Theoretical analysis and numerical study of natural convection inside combined solar chimney

Abstract

Solar chimneys may induce natural ventilation through solar radiation. However, sufficient theoretical studies are needed as a basis to fully exploit passive design in practical green building design. In this work, we investigate the heat transfer properties of turbulent natural convective flows in a combined solar chimney with a thermal flux at the absorption wall by means of theoretical analysis and numerical simulations. Two different flow patterns have been found, one with a clear thermal boundary layer flow pattern and the other without, based on high Rayleigh numbers. For flow development in these two flow regimes, the transient scaling analysis is performed separately and the control mechanism for each phase is presented. Some new scale relationships are established to characterize the ventilation performance of solar chimneys, including thermal boundary layer thickness δ_T, velocity v_T, mass flow m, and so on. For the distinct thermal boundary layers, δ_T,s ~ HΓ^2/5/Bo^1/5κ^2/5, v_T,s ~ Bo^2/5κ^4/5Γ^1/5/H, m ~ ρBo^1/5κ^2/5Γ^3/5. For nonobvious thermal boundary layers, v_T,f ~ Bo^1/3κ/H^2/3W^1/3, m ~ ρBo^1/3κ/A^2/3. The important scale relationships are validated using corresponding numerical simulation data, such as the mass flow rate scale M ~ (Γ/κ)^3/5Bo^1/5 in the distinct thermal boundary layer flow state, and so on. The air changes per hour and heat exchange efficiencies are calculated for a solar chimney with a fixed height-to-width ratio to provide a basis for the design of a solar chimney.

INTRODUCTION

Energy consumption in buildings has been rising every year in response to rapid social development. There has been a lot of interest in finding practical ways to reduce the energy consumption of buildings and enhance their internal thermal environment. An efficient passive design strategy for buildings with zero energy use is to provide natural ventilation using solar chimneys.¹ Miyazaki et al.² found that natural ventilation by solar chimneys not only optimizes the indoor environment but also reduces the energy consumption of the building. It can even save 50% of energy per year compared to mechanical ventilation.

Hot pressure ventilation is used in solar chimneys to reduce the consumption of energy. One of the most frequently used sources of renewable energy is solar energy. The study³ found that buildings using solar chimneys can reduce cooling loads by 30%–100% in summer and heating loads by 46%–56% in winter. Therefore, it is beneficial to use solar chimney enhanced ventilation technology to reduce building energy consumption.

Scaling analysis has been extensively utilized in the past to examine convective flows with various geometries and boundary restrictions. The scale analysis of the transient natural convection in a rectangular side heating cavity with the aspect ratio A < 1 shows that under a fixed aspect ratio, the flow is dominated by heat conduction before the Rayleigh number (Ra) exceeds the critical Ra.⁴ The numerical simulation and scale analysis of the triangular cavity with inclined wall cooling and underside heating show that the transient phenomenon begins with the cooling of the inclined wall, and there are both hot and sticky layers on the wall.^5,6 Lei and Patterson⁷ investigated potential flow patterns caused by solar radiation-induced unsteady natural convection in a small slope triangular cavity. The unsteady natural convection boundary layer of a fluid with Pr > 1 was scaled by Lin et al.⁸ It was found that the boundary layer's early development was one-dimensional as a result of the influence of pure heat conduction. In the steady state, the development of the boundary layer becomes two-dimensional (2D) as convection dominates. By analyzing the heat transfer characteristics of the upslope flow and the unsteady roof plume in the triangular cavity where periodic heating is applied to the roof, it is found that the development of the upslope flow includes two stages: heat conduction and convection, and the heating time has an impact on the roof flow.⁹ For a triangular cavity with linearly increasing temperature applied on the inclined wall, the boundary layer reaches a quasi-steady state before the heating is completed, and the thickness of the thermal boundary layer first increases and then decreases.¹⁰ The scale analysis of the natural convective thermal boundary layer near the suddenly heated inclined plate has obtained the scale relationship between the maximum velocity in the boundary layer, the time when the boundary layer reaches the steady state, the thickness of the thermal layer and the thickness of the viscous layer, and found that the fluid flow state can be divided into three stages: the initial stage, the transition stage and the steady stage.¹¹ The findings of scale analysis in the above studies^4–11 were verified with numerical results. For solar chimney, He¹² and Rakesh and Lei¹³ conducted scaling analysis of laminar natural convection in a vertical solar chimney. It was discovered that the boundary layers in the chimney can be classified into two types: distinct thermal boundary layers and indistinct thermal boundary layers. Thermal boundary layer thicknesses, inner and outer velocity boundary layer thicknesses, and the amount of time required for the boundary layer development to attain a stable state were obtained, representing transient flows. By comparing the results of direct numerical simulations, the accuracy of the dimensional analysis is confirmed.

Numerical simulation and experimentation have been used to study the boundary layer close to the solar chimney's absorption wall. Chen et al.¹⁴ and Li et al.¹⁵ established a rectangular chimney model, laying an electric heating film on a high-density plate and changing the voltage to change the heat flux. Researchers have looked at the temperature and speed characteristics in vertical solar chimney channels of various widths. They found a temperature and velocity boundary layer near the absorber wall of the chimney. The photovoltaic (PV) Trombe wall was numerically simulated using the Renormalization-group (RNG) k–ε model by Lei et al.¹⁶ and Su et al.,¹⁷ and the results showed that a thermal boundary layer exists close to the absorb wall of the chimney. By varying the width of the chimney, the optimal ratio of height to width (aspect ratio) of the chimney is obtained when the ventilation volume achieves its highest value. To characterize the ventilation volume and heat transfer characteristics, the fitting formulas of Nusselt number (Nu) and Reynolds number (Re) as a function of Rayleigh number (Ra) were provided. Imran et al.¹⁸ conducted field tests and 2D numerical simulation studies on the inclined solar chimney model. In the chimney passage, the temperature profiles at various heights along the breadth of the chimney are evaluated. It was found that there is a temperature boundary layer at the absorber wall of the chimney.

In addition, there are other related studies of various types of solar chimneys. Khedari et al.¹⁹ did an experimental investigation on the rooftop solar chimney, in addition, Zhao,²⁰ Ramadan Bassiouny and Koura²¹ examined the ventilation effect of vertical solar chimney through CFD numerical simulation. All of the findings indicated that when the chimney's aspect ratio rises, the ventilation capacity initially rises and subsequently falls. Xue and Su²² made an analysis of the flow field characteristics inside the solar chimney by utilizing the computational simulation method, and proposed a gradually shrinking chimney channel structure that can effectively decrease the outlet flow inversion and improve the natural ventilation effect. Vertical solar chimneys were the subject of experimental investigation by Burek and Habeb.²³ Different solar radiation intensities were controlled by using electric heating pads. The ventilation performance under different solar radiation intensity and chimney width was investigated. The quantitative relationship between chimney ventilation rate and solar radiation intensity, and chimney width was obtained. For combined solar chimneys, Yang et al.²⁴ and Duan²⁵ studied the influence of different widths, inclined section lengths, and outdoor wind speed on its ventilation performance. It is shown that the ventilation capacity is enhanced as the width and length of the tilt profile increase.

