Mathematical Modeling and Optimization of Vanadium-Titanium Black Ceramic Solar Collectors

Abstract

The vanadium-titanium black ceramic (VTBC) coating on all-ceramic solar collectors has both high absorptance (0.94) and high emissivity (90%). However, the thermal conductivity of ceramic is very low (1.256 W/mK). To improve the heat collection efficiency of VTBC solar collectors, this paper establishes a mathematical model based on the energy-conservation relationships under steady-state conditions and creates a corresponding computer program. Key parameters for VTBC solar collectors include the heat-removal factor, effective transmittance-absorptance product for the absorber, total heat loss coefficient, etc. Then, via experimental testing, this study proposes a reference model for domestic VTBC solar collectors in a cold location (η = 0.89 − 2.20T_m*). Last, this work analyzes the influences of fin design and transparent cover design on VTBC solar collectors individually, using the created computer program. Results show that the most effective optimization method is to increase the transmittance of the transparent cover. By increasing the transmittance from 0.93 to 0.96, this study creates an optimized VTBC solar collector theoretical model (η = 0.92 − 2.20T_m*).

1. Introduction

To solve the problems of environmental pollution, fossil-fuel energy exhaustion, and climate warming, scholars have attempted to create ways to use renewable and sustainable energies in buildings. Solar energy is considered the alternative energy source with the greatest potential to replace conventional fossil-fuel energy [1,2]. At present, the application of conventional evacuated-tube collectors and metal-plate solar collectors has three main problems [3,4,5,6]. First, the combination of collectors and buildings is still not adequate in terms of visual esthetics or structure [7,8]. Second, the service life of conventional collectors is only 15–20 years, which does not match the long life of buildings [9]. For an evacuated-tube collector, the primary factors determining its service life are the destruction of the vacuum space and the damage to the glass, while for a metal absorber collector, efficiency attenuation is usually caused by the deterioration of the absorptive coating [10,11]. Third, because the main disadvantage of solar energy collection is related to distribution, large-scale applications of solar energy remain expensive. Therefore, the cost of conventional collectors is difficult to reduce.

Ceramics are inexpensive civil engineering materials that have been widely used for a long time. Such materials possess good thermal properties and temperature-stress stability and are suitable raw materials for solar collectors. To overcome the abovementioned shortcomings of conventional collectors, vanadium-titanium black ceramic (VTBC) solar collectors were invented in 2006 [12]. The appearance and a partial section of a typical VTBC solar absorber, the core component of a collector, are shown in Figure 1. However, there are also disadvantages of VTBC solar collectors. On the one hand, the emissivity of the absorber coating, made from vanadium extraction tailings, is as high as approximately 90% [13]. On the other hand, the thermal conductivity of ceramic is as low as approximately 1.3 W/mK, only approximately 1/300 that of copper [14]. These features of the materials have negative effects on the performance of VTBC solar collectors.

Compared with other solar collectors, VTBC solar collectors are roughly equivalent in terms of collection efficiency. For example, based on previous studies, the relationship between collection efficiency (η) and irradiation intensity (I) of a typical VTBC collector, as illustrated above, is shown in Figure 2. However, the efficiencies are not only a function of irradiance but also influenced by other ambient conditions and system designs. A literature review showed that the efficiency values of VTBC collectors obtained under laboratory conditions range from 39% to 65% [15,16,17,18]. Moreover, VTBC collectors have been used in construction projects, such as residences, greenhouses, and educational buildings (Figure 3). Under these practical conditions, the efficiency values range from 24% to 57% [12,19,20,21,22,23]. Among these, more than 80% of the testing results are within the range of 40–50%.

However, previous studies have mostly focused on the manufacture or application of VTBC collectors, rather than deeply investigating their thermal mechanism [24]. Beyond these studies, Żukowski [25] presented the results of laboratory testing and numerical simulations of VTBC absorbers without considering the effects caused by transparent cover and insulation materials. Ma [26] conducted experiments on the dynamic thermal performance of VTBC collectors under multiple factors but offered no optimization proposal.

