Performance Analysis of a Natural Convection Solar Air Heater Operating in Ho Chi Minh City

Figure 1 shows a schematic diagram of a natural convection solar air heater used in this analytical study. The diagram illustrates how air flows through the collector, which consists of a glass cover, an absorber plate, and insulation. The device is tilted at an angle θ relative to the ground. Solar radiation, represented by the vector I, strikes the tilted glass surface. The diagram also indicates the inlet and outlet air positions with elevations z₁ and z₂, respectively. Air enters the channel at the lower end and exits at the higher end, driven by natural convection [17, 18].

2.1 Energy balance equations

• Energy balance equation for glass cover:

$\begin{array}{r}\alpha_g I_t=h_w\left(T_g-T_a\right)+h_{r, g, s}\left(T_g-T_s\right)+ \\ h_{c, g, f}\left(T_g-T_f\right)+h_{r, g, p}\left(T_g-T_p\right)\end{array}$             (1)

where, I_t is the average radiation on tilted surface, $\alpha_g$ is absorptivity of glass cover, h_w is convection heat transfer coefficient due to wind, T_gis mean temperature of glass, T_a is ambient temperature (inlet temperature), h_r,g,sis radiation heat transfer coefficient between glass and sky, h_r,g,p is radiation heat transfer coefficient between glass and absorber plate, h_c,g,f  is convection heat transfer coefficient between glass and airflow, T_f  is mean temperature of air, T_p is mean temperature of absorber plate, T_s is sky temperature.

Figure 1. The natural convection solar air heater studied analytically and experimentally in the current work

• Energy balance equation for the airflow:

$Q=L b h_{c, g f}\left(T_g-T_f\right)+L b h_{c, p, f}\left(T_p-T_f\right)$             (2)

$Q=\dot{m} c_p\left(T_o-T_a\right)$          (3)

where, Q is useful heat gain, L is collector length, b is collector width, $\dot{m}$ is air mass flow rate, c_p  is specific heat of air, T_o is outlet air temperature, h_c,p,f  is convection heat transfer coefficient between the plate and airflow.

• Energy balance equation for absorber plate:

$\alpha_p \tau_g I_t=h_{c, p, f}\left(T_p-T_f\right)+h_{r, g, p}\left(T_p-T_g\right)+h_b\left(T_p-T_a\right)$        (4)

where, a_p is the absorptivity of the plate, $\tau_g$ is the transmissivity of the glass cover, and h_b is the convection heat transfer coefficient between the plate and the environment (heat loss through the insulation).

2.2 Induced air flow rate

Applying Bernoulli’s equation to the airflow channel from the inlet (location 1) to the outlet (location 2), as shown in Figure 1 [19]:

$\begin{aligned} & z_1 \rho_1 g+p_1+\frac{\rho_1 v_1^2}{2}=z_2 \rho_2 g+p_2+\frac{\rho_2 v_2^2}{2}+ f \frac{L}{D_h} \frac{\rho v^2}{2}+K_1 \frac{\rho_1 v_1^2}{2}+K_2 \frac{\rho_2 v_2^2}{2}+K_3 \frac{\rho v^2}{2}\end{aligned}$             (5)

Since both the inlet and outlet vents are open to the atmosphere, the pressure at these points is equal to atmospheric pressure (p₁ = p₂). The air enters the inlet from the atmosphere, which is considered an infinite reservoir with a negligible initial velocity (v₁ = 0). Hence, Eq. (5) is reduced to:

$z_1 \rho_1 g-z_2 \rho_2 g=\frac{\rho_2 v_2^2}{2}+f \frac{L}{D_h} \frac{\rho v^2}{2}+K_2 \frac{\rho_2 v_2^2}{2}+K_3 \frac{\rho v^2}{2}$             (6)

where, z is elevation, ρ is air density, g is gravitational acceleration, f is friction factor, D_his hydraulic diameter, K₂ and K₃ are the loss coefficients at the outlet and other losses of the channel, respectively. K₂ = 1 and K₃ = 1.1 [20].

