**4. System heat transfer mechanism in the model**

A complex differential analysis is required to fully analyze the multi-flapped collector due to shadows and the transient heat flow of flaps at different potentials (Figure 3). For instance, when the sun is between the azimuth of east and south, wings 1 and 2 receive full-area radiation at different incident angles, while wing 3 receives partial-area direct radiation due to the shadow of wing 2 cast on it. The case is reverse when the sun is between the azimuths of south and west, the critical time of changeover. When the sun is over the azimuth of 0°, wings 1 and 3 receive full-area direct radiation at different incident angles, and wing 2 receives partial direct radiation. This only happens intermittently (Figure 3).

**Figure 3.** Azimuth positions of the solar insolation with respect to the three absorber surfaces

The analysis of a tri-axially configured surface is cumbersome because of its distinct features compared to an ordinary flat-plate model; to overcome this, an assumption is made that the collector absorber material is a highly conductive metal such that heat due to absorbed radiation is evenly distributed in all the flaps. This implies that solar radiation incident on the collector is such that there is no temperature gradient anywhere in the collector (i.e. the absorber is isothermal at any point in time).

#### **4.1. Assumptions on analytical methods**

In the course of analysis of this solar collector, a number of overriding assumptions were made which made the correlation of analyzed values comparable to experimental data values. Experimental data for this type of solar collector at the study location (Nsukka, Enugu state, Nigeria) was not available; however, such correlations provide a complete analytical method. Such assumptions considered a vertical tri-wing absorber collector heated by insolation to a temperature Tp with free convection boundary layer effects and also considered a zero air stream velocity which could be increased to some maximum value and then decreased to zero again due to free stream conditions, provided the gap between the plate and the cover is much greater than the boundary layer thickness.

To effectively model flow within the solar chimney, the analytical method employed required the following steps:


#### **4.2. Solar insolation data at experimental setup location**

**4. System heat transfer mechanism in the model**

30 Solar Radiation Applications

radiation. This only happens intermittently (Figure 3).

absorber is isothermal at any point in time).

**4.1. Assumptions on analytical methods**

greater than the boundary layer thickness.

**Figure 3.** Azimuth positions of the solar insolation with respect to the three absorber surfaces

The analysis of a tri-axially configured surface is cumbersome because of its distinct features compared to an ordinary flat-plate model; to overcome this, an assumption is made that the collector absorber material is a highly conductive metal such that heat due to absorbed radiation is evenly distributed in all the flaps. This implies that solar radiation incident on the collector is such that there is no temperature gradient anywhere in the collector (i.e. the

In the course of analysis of this solar collector, a number of overriding assumptions were made which made the correlation of analyzed values comparable to experimental data values. Experimental data for this type of solar collector at the study location (Nsukka, Enugu state, Nigeria) was not available; however, such correlations provide a complete analytical method. Such assumptions considered a vertical tri-wing absorber collector heated by insolation to a temperature Tp with free convection boundary layer effects and also considered a zero air stream velocity which could be increased to some maximum value and then decreased to zero again due to free stream conditions, provided the gap between the plate and the cover is much

A complex differential analysis is required to fully analyze the multi-flapped collector due to shadows and the transient heat flow of flaps at different potentials (Figure 3). For instance, when the sun is between the azimuth of east and south, wings 1 and 2 receive full-area radiation at different incident angles, while wing 3 receives partial-area direct radiation due to the shadow of wing 2 cast on it. The case is reverse when the sun is between the azimuths of south and west, the critical time of changeover. When the sun is over the azimuth of 0°, wings 1 and 3 receive full-area direct radiation at different incident angles, and wing 2 receives partial direct

> The experimental setup was located at Nsukka, Nigeria, longitude 7° 23' 45" E and latitude 6° 51' 24" N [10], and the data acquisition method was adopted from the works of [11]. Solar data of the year and declination of that day were taken. This data was used as a case study (data presented in another chapter of this book) with the assumption that insolation recorded for a particular month is approximately the same for the same month for every other year. Also obtained from the literature were the measurement of instantaneous values of total insolation IT against diffuser Id and beam Ib insolations. From the instantaneous values, an expression relating to the total and beam insolations was deduced by the method of least squares for a third-order polynomial that fits the distribution of data. The polynomial is deduced as

$$\mathbf{I}\_{\rm b} = 0.01524 \mathbf{I}\_{\rm r}^3 - 0.0152 \mathbf{I}\_{\rm r}^2 + 0.988 \mathbf{I}\_{\rm r} - 0.9882 \left( \frac{\text{MJ}}{\text{m}^2} \right) \text{for } \mathbf{I}\_{\rm r} > 0.22 \left( \frac{\text{MJ}}{\text{m}^2} \right) \tag{6}$$

And with linear regression of natural logarithm of data distribution,

$$\mathbf{I}\_{\mathbf{b}} = \mathbf{e}^{(1.761 \text{lnl}\_{\Gamma} - 1.761)} \left(\frac{\mathbf{K} \mathbf{J}}{\mathbf{m}^2}\right) \text{for } \mathbf{I}\_{\Gamma} > 220 \left(\frac{\mathbf{K} \mathbf{J}}{\mathbf{m}^2}\right) \tag{7}$$

$$\mathbf{I}\_{\mathbf{d}} = \mathbf{I}\_{\mathbf{T}} - \mathbf{I}\_{\mathbf{b}} \tag{8}$$

where IT = total insolation, Ib = direct beam insolation and Id =diffuse insolation.

These parameters were used to compute instantaneous values of hourly insolation for the day of the year when data were taken.

Solar radiation on a tilted collector is made up of the beam radiation component, the diffuse radiation component and the ground-reflected diffuse radiation components. Hence, the total hourly solar radiation on a tilted surface is the summation of the three components:

$$\mathbf{I}\_{\rm c} = \mathbf{I}\_{\rm bc} + \mathbf{I}\_{\rm dc} + \mathbf{I}\_{\rm dgc} \tag{9}$$

where subscript c denotes solar radiation on the collector. Furthermore,

$$\mathbf{I}\_{\rm c} = \mathbf{I}\_{\rm bc} \mathbf{R}\_{\rm b} + \mathbf{I}\_{\rm d} \left[ \left( \mathbf{1} + \cos \mathfrak{R} \right) / \mathfrak{Z} \right] + \left( \mathbf{I}\_{\rm d} + \mathbf{I}\_{\rm b} \right) \mathfrak{p} \left[ \mathbf{1} / \mathfrak{Z} \left( \mathbf{1} - \cos \mathfrak{R} \right) \right] \tag{10}$$

For the case study, considering the collector being a vertical wall and no vegetation covering the ground, β=90° and ρ=0.2. Substituting these conditions into the equation of the total hourly radiation on the collector gives

$$\mathbf{I}\_{\rm c} = \mathbf{I}\_{\rm bc} \mathbf{R}\_{\rm b} + 0.5 \mathbf{I}\_{\rm d} + 0.1 \mathbf{(I}\_{\rm d} + \mathbf{I}\_{\rm b}) \mathbf{(R}\_{\rm b} + 0.1) \mathbf{I}\_{\rm b} + 0.6 \mathbf{I}\_{\rm d} \tag{11}$$

Both sides of each wing are exposed to diffuse radiation, while only one side of the wings is exposed to direct beam radiation depending on the area factor receiving this direct radiation. Hence, the total radiation aimed at each wing of the collector is

$$\mathbf{I}\_{\rm cl} = \forall \mathbf{I}\_{\rm b} \mathbf{R}\_{\rm bl} + \mathcal{D} \left[ 0.5 \mathbf{I}\_{\rm d} + 0.1 \mathbf{(I}\_{\rm d} + \mathbf{I}\_{\rm b}) \right] = \left( \forall \mathbf{R}\_{\rm b} + 0.2 \right) \mathbf{I}\_{\rm b} + 1.2 \mathbf{I}\_{\rm d} \tag{12}$$

Thus, the total radiation aimed at the whole collector in the chimney is

$$\mathbf{I}\_{\mathbf{c}} = \mathbf{I}\_{\mathbf{c}1} + \mathbf{I}\_{\mathbf{c}2} + \mathbf{I}\_{\mathbf{c}3} \tag{13}$$

Numbered subscripts 1, 2 and 3 denote wing positions.

