is mathematically calculated according to the recommendations from [43]), ṁ_w [kg/s] is the mass flow rate of the water, T_w,out [K] is the outlet temperature of the water, and T_w,in [K] is the inlet temperature of the water.

3.3. Multi-Criteria Decision-Making Method

The optimal solar design (dimensioning and selection) of the presented GTCs (Figure 1) is determined using three groups of indicators: geometric (mass m [kg], surface occupation SO [m²], total surface occupation TSO [m²], volume occupation VO [m³]), economic (manufacturing cost CM [EUR] and exploitation cost CE [EUR]), and ecological (embodied energy E_emb [kWh] and greenhouse gas emissions e_CO2 [kg]).

A multidisciplinary evaluation of GTC performance can be conducted using various multi-criteria analysis (MCA) methods [47]. Due to its simplicity and accuracy, the multi-criteria decision-making (MCDM) method with simple additive weighting (SAW) was chosen for this paper [48,49].

The MCDM method assigns a specific percentage criterion of importance, i.e., weight of significance (w_c [%] and w_sc [%]), to all indicators used (

Table 2. The weight of importance to all used indicators.

Criterion	Geometric				Economic		Ecological		C-sum
wc	14				50		36		100
Sub-criterion	m	SO	TSO	VO	CM	CE	Eemb	eCO2	SC-sum
wsc	5	5	2	2	35	15	18	18	100

).

By defining w_c and w_sc values, all included indicators (in this case, eight of them, as shown in Table 2) are normalized to a dimensionless level. Only then can their classification in descending or ascending order, depending on the desired objective (the objective function), be performed. In this case, the objective function is minimizing heat losses in GTCs. The following figure (Figure 8) illustrates the procedure for selecting the mentioned objective function.

In the first two steps, numerous values of all sub-criteria, i.e., indicators (k₁ values: m, SO, TSO, VO, CM, CE, E_emb, and e_CO2) and specific indicators (k₂ values: Q_loss/m, Q_loss/SO, Q_loss/TSO, Q_loss/VO, Q_loss/CM, Q_loss/CE, Q_loss/E_emb, and Q_loss/e_CO2), are defined for all GTC types. In the third step, k₃ values are calculated as in Equation (13), while in the fifth step, values k₅ are a function of k₂, k₃, and k₄, as shown Figure 8.

$$k_3={\left({\displaystyle \frac{Q_{loss}}{m}}\right)}_{SGTCa}+{\left({\displaystyle \frac{Q_{loss}}{m}}\right)}_{SGTCv}+{\left({\displaystyle \frac{Q_{loss}}{m}}\right)}_{DGTCa}+{\left({\displaystyle \frac{Q_{loss}}{m}}\right)}_{DGTCv}$$

(13)

Equation (13) represents the sum of Q_loss/m specific indicators for an SGTC with an air layer, SGTC with a vacuum layer, DGTC with an air layer, and DGTC with a vacuum layer. Based on this equation, the remaining seven equations can also be developed (for the other specific indicators).

Determining the k₅ values in the fifth step is shown in the example of the SGTC with an air layer taking into account specific indicator Q_loss/m (Equation (14)):

$$k_5=k_4{\displaystyle \frac{k_2}{k_3}}=0.05{\displaystyle \frac{{\left(\frac{Q_{loss}}{m}\right)}_{SGTCa}}{{\left(\frac{Q_{loss}}{m}\right)}_{SGTCa}+{\left(\frac{Q_{loss}}{m}\right)}_{SGTCv}+{\left(\frac{Q_{loss}}{m}\right)}_{DGTCa}+{\left(\frac{Q_{loss}}{m}\right)}_{DGTCv}}}$$

(14)

In the final stage, the values obtained from the fifth step are classified (sixth step, k₆ values) to identify the minimum value, which represents the objective function. This process concludes the calculation.

The subsequent table (

Table 3. Geometric, economic, and ecological indicators for analyzed GTC constructions [50,51,52].