Mandal et al.²⁶ used artificial neural networks (ANN) to simulate how power generation performance changed under various conditions with regard to solar chimney power generation. They found that reducing the height of the collector inlet and increasing the diameter of the chimney can significantly improve the power generation efficiency, but the cost of increasing the diameter of the chimney is higher. It is discovered that geometric optimization is essential to the efficiency of solar chimney power generation (SCPPs) by examining the experimental procedures and advancements of SCPPs.²⁷ When the ground sloped absorber (γ = 0.6°) and divergent chimney (ϕ = +0.75°), the power generation can be increased by 80%.²⁸ Stair-shaped absorbers can increase power generation capacity by about 80%.²⁹ The wave absorber can increase power generation capacity by about 58.61%.³⁰ Studies conducted on a SCPP indicate that the diameter of the chimney should be 50 cm, and the entrance height should be 7 cm, to reach the highest power generating capacity.³¹

According to earlier research,^32,33 a vertical solar chimney will experience backflow when the chimney width is large. The combined solar chimney combines vertical and inclined solar chimneys. Effectively avoiding outlet backflow and improving ventilation and heat exchange efficiency. In addition, the combination of chimneys and building facades has the advantages of esthetics and a small footprint. Therefore, this study focuses on the combined solar chimney as the research object.

Scale analysis methods have been widely used in the study of natural convective flow and heat transfer,^4–11 but most focus on closed cavities and flat plates. The analysis of transient convection inside a combined solar chimney is still not clear enough. Additionally, because solar radiation and ventilation openings are present, the instantaneous flow and heat transfer characteristics of solar chimney systems are quite complicated. Previous research (such as Tang and colleagues^1,12,13,22) used steady-state numerical simulation and simple k–ε turbulence models, which are unable to fully represent the instantaneous turbulent flow and heat transfer characteristics in solar chimneys.

Therefore, the combined solar chimney is used in this study as the research object, and scale analysis is used to discuss the development process of the transient flow in the structure. Based on the RNG k–ε turbulence model, the relevant numerical simulation is performed. The control mechanism of each stage of flow development is discussed, and the theoretical relationships of airflow temperature, velocity, and mass flow rate, which characterizes the ventilation performance of the solar chimney, is derived. Air change per hour is used as the evaluation index to demonstrate how applicable solar chimneys are in real-world engineering.

The RNG k–ε turbulence model adopted in this study has high accuracy in eddy current simulation. It can more accurately depict the solar chimney's turbulent flow and heat transmission properties. Mutual verification the result of numerical simulation and theoretical analysis to make the research more comprehensive and reliable. This investigation expands the theoretical research idea for the analysis of turbulent convection heat transfer characteristics in an open cavity. It has a certain theoretical significance for solar chimney engineering design.

SCALING ANALYSIS

Figure 1A depicts the physical representation of the combined solar chimney natural ventilation system, which consists of a vertical air channel and a slanted air channel. The left side wall is a glass plate (separated from the outside world), and the right side wall is an absorbing wall. The width of the chimney W is horizontal distance among the absorbing wall and the glass plate, l is the length of the chimney in the sloping section, H is the height of the vertical section, and h is the height of the bottom entrance of the chimney. Among them, the inclination angle of the chimney inclined section α = 45°, and other dimensions of the chimney are explained in Table 3. The sun serves as a heat source, absorbing solar energy through its vertical and inclined endothermic walls, which then heat the air inside the chimney. The air inside and outside the chimney produces a temperature difference, which causes a density difference, forming a hot pressure to provide buoyancy for air flow. The effect of thermal buoyancy, the channel's flow rises and exits at its top outlet into the surrounding environment.

Figure 1. Combined solar chimney model. (A) Schematic diagram. (B) Boundary conditions.

Figure 1B shows the 2D boundary conditions of the solar chimney model. The glass plate is the radiation boundary (∂T/∂x = 0), the absorbing wall is the fixed heat flux boundary (q ~ k∂T/∂x), the bottom inlet is the pressure inlet, and the top outlet is the pressure outlet (∂T/∂y = 0), inlet and outlet turbulence intensity I = 2%. Ambient temperature T₀ = 298.15 K. Nouanégué et al. demonstrated that the influence of wall thickness on ventilation effect was negligible compared with Ra and chimney geometry parameters.³⁴ The effect of wall thickness is not taken into account in this paper. Furthermore, the following assumptions are made: the external environment is steady, wall heat storage is neglected, and the gas in the model is an incompressible Newtonian fluid.

The continuity equation, Navier–Stokes equations based on the Boussinesq approximation, and the energy equation all control the flow through the solar chimney channel.

The continuity equation is: [Image Omitted. See PDF]

The momentum equation is: [Image Omitted. See PDF] [Image Omitted. See PDF]where u and v are the the velocity in x and y directions, p is pressure, ρ is the air density, v′ = v_t + v, v′ is the sum of turbulent and laminar viscosity coefficients, v_t, v is the turbulence and laminar viscosity coefficient, g is the acceleration of gravity in the y direction, β is the coefficient of air expansion, T₀ is the initial temperature.

The energy equation is: [Image Omitted. See PDF] [Image Omitted. See PDF]where Γ is the generalized diffusion coefficient, v/Pr is the caused by molecular diffusion, v_t/σ_t is the caused by turbulent fluctuations, S_T is the viscous dissipation term.

The k-equation is: [Image Omitted. See PDF]

The ε-equation is: [Image Omitted. See PDF]where G_k is the term for the creation of turbulent kinetic energy caused by an average velocity gradient, G_b is the buoyancy-based turbulent kinetic energy generation, Y_m is the contribution of pulsation dissipation to dissipation rate in compressible turbulent flow, σ_k and σ_ε are the k and ε turbulent Prandtl number respectively, S_k and S_ε are source term. [Image Omitted. See PDF] [Image Omitted. See PDF]where η is the time scale proportion of turbulence to the average tension.³⁵ The constants in the RNG k–ε model are: $${C}_{\mu }^{^{\prime} }$$ = 0.0845, $${C}_{1\varepsilon }$$ = 1.42, $${C}_{2\varepsilon }$$ = 1.68, $${\sigma }_{k}$$ = 1.393, $${\eta }_{0}$$ = 4.38.

Fluid heat transfer depends on three dimensionless control parameters, namely Rayleigh number (Ra), Prandtl number (Pr), the aspect ratio of chimney (A), and length ratio of inclined section to vertical section (l/H). Ra, Pr, and A are defined as follows: [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]

Then, scaling analysis is performed using Equations (1)–(4). Two possible cases with and without obvious thermal boundary layers are considered, respectively. Scaling analysis is applied to calculate the scales of thermal boundary layer thickness, time to reach full development, temperature, speed, and steady-state rate of mass flow.