#### **5. Optical properties of glazing and absorption of solar radiation**

The determination of collector performance requires the evaluation of the amount of solar radiation actually absorbed by the collector through glazing. Optical properties (transmittance, reflectance and absorptance) of the glaze cover, affects the absorption of solar radiation. These optical properties depend on the thickness of glazing and the refractive index and extinction coefficient of the glaze material. At a particular zenith angle of the sun, there are different incident angles of radiation on the circumference of the cylindrical glazing of the collector, unlike the flat-plate collector which has one uniform incident angle of radiation throughout the surface of glaze. Due to this feature, the evaluation of optical properties of a cylindrical glaze is complex. Though the glaze is cylindrical, the absorber plate is flat; hence, it was assumed that optical properties of a flat glass glazing are approximate to that of a cylindrical glass glazing. According to [12], expressions for reflection of unpolarized radiation passing from a medium with refractive index n1 to another with refractive index n2 are

Steady State Thermal Analysis of a Tri-Wing Solar Chimney http://dx.doi.org/10.5772/59721 33

$$\mathbf{r}\_r = \frac{\sin^2 \left(\boldsymbol{\theta}\_2 - \boldsymbol{\theta}\_1\right)}{\sin^2 \left(\boldsymbol{\theta} + \boldsymbol{\theta}\_1\right)}\\\text{and } \mathbf{r}\_i = \frac{\tan^2 \left(\boldsymbol{\theta}\_2 - \boldsymbol{\theta}\_1\right)}{\tan^2 \left(\boldsymbol{\theta}\_2 + \boldsymbol{\theta}\_1\right)}\tag{14}$$

where θ1 and θ2 are the angles of incidence and refraction. The average of reflectance of unpolarized radiation [12] is then

$$\mathbf{r} = \mathbf{0}.\mathbf{5}\left(\mathbf{r}\_{\perp} + \mathbf{r}\_{\parallel}\right) \tag{15}$$

where r⊥. and r<sup>∥</sup> represents perpendicular and parallel reflectance components of the unpo‐ larized radiation. The angle of refraction, θ2, is related to the refractive indices and the incident angle by Snell's law expressed as

$$\boldsymbol{\Theta}\_{2} = \sin^{-i} \left[ \frac{\mathbf{n}\_{1}}{\mathbf{n}\_{2} \left( \sin \theta\_{1} \right)} \right] \tag{16}$$

For air, n1 is approximately equal to 1. Consequently, the expressions for transmission of unpolarized radiation passing through a slab of glass from one interface to another interface and neglecting absorption decomposed to perpendicular and parallel components are as follows:

For a perpendicular component,

c bc dc dgc II I I =++ (9)

I I R I (1 cos / 2] I I 1 / 2 1 cos c bc b d = + + b + +r - b <sup>é</sup> ) ( d b ) é ù ( ) <sup>ë</sup> ë û (10)

c bc b d ( db b )( ) b d I I R 0.5I 0.1 I I R 0.1 I 0.6I = ++ + + + (11)

ë û (12)

c c1 c2 c3 II I I =++ (13)

For the case study, considering the collector being a vertical wall and no vegetation covering the ground, β=90° and ρ=0.2. Substituting these conditions into the equation of the total

Both sides of each wing are exposed to diffuse radiation, while only one side of the wings is exposed to direct beam radiation depending on the area factor receiving this direct radiation.

ci b bi <sup>d</sup> ( d b ) ( <sup>b</sup> ) b d I I R 2 0.5I 0.1 I I R 0.2 I 1.2I =" + + + = " + + é ù

**5. Optical properties of glazing and absorption of solar radiation**

from a medium with refractive index n1 to another with refractive index n2 are

The determination of collector performance requires the evaluation of the amount of solar radiation actually absorbed by the collector through glazing. Optical properties (transmittance, reflectance and absorptance) of the glaze cover, affects the absorption of solar radiation. These optical properties depend on the thickness of glazing and the refractive index and extinction coefficient of the glaze material. At a particular zenith angle of the sun, there are different incident angles of radiation on the circumference of the cylindrical glazing of the collector, unlike the flat-plate collector which has one uniform incident angle of radiation throughout the surface of glaze. Due to this feature, the evaluation of optical properties of a cylindrical glaze is complex. Though the glaze is cylindrical, the absorber plate is flat; hence, it was assumed that optical properties of a flat glass glazing are approximate to that of a cylindrical glass glazing. According to [12], expressions for reflection of unpolarized radiation passing

where subscript c denotes solar radiation on the collector. Furthermore,

Hence, the total radiation aimed at each wing of the collector is

Numbered subscripts 1, 2 and 3 denote wing positions.

Thus, the total radiation aimed at the whole collector in the chimney is

hourly radiation on the collector gives

32 Solar Radiation Applications

$$\left| \tau\_{\perp} = \left| \frac{1 - \tau\_{\perp}}{1 + \tau\_{\perp}} \right| \right. \tag{17}$$

For a parallel component,

$$\boldsymbol{\tau}\_{\parallel} = \begin{bmatrix} \mathbf{1} - \boldsymbol{\tau}\_{\parallel} \\ \mathbf{1} + \boldsymbol{\tau}\_{\parallel} \end{bmatrix} \tag{18}$$

where *τ*⊥ and *τ*∥ represent the perpendicular component and the parallel component of transmittance of unpolarized radiation for single cover glaze.

The average transmittance of these two components is

$$
\tau\_r = \frac{\left(\tau\_\perp + \tau\_\parallel\right)}{2} \tag{19}
$$

In order to account for absorption, a loss factor was introduced by Bourguer's law:

$$
\pi\_2 = e^{-\{kL / \cos \theta\_2\}} \tag{20}
$$

where k is the extinction coefficient (which varies from 4m-1 for white glass to 32m-1 for green glass). L is the thickness of the glass. In this study, the thickness of cover glass was taken to be 2.5mm, the extinction coefficient 5m-1 and the refractive index 1.526. The product k.L = 0.0025m x 5m-1 = 0.0125.

Some of the radiation passing through the glaze and striking the absorber plate is reflected back to the cover system. However, not all of this radiation is lost, since some of it is reflected back to the plate. The multiple reflections and absorptions between the plate and the cover is the greenhouse effect. To account for this greenhouse phenomenon, the actual fraction of incident radiation absorbed by the plate is called the transmittance-absorptance product, (τα). tThis is reasonably approximated as

$$\mathbf{A}\left(\pi a\right) = \mathbf{A}\left(\pi\right)a\_p\tag{21}$$

The constant A ranges from 1.01 to 1.02, but for conservativeness, 1.01 is preferred:

$$(\tau \alpha) = 1.01(\tau)\alpha\_p \tag{22}$$

For transmittance of diffuse radiation, the effective incident angle θ<sup>1</sup> for vertical collectors is 59.5° for both ground and sky diffuse radiation. Hence, the effective refractive angle θ<sup>2</sup> of diffuse radiation from Snell's law is

$$\boldsymbol{\Theta}\_{2} = \sin^{-1}\left[\frac{\mathbf{n}\_{1}}{\mathbf{n}\_{2}}(\sin \theta\_{1})\right] \tag{23}$$

Likewise, the reflection components are

$$\mathbf{r}\_{\perp} = \frac{\sin^{2}(\theta\_{2} - \theta\_{1})}{\sin^{2}\left(\theta\_{2} + \theta\_{1}\right)}\tag{24}$$

$$\mathbf{r}\_{\parallel} = \frac{\tan^2(\theta\_2 - \theta\_1)}{\tan^2(\theta\_2 + \theta\_1)}\tag{25}$$

The transmittance components are

( ) 2 *<sup>r</sup>* ^ <sup>+</sup> <sup>=</sup> <sup>P</sup> t t

<sup>2</sup> [/ ]

where k is the extinction coefficient (which varies from 4m-1 for white glass to 32m-1 for green glass). L is the thickness of the glass. In this study, the thickness of cover glass was taken to be 2.5mm, the extinction coefficient 5m-1 and the refractive index 1.526. The product k.L = 0.0025m

Some of the radiation passing through the glaze and striking the absorber plate is reflected back to the cover system. However, not all of this radiation is lost, since some of it is reflected back to the plate. The multiple reflections and absorptions between the plate and the cover is the greenhouse effect. To account for this greenhouse phenomenon, the actual fraction of incident radiation absorbed by the plate is called the transmittance-absorptance product, (τα).

> ( ) ( ) <sup>p</sup> τ Aτ a

The constant A ranges from 1.01 to 1.02, but for conservativeness, 1.01 is preferred:

( ) ( ) <sup>p</sup>

 t a

For transmittance of diffuse radiation, the effective incident angle θ<sup>1</sup> for vertical collectors is 59.5° for both ground and sky diffuse radiation. Hence, the effective refractive angle θ<sup>2</sup> of

> ( ) <sup>i</sup> <sup>1</sup> 2 1 2 n sin sin n - é ù q = ê ú <sup>q</sup>

ë û

( )

2 1

2 1

2 1

2 1

2

sin ( – ) <sup>r</sup> sin ^

2

2

tan ( – ) <sup>r</sup> tan ( )

2

ta

 a

q

*kL cos e* - =

In order to account for absorption, a loss factor was introduced by Bourguer's law:

(19)

(20)

= (21)

= 1.01 (22)

q q <sup>=</sup> q +q (24)

q q <sup>=</sup> q +q <sup>P</sup> (25)

(23)

t

2

t

x 5m-1 = 0.0125.

