## Abstract

This study presents the geometrical optimization of a solar air heater. A reasonable working range of geometrical parameters: length 1–3 m, width 0.1–0.5 m, and height 0.005–0.05 m (or aspect ratio 1–20), is investigated as per general applications and installation constraints. The effect of these geometrical parameters on thermal (heat flux and outlet air temperature) and hydraulic (air velocity, mass flowrate, pressure loss and pumping power, and Bejan number) parameters are theoretically studied. Later, a numerical investigation is carried out with an optimum geometry of SAH-duct (length 2.44 mm and width 0.3 m), for aspect ratio 4–18 and Re 3000–15,000. An experimental study is also done to validate numerical results. The heat transfer and frictional loss do not show a significant change in the values for the aspect ratio 8–12.

## 1 Introduction

A solar air heater (SAH) absorbs solar radiation on the absorber plate, which in turn heats the air passing over it, and the heated air can be used for various industrial, household, agricultural, and building ventilation applications. The medium aspect ratio (AR = W/H) SAH-duct can be assumed as an asymmetrically heated type heat exchanger where heat is applied only on the absorber plate, and the remaining sides are insulated. In the SAH, air is a working fluid that flows in a low-to-medium turbulent regime when the heat transfer is mainly by the forced convection for a low-to-medium temperature rise and mass flowrate applications.

The SAH has low thermal efficiency due to the low heat transfer coefficient (h) of air with the absorber plate. Many techniques have been investigated for the heat transfer improvement in the SAH-duct such as multi-passing, multi-glazing, the selective absorber layer, fins, baffles, blocks, and rib-roughness, which have been discussed in detail by Chhaparwal et al. [1]. Does smooth SAH-duct have an optimum length (L), width (W), and aspect ratio (AR) corresponding to maximum possible heat transfer at reasonable pumping power? Most of the authors have taken SAH-duct dimensions arbitrarily, as per the installation-space constraints or standard size of sheets available in the market.

Only a few studies are done on optimization of a smooth SAH-duct. Hence, the general studies related to a rectangular duct design (other than a SAH-duct) are also included in the literature survey. Charters [2] proposed a design procedure for asymmetrically heated rectangular SAH-duct, based on technical and economic grounds, see Eq. (1).
$AR0.8(1+AR)0.2=QL0.0182kΔTpfPr0.4(μH2mf)0.8$
(1)
The present study also uses a similar technical based methodology. Hollands and Shewen [3] proposed the concept of short-length SAH. They found that for constant pressure loss and mass flowrate per unit collector area (G), shorter length SAH-duct gave a maximum value of heat transfer coefficient. Bhargava et al. [4] reported an increase in outlet air temperature (To) with SAH-duct length, but instantaneous and average thermal efficiency were decreased. Husain et al. [5] found that outlet temperature and heat gain by water increases with length at constant water-velocity and absorber plate area. The outlet temperature was decreased with an increase in the height of the duct.
The AR of a rectangular cross section, a SAH-duct, is most important parameter to decide its thermal-hydraulic performance. It is also the most studied parameter in a rectangular duct-design. The aspect ratio has been investigated in gas turbine blade-cooling and micro-channel in literature, but the significant study is missing in the context of a SAH. Rohsenow and Hartnett [6] compiled significant work by many thermodynamists [7,8] on internal duct flow for the laminar and turbulent regime. For the calculations of friction factors (f) and Nusselt numbers (Nu) in fully developed laminar flow in rectangular ducts, the diameter is defined by Eq. (2) to consider the shape effect [8]
$Dl=23Dh+1124AR(2−AR)$
(2)

The studies by Hartnet and Irvine [7] suggested that f and Nu increase with the increase in AR. Murata and Mochizuki [9] investigated the effect of a cross-sectional AR on turbulent heat transfer in an orthogonally rotating rectangular smooth duct of gas turbines for effective cooling. It was found that the relative intensity of the direct and indirect influences of Coriolis force that causes heat transfer depends on the AR. Chang and Lin [10] did experiment to study AR effect on steady and unsteady longitudinal vortex flow in mixed convection of air in the bottom heated horizontal rectangular duct. The different value of AR causes variation in the profile and intensity of vortex flow around the cross section of the duct. Chang et al. [11] studied the AR effect on the thermal and hydraulic performance of air-water flow at intermittent slug and slugged annular flow conditions where decreasing AR increased both heat transfer and pressure drop. Etemad and Mujumdar [12] numerically showed that AR has a significant effect on simultaneously developing flow through a rectangular channel. Han and Park [13] experimentally studied the effect of AR on the distributions of the local heat transfer coefficient for developing flow in short rectangular channels used for turbine blade cooling. Time-averaged secondary flow due to instantaneous vortical motion decides heat and momentum transfer in the rectangular duct. Choi and Park [14] applied large eddy simulation and showed that with increasing AR, the effects of the hot-sweep flow of the clockwise and counter-clockwise rotating vortices become equally dominant near the wall-normal bisector of the ducts. Lee and Garimella [15] numerically studied laminar convective heat transfer in the micro-channels of different AR, where the average Nusselt number was increased with an increase in the height of the channel.