Main Components
Element	V	m	SO	TSO	VO	CM	CE	Eemb	eCO2
ABS	0.000253	0.683	0.08	0.184	0.00049	25	13	1.81	-
IGT	0.000807	1.775	0.088	0.276	0.0076	76	38	2.29	-
EGT	0.000943	2.075	0.1024	0.322	0.0103	91	46	2.68	-
Glass tube collector types
Model		m	SO	TSO	VO	CM 1	CE 2	Eemb	eCO2 3
SGTC with an air layer		2.458	0.088	0.276	0.0076	101	51	4.1	1303
SGTC with a vacuum layer						111	56		809
DGTC with an air layer		4.533	0.1024	0.322	0.0103	192	96	6.78	2215
DGTC with a vacuum layer						221	111		1418

¹ The CM [EUR] for SGTC with a vacuum layer increase by 10% compared to SGTC with an air layer. The same costs for DGTC with a vacuum layer increase by 15% compared to DGTC with an air layer. ² In all cases, the exploitation costs CE [EUR] during the 20 years (adopted working time) are 50% of the CM. The same time period is taken for the calculation of ecological indicators E_emb and e_CO2. ³ CO₂ emission cannot be determined without knowing the E_loss value (Equation (16), Chapter 4).

) presents the adopted geometric indicators (as per the data in Table 1), along with the economic and ecological indicators. The ecological indicators include embodied energy (Equation (15)) and greenhouse gas emissions (Equation (16)) [50] for the analyzed GTC constructions.

$$E_{emb}={\displaystyle \frac{m{E}_{emb,spec}}{20}}$$

(15)

$$e_{CO2}=e_{CO2,spec}{E}_{pry}=e_{CO2,spec}{K}_{pry}{E}_{fin}=e_{CO2,spec}{K}_{pry}{\displaystyle \frac{E_{loss}time}{1000{\eta }_{el}}}$$

(16)

where [47] E_emb,spec [kWh/kg] is the specific embodied energy (E_emb,spec = 53 kWh/kg for aluminum [51], and E_emb,spec = 25.8 kWh/kg for glass [52]), e_CO2,spec [kg/kWh] is the specific CO₂ emission (For electricity [50], e_CO2,spec = 0.53 kg/kWh, while K_pry = 2.5), E_pry [kWh] is the primary energy consumption, K_pry [-] is the primary energy transformation coefficient¹¹, E_fin [kWh] is the final energy consumption, time [h] is the working time⁸, η_el [-] is the efficiency of the electric boiler (η_el = 0.98 for electric boiler).

Equation (16) is used when thermal energy needs to be provided from another energy source. In this case, it is an electric boiler because it is one of Serbia’s most common heat-energy generators [50].

4. Results and Discussion

Figure 9 shows Q_loss and t_igt in the SGTC with an air layer (Figure 9a) and SGTC with a vacuum layer (Figure 9b) for previously adopted t_abs, t_o and W values.

All diagrams (Figure 9) were formed by varying the following three values: t_abs (40 °C, 50 °C, 60 °C, 70 °C, 80 °C and 90 °C)—first variable, t_o (10 °C, 20 °C, and 30 °C)—second variable, and W (1 m/s, 3 m/s, and 5 m/s)—third variable.

Figure 9 shows that the heat losses in the SGTCs (SGTC with an air and SGTC with a vacuum layer) increase with the increase in the temperature of the absorber and the increase in the wind speed. A higher absorber temperature increases the temperature difference with the environment, so the driving force of the heat losses increases. When the wind is faster, convective component Equation (6) is higher, and heat losses increase. Wind speed also determines the temperature of the glass (these values are proportional). Contrary to that, higher ambient temperature causes the opposite effect. When the ambient temperature increases, the temperature difference between the environments decreases [53,54].