34 Solar Radiation Applications

tThis is reasonably approximated as

diffuse radiation from Snell's law is

Likewise, the reflection components are

$$
\pi\_\perp = \frac{\left(1 - \pi\_\perp\right)}{\left(1 + \pi\_\perp\right)}\tag{26}
$$

$$\pi\_{\parallel} = \frac{\left(1 - \tau\_{\parallel}\right)}{\left(1 + \tau\_{\parallel}\right)} \tag{27}$$

Cover absorptance *τa* was evaluated with the relation

$$\mathbf{\tau}\_{\mathbf{a}} = \mathbf{e}^{-\text{was }\text{ev}\theta\_2} \tag{28}$$

Hence, the resultant transmittance for diffuse radiation is

$$
\boldsymbol{\pi}\_{\rm{d}} = \boldsymbol{\pi}\_{\rm{r}} \cdot \boldsymbol{\pi}\_{\rm{a}} \tag{29}
$$

Since all necessary optical properties of the glass glazing have been evaluated, the amount of solar radiation actually absorbed by the collector can be easily deduced. The total incident radiation aimed at each wing of the collector is

$$\mathbf{I}\_{\rm cl} = \forall \mathbf{I}\_{\rm b} \mathbf{R}\_{\rm b\%} + 2 \left[ \mathbf{0}.5 \mathbf{I}\_{\rm d} + \mathbf{0}.1 \mathbf{(I}\_{\rm d} + \mathbf{I}\_{\rm b}) \right] \tag{30}$$

the total absorbed solar radiation of each wing is

$$\mathbf{S}\_{\mathrm{i}} = (\tau \alpha)\_{\mathrm{b}} \,\forall \mathbf{I}\_{\mathrm{b}} \mathbf{R}\_{\mathrm{b}l} + \mathcal{Q}(\tau \alpha)\_{\mathrm{b}} \left[ 0.5 \mathbf{I}\_{\mathrm{d}} + 0.1 \mathbf{I} \left( \mathbf{I}\_{\mathrm{d}} + \mathbf{I}\_{\mathrm{b}} \right) \right] \tag{31}$$

$$\mathbf{R} = (\texttt{\tau}\boldsymbol{\alpha})\_{\mathrm{b}} \,\,\forall \mathbf{I}\_{\mathrm{b}} \,\mathbf{R}\_{\mathrm{bl}} + (\texttt{\tau}\boldsymbol{\alpha})\_{\mathrm{b}} \left[\mathbf{I}\_{\mathrm{d}} + \mathbf{0}.\mathbf{2}\left(\mathbf{I}\_{\mathrm{d}} + \mathbf{I}\_{\mathrm{b}}\right)\right] \tag{32}$$

The mean absorbed solar radiation, S, of the whole collector plate is evaluated by the relation

$$\mathbf{S} = \mathbf{1} / \Im \left[ \mathbf{S}\_1 + \mathbf{S}\_2 + \mathbf{S}\_3 \right] \tag{33}$$

#### **6. The collector efficiency factor and the collector loss coefficient**

Considering the unusual configuration of the tri-wing absorber plate, a section of it was reduced to a vertical flat-plate collector over both sides of the plate without back insulation. This implies that a wing of the absorber can be treated as a flat-plate air heater with flow over both sides of the plate. The determination of the collector efficiency factor F′ and the collector loss coefficient UL provided the basis for the analytical prediction of collector performance when used in the Hottel-Whillier-Bliss equation [12, 13]:

$$\mathbf{q}\_u = \mathbf{F}\left[\mathbf{S} - \mathbf{U}\_1 \left(\mathbf{T}\_r - \mathbf{T}\_a\right)\right] \tag{34}$$

The collector efficiency factor F' is defined as the ratio of the actual useful energy gain to the useful gain that would result if the collector absorbing surface had been at the local fluid temperature, that is

$$\mathbf{F}^{\cdot} = \frac{\mathbf{q}\_u}{\left[\mathbf{S} - \mathbf{U}\_l \left(\mathbf{T}\_r - \mathbf{T}\_a\right)\right]} \tag{35}$$

where qu is the actual useful heat collector rate, S is absorbed radiation and Tr is average fluid temperature.

The collector loss coefficient Ul is defined as the lumped overall heat loss value of the whole collector to the operating temperature difference of the collector, that is

$$\mathbf{U}\_1 = \mathbf{E}\_{\text{loss}} (\mathbf{T}\_p - \mathbf{T}\_a) \tag{36}$$

where Eloss= heat loss, Tp = absorber plate temperature and Ta = ambient temperature of surrounding air.

To conform to the performance equation of a flat-plate collector, the configuration of the chimney collector is transformed to suit the equation. Criteria for the transformation are as follows:


#### **6.1. Electrical analogy**

**6. The collector efficiency factor and the collector loss coefficient**

when used in the Hottel-Whillier-Bliss equation [12, 13]:

temperature, that is

36 Solar Radiation Applications

temperature.

surrounding air.

plate.

plate.

collector.

follows:

Considering the unusual configuration of the tri-wing absorber plate, a section of it was reduced to a vertical flat-plate collector over both sides of the plate without back insulation. This implies that a wing of the absorber can be treated as a flat-plate air heater with flow over both sides of the plate. The determination of the collector efficiency factor F′ and the collector loss coefficient UL provided the basis for the analytical prediction of collector performance

> ( ) ' <sup>u</sup> lr a q FS UT T =- - é ù

The collector efficiency factor F' is defined as the ratio of the actual useful energy gain to the useful gain that would result if the collector absorbing surface had been at the local fluid

where qu is the actual useful heat collector rate, S is absorbed radiation and Tr is average fluid

The collector loss coefficient Ul is defined as the lumped overall heat loss value of the whole

where Eloss= heat loss, Tp = absorber plate temperature and Ta = ambient temperature of

To conform to the performance equation of a flat-plate collector, the configuration of the chimney collector is transformed to suit the equation. Criteria for the transformation are as

**1.** The total area of the tri-wing absorber plate is equal to the total area of the flat absorber

**2.** The height of the chimney collector is the same as the height of the resulting flat absorber

**3.** The area of the circular inlet column of the chimney collector and the rectangular area inlet column of the resulting flat-plate collector are equal throughout the height of the

( ) ' <sup>u</sup> lr a

collector to the operating temperature difference of the collector, that is

<sup>q</sup> <sup>F</sup>

ë û (34)

S UT T <sup>=</sup> é ù - - ë û (35)

U E (T T ) l loss p a = - (36)

According to [12], the usual procedure of deriving F' and Ul is by converting a schematic diagram of the collector system to a thermal network of electrical analogy and analyzing the resulting circuit. A schematic diagram with equivalent thermal network is shown in Figure 4.

**Figure 4.** Schematic diagram with equivalent thermal network

At the leading edge of the flat absorber plate, airflow is halved to flow on both sides of the plate. Absorbed solar energy heats up the plates to a temperature Tp, and energy is transferred to fluid at temperature Tr through the convection heat transfer coefficient h2 and to the cover glass through the linearized radiation heat transfer coefficient h1. Energy is also transferred to the cover glass from the fluid though the convective heat transfer coefficient ht , and finally energy is lost to the ambient air through the combined convection and radiation coefficient Ut from the cover glass.

#### **7. Energy balance equations**

Energy balance equations on the cover, the plates and the fluid are derived as follows:

$$2\,\mathrm{U}\_{\mathrm{t}}\left(\mathrm{T}\_{\mathrm{a}}-\mathrm{T}\_{\mathrm{c}}\right) + 2\,\mathrm{h}\_{\mathrm{1}}\left(\mathrm{T}\_{\mathrm{p}}-\mathrm{T}\_{\mathrm{c}}\right) + 2\,\mathrm{h}\_{\mathrm{2}}\left(\mathrm{T}\_{\mathrm{r}}-\mathrm{T}\_{\mathrm{c}}\right) = 0\tag{37}$$

$$\mathbf{S} + 2\mathbf{h}\_1 \left(\mathbf{T}\_c - \mathbf{T}\_p\right) + 2\mathbf{h}\_2 \left(\mathbf{T}\_r - \mathbf{T}\_p\right) = \mathbf{0} \tag{38}$$

$$2\mathbf{h}\_1\left(\mathbf{T}\_c - \mathbf{T}\_r\right) + 2\mathbf{h}\_2\left(\mathbf{T}\_p - \mathbf{T}\_r\right) = \mathbf{q}\_u\tag{39}$$

Simplifying and expanding the expressions as a function of heat transfer coefficients and temperatures yields

$$\mathbf{U\_t T\_a - U\_t T\_c + h\_r T\_p - h\_r T\_c + h\_t T\_r - h\_t T\_c = 0} \tag{40}$$

$$\mathbf{0}.5\mathbf{S} + \mathbf{h}\_1 \mathbf{T}\_c - \mathbf{h}\_1 \mathbf{T}\_p + \mathbf{h}\_2 \mathbf{T}\_r - \mathbf{h}\_2 \mathbf{T}\_p = \mathbf{0} \tag{41}$$

Equations (40) and (41) can be expressed as

$$\mathbf{A} = \mathbf{B} \mathbf{T}\_{\mathbf{c}} = \mathbf{T}\_{\mathbf{p}} \tag{42}$$

$$\mathbf{D} = \mathbf{E} \mathbf{T}\_p = \mathbf{T}\_c \tag{43}$$

where A=Ut Ta + h1Tr , B=Ut + hr + h1, C=hr , D= 0.5S + h2Tr and E=hr + h2.

