Periodic analysis of Roof Radiation Trap System: A passive heating concept

Applied Energy 13 (1983) 165-181

Periodic A n a l y s i s o f R o o f Radiation Trap S y s t e m : A Passive H e a t i n g Concept M. S. Sodha, N. K. Bansal? and Sant Ram Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi--110 016 (India)

S UMMA R Y The thermal flux coming into an air-conditioned space through a 'Roof Radiation Trap System' has been evaluated. Numerical calculations have been performed corresponding to the meteorological data on 11 January at Boulder, Colorado (USA), a harsh North American winter day. Subsequent parametric studies using the simulation indicated that the critical parameters were the width of the air gap and that of the concrete roof through which the thermal flux passes into the room.

NOMENCLATURE A Ao A1 a

Bo B1

bo b.

Area of the enclosed air, m 2 Constant, °C Constant, °C m- 1 Absorptance of the concrete surface, dimensionless Constant, °C Constant, °C m- 1 Zeroth harmonic in the Fourier expansion of solar insolation, W / m 2 nth harmonic in the Fourier expansion of solar insolation, W/m 2

~" Presently Alexander Von Humboldt Fellow with the program group for systems research and technological development, Nuclear Research Centre Juelich (West Germany). 165

Applied Energy 0306-2619/83/0013-0165/$03.00 4)Applied Science Publishers Ltd, England, 1983. Printed in Great Britain

166 C1

C2 C3

Cp Do D1 h hi h2 hrr hr g 1

K2 K3

Lx Z 2

L3 MA n

Re

s(t) s'(t) T

rsA Tso

M. S. Sodha, N. K. Bansal, Sant Ram

Specific heat of concrete, J/kg °C Specific heat of insulation, J/kg °C Specific heat of concrete, J/kg °C Specific heat of air, J/kg °C Constant, °C Constant, °C m - 1 Convective heat transfer coefficient from the insulation layer to the ambient environment, W/m 2 °C Convective heat transfer coefficient from the concrete to the air, W/m 2 °C Convective heat transfer coefficient from the insulation to the air, W/m 2 °C Convective heat transfer coefficient from the concrete layer to the air-conditioned space, W/m 2 °C Radiative heat transfer coefficient from the concrete layer to the insulating layer, W/m 2 °C Thermal conductivity of the first concrete layer, W/m °C Thermal conductivity of the insulation, W/m °C Thermal conductivity of the second concrete layer, W/m °C Thickness of concrete layer facing the air-conditioned space, m Thickness of the insulating layer, m Thickness of the concrete layer over the insulating layer, m Mass of the enclosed air, kg Number of harmonics Real part Solar intensity on the glazing at any time t, W/m 2 Solar intensity on the horizontal blackenediconcrete roof at x 1 = 0 (W/m 2) Temperature distribution inside the insulating layer, °C Ambient temperature, °C Zeroth harmonic in the Fourier expansion of the ambient temperature, °C nth harmonic in the Fourier expansion of the ambient temperature, °C Solair temperature, °C Zeroth harmonic in the Fourier expansion of the solair temperature, °C

Periodic analysis of roof radiation trap system

Tsn Ul X

(= K/pC) ~c

OA OAO

OAn OR 6O "Cg

Pl Pz P3

2, 2~ Vn Vn t

167

nth harmonic in the Fourier expansion of the solair temperature, °C Heat transfer coefficient from the enclosed air to the ambient environment through the glazing, W/m 2 °C Coordinate, m Thermal diffusivity of the concrete layer, m2/s Absorptance of the insulating layer, dimensionless Temperature of the enclosed air, °C Zeroth harmonic in the Fourier expansion of the enclosed air temperature, °C nth harmonic in the Fourier expansion of the enclosed air temperature, °C Temperature of the air-conditioned space (i.e. the room), °C 2rt/(period), h - 1 Transmittance of glazing, dimensionless Density of first layer of concrete, kg/m 3 Density of insulating layer, kg/m 3 Density of second layer of concrete, kg/m 3 Constant, °C Constant, °C Reflectance of the insulating surface, dimensionless Constant, °C Constant, °C Constant, °C Constant, °C Emissivity of the concrete layer 1.

