Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy

*Nguyen Van Hap and Nguyen Minh Phu*

## **Abstract**

Multipass air collectors are commonly used because they produce higher air temperature than that of a single-pass one due to reduced top heat loss. In this chapter, two mathematical models of convection and radiant heat transfer in a double-pass solar air heater were presented. They included an average temperature model and a model of temperature variation along the airflow direction. The method for solving these two mathematical models was reported. The average temperature model was solved by dealing with a system of linear algebraic equations, whereas the other model was derived as ordinary differential equations and solved by a numerical integration. The calculation programs were developed in EES software. The computation time of temperature variation model was about 0.9 s, but that of the average temperature model was negligible. Outcomes from two solutions were almost identical. The largest error of the outlet air temperature was 2.1%. The models are applicable to multipass collectors with or without recycling airflow.

**Keywords:** temperature gradient, numerical integration, multiple-pass heat exchanger, convection heat transfer, radiation heat transfer

## **1. Introduction**

Energy conversion is a current concern for developing countries because the industrial development is premium strategic target and the traditional fuel is increasingly scarce. Solar thermal energy conversion is deployed massively in countries near the equator due to the availability and stability of solar intensity. Converting thermal radiation into hot air is widely applied in drying agriculture, space heating, desiccant regeneration, timber seasoning, and natural ventilation. A solar air heater (SAH) has a simple structure and is easily made from local materials [1]. However, it has the disadvantage that the convection heat transfer coefficient (HTC) of the air is low, and the heat loss is considerable. Therefore, solutions to enhance heat transfer are proposed and applied. It may be the addition of inserts into the SAH duct to remove the viscous sublayer close to the absorption plate. They can be vortex generators [2–4], baffles [5–7], fins [8], or ribs [9]. Increasing the number of air passes in the SAH duct is also a measure to reduce heat loss due to the high

temperature of the absorber plate. SAH having two, three, or four passes with or without airflow recycling has demonstrated a high thermohydraulic performance. The solution method for the mathematical model of a multiple-pass SAH has always been of interest to researchers since the SAH has several glasses, absorber plate, back plate, and multiple airflows. There are two commonly used analytical models: local solution and mean solution. The local solution establishes ordinary differential equations (ODEs) along with the air temperature boundary conditions. The mean solution approximates the temperature gradients to the temperature differences in each air pass to form a system of linear algebraic equations. It is obvious that the mean solution is more straightforward than the local solution and able to calculate by hand. However, the local solution can predict the temperature of fluid flow and heat exchanger surfaces along the collector length. From these temperature profiles, temperature cross or temperature meet phenomena can be detected and local corrections can be made. **Table 1** presents literature review on solutions studied on multiple-pass SAH during the last 15 years. Both the solutions are almost equally used by researchers. Typically, there are four research groups on the analytical model of multiple-pass SAH. The research group of Ho et al. employed the local solution. Meanwhile, the research teams of Velmurugan et al. and Matheswaran et al. applied the mean solution. Our research group (Phu et al.) used both the solutions.


#### **Table 1.**

*Analytical studies on the multiple-pass SAH.*

*Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy DOI: http://dx.doi.org/10.5772/intechopen.105133*

From the above extensive literature review, there has not been a study on comparing the heat transfer model for a multiple-pass SAH using both the solutions. Hence, in this chapter, both the solutions are applied to a typical double-pass SAH to realize the mathematical model, the solution method, and the result. From the analysis in this chapter, it can be developed for a variety of air collectors and further research on hydraulics and energy performance.

## **2. Model description**

**Figure 1** depicts the physical model of a double-pass solar air heater. It consists of a glass cover, an absorber plate, and a back plate. Solar radiation penetrates the glass to the absorber plate and heats the absorber plate. Airflow travels over the plate surface and receives heat. In addition, the air can receive more heat from the glass and back plate because these surfaces absorb thermal radiation from the absorber plate. The air moves from the top to the bottom channel forming two passes. The following subsection presents the five-temperature calculation models of the double-pass SAH, including glass (*Tg*), air in first pass (*Tf*1), absorption plate (*Tp*), air in second pass (*Tf*2), and the back plate (*Tb*).