Flow regime with an obvious thermal boundary layer

The solar chimney's absorbing wall is equivalent to the heat source with constant q heat flow. In the initial stage, the difference of temperature will cause the transfer of heat from the absorbing wall to the ambient fluid, thus forming a thermal boundary layer of thickness δ_T near the absorbing wall. Heat flux between the ambient fluid and the heat absorption wall is q ~ kΔT/δ_T. δ_T is the thermal boundary layer thickness, k is the thermal conductivity of fluid, and ΔT is temperature difference between the environment fluid and the heat absorption wall⁹: [Image Omitted. See PDF]

The unsteady term in Equation (4) is O[ΔT/t], O[vΔT/H] is the convection term, and the thermal conductivity term is O[ΓΔT/δT²]. The unsteady term and the convection term in Equation (4) are compared as follows: [Image Omitted. See PDF]

Since the time is short enough, H/vt » 1, the equilibrium at this time is among the unsteady and the thermal conductivity terms, and then the thickness δ_T of the thermal boundary layer is as follows⁴: [Image Omitted. See PDF]

Scale in Equation (15) remains until the convective term plays a dominant role (or reaches the quasi-steady stage).

In Equation (3), where the unsteady term is O[v/t], the convection term is O[v²/H], the viscous term is O[v'v/δ_T²], and the buoyancy term is O[gβΔT], then the ratio between the unsteady term and the convection term is: [Image Omitted. See PDF]

The proportion of the unsteady to the viscous terms is: [Image Omitted. See PDF]

Scale (16) shows that the unstable term is significantly bigger than the convective term in the case of a very short time. Scale (17) shows that the proportion of the unstable and viscous term is related to the turbulent Prandtl number σ_t and Pr. For σ_t » 1, Pr » 1, the equilibrium is among the viscous term and the buoyant term. When Pr « 1, the equilibrium is among the unstable and the buoyancy terms.¹² For Pr = 0.71, the speed scale v_T at the vertical in the thermal boundary layer is obtained by balancing the unstable and buoyant terms in the vertical momentum equation: [Image Omitted. See PDF]

According to Bejan,³⁶ the product of Ra and Pr is called Boussinesq number, namely Bo = RaPr.

Then the above scale is: [Image Omitted. See PDF]

With the evolution of time, the flow enters the convective stage. Based on the equilibrium among the convective and thermal conductivity terms in Equation (4), The period of time needed for the boundary layer to develop and achieve the quasi-steady state can be obtained: [Image Omitted. See PDF]

The thermal boundary layer thickness δ_T,s at time t_s is: [Image Omitted. See PDF]

Then, the temperature improvement scale ΔT in the boundary layer during the quasi-steady state is: [Image Omitted. See PDF]

In the quasi-steady state, the vertical speed scale v_T,s of the boundary layer is: [Image Omitted. See PDF]

After reaches quasi-steady state, mass flow rate m is: [Image Omitted. See PDF]

Inside layer and outside layer are the two divisions of the velocity boundary layer. The inner velocity boundary layer is controlled by inertial-viscous equilibrium in the close-wall region, represented by the subscript i, while the outside speed boundary layer is controlled by the buoyancy-viscous equilibrium. In the inside velocity boundary layer, the balance among the unstable and the viscous terms in Equation (3) determines the inner speed boundary layer thickness δ_vi³⁷: [Image Omitted. See PDF]

According to scales (21) and (22), the thickness δ_vi,s of the inside boundary layer in the quasi-steady stage can be obtained: [Image Omitted. See PDF]

When the boundary layer develops to the quasi-steady stage, for Pr = 0.71, the outer velocity boundary layer and the thermal boundary layer have the same thickness,³⁶ as shown in Figure 2B. Therefore, the thickness of the outside speed boundary layer thickness scale δ_vo equal to the thermal boundary layer thickness scales δ_T,S namely [Image Omitted. See PDF]

Figure 2. Schematic diagram of the initial and quasi-steady stages of boundary layer development. (A) In the initial stage. (B) In the quasi-steady stage.

Consistent with the conclusions drawn by He¹² and Rakesh Khanal,¹³ the above scaling correlations is only applicable for δ_T,s < W. Only if the thickness of the thermal boundary layer at any place in the vertical direction of the chimney meets δ_T,s < W, then can the development of thermal boundary layer be free from restriction and interference as in large space. If δ_T,s > W, the thermal boundary layer cannot free development, which is the characteristic of natural convective heat transfer in finite space. Therefore, the requirement for the presence of an obvious thermal boundary layer can be obtained from δ_T,s < W, as shown in scale (28) [Image Omitted. See PDF]

According to δ_T,s > W, the interval without obvious thermal boundary layer flow regime can be obtained as follows: [Image Omitted. See PDF]

According to the above scaling analysis and the study of Rakesh Khanal,¹³ there are two possibilities:

• 1.

  As Ra > A⁵Γ²/Prκ², the of the thermal boundary layer's thickness in the stable stage is smaller than the width of the chimney, and an obvious thermal boundary layer flow pattern dominated by convection is formed in the quasi-steady stage.

• 2.

  When Ra < A⁵Γ²/Prκ², the thermal boundary layer thickness extends over the whole breadth of the chimney before the flow reaches the convection heat conduction equilibrium. A flow state without obvious thermal boundary layer dominated by thermal conductivity is formed. These important scale relationships of this flow pattern will be discussed in Section 2.2.

Flow regime without an obvious thermal boundary layer

When Ra is less than the critical value, the thermal boundary layer will finally grow to the entire breadth of the chimney until thermal conductivity and convection are balanced. This situation often occurs when the width of a solar chimney is small or the heat absorbed by the absorbing wall is small. In this state of flow, the thermal boundary layer thickness scale reaches δ_T ~ W at t ~ t_w¹³: [Image Omitted. See PDF]

When the thermal boundary layer thickness δ_T reaches the chimney width W, the time scale t_w is obtained: [Image Omitted. See PDF]

Scale (28) may additionally be obtained by contrasting different time scales. When the time scale t_s in scale (20) is smaller than the time scale t_w, a distinct thermal boundary layer will appear.¹³

From scales (13) and (19), the corresponding temperature growth scale ΔT_w in the chimney and vertical velocity scale v_T,w of the boundary layer can be obtained: [Image Omitted. See PDF] [Image Omitted. See PDF]

So the viscous term in the vertical momentum Equation (3) is O[v'v/W²]. When t~t_w, the proportion between the unsteady and the convective terms show that the former is more significant than the latter as H > vt. According to the proportion of unsteady to viscous terms, the unstable term is more significant than the viscous term as Pr = 0.71. Therefore, the equilibrium at t ~ t_w is controlled by unsteady-buoyancy mechanism.

