Theoretical investigation of the temperature characteristics and output parameters of an industrial crystalline silicon solar cell with a microfluidic cooling system

Operating temperature is a key factor affecting the output power of a crystalline silicon solar cell (c-Si SC). Based on solving basic semiconductor equations, Maxwell equations and heat flow equations by finite difference method, this work has theoretically investigated the influences of microfluidic cooling system on the temperature distributions and output parameters of an industrial c-Si SC packaged as a module form, where the cooling system is installed the interface between the aluminum electrode and back ethylene vinyl acetate layer and the cooling medium flowing through microchannel is selected as water. Under the influences of the cooling system, the back surface temperature of the c-Si SC is fixed at about water temperature and the front surface temperature will decrease with the water temperature decreasing. The temperature distributions of c-Si active layer and aluminum layer decrease approximately linearly along the two layer’s thickness directions. The cooling effects of the cooling system are mainly determined by water temperature and when the filling factor of microchannel is larger than 0.1, its influences on the temperature distribution of the c-Si SC are small. In the environmental temperature range from −15 °C to 60 °C, the increase rates of Voc, FF and η per degree Celsius with the water temperature decreasing are about 0.0014 V, 0.036% and 0.05%, respectively.


Introduction
Photovoltaic (PV) electricity generation has become one of the most important ways to obtain electrical energy. The global energy review 2021 issued by IEA (International Energy Agency) indicates that in 2021, PV electricity generation has accounted for 12.05% of global renewable electricity generation and 3.49% of global electricity generation [1]. Recent years, different schedules for carbon dioxide emissions have been set by many countries, such as China [2], Canada [3], Mexico [4], etc, so spurred on by soaring demand for low-carbon electricity generation, the demand of PV electricity generation will continue to rapidly grow in future.
One of the challenges of exploiting solar energy via PV technology is that the photoelectrical conversion process of a PV module is very sensitive to nearly all of environmental variables, such as season [5], latitude [6], cloud [7], dust or shelter [8], rain [9], wind [10], temperature [11] and humidity [12]. Because environmental variables are volatile, the output power of a PV module is often unstable and fluctuant. For this reason, evaluating the impacts of different environmental variables on the output power and predicting the law of variation of the output power with different environmental variables have become two very active research areas in the field of PV research since the 1970s [13,14]. Until now, large amounts of empirical, theoretical and experimental formulas about the relations between the output power of a PV module and environmental variables have been proposed and established [7,9,10,[12][13][14].
2. Model, method and material properties 2.1. The structure model of PV module and c-Si SC A standard c-Si SC module is often composed of a front cover glass layer, c-Si SC layer, tedlar layer and two EVA layers between the glass layer and c-Si SC layer and between the c-Si SC layer and tedlar layer, respectively [32,33]. Figure 1(a) gives a schematic diagram of such a module, where T1, T2, T3, T4 and T5 denote the thicknesses of glass layer, EVA layer between the glass layer and c-Si SC layer, c-Si SC layer, EVA layer between the c-Si SC layer and tedlar layer, and tedlar layer, respectively. Here, the c-Si SC layer in figure 1(a) is chosen as a PERC (passivated emitter rear cell) c-Si SC, which can be further represented by an enlarged structure diagram [34,35], as shown by figure 1(b). The legends of c-Si, SiN x , Silver, SiO 2 and aluminum denote the materials of the active layer, antireflective coating layer, electrode on the front surfaces, passive layer, electrode on the back surface of the c-Si SC, respectively and the legend of microchannel, which is rectangular, represents the channel through which microfluidic medium can flow. In figure 1(b), H1 and H4+H5 represent the heights of the electrodes on the front and back surface, respectively and H2 and H4 represent the heights of the texture structures on the front and back surfaces, respectively. The thicknesses of antireflective coating layer and passive layer are denoted by H6 and H7, respectively and the height of microchannel is represented by H8. The thickness of c-Si active layer is equal to the sum of H2, H3 and H4. The thickness of c-Si SC layer T3 in figure 1(a) is equal to the sum of H1, H2, H3, H4 and H5. The widths of the c-Si SC periodic unit and silver electrode on the front surface are denoted by L1 and L2, respectively. The widths of aluminum electrode holes on the left and right hands of the periodic unit are represented by L3 and L5, respectively. The widths of aluminum electrode hole at the center of the c-Si SC periodic unit and microchannel are represented by L6 and L4, respectively. The angle of texture structure is denoted byθ and the filling factor of microchannel (f C ) is defined as L4/L1. In order to expediently describe the geometric parameters of the c-Si SC, we have established a Cartesian coordinate system in figure 1(b), where the origin is set at the upper left corner of the c-Si SC.