Solving equations (42) and (43) simultaneously and re-substituting values into equations (40) and (41),

$$\mathbf{T\_p = \frac{\mathbf{AC} + \mathbf{BD}}{\left[\mathbf{BE} - \mathbf{C}\right]^2}} = \frac{(\mathbf{U\_t T\_a + h\_1 T\_r})\mathbf{h\_r + \left(\mathbf{U\_t + h\_r + h\_1}\right)0.5\mathbf{S} + h\_2 T\_r}{(\mathbf{U\_t + h\_r + h\_1)(h\_r + h\_2) - h\_r}^2} \tag{44}$$

$$\mathbf{T\_p = \frac{AE + CD}{\left[\mathbf{BE - C^2}\right]} = \frac{(\mathbf{U\_t T\_a + h\_1 T\_r})(\mathbf{h\_r + h\_2}) + (\mathbf{0.5S + h\_2 T\_r})\mathbf{h\_r}}{(\mathbf{U\_t + h\_r + h\_1})(\mathbf{h\_r + h\_2}) - \mathbf{h\_r}^2} \tag{45}$$

Subtracting Tr from both sides of equations (44) and (45),

$$\mathbf{T\_p} - \mathbf{T\_r} = \frac{\mathbf{U\_t h\_r} \left(\mathbf{T\_a} - \mathbf{T\_r}\right) + 0.5 \mathbf{S} \left(\mathbf{U\_t} + \mathbf{h\_r} + \mathbf{h\_t}\right)}{\left(\mathbf{U\_t} + \mathbf{h\_r} + \mathbf{h\_t}\right) \left(\mathbf{h\_r} + \mathbf{h\_2}\right) - \mathbf{h\_r}^2} \tag{46}$$

$$\mathbf{T\_c - T\_r = \frac{\left(\mathbf{U\_t h\_r + U\_t h\_2\right)\left(T\_a - T\_r\right) + 0.5S\mathbf{h\_r}}{\left(\mathbf{U\_t + h\_r + h\_1}\right)\left(\mathbf{h\_r + h\_2}\right) - \mathbf{h\_r}^2}}\tag{47}$$

Substituting equations (46) and (47) into equation (39),

$$0.5\mathbf{q}\_u = \frac{0.5\mathbf{S}\left(\mathbf{h}\_r\mathbf{h}\_r + \mathbf{U}\_r\mathbf{h}\_2 + \mathbf{h}\_r\mathbf{h}\_2 + \mathbf{h}\_1\mathbf{h}\_2\right) + \mathbf{U}\_t\left(\mathbf{h}\_r\mathbf{h}\_r + \mathbf{h}\_r\mathbf{h}\_2 + \mathbf{h}\_1\mathbf{h}\_2\right) + \left(\mathbf{T}\_a - \mathbf{T}\_r\right)}{\left(\mathbf{U}\_t + \mathbf{h}\_r + \mathbf{h}\_1\right)\left(\mathbf{h}\_r + \mathbf{h}\_2\right) - \mathbf{h}\_r^2} \tag{48}$$

Rearranging equation (46),

**7. Energy balance equations**

38 Solar Radiation Applications

temperatures yields

where A=Ut

(40) and (41),

Ta + h1Tr

Equations (40) and (41) can be expressed as

, B=Ut + hr + h1, C=hr

Energy balance equations on the cover, the plates and the fluid are derived as follows:

Simplifying and expanding the expressions as a function of heat transfer coefficients and

2U T T 2h T T 2h T T 0 ta c 1p c 2r c ( -+ -+ -= ) ( ) ( ) (37)

S 2h T T 2h T T 0 + -+ -= 1c p 2r p ( ) ( ) (38)

2h T T 2h T T q 1c r 2p r u ( -+ -= ) ( ) (39)

UT UT hT hT hT hT 0 ta tc rp rc 1r 1c - + -+-= (40)

1c 1p 2r 2p 0.5S h T h T h T h T 0 +-+- = (41)

A BT T c p = -= (42)

D ET T p c = -= (43)

(44)

(45)

, D= 0.5S + h2Tr and E=hr + h2.

ta 1r r t r 1 ( ) 2 r

t r 1r 2 r

ta 1r r 2 2r r

t r 1r 2 r

Solving equations (42) and (43) simultaneously and re-substituting values into equations

p 2 2

p 2 2

AC BD (U T h T )h U h h 0.5S h T <sup>T</sup> BE C (U h h )(h h ) h <sup>+</sup> + + ++ + = = é ù - ++ + - ë û

AE CD (U T h T )(h h ) (0.5S h T )h <sup>T</sup> BE C (U h h )(h h ) h <sup>+</sup> + ++ + = = é ù - ++ + - ë û

$$\mathbf{q}\_{\rm u} = \frac{\mathbf{S} \left( \mathbf{h}\_{\rm i} \mathbf{h}\_{\rm r} + \mathbf{U}\_{\rm i} \mathbf{h}\_{\rm 2} + \mathbf{h}\_{\rm r} \mathbf{h}\_{\rm 2} + \mathbf{h}\_{\rm i} \mathbf{h}\_{\rm 2} \right)}{\left( \mathbf{U}\_{\rm t} + \mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm i} \right) \left( \mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm 2} \right) - \mathbf{h}\_{\rm r}^{2}} - \frac{2 \mathbf{U}\_{\rm t} \left( \mathbf{h}\_{\rm 1} \mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm r} \mathbf{h}\_{\rm 2} + \mathbf{h}\_{\rm 1} \mathbf{h}\_{\rm 2} \right) + \left( \mathbf{T}\_{\rm a} - \mathbf{T}\_{\rm r} \right)}{\left( \mathbf{U}\_{\rm t} + \mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm 1} \right) \left( \mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm 2} \right) - \mathbf{h}\_{\rm r}^{2}} \tag{49}$$

Comparing equation (49) with the Hottel-Whillier-Bliss equation, qu =F' S−UL(Tr−Ta) , it can be deduced that

$$\mathbf{F}' = \frac{\mathbf{h}\_{\mathrm{i}}\mathbf{h}\_{\mathrm{r}} + \mathbf{U}\_{\mathrm{i}}\mathbf{h}\_{\mathrm{2}} + \mathbf{h}\_{\mathrm{r}}\mathbf{h}\_{\mathrm{2}} + \mathbf{h}\_{\mathrm{i}}\mathbf{h}\_{\mathrm{2}}}{\left(\mathbf{U}\_{\mathrm{i}} + \mathbf{h}\_{\mathrm{r}} + \mathbf{h}\_{\mathrm{i}}\right)\left(\mathbf{h}\_{\mathrm{r}} + \mathbf{h}\_{\mathrm{2}}\right) - \mathbf{h}\_{\mathrm{r}}\,^{2}} \tag{50}$$

$$\mathbf{U}\_{\rm L} = \frac{2\mathbf{U}\_{\rm t} \left(\mathbf{h}\_{\rm l}\mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm r}\mathbf{h}\_{\rm 2} + \mathbf{h}\_{\rm 1}\mathbf{h}\_{\rm 2}\right) + \left(\mathbf{T}\_{\rm a} - \mathbf{T}\_{\rm r}\right)}{\left(\mathbf{h}\_{\rm r}\mathbf{h}\_{\rm r} + \mathbf{U}\_{\rm r}\mathbf{h}\_{\rm r} + \mathbf{h}\_{\rm r}\mathbf{h}\_{\rm 2}\right) + \mathbf{h}\_{\rm 1}\mathbf{h}\_{\rm 2}}\tag{51}$$

It has been demonstrated experimentally that h1 =h2 =h according to Duffie and Beckman; therefore, equations (47) and (48) reduce to

$$\mathbf{F}' = \frac{\mathbf{h}\left(\mathbf{U}\_{\mathrm{t}} + 2\mathbf{h}\_{\mathrm{r}} + \mathbf{h}\right)}{\left(\mathbf{U}\_{\mathrm{t}} + \mathbf{h}\_{\mathrm{r}} + \mathbf{h}\_{\mathrm{t}}\right)\left(\mathbf{h}\_{\mathrm{r}} + \mathbf{h}\_{\mathrm{z}}\right) - \mathbf{h}\_{\mathrm{r}}^{2}} \text{ and } \mathbf{U}\_{\mathrm{L}} = \frac{2\mathbf{U}\_{\mathrm{t}}\left(2\mathbf{h}\_{\mathrm{r}} + \mathbf{h}\right) + \left(\mathbf{T}\_{\mathrm{a}} - \mathbf{T}\_{\mathrm{r}}\right)}{\left(\mathbf{U}\_{\mathrm{t}} + 2\mathbf{h}\_{\mathrm{r}}\right) + \mathbf{h}}\tag{52}$$