The thermal boundary layer thickness for t > t_w is equal to the chimney's breadth. At this point, heat conduction is still dominant, and convection is not enough to exit the heat released by the absorbing wall to the thermal boundary layer. Therefore, the temperature of the whole chimney passage increases with the increase of time. Through energy balance and κ = k/ρc_p, it can be obtained that the temperature increase ΔT_w⁺ in the chimney is [Image Omitted. See PDF] [Image Omitted. See PDF]

The above correlation shows that when the thermal boundary layer covers the entire breadth of the chimney, the temperature rises linearly with passing time. Due to the rise in buoyancy, the airflow in the chimney accelerates as the temperature in the chimney rises. According to the ratio of the unsteady term to the viscous term in the vertical momentum Equation (3), when t > (Γ/v′)t_w, the viscous term is more significant than the unstable term. Consequently, for t_w < t < (Γ/v′)t_w, the flow is controlled by the unsteady-buoyancy mechanism, namely [Image Omitted. See PDF]

The boundary layer vertical velocity scale v_T under unsteady-buoyancy equilibrium is obtained: [Image Omitted. See PDF]

The velocity scale (37) describes thermal conductivity, where the velocity increases with t² increasing. This velocity scale is applied until t = t_f, when there is a balance among the heat absorbed by the endothermic wall and the heat taken away by convection in the chimney. As a result, the stable state time scale t_f can be obtained through the balance of energy: [Image Omitted. See PDF] [Image Omitted. See PDF]

The time scale shown above is accurate for t_f < (Γ/v′)t_w, that is, for Ra < A⁵Γ²/Prκ² < A⁵ν‘³Prκ³. When the diffusion time t_f is reached, the flow is controlled by unsteady-buoyancy mechanism. The scales corresponding to temperature difference ΔT_w⁺,_f, velocity v_T,f and mass flow rate m are: [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]

Summary of the scaling analysis

Figure 3 depicts several flow patterns for the development of the flow in the solar chimney. Depending on Ra, there may be two primary flow regimes in the solar chimney. Air flow is regulated by convection when the thermal boundary layer thickness, δ_T,s < W, is less than the chimney's breadth in a quasi-steady state. Conduction, which lacks an obvious thermal boundary layer, dominates this flow when the thickness of the thermal boundary layer is equal to the width of the chimney channel (δ_T ~ W).

Figure 3. Schematic diagram of possible flow patterns.

The detailed transient characteristics, control mechanisms, and crucial dimensional correlations of fluid development are shown in Table 1. Use “'” to represent the corresponding dimensionless scale relationship. For example, (19) represents the dimensionless scale relation, and (19') represents the corresponding dimensionless scale relation. The dimensionless scales in Table 1 use the following scales: the velocity scale κ²/HΓ, the temperature scale qH/k, the time scale H²/Γ, and the mass outflow rate scale ρκ. In Table 1, V_T is the dimensionless velocity scale, τ is the dimensionless time, θ is the dimensionless temperature, and M is the dimensionless mass flow rate. For the transient flow that forms a boundary layer on the absorb wall with constant heat flux applied, it mainly passes through the initial stage, the transition stage and the full development stage (this is also verified in the analysis of Nu in Section 3). The wall temperature, thermal boundary layer's thickness, velocity boundary layer's thickness, vertical speed in the boundary layer, and the amount of time needed for the boundary layer to form and reach a quasi-steady state are the main parameters to express the instantaneous flow characteristics. The parameters representing the instantaneous flow properties and the qualitative relationship between these parameters and the control parameters are obtained in the scaling analysis at each stage. The plausibility of these obtained qualitative relations will be mutually verified with numerical simulation results in Section 3.

Table 1 Flow development and important scale correlations under different control mechanisms.

Regime	Time	Control mechanisms	Important scale correlations
			Dimensional scale correlations		Dimensionless scale correlations
With an obvious thermal boundary layer $$Ra\gt \frac{{A}^{5}{\Gamma }^{2}}{\text{Pr}{\kappa }^{2}}$$	$$t\lt {t}_{s}\lt {t}_{w}$$	Unsteady-buoyancy equilibrium dominated by thermal conductivity	$${{\rm{v}}}_{T}\sim \frac{Bo{\kappa }^{2}{\Gamma }^{1/2}}{{H}^{4}}{t}^{3/2}$$	(19)	$${V}_{T}\sim Bo{\tau }^{3/2}$$	(19')
	$$t\ge {t}_{s}$$	Thermal conduction-convection equilibrium with an obvious thermal boundary layer	$${t}_{s}\sim \frac{{H}^{2}}{B{o}^{2/5}{\kappa }^{4/5}{\Gamma }^{1/5}}$$	(20)	$${\tau }_{s}\sim {\left(\frac{\Gamma }{\kappa }\right)}^{4/5}B{o}^{-2/5}$$	(20')
			$$\Delta T\sim \frac{qH{\Gamma }^{2/5}}{kB{o}^{1/5}{\kappa }^{2/5}}$$	(22)	$$\theta \sim {\left(\frac{\Gamma }{\kappa }\right)}^{2/5}B{o}^{-1/5}$$	(22')
			$$\begin{array}{c}m\sim \rho {{\rm{v}}}_{T,s}{\delta }_{T,s}\\ \sim \rho B{o}^{1/5}{\kappa }^{2/5}{\Gamma }^{3/5}\end{array}$$	(24)	$$M\sim {\left(\frac{\Gamma }{\kappa }\right)}^{3/5}B{o}^{1/5}$$	(24')
Without an obvious thermal boundary layer $$Ra\lt \frac{{A}^{5}{\Gamma }^{2}}{\text{Pr}{\kappa }^{2}}$$	$$t\sim {t}_{w}$$	Unsteady-buoyancy equilibrium dominated by thermal conductivity	$${{\rm{v}}}_{T}\sim \frac{Bo{\kappa }^{2}{\Gamma }^{1/2}}{{H}^{4}}{t}^{3/2}$$	(19)	$${V}_{T}\sim Bo{\tau }^{3/2}$$	(19')
			$${t}_{w}\sim \frac{{W}^{2}}{\Gamma }$$	(31)	$${\tau }_{w}\sim {A}^{-2}$$	(31')
	$${t}_{w}\lt t\lt {t}_{f}$$		$$\Delta {T}_{{w}^{+}}\sim \frac{q\kappa }{kW}t$$	(36)	$${\theta }_{{w}^{+}}\sim \frac{\kappa }{\Gamma }A\tau $$	(36')
	$$t\ge {t}_{f}$$	Thermal conduction-convection equilibrium without an obvious thermal boundary layer	$$\begin{array}{c}m\sim \rho {{\rm{v}}}_{T,f}W\\ \sim \rho \frac{\kappa B{o}^{1/3}}{{A}^{2/3}}\end{array}$$	(42)	$$M\sim {\left(\frac{Bo}{{A}^{2}}\right)}^{1/3}$$	(42')

VERIFICATION OF NUMERICAL SIMULATION AND SCALING ANALYSIS

In this section, ANSYS Fluent is used for numerical simulation to verify the important scale relationships in Table 1. The 2D model is selected on the basis that 2D simulation can capture the physical characteristics of air flow in the chimney passage, and the difference in results is negligible. This conclusion has been confirmed by the studies of Gong et al.,³⁸ Pasut et al.,³⁹ and Zamora et al.⁴⁰ The dimensionless form corresponding to Equations (1)–(4) is as follows: [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF] [Image Omitted. See PDF]where X = x/H, Y = y/H, τ = t/(H²/Γ), θ = (T − T₀)/(qH/k), U = u/(κ²/HΓ), V = v/(κ²/HΓ), P = p/(κ²/HΓ)².