The values of T1, T2, T4 and T5 in figure 1(a) are selected as 3 × 10 −3 m, 500 × 10 −6 m, 500 × 10 −6 m and 1 × 10 −4 m, respectively, as those used in [36]. The values of H1, H2, H3, H4, H6 and H7 in figure 1(b) are chosen as 0.5 μm, 7 μm, 186 μm, 7 μm, 0.08 μm and 0.2 μm, respectively, as those used in our previous work [35]. The H5 value can vary from about 20 μm to about 100 μm [37,38]. Here, we select the value of H5 as 41 μm. The fabrication of a microchannel with larger height-width ratio can be realized by nanoimprint lithography and reactive ion etching, as shown by [39]. From a cooling effect perspective, the smaller spacing between the microchannel and c-Si active layer, will help to enhance the cooling effects of the microchannel, but the manufacturing costs of the microchannel will rapidly increase with the height of microchannel increasing, especially for fabricating a microchannel with a larger height-width ratio. Considering the fabricating costs and cooling effects, we select the height-width ratio as 0.6 when the microchannel width is 5 μm, i.e. H8 = 3 μm. By using the H1, H2, H3, H4 and H5 values, the T3 value in figure 1(a) can be easily obtain, i.e. T3 = 200.5 μm. The L1, L2, L3, L5 and L6 values are selected as 10 μm, 0.15 μm, 0.075 μm, 0.075 μm and 0.15 μm, respectively, also as those used in our previous work [35]. The L4 value will vary with the f C . The θ value is 54.46°, which is equal to the vertex angle of the ideal textured pyramid of a industrial c-Si SC.
The process parameters are also adopted as those we have used in our previous work [35], i.e. the minority carrier lifetime of c-Si substrate with doping boron concentration equal to 5 × 10 17 /cm 3 is 1 ms and the pn junction is generated by phosphorus diffusion process with diffusion temperature, diffusion time and phosphorus atmosphere concentration equal to 900°C, 10 min and 5 × 10 20 /cm 3 , respectively. The ohmic contacts have been formed between aluminum layer and c-Si active layer and between silver layer and c-Si active layer.

Simulation method
The output parameters of the c-Si SC under different environmental conditions can be obtained by solving basic semiconductor device equations, Maxwell equations and heat flow equation, where the heat flow equation is given by [15,35], where C and κ are the thermal capacity per unit volume and thermal conductivity, respectively and T is Kelvin temperature. Q s in equation (1) denotes the heat generated in per unit volume, which can be expressed as the following, where h, v and E g denotes Plank constant, incident light frequency and energy band width, respectively. The term of (hv-E g ) represents the heat released by the hot carrier energy relaxation process due to hv > E g and the terms of Q J , Q C and Q PT denote Joule heat, heat absorbed or released in the generation and recombination process of carriers and heat generated by Peltier and Thomson effects, respectively. The Q J , Q C and Q PT terms, which are related with carrier transport process, can be expressed as the following [15], where q represents the unite charge and the corner marks of n and p denote electron and hole, respectively. J, μ, E and P denote current density, mobility, quasi-Fermi level, and thermoelectric power, respectively. R and G represent the recombination and generation rates of carriers, respectively. Finite difference method is used to solve basic semiconductor device equations, Maxwell equations and heat flow equation under different boundary conditions and the computational details, including meshing scheme, boundary conditions and solving order, can be found in our previous works [15]. In order to obtain the impurity distribution in the c-Si active layer of figure 1(b), we first use the complementary error function to describe the phosphorus dopant distribution and then interpolate the phosphorus distribution into the c-Si active layer by linear interpolation method. Under the diffusion condition given by section 2.1, the junction depth of phosphorus dopant is about 0.8 μm. The grid resolution is set to be λ min /20, where λ min is the minimum wavelength of incident light. Air mass 1.5 global tilted irradiance data with the wavelength range from 300 nm to 1200 nm and the refractive indexes of all materials are given by professional website [40].