$z_1 \rho_1 g-z_2 \rho_2 g=2 \frac{\rho_2 v_2^2}{2}+f \frac{L}{D_h} \frac{\rho v^2}{2}+1.1 \frac{\rho v^2}{2}$         (7)

Using Boussinesq approximation: $\rho_1=\rho \beta \mathrm{T}_a$ and $\rho_2=\rho \beta \mathrm{T}_o$. And assume v₂ = v:

$g L \sin (\theta) \beta\left(\mathrm{T}_o-\mathrm{T}_a\right) \rho=\frac{1}{2}\left(2 \rho \beta \mathrm{~T}_o v^2+f \frac{L}{D_h} \rho v^2+1.1 \rho v^2\right)$              (8)

where, $\beta$ is the thermal expansivity of air, $\beta=1 / T_f$ in which $T_f=0.25 T_a+0.75 T_o [20], \theta$ is the inclination angle of the collector. Therefore, air velocity inside the channel can be estimated as:

$v^2=2 g \beta L \sin (\theta)\left(T_o-T_a\right)\left(1.1+f L / D_h+2 \beta T_o\right)^{-1}$            (9)

Hydraulic diameter:

$D_h=\frac{4 H b}{2(H+b)}$         (10)

The friction factor (f) for laminar and turbulent flows is calculated by the following correlations, respectively:

$f=1.906(G r / P r)^{1 / 12}$               (11)

$f=1.368(G r / P r)^{1 / 11.9}$           (12)

where, Pr is the Prandtl number of air.

Grashof number:

$G r=g \beta\left(T_p-T_g\right) L^3 \frac{\rho^2}{\mu^2}$            (13)

2.3 Solar radiation on the inclined collector

The Liu and Jordan model is an isotropic model that simplifies solar radiation by assuming diffuse radiation is uniformly distributed across the sky. It is a foundational method for estimating the solar radiation incident on a tilted surface by using global horizontal radiation data. This model is particularly useful for predicting solar energy in cloudy weather conditions and has been applied in various studies to determine the optimal tilt angle for solar collectors. Average radiation on a sloped surface is evaluated by the method of Liu and Jordan as [21, 22]:

$I_t=I\left(1-I_d / I\right) R_b+I\left(I_d / I\right) \frac{1+\operatorname{Cos}(\theta)}{2}+I \rho_g \frac{1-\operatorname{Cos}(\theta)}{2}$              (14)

where, R_b is the ratio of the average beam radiation on the tilted surface to that on a horizontal surface, I_d  is diffuse radiation, $\rho_g$ is ground reflectance.

$R_b=\frac{\operatorname{Cos}(\phi-\theta) \operatorname{Cos}(\delta) \sin \left(\omega_s^{\prime}\right)+\left(\pi \omega_s^{\prime} / 180\right) \sin (\phi-\theta) \sin (\delta)}{\operatorname{Cos}(\phi) \operatorname{Cos}(\delta) \sin \left(\omega_s\right)+\left(\pi \omega_s / 180\right) \sin (\phi) \sin (\delta)}$              (15)

where, $\phi$ is latitude, w_s is the sunset hour angle, $\omega _{s}^{'}$ is the sunset hour angle for the tilted surface, $\delta$ is solar declination.

$\delta=23.45 \sin \left(360 \frac{284+n}{365}\right)$        (16)

where, n is the day number of the year.

$\omega_s=\arccos \left(-\sin (\phi) \frac{\sin (\delta)}{\cos (\phi) \cos (\delta)}\right)$           (17)               

$\omega_s^{\prime}=\min \left[\begin{array}{l}\arccos (-\tan (\phi) \tan (\delta)) \\ \arccos (-\tan (\phi-\theta) \tan (\delta))\end{array}\right]$               (18)

I_d/I in Eq. (14) is a function of the clearness index $\bar{K}_T$ as:

$I_d / \mathrm{I}=1.391-3.56 \bar{K}_T+4.189 \bar{K}_T^2-2.137 \bar{K}_T^3$            (19)

for $\omega_{\mathrm{s}}<81.4^{\circ}$:

$I_d / I=1.311-3.022 \bar{K}_T+3.427 \bar{K}_T^2-1.821 \bar{K}_T^3$           (20)

for $\omega_{\mathrm{s}}>81.4^{\circ}$:

$\bar{K}_T=I / I_o$       (21)

where, I_o is the extraterrestrial radiation on a horizontal surface.