To improve the heat-collection efficiency of all-ceramic solar collectors, this paper reports on the will building of a mathematical model of a VTBC solar collector, analyzing its heat-transfer process via theoretical computing. Then, this paper reports on the observed performance of a typical domestic VTBC solar collector via an experimental study. After a sensitivity analysis of the parameters, this paper presents the optimized collector design by varying different parameters individually [27].

2. Theory

The working principle of VTBC solar collectors is similar to that of metal plate collectors. First, part of the solar energy radiates on the absorber through the transparent cover, while the other part is absorbed by the cover or reflected back into the sky. Second, part of the solar energy reaching the absorber is absorbed and converted into thermal energy, while the other part is reflected by the plate back to the cover. Third, the heat transfer fluid flows into the flow passages from the inlet of the collector and is heated by thermal energy. Fourth, after the temperature rises, the fluid medium flows out from the outlet of the collector containing useful thermal energy. In such a heat-transfer cycle, the incident solar radiation energy is gradually stored in heat storage equipment; however, the cover, end walls, and backplate of the collector continuously lose heat to the ambient environment. This cycle proceeds until the absorber temperature reaches a steady-state point [28].

2.1. Energy Conservation Relationship

According to the law of energy conservation, the internal energy increase in the collector per unit time (Q_s) is equal to the difference between the solar radiant energy projected onto the aperture surface of the collector (IA_a) and the optical loss (Q_l,o), heat loss (Q_l,h), and useful energy outputted by the collector (Q_u) (Equation (1)). In this process, Q_l,o includes the optical energy reflected not only by the transparent cover but also by the absorber (Figure 4).

Q_s = IA_a − Q_l,o − Q_l,h − Qu = (ṁC)dT_abs/dt

(1)

Under unsteady-state conditions, such as early in the morning, the absorber temperature (T_abs) increases. All components of the collector continuously absorb heat and store energy. When the sun sets in the evening, T_abs drops, and the components of the collector continuously release heat and energy. Under steady-state conditions, Q_s is 0. Therefore, for the convenience of analysis, unsteady-state conditions are not considered in this study. At this point, Q_u can be expressed by Equation (2).

Q_u = IA_a − Q_l,o − Q_l,h = C_pq_m (T_f,o − T_f,i)

(2)

On the one hand, Q_l,o is mainly related to the transmittance of the transparent cover (τ) and the absorptance of the absorber coating (α) [29]. In Equation (3), (τα)_e is the effective transmittance–absorptance product for the absorber, which will be further illustrated in Section 2.2.

Q_l,o = IA_a [1 − (τα)_e]

(3)

On the other hand, when T_abs is higher than the ambient temperature (T_a), part of the solar radiation energy absorbed by the collector is lost to the environment. Since T_abs is the highest among all parts of the collector, Q_l,h can be expressed by the difference between T_abs and T_a (Equation (4)).

Q_l,h = A_aU_L (T_abs − T_a)

(4)

2.2. Effective Transmittance-Absorptance Product for the Absorber

Transmittance (τ) relates to the solar radiation energy passing through the transparent cover to the surface of the absorber while absorptance (α) relates to the radiation energy absorbed by the absorber. According to Figure 5 [30], these radiation transmission, absorption, and reflection processes are repeated continuously between the transparent cover and the absorber. The solar radiant energy absorbed by the absorber in this process can be summed, and the transmittance-absorptance product for the absorber (τα) can be calculated using Equation (5). In this equation, τ and α are both measured values. Among these, τ is 90–96% for the commonly used ultraclear tempered glass [31]. This study uses an average value, 93%. In addition, the α value of a VTBC solar collector is 0.94 [24]. The reflectivity of the transparent cover to scattered radiation (ρ_d) is related to the extinction coefficient (K) and the travel length (L) of solar radiation.