Material properties related with temperature
The variations of c-Si material properties with temperature are only considered the band gap, valence and conduction band densities of states, mobilities of electrons and holes, as following [15,35], where the subscripts of c and v in equation (7) denote conduction and valence bands, respectively and the subscripts of n and p in equation (8) denote electron and hole, respectively. E g (300) in equation (6) represents the width of band gap at 300 K. μ n and μ p denote the electron and hole mobilities, respectively and N c,300 and N v,300 are the effective state densities of the conduction and valence band at 300 K, respectively. N d denotes the local doping concentration. In our simulation, we select these parameters as those we have used in our previous work [15], i.e. E g (300) = 1.08 eV; E gα = 4.73 × 10 −4 , E βα = 636, α n = α p = −0.57, β n = β p = −2.33, γ n = γ p = 2.546, μ n2 = 1252 cm 2 /V·s, μ n2 = 407cm 2 /V·s, N c,300 = 2.8 × 10 19 cm −3 , N v,300 = 1.04 × 10 19 cm −3 , μ n1 = 88cm 2 /V·s, μ p1 = 54.3cm 2 /V·s, N n0 = 1.432 × 10 17 cm −3 and N n0 = 2.67 × 10 17 cm −3 . The variations of the values of κ and C with temperature for different materials can be represented by [35], where κ 300 and C 300 denote thermal conductivity and thermal capacity at 300 K, respectively and the parameters of α, β and C 1 are material constants. The values of these parameters are listed in table1.

Simplifying the heat transfer process by equivalent thermal resistance network
During encapsulating a c-Si SC module under the vacuum and heating conditions, the glass and tedlar layers are both tightly bond with c-Si SC layer by EVA layer and after encapsulated, these layers will become a multilayer solid structure. Some theoretical results based on molecular dynamics simulation have shown that the interfacial thermal resistance at a solid-solid interface is generally in the range from 6×10 −10 m 2 K W −1 to 1×10 −7 m 2 K W −1 [41,42]. According to the theoretical relations between thermal conductivity, layer thickness and thermal resistance, the thermal resistances of the glass layer (R Glass ) , EVA layer (R EVA ) and tedlar layer (R Tedlar ) are 17×10 −4 m 2 K W −1 , 14×10 −4 m 2 K W −1 and 5×10 −4 m 2 K W −1 , respectively. Due to the thermal resistance of each layer in series with its interfacial thermal resistance and the large difference of the orders of magnitude of the thermal resistances between each layer and its corresponding interfacial thermal resistances, the interfacial thermal resistances between the glass and EVA layers, between EVA and c-Si SC layers and between EVA and tedlar layers, will be omitted in our following discussion. The thermal resistant network, as shown by figure 2(a), can be used to describe the heat transfer process between the heat source (i.e. the c-Si SC) and surroundings, where the heat transfer process between EVA and c-Si SC layers can be realized by two routes, i.e. the direct contact heat transfer between the heat source and surroundings and the indirect heat transfer between the heat source and surroundings through the cooling system. In figure 2(a), T Cell , T Front , T Back and T Ch denote the temperature values of the c-Si SC, surroundings situated at the front surface of the PV module, surroundings situated at the back surface of the PV module and cooling medium within the microchannel, respectively. R Fconv and R Bconv represent the contact thermal resistances between the front and back surfaces of the PV module and surroundings, respectively, which are   [35,36]. generated by convection process. R Ch represents the contact thermal resistance between the back surface of the c-Si SC and cooling medium. According to the computational rule of thermal resistant, we can further simplify the thermal resistance network of figure 2(a) as figure 2(b), where R Fequiv and R Bequiv denote the equivalent thermal resistances between the front and back surfaces of the c-Si SC and surroundings, respectively. As shown by figure 2(b), the heat transfer process of the PV module can be regarded as a three-port thermal resistance network with its center at the c-Si SC. By using the R Fequiv and R Bequiv parameters, the thermal boundary conditions required in solving equation (1) can be set to the physical