INTRODUCTION

The roof radiation trap concept developed by Givoni 1 for providing passive heating consists of a gap between the concrete roof of the building and an insulating layer; the southern side of the building being glazed and provided with a hinged insulating panel, internal or external, which can be closed or opened according to the need. During the day time, when solar radiation is incident on the south wall, the southern hinged insulating panel is opened. Solar radiation penetrates through the glazing and is absorbed by the concrete roof. Part of the absorbed energy is conducted through the roof and part is transferred to the air in the space between the roof and the fixed insulating layer. The heated air in the space is then

M. S. Sodha, N. K. Bansal, Sant Ram

168

blown by a fan to gravel storage. During night time the hinged insulating panel on the southern side is closed to minimise thermal losses through the glazing. In the present paper we have developed a periodic analysis of the roof radiation trap concept. This has yielded expressions for calculating (i) the heat flux conducted into the living space, which is assumed to be at a constant temperature, OR, as a result of air conditioning, and (ii) the temperature of the air in the space between the concrete roof and the fixed insulating layer. For a numerical appreciation of the analysis we have performed calculations corresponding to the daily variations of solar intensity and ambient temperature on 11 January at Boulder, Colorado (USA), which experiences the harsh North American winter climate.

2.

THEORY

The configuration of the system is shown in Fig. 1. The temperature distribution, O(x 1, t) inside the roof material is given by the solution of the one-dimensional heat conduction equation, i.e.,

~20(xl,t)

1 O0(x~,t)

~x~

~

~t

(1)

Insulation , .:

-

,.

.

.

,

-

•

.

-

./...

.

.

"_

.:-

-

, ,~

-'-_,.

,--

~

,-

".

, .- .. f -

-..

-

-.. .

".

.

.-

, ,X2

. -. . . .

X2=L2 6

~

-- Conerote

i!~i: ::i,i,!!,!., :i ~~!~~i ,:.:::~:;. :::~i,~.i~! ~,:i~::!!~:.i. ~!i:~i.i::i:~::! :i:::i~:! =° . . . . . . . .

.

. , . .

. . . . .

::

''.2":-.','.':'.'...~

Flux Room 9R (20"C) Fig.

1.

Configuration

of the

roof

radiation

trap.

XI=L

I

Periodic analysis of roof radiation trap system

169

with the following boundary conditions azgS'(t) = - K 1 80(x1' t) x, ~Xl =0 + h l [ O ( x l , t)l~, =o - 0A(t)] + e A R ( 1 - 2ra)

(2)

and a O ( x 1, t ) ~1 =

-K1

t~x1

= hrr[O(Xl

L1

'

t)l~, =L, - OR]

(3)

The temperature distribution inside the fixed insulating layer is also given by the solution of the one-dimensional heat conduction equation as eqn. (1) subject to the following boundary conditions (xz, K2 T~

t). ~ = o

=h[TsA(t)- T(x2, t)l~=o]

(4)

and -K2

c3T ( x z , t) ~,~ t3x2

=L2

= h2[T(x2, t)lx2=L~ - 0A(t)] -- eAR(1

-

'~r)

(5)

T(x2, t) being the temperature distribution inside the insulation. The solair temperature, TsA(t), in eqn. (4) is given by Tsk(t) = Ta(t ) q ctcS(t) h

e AR h

(6)

and AR, the amount of long wave radiation emitted by the concrete surface, is calculated from the linearised expression e A R = hr[O(x 1, t)lx, = o - T ( x z , t)lx2=L2]

(7)

The temperature of the enclosed air is governed by the following equation M A C p OOA(t) .4 Ot

hl[O(Xl, t)lx, =o - 0A(t)] + h2[T(x2, t)lx2=L2 - 0A(t)] -- UI[OA(t) -- Ta(t)]