The above literature review suggests that all the SAH-duct geometrical parameters (L, W, H, and AR) are not studied simultaneously in a single study. The AR is the most important parameter but only a few detailed investigations are available for asymmetrically heated SAH-duct when compared to an all-sides-heated symmetrically heated rectangular duct as shown in Fig. 1. The AR-effect studies are mostly done in the laminar regime, and only a few studies are investigated in a turbulent regime. The effect of width is not reported in the literature.

All these research gaps motivated to investigate all the geometrical parameters for the same boundary conditions. First of all, a parametric study is done to plot design curves that can help to choose optimum dimensions as per the applications of SAH and installation-space constraints. The design curves for L, W, and AR are plotted based on the thermal-hydraulic performance of the SAH-duct. Then, a numerical study is conducted with the help of the CFD tool ansys fluent to investigate the effect of AR in detail. Finally, the numerical results of AR-study are validated with an experimental study on a SAH-setup fabricated at the university campus.

## 2 Methodology and Result Analysis

### 2.1 Parametric Study.

In parametric study length L, width W, and height H of SAH-duct are varied as shown in Table 1 at constant heat flux in transitional flow regime. The results are obtained in terms of Bejan number (Be), outlet air temperature To, and absorber plate temperature Tp. These are calculated by Standard formulae for friction factor, Nusselt number, heat transfer coefficient, and pressure loss as given by Eqs. (3)(8). For general applications, space available for installation and environmental conditions, the air temperature rise is up to 20 K and pressure loss is up to 100 Pa.

#### 2.1.1 Hydraulic Performance Analysis.

The hydraulic performance of the SAH can be interpreted in terms of Bejan number (Be). It is very useful dimensionless number that has ability to represent heat transfer, mass transfer, momentum transfer, etc., with suitable changes in its formulae. Equations (3)(6) show that Be = f(W, L, AR, Re). Figures 2(a)2(c) show the variation of Be with W, L, AR, and Re, where Be increases due to an increase in the pressure loss with increment in the values of all these parameters except W. These results show that for L 1–3 m, the effect of AR and W is less as compared to L for more than 3 m. Larger L causes high pressure drop. Figure 2(a) shows that the variation in Be reduces at higher AR (>10) values. In Fig. 2(b) at constant AR 15, Be is very high at lower values of W, because area of cross section decreases at lower W values which increases velocity of the flow and pressure loss is directly proportional square of the velocity. Hence, lower W values are avoided but larger W reduces velocity as effective efficiency($ηeff$) is function of the velocity of flow. The W values 0.2–0.3 m are suggested for optimum $ηeff$ from hydraulic performance consideration. Figure 2(c) shows that Be increases with Re, and rate of increment is higher after Re 15,000. For general SAH applications Re > 15,000 is not used.
$Be=ΔPL2μν$
(3)
$ΔP=fDρLV22Dh$
(4)
$fD=0.079Re−1/4$
(5)
$Dh=2WAR+1$
(6)
$Nu=hDhk=0.023Re0.8Pr0.4$
(7)
$h=mCp(To−Ti)WL(Tp−Tf)$
(8)