#### **2.1 Local solution**

In this solution, the temperature of the collector components is a function of *x*-direction, i.e.,*T*(*x*). The energy balance for the glass cover is shown in Eq. (1). Solar energy arriving at the collector (*I*) and absorbed by the glass is balanced with the heat exchanged by convection and radiation above the glass surface, convection to the air in the first pass, and radiation to the absorber plate [24]:

$$\ln \mathbf{I}\_{\mathbf{g}} + h\_w \left( T\_a - T\_\mathbf{g}(\mathbf{x}) \right) + h\_{\text{ra}} \left( T\_{\text{sky}} - T\_\mathbf{g}(\mathbf{x}) \right) + h\_{r\mathbf{g},\mathbf{p}} \left( T\_p(\mathbf{x}) - T\_\mathbf{g}(\mathbf{x}) \right) + h\_{f1\mathbf{g}} \left( T\_{f1}(\mathbf{x}) - T\_\mathbf{g}(\mathbf{x}) \right) = \mathbf{0}. \tag{1}$$

The interpretation of the symbols can be found in **Table 2** and **Figure 1**. The heat transfer coefficients (*h*) can be viewed in Eqs. (16)–(20). The variation of the air temperature in the first pass in the direction of flow is described by Eq. (2). The

**Figure 1.** *Double-pass solar air heater.*


#### **Table 2.**

*Input parameters.*

change in air temperature is due to the convective heat exchange with the glass cover and the absorber plate [25]:

$$\frac{d\mathcal{T}\_{f1}(\mathbf{x})}{d\mathbf{x}} = \frac{\mathcal{W}h\_{f1,p}\left(T\_{\mathcal{S}}(\mathbf{x}) - T\_{f1}(\mathbf{x})\right) + \mathcal{W}h\_{f1,p}\left(T\_p(\mathbf{x}) - T\_{f1}(\mathbf{x})\right)}{\dot{m}c\_p}.\tag{2}$$

Like the glass cover, the heat balance for the absorber plate is expressed as follows:

$$\begin{split} & \left( \operatorname{I\tau}\_{\mathfrak{g}} a\_{\mathfrak{p}} + h\_{f1,\mathfrak{p}} \left( T\_{f1}(\mathfrak{x}) - T\_{\mathfrak{p}}(\mathfrak{x}) \right) + h\_{f2,\mathfrak{p}} \left( T\_{f2}(\mathfrak{x}) - T\_{\mathfrak{p}}(\mathfrak{x}) \right) \right. \\ & \left. + h\_{\mathfrak{r}\mathfrak{g},\mathfrak{p}} \left( T\_{\mathfrak{g}}(\mathfrak{x}) - T\_{\mathfrak{p}}(\mathfrak{x}) \right) + p \mathbf{x} h\_{r\mathfrak{b},\mathfrak{p}} \left( T\_{\mathfrak{b}}(\mathfrak{x}) - T\_{\mathfrak{p}}(\mathfrak{x}) \right) = \mathbf{0}. \end{split} \tag{3}$$

The variation of air temperature in the second pass is

$$\frac{dT\_{f2}(\mathbf{x})}{d\mathbf{x}} = \frac{\mathsf{W}h\_{f2,b}\left(T\_b(\mathbf{x}) - T\_{f2}(\mathbf{x})\right) + \mathsf{W}h\_{f2,p}\left(T\_p(\mathbf{x}) - T\_{f2}(\mathbf{x})\right)}{\dot{m}c\_p}.\tag{4}$$

The energy balance of the back plate is

$$h\_{r,b,p}\left(T\_b(\mathbf{x}) - T\_p(\mathbf{x})\right) + h\_{f2,b}\left(T\_b(\mathbf{x}) - T\_{f2}(\mathbf{x})\right) + h\_b(T\_b(\mathbf{x}) - T\_a) = \mathbf{0}.\tag{5}$$

*Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy DOI: http://dx.doi.org/10.5772/intechopen.105133*

In this model, the air temperature equations are ordinary differential equations (ODEs); boundary conditions for ODEs (2) and (4) are [25]

$$T\_{f^{\mathbf{1}(x=0)}} = T\_{\mathfrak{a}} \tag{6}$$

$$T\_{f1(\mathbf{x}=L)} = T\_{f2(\mathbf{x}=L)}.\tag{7}$$

#### **2.2 Mean solution**

In this solution, the collector temperature components (*Tg*,*Tf*1,*Tp*,*Tf*2, and *Tb*) are the mean temperatures. Like the five governing equations in the local solution, the five corresponding governing equations in the mean solution are written as follows:

Glass cover:

$$I a\_{\rm g} + h\_w \left( T\_a - T\_{\rm g} \right) + h\_{\rm ru} \left( T\_{\rm sky} - T\_{\rm g} \right) + h\_{r\rm g, p} \left( T\_p - T\_{\rm g} \right) + h\_{f1\_{\rm g}} \left( T\_{f1} - T\_{\rm g} \right) = \mathbf{0}. \tag{8}$$

The airflow in the first pass, spatial derivative of temperature is discretized as (17):

$$\frac{T\_{f1,o} - T\_a}{L} = \frac{\mathcal{W}h\_{f1,p}\left(T\_g - T\_{f1}\right) + \mathcal{W}h\_{f1,p}\left(T\_p - T\_{f1}\right)}{\dot{m}c\_p}.\tag{9}$$

Absorber plate:

$$\begin{aligned} \left(\mathbf{I}\tau\_{\mathbf{g}}a\_{\mathbf{p}} + h\_{f1,\mathbf{p}}\left(T\_{f1} - T\_{\mathbf{p}}\right) + h\_{f2,\mathbf{p}}\left(T\_{f2} - T\_{\mathbf{p}}\right) + h\_{r\mathbf{g},\mathbf{p}}\left(T\_{\mathbf{g}} - T\_{\mathbf{p}}\right) + h\_{r,b,\mathbf{p}}\left(T\_b - T\_{\mathbf{p}}\right) = \mathbf{0}. \end{aligned} \tag{10}$$

The air in the second pass:

$$\frac{T\_o - T\_{f1,o}}{L} = \frac{\mathcal{W}h\_{f2,b}\left(T\_b - T\_{f2}\right) + \mathcal{W}h\_{f2,p}\left(T\_p - T\_{f2}\right)}{\dot{m}c\_p} \tag{11}$$

Back plate:

$$h\_{r,b,p}(T\_b - T\_p) + h\_{f2,b}(T\_b - T\_{f2}) + h\_b(T\_b - T\_a) = 0.\tag{12}$$

Two new variables (*Tf*1,*<sup>o</sup>* and *To*) can be resolved by the arithmetic mean of terminal temperatures [16]:

$$T\_{f1} = \mathbf{0.5} (T\_a + T\_{f1,o}) \tag{13}$$

$$T\_{f2} = \mathbf{0.5} (T\_{\sigma} + T\_{f1,\sigma}).\tag{14}$$

#### **2.3 Common equations**

The common equations of the two models above are presented in this section. The sky temperature (*Tsky*) is calculated as follows [22]:

$$T\_{\rm sky} = 0.0552 T\_a^{1.5}.\tag{15}$$

The convective heat transfer coefficient of the wind on the glass cover is written as follows [17]:

$$h\_w = \text{5.7} + \text{3.8V}\_{wind}.\tag{16}$$

The radiant heat transfer coefficient of the glass cover and the sky is

$$h\_{ra} = \sigma e\_{\rm g} \left( T\_{\rm g}^{\ 2} + T\_{\rm sky}^{\ 2} \right) \left( T\_{\rm g} + T\_{\rm sky} \right). \tag{17}$$