It is necessary to outline expressions of heat coefficients as functions of F' and UL. These heat coefficients are expressed as

$$\mathbf{U}\_t = \left(\mathbf{h}\_{\text{wind}} + \mathbf{h}\_{\text{rea}}\right) \tag{53}$$

where hrea is the radiative loss coefficient between the cover and ambient air expressed as

$$\mathbf{h}\_{\rm raa} = \mathbf{e}\_{\rm c} \mathbf{a} \left( \mathbf{T}\_{\rm c}^2 + \mathbf{T}\_{\rm s}^2 \right) \left( \mathbf{T}\_{\rm c} - \mathbf{T}\_{\rm s} \right) \tag{54}$$

where glass cover emittance εc=0.88, Stefan-Boltzmann constant α=5.6 <sup>x</sup> <sup>10</sup>−<sup>1</sup> w / m<sup>2</sup> k4 and sky temperature Ts=0.0552Ta 1.5.

The convective atmospheric heat loss coefficient due to wind, hwind, is dependent on the prevailing wind velocity given by [14] as

$$\mathbf{h}\_{\text{wind}} = \mathbf{5.8} + \mathbf{3.8v} \tag{55}$$

where v (ms−<sup>1</sup> ) is the wind velocity (2 ms<sup>−</sup><sup>1</sup> ) for the study location); thus,

$$\mathbf{h}\_{\text{wind, location}} = \mathbf{5.8} + \mathbf{3.8x2} = \mathbf{13.4} \le \mathbf{/m}^2 \mathbf{c} \tag{56}$$

The radiative heat loss coefficient between the plate and the cover after linearization is given as

$$\mathbf{h}\_{\rm r} = \frac{\sigma}{\left(\varepsilon\_{\rm c}^{-\rm h} + \varepsilon\_{\rm p}^{-\rm h} - \mathbf{h}\right)\left(\mathbf{T}\_{\rm c}^{2} + \mathbf{T}\_{\rm p}^{2}\right)\left(\mathbf{T}\_{\rm c} - \mathbf{T}\_{\rm p}\right)}\tag{57}$$

where absorber plate emittance ε<sup>p</sup> =1−α<sup>p</sup> =0.1.

#### **8. Heat transfer and collector performance analysis**

A review of the principles and theories governing natural free convection heat transfer [4] is done for the determination of other collector performance parameters. Thus, considering a vertical absorber plate of one wing of the collector heated by insolation to a temperature Tp, a free convection boundary layer is formed. The boundary layer is such that at the wall of the plate, the velocity of air stream is zero. This increases to some maximum value and then decreases to zero due to free stream conditions, provided the gap between the plate and the cover is much greater than the boundary layer thickness. The equation of motion for such a system is derived from the Navier-Stokes momentum equation of fluid flow given by [15]:

$$\mathbf{q}\left(\frac{\partial \mathbf{u}}{\partial \mathbf{x}} + \mathbf{v}\frac{\partial \mathbf{u}}{\partial \mathbf{y}}\right) = -\mathbf{q}\mathbf{g} + \frac{\mu \hat{\sigma}^2 \mathbf{u}}{\partial \mathbf{y}^2} \tag{58}$$

where ρ is the density of air; x is the displacement in the vertical direction, the direction of buoyant flow; y is the displacement perpendicular to the direction of buoyant flow; U is the velocity component in the x -direction; v is the velocity component in the y-direc‐ tion; p is air pressure; μ is the dynamic viscosity of air and g is acceleration due to gravity of value 9.18 (ms<sup>−</sup><sup>2</sup> ).

Equation (58) can be expressed as

It is necessary to outline expressions of heat coefficients as functions of F'

where glass cover emittance εc=0.88, Stefan-Boltzmann constant α=5.6 <sup>x</sup> <sup>10</sup>−<sup>1</sup>

where hrea is the radiative loss coefficient between the cover and ambient air expressed as

( )( ) 2 2

The convective atmospheric heat loss coefficient due to wind, hwind, is dependent on the

The radiative heat loss coefficient between the plate and the cover after linearization is

A review of the principles and theories governing natural free convection heat transfer [4] is done for the determination of other collector performance parameters. Thus, considering a vertical absorber plate of one wing of the collector heated by insolation to a temperature Tp, a free convection boundary layer is formed. The boundary layer is such that at the wall of the plate, the velocity of air stream is zero. This increases to some maximum value and then

( )( )( ) <sup>r</sup> h h 22 c p c pcp <sup>σ</sup> <sup>h</sup>

coefficients are expressed as

40 Solar Radiation Applications

temperature Ts=0.0552Ta

where v (ms−<sup>1</sup>

given as

1.5.

) is the wind velocity (2 ms<sup>−</sup><sup>1</sup>

prevailing wind velocity given by [14] as

where absorber plate emittance ε<sup>p</sup> =1−α<sup>p</sup> =0.1.

**8. Heat transfer and collector performance analysis**

and UL. These heat

w / m<sup>2</sup> k4

and sky

U h h t wind rea = + ( ) (53)

rea c c s c s h T TTT =ea + - (54)

wind h 5.8 3.8v = + (55)

) for the study location); thus,

2 wind, location h =+ = ° 5.8 3.8x 2 13.4 w / m c (56)

ε ε hT T T T - - <sup>=</sup> +- + - (57)

$$\left(\mathbf{U}\frac{\partial\mathbf{u}}{\partial\mathbf{x}} + \mathbf{v}\frac{\partial\mathbf{u}}{\partial\mathbf{y}}\right) = \frac{\partial\mathbf{p}}{\partial\mathbf{x}} - \rho\mathbf{g} + \mu\frac{\partial^2\mathbf{u}}{\partial\mathbf{y}^2} \tag{59}$$

where <sup>∂</sup> <sup>p</sup> <sup>∂</sup> <sup>x</sup> =g. ρ*∞* and ρ*∞* is the free stream air density.

The density difference can be expressed in terms of the volume coefficient of expansion, β, defined by

$$\mathbf{I} = \frac{\mathbf{1}\left(\frac{\partial \mathbf{V}}{\partial \mathbf{T}}\right)\_p}{\mathbf{V}\left(\frac{\partial \mathbf{T}}{\partial \mathbf{T}}\right)\_p} = \frac{\mathbf{1}}{\mathbf{V}\_o} \left[\frac{\mathbf{V} - \mathbf{V}\_o}{\mathbf{T} - \mathbf{T}\_o}\right] = \left[\frac{\mathbf{p} - \mathbf{p}\_o}{\mathbf{T} - \mathbf{T}\_o}\right] \tag{60}$$

Substituting equation (59) into (58) brings the momentum equation to

$$\log\left(\mathbf{U}\frac{\partial\mathbf{u}}{\partial\mathbf{x}} + \mathbf{V}\frac{\partial\mathbf{u}}{\partial\mathbf{y}}\right) = \mathbf{q}\mathbf{g}\beta\left(\mathbf{T} - \mathbf{T}\_{\circ}\right) + \mu\frac{\partial^2\mathbf{u}}{\partial\mathbf{y}^2} \tag{61}$$

integrating equation (61) with respect to y from the boundary layer thickness δ to the plate surface; since the velocity in the flow direction u is much greater than the velocity component perpendicular to the flow direction v, hence v tends to zero compared to u. The integral momentum equation becomes

$$\frac{\text{d}}{\text{d} \text{x}} \left[ \int\_{0}^{\text{s}} \text{�} \mathbf{u}^{2} \text{d} \mathbf{y} \right] = \underset{0}{\text{�}} \text{q} \otimes \mathbf{g} \left( \text{T} - \text{T}\_{\text{o}} \right) \text{d} \mathbf{y} - \mu \frac{\partial \mathbf{u}}{\partial \mathbf{y}} = \mathbf{0} \tag{62}$$

The functional relation between the temperature distribution and the boundary layer thickness is needed to solve equation (61). From the prevailing conditions of the system, temperature T=Tp at y=0, T=T*∞* at y=δ and dt dy =0 at y=δ, a parabolic temperature function can be assumed to represent the temperature profile of the system with respect to y.