$$\left(\tau \alpha \right) = \tau \alpha {\displaystyle \sum }_{\mathrm{n} = 0}^{\infty } [\left(1 - \alpha \right) {\rho }_{\mathrm{d}}{]}^n = \frac{\tau \alpha }{1 - \left(1 - \alpha \right){\rho }_{\mathrm{d}}}$$

(5)

Furthermore, for the whole collector, part of the solar radiation absorbed by the transparent cover is not lost. This is because this part of the energy increases the cover temperature, and so reduces the heat loss transferred from the absorber to the cover. This process equals the increase in τ. The effective transmittance–absorptance product for an absorber ((τα)_e) with N layers of transparent covers can be calculated according to Equation (6). Among these, a_i is a constant value, as shown in

Table 1. The values of α_i.

Number of Layers of Transparent Covers	ai	Emissivity of Absorber (εp)
		0.95	0.50	0.10
1	a1	0.27	0.21	0.13
2	a1	0.15	0.12	0.09
	a2	0.62	0.53	0.40
3	a1	0.14	0.08	0.06
	a2	0.45	0.40	0.31
	a3	0.75	0.67	0.53

[32]. The emissivity of a VTBC surface (ε_p) is 0.9 [15]. The value of K for a single ultraclear cover is 4/m.

$${\left(\tau \alpha \right)}_{\mathrm{e} }= \left(\tau \alpha \right){ + \alpha }_{1 }\left({1 - \mathrm{e}}^{{-K}_1{L}_1}\right){ + \alpha }_2{\tau }_1 \left({1 - \mathrm{e}}^{{-K}_2{L}_2}\right){ + \alpha }_3{\tau }_1{\tau }_2\left({1 - \mathrm{e}}^{{-K}_3{L}_3}\right) +\dots$$

(6)

The value of L_i [33] should be calculated according to the thickness of the corresponding cover layer (δ_c) and the incidence angle of solar radiation (θ_i) (Equation (7)). For a VTBC solar collector under steady-state outdoor conditions and with a single-layer ultraclear glass cover [34], (τα)_e is primarily related to δ_c. In Equation (8), the values of sinθ_i and r are 1 and 1.52, respectively.

$$L_{\mathrm{i}} = \frac{{\mathsf{\delta }}_{\mathrm{c}}}{\sqrt{1 - (\frac{{\sin \theta }_{\mathrm{i}}}{\mathrm{r}})}}$$

(7)

$${\left(\tau \alpha \right)}_e = 0.925 \times 0.94 + 0.27 [1 - \exp \left({-4L}_{\mathrm{i}}\right)] = 0.8695 + {0.27 (1 - \mathrm{e}}^{-{6.9\delta }_{\mathrm{c}}})$$

(8)

2.3. Heat Loss Analysis

The total heat loss of a VTBC solar collector (Q_l,h) is composed of heat loss via the top (Q_t), bottom (Q_b), and edge walls (Q_e) (Equation (9)–(12), Figure 6) [35].

Q_l,h = Q_t + Q_b +Q_e

(9)

Q_t = A_aU_t (T_abs−T_a)

(10)

Q_b = A_aU_b (T_abs−T_a)

(11)

Q_e = A_aU_e (T_abs−T_a)

(12)

By substituting Equations (9)–(12) into Equation (4), Equation (13) can be obtained [36]. U_L, U_t, U_e, and U_b refer to the heat loss coefficient of the total collector, the top, the end walls, and the bottom, respectively.