boundaries of the c-Si SC. The values of R Fconv and R Bconv per square meter, which can be obtained by using Sharples model [36] and the relations between the heat resistances and heat transfer coefficients of the front and back surfaces of the PV module, are about 0.1839 K W −1 and 0.1343 K W −1 under natural convection atmospheric conditions (i.e. v w = 0, where v w denotes wind speed), respectively. According to the computational rule of thermal resistant, the values of R Fequiv and R Bequiv per square meter will be 0.1870 m 2 K W −1 and 0.1362 m 2 K W −1 , respectively. The value of R Ch can be affected by many factors, such as the cross-section, inlet and outlet shapes, sizes, engineering tolerance and inadequacies of the microchannel and the flow rate, compound ingredient, viscosity and thermophysical changes of the fluid in the microchannel. Traditionally, cooling liquids are often selected as water, ethylene glycol and oil. Because the thermal conductivity of water is larger than those of ethylene glycol and oil, the contact thermal resistance R Ch of water will be lower than those of oil and ethylene glycol and the most common cooling liquids used in microchannel cooling system are water or water with nanoparticles, such as Al 2 O 3 /water, SiN/water, SiC/water, Cu/water, Ag/water and Au/water [43]. Although the contact thermal resistance of water with nanoparticles in microchannel is generally lower than that of water, its stability is poor under certain environmental conditions. In our simulation, we select the cooling liquid flowing through microchannel as water. The thermal conductivity and thermal capacity of water are equal to 0.59 W m −1 K −1 and 4.2 × 10 3 J −1 kg·K, respectively and the R Ch values at the interface between the wall of microchannel and water with different flow velocities can vary in the range from 2 × 10 −6 m 2 K W −1 to 10 −4 m 2 K W −1 [44]. In general, the larger the flow velocity, the smaller the R Ch values will be. In our following discussion, the value of R Ch per square meter is selected as 10 −6 m 2 K W −1 .

Results and discussion
In this section, the effects of the microfluidic cooling system on the temperature distribution of the c-Si SC will be first discussed, then the temperature variations of the c-Si SC with microchannel parameters under different environmental conditions will be analyzed and finally the effects of the microfluidic cooling system on the output parameters of the c-Si SC will be evaluated. Here, we assume that T Front = T Back .The range of T Front or T Back is from −15°C to 60°C. The range of ΔT is from 0°C to 10°C, where ΔT = T Back -T Ch .
3.1. The temperature distribution features of the c-Si SC with the cooling system In our previous works [11,15], we have shown that as long as the environmental temperatures in the regions near the front and back surfaces of a c-Si SC without any artificial cooling systems are equal, the temperature distributions of the c-Si SC under different environment conditions will be nearly uniform and have almost nothing to do with the heat transfer coefficients between the c-Si SC and external environments. As far as the equivalent thermal resistant network model shown by figure 2(b) is concerned, the temperature distribution of the c-Si SC without a microfluidic cooling system will also be uniform, when T Front = T Back , but when a microfluidic cooling system is installed on the interface between the back surface of aluminum layer and EVA layer, the temperature distributions of the c-Si SC will be influenced by the cooling system. Figure 3 is a typical temperature distribution of the c-Si SC with a microfluidic cooling system, where T Back = 25°C, ΔT = 6 ℃ and f C = 0.5, respectively. According to figure 3(a), the temperature values away from the front surface texture region will decrease with increasing the Y values in the Y direction and have little change in the X direction. The temperatures for all points inside the c-Si SC are below T Back . According to figures 3(b) and (c), the obvious temperature changes in the X direction appear in the front-surface texture region and adjacent region of microchannel. The highest and lowest temperatures appear on the front surface of c-Si active layer and boundary between aluminum layer and microchannel, respectively.