When t_o = 10 °C and W = 1 m/s (Figure 9a), heat losses in the SGTC with an air layer are between 17.17 W (for t_abs = 40 °C) and 56.27 W (for t_abs = 90 °C). The heat losses are higher 8.54–9.25% (W = 3 m/s) and 13.17–14.49% (W = 5 m/s) for the same ambient temperature. If the second variable is t_o = 20 °C, the heat losses are within the following limits: 11.14–49.37 W (W = 1 m/s), 12–53.65 W (W = 3 m/s), and 12.47–56.11 W (W = 5 m/s). Maximum (for t_abs = 90 °C) heat loss values are the lowest when t_o = 30 °C: 42.37 W (W = 1 m/s), 45.82 W (W = 3 m/s), and 47.81 W (W = 5 m/s). The highest and lowest internal glass tube temperatures for the SGTC with an air layer are (Figure 9a): t_igt = 47.5 °C (for t_o = 30 °C, t_abs = 90 °C, and W = 1 m/s), and t_igt = 14.49 °C (for t_o = 10 °C, t_abs = 40 °C, and W = 5 m/s).

The calculation algorithm (Figure 4) for the SGTC with a vacuum layer is simpler due to the elimination of the convective component of Equation (4). The same effect explains the reduction of the heat losses (Figure 9b). For example, from 64.42 W to 27.11 W (t_o = 10 °C, t_abs = 90 °C, and W = 5 m/s) is the best case scenario. In this case, the value t_igt is in the following range: t_igt = 11.87 °C (for t_o = 10 °C, t_abs = 40 °C, and W = 5 m/s) and t_igt = 39.3 °C (for t_o = 30 °C, t_abs = 90 °C, and W = 1 m/s).

Analogous to the previous figure (Figure 9), the next figure (Figure 10) shows Q_loss and t_igt in the DGTC with an air layer (Figure 10a) and DGTC with a vacuum layer (Figure 10b) for previously adopted t_abs, t_o and W values.

When it comes to the DGTCs (DGTC with an air layer and DGTC with a vacuum layer), the conclusions are the same as in the case of the SGTCs (Figure 10): heat losses are proportional to the absorber temperature, heat losses are proportional to the wind speed, and heat losses are inversely proportional to the ambient temperature.

For the DGTC with an air layer (Figure 10a), heat losses are the highest (Q_loss = 47.14 W) when t_o = 10 °C, t_abs = 90 °C, and W = 5 m/s and the smallest (Q_loss = 12.65 W) when t_o = 10 °C, t_abs = 40 °C, and W = 1 m/s. The internal glass tube temperatures are between t_igt = 23.4 °C (t_o = 10 °C, t_abs = 40 °C, and W = 5 m/s) and t_igt = 59.8 °C (t_o = 30 °C, t_abs = 90 °C, and W = 1 m/s). In contrast, the external glass tube temperatures are between t_egt = 12.8 °C and t_egt = 42.3 °C (the same boundary conditions apply as for t_igt).

The highest Q_loss = 23.35 W (t_o = 10 °C, t_abs = 90 °C, and W = 5 m/s) and the smallest Q_loss = 6.48 W (t_o = 10 °C, t_abs = 40 °C, and W = 1 m/s) heat losses in the DGTC with a vacuum layer can be seen in Figure 10b. Compared with the DGTC with an air layer, heat losses in the DGTC with a vacuum layer are on average 1.92 times lower (maximum is 2.07 times, minimum is 1.78 times). The internal glass tube temperatures are between t_igt = 18.6 °C and t_igt = 51.8 °C. The external glass tube temperatures are between t_egt = 11.4 °C and t_egt = 37.4 °C.

The diagrams in Figure 10 are interesting because of another phenomenon. Namely, it can be seen that the analytical calculation results are missing when t_o = 20 °C (t_abs = 40 °C). They are also missing when t_o = 30 °C (t_abs = 40 °C and t_abs = 50 °C). This anomaly is characteristic of the triple iterative calculation algorithm for DGTCs (Figure 6). As the ambient temperature increases, the convergence of results for values of tabs close to that is impossible. Temperatures t_abs, t_igt, and t_egt are approaching, while the control equations tend to divagation. It proves that DGTCs operate at higher operating temperatures than SGTCs.

Experimental validation of the heat losses analytical model for the SGTC with an air layer is shown in Figure 11 (for t_o = 20 °C) and Figure 12 (for t_o = 30 °C).