#### 2.1.2 Thermal Performance Analysis.

For the required application To is the desired output parameter. Heat transfer from the absorber plate can be reported by change in its temperature Tp. This makes To and Tp as the important evaluation parameters of the thermal performance. From Eqs. (6), (7), and (8), it can be said that To and Tp = f(L, W, AR, Re). Figure 3(a) shows an initial steep rise in To up to AR 5, then variation reduces significantly for AR > 15. Figure 3(b) shows steep decline in To up to W 0.2 then variation reduces significantly for W > 0.5. The higher AR and lower W values reduce the cross-sectional area of the SAH-duct and bring heated absorber plate closer to the main centerline fluid flow, hence To increases. Higher TP means low heat transfer from absorber plate and vice-versa. Figures 3(c) and 3(d) show that Tp decreases with an increase in the length and width of duct because fluid flow gets more time and space of contact with the heated plate. Tp also decreases with an increase in the AR of the duct due to increased closeness of absorber plate with the centerline fluid flow. Figure 3(e) shows To and Tp decreases with Re at constant W, L, and AR. Increment in Re increases turbulence and velocity of the flow for constant area of cross section. To decreases with Re because a fixed amount of fluid gets less time with the heated absorber plate. While Tp decreases with Re because enhanced turbulence causes more mixing of hot and cold fluid which in turn increases heat transfer from the heated plate which lowers its temperature. Here, it is interesting to note that To shows greater value for W 0.1–0.2 m (Fig. 3(b)) but the corresponding high values of Be Fig. 2(b) demotivate to use such low value of W. This thermal analysis of fluid flow through SAH-duct complement the findings from its hydraulic analysis in Sec. 2.1.1.

### 2.2 Numerical Setup.

Above theoretical study suggests that SAH gives required output for L 1–3 m and W 0.2–0.3 m for the available space constraint and general applications. However, the change in the values of AR does not affect space requirement but it significantly affects the flow and heat transfer characteristics around the cross section of SAH duct. This motivates us to carry out a detailed numerical and experimental analysis of AR-effect on a smooth asymmetrically heated SAH-duct in turbulent flow regime. For AR below 4, Nu (heat transfer) drastically decreases and for AR above 20, the f (pumping power) is too high. Hence most of the researchers have taken AR value arbitrarily between 4 and 20 only. How does Nu and f changes in this range? Does it have any significant impact on the performance of SAH? Yes, it has significant effect on Nu and f for this AR range in the laminar flow regime. Is it true for turbulent regime also? This section tries to solve these questions by a numerical study. An asymmetrically and symmetrically heated rectangular duct with constant heat flux and constant wall temperature conditions in a laminar flow regime has been studied by Schmidt [16]. Figure 4(a) shows the variation of Nu with AR when one or more walls are heated at constant wall temperature conditions. The results by various researchers for different cross-sectional duct at different thermal-physical conditions are compiled by Shah and London [17]. They developed expression for friction factor Eq. (9) which was plotted in Fig. 4(b). All these results show that the duct aspect ratio has significant effect on heat transfer and pressure loss in the laminar flow in an asymmetrically heated rectangular duct.
$fRe=−8c1(W/2)2um(1+AR)2$
(9)

However, for the turbulent flow as the velocity profile is not well-defined hence, AR-effect is unknown. So, a numerical study is carried out to see the effect of AR on thermal-hydraulic performance of an asymmetrically heated solar air heater duct, where three sides are insulated and constant wall heat flux is applied only on the top surface.