In order to further analyze the variation laws of the temperature distribution of the c-Si SC with a microfluidic cooling system, the temperature profiles along three profile lines of figure 3(a) are drawn in figure 4(a), where the three profile lines are the lines parallel to the Y axis and passing through the points, (X = 0 μm,Y = 0 μm) , (X = 2 μm,Y = 0 μm) and (X = 4 μm,Y = 0 μm), respectively. In figure 4(a), the regions denoted by R A and R C represent the front and back texture regions of the c-Si SC, respectively and the region denoted by R B denotes the internal region of the c-Si SC. The region denoted by R D represents the region of aluminum layer with the same Y values as those of the microchannel region. As shown by figure 4(a), under the influences of the microfluidic cooling system, except for the temperatures of R A , R C and R D regions, the temperature of R B region will decrease nearly linearly with increasing the Y values and furthermore the temperature drop rate per unit Y value in R B region is larger than that in the region between R C and R D regions. The highest temperature of the c-Si SC, which appears on the front surface of the c-Si active layer, is about one degree Celsius below T Back , whereas the lowest temperature, which appears in the boundary layer adjacent to the microchannel region, is very close to T Ch . Overall, under the influences of a microfluidic cooling system, the SC temperature can be effectively limited within the range from environmental temperature (as shown by the line L E in figure 4(a), where L E represent environmental temperature) to the temperature of cooling water flowing through the microchannel (as shown by the line L F in figure 4(a), where L F represent cooling water temperature). The temperatures of the c-Si active and aluminum layers will all decrease linearly with increasing the Y value.
In order to clearly observe the variations of the temperature profiles in the R A , R B , R C and R D regions, the enlarged figures of four regions are drawn in figures 4(b)-(e), respectively. The figure 4(b) shows that because most of the heat released by the hot carrier energy relaxation process and the recombination process of photongenerated carriers is largely concentrated in the R A region [15,45], the temperature profiles denoted by X = 2 μm and X = 4 μm have distinct nonlinear characteristics in the R A region, whereas because the the temperature profile denoted by X = 0 μm is out of the front surface texture region, its temperature profile is linear. Figure 4(c) shows that the temperature drop rates per unit Y value along three profile lines are basically identical and the temperature differences between the three different profile lines are very small at the same Y  value. Figure 4(d) shows that influenced by the SiO 2 passive layer, the temperature profiles denoted by X = 2 μm and X = 4 μm appear similar temperature jump phenomena in the Y value range from 193 μm to 203 μm, whereas the temperature profile denoted by X = 0 μm appears a transit region with temperature slowly decreasing with increasing the Y value in the same Y value range. Figure 4(e) shows that the temperature profiles are strongly influenced by the microchannel region in the Y value range from 235 μm to 241 μm. According to the c-Si SC structure shown by figure 1(b) and the coordinate range of the R A , R C and R D regions along the Y direction, the regions with nonlinear temperature variations depend not only on the positions of heat and cold sources (see figure 4(b) and figure 4(e)) but also on the thermal properties of the materials in the c-Si SC. Figure 4(a) has shown that the temperature profile along a profile line parallel to the Y axis can visually retain more detail information about the temperature distributions of the c-Si SC under different conditions. In our following discussion, the temperature profile is selected as the temperature profile along the line parallel to the Y axis and passing through the point (X = 2 μm,Y = 0 μm). Figure 5(a) gives the variations of the temperature profiles with ΔT, where T Back = 25°C and f C = 0.5. As shown by figure 5(a), except for the temperature profile with ΔT = 0°C, other temperature profiles show the same variation laws as those obtained from figure 4(a), i.e. for each temperature profile, there exist two linear variation regions in