(8)

M. S. Sodha, N. K. Bansal, Sant Ram

170

The periodic solutions of the one-dimensional equations for the temperature distributions 0 and T are governed by the equations O(xx,t)

=A o + A l x

1

(1"3

+Re y , [2.exp (~.xl) + 2'.exp ( - ~ . x l ) ] e x p ( i n m t )

(9)

n=l

and

T(x2, t)

= B o + BIX 2

+ Re ~

[v. exp (~'.x2) + v'.exp ( - ~'.x2)] exp (imot)

(10)

n=l

respectively, where

~.=(l+i)[

2K, .~

(ll)

and ~no3p2C2~ l/2

~'.--(1 +i) [ 2K 2 J

(12)

The temperatures of the enclosed air, the ambient and also the solar intensity and solair are represented by a Fourier series of period 24 hours, viz.,

0A(/)

= 0A0 +

Re ~

OA.exp(intot )

(13)

T~.exp(inmt)

(14)

n=l oo

Ta(t ) = Tao +

Re ~ n=l

S(t) = b o +

Re ~

b.exp(imot)

(15)

n=l

and TsA(t ) = Tso "4-Re

Z n=l

Ts. exp (inoot)

(16)

l_

[

V

h,(I - ; 0

V

; O l e x p ( - ~ ' . L 2)

(h - K2a'.) [K2~',-hz-h,(I-

(h + K2~t'.) -{Kzo(,+hz+h,(l-A,)}exp(ot'nL2)

h2

-h~

-Lh'+h2+U'+

.4

OAo Lc J

[

J|

0,,..

v'.

v~

2"

~,

l_hloo oBI IAol A c2c.c

- h~(I - 2,a) exp ( - a'.L 2 )

0

-- h,(I - 2,a) exp (~t'.L2)

hr(l-2,)

(h, + K~t.~ exp (~t.L~)

h 2 exp ( - ~t'.L2 )

- [ h 2 L 2 + (I - 2,)h,L 2 + K2]

K2

0

0

h ~ + h,(l - ,~ra) + K~ ~t.

( h , - Kaat.) exp ( - oqLt)

ht + h,(l - ) . , a ) - Kla .

h 2 exp (~t'.L2)

h2L 2

- h ~ ( l - 2~)L 2

0

hI

- { h : + h,(l - 2,)}

h

0

0

0

(h,,L~ + K~)

h,(l - ~.,)

h2

- h , ( I - 2,a)

0

-K l

hi

ht(l - ~.ra)

ht

I

=

C.s

C.,

CR3

C.2

c,, (18)

(17)

.-,..I

-,L

t..,

172

M. S. Sodha, N. K. Bansal, Sant Ram

Substituting the eqns (7), (9), (10), (13), (14), (15) and (16) into eqns (2), (3), (4), (5) and (8) and equating the coefficients of equal powers of exp ( i n , t ) leads to the two matrix equations for the determination of the unknown constants, eqn. (17) for the time-independent part and eqn. (18) for the time-dependent part. The various constants on the right-hand side of eqns (17) and (18) are defined as (19)

C 1 = - U 1Tao C 2 =

Tgabo

(20)

C a = h,.rO R

(21)

C4 = h T s o

(22)

C5 = 0

(23)

C.1 = - U 1Ta.

(24)

C.2 = Tgab.

(25)

C.3 = 0

(26)

C.4 = h Ts.

(27)

C.5 = 0 (28) In order to introduce a time delay between the attainments maximum solar intensity and maximum temperature in the gap, it may be desirable to have a concrete layer above the fixed insulating layer (Fig. 2). In this Concrete

X3=O

X3 = L 3 j X 2

=0

X 2 =L Z --Insukztion

--Concrete

X 1 =0

X1 = L1

Flux Room OR (20 *C)

Fig. 2.

Configuration of the roof radiation trap with a concrete layer on the insulating surface.