U_L = U_t + U_b + U_e (A_e/A_a)

(13)

As shown in Figure 7, to calculate these heat loss coefficients, it is necessary to define the heat transfer process of the collector in operation, which mainly includes (1) the radiative (r) and convective (conv) heat transfer between the outer surface of the transparent cover (c1) and the ambient environment (a); (2) the radiative (r) and convective (conv) heat transfer between the inner surface of the transparent cover (c2) and the absorber (abs); (3) the conductive (cond) heat transfer between the absorber (abs) and the bottom of the collector (b); and (4) the radiative (r) and convective (conv) heat transfer between the bottom (b) and edge walls (e) and the ambient environment (a) [37].

Figure 7 can be simplified in terms of the thermal resistance between the two sides (Figure 8). According to this simplified figure, empirical formulas for calculating the heat-loss coefficient at the top of the collector (U_t) are proposed as Equation (14) [38,39,40]. When the average temperature of the hot plate is between T_a and 200 °C, the error of the formulas is no more than 0.3 W/m²K [41].

$$U_{\mathrm{t}}=[\frac{N}{\frac{c}{T_{\mathrm{abs}}} {\left(\frac{T_{\mathrm{abs} }{- T}_{\mathrm{a}}}{N+f}\right)}^{\mathrm{e}}}+\frac{1}{h_{\mathrm{w}}}{]}^{-1}+\frac{{\sigma (T}_{\mathrm{abs}}{+T}_{\mathrm{a}}{) (T}_{\mathrm{abs} }^2{+T}_{\mathrm{a}}^2)}{{{(\epsilon }_p+{0.00591Nh}_{\mathrm{w}})}^{-1}+\frac{2N+f - 1+{0.133\epsilon }_{\mathrm{p}}}{{\epsilon }_{\mathrm{c}}}- N}$$

(14)

where

$$c =\{\begin{array}{c}520 (1-{0.000051\phi }^2), 0° < \phi < 70°\\ 392.6, 70° \le \phi < 90°\end{array}$$

f = (1 + 0.0892h_w − 0.1166h_wϵ_p) (1 + 0.07866N)

$$h_{\mathrm{w}} = 5.7 + 3.8w$$

e = 0.43 (1 − 100/T_abs)

The heat loss at the bottom of the collector depends on the thermal resistances R₄ of the insulation layer and R₅ to the ambient environment. As R₄ is much larger than R₅, its bottom heat-loss coefficient can be approximately expressed by U_b = λ_b/δ_b. The heat loss of the edge walls (U_e) of the collector is relatively small and can be considered 2% of U_t [15].

2.4. Heat-Transfer Analysis

The heat-transfer performance of a VTBC solar collector can be evaluated by its instantaneous efficiency (η) and average efficiency. Key steps for obtaining these parameters are the calculations of the absorber temperature (T_abs) and the mean fluid temperature (T_f).

2.4.1. Efficiency Factor

After calculating the total heat-loss coefficient of the collector (U_L) according to Equation (13), Q_l,h can be calculated using Equation (4). However, the heat loss and the temperature difference between the collector and the ambient environment (T_a) are proportional, and so the absorber temperature is not easy to determine. Therefore, the practicability of Equation (4) is limited to some extent. In contrast, the inlet (T_f,i) and outlet (T_f,o) fluid temperatures of the collector are easier to measure. Thus, the performance of the collector can also be reflected by T_f,i and T_f,o, as shown in Equation (15) [42].

Q_l,h = A_aF’U_L (T_f − T_a)

(15)

where

T_f = 0.5 (T_f,i + T_f,o)

For the VTBC solar collector shown in Figure 1, its efficiency factor (F’) can be calculated according to Equation (16) [43]. The thermal conductivity of the VTBC fin (λ_abs) is 1.256 W/mK. Therefore, F’ is a physical quantity related to the geometric structure of the collector and is constant for a collector with a certain structure and fluid flow. The F’ of a good flat-plate solar collector is generally 0.9–1.0 [44].

$$F^{\prime } = \frac{{1/U}_L}{W \{\frac{1}{U_{L }\left[D_{\mathrm{o}} + \left({W - D}_{\mathrm{o}}\right) F\right]} + \frac{1}{{\mathrm{h}}_{\mathrm{i}}{\mathsf{\pi }D}_{\mathrm{i}}}\}}$$

