Effect of Tube Number on Critical Heat Flux and Thermal Performance in Linear Fresnel Collector Based on Direct Steam Generation

In this paper, the approach of general modeling is based on energy balance about heat element collector like other studies. This energy balance involves direct solar radiation, radiation losses from flat mirror, heat collector element, heat losses from heat collector element and input heat to heat transfer fluid that is visible in Figure 1. On the other side, heat transfer circuit of this process is observable Figure 2.

1.png

Figure 1. Fresnel model [2]

2.png

Figure 2. Heat transfer circuit for Fresnel system

According to this circuit, after entering the sun heat flux to the mirror, radiation heat flux reflected from mirror to external surface of glass that its amount is $I_{1}$ . Percentage of the radiation heat flux as conduction heat flux passes from glass wall and residual radiation heat flux according to glass reflection coefficient as convection heat flux enters to the environment. The environment inside the cavity is relatively vacuum.

In the following, transitional radiation heat flux in the amount of $I_{2}$  enter to external surface of tubes inside of cavity that it was divided between tubes. In this section, according to the reflection coefficient of tube, a part of that heat flux is reflected and exchanged radiation and convection heat flux with glass internal surface. We called off the radiation heat flux between glass and tubes because the amount of this heat flux is very low. The remaining amount of heat flux transient from tubes wall as conduction heat flux and heat enter to heat transfer fluid as convection heat flux.

If number of tubes increase to N so convection heat flux from tube to glass and radiation heat flux from mirror to tube divide to number of tubes.

2.1 Radiation heat flux from mirror to glass and tube

This section expresses the equations about radiation heat flux from mirror to glass ( $\dot{q}_{g, s . r a d}$ ) and tube ( $\dot{q}_{a, s . r a d}$ ) [2]:

$\dot{q}_{g, s . r a d}=\mathrm{I}_{s u n} \cdot \beta . \alpha_{g} . \mathrm{W}$              (1)

$\dot{q}_{a, s . r a d}=\frac{1}{N} \mathrm{I}_{s u n} \cdot \beta \cdot \gamma \cdot \alpha_{\mathrm{a}} \cdot \mathrm{W}$          (2)

In the above equations $W, N, I_{s u n}, \alpha_{a}, \alpha_{g}, \beta \text { and } \gamma$  respectively show width of total of mirrors, number of tubes, the amount of sun irradiation, absorption coefficient of tube, absorption coefficient of glass and reflection and transient coefficient of glass.

2.2 Conduction heat flux in glass and tube wall

The heat passes from tubes and glass wall in the form of conduction heat flux when radiation heat flux receives to external surface of glass and tubes. This heat flux has direct connection with glass and tube thermal conductivity coefficient [14]. According to glass and tube material, conduction heat flux has high value when this coefficient has high value and at the result, it effects in coefficient of heat transfer for fluid and the convection heat flux from absorber to fluid increases eventually.

2.3 Convection heat flux from glass to environment

As soon as the radiation heat flux reaches to glass’s wall, the heat transfers from that and ultimately enters to cavity as convection heat flux. According to glass reflection coefficient, remaining of this flux reflects from glass and is transferred to the ambient in the form of convection heat transfer ( $\dot{q}_{g-e, \text { conv }}$ ). In the following, equations of this section are presented [2, 10]:

$\dot{q}_{g-e . c o n v}=h_{e} \cdot w_{c}\left(T_{g . e x}-T_{a m b}\right)$             (3)

The value of heat transfer coefficient is calculated the following equations.

$h_{e}=k_{e} \cdot N u_{e} / w_{c}$             (4)

$N u_{e}=0.15 R a^{\frac{1}{3}}$                (5)

$R a=\left(g \cdot \beta_{e}\left(T_{g, e x}-T_{a m b}\right) w_{c}^{3}\right) /\left(v_{e} \cdot \alpha\right)$       (6)

In the above equations $h_{e}, w_{c}, T_{g, e x}, T_{a m b}, k_{e}, N u_{e}, R a, g, \beta_{e}, v_{e} \text { and } \alpha$ respectively show heat transfer coefficient, length of bottom trapezoidal base, temperature of external surface of glass, temperature of ambient, thermal conductivity coefficient of ambient air, Nusselt number about ambient, Rayleigh number, gravitational acceleration, volumetric thermal expansion coefficient about air, kinematic viscosity about air and thermal diffusivity.