The radiant heat transfer coefficient of the absorber plate and the back plate is

$$h\_{r,b,p} = \sigma \left( T\_b{}^2 + T\_p{}^2 \right) \frac{T\_b + T\_p}{\mathbf{1}/\varepsilon\_b + \mathbf{1}/\varepsilon\_p - \mathbf{1}}.\tag{18}$$

The radiant heat transfer coefficient of the absorber plate and the glass cover is

$$h\_{r\_\text{g},p} = \sigma \left(T\_\text{g}^{\;2} + T\_p^{\;2}\right) \frac{T\_\text{g} + T\_p}{\mathbf{1}/\varepsilon\_\text{g} + \mathbf{1}/\varepsilon\_p - \mathbf{1}}.\tag{19}$$

It should be noted that in the local solution, the temperature of the collector components varies in the *x*-direction. Hence, the radiant heat transfer coefficients also change with the *x*-direction. The forced convection heat transfer coefficients of the air in the passes and surfaces (*hf*1,*g*, *hf*1,*p*, *hf*2,*p*, and *hf*2,*b*) is calculated from the following equation [22]:

$$h = 0.018 \, Re^{0.8} Pr^{0.4} k / D\_\varepsilon \tag{20}$$

where *Re* is the Reynolds number, *Pr* is the Prandtl number, and *De* is the hydraulic diameter. These quantities can be estimated as

$$\text{Re} = \rho \mathbf{D}\_{\text{e}} \mathbf{V} / \mu \tag{21}$$

$$Pr = \mu c\_p / k \tag{22}$$

$$D\_{\epsilon} = \frac{4WD}{2(W+D)}.\tag{23}$$

The air velocity in a pass (*V*) and air mass flow rate (*m*\_ ) are correlated by

$$
\dot{m} = \text{WD}\rho V.\tag{24}
$$

The heat transfer coefficient from the back plate to the surroundings is evaluated as follows:

$$h\_b = k\_i/t\_i. \tag{25}$$

The heat gain of air received when passing through the collector is

$$Q = \dot{m}c\_p(T\_o - T\_a). \tag{26}$$

The thermal efficiency of the collector is

$$
\eta\_{th} = \frac{Q}{LWH}.\tag{27}
$$

*Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy DOI: http://dx.doi.org/10.5772/intechopen.105133*

### **2.4 Solution strategy and validation**

**Table 2** presents the fixed parameters to be entered into the mathematical models to calculate the five temperatures in the double-pass solar air heater. For the local solution, the mathematical model consists of ODEs, so a certain understanding of numerical methods and computer skills is required. In our study, we use the integral function in EES software (F-chart software). The program will calculate iteratively until the difference between the temperatures *Tf*1(*<sup>x</sup>* <sup>=</sup> *<sup>L</sup>*) and *Tf*2(*<sup>x</sup>* <sup>=</sup> *<sup>L</sup>*) is very small. For the mean solution, Eqs. (8)–(12) can be rearranged into a system of linear algebraic

**Figure 2.** *Validation of local air temperature in local solution (a) [25] and collector efficiency in mean solution (b) [24].*

equations that can be solved by hand or by applying an iterative method. We also write the code in EES software because of its convenience in solving a system of equations. The computation time of the local solution is about 0.9 s. Meanwhile, the results can be obtained immediately for the mean solution. The validation of both the solutions can be seen in our previous studies [24, 25]. **Figure 2a** presents a comparison of the air temperature distribution along the collector length between the local solution and that of Ramani et al. [11]. **Figure 2b** confirms the calculation results of the collector using mean solution with the published data by Ramadan et al. [30]. It is clear that there is good agreement between the present study and the published results.

## **3. Results and discussion**

**Figure 3** presents the fluid temperature through the two approaches. For the local solution (curves), it is possible to clearly observe that the temperature profiles rise along the flow direction. The air temperature leaving the first pass coincides with the air temperature entering the second pass, i.e., the temperatures at *x* = 2 m. This proves that the iterative solution has high accuracy. For the mean solution (horizontal lines), the air temperature in a pass is an average value. It is obviously deduced that the mean temperature is in the middle of the temperature distribution of the local solution, which confirms the accuracy of both the approaches.