Where X, Y are dimensionless coordinates; U, V are dimensionless velocity in the X and Y directions; P is dimensionless pressure; θ is dimensionless temperature; τ is dimensionless time.

The RNG k–ε model is able to analyze turbulent vortices more accurately and produce more accurate calculation results because of the vortex areas that are present inside the chimney.⁴¹ Therefore, the RNG k–ε model is chosen in the simulation. A radiation model needs to be added to the calculation model because there is also considerable radiation heat transfer on the inner surface of the glass plate in the heat-collecting face of the heat-absorbing plate.⁴² The DO model is selected in this case. In the simulation, enhanced wall treatment is used.

In this study, the initial flow field is set to be isothermal static, that is, T = T₀. The boundary conditions for the numerical simulation are displayed in Figure 1B and are as follows: the inlet is the pressure inlet (P = 0), the outlet is the pressure outlet (P = 0, ∂θ/∂Y = 0, ∂U/∂Y = 0, ∂V/∂Y = 0, θ = 0), the heat absorbing wall is a fixed heat flow boundary (U = V = 0, ∂θ/∂X = 1), and the other walls are non-slip walls (U = V = 0, ∂θ/∂X = 0).

The finite volume method is used to discrete the governing equations. The diffusion term is discretized by the second-order central difference scheme, the convection term by the second-order upwind scheme, and the transient term by the second-order implicit time advance scheme. The SIMPLE algorithm is used to calculate the coupling between pressure and velocity. In addition to the energy equation, the convergence iteration residual of all equations is 10⁻⁶, and the energy equation is 10⁻⁸.

For this study, the following assumptions and simplifications are made to simplify the computation process: (1) The outdoor environment (solar radiation intensity and outdoor air temperature) does not change with time; (2) The flow in the chimney passage is turbulent; (3) Assume that the gas in the chimney passage is an incompressible Newtonian fluid; (4) The thickness of each wall of the chimney is ignored; (5) The calculation is simplified by Boussinesq hypothesis, and the density in the buoyancy term of the vertical momentum equation is considered as a function of temperature, while the density in other terms is a constant.

The numerical simulation and experimental study are contrasted to confirm the applicability of the turbulence model and boundary conditions. A physical model of a vertical solar chimney in 3D is built, with channel dimensions of 1025 mm in height, 925 mm in length, and 20 mm in width. Simulation results under different solar radiation intensities are compared with experiment data,²³ as shown in Table 2. The ventilation volume increases with the increase in solar radiation intensity. The overall trend of the numerical simulation results is consistent with the experimental results, and the deviation is less than 12%. Within the allowable range of error, the validity of the turbulence model and calculation method used in this paper are fully demonstrated.

Table 2 Validity verification of turbulence model.

Solar radiation intensity (W/m2)	The ventilation of the test23 Me	The ventilation of numerical simulation Mc	Deviation (Me−Mc)/Me (%)
200	0.989	0.904	8.57
400	1.501	1.365	9.06
600	1.794	1.590	11.37
800	1.901	1.767	7.05
1,000	2.161	1.914	11.39

The reason for the error between numerical simulation and experiment are: (1) The experiment cannot exclude the interference of external factors such as wind, temperature, and humidity, while numerical simulation only involves hot pressure ventilation and will not be affected by external factors; (2) In the numerical simulation, some simplifications were made to the physical model, without considering the thickness of the heat absorbing plate. In the experiment, the heat-absorbing plate had a certain thickness; (3) In CFD simulation, mathematical models are established based on assumptions and simplifications, which cannot fully accurately reflect the complexity of flow and heat transfer in experiments.

Figure 4 shows the grid diagram of the combined solar chimney. The grid is encrypted at the inlet, outlet, and wall surface. The grid system is divided into structured grids and four sets of grids and time steps are used to verify the independence of the grids and time steps, ensuring the accuracy of the numerical calculations. As indicated in Figure 5A,B, the time series of dimensionless outlet mass flow rate when A = 60.0 and Ra = 2.42 × 10¹³, and the dimensionless velocity distribution along the chimney width at upper end of the chimney inlet when A = 7.5 and Ra = 9.47 × 10¹⁰ is in steady state are, respectively, illustrated. As seen in Figure 5A, in the short time after flow start-up, with the increase in mesh number, the time to reach the full development stage is longer. This is due to the computational complexity and data transmission between grid nodes both increase with the increase in grid number, while the computational efficiency decreases. This increases the time required for computing to reach the full development stage. It is important to note that the minus velocity at X = 0.133 shown in Figure 5B is cause of the fact that the airflow is a vertical channel connected to the chimney by a horizontal inlet. The direction of airflow will shift by 90°, and its flow state will alter dramatically. Moreover, the negative velocity will accelerate the flow of air into the chimney, enhancing the ventilation capacity. As can be seen from Figure 5, the numerical results obtained with the four grid systems and time steps are not significantly different during the fully developed stage, and the error is less than 2% after quantitative calculations. To obtain accurate simulation results within limited computing resources, grid 120 × 5200 and time step τ = 4.0 × 10⁻⁵ are selected for calculation. All different case models have been verified for grid and time-step independence.

Figure 4. Grid diagram in the model calculation domain.

Figure 5. Grid and time step independence tests. (A) Time series of dimensionless outlet mass flow rate calculated by different grids at A = 60, Ra = 2.42 × 1013. (B) The dimensionless velocity distribution of the upper end of the chimney inlet along the width of the chimney at A = 7.5, Ra = 9.47 × 1010 in steady state.

As shown in Table 3, the cases involved in the calculation of this paper are mainly considered with air as the fluid medium (Pr = 0.71). Variations in the Ra number and the ratio A of the vertical height of the solar chimney to the width of the chimney have been considered. Solar chimneys can be applied to residential buildings, dormitories, and public buildings, so this paper selects three different aspect ratios of A = 7.5, 60.0, and 75.0 for analysis.

Table 3 Parameter setting of numerical simulation cases.