2.4 Convection heat flux from tube to fluid

After passing of conduction heat flux from tube wall, the heat in the form of convection heat transfer ( $\dot{q}_{a-f, c o n v}$ ) is transferred to fluid and increases the fluid temperature.

In the following, we express equations about convection heat flux for three section of fluid flow, including single-phase, sub-cooled and two phase.

2.5 Single-phase fluid flow section

Heat flux from tube to fluid is calculated by Eq. (7) [2, 14]:

$\dot{q}_{a-r, c o n v}=h_{f} \pi D_{a, i n}\left(T_{a, i n}-T_{m}\right)$                (7)

Parameter of $T_{m} \text { equals } T_{m}=\frac{T_{i n}+T_{o u t}}{2}$ and Nusselt number in the heat transfer coefficient equation is calculated by the following equation.

$N u_{f}=0.023\left(R e_{f}\right)^{0.8}\left(P r_{f}\right)^{0.4}$                      (8)

In the above equations $h_{f}, T_{a, i n}, T_{m}, T_{i n}, T_{o u t}, N u_{f}, R e_{f} \text { and } P r_{f}$ respectively show heat transfer coefficient of fluid, temperature of internal surface of tube, average temperature of fluid, inlet temperature of fluid, outlet temperature of fluid, Nusselt number, Reynolds number for fluid and Prandtl number for fluid.

2.6 Sub-cooled boiling section

Whenever the temperature of internal surface of the tube is reached to saturation temperature of fluid, the heat transfer fluid enters to sub-cooled section. Convection heat flux from tube to fluid is calculated as follows by Forster-Zubber [15]:

$\dot{q}_{n b}=h_{n b} \pi D_{a, i n}\left(T_{a, i n}-T_{o u t}\right)$               (9)

The value of heat transfer coefficient is calculated by Eq. (10).

$h_{n b}=0.00122 \frac{k_{f}^{0.79} C_{p, f}^{0.45} \rho_{l}^{0.49}}{\sigma^{0.5} \mu_{f}^{0.29} h_{l v}^{0.24} \rho_{v}^{0.24}}\left(T_{a, i n}\right.$

$\left.-T_{\text {out}}\right)^{0.24} \Delta P_{\text {sat}}^{0.75}$              (10)

In the above equations $h_{n b}, \pi, D_{a, i n}, T_{a, i n}, T_{o u t}, k_{f}, C_{p, f}, \rho_{l}, \sigma, \mu_{f}, h_{l v}, \rho_{v}$ and $\Delta P_{s a t}$  respectively show heat transfer coefficient, Pi number, internal diameter of tube, external diameter of tube, thermal conductivity coefficient of fluid, specific heat of fluid in constant pressure, density of fluid in liquid mode, Stephen-Boltzmann coefficient, viscosity of fluid, enthalpy, density of fluid in vapor mode and difference of fluid pressure.

2.7 Two-phase fluid flow section

As soon as the fluid temperature reaches saturation temperature, the heat transfer fluid enters the two-phase section. The difference between this section and other sections is in the type of fluid flow and different method of operation about fluid. In this section, fluid temperature remains constant and only vapor quality changes in length of tube until critical heat flux. For this section, Shah Equation is used [16] as follows.