Therefore, let the function be of the form

$$\mathbf{T} = \mathbf{A}\mathbf{y}^2 + \mathbf{B}\mathbf{y} + \mathbf{C} \tag{63}$$

Substituting the conditions of flow into equation (61) gives

$$\mathbf{T} - \mathbf{T}\_p = \mathbf{A}\mathbf{y}^2 - \mathbf{A}\mathbf{n}\delta\mathbf{y} \text{ and } \mathbf{T}\_p - \mathbf{T}\_o = \mathbf{2}\mathbf{A}\delta^2 - \mathbf{2}\delta^2 = \mathbf{A}\delta^2\tag{64}$$

Also the expression for the velocity profile is necessary to solve the integral momentum equation which can be deduced from the following flow conditions: u=0 at y=0, u=0 at y=δ and <sup>∂</sup><sup>2</sup> <sup>u</sup> <sup>∂</sup> <sup>y</sup><sup>2</sup> = −βg(T−T*∞*) / v at y=0. These four flow conditions can be fitted into a cubic equation of the form

$$\frac{\mathbf{u}}{\mathbf{u}\_{\times}} = \mathbf{A}\mathbf{y}^{3} + \mathbf{B}\mathbf{y}^{2} + \mathbf{C}\mathbf{y} + \mathbf{D} \tag{65}$$

where ux is a fictitious velocity which is a function of displacement in the x -direction. Applying the conditions of flow into the cubic equation gives

$$\begin{aligned} \mathbf{D} &= \mathbf{0}, \\ \mathbf{C} &= \left[ \frac{\mathfrak{Bg} \left( \mathbf{T}\_{\mathrm{p}} - \mathbf{T}\_{\mathrm{ox}} \right) \delta}{4 \mathbf{v}} \right], \\ \mathbf{B} &= \left[ -\frac{\mathfrak{Bg} \left( \mathbf{T}\_{\mathrm{p}} - \mathbf{T}\_{\mathrm{ox}} \right)}{2 \mathbf{v}} \right], \\ \mathbf{A} &= \left[ -\frac{\mathfrak{Bg} \mathfrak{Bg} \left( \mathbf{T}\_{\mathrm{p}} - \mathbf{T}\_{\mathrm{ox}} \right)}{4 \mathbf{v} \delta} \right], \end{aligned} \tag{66}$$

Substituting these expressions into equation (65) and evaluating yields

The functional relation between the temperature distribution and the boundary layer thickness is needed to solve equation (61). From the prevailing conditions of the system, temperature

2 22 2

Also the expression for the velocity profile is necessary to solve the integral momentum equation which can be deduced from the following flow conditions: u=0 at y=0, u=0 at y=δ

3 2

<sup>u</sup> Ay By Cy D

where ux is a fictitious velocity which is a function of displacement in the x -direction. Applying

( )

¥

ü ï ï

ï ï ï ï ý ï ï ï ï ï

p

é ù b -d <sup>=</sup> ê ú ê ú ë û é ù b - = -ê ú ê ú ë û é ù

gT T C , 4v

gT T B , 2v

βgT T <sup>A</sup>

( )

p

( )

¥

¥

p

4v

<sup>ï</sup> ë ûþ

b - = -ê ú ê ú d

x

D 0,

=

u

the conditions of flow into the cubic equation gives

<sup>∂</sup> <sup>y</sup><sup>2</sup> = −βg(T−T*∞*) / v at y=0. These four flow conditions can be fitted into a cubic equation

p p T T Ay An y and T T 2A 2 A ¥ - = - d - = d-d= d (64)

to represent the temperature profile of the system with respect to y.

Substituting the conditions of flow into equation (61) gives

dy =0 at y=δ, a parabolic temperature function can be assumed

<sup>2</sup> T Ay By C = ++ (63)

= + ++ (65)

(66)

T=Tp at y=0, T=T*∞* at y=δ and dt

42 Solar Radiation Applications

and <sup>∂</sup><sup>2</sup> <sup>u</sup>

of the form

Therefore, let the function be of the form

$$\frac{\mathbf{u}}{\mathbf{u}\_{\times}} = \frac{\left[\frac{\beta \mathbf{g} \,\mathrm{s}^2 \left(\mathbf{T}\_p - \mathbf{T}\_{\times}\right)}{4 \mathbf{u}\_{\times} \mathbf{v}}\right] \mathbf{y}}{\left(1 - \frac{\mathbf{y}}{\mathbf{S}}\right)^2} \tag{67}$$

The terms involving (T<sup>p</sup> −T*∞*), δ<sup>2</sup> and ux may be incorporated into the function ux so that the velocity profile can be reduced to

$$\frac{\mathbf{u}}{\mathbf{u}\_{\times}} = \frac{\mathbf{y} / \delta}{\left(1 - \frac{\mathbf{y}}{\delta}\right)^2} \tag{68}$$

Now the momentum equation of equation (61) when solved has its terms reduced to

$$\begin{aligned} \stackrel{\circ}{\mathbf{j}} : \mathbf{u}^2 \mathbf{d} \mathbf{y} &= \stackrel{\circ}{\mathbf{j}} \frac{\mathbf{u}\_\star}{\delta^2} \left[ (\mathbf{y}^2 - \frac{4\mathbf{y}^3}{\delta} + \frac{6\mathbf{y}^4}{\delta^2} - \frac{4\mathbf{y}^5}{\delta^3} + \frac{\mathbf{y}^6}{\delta^4} \right] \mathbf{d} \mathbf{y} \\ &= \frac{\mathbf{u}\_\star^2}{\delta^2} \left[ \frac{\mathbf{y}^3}{3} - \frac{\mathbf{y}^4}{\delta} + \frac{5\mathbf{y}^6}{\delta^3} + \frac{\mathbf{y}^7}{\delta^4} \right] \mathbf{y}\_\circ^3 \\ &= \mathbf{u}\_\star^{-2} \delta (\frac{1}{3} - \frac{6}{5} - \frac{2}{3} + \frac{1}{7}) \\\\ &\stackrel{\circ}{\to} \stackrel{\circ}{\to} \mathbf{u}^2 \mathbf{d} \mathbf{y} = \frac{\mathbf{u}\_\star^2 \delta}{1.05} \end{aligned} \tag{70}$$

Solving equation (69) in terms of temperature variables,

$$\int\_{0}^{\delta} \mathbf{j} \cdot \left(\mathbf{T} - \mathbf{T}\_{\alpha}\right) d\mathbf{y} = \int\_{0}^{\delta} \mathbf{j} \left(\mathbf{T}\_{\mathrm{p}} - \mathbf{T}\_{\alpha}\right) \left(\mathbf{1} - \frac{\mathbf{2}\mathbf{y}}{\delta} + \frac{\mathbf{y}^{3}}{\delta^{2}}\right)\_{0}^{\delta} = \left(\mathbf{T}\_{\mathrm{p}} - \mathbf{T}\_{\alpha}\right) \delta(\mathbf{1} - \mathbf{1} + \frac{\mathbf{1}}{\mathbf{3}}) \tag{71}$$

$$\implies \prod\_{0}^{\delta} \colon \left(\mathbf{T} - \mathbf{T}\_{\circ \circ}\right) \mathbf{d} \mathbf{y} = \frac{1}{3} \left(\mathbf{T}\_{\circ} - \mathbf{T}\_{\circ \circ}\right) \delta \tag{72}$$

$$\begin{aligned} \frac{\delta \mathbf{u}}{\delta \mathbf{y}} \Big|\_{\mathbf{y}=0} &= \frac{\mathbf{Ux}}{\delta} \Big( \mathbf{y} - \frac{2 \mathbf{y}^2}{\delta} + \frac{\mathbf{y}^3}{\delta^2} \Big) \Big|\_{\mathbf{y}=0} = \frac{\mathbf{Ux}}{\delta} \Big( \mathbf{1} - \frac{4 \mathbf{y}}{\delta} + \frac{3 \mathbf{y}^2}{\delta^2} \Big) \Big|\_{\mathbf{y}=0} \\ &\Rightarrow \frac{\delta \mathbf{u}}{\delta \mathbf{y}} \Big|\_{\mathbf{y}=0} = \frac{\mathbf{Ux}}{\delta} \end{aligned} \tag{73}$$

Substituting equations (67), (68) and (70) into equation (61),

$$\frac{1}{1.05} \left( \frac{\text{d}}{\text{d} \text{x}} \text{u}\_{\text{x}}^{\text{-2}} \right) = \frac{1}{3} \text{g} \Re \left( \text{T}\_{\text{p}} - \text{T}\_{\text{o}} \right) \delta - \frac{\text{vx}}{\text{g}} \tag{74}$$

where v<sup>=</sup> <sup>μ</sup> <sup>ρ</sup> is known as kinematic viscosity.