Periodic analysis of roof radiation trap system

173

case the concrete above the fixed insulating layer will have a t e m p e r a t u r e distribution given b y ~(X3,

t) = D O + DIX 3 ~ r 3

+ R e ~-a [~.exp(fl.x3) + ~ ' . e x p ( - f l . x 3 ) ] e x p ( i n o o t )

(29)

n=l

where

)no,)p3C3~ 1/2 [ 2K 3 J

fl.=(l+i)

(30)

The energy balance e q u a t i o n s for this configuration m a y be written as

aO(xl,t) rgaS'(t) = - K 1

x, =o

8xl

+h~[O(xx,t)l~,=o-OA(t)]+eAR(1

-- Kx

-2ra )

8 0 ( x 1, t ) x l =L~ = hrr[O(xl' t)lx~ =LI -- OR] 8x I

M A C p 630A(/) A Ot - hl[O(xx'

t)lxl

=o

t ) x~ = o = h [ T s A ( t )

- K 3 0~b(x3, t) ~,3= = ~X3 L3

-- K 2

(32)

- 0A(/)]

+ h2[T(x2, t)lx2 =L~ -- 0A(t)] -- e l [0A(t) -- Ta(t)]

K3aq~(x3,ox3

(31)

- ok(x3, t ) l x ~ = o ]

0 T(x2, t) ~,~= o 0X2

(33) (34)

(35)

- K 2 63T(x2,ox2 t) :,2 =L2 = h2[T(x2' t)lx2_-L2 -- 0A(t)] -- eAR(1 - 2r)

(36)

and T(x2, t)[~,2= o = q~(x3, t)lx3 =I.3

(37)

0

0 0

0 L 3

l

ht

{ht + K t a - + h,(I - 2,a)}

(hr - Kiot,,)exp(-a.L 0

0

0

h,(l - ) . , )

0

ht - Kta . + hr(l - ).ra)}

(h. + Ktot,)exp(ot.Lt)

0

0

h,(1-2~)

0

exp (//.L3)

0

K3/~ , ¢xp (fl.L s )

(h + Kzat',)

0

0

0

0

ht

0 0

0

h,(l - ) . ~ )

exp ( - # . L 3 )

0

- Ksfl . exp ( - / ] . L ~ )

(h - K2a',,)

0

0

-K 3

0 0

0

0

0

K3

h

-K t

(K 1 + h~L I)

h.

0

0 0

0

h I + h , ( l - 2ra)}

0

ht

h2

-

c~ I

D,

-1

-1

0

h~

0 {h z - K2a~, + h,(l - 2,) } cxp ( - a'.L 2)

0

0

0

K2ot'"

-hz

- ( h t + h 2 + U~ +M^Crint°l" --S---

_0^o.l L_C~J

0

- ).ra) ¢xp ( - a~L2)

h2¢xp ( - ~ t ' L 2 )

0

K2a', -

0

h2

C; I

c.' I

Do

C~ I

s,

=

oil

At Bo

0

-h,(l

- 2,)L2]

0

C; I

Ao

0

- ).,.a ) ¢xp (~'.L 2)

h 2 ¢xp (~t~,L2)

0

-[K2+hzLz+h,(l

K2

0

0

-hi

0

-(ht+h2+Ut)

h2Lx h,(l - ).,a)L 2

- {h 2 + K:t', + h,(l - 2 r ) l c x p (~(,L a )

- h,(l

- 1

- [h2 + h~(l - ,;.,)]

0

0

0

h,(l - Ara)

vn

"-...I 4~

I

(39)

¢~,

c-'~ I

~2

(38)

Periodic analysis of roof radiation trap system

175

The unknown constants in this case are given by eqns (38) for the timeindependent part, and (39) for the time-dependent part. The constants on the right-hand side of eqns (38) and (39) are defined as C'1 = - U, Tao

(40)

C '2 = z , a b o

(41)

C '3 = h,rO R

(42)

C a = h Ts0

(43)

t

t

t

(44)

C" 1 = - U , Ta.

(45)

C~2 = z , a b .