Case	Symbol	Line type	H (m)	l/H	A	q (W/m2)	Ra $$\left(R{\rm{a}}\gt \frac{{A}^{5}{\Gamma }^{2}}{\text{Pr}{\kappa }^{2}}\right)$$	Ra $$\left(Ra\lt \frac{{A}^{5}{\Gamma }^{2}}{\text{Pr}{\kappa }^{2}}\lt \frac{{A}^{5}{\nu ^{\prime} }^{2}}{\text{Pr}{\kappa }^{3}}\right)$$
Cy1	○	──	3.0	0.67	7.5	2.34 × 101	7.60 × 1012
Cy2	×	----				7.35 × 101	2.39 × 1013
Cy3	□	…….				1.13 × 102	3.67 × 1013
Cy4	▷	…				2.36 × 102	7.67 × 1013
Cy5	✰	——				3.64 × 102	1.18 × 1014
Cn1	▼	──	12.0	0.20	60.0	1.20 × 10−3		3.64×108
Cn2	■	——				1.60 × 10−3		4.85 × 108
Cn3	●	----				2.40 × 10−3		7.27 × 108
Cn4	▲	…….				3.20 × 10−3		9.70 × 108
Cy6	◇	──				2.34 × 101	1.95 × 1015
Cy7	Δ	…				2.36 × 102	1.97 × 1016
Cn5	▶	…	15.0		75.0	2.00 × 10-3		1.48 × 109
Cn6	◆	──				2.80 × 10-3		2.07 × 109
Cn7	◀	——				3.40 × 10-3		2.54 × 109
Cn8	★	──				4.00 × 10-3		2.96 × 109
Cy8	◁	----				2.34 × 101	4.75 × 1015
Cy9	▽	…….				2.36 × 102	4.80 × 1016

The Nusselt number is an essential dimensionless number that reflects the convective heat transfer capacity during the sudden application of heating flux on the inner wall of a combined solar chimney cavity. Figure 5 shows the results of 2D numerical simulations of the mean Nu time series of the absorbing wall. The mean Nu is defined as follows: [Image Omitted. See PDF]

In Equation (49), for the chimney vertical section L = H and for the chimney inclined section L = l.

With time, the mean Nu of the absorbing wall first rises and then falls noticeably, followed by an oscillating state and then a stable state, as seen in Figure 6. It can be broken down into three phases: the initial phase, the transition phase, and the quasi-steady phase. After the onset of the flow, an extremely thin vertical boundary layer forms on the endothermic wall and rapidly develops due to the large temperature difference among the endothermic wall and the air near the wall. In this developing phase of the boundary layer, heat transfer is dominated by the internal fluid due to strong viscous forces and weak convection. As time goes on, the boundary layer will keep increasing during the rise of the airflow, and the boundary layer will sufficiently develop until a stable state is reached.

Figure 6. Time series of the average Nusselt number of the absorbing wall under Cy3 (A = 7.5, Ra = 1.34 × 1011).

Thermal flow structures under different flow patterns

The flow development in the solar chimney is analyzed in Section 2, where there may be flow states with or without a distinct thermal boundary layer. These flow control mechanisms are illustrated in Figure 3. To further understand the flow structure under each flow state and explain the mechanism of flow and heat transfer characteristics in the chimney, the isotherms and streamlines in the chimney passage under the two flow states are shown in Figures 7 and 8.

Figure 7. Evolution of temperature contours and streamlines in the combined solar chimney over time (case: Cy3). (A) τ = 1.60 × 10−4, (B) τ = 4.81 × 10-4, (C) τ = 8.82 × 10-4, (D) τ = 5.73 × 10-3, and (E) τ = 0.67.

Figure 8. Isotherm and streamlines of combined solar chimney in full development stage under different Ra. (A) Ra = 3.64 × 108, (B) Ra = 4.85 × 108, (C) Ra = 7.27 × 108, and (D) Ra = 9.70 × 108.

To more intuitively understand the transient development process of natural air convection in the combined solar chimney, isotherms and streamlines at different times in the channel are given in Figure 7 when there is an obvious thermal boundary layer flow pattern (i.e., the typical case of high Ra flow state Ra > A⁵Γ²/Prκ²). The initial state of the flow field in the channel is isothermal static. A fixed heat flux boundary condition is imposed on the endothermic wall, leading to the formation of a thermal boundary layer near it. Due to heat conduction, the thickness of this thermal boundary layer gradually increases over time, its thickness is indicated by the scale (15), as depicted in Figure 7A. Under the dominant influence of thermal buoyancy, the fluid within the leading edge of the thermal boundary layer undergoes acceleration along the endothermic wall, giving rise to the formation of eddy currents within this layer. The presence of these vortices leads to convective dissipation of heat in the boundary layer. Figure 7B illustrates the morphology of the thermal boundary layer, with flow transitioning in this region. With the progressive advancement of the flow, a significant vortex with counter-rotating motion is generated at the outer periphery through the interaction of vortex clusters within the boundary layer, as depicted in Figure 7C. The boundary layer exhibits turbulent characteristics. The vortex acquires energy from the convective thermal boundary layer, leading to a gradual enlargement in its size, as depicted in Figure 7D. Once an equilibrium is reached between the thermal boundary layer and the channel flow, that is, after the time reaches scale (20), the flow reaches the full development stage. At this time, the thermal boundary layer thickness, temperature, wind speed and mass flow rate are represented by scales (21)–(24), respectively. The thermal boundary layer morphology and streamline distribution in the fully developed stage are shown in Figure 7E.

Without distinct thermal boundary layer flow pattern (i.e., the typical case of low Ra flow state Ra < A⁵Γ²/Prκ²), the isotherm and streamlines diagram in the chimney passage at the full development stage are illustrated in Figure 8. For this flow state, the thermal boundary layer extends to the full chimney width before the flow reaches full development, which is represented by (31), where the equilibrium is controlled by the unsteady-buoyancy equilibrium. After that, the thermal boundary layer continues to develop under the unsteady-buoyancy control mechanism dominated by thermal conductivity, and the convection effect gradually increases, and is controlled by the heat conduction-convection equilibrium mechanism when it reaches the full development stage. It can be clearly seen from the isotherm in Figure 8 that the thermal boundary layer covers the entire width of the chimney, so for the case of low Ra, there is without distinct thermal boundary layer inside the chimney. The streamlines are evenly distributed parallel to the wall, indicating that the flow is fully developed.

