Optimum Inclination Angles of Booster Mirrors and Solar Radiation Availability on the Horizontal and Inclined Box Type Solar Cookers

doi:10.4236/jpee.2013.15008

Paper Menu >>

Journal Menu >>

Journal of Power and Energy Engineering, 2013, 1, 53-57

http://dx.doi.org/10.4236/jpee.2013.15008 Published Online October 2013 (http://www.scirp.org/journal/jpee)

Optimum Inclination Angles of Booster Mirrors and Solar

Radiation Availability on the Horizontal and Inclined Box

Type Solar Cookers

V. P. Sethi1, K. Sumathy2, D. S. Pal3

1Department of Mechanical Engineering, Punjab Agricultural University, Ludhiana, Punjab India; 2Department of Mechanical Engi-

neering, North Dakota State University, Fargo, USA; 3Department of Mathematics, Statistics and Physics, Punjab Agricultural Uni-

versity, Ludhiana, Punjab India.

Email: [email protected]

Received October 2013

ABSTRACT

Mathematical relations are developed to compute optimum inclination angle of booster mirror for horizontally placed

cooker (λ) and for optimally inclined cooker (ψ) during all months (selected day) of the year at 30˚N latitude for max-

imizing the reflected component of solar intensity onto the absorber plate of the cooker. A solar radiation model is also

developed and used to compute the ratio of various solar intensities on horizontal, inclined and normal surface of the

absorber plate for all months at 30˚N latitude. These ratios give a clear indication of greater solar rad iation availability

on the optimally inclined cooker as compared to the horizontally placed cooker for faster cooking especially during

winter months when solar radiation capture is small. Experimental validations have also been performed to access the

accuracy of the developed relations and model.

Keywords: Solar Radiation; Solar Cooker; Optimum Inclination; Booster Mirror

1. Introduction

Many important modular studies have been performed on

box type solar cookers till date to optimize their perfor-

mance. Transient analysis was performed to get the over-

all thermal performance of the box type solar cooker by

[1]. A box type solar cooker which could perform well in

clear sunny d ays was developed [2]. An obliqu e pan was

also designed for putting the food material in tilted posi-

tion. Parametric study of box type solar cooker was also

performed with and without booster mirrors called ref-

lectors [3]. Thermal performance of box type solar cook-

er was tested and a test procedure was developed to test

its performance using two figures of merit F1 and F2 [4].

An improved box type solar cooker with tilted absorb-

ing surface was developed but the problem of placing the

food material could not be solved [5]. Methods of testing

and evaluating the advanced version of the box-type so-

lar cooker were also developed [6]. A model for predic-

tion of the cooking power of a solar cooker based on

three controlled parameters (solar intercept area, overall

heat loss coefficient, and absorber plate thermal conduc-

tivity) and three uncontrolled variables (insolation, tem-

perature difference, and load distribution) was developed

[7]. The performance of solar cookers by analyzing the

previously collected data was also evaluated [8]. The role

of cooking vessel inside the cooker was established tak-

ing into consideration its lid and the bottom surface [9,

10]. Cooking vessel design in cylindrical shape of box

type was also improved and its heat transfer from the lid

was also made faster to the food material placed inside

the vessel [11,12]. A comparative experimental study of

a box type solar cooker with two different cooking ves-

sels was made [13]. Fins are shown to improve the heat

transfer from the internal hot air of the cooker towards

the interior of the vessel where the food to be cooked is

placed. An optimally inclined box type solar cooker with

modified cooking vessel design was presented [14]. The

performance of optimally inclined cooker was consis-

tently better as compared to conventional horizontally

placed cooker in terms of higher solar radiation availabil-

ity, absorber plate temperature, higher chamber tempera-

ture and lesser cooking time.

The review reveals that various researchers around the

world have developed many improved solar cooker de-

signs either by increasing the aperture (solar interception

area), by using multiple reflectors, by changing the incli-

nation of the solar cookers, by reducing the overall heat

transfer coefficient, by increasing the conductive heat

transfer of absorber plate and by modifying the cooking

Optimum Inclination Angles of Booster Mirrors and Solar Radiation Availability

on the Horizontal and Inclined Box Type Solar Cookers

vessel design. In this study, a mathematical model is de-

veloped to compute the optimum inclination angle of the

booster mirror for horizontally placed and inclined cook-

er for maximizing the reflected component of the solar

radiation for better performance during all months and

selected latitudes.