Experimental heat losses were determined using Equation (12), as previously described. The Q_loss values were obtained from a four-month measurement period (15 July to 15 October 2021), ensuring that the experimental values for t_o and W align with the corresponding values used in the mathematical (analytical) model. This alignment provided the necessary conditions to validate the model for initial research on glass tube collectors (GTCs). The validated model is designed to function without requiring additional inputs, such as solar radiation intensity or mass flow rate, while maintaining accuracy within the desired limits.

As illustrated in Figure 11, the experimental data points grouped between t_abs = 70 °C and t_abs = 90 °C (W = 1 m/s and W = 3 m/s) and between t_abs = 60 °C and t_abs = 90 °C (W = 2 m/s). Similar phenomena are shown in Figure 12. If it is known that the absorber temperature in classic single-glazed solar collectors is between 30 and 80 °C [55], it can be concluded that the presented method has accuracy within satisfactory limits. For more detailed results, it is necessary to develop mathematical models with all the specificities that characterize the solar construction.

After the theoretical analysis of the thermal performance of GTCs and experimental validations, the results of the MCDM method will be presented in detail below to determine the optimal design of GTCs.

Following Figure 9 and Table 3,

Table 4. Specific geometric, economic, and ecological indicators.

Specific Indicator	SGTC		DGTC		Sum
	Air Layer	Vacuum Layer	Air Layer	Vacuum Layer
${\displaystyle \frac{Q_{loss}}{m}}$	7.84	4.87	7.23	4.63	24.56
${\displaystyle \frac{Q_{loss}}{SO}}$	218.98	136.02	319.92	204.88	879.8
${\displaystyle \frac{Q_{loss}}{TSO}}$	69.82	43.37	101.74	65.16	280.08
${\displaystyle \frac{Q_{loss}}{VO}}$	2535.53	1575	3180.58	2036.89	9328
${\displaystyle \frac{Q_{loss}}{CM}}$	0.19	0.11	0.17	0.09	0.56
${\displaystyle \frac{Q_{loss}}{CE}}$	0.38	0.21	0.34	0.19	1.12
${\displaystyle \frac{Q_{loss}}{E_{emb}}}$	4.7	2.92	4.83	4.48	16.93
${\displaystyle \frac{Q_{loss}}{e_{CO2}}}$	0.01	0.01	0.01	0.01	0.06

defines specific geometric, economic, and ecological indicators for optimal GTC type selection (k₂ values). In this phase (for t_o = 20 °C and W = 3 m/s), the values of heat losses (Q_loss with e_CO2, Equation (16), Table 3) were used when t_abs = 50 °C (SGTC with an air layer), t_abs = 60 °C (SGTC with a vacuum layer), t_abs = 80 °C (DGTC with an air layer), and t_abs = 90 °C (DGTC with a vacuum layer)—typical average values (Figure 9, Figure 10). In the same table (Table 4), the last column provides an insight into the values k₃ (Equation (13)).

Values of k₄ can be seen in Table 2. Values of k₅ (Equation (14)) and k₆ (ranking of indicators) from Figure 8 are shown in

Table 5. Ranking of the specific geometric, economic, and ecological indicators.

Specific Indicator	SGTC		DGTC		Rank SGTC with a Vacuum Layer
	Air Layer	Vacuum Layer	Air Layer	Vacuum Layer
${\displaystyle \frac{Q_{loss}}{m}}$	0.0160	0.0099	0.0147	0.0094	2
${\displaystyle \frac{Q_{loss}}{SO}}$	0.0124	0.0077	0.0182	0.0116	1
${\displaystyle \frac{Q_{loss}}{TSO}}$	0.0050	0.0031	0.0073	0.0047	1
${\displaystyle \frac{Q_{loss}}{VO}}$	0.0054	0.0034	0.0068	0.0044	1
${\displaystyle \frac{Q_{loss}}{CM}}$	0.1184	0.0669	0.1058	0.0589	2
${\displaystyle \frac{Q_{loss}}{CE}}$	0.0505	0.0286	0.0456	0.0253	2
${\displaystyle \frac{Q_{loss}}{E_{emb}}}$	0.0500	0.0310	0.0514	0.0477	1
${\displaystyle \frac{Q_{loss}}{e_{CO2}}}$	0.0450	0.0450	0.0450	0.0450	1

. The specific indicators with the lowest values are a goal function because the aim is to minimize heat losses in GTCs.