(46)

C~3 = 0

(47)

C,~4 = h Ts.

(48)

~5 = ~6 = ~7 = 0

(49)

C5 = C6 = C7 = 0

The constants in eqns (17), (18), (38) and (39) have been obtained by numerically inverting the matrices. The numerical calculations have been performed on an ECIL micro 78 computer.

3. R E S U L T S A N D D I S C U S S I O N

Numerical calculations for the heat flux coming into the living space maintained at constant temperature and for the time variation of the air temperature in the space between the fixed insulating layer and the roof have been made assuming the daily variation of the ambient temperature and solar intensity on 11 January at Boulder (i.e. at 40.02 ° N latitude), 2 Colorado (USA), which experiences a harsh North American winter climate. The solar radiation incident on the south vertical glazing has been calculated from the data given on a horizontal plane using standard expressions.3 The Fourier coefficients of the ambient temperature, solair temperature, solar intensity on horizontal and south-facing vertical

M. S. Sodha, N. K. Bansal, Sant Ram

176

TABLE 1 Fourier Coefficients o f the A m b i e n t T e m p e r a t u r e D a t a for 11 J a n u a r y at Boulder, Colorado (USA)

n a', (°C)

0

1 8.8644

2 1.955 0

4.28417

0.387716

3 1.778 5

4 1.887 8

5 0.238 59

6 1.014 52

5-323126

5-005633

4.658817

- 1.608 33

a~(rad.)

5.24096

TABLE 2 Fourier Coefficients of the Solair T e m p e r a t u r e D a t a for 11 J a n u a r y at Boulder, C o l o r a d o (USA)

n

0

b'n (°C)

1 15.15299

2 7.581 2

3 3.31903

4 2.2145

5 0.33042

6 1.62151

0.387 611

4.244 29

5.665 102

2.219 38

4.908 35

2.975 ~'n (rad.)

3.883 01

TABLE 3

Fourier Coefficients of the Solar I n s o l a t i o n o n H o r i z o n t a l Surface for 11 J a n u a r y at Boulder, C o l o r a d o (USA) n b" (W/m 2)

0

1 272.4977

2 171.2317

3 68.73078

4 18.5657

5 23.365 1

6 17.8894

157.518 ~kn(rad.)

3.46144

0.62108

3.95388

0.03497

2.70281

0.13098

TABLE 4

F o u r i e r Coefficients of the Solar Insolation o n a S o u t h Facing Vertical Surface for 11 J a n u a r y at Boulder, C o l o r a d o (USA) n a~ (W/m z)

0

1 539.75

2 336-08

3 118.288

4 32.45

5 80.347

6 75.62

311.65 ~b, (rad.)

3.429 69

0.605 56

4.179 44

3.900 625

1.657 095

5.539 49

Periodic analysis of roof radiation trap system

177

surfaces are given in Tables 1, 2, 3 and 4 respectively. The values of other physical constants used in the calculations are: h = 22" 1 W / m z °C h 1 = 8"29 W / m a °C h 2 = 8.20 W / m 2 °C h r = 6.67 W / m z °C hrr = 8"29 W / m 2 ° C

Pl = 1855-3 kg/m 3 C 1 = 795.5 J/kg °C K 1 =

0.721 W / m °C

P2 = 120.1 kg/m 3 C 2 = 1980.0 J/kg °C K 2 = 0.04 W / m °C P3 = 1855.3 kg/m 3 C 3 = 795.5 J/kg °C K a = 0 - 7 2 1 W / m °C A =9.0m z a =0.9 )~r = 0.9 OR =

20 °C

rg = 0 . 8 5 Cp = 1006.0 J/kg °C The values of heat transfer coefficients have been obtained using standard expressions. 4,5 In Fig. 3 we have shown the hourly variation of the heat flux coming into the air-conditioned space as a function of various air gap widths, keeping the thicknesses o f the concrete and insulation, i.e., L 1 and L 2, constant and equal to 0.30 m. In contrast to the irregular variation in the solair temperature, the heat flux shows a smooth periodic variation. It is observed that the increase in the width of air gap helps to achieve a better performance. Figure 3 also shows the comparative performance of the configurations in Figs 1 and 2. The presence of the concrete layer on the