The energy equation for free convection of the collector system is expressed as

$$\mathbf{q}\mathbf{C}\_{p}\left(\frac{\mathbf{U}\boldsymbol{\partial}\mathbf{T}}{\partial\mathbf{x}} + \mathbf{V}\frac{\mathbf{U}\boldsymbol{\partial}\mathbf{T}}{\partial\mathbf{y}}\right) = \mathbf{k}\frac{\partial^{2}\mathbf{T}}{\partial\mathbf{y}^{2}}, \text{ or} \left(\frac{\mathbf{U}\boldsymbol{\partial}\mathbf{T}}{\partial\mathbf{x}} + \mathbf{V}\frac{\mathbf{U}\boldsymbol{\partial}\mathbf{T}}{\partial\mathbf{y}}\right) = \mathbf{a}\frac{\partial^{2}\mathbf{T}}{\partial\mathbf{y}^{2}}\tag{75}$$

where α<sup>=</sup> <sup>k</sup> ρCp is called the thermal diffusivity. The integral form of equation (72) with velocity tending to zero is

$$\frac{\text{d}}{\text{d} \mathbf{x}} \Big|\_{0}^{\delta} \colon \mathbf{U} \Big(\mathbf{T} - \mathbf{T}\_{\text{o}}\big) \text{dy} = \mathbf{a} \frac{\text{d} \mathbf{T}}{\text{dy}} \Big|\_{\text{y}} = \mathbf{0} \tag{76}$$

From equation (67), it can be deduced that the relation of the order of magnitude between ux and δ is ux −δ<sup>2</sup> . Inserting this relation into equation (68) and solving yields δ−x1/2 . Hence, it can be assumed that

$$\mathbf{u}\_{\chi} = \mathbf{A} \mathbf{x}^{1/2} \text{ and } \boldsymbol{\delta} = \mathbf{B} \mathbf{x}^{1/4} \tag{77}$$

Introducing these expressions into equations (67) and (74) yields

$$\frac{5}{240} \mathbf{A}^2 \mathbf{B} \mathbf{x}^{1/4} = \mathbf{g} \Re \left( \mathbf{T} - \mathbf{T}\_\circ \right) \left( \frac{\mathbf{B} \mathbf{x}^{1/4}}{\mathbf{3}} \right) - \left( \frac{\mathbf{A}}{\mathbf{B}} \right) \mathbf{v} \mathbf{x}^{1/4} \tag{78}$$

Solving for A and B in the above equation,

Steady State Thermal Analysis of a Tri-Wing Solar Chimney http://dx.doi.org/10.5772/59721 45

$$\mathbf{A} = 5.17\mathbf{v} \left( 0.952 + \mathbf{Pr} \right)^{-99} \left[ \mathbf{g} \mathbf{f} \left( \mathbf{T} - \mathbf{T}\_{\circ} \right) \frac{\mathbf{x}^{3}}{\mathbf{v}^{2}} \right]^{\circ \approx 99} \left[ \mathbf{Pr} \right]^{\circ \approx 99} \tag{79}$$

$$\mathbf{B} = \mathbf{3.93} \mathbf{Pr}^{-\frac{1}{4}} \left[ \mathbf{g} \boldsymbol{\beta} \left( \mathbf{T} - \mathbf{T}\_{\circ} \right) \frac{\mathbf{x}^{3}}{\mathbf{v}^{2}} \right]^{-\times 3.} \left[ \mathbf{Pr} \right]^{-\text{r}3.} \tag{80}$$

Substituting the expression for A into equation (77) gives the fictitious velocity ux at any point x as

$$\mathbf{u}\_{\times} = 5.17\,\mathrm{v} \left( 0.952 + \mathrm{Pr} \right)^{-\frac{1}{2}} \mathrm{(Gr\_{\times})}^{-r} \tag{81}$$

where Grx is the Grashof number at point x, this is given by

2 3 2

(73)

. Hence, it

y 0 2 2 y 0 y 0

= ==

æ öæ ö = - + = -+ ç ÷ç ÷ è øè ø

y 0

=

δu Ux <sup>|</sup> δy δ

( ) <sup>2</sup> x p 1d 1 vx u gT T 1.05 dx 3 ¥

p 2 2 UT UT T UT UT T ρC V k , or V α xy xy y y æ öæ ö ¶ ¶¶ ¶ ¶¶ ç ÷ç ÷ += +=

ç ÷ = b - d- <sup>è</sup> <sup>ø</sup> <sup>d</sup> (74)


<sup>x</sup> u Ax and Bx = d= (77)

æ ö æ ö =b - - ç ÷ ç ÷ è ø è ø (78)

2 2

ρCp is called the thermal diffusivity. The integral form of equation (72) with velocity

¶¶ ¶¶ ¶ ¶ è øè ø (75)

δu Ux 2y y Ux 4y 3y |y |1 | δy δ δ δ δ δ δ

Þ =

The energy equation for free convection of the collector system is expressed as

( ) 0 y <sup>d</sup> dT : U T T dy 0 dx dy ¥

From equation (67), it can be deduced that the relation of the order of magnitude between ux

1/2 1/4

( ) 1/4 <sup>5</sup> 2 1/4 Bx A 1/4 A Bx g T T vx 240 3 B ¥

. Inserting this relation into equation (68) and solving yields δ−x1/2

d

Introducing these expressions into equations (67) and (74) yields

Solving for A and B in the above equation,

Substituting equations (67), (68) and (70) into equation (61),

<sup>ρ</sup> is known as kinematic viscosity.

where v<sup>=</sup> <sup>μ</sup>

44 Solar Radiation Applications

where α<sup>=</sup> <sup>k</sup>

tending to zero is

and δ is ux −δ<sup>2</sup>

can be assumed that

æ ö

$$\mathbf{Gr}\_{\mathbf{x}} = \mathbf{g}\beta \left(\mathbf{T} - \mathbf{T}\_{\kappa}\right) \frac{\mathbf{x}^{3}}{\mathbf{v}^{2}} \tag{82}$$

The resultant expression for boundary layer thickness from equation (75) is

$$\frac{\delta}{\chi} = \text{Bx}^{-\text{Re}} = 3.93 \text{Pr}^{-\frac{1}{2}} \left( 0.952 + \text{Pr} \right)^{\frac{1}{4}} \left( \text{Gr}\_{\text{x}} \right)^{-\frac{1}{4}} \tag{83}$$

The heat transfer coefficient may be evaluated using equation (62)

$$\mathbf{q}\_p = -\mathbf{k} \mathbf{A} \frac{\mathbf{d} \mathbf{T}}{\mathbf{d} \mathbf{y}} = \mathbf{k} \mathbf{A} \left(\frac{2}{\delta}\right) \left(\mathbf{T}\_p - \mathbf{T}\_o\right) = \mathbf{h} \mathbf{A} \left(\mathbf{T}\_p - \mathbf{T}\_o\right) \tag{84}$$

where it was derived that h= 2k <sup>δ</sup> . Hence, the dimensionless equation for the heat transfer coefficient becomes the expression for Nusselt number Nux :

$$\mathrm{Nu}\_{\mathrm{x}} = 0.508 \mathrm{Pr}^{-\frac{1}{2}} \left( 0.952 + \mathrm{Pr} \right)^{-\frac{1}{4}} \left( \mathrm{Gr}\_{\mathrm{x}} \right)^{-\frac{1}{4}} \tag{85}$$

In the case study, it was assumed that solar heating of the collector is one of the constant heat flux conditions. In such a case, a modified Grashof number is introduced:

$$\mathbf{Gr}\_{\mathbf{x}}{}^{\*} = \mathbf{Gr}\_{\mathbf{x}}.\mathbf{Nu}\_{\mathbf{x}} = \frac{\mathbf{g}\mathfrak{B} \times \mathbf{^{4}q}\_{\mathbf{p}}}{\mathbf{k}\mathbf{v}^{2}}\tag{86}$$

where qp is the plate heat flux. The local heat transfer coefficient is correlated as

$$\text{Nu}\_{\text{x}} = \frac{\text{hx}}{\text{k}} = 0.6 \left( \text{Gr}\_{\text{x}} \text{Pr} \right)^{0.2}, \ 10^{5} < \text{Gr}\_{\text{x}} \text{"<} < 10^{11}, \text{q}\_{\text{p}} = \text{constant} \tag{87}$$

For the turbulent range, the average heat transfer coefficient for the constant heat flux cause is obtained from

$$\mathbf{h} = \frac{1}{\mathbf{L}} \int\_0^1 \mathbf{h}\_\mathbf{x} \mathbf{d}\_\mathbf{x} = \mathbf{1}.25 \mathbf{h}\_\mathbf{x} \tag{88}$$

It has been established that during distribution the local heat transfer coefficient is essentially with x when turbulent free convection is encountered. In such a case, the average heat transfer coefficient is h=hx. The mean velocity of air stream due to natural convection was obtained from evaluating the mean value of velocity distribution of equation (64) as

$$\mathbf{u} = \mathbf{u}\_{\times} \left[ \left[ \frac{\mathbf{y}}{\mathfrak{S}} \left( 1 - \frac{\mathbf{y}}{\mathfrak{S}} \right) \right] \mathbf{d} \left( \frac{\mathbf{y}}{\mathfrak{S}} \right) = 0.0833 \,\mathrm{U}\_{\times} \tag{89}$$