(16)

where

h_i = (1430 + 23.3T_f − 0.048T_f²) w_w^0.8D_i^−0.2

$$w_{\mathrm{w} }= \frac{q_{\mathrm{m}}}{\frac{{\mathsf{\pi }D}_{\mathrm{i}}^2}{4}{\rho }_{\mathrm{w}}\mathrm{int}(\frac{1}{W})}$$

$$F = \frac{\tan \mathrm{h}[\frac{m \left({W - D}_{\mathrm{o}}\right)}{2}]}{\frac{m \left({W - D}_{\mathrm{o}}\right)}{2}}$$

$$m = \sqrt{\frac{U_L}{{\lambda }_{\mathrm{abs}}\delta }}$$

2.4.2. Heat-Removal Factor

After introducing F’, U_L can be calculated based on T_f. However, although T_f can be measured, it is not easy to control in actual tests. This is because T_f,o varies with solar conditions. In contrast, T_f,i is easier to measure. Therefore, in this study, U_L is defined according to T_f,i. Then, the heat removal factor (F_R) of the collector is introduced (Equations (17) and (18)) [45]. F_R is a dimensionless parameter that comprehensively reflects the influence of the heat-transfer performane of the collector and the fluid. It is not only related to the geometric structure and heat transfer characteristics of the collector but also affected by the mass flow rate of the fluid (q_m), the specific heat capacity of the fluid (C_p), and the aperture area of the collector (A_a).

Q_l,h = A_aF_RU_L (T_f,i − T_a)

(17)

$$F_{\mathrm{R}} = \frac{q_{\mathrm{m}}{c}_{\mathrm{p}}}{A_{\mathrm{a}}{U}_{\mathrm{L}}} [1 - \exp \frac{A_{\mathrm{a}}{U}_{\mathrm{L}}{F}^{\prime }}{q_{\mathrm{m}}{c}_{\mathrm{p}}}]$$

(18)

2.4.3. Absorber Temperature and Mean Fluid Temperature

Then, T_abs and T_f can be obtained according to Equations (19) and (20), respectively [43].

$$T_{\mathrm{abs}}{ = T}_{\mathrm{f},\mathrm{i}} + \frac{Q_{\mathrm{u}}}{A_{\mathrm{a}}{F}_{\mathrm{R}}{U}_{\mathrm{L}}}{ (1 - F}_{\mathrm{R}})$$

(19)

$$T_{\mathrm{f}}{ = T}_{\mathrm{f},\mathrm{i}} + \frac{Q_{\mathrm{u}}}{A_{\mathrm{a}}{F}_{\mathrm{R}}{U}_{\mathrm{L}}} (1 - \frac{F_{\mathrm{R}}}{F^{\prime }})$$

(20)

2.4.4. Collection Efficiency

Instantaneous efficiency (η) is the thermal performance of a collector at a specific time and can be expressed as Equation (21) [46,47]. If it is assumed that U_L is a constant that does not change with temperature, η can be expressed as Equation (22). When η is 0, T_abs and T_f are equal to T_f,i, which means that the collector reaches its highest operating temperature. This temperature depends mainly on (τα)_e and U_L. Therefore, increasing (τα)_e or decreasing U_L are effective ways to improve the operating temperature.

$$\eta = \frac{Q_{\mathrm{u}}}{A_{\mathrm{a}}I}{ = F}_{\mathrm{R}} \left[{\left(\tau \alpha \right)}_{\mathrm{e}} {- U}_L\frac{T_{\mathrm{f},\mathrm{i}} {- T}_{\mathrm{a}}}{I}\right]$$