M. S. Sodha, N. K. Bansal, Sant Ram

178

FLUX : FOR CONFIGURATION 1 WITH Ll=0.30m AND L2=030m

I : 36 k ~

FOR CONFIGURATION 2 WITH LI =0.3m, L2 = 00"/6m AN(] L3:0 23rn

k I~ I ~ F'-. ~

I

AIRGAP:122m A,RGAP = IS2rn Ill AIRGAP= Ia3m ENCLOSEDAIR TEMPERATURE II

. ......

. ~ . _ / , . ~ /d) .o--,-,,\\. / 4'" ,.. ....... /,~/ , , ~ ,..f----~_

~'L "--,\

\ \ ,,~.. "-.,'-q, \ J.__~

/

30

20 ~-

"',. \

x,d/.'/

/

/ "-~.'u~.~.~...,__.

L,..-"---."~-.~,,.",,'~./;'~,,, J / / , , ' - " ~ < ~ 3

5

~,

"~\

11 ~

/17 /

\\\\ \\\ \\\ \\\ \x x \\ -12

\

\

x

I

21

zU,.l 23

25

iii ii//i II /I i11 /I /

\

/19 /

io

\N\ \ \\\.,. ".,.fill i i /7 i /ii /

/

"rIME (Hr]

Fig. 3. Variations of (i) the heat flux going into the air-conditioned space and (ii) the enclosed air temperature with time, keeping L~, L 2 and L 3constant but for various widths of air gap. insulation surface, though decreasing the average heat flux going into the air-conditioned space, helps in load levelling. Figure 4 shows the effect o f changing the width of the concrete roof, Ll, keeping the insulation thickness, L2, and the width of the air gap constant. As expected, the increase of Ll not only helps in load levelling but also reduces the average heat flux coming into the air-conditioned space. Similarly Fig. 5 shows the hourly variation o f the heat flux coming into the r o o m as a function o f various insulation thicknesses, keeping the thickness of the concrete roof, L ~, and the width o f the air gap constant. It is seen that the increase in the insulation thickness has a marginal effect on the performance. -. . . . . . . . .

Periodic analysis of roof radiation trap system

F

~

AIR f~ko: 1.S3m L2 : 0.]0 m

"

~./

•

100

/"'~ ,

E

! I

\N

I L 1 = 0.1Sin II L1 = 0.23m Ill L1 = O.30m ---- -- SOl.AIRTEHI~RATU~ ENELOSEDAR TEHPERATUI~ FLUX

3O 2sg

\

\\

I I I I

80

179

\

"~;~. N

60

q

/

g

20

~3-.

"% 5

7_-----.-"~l..~

11

/ ~

/

15

\'

' x

/."f'~\

~-

•

.j

m

i

/I r

Fig. 4. Variations of the heat flux going into the air-conditioned space, solair temperature and the enclosed air temperature with time, keeping the air gap and L2 constant and varying the thickness L 1 (configuration 1).

Figure 6 shows the comparative performance of a 'Roof Radiation Trap System' with respect to that of a 'Single Glazed R o o f ' of thickness 0-30 m. It is seen that a 'Roof Radiation Trap System' performs better than a 'Single Glazed R o o f ' even with an air gap of width 0.30 m.

4.

CONCLUSION

The thermal performance of the 'Roof Radiation Trap System' has been evaluated in terms of heat flux coming into an air-conditioned

I~

- -

FLUX

. . . . .

ENCLOSED AIR TEMPERATURE

35

AlP GAP: 1.83m LI : 0.30m

30

I II

~

L2 L2

: 0.0"/, :0.1SIn

//

//7

"

25.

50 i

15

30 ./L2=023m

"

5

,~

II

J

10

~ ' - ' ~ . . =

TIME (Hr