2. Description of Horizontal and Inclined

Box Type Solar Cookers

Two identical box type solar cookers of length 580 mm,

width 300 mm and height 155 mm each were fabricated

using galvanized iron sheet of 0.8mm thickness. One

cooker was kept in horizontal position on ground (Figure

1) while the other cooker was kept on an optimally in-

clined frame (Figure 2) in inclined pos ition. Each cooker

has two top glass covers of 4 mm thickness and a booster

mirror with a provision of altering the angle of inclina-

tion. A round shape cylindrical cooking vessel of 175 mm

diameter and 52 mm depth was placed inside the hori-

zontally placed cooker. The designed parallelepiped cook-

ing vessel was placed horizontally inside the inclined

cooker for greater heat transfer to the food material.

3. Mathematical Model for Optimum Tilt

Angle of the Booster Mirrors

3.1. Horizontally Placed Cooker

Optimum tilt angle of the booster mirror is computed for

all months and given latitude for maximizing the reflect-

ed radiation component onto the absorber plate.

In Figure 1, at the top edge A of the booster mirror

Figure 1. Horizontal cooker with booster mirror at opti-

mum inclined angle λ.

Figure 2. Inclined cooker with booster mirror at optimum

inclined an gle ψ.

λ + θz + i = 90˚ (1)

also

90CAB r∠ +=

(2)

from Equations (1) and (2)

CAB

λθ

∠=+

(3)

If ray Ir1 strikes the top of the inclined reflector mirror

at an angle λ + θz, then the line AC formed by reflected ray

Rr1 reac hi ng t h e ed ge of t he uppe r gl ass c ove r al so ha s the

same internal angle λ + θz, then in triangle ABC;

, 90

90 2

CAB ABC

and ACB

θλ λ

λθ

∠=+∠= ±

∠= −−

Also mathematically we know that

sin() sin(902)sin(90)

BC ABAC

θλ λθ λ

°°

= =

+−− ±

(4)

Since AB= BC = W (width of the absorber plate)

sin() sin(90 2)

θλ λθ

+−−

(5 )

Equation (5) gi ves the value of optimum tilt angle of the

booster mirror λ for horizontally placed cooker ;

90 2

−

(6)

3.2. Optimum Tilt Angle of the Booster Mirror

for Inclined Cooker

Optimum tilt of the booster mirror (ψ) in case of north

facing reflector (Figure 2) of inclined solar water heater

was give n by [ 15] and is used for inclined cooker as

ψ = (π – β - 2ϕ + 2δ)/3 (7)

The declination angle “δ” is computed using the given

Equation (8)

360(284 )

23.45sin 365



=



(8)

Booster

mirror angle λ

15.5 cm

8cm

43 cm

W=58cm

90±λ

θz

+λ

N o rmalto

t hep l aneof

reflection

Incident

ray,I

R ef lecte

dray, R

Glass wool

insulation

-λ+λ

North facing booste r

mirror

Ab so rbe r

plate

Parallelepiped

cooking v esse l

Double glass

co ve r

+ψ

-ψ

Inc ident

radiation

β, Inclined stand

Double ste p

frame

Optimum Inclination Angles of Booster Mirrors and Solar Radiation Availability

on the Horizontal and Inclined Box Type Solar Cookers

where “n” is the number of days starting from January 1.

4. Solar Radiation Model

Variation of effective width of the sun rays intercepted

by the absorber plate in horizontal and inclined position

of the cooker is shown in Figures 3(a) and (b).

In ΔABC

∠= −

cos(90 )

=×−

(9)

Where αs can be computed using Equation (15).

Hourly solar radiation incident on the inclined surface

of the absorber plate depends upon the time of the day i.e.

the hour angle ω (zero at noon, negative in the morning

and positive in afternoon, varies by 15˚ after each hour),

nth day of the year (starts from Januar y 1) i.e. declination

angle δ, altitude angle αs with horizontal or zenith angle

θz with vertical and surface azimuth angle γ (in northern

hemisphere, it is zero for south facing surfaces, 180˚ for

north facing surfaces, −90˚ for east facing and +90 for

west facing surfaces), latitude angle

of a place and tilt

angle

of the s urface with horiz ont a l.