$\dot{q}_{n b}=h_{t p} \pi D_{a, i n}\left(T_{a, i n}-373.15\right)$             (11)

$h_{t p}=h_{L}\left[(1-x)^{0.8}+\frac{3.8 x^{0.76}(1-x)^{0.04}}{p r^{0.38}}\right]$            (12)

$h_{L}=0.023 R e_{l}^{0.8} P r_{l}^{0.4} \frac{k_{l}}{D_{a, i n}}$                 (13) 

In the above equations $h_{t p}, \pi, D_{a, i n}, T_{a, i n}, h_{L}, x, p r, R e_{l}, P r_{l} \text { and } k_{l}$ respectively show heat transfer coefficient in two-phase mode, Pi number, internal diameter of tube, temperature of internal surface of tube, heat transfer coefficient in convection mode, vapor quality percent, ratio of saturation pressure to critical pressure, Reynolds number in liquid mode, Prandtl number in liquid mode and thermal conductivity coefficient in liquid mode.

2.8 Energy balance inside of fluid

Energy balance’s equations for inside of absorber tube change because mode of fluid convert from single-phase to sub-cooled and then to two-phase. It is noteworthy that energy balance’s equations define based on enthalpy and vapor quality in two-phase section while they are same for single-phase [2] and sub-cooled [2] sections and explain based on temperature difference of fluid in the beginning and end of fluid flow path.

2.9 Energy balance for single-phase and sub-cooled boiling sections

Since in the single-phase and sub-cooled sections, vapor quality equals to zero, specific heat and mass flow rate are two important parameters in balance of energy. Eq. (14) shows balance of energy in these sections.

$\dot{q}_{a-f, c o n v}=\frac{\dot{m} C_{p}\left(T_{\text {out}}-T_{\text {in}}\right)}{l}$      (14)

In the above equation $\dot{m}, C_{p}, T_{\text {out}}, T_{\text {in}} \text {and } l$ respectively show mass flow rate of fluid, specific heat in constant pressure, and outlet temperature of fluid, inlet temperature of fluid and length of each element from tube.

2.10 Energy balance for two-phase flow section

In this stage, fluid is entered to two-phase flow section and we see combine of liquid and vapor. As a result, important parameter in balance of energy inside of fluid is fluid enthalpy. Eq. (15) expresses this balance of energy.

$\dot{q}_{a-f, c o n v}=\frac{m_{p} h_{f g}}{l}$       (15)

In the above equation $m_{p}$ is fluid mass in each element that calculated by bottom equation.

$m_{p}=m_{\text {total}}\left(x_{\text {out}}-x_{\text {in}}\right)$              (16)

Also $m_{\text {total }}$ is total mass of flow that is calculated by $m_{\text {total}}=\rho V$ .

In the above equations $m_{p}, h_{f g}, m_{\text {total}}, x_{\text {out}}, x_{\text {in}}, \rho \text { and } V$ respectively show mass fluid in each element, enthalpy, total mass fluid, vapor quality in end of element, vapor quality in first of element, density of fluid and volume of fluid.

2.11 Convection heat flux from tube to glass

By entering the radiation heat flux to external surface of tube, some of this heat flux passed from tube wall and the remainder of it in the form of convection heat transfer $\left(\dot{q}_{a-g, c o n v}\right)$ go to glass. The following equation shows this heat flux [15, 17]:

$\dot{q}_{a-g . c o n v}=\frac{1}{N} h_{c} l\left(T_{a, e x}-T_{g, i n}\right)$             (17）

In the above equation $N, h_{s}, l, T_{a, e x} \text { and } T_{g, i n}$ respectively show number of tube, heat transfer coefficient about cavity, length of element, temperature of external surface of tube and temperature of internal surface of tube.

2.12 Balance of energy in set of absorber

Actually, the first law of thermodynamic used as a helpful tool for more problem of heat transfer [10]. For the expression of this law, first we should specify the control volume. In this model, we can consider two control volume include control volume for glass and control volume for tube. Eqns. (18) and (19) respectively are equations about conversion of energy in glass and tube.