#### 2.2.1 Governing Equation.

The governing equations—partial differential equations, which are derived based on mass, momentum, and energy conservation [18]. These equations govern the computational domain of the problems and vary as per the nature of the problem. The SAH is a case of forced convective heat transfer by an incompressible steady-state turbulent flow through one principal wall heated rectangular duct. The $k−ε$ model introduces two new variables into the system of equations. Where, k is the turbulence kinetic energy and is defined as the variance of the fluctuations in velocity and $ε$ is the turbulence eddy dissipation (the rate at which the velocity fluctuations dissipate). The continuity equation is then
$∂ρ∂t+∂∂xj(ρUj)=0$
(10)
and the momentum equation becomes
$∂ρUi∂t+∂∂xj(ρUiUj)=−∂p∂xi+∂∂xj[μeff(∂Ui∂xj+∂Uj∂xi)]$
(11)
where μeff is the effective viscosity accounting for turbulence.
The $k−ε$ model, like the zero-equation model, is based on the eddy viscosity concept, so that μeff = μ+ μt and μt is the turbulence viscosity. The $k−ε$ model assumes that the turbulence viscosity is linked to the turbulence kinetic energy and dissipation via the relation
$μtCμρk2ε$
(12)
where Cμ is a constant. The values of k and $ε$ come directly from the differential transport equations for the turbulence kinetic energy and turbulence dissipation rate:
$∂ρk∂t+∂∂xj(ρUjk)=∂∂xj[(μ+μtσk)∂k∂xj]+Pk−ρε+Pkb$
(13)
$∂ρε∂t+∂∂xj(ρUjε)=∂∂xj[(μ+μtσε)∂ε∂xj]+εk(Cε1Pk−Cε2ρε+Cε3Pεb)$
(14)
where $Cε1,Cε2,σk,σε$ are constants. The starting point of the present formulation is the $k−ω$ model developed by Wilcox. It solves two transport equations: one for the turbulent kinetic energy, k, and one for the turbulent frequency, $ω$. The stress tensor is computed from the eddy-viscosity concept. k-equation is given by
$∂ρk∂t+∂∂xj(ρUjk)=∂∂xj[(μ+μtσk)∂k∂xj]+Pk−β′ρεω+Pkb$
(15)
$ω$-equation is given by
$∂ρω∂t+∂∂xj(ρUjω)=∂∂xj[(μ+μtσω)∂ω∂xj]+αωkPk−βρεω2+Pωb$
(16)
In addition to the independent variables, the density, $ρ$, and the velocity vector, U, are treated as known quantities from the Navier–Stokes method, whereas Pk is the production rate of turbulence. The model constants are given by $β′=0.09$, α = 5/9, $β=0.075$, $σk=2$, and $σω=2$. The shear stress transport (SST) k$−ω$ transport equations are similar to the standard k$−ω$ equations as discussed previously the set of equations has blending functions which activates k$−ω$ model near wall boundary layer and k$−ϵ$ model in the center-line main flow away from the wall.

#### 2.2.2 Computational Domain.

The computational domain consists of fluid zone (air) takes the shape and size of SAH-duct and solid walls (aluminum) of zero thickness to neglect conductance resistance.

In numerical analysis, height H is varied from 16.6 mm to 75 mm at constant width W 300 mm, to obtain aspect ratio 4:1 to 18:1. The complete length of SAH-duct L is 2440 mm, which includes entrance section 640 mm $5W×H$ to have developed flow in test section of 1500 mm and exit section of 300 mm $2.5W×H$ as per height 25 mm to avoid exit-effect [19].

A three-dimensional geometry is modeled in ansys workbench and then imported in ICEM CFD meshing software. Figure 5 shows the non-uniform meshing where large number of elements are kept near the surface to capture the velocity gradient effect. The lateral sides and bottom plate have adiabatic boundary condition.

Uniform heat flux of 1000 W/m2 is applied over the absorber plate, and Re is varied from 3000 to 15,000. At outlet, zero static pressure boundary condition is applied. All different boundary conditions and geometrical parameters are summarized in Table 2. Mesh independence test is conducted to find optimum mesh size to keep balance between desired accuracy and available computing power of the work-station. Figures 6(a) and 6(b) show that after approximate three million mesh elements of tetrahedral shape, the value of f and Nu become constant; hence, this mesh-size is used for further analysis.

The numerical results (at AR 4) are validated within the acceptable limits for SST k$−ω$ model with the correlations given by Dittus–Boeltier and Blassius for Nu and f, respectively, as shown in Fig. 7. Hence, $k−ω$ model is used for all further simulations.

By changing the height of the rectangular duct, aspect ratio is changed to 4, 6, 8, 10, 12, 14, 16, and 18. Each aspect ratio is studied for the range of Re 3000–15,000. The results are summarized by the Fig. 8(a) for friction factor and Fig. 8(b) for Nusselt number.

The value of f and Nu decreases and increases, respectively, with an increase in Re. At constant Re, the value of f and Nu changes with AR, and the variation is significant for Re 3000–8000. As aspect ratio increases, height of the duct decreases, less volume of space is available for the air to flow through the duct which increases velocity of the flow. Therefore, pressure loss increases, and subsequently, friction factor increases which is in agreement with a past study [16]. Nusselt number increases with AR because more mixing of hot air with the relatively cold air takes place. However, the variation in friction factor and Nusselt for turbulent flow is less than the laminar flow. Because, turbulence causes sufficient movement of air molecules for heat gain as compared to the laminar flow. This reduces importance of AR, however, in asymmetrical heating even for turbulent flow the AR-effect can not be neglected.