The final ranking of glass tube collectors (GTCs), derived using the multi-criteria decision-making (MCDM) method, is detailed in

Table 6. Final ranking of GTCs.

Specific Indicator	SGTC		DGTC
	Air Layer	Vacuum Layer	Air Layer	Vacuum Layer
Final results	0.30266	0.19563	0.29481	0.20690
Final ranking	4	1	3	2

. The analysis reveals that simpler designs, such as the single-glazed tube collector (SGTC) with a vacuum layer, frequently emerge as superior options based on the primary geometric, economic, and ecological criteria (refer to Table 6). The double-glazed tube collector (DGTC) with a vacuum layer ranks second in the evaluation. Conversely, the single-glazed tube collector (SGTC) with an air layer occupies the lowest position according to the MCDM method results.

Vacuum technology leads to increased embodied energy investments and, consequently, higher production costs. This chain reaction also results in higher maintenance costs. However, it significantly reduces heat losses by 37.88% compared to single-glazed tube collectors (SGTCs) with an air layer and by 35.96% compared to double-glazed tube collectors (DGTCs) with an air layer.

Heat losses in solar collectors represent the portion of energy that is not utilized and is instead dissipated into the environment. Although more advanced solar collector designs often exhibit higher thermal (useful) power, this does not necessarily imply a reduction in heat losses or an increase in thermal efficiency. In many cases, as the thermal power of solar collectors increases, so do the heat losses, indicating a greater amount of unused energy. Consequently, such technical challenges must be addressed with careful, multidisciplinary approaches.

5. Conclusions

In the first part of this paper, the heat losses of four types of glass tube collectors (GTCs) are analyzed mathematically: the single-glazed glass tube collector with an air layer, the single-glazed glass tube collector with a vacuum layer, the double-glazed glass tube collector with an air layer, and the double-glazed glass tube collector with a vacuum layer. Appropriate mathematical models and calculation algorithms were developed, employing double iteration for single-glazed collectors and triple iteration for double-glazed collectors. The results indicate that the minimum and maximum heat loss values Q_loss are as follows: 5.29 W and 64.42 W for the single-glazed collector with an air layer, 2.74 W and 27.11 W for the single-glazed collector with a vacuum layer, 12.65 W and 47.14 W for the double-glazed collector with an air layer, and 6.48 W and 23.35 W for the double-glazed collector with a vacuum layer, depending on the ambient temperature (t_o) and wind speed (W). It was concluded that heat losses are proportional to the absorber temperature and wind speed while inversely proportional to the ambient temperature.

In the second part of the paper, the theoretical results of heat losses are compared with experimental measurements obtained over a four-month period. The experimental values, which align with the theoretical values, were assessed according to two criteria: ambient temperature and wind speed. The agreement between the theoretical and experimental models indicates that the developed calculation algorithms are validated experimentally.

In the third part of the paper, three primary groups of indicators, along with their corresponding sub-indicators, are defined to determine the optimal design of GTCs using the multi-criteria decision-making (MCDM) method with simple additive weighting (SAW). These groups include geometric indicators (mass, surface occupation, total surface occupation, volume occupation), economic indicators (production costs and exploitation costs), and ecological indicators (embodied energy and greenhouse gas emissions). The multidisciplinary analysis reveals that the single-glazed collector with a vacuum layer is the most favorable option, while the single-glazed collector with an air layer is the least favorable.

For example, the literature often neglects the inclusion of economic and ecological factors in assessing the thermal performance of solar collectors, which challenges the justification of many solar applications.

The primary objective of the proposed methodology is to offer a novel perspective for future research on the shortcomings of solar collectors and to underscore the significance of a comprehensive approach. The paper aims to provide future research with a new viewpoint on these issues by applying a simple and efficient multidisciplinary method.