$\begin{aligned} I_o= & \frac{G_{S C}}{\pi}\left(1+0.033 \operatorname{Cos} \frac{360 n}{365}\right) \times \left(\operatorname{Cos}(\delta) \operatorname{Cos}(\phi) \sin \left(\omega_s\right)+\frac{\pi \omega_s}{180} \sin (\delta) \sin (\phi)\right)\end{aligned}$      (22)      

where, G_SC is the solar constant, $G_{S C}=1367 \mathrm{~W} / \mathrm{m}^2$.

2.4 Other equations

Heat transfer coefficient due to wind is estimated by the McAdam equation [23]:

$h_w=5.7+3.8 V_w$           (23)

where, V_wis wind velocity.

Sky temperature [24]:

$T_s=0.0552 T_a^{1.5}$                (24)          

where, T_ais the ambient temperature.

The heat transfer coefficient between the air flow in the channel and the glass (h_c,g,f), and the heat transfer coefficient between the air flow in the channel and the absorber plate (h_c,p,f), are supposed equal and are given by [20]:

$h_{c, g, f}=\left(0.68+0.67 \frac{R a^{0.25}}{\left(1+(0.492 / P r)^{9 / 16}\right)^{4 / 9}}\right) k / L$         (25)

for laminar flow (Ra < 10⁹):

$h_{c, g, f}=\left(0.825+0.39 \frac{R a^{1 / 6}}{\left(1+(0.492 / P r)^{9 / 16}\right)^{8 / 27}}\right)^2 k / L$              (26)

for turbulent flow (Ra > 10⁹).

where, Ra is the Rayleigh number (Ra = GrPr), k is the thermal conductivity of air.

The radiation heat transfer coefficient between the glass and the sky:

$h_{r, g, s}=\sigma \varepsilon_g\left(T_g^2+T_s^2\right)\left(T_g+T_s\right)$       (27)

where, $\varepsilon_g$ is the emissivity of glass, $\sigma$ is Stefan’s constant.

The radiation heat transfer coefficient between the glass and the absorber plate:

$h_{r, g, p}=\sigma\left(T_g^2+T_p^2\right) \frac{T_g+T_p}{1 / \varepsilon_g+1 / \varepsilon_p-1}$        (28)

where, $\varepsilon_p$ is the emissivity of the absorber plate.

Convection heat transfer coefficient between the plate and the environment:

h_b = k_i/t_i                  (29)

where, t_i is the insulation thickness, and k_i is the thermal conductivity of insulation.

Induced air mass flow rate:

$\dot{m}=\rho v b H$              (30)

where, H is the air channel height.

Thermal efficiency of the natural solar air heater:

$\eta=\frac{Q}{L b I_t}$           (31)

Temperature effectiveness [25]:

$\varepsilon=\frac{T_o-T_a}{T_p-T_a}$         (32)

Table 1 shows the parameters entered into the simulation program. The radiation to the inclined collector is calculated from the monthly average horizontal radiation at Ho Chi Minh City, given in Table 2. This data is averaged from the weather data distributed with TRNSYS simulation software. The mathematical model above is written in EES (F-chart software), which is good at dealing with nonlinear algebraic equations.

Table 1. Input parameters

Table 2. Average total radiation on a horizontal surface (I) in Ho Chi Minh City

Figure 4 illustrates the average solar radiation on the inclined collector (I_t) as a function of the inclination angle (θ) for four typical months: January, March, September, and November. The tilted surface radiation is calculated using the Liu and Jordan model, which is described in Section 2.3. The input for the model is the monthly average total radiation on a horizontal surface (I), which is provided in Table 2. The graph shows that for each month, there is an optimal inclination angle that maximizes the solar radiation received. For January, the maximum radiation is around a 35° inclination angle, while for March and November, the peak is at a slightly lower angle. For September, the radiation is generally lower, and the optimal angle is less pronounced. These results demonstrate how the optimal tilt angle for the collector changes depending on the month due to the sun's position.