The volumetric flow rate is

$$\mathbf{V}\_{\rm o} = \mathbf{C}\_{\rm d} \mathbf{A}\_{\rm o} \mathbf{U} \tag{90}$$

The coefficient of discharge Cd is taken as 0.6 as adapted from [1]. Then the mass flow rate is obtained from the expression

$$
\dot{\mathbf{m}} = \mathbf{p} \mathbf{V}\_o \tag{91}
$$

In the previous section, expressions for the overall collector heat loss coefficient UL and the collector efficiency factor F', which are very important performance parameters, were derived. These parameters are applied in the evaluation of useful energy rate extracted from the collector. To incorporate the flow rate and express the system energy equation using the collector inlet temperature, another performance parameter is introduced as the collector heat removal factor. The collector heat removal factor FR is defined as the ratio of actual useful heat collector rate to useful heat collector rate attainable with the entire collector surface at the inlet fluid temperature. This is stated mathematically as

$$\mathbf{F\_R} = \frac{\mathbf{mC\_p} \left(\mathbf{T\_o} - \mathbf{T\_i}\right)}{\mathbf{A\_c} \left[\mathbf{S} - \mathbf{U\_L} \left(\mathbf{T\_i} - \mathbf{T\_a}\right)\right]} = \left[\frac{\mathbf{mC\_p}}{\mathbf{A\_c} \mathbf{U\_L}}\right] \left|1 - \mathbf{e}^{-\frac{\mathbf{A\_c U\_L F}{\mathbf{mC\_p}}}{\mathbf{mC\_p}}}\right|\tag{92}$$

Thus, the energy equation of the system becomes

4

hx Nu 0.6 Gr Pr , 10 Gr 10 ,q constant k = = << = (87)

<sup>b</sup> = = (86)

<sup>L</sup> = = ò (88)

é ù æ ö æö =- = ê ú ç ÷ ç÷ ë û è ø èø <sup>ò</sup> (89)

V CAU o do = (90)

m V o & = r (91)

kv

\* p x xx 2 g xq Gr Gr .Nu

0.2 5 \* 11

For the turbulent range, the average heat transfer coefficient for the constant heat flux cause

xx x

It has been established that during distribution the local heat transfer coefficient is essentially with x when turbulent free convection is encountered. In such a case, the average heat transfer coefficient is h=hx. The mean velocity of air stream due to natural convection was obtained

x x

The coefficient of discharge Cd is taken as 0.6 as adapted from [1]. Then the mass flow rate is

In the previous section, expressions for the overall collector heat loss coefficient UL and the collector efficiency factor F', which are very important performance parameters, were derived. These parameters are applied in the evaluation of useful energy rate extracted from the collector. To incorporate the flow rate and express the system energy equation using the collector inlet temperature, another performance parameter is introduced as the collector heat removal factor. The collector heat removal factor FR is defined as the ratio of actual useful heat

yy y u u 1 d 0.0833U δδ δ

where qp is the plate heat flux. The local heat transfer coefficient is correlated as

x x xp

1

0 <sup>1</sup> h h d 1.25h

from evaluating the mean value of velocity distribution of equation (64) as

1

0

( )

is obtained from

46 Solar Radiation Applications

The volumetric flow rate is

obtained from the expression

$$\mathbf{Q}\_u = \mathbf{2} \mathbf{A}\_c \mathbf{F}\_\mathbf{R} \left[ \mathbf{S} - \mathbf{U}\_\mathbf{L} \left( \mathbf{T}\_\mathbf{i} - \mathbf{T}\_\mathbf{a} \right) \right] \tag{93}$$

The efficiency of the whole collector system η defined as the ratio of the useful heat extracted from the collector Qu to the total incident solar radiation on the collector is mathematically expressed as

$$\eta = \frac{\mathbf{Q}\_u}{\mathbf{A}\_c \mathbf{I}\_\Gamma} = 2\mathbf{F}\_\mathbf{R} \left[ \left( \tau \alpha \right)\_c - \frac{\mathbf{U}\_\mathbf{L}}{\mathbf{I}\_\Gamma} \left( \mathbf{T}\_\mathbf{i} - \mathbf{T}\_\mathbf{a} \right) \right] \tag{94}$$

where (τα) is the effective transmittance-absorptance product of the collector glazing,

$$\left(\mathsf{\tau}\alpha\right)\_{\mathsf{c}} = \frac{\mathsf{S}\_{\mathsf{T}}}{\mathsf{I}\_{\mathsf{T}}} \tag{95}$$

where temperatures T1 and Ta are the collector's inlet and outlet temperatures, respectively.

#### **9. Results and discussions**

The modelled parameters from the analysis of the momentum equation provide a platform to analyze the thermal behaviour on each wing based on the use of a selective surface. This was used to graphically illustrate the heat flow pattern in the chimney and the radiative effects on the emitting and receiving surfaces. At any position of the sun, only two wings of the collector receive full-area direct radiation. At a smaller zenith angle, a larger portion of the farthest wing is lightened, while a smaller portion is lightened at a higher zenith angle. Figure 5 gives the direct beam area ratio ∀ for the three wings of the collector with respect to midpoint solar time.

**Figure 5.** Direct beam area ratio ∀ for the three wings

The shading of the wings was taken into account to evaluate the actual direct beam radiation absorbed by each wing so that the average radiation absorbed by the whole collector can be deduced. To investigate the shape of the lightened area of a wing at any time interval, the zenith angle and the hour angle are required. The shape of the lightened region of the shadowed wing is always triangular depending on the zenith angle of the sun. Five variants of the shadowed wing model are shown in Figure 6.

**Figure 6.** Variants of the shadowed wing model

Figure 7 shows the graph of temperature elevation and mass flow rate and insolation which points out that increase in temperature elevation is not always consequent on increase in the mass flow rate as illustrated by the negative gradient of mass flow rate between 11.30 am and 1.30 pm compared to the positive gradient of temperature elevation within the same insolation range.

**Figure 7.** Graph of temperature elevation and mass flow rate and insolation

#### **10. Conclusion**

**Figure 5.** Direct beam area ratio ∀ for the three wings

48 Solar Radiation Applications

**Figure 6.** Variants of the shadowed wing model

of the shadowed wing model are shown in Figure 6.

The shading of the wings was taken into account to evaluate the actual direct beam radiation absorbed by each wing so that the average radiation absorbed by the whole collector can be deduced. To investigate the shape of the lightened area of a wing at any time interval, the zenith angle and the hour angle are required. The shape of the lightened region of the shadowed wing is always triangular depending on the zenith angle of the sun. Five variants

Figure 7 shows the graph of temperature elevation and mass flow rate and insolation which points out that increase in temperature elevation is not always consequent on increase in the The result shows that modelling buoyant airflow within the chimney and the variation of air temperature elevation with insolation is minimal while the chimney has better efficiency at lower values of solar radiation. Thus, the method of analysis of thermal performance for this type of solar chimney has been accomplished. Nevertheless, a method of complete analytical evaluations that will give satisfactory results can be achieved by obtaining a table of correlation factor from experimental data values to analytical values of this kind.

#### **Author details**


#### **References**


**References**

50 Solar Radiation Applications

pp 81-90.

[1] Ekechukwu, O. V. and Norton, B. (1997), "Design and measured performance of a so‐ lar chimney for natural circulation solar-energy dryers", Renewable energy, Vol. 10,

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[4] Ong, K. S. (2001), A Mathematical Model of a Solar Chimney. Monash University

[5] Shiv Lal, S.C. Kaushik, P.K. Bhargav, 2013. Solar chimney: A sustainable approach for ventilation and building space conditioning. *International Journal of Development and Sustainability Online ISSN: 2168-8662* – *www.isdsnet.com/ijds Volume 2 Number 1*

[6] Alter, L. (2011), "The Trombe wall: Low tech solar design makes a comeback", availa‐ ble at:http://www.treehugger.com/sustainable-product-design/the-trombe-wall-low-

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[8] Zhai, X., Dai, Y. and Wang, R. (2005), "Comparison of heating and natural ventilation in a solar house induced by two roof solar collectors", Applied Thermal Engineering,

[9] Hirunlabh, J., Khedari, J. and Bunnag T. (1997), "Experimental study of a roof solar collector towards the natural ventilation of new houses", Energy and Buildings, Vol.

[11] Onyeogu and Ibekwe, (2001). Measuring insolation data at Nsukka using a flat plate collector. Unpublished MSc thesis submitted to Dept of Mech Engr University of Ni‐

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[13] Bello R. S. and Odey S. O. (2009), "Development of hot water solar oven for low tem‐ perature thermal processes", Leonardo Electronic Journal of Practices and Technolo‐ gies, No. 14, 73-84 Romania. ISSN: 1583-1078 URL: www.lejpt.academicdirect.org/

[10] Wikipedia, 2014. Mach number. http://en.wikipedia.org/wiki/Mach\_number.

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**Chapter 3**