### 2.3 Experimental Setup.

An experimental study is carried out to validate the numerical results of AR study. The length test section, entry, and exit section are provided as 1500 mm, 640 mm, and 300 mm, respectively, as per the ASHRAE standards 93–77, see Fig. 9. The height of SAH-duct is changed by standard size of sun-mica sheet of 2 mm, wooden sheet of 3 mm, 6 mm, 12 mm, and 18 mm thickness available in market. These wooden sheet are placed over the foot-steps of SAH-duct in such a way that AR of 6, 10, 15, and 20 is obtained as shown in Figs. 10 and 11. An electric heater plate of area 1500 mm × 330 mm is fabricated by series and parallel loops of heating (nichrome) wire on a 5-mm thick asbestos sheet. The nichrome wiring is done over mica sheet to get uniform heat flux on absorber plate. The heat flux can be varied from 0 to 1000 W/m2 by a three phase variable transformer of 10 A rating.

The heater plate is packed by 80 mm glass wool layer to reduce top heat losses. The plaster of Paris fills in the gap and prevent wood from catching fire. Twenty-five washer type FeK thermocouples measure average temperature of test section plate while three thermocouples placed inlet and outlet of the duct measure air temperature, see Fig. 12. Six digital eight-channel temperature indicators show the output of the thermocouples.

A 2-hp blower with voltage regulator gives variable mass flowrate (or Reynolds number). A calibrated venturi meter with a U-tube manometer is connected in between blower-outlet and SAH-inlet to measure the mass flowrate. The pressure drop across the test section is measured with the help of micro-manometer having least count of 0.01 mm of water. The complete fabricated setup with all measuring instruments and parts with test section is shown in Fig. 13. It takes approximate 2 h to achieve steady-state (when there is no change in pressure and temperature readings for 10–15 min). The numerical results are in agreement with the experimental results as shown in Figs. 14(a) and 14(b). The general explanation for the experimental results is similar to the previously discussed numerical results. The deviation in results cannot be stated in a single percentage value. The deviation in both types of studies varies with Reynolds number. However, the range of deviation lies within 3–8%. In numerical studies beside the absorber plate, all three sides are smooth surfaces. While, in the experimental setup, these surfaces are made of wood with a sunmica-sheet covering, which cannot give a perfectly smooth surface. The experimental value of friction factor is more than its numerical value. In numerical studies, it is assumed that there is zero heat loss outside the setup, while in the experimental setup, glass wool and plywood cannot provide perfect thermal insulation.

## 3 Conclusion

This study presents a systematic way to choose dimensions of a solar air heater. It shows that length, width, and aspect ratio have significant effect on its performance. Length and width should be chosen as per outlet air temperature requirement. Increase in length and width gives more surface for heat flux, it also increases pumping power and decreases velocity of air which reduces its efficiency. Hence, length and width should be chosen considering both the contradictory effects. For general applications and installation areas length 2–3 m and width 0.2–0.3 m serves the purpose. Selection of aspect ratio is an important step after optimum length and width have been decided. For the AR < 4, velocity is too low and as flow is in transitional regime, the heat transfer is also reduced. For AR > 15, flow through duct needs high pumping power. Hence, AR 8:1 to 12:1 fulfil functional requirement of solar air heater with insignificant difference in its performance.

## Acknowledgment

This study has been carried out as part of doctoral-thesis work on friction and heat transfer analysis of solar air heater in the Mechanical engineering department at Manipal University, Jaipur, India. The authors wish to record their indebtedness to the institute for opportunities, facilities, doctoral fellowship, and endowment fund EF/2016/05-03 afforded for carrying out the research.

## Data Availabilty Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. The authors attest that all data for this study are included in the paper.

## Nomenclature

• h=

heat transfer coefficient, W/mK

•
• i=

inlet property

•
• l=

laminar equivalent diameter

•
• o=

outlet property

•
• D=

diameter, mm

•
• G=

mass flowrate, kg/sm2

•
• H=

height of duct, mm

•
• I=

solar intensity, W/m2

•
• L=

length of duct, mm

•
• P=

pressure, Pa

•
• Q=

heat flux, W/m2

•
• T=

temperature, K

•
• V=

cross-sectional averaged velocity, m/s

•
• W=

width of duct, mm

•
• AR=

aspect ratio, W/H

•
• Be=

Bejan number

•
• Nu=

Nusselt number

•
• PP=

pumping power, W

•
• Re=

Reynolds number

•
• St=

Stanton number

•
• TRP=

temp rise parameter, ΔT∕I

•
• μ=

dynamic viscosity, kg/ms

•
• ρ=

density, kg/m3s

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