(21)

$$\eta = {\left(\tau \alpha \right)}_{\mathrm{e}} {- U}_L\frac{T_{\mathrm{abs} }{- T}_{\mathrm{a}}}{I} = F^{\prime } \left[{\left(\tau \alpha \right)}_{\mathrm{e}} {- U}_L\frac{T_{\mathrm{f}} {- T}_{\mathrm{a}}}{I}\right]$$

(22)

If A = F_R (τα)_e, B = F_RU_L, T_m* = (T_f − T_a)/I, then η = A − BT_m* [48]. For a given solar collector, A is the intercept efficiency, a constant that mainly depends on the optical characteristics of the transparent cover and the absorber. It represents the highest efficiency that the collector can theoretically achieve. B is the heat loss coefficient based on the inlet temperature and a constant that mainly depends on the collector structure and thermal insulation design. Theoretical analysis and experience show that the thermal performance of a collector can be roughly evaluated when A and B are obtained.

3. Methods

Using the abovementioned mathematical model of VTBC solar collectors, it is possible to conduct an optimization design theoretically by altering some influencing factors. This computation aims to obtain η and evaluate the performance of VTBC solar collectors with certain structures. Based on Equations (21) and (22), U_L and other parameters are required when calculating η. According to Equations (9)–(14), U_L is a function of T_abs. However, T_abs is not a constant parameter known in advance. Therefore, the calculation of U_L should adopt a numerical solution method. The computing method designed in this study assumes T_abs* based on experience at the beginning and calculates it gradually according to the equations above until the calculated T_abs is obtained (Figure 9). If the absolute value of the difference between T_abs* and T_abs is less than 10⁻⁶, it can be considered to meet the accuracy requirements [28]. Otherwise, the calculated value is iterated as the second assumed value until T_abs that meets the requirement of precision is obtained. Finally, η of the collector is calculated according to this value. To facilitate and accelerate the iterative computing progress, a computer program was created.

4. Experimental Study of a Reference Collector

The key to improving the thermal performance of solar collectors lies in increasing (τα)_e and reducing U_L. For a VTBC solar collector, the absorptance (α) and emissivity (ε_p) of the absorber are fixed values, while other factors vary according to different collector designs. To observe the performance of a typical domestic VTBC solar collector, the authors conducted an experimental study and analyzed the relationships between the collector performance and the ambient conditions.

4.1. Experimental Setup

A reference VTBC solar collector was designed, tested, and analyzed by the abovementioned mathematical model (Figure 10 and Figure 11 and

Table 2. Input parameters of the reference collector model.

Item		Value	Unit
Solar collector	Length (l)	2.1	m
	Width	0.7	m
	Thickness	0.18	m
	Aperture area (Aa)	1.47	m2
	Inclination angle ($\phi$)	42	°
VTBC absorber	Fin width (W)	0.04	m
	Fin thickness (δ)	0.004	m
	Emissivity (εp)	0.90	-
	Absorptivity (α)	0.945	-
	Heat-conduction coefficient	1.256	W/mK
	Outer diameter of passage (Do)	0.025	m
	Inner diameter of passage (Di)	0.017	m
Transparent cover	Layer amount (N)	1	-
	Thickness (δc)	0.0032	m
	Transmittance (τ)	0.93	-
	Refractive index (r)	1.52	-
	Emissivity (εc)	0.94	-
Insulating layer	Thickness (δb)	0.05	m
	Heat-conduction coefficient (λb)	0.03	W/mK

) [49]. The experiment was conducted from 9 a.m. to 4 p.m. on 10 to 12 April on the campus of Shandong Jianzhu University in a cold area of China (E: 117.20, N: 36.68). The sampling time interval was 5 min. The inlet fluid temperature (T_f,i) and fluid speed in the passage (W_w) remained constant at 20 °C and 0.707 kg/s, respectively.