Zenith angle of sun on the inclined absorber plate θi is

given by [ 1 6]

cos[sin (sin coscos cos cossin )

cos (coscoscossin cos sin )

cos sinsinsin]

ifg

θδ βδωβ

δωβ δβ

δ ωβ

−

= +

+−

(10)

Zenith angle of the sun with vertical (θz) and solar al-

titude angle (αs) with horizontal at any time of the day

and for any day of the year can be determined at any

specific latitude location is given b y [16].

(a) (b)

Figure 3. Variation of effective width of the sun rays inter-

cepted by the absorber plate in (a) horizontal and (b) in-

clined position.

cos(cos .cos .cossin.sin)

θφδ ωδφ

−

= +

(11)

ω = 15˚ (tsolar – 12) (12)

The hour angle is 15˚ times the number of hours from

solar noon. It is negative before noon, zero at noon and

positive after 12.

Intensity of extra terr es trial radiatio n Iext measure d on a

plane normal to the radiation on the nth day of the year,

360

10034cos365

ext sc

II .





= +







(13)

Value of direct normal solar radiation in terrestrial re-

gion depe nds upon t urbidity fa c tor Tr of atmosphere [17].

exp 0.9 9.4sin

n exts



−

= ×



(14)

Tr is known for different months and for different re-

gions [18].

αs = 90 − θz (15)

'cos

bn i

(16a)

cos

hb nz

(16b)

The amount of diffuse radiation available on the in-

clined surface can be known as (Tiwari, 2006)

cos 1cos/ 2

sin

bi b

ext sext

θβ





= +−











(17)

1( )cos

dext nz

I II

= −

The reflected component of total radiation is then com-

puted as

sin/2

R gh

I rI

(1 8)

Where “r” is the ground reflectivity (0.3)

II= +

(1 9)

On the horizontal surface of the absorber plate at solar

noon.

Total solar radiation falling on the inclined surface of

the absorber plate at solar noon is given by

Ii = Ib' + Id' + IR' (20)

A computer program in C++ is developed and used for

comput i ng the solar r adiati on flux on various surfa c e s.

5. Results and Discussion

5.1. Optimum Tilt Angles

Optimum tilt angles computed for winter and summer

months at 30˚N latitude are shown in Table 1. It shows

Absorber

plate in

inclined

position

Beam radiation

on inclined

surface (Iib)

Beam radiation

on normal

surface (In)

Beam radiation

on horizontal

surface (Ih)

Optimum Inclination Angles of Booster Mirrors and Solar Radiation Availability

on the Horizontal and Inclined Box Type Solar Cookers

that for horizontal cook er, optimum inclination angle λ is

3.3˚, 1.2˚ and 8.2˚ outwards (+) from the vertical position

(λ = 0˚) in October, February and March. It becomes

2.08˚, 5.6˚ and 4.2˚ inwards (−) from the vertical position

during November, December and January due to greater

solar zenith angle (θz). During summer, optimum inclina-

tion angle λ is 20.8 ˚, 11.2˚, 6.8˚, 8.9 ˚, 16.7˚ and 27.6˚. In

the case of inclined cooker, the optimum inclination an-

gle ψ in winter months is 18.5˚, 12.1˚, 9.4˚, 11.0˚ 16.3˚

and 23.3˚. This variation is much higher in summer

months as 41.5˚, 47.6˚, 50.6˚, 49.2˚, 44˚ and 36.2˚ for

maximization of reflected component.

5.2. Solar Radiation Capture Ratios

The ratio Wh/Wi is 0.62, 0.73, 0.84, 0.76, 0.65 and

0.60 during January, February, March, October,

November and December which shows that inter-

cepted width for solar radiation capture is much

smaller for horizontal cooker as compared to in-

clined cooker. The ratio of solar radiation intensity

available on inclined to horizontal cooker (Ii/Igh) is

1.51, 1.33, 1.15, 1.26, 1.45 and 1.66 during January,

February, March, October, November and Decem-

ber which sho ws that in clined cook er receives much

higher solar radiation intensity as compared to ho-

rizontal cooker in winter months. The variation of

horizontal versus normal surface and inclined ver-

sus normal surface also shows that inclined cooker

receives almost maximum possible radiation during

all months of t he year as the ratio is al ways close to

one. Whereas for horizontal cooker this variation is

0.59 to 1.00 during winter and summer months.