Figure 4. Solar radiation on the inclined collector for typical months

Figure 5 qualitatively shows that the induced air mass flow rate is influenced by both the air gap height (H) and the collector inclination angle (θ) in February with L = 1 m. Quantitatively, the results indicate that a larger air gap height leads to a higher mass flow rate. At a 65° inclination, the mass flow rate peaked at roughly 0.0043 kg/s for H = 100 mm. This exceeded rates for H = 65 mm (0.0042 kg/s) and H = 30 mm (0.0040 kg/s). A larger air channel provides less resistance to the airflow, allowing a greater volume of air to circulate for the same driving force of natural convection. The graph also shows that for each air gap height, there is an optimal inclination angle that maximizes the flow rate, which for all three cases appears to be between 60° and 70°. This is because the natural convection driving force, which is a function of the temperature difference and the vertical height difference (z₂−z₁), is maximized at this angle, while frictional losses also play a role.

Figure 5. Effect of air gap height and collector inclination on induced air flow rate

Figure 6. Effect of air gap height and collector inclination on temperature effectiveness

Figure 6 demonstrates the relationship between temperature effectiveness ($\varepsilon$), air gap height (H), and collector inclination angle (θ). The graph shows that a smaller air gap height results in a higher temperature effectiveness. The curve for H = 30 mm is consistently above the curves for H = 65 mm and H = 100 mm, indicating a greater temperature effectiveness. For a given amount of heat transferred to the air, a smaller air channel forces the air to absorb the heat more intensely, leading to a larger temperature increase. As a result, the outlet air temperature (T_o) becomes significantly higher relative to the ambient temperature (T_a) and the absorber plate temperature (T_p), as defined by the temperature effectiveness formula (32). Temperature effectiveness drops as the inclination angle increases. This occurs primarily because the natural convection airflow rate decreases. As the angle of inclination increases, the component of gravity that drives the buoyant air upward along the channel decreases, reducing the air velocity. A lower air velocity results in a lower mass flow rate, which in turn reduces the amount of useful heat gained by the air for a given solar radiation input. Additionally, as the inclination angle increases, the solar radiation absorbed by the collector may also decrease after a certain optimal angle is reached, further lowering the overall heat transfer o the air.

Figure 7. Effect of air gap height, month and collector inclination on thermal efficiency

Figure 8. Effect of collector length (L) when fixing H = 65 cm

Figure 7 demonstrates the effect of air gap height (H), month, and collector inclination (θ) on thermal efficiency (η). The graphs for February, March, October, and December all show a consistent trend: for a given month and inclination angle, a larger air gap height (H) leads to a higher thermal efficiency. In the February graph, the thermal efficiency for H = 100 cm is significantly higher than for H = 30 cm. As seen in Figure 5, a larger air gap height leads to a higher air mass flow rate, which in turn increases the useful heat gain. While a larger H results in a lower temperature effectiveness (as per Figure 6), the increased mass flow rate from the larger channel size outweighs this effect, leading to a higher overall thermal efficiency. The graphs also show that for each case, there is an optimal inclination angle that maximizes the thermal efficiency, which varies with the month due to changes in solar radiation. Figure 7 shows that the optimal inclination angle for thermal efficiency varies significantly by month. For February and March, the thermal efficiency peaks at an inclination angle between 50° and 60° degrees for all air gap heights (H = 30 cm, H = 65 cm, and H = 100 cm). This suggests that for months with a higher sun angle, a steeper collector tilt is required to achieve the best performance. In contrast, for October and December, the optimal inclination angle is lower, typically between 40° and 50°. This is consistent with the sun's lower position in the sky during these months, where a shallower angle is needed to capture a greater amount of solar radiation. The graphs clearly show that to maximize the thermal efficiency of the solar air heater, the inclination angle should be adjusted seasonally to align with the sun's position.

Figure 8 examines the impact of collector length (L) on thermal efficiency (η) for a fixed air gap height of H = 65 cm across several months (February, March, October, and December). It shows that a shorter collector length leads to a higher thermal efficiency. For any given month and inclination angle, the thermal efficiency for a collector length of L = 1 m is the highest, followed by L = 1.5 m, and then L = 2 m. In February, at an inclination angle of 60°, the thermal efficiency is highest for L = 1 m, and lowest for L = 2 m. This is because a longer channel increases frictional losses, which reduces the induced air mass flow rate for a given temperature difference. While a longer collector can absorb more solar energy, the reduced air flow rate due to increased channel friction means that the air heats up less effectively along its path, leading to a lower overall thermal efficiency. As with the other figures, each graph also shows an optimal inclination angle for thermal efficiency, which varies by month.