4.2. Experimental Results

Based on the experiment, the dependencies among I, T_a, and T_f,o are illustrated below, although the focus of this paper is to discuss the effects of collector structure design on thermal performance. The trend lines in Figure 12 show that the performance of the tested collector is positively related to I and T_a (Figure 12). On the one hand, with an increase in I, Q_u of the collector increases rapidly, which in turn increases η. On the other hand, with an increase in T_a, the temperature difference between T_a and T_abs decreases, Q_l,h decreases, and η increases. However, the influence of wind is not great, as the average value of w is only 2.8 m/s. In general, the instantaneous efficiency equation of the testing collector is η = 0.89 − 2.20T_m*.

5. Mathematical Optimization

This section reports on sensitivity analyses of the effects of the design parameters on the performance of the typical VTBC solar collector in order to conduct optimization. The influence of each parameter is shown in the figures below, representing its influence on T_f, T_abs, Q_l,h, Q_l,o, Q_u, F, F’, F_R, and η. Although external parameters such as ambient and fluid conditions also have impacts on the performance of the collector, this section focuses on the designs of the fin and the transparent cover. Therefore, in this optimization study, T_a, I, w, T_f,i, and q_m still under the average conditions of the above experiment, which are 11.8 °C, 864 W/m², 2.8 m/s, 20 °C, and 0.0707 kg/s, respectively.

5.1. Influence of Fin Design

For the VTBC solar collector described in Figure 1, the most important design parameters of the fin are width (W) and thickness (δ).

5.1.1. Fin Width

In this scenario, W is the only variable (Figure 13). The value of W is no less than D_o, and it has a great influence on the thermal performance of the collector. First, an increase in W increases not only the absorber area parallel to the transparent cover but also the heat-transfer distance to the fluid passage. Because the thermal conductivity of VTBC is poor, T_f shows no clear dependence on W. However, as δ is small and remains unchanged in this scenario, T_abs has a linear relationship with W. Second, W has little effect on Q_l,o, but has negative impacts on Q_l,o and Q_u. Third, although an increase in W also leads to decreases in F, F’, and F_R, there is a peak value of η. When W increases from 30 to 50 mm, η increases by 1.55%; when W increases from 50 to 80 mm, η decreases by 9.01%.

5.1.2. Fin Thickness

In this scenario, δ was the only variable (Figure 14). First, an increase in δ greatly influenced T_f and T_abs. Second, change in δ had little effect on energy loss and output. Third, overall, F, F’, and F_R all increased slightly with an increase of δ, while η hardly changed. This indicates that for the optimal design of a VTBC solar collector, a change in δ does not significantly affect its thermal performance.

5.2. Influence of Transparent Cover Design

For the VTBC solar collector described in Figure 1, the most important design parameters for the transparent cover are number of layers (N), thickness (δ_c), transmittance (τ), and refractive index (r).

5.2.1. Number of Transparent Layers

In this scenario, N was the only variable (Figure 15). First, with an increase in N, T_f, and T_abs both decreased. The change range of 1 to 2 was much greater than that of 2 to 3. Second, an increase in N had little effect on Q_l,h and Q_u. Third, N had little influence on F, F’, and F_R. Moreover, an increase in N had some effect on (τα)_e and η. When N increased from 1 to 2, η increased by 5.17%; however, when N increased from 2 to 3, η decreased by 2.83%. Therefore, N = 2 is the best option for this model.

5.2.2. Thickness of the Transparent Cover

In this scenario, δ_c was the only variable (Figure 16). First, when δ_c increased, T_f and T_abs both increased. Second, δ_c has little effect on energy loss and output. Third, δ_c has little effect on F, F’, and F_R but had some positive effect on η. When δ_c increased from 3 to 12 mm, η increased by 1.82%. In general, increasing δ_c had little effect on the performance of the collector.