$\dot{q}_{g, s . r a d}+\left(N . \dot{q}_{a-g . c o n v}\right)-\dot{q}_{g-e, c o n v}=0$       (18)

$\dot{q}_{a, s, r a d}-\dot{q}_{a-f, c o n v}-\dot{q}_{a-g . c o n v}=0$            (19)

Also, two balance of energy are established for surfaces of glass and tube that come in Eqns. (20) and (21).

$\dot{q}_{g, \text {cond}}+\dot{q}_{a-g . c o n v}=0$                    (20)

$\dot{q}_{a, \text {cond}}-\dot{q}_{a-f . c o n v}=0$                   (21)

In this study, the validation is done based on Sahoo and colleagues’ research work [2]. Sahoo did hydrothermal analysis for fluid in linear Fresnel concentrator. This analysis is for two section of fluid flow embrace single-phase and two-phase. Also, for hydrothermal analysis was used from one dimensional model in both sections. It is worth mentioning, for solution of this one dimensional model, has used from solution of energy balance equations. In Table 1, geometrical and thermal properties was shown.

The used fluid is water for validation that its result shown in Figure 4. Difference of distribution of bulk fluid temperature in the first path and in two-phase section is insignificant and only exist difference of 1.9 percent than Sahoo’s results in the start of two-phase flow. In this validation, in single-phase section exerted pressure was enhanced so that eventually it becomes equal to 1.4 bar in two-phase section.

According to this, saturation temperature of fluid increased in two-phase section. Since, used fluid is water and as we know the water after passing of sub-cooled section entered to two-phase flow, then fluid temperature is constant in this section and only vapor quality changed in length of fluid flow path. So, fluid temperature was constant and equal saturation temperature of fluid (373.15 Kelvin).

4.png

Figure 4. Comparison of distribution of bulk fluid temperature in length of path in done analysis by Sahoo et al. [2]

Table 1. geometrical and thermal properties about set of linear Fresnel concentrator [2]

In this research, the heat transfer analysis is as regards a model of the linear Fresnel collector in two modes of single and double tubes for three sections of single-phase, sub-cooled and two-phase.

The distribution of fluid temperature, internal surface temperature of tube and vapor quality is studied in this research. With due to attention to the type of system that is considered from type of direct steam generation, the water is used as a heat transfer fluid. So, geometry of cavity is considered trapezoidal type.

In this study was used from performing code calculation and in every section of fluid regime, iterative method based on Gauss-Seidel approach was entered by “While” construct in code calculation. The scrutinizing of critical heat flux (CHF) in nanofluids that is the most important part of modeling is obtained from the Look up Table (LUT) method [19] and this approach has been distinguished current research rather than other works. In this table, the vapor quality is presented in the order of the heat flux entered into the fluid. The obtained results of convective heat flux in two-phase section of fluid flow in each element were contrasted with Look up Table’s heat fluxes. The location of CHF is shown when the acquired convective heat flux been larger than corresponding element in Look up Table’s data. In the following, distribution of internal surface temperature of tube, distribution of fluid temperature, and distribution of vapor quality are investigated.

The most important target of this research is the comparison of the incidence of critical heat flux in different modes. First, comparison is done for complexes of single tube and double tubes by mass flow rate of fluid equals 0.23 that in complex of double tubes there is an increase of 21.6 percent in length of the incidence of critical heat flux, a decrease of 1.6 in the distribution of internal surface temperature of tube and a decrease of 9 percent of vapor quality than complex of single tube. Consequently, the analysis for three different of mass flow rate including 0.23, 0.2 and 0.19 in complexes of single tube and double tubes are done separately. As a result, there is an increase of 3.8 percent in length of critical heat flux in mass flow rate equals 0.2 than 0.23 and so there is an increase of 4.2 percent in length of critical heat flux in mass flow rate equals 0.19 than 0.23 for the complex of single tube. So, for the complex of double tubes, the increase of 2.2 percent in length of critical heat flux in mass flow rate equals 0.2 than 0.23 and an increase of 3 percent in length of critical heat in mass flow rate equals 0.19 than 0.23 can be seen.