5.3. Experimental Validation

The Figure 4 shows that the measured and predicted

values of solar radiation intensities match well for hori-

zontal as well as for inclined surface of absorber plate

within 5% of the standard deviation. It shows that the

developed model is accurate enough for making correct

predictions for solar radiation availabilities at optimized

inclination angles.

Finally it can be concluded that inclined cooker has

much better solar radiation capture during winter months

for efficient cooking as compared to horizontally placed

cooker at 30˚N latitude.

Table 1. Optimum tilt angles computed for horizontal (λ) and inclined cooker (ψ) at 30˚N latitude during all months of the

year.

Winter Month and date, (n) Latitude (˚ N) 30 Summer Month and date, (n) Latitude (˚N) 30

θz λ ψ θz λ ψ

Oct. 15, (288) 40.1 03.3 18.5 Apl, 15, (105) 20.8 16.1 41.5

Nov. 15, (319) 49.3 −2.08 12.1 May 15, (135) 11.2 22.5 47.6

Dec. 15, (349) 53.5 −5.6 9.4 June 15, (166) 6.8 25.5 50.6

Jan. 15, (15) 51.3 −4.2 11.0 July 15, (196) 8.9 24.0 49.2

Feb. 15, (46) 43.2 1.2 16.3 Aug. 15, (227) 16.7 18.8 44.0

Mar. 15, (74) 32.7 8.2 23.3 Sept. 15, (258) 27.6 11.4 36.2

Table 2. Shows the ratios of intercepted widths for horizontal (Wh) and inclined cooker (Wi) and solar radiation intensities on

Horizontal (Igh), Inclined (Ii) and normal (In) surfac es for horizontal and incli ned cookers at 30˚N latitude during all months

of year.

Month and date, (Tr) Latitude 30˚N Month and date,(Tr) Latitude 30˚N

Wh/Wi Ii/Igh Ihb/In Ib׳/In Wh/Wi Ii/Igh Ihb/In Ib׳/In

Jan. 15, (3.1) 0.62 1.51 0.62 0.99 July 15, (4.3) 1.00 1.00 0.99 0.99

Feb. 15, (3.2) 0.73 1.33 0.73 1.00 Aug. 15, (4.2) 0.95 1.04 0.96 0.99

Mar. 15, (3.5) 0.84 1.15 0.84 1.00 Spt. 15, (3.9) 0.88 1.08 0.88 0.97

Apl, 15, (3.9) 0.93 1.05 0.93 1.00 Oct. 15, (3.6) 0.76 1.26 0.76 0.98

May 15, (4.1) 0.98 1.02 0.98 1.00 N ov. 15, (3.3) 0.65 1.45 0.65 0.99

June 15, (4.2) 1.00 1.00 0.99 0.99 Dec. 15, (3.1) 0.60 1.66 0.59 0.99

Optimum Inclination Angles of Booster Mirrors and Solar Radiation Availability

on the Horizontal and Inclined Box Type Solar Cookers

Figure 4. Measured and predict ed sol ar r adi ation i nte nsit ie s

on the horizontal and inclined surfaces of absorber plate on

15th March, 2012.

REFERENCES

[1] Y. S. Yada v and G. N. Tiwari, “Transient Analytical St u-

dy of Box-Type Solar Cooker,” Energy Conversion and

Management, Vol. 27, No. 3, 1987, pp. 121-125.

http://dx.doi.org/10.1016/0196-8904(87)90064-1

[2] P. C. Pande and K. P. Thanvi, “Design and Development

of a Solar Cooker for Maximum Energy Capture in

Stationary Mode,” Energy Conversion and Management,

Vol. 27, 1987, pp. 117-120.