the Y value ranges from about 0 μm to about 200 μm and from about 200 μm to about 240 μm, which are corresponding to the c-Si active and aluminum layer, respectively and the temperature of each temperature profile with ΔT > 0°C will decrease with the Y value increasing. In addition, figure 5(a) also shows that with the ΔT value increasing, the overall temperature profile will move toward the lower temperature range and the temperature profile with a larger ΔT value will have a larger temperature drop rate per unit Y value, which means that the larger the ΔT value, the more obvious the cooling effects of the microfluidic cooling system on the c-Si SC will be. Figure 5(b) gives the variations of the temperature gradients (i.e. δT/δY ) of the c-Si active layer and aluminum layer with ΔT, where the profile line is also selected as the line parallel to the Y axis and passing through the point (X = 2 μm,Y = 0 μm) and the T Back values are selected as −15°C, 25°C and 60°C. As shown by figure 5(b), the δT/δY values in both the c-Si active layer and aluminum layer linearly decrease with the ΔT value increasing and the T Back variation has a more obvious impact on the δT/δY value of aluminum layer than that of the c-Si active layer. In addition, under the same ΔT value, the δT/δY value of the c-Si active layer are smaller than that of aluminum layer.

The temperature distribution variations of the c-Si SC with T Ch
In fact, figure 5(a) also shows that larger temperature differences will appear in both the c-Si active layer and aluminum layer by using the microfluidic cooling system. In order to evaluate the temperature differences, we also extract the T values from the front and back surfaces of the c-Si active layer and from the back surface of the aluminum layer. The results show that with the ΔT value increasing from 0°C to 10°C, the T values of the front and back surfaces of the c-Si active layer will drift from the T Back value by 0°C∼−1.9°C and 0 ℃ ∼−8.5°C, respectively and the T values of the back surface of the aluminum layer will drift from the T Back value by 0°C∼−9.5°C. The temperature differences between the front and back surfaces of the c-Si active layer and between the back surface of the c-Si active layer and back surface of the aluminum layer are about 0°C∼−6.6°C and 0°C∼−1°C, respectively.

The temperature distribution variations of the c-Si SC with f C
Considering that the temperature distributions on the profile lines used by figure 5(a) will be influenced not only by the direction of heat flow but also by the distance between the profile line and the left margin of microchannel, so in order to eliminate the influences of such a man-made factor, here the profile line is selected as the line parallel to the Y axis and passing through the point (X = 5 μm,Y = 0 μm). Figure 6(a) is the variations of the temperature profiles of the c-Si SC with f C . As shown by figure 6(a), the basic features of the temperature profiles denoted by different f C values are similar with those given by figure 4(a) and figure 5(a). Due to the larger thermal conductivities of aluminum and c-Si material, the microchannel with the f C value larger than zero can limit the temperature of the region adjacent to the back surface of aluminum layer to about the cooling water temperature and establish approximately linear temperature distributions inside both aluminum layer and c-Si active layer, as shown by the curves with f C > 0 in figure 6(a). Figure 6(a) also shows that when the f C value decrease from 0.1 to 0, the front surface temperature will gradually approach environmental temperature (i.e. the temperature profile with f C = 0), which means that the cooling effects of the cooling system become gradually weak and when the f C value increase from 0.1 to 0.9, the temperature profiles denoted by different f C values are close to each other. In addition, the variations of the temperature profiles with the f C values increasing do not show significant regularities. For example, except for the temperature profiles with f C = 0, in the Y value range from 180 μm to 200 μm, the temperature profile curves denoted by f C = 0.05, f C = 0.1 and f C = 0.3 are above others, whereas in the Y value range from 0 μm to 20 μm, the temperature profile curves denoted by f C = 0.05, f C = 0.1, f C = 0.6 and f C = 0.9 are very close and above others.