Verification of transient temperature and velocity scales

The dimensionless scaling correlations given in Table 1 are validated using numerical simulation findings in this and the following section, hence validating the dimensional analysis. Scale (19′) indicates a rise in thermal boundary layer velocity during the initial stages of development of the flow. The effective range of this scale can be up to t_s in the case of a thermal boundary layer that is well-defined and up to t_w in the absence of such a layer. Figure 9A illustrates the relation between V_T/Bo and (τ/τ_s)^3/2 when there is an obvious thermal boundary layer in stable state. As expected, there is a linear relationship between V_T/Bo and (τ/τ_s)^3/2 in the range of τ < τ_s, can be seen from the enlarged part 0 < (τ/τ_s) < 1.5 in Figure 9A, which means that the formation of thermal boundary layer near the absorbing wall is controlled by heat conduction. Figure 9B shows the relationship between V_T/Bo and (τ/τ_w)^3/2 without distinct thermal boundary layer. The graphic demonstrates that the velocity data in various circumstances increases linearly with time when τ < τ_w, demonstrating the validity of the transient velocity scale.

Figure 9. Verification of vertical velocity scale in the thermal boundary layer. (A) The velocity scales of the initial stage with distinct thermal boundary layer: VT/Bo and (τ/τs)3/2. (B) The velocity scales of the initial stage without distinct thermal boundary layer: VT/Bo and (τ/τw)3/2.

For t > t_w, the temperature in the chimney passage rises linearly with time as indicated by the scale (35′) until it approaches stable state. This scaling is verified by numerical simulation data shown in Figure 10. As shown in Figure 10, for τ/τ_w > 1, the temperature scale under different Ra increases linearly and gradually flattens out near the steady state.

Figure 10. Temperature scale before the flow reaches steady state in cases without distinct thermal boundary layer: θw+/(κA/Γ) and τ/τw.

Verification of mass flow rate scale and temperature scale in a fully developed state

For the thermal boundary layer flow pattern of a solar chimney with an obvious thermal boundary layer, the mass rate of flow scale in steady-state is determined by (24′). As shown in Figure 11A, the numerical simulation results are contrasted to the scaling analysis results. It can be seen that the fit to the results of the numerical simulations lies approximately on a straight line. The mass flow rate of numerical calculation shows a linear relationship with scale prediction. Scale (24′) can therefore accurately represent the steady-state mass rate of flow of solar chimney with an obvious thermal boundary layer flow regime. For the flow regime without obvious thermal boundary layer, the mass flow rate scale in steady-state is determined by (42′). The results of the numerical simulation at different Ra are contrasted with the scaling analysis results. Figure 11B clearly shows the linear relationship among the results of numerical simulation and scale analysis. It is confirmed that the scale (42′) is correct in predicting the mass flow rate without a distinct thermal boundary layer flow state. Figure 12 verifies the validity of temperature scale (22′) when there is an obvious thermal boundary layer flow reaching stable state. Figure 12 shows the correlation between θ/(Γ/κ)^2/5 and Bo^−1/5 at different points under different cases in steady-state. As expected, the fit results at different points all follow a linear relationship.

Figure 11. The mass flow rate scale verification of boundary layer near absorb wall in steady-sate. (A) Steady-state mass rate of flow scale with distinct thermal boundary layer: M/(Γ/κ)3/5 and Bo−1/5. (B) Steady-state mass rate of flow scale without distinct thermal boundary layer: M and (Bo/A)1/3.

Figure 12. The temperature scale with distinct thermal boundary layer in steady-sate.

ANALYSIS OF AIR CHANGES PER HOUR AND HEAT EXCHANGE PERFORMANCE

Solar chimneys are a technology that harnesses solar energy to improve natural ventilation. It is necessary to apply the energy saving and environmental protection methods of solar chimneys to practical engineering. Therefore, ventilation frequency and air exchange are introduced in this study to investigate the suitability of solar chimneys in practical projects.

The number of air changes is a commonly used indicator for measuring the supply air volume in air conditioning engineering, and is an important indicator for evaluating the performance of solar chimneys.^43–45 Its definition is as follows⁴⁶: [Image Omitted. See PDF]where n is the ventilation rate; G is the ventilation quantity, G = 3600 m/ρ; V_room is the volume of room.

According to the mass flow rate scales (24) and (41) of the full development stage of the two boundary flow regimes. Ventilation volume G in the air change per hour formula can be expressed on the following scale: [Image Omitted. See PDF]

Then the air change per hour n is: [Image Omitted. See PDF]

Taking the solar chimney with A = 7.5 calculated in this paper as an example, it is assumed that the length × width × height of the room is 4.0 m × 4.5 m × 3.0 m when it is applied to a residence. According to the atmospheric meteorological parameters of Shijiazhuang City, Hebei Province, China, the air density is 1.205 kg/m³. Shijiazhuang is a cold winter and hot summer area with four distinct seasons. Spring and autumn are transitional seasons, suitable for natural ventilation by a solar chimney. The number of air changes is approximately: [Image Omitted. See PDF]

When the solar chimney is applied to the underground garage. Take two standard parking spaces in the underground garage, with a size of 4.8 m × 2.0 m × 1.8 m, and a height of 3.0 m, then the air changes times are approximately: [Image Omitted. See PDF]

Article 4.2 of the Indoor Air Quality Standard⁴⁷ and Article 3.1.9 of the Design Code for Heating, Ventilation, and Air Conditioning⁴⁸ stipulate that for residential buildings, the minimum fresh air volume required by indoor personnel should be greater than or equal to 30 m³/h, that is, the minimum number of ventilation and air changes per person per hour is 0.56 times/h. Article 8.1.5 of the Code for Fire Protection Design of Garages, Repair Garages and Parking Lots⁴⁹ stipulates that the number of ventilation changes per hour is six to eight times.

Comparing the number of ventilation changes obtained in this study with the requirements in codes and standards, it is found that the ventilation capabilities of this solar chimney can meet the comfort and health requirements of residential buildings and underground garages in the code.^47–49

It is evident from the scale analysis results and the air changes per hour that chimney width, height, and Ra are the primary parameters influencing chimney ventilation. Flow conditions at Ra > A⁵Γ²/Prκ² are more suitable for ventilation applications. In the range studied in this paper, C_y9 (H = 15.0, l/H = 0.2, A = 75.0, Ra = 1.75 × 10¹⁴) has the best ventilation performance.

An essential metric for assessing the heat exchange capabilities of a solar chimney is heat exchange efficiency.^50,51 It represents the ability of a solar chimney to convert solar energy into usable heat, thereby removing indoor heat. The following formulas can be used to compute the heat released by natural ventilation⁵²: [Image Omitted. See PDF]where Q_out is the natural ventilation heat dissipation (J); C_v is the constant specific heat capacity (J/(kg·K)); T is the steady-state outlet temperature (K); T₀ is the initial temperature (K); Q₀ is the heat absorbed by the absorbing wall (J), Q₀ = 3 600q(H + L).

In the above equation, Q₀ = 3600q(H + L). The heat exchange efficiency is: [Image Omitted. See PDF]

As well as the number of air changes per hour, when the combined solar chimney with A = 7.5 is applied to residential and parking lots, the heat exchange efficiency ranges from 63.33% to 98.44%.