**Figure 4** is to compare the temperatures of the heat transfer surfaces obtained by the two solutions. The glass and back plate temperatures of the two methods are quite similar. However, the absorber plate temperature of the mean solution is insignificantly higher than that of the local solution. Observing the absorber plate temperature profile in the local solution, the maximum temperature is at *x* = 0.7 m. This is because the two air currents across the surface of the absorber plate are in opposite directions. This complexity leads to small deviations between the two approaches in predicting the absorber plate temperature. The maximum deviation of the absorber plate

**Figure 3.** *Air temperature in passes for mean solution (horizontal lines) and local solution (curves).*

*Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy DOI: http://dx.doi.org/10.5772/intechopen.105133*

**Figure 4.** *Temperature of plates and glass cover for mean solution (horizontal lines) and local solution (curves).*

temperature is 2.6%, which occurs at *x* = 2 m. The glass cover and back plate temperatures increase with the airflow direction, as expected. It is suggested that the back plate insulation should be as well as possible, i.e., increasing *ti* and decreasing *ki*. Thereby, the heat loss to the environment can be reduced, and the back plate temperature approaches that of the absorber plate.

The comparison of radiant heat transfer coefficient (HTC) is shown in **Figure 5**. In terms of magnitude, the radiant heat transfer coefficient between the absorber plate and the back plate (*hr,b,p*) is the largest, and the radiant heat transfer coefficient between the glass and the sky (*hra*) is the smallest. These are because the temperature of the absorber plate is the highest and the temperature of the glass is the lowest. The

**Figure 5.** *Radiant heat transfer coefficients for mean solution (horizontal lines) and local solution (curves).*

**Figure 6.** *Outlet air temperature and thermal efficiency for local and mean solutions.*

error in the prediction of the radiant heat transfer coefficient between the absorber plate and the glass cover is the largest. This can be explained by the highest difference between the glass and absorber plate temperatures compared to other temperature differences (**Figure 4**).

Expanding the investigation, the influence of air mass flow rate in the range 0.01–0.1 kg/s on the outlet temperature and the thermal efficiency can be seen in **Figure 6**. As the flow increases, the temperature is reduced because the heater is well cooled. The increased flow rate increases the convection heat transfer rate, thus increasing the thermal efficiency. It can be concluded that the error in predicting the outcome parameters of the two methods is negligible. The air temperature of the mean solution is higher than that of the local solution, resulting in a higher thermal efficiency of the mean solution. The highest deviation of 2.1% is observed at the lowest airflow rate. The deviation reduced sharply with the flow rate. However, this difference is not of concern for a practical application.

## **4. Conclusion**

Two analytical models for calculating heat transfer in a double-pass solar air heater were presented in this chapter. These included local solution and mean solution, which were commonly used to predict the performance of multiple-pass solar heaters. Some comparative results of the two models were drawn as follows:


*Heat Transfer in Double-Pass Solar Air Heater: Mathematical Models and Solution Strategy DOI: http://dx.doi.org/10.5772/intechopen.105133*

• The mean solution predicted a slightly higher temperature than that of the local solution.

Both the models can be easily customized to predict heat transfer in a solar air heater with various modifications, such as multiple-pass collector, finned or ribbed or baffled absorber plate, and airflow recirculation as well.

## **Acknowledgements**

The first author acknowledges the support of time and facilities from Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for this study.

## **Conflict of interest**

The authors declare no conflict of interest.

## **Author details**

Nguyen Van Hap1,2 and Nguyen Minh Phu<sup>3</sup> \*

1 Faculty of Mechanical Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Viet Nam

2 Viet Nam National University, Ho Chi Minh City, Viet Nam

3 Faculty of Heat and Refrigeration Engineering, Industrial University of Ho Chi Minh City, Ho Chi Minh City, Viet Nam

\*Address all correspondence to: nguyenminhphu@iuh.edu.vn

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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## **Chapter 3**