By using the same method as that used in figure 5(b), the δT/δY values of the c-Si active layer and aluminum layer can be obtained. The calculation results show that when the f C values increase from 0.1 to 0.9, the δT/δY values of the c-Si active layer and aluminum layer will vary within the narrow ranges of −0.037±0.001°C/μm and −0.022 ± 0.001°C μm, respectively. The effects of the variations of the δT/δY values with f C are equivalent to superpose the δT/δY value determined by a specific ΔT value by a small perturbation determined by a specific Δf C value.
The irregular variations of the temperature profile curves with f C are the manifestation of the interaction between the cooling effects of the microfluidic cooling system and the heating effects of c-Si active layer generated by different heat generation mechanisms. Due to the high conductivity of aluminum material, when the f C value is larger than 0.1, the temperature values on the lower surface of aluminum layer will not decrease obviously with increasing the f C values. Figure 6(b) gives the variations of the total heat power of different heat generation mechanisms with f C . As shown by figure 6(b), when the f C value increases from 0.1 to 0.9, the total heat powers denoted by different f C values have little change in the whole Y value range, but by the partial enlarger versions of figure 6(b), we can see that the total heat powers actually fluctuate with the f C variation. Under the dual function of the cooling and heating effects, the irregular variations of the temperature profile curves with f C appear.
3.3. Evaluating the improvement of the output parameters of the c-Si SC by using the microfluidic cooling system The output parameters of a c-Si SC mainly include the short circuit current, open circuit voltage, filling factor and efficiency, which are often denoted by Jsc, Voc, FF and η, respectively. Considering the influences of the variations of f C on the temperature distributions of the c-Si SC are very small, here we only consider the influences of the variations of ΔT on the output parameters.  Figure 7 gives the output parameters varying with ΔT. As shown by figure 7, when ΔT >0°C, the values of Jsc, Voc, FF and η at the same ΔT value will all decrease with the T Back value increasing, but because the Jsc value is negative, its absolute value increases. The values of Voc, FF and η at the same T Back value will increase with the ΔT value increasing, but the Jsc absolute values at the same T Back values will first decreases and then increase.

The variations of the output parameters of the c-Si SC with ΔT
The minimum values of Voc, FF and η, which all lie in the region with T Back ≈60°C and ΔT≈0°C, are 0.6118 V, 81.8372% and 17.9564%, respectively, whereas the minimum value of Jsc, which lies in the region with T Back ≈60°C and ΔT≈10°C, is 29.7372 mA cm −2 . The maximum values of Voc, FF and η, which all lie in the region with T Back ≈−15°C and ΔT≈10°C, are 0.7614 V, 87.3831% and 23.6929%, respectively, whereas the maximum absolute value of Jsc, which lies in the region with T Back ≈−15°C and ΔT≈0°C, is 29.6548 mA cm −2 . According to figure 5(a), the temperature values on the front and back surface of the c-Si SC will be close to environmental temperature and cooling water temperature, respectively and when the ΔT value increases from 0°C to 10°C under the same T Back value, the temperature values on the back surface will be confined to cooling water temperature, while the temperature values on the front surface will only has a minor reduction, which means that the average temperature of the c-Si SC will be influenced not only by cooling water temperature, but also by environmental temperature . Due to the Voc, FF and η values increasing with the average temperature decreasing and the absolute value of Jsc decreasing with the average temperature decreasing, the variations of the Jsc,Voc, FF and η values with ΔT in figure 7 agree with the variations of the average temperature of the c-Si SC.  contour lines for δJsc and δη are unequal under different T Back values and the variation trends of δJsc and δη under a specific T Back value will decrease and increase, respectively. Figure 8 also shows that with the ΔT value increasing, the rangeability of δJsc will be at least an order of magnitude smaller than those of δVoc, δFF and δη, which is in agreement with other results related with the variations of the output parameters with temperature [16,46]. According to [16], when the temperature of c-Si SC decrease, photocurrent and reverse saturation current will remain basically unchanged and decrease, respectively, so the variations of the δJsc values with ΔT are mainly determined by reverse saturation current. Because reverse saturation current, which has a nonlinear relation with the band gap, valence-and conductionband densities of states of c-Si material, will decrease rapidly with the temperature of c-Si SC decreasing, the contour lines of δJsc in the range of ΔT from 0°C to 10°C have two distinct regions along the direction of Δ T axis, i.e. the regions with the ΔT values in the ranges from 0°C to 4°C, where the δJsc values are smaller than 10 −4 mA cm −2 and the region from 4°C to 10°C, where the δJsc values can vary in the range from 10 −4 mA cm −2 to 0.11 mA cm −2 .