http://dx.doi.org/10.1016/0196-8904(87)90062-8

[3] B. A. Jubran and M. A. Alsaad, “Parametric Study of

Box-Type Solar Cooker,” Energy Conversion and Mana-

gement, Vol. 32, 1987, pp. 223-234.

http://dx.doi.org/10.1016/0196-8904(91)90126-4

[4] S. C. Mullick, T. C. Khandpal and A. K. Saxena, “Ther-

mal Test Procedures for Box Type Solar Cooker,” Solar

Energy, Vol. 39, 1987, pp. 353-360.

http://dx.doi.org/10.1016/S0038-092X(87)80021-X

[5] N. M. Nahar, “Performance and Testing of an Improved

Hot Box Solar Cooker,” Energy Conversion and Mana-

gement, Vol. 30, 1990, pp. 9-16.

http://dx.doi.org/10.1016/0196-8904(90)90051-Y

[6] M. Grupp, P. Montagne and M. Wackennagel, “A Novel

Advanced Box Type Solar Cooker,” Solar Energy, Vol.

47, 1991, pp. 107-113.

http://dx.doi.org/10.1016/0038-092X(91)90041-T

[7] P. A. Funk and D. L. Larson, “Parametric Model of Solar

Cooker Performance,” Solar Energy, Vol. 62, No. 1, 1998,

pp. 63-68.

[8] P. A. Funk, “Evaluating the International Standard Pro-

cedure for Testing Solar Cookers and Reporting Perfor-

mance,” Solar Energy, Vol. 68, No. 1, 2000, pp. 1-7.

[9] R. A. V. Narasimha and S. Subramanyam, “Solar Cookers-

Part I: Cooking Vessel on Lugs,” Solar Energy, Vol. 75,

No. 3, 2003, pp. 181-185.

[10] R. A. V. Narasimha and S. Subramanyam, “Solar Cookers-

Part II: Cooking Vessel with Central Annular Cavity,”

Solar Energy, Vol. 78, No. 1, 2005, pp. 19-22.

[11] A. R. Reddy and R. A. V. Narasimha, “Prediction and

Experimental Verification of Performance of Box Type

Solar Cooker-Part I. Cooking Vessel with Central Cylin-

drical Ca vity,” Energy Conversion and Management, Vol.

48, No. 7, 2007, pp. 2034-2043.

[12] A. R. Reddy and R. A. V. Narasimha, “Prediction and

Experiemental Verification of Performance of Box Type

Solar Cooker-Part II. Cooking Vessel with Depressed

Lid,” Energy Conversion and Management, Vol. 49, 2008,

pp. 240-246.

http://dx.doi.org/10.1016/j.enconman.2007.06.020

[13] A. Harmim, M. Boukar and M. Amar, “Experimental Stu-

dy of a Double Exposure Solar Cooker with Finned

Cooking Vessel,” Solar Energy, Vol. 82, No. 4, 2008, pp.

287-289.

[14] V. P. Sethi and S. Kumar, “Inclined Box Type Solar

Cooker Using Modified Absorber Plate and Ne w Cooking

Vessel Design,” Proceedings of International Conference

on Recent Innovations in Technology, Kottayam, Kerala,

12-14 January 2012, pp. 233-237.

[15] S. P. Sukhatme, “Solar Energy, Principles o f Thermal Col-

lection and Storage,” 6th Edition, Tata McGraw-Hill Pub-

lishing Company Limited, New Delhi, 1990.

[16] J. A. Duffie and W. A. Beckman, “Solar Engineering of

Thermal Processes,” John Wiley and Sons, New York,

1991.

[17] G. N. Tiwari, “Solar Energy Fundamentals, Design, Mo-

delling and Applications,” Narosa Publishing House, New

Delhi, 2006.

[18] N. K. Bansal, M. Kleeman and M. Meliss, “Renewable

Energy Sources and Conversion Technology,” Tata Mc-

Graw Hill Publishing Co., New Delhi, 1990.

200

400

600

800

1000

1200

Solar Intensity (W/m

)

Time of the day

Measured intensity on horizontal surface

Predicted intensity on horizontal surface

Measured intensity on inclined surface

10:00

10:30

11:00

11:30

12:00

12:30

1 :00

1 :30

2 :00

2 :30

3 :00

3 :30

4 :00