CONCLUSION

This research uses dimensional analysis to investigate the flow and heat transfer control mechanism of transient natural convection in a combined solar chimney with a fixed heat flux on the absorbing wall. Numerical simulation results are provided to verify the scaling relations.

There may be two flow regimes depending on Ra for a solar chimney that presents a fixed heat flux to the absorption wall. For Ra > A⁵Γ²/Prκ², the flow with obvious thermal boundary layer is formed under the strong convection. Some new scale relationships for this flow state are obtained through scaling analysis, including temperature, vertical velocity, mass flow and characteristic time scales in the thermal boundary layer. For example, the characteristic time scale for reaching the full development stage is expressed by t_s ~ H²/Bo^2/5κ^4/5Γ^1/5, and the thermal boundary layer thickness in the fully developed stage is expressed by δ_T,s ~ HΓ^2/5/Bo^1/5κ^2/5. This flow state is more suitable for ventilation applications.

For Ra < A⁵Γ²/Prκ², a flow pattern without an obvious thermal boundary layer is generated. Several new scaling relations are obtained by scaling analysis, including temperature, vertical velocity, mass flow, and characteristic time in the thermal boundary layer. For example, the mass flow scale is expressed by m ~ ρκBo^1/3/A^2/3. For both flow regimes, the ventilation properties strongly depend on Ra.

Finally, the ventilation and heat exchange properties of the chimney are analyzed. It is of theoretical interest to apply the solar chimney with aspect ratio A = 7.5 to practical engineering, which can fulfill the ventilation requirements of an ordinary residential house with two standard parking spaces and an underground garage.

LIMITATIONS, IMPROVEMENTS AND FUTURE DIRECTION

This study simplifies the physical model in terms of dimension and boundary conditions, ignoring the development of flow heat transfer in spanwise space and the change of the external environment with time. As a result, there are some temporal and spatial limitations on the outcomes of theoretical analysis and numerical simulation. In future studies, 3D physical models that are closer to actual engineering can be considered, and more complex environmental conditions can be considered; such as the variations in medium temperature and thermal radiation conditions throughout time. Further, we will consider numerical and experimental studies that are closer to engineering reality.

NOMENCLATURE

• A
• solar chimney aspect ratio
• Bo
• Boussinesq number
• c_p
• specific heat capacity (J·kg⁻¹·K⁻¹)
• g
• gravitational acceleration (m·s⁻²)
• G
• ventilation quantity (kg)
• H
• vertical chimney length (m)
• k
• thermal conductivity coefficient (W·m⁻¹·K⁻¹)
• l
• length of inclined section of chimney (m)
• m
• air mass flow rate in steady state (kg·s⁻¹)
• M
• dimensionless air mass flow rate in steady state
• n
• ventilation rate (times·h⁻¹)
• Nu
• Nusselt number
• p
• pressure (Pa)
• P
• dimensionless pressure
• Pr
• Prandtl number
• q
• heat flux of absorbing wall (W·m⁻²)
• Q₀
• energy rate absorbed by the absorbing wall (J)
• Q_out
• energy rate removed by natural ventilation (J)
• Ra
• Rayleigh number
• t_f
• time scaling of heat conduction and convective equilibrium in the absence of an obvious thermal boundary layer (s)
• t_s
• time scaling of heat conduction and convective equilibrium with an obvious thermal boundary layer (s)
• t_w
• time scaling at thermal boundary layer thickness reaches the width of the chimney without obvious thermal boundary layer flow regime (s)
• T₀
• initial temperature (K)
• u,v
• dimensional velocity in the x and y directions (m·s⁻¹)
• U,V
• dimensionless velocity in the X and Y directions
• v_T
• thermal boundary layer velocity scaling for nonstationary terms and buoyancy equilibria with an obvious thermal boundary layer (m·s⁻¹)
• v_T,f
• thermal boundary layer velocity scaling for heat conduction and convective equilibrium in the absence of an obvious thermal boundary layer (m·s⁻¹)
• v_T,s
• thermal boundary layer velocity scaling for convection and heat conduction equilibrium with an obvious thermal boundary layer (m·s⁻¹)
• v_T,w
• velocity scaling of the thermal boundary layer thickness up to the chimney width in the absence of an obvious thermal boundary layer flow regime (m·s⁻¹)
• V_T
• dimensionless velocity scaling
• W
• solar chimney width (m)
• x、y
• dimensional coordinates
• X、Y
• nondimensional coordinate
• β
• thermal expansion coefficient (1·K⁻¹)
• δ_T
• thermal boundary layer thickness (m)
• δ_T,s
• thermal boundary layer thickness with an obvious thermal boundary layer heat conduction and convective equilibrium (m)
• δvi
• inner velocity boundary layer thickness of obvious thermal boundary layer (m)
• δ_vi,s
• steady inner velocity boundary layer thickness with an obvious thermal boundary layer (m)
• δ_vo,s
• steady outer velocity boundary layer thickness with an obvious thermal boundary layer (m)
• ΔT
• temperature difference (K)
• ΔT_w
• temperature difference when the thickness of the thermal boundary layer reaches the width of the chimney without an obvious thermal boundary layer (K)
• ΔT_w⁺
• temperature difference after thermal boundary layer thickness reaches the width of the chimney without an obvious thermal boundary layer (K)
• ΔT_w⁺,_f
• temperature difference of heat conduction and convection equilibrium without an obvious thermal boundary layer (K)
• κ
• thermal diffusion coefficient (m²·s⁻¹)
• v
• kinematic viscosity coefficient (m²·s⁻¹)
• v_t
• turbulence viscosity coefficient (m²·s⁻¹)
• v′
• sum of turbulent and laminar viscosity coefficients (m²·s⁻¹)
• ρ
• air density (kg·m⁻³)
• τ
• dimensionless time
• τ_s
• dimensionless time scaling of heat conduction and convective equilibrium with an obvious thermal boundary layer
• τ_w
• dimensionless time scaling at thermal boundary layer thickness reaches the width of the chimney without obvious thermal boundary layer flow regime
• θ
• dimensionless temperature
• Γ
• generalized diffusion coefficient

AUTHOR CONTRIBUTIONS

Huimin Cui: Methodology, conceptualization, supervision, writing-review and editing. Mengjiao Han: Data curation, writing-original draft, writing-review and editing. Jitao Zhang: Writing-review and editing. Zhiming Han: Supervision, writing-review and editing. Feng Xu: Supervision, writing-review and editing. Qingkuan Liu: Writing-review and editing.

ACKNOWLEDGMENTS

The study is financially supported by Natural Science Foundation of Hebei Province (E2022210069), Natural Science Foundation of Hebei Province Innovative Research Group Project (E2022210078), National Natural Science Foundation of China (11802186), and Central Leading Local Science and Technology Development Fund Project (236Z5410G); High Talents Project of Hebei Province (Hebei Office [2019] No. 63).

CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT

Data will be made available on request.