The increase rates of δVoc, δFF and δη under different environmental temperatures are about 0.014 V/°C, 0.036%/°C and about 0.05%/°C in the ΔT range from 0°C to 10°C, respectively and the increase rates of δJsc are smaller than 1×10 −3 mA cm −2 .°C in the ΔT range from 0°C to 4°C and about 0.002 mA cm −2 in the ΔT range from 4°C to 10°C. In a wider range of ΔT from 0°C to 30°C, we have further given the variations of η and δη with ΔT under three different conditions, i.e. T Back = −15°C, 25°C and 60°C. Figure 8(e) shows that with the ΔT value increasing , the η value will basically keep the trend of linear increase in the ΔT range from 0°C to 30°C and the spacing between two efficiency lines with different T Back values remains unchanged.  values show three close lines in figure 8(b). Figures 8(e) and (f) also shows that all conclusions we have obtained in the ΔT range from 0°C to 10°C can be generalized to a wider ΔT range.

Conclusions
The temperature distributions and output parameters of an industrial c-Si SC with a microfluidic cooling system under natural convection atmospheric conditions have been analyzed and discussed, where the c-Si SC are encapsulated as a standard c-Si SC module and the cooling system are installed directly on the back surface of aluminum electrode. The conclusions are as following, (1)For the microfluidic cooling system, the cooling medium temperature has greater impact on the temperature distribution of the c-Si SC than the filling factor f C . As long as the f C value is larger than 0.1, the variations of f C will have a small impact on the temperature distribution of the c-Si SC. The lower the cooling medium temperature, the lower the operating temperature of the c-Si SC.
(2)By using such a cooling system in the range of environmental temperature from −15°C to 60°C, the operating temperature of the c-Si SC can be limited within the range from environmental temperature to cooling medium temperature. With the cooling medium temperature decreasing from environmental temperature to lower 10°C than environmental temperature, the increase of δVoc, δFF and δη per degree Celsiu are about 0.014 V/°C, 0.036%/°C and 0.05%/°C, respectively.
(3)The temperature distributions along the profile line in the thickness direction can be divided into two kinds of regions, i.e. the regions with linear and nonlinear temperature variations. The regions with linear temperature variations mainly appear in the c-Si active layer away from the front and back surface texture structures and aluminum layer away from the back surface texture structure and cooling system. The regions with nonlinear temperature variations mainly appear in the regions adjacent to the interfaces between antireflective coating layer and c-Si active layer, between c-Si active layer and aluminum layer and between aluminum layer and cooling system. The nonlinear temperature variations in the interface regions are mainly caused by the differences between the thermal conductivities of two materials formed the interface.
(4)The rectangular microchannel is used to guide cooling water. Compared to the total thickness of aluminum layer, the height of microchannel is very small, so the shape variations of microchannel will have little impact on the temperature distribution of c-Si active layer, especially for the microchannel with a larger f C value.
(5)Due to the fabricating and maintaining costs of photovoltaic modules increasing and the operating temperature of c-Si SC under non-concentrating conditions relatively low, various auxiliary cooling systems designed for terrestrial non-concentrating photovoltaic modules are mainly in the theoretical and experimental stage and are seldom used in practical modules. From the point view of applications, these cooling system are promising to be first applied to those modules installed in tropical regions.