5.2.3. Transmittance of the Transparent Cover

In this scenario, τ was the only variable (Figure 17). First, T_f and T_abs increased linearly with τ. Second, Q_l,h was not affected by τ, while Q_l,o and Q_u were. Third, τ had little influence on F, F’, and F_R, but has a great influence on η. Based on a literature review, for ultraclear glass τ can be as large as 0.96 [50,51]. When τ increased from 0.82 to 0.96, η increased by 17.42%. Therefore, the use of a transparent cover with a high τ is crucial for collection efficiency.

5.2.4. Refractive Index of the Transparent Cover

In this scenario, r was the only variable (Figure 18). From the perspective of transparent cover materials, the average r values of organic materials, ordinary glass materials, and transparent plastic materials were approximately 1.49, 1.52, and 1.59, respectively [52]. Increasing r had a slightly negative effect on T_abs and η but had almost no effect on the other indicators. In general, the effect of r on collector performance was not great.

5.3. Optimization Results

Effective ways to improve the efficiency of the VTBC solar collector were to increase τ, N, and δ_c. Among these, the most useful and low-cost method was to use a transparent cover with a high τ (

Table 3. Sensitivity analyses of τ, N, and δ_c.

Key Parameter	Original Value	Optimized Value	Increase of η
τ	0.82	0.96	17.42%
N	1	2	5.17%
δc	3 mm	12 mm	1.82%

). Therefore, this study increased τ to 0.96 individually by replacing the normal glass cover with an ultraclear glass cover. According to the calculation results, the instantaneous efficiency equation of the optimization model was η = 0.92 − 2.20T_m*.

6. Discussion

The thermal performance of solar collectors is also affected by external factors, which include ambient conditions (T_a, I, and w) and system situations (T_f,i, q_m, and A_a). Among these, the latter are more controllable in system operation. In Figure 19a, T_f,i was the only variable. For this optimization model, T_f,i has an optimum value (approximately 20 °C). With a continuous increase in T_f,i, T_abs also increased. Therefore, the temperature difference between T_abs and T_a increased, which led to an increase in Q_l,h, and a decrease in η. In Figure 19b, q_m was the only variable. T_f and T_abs decreased as q_m increased but gradually became stable. Although an increase in q_m did not affect Q_l,o, it had a small positive effect on Q_l,h and η. When q_m increased from 0.03 kg/s to 0.10 kg/s, η increased by 3.97%. In Figure 19c, A_a is the only variable. As q_m remained unchanged when A_a continues to increase; the working medium cannot transfer the heat absorbed to the outlet in a timely manner, resulting in more energy loss. That is, the higher the A_a is, the lower the η. In this scenario, when Aa increased from 1 to 8 m², η decreased by 7.52%. Although these parameters can influence the performance of the VTBC collector, their effects were far smaller than that of the collector’s structural design.

7. Conclusions

From theoretical and experimental points of view, this paper has analyzed the heat-transfer process and thermal performance of VTBC solar collectors and optimized the collector design according to the effects of different parameters. First, in terms of heat-transfer analysis, the performance of VTBC solar collectors can be evaluated by factors such as F’, F_R, and η. Although the analysis theories are similar to those for traditional metal flat-plate solar collectors, ceramic materials have unique characteristics. On the one hand, the VTBC coating has both high absorptance (α = 0.94) and high emissivity (ε_p = 90%). On the other hand, the thermal conductivity of VTBC is very low (λ_abs = 1.256 W/mK). Second, in terms of optimization, this paper set up a thermal-performance calculation method under steady-state conditions and a corresponding computer program. Via experimental testing, this study has proposed a reference model (η = 0.89 − 2.20T_m*). Then, the study has analyzed the influence of fin design and transparent cover design on the collector using the mathematical model and varied the most sensitive parameter τ individually to create the optimization model (η = 0.92 − 2.20T_m*).

This paper presents a fast and low-cost method to optimize VTBC collector design under steady-state conditions based on mathematical modeling. In future studies, more work should be done to verify the accuracy of the model via optimized absorber manufacturing and corresponding experimental testing under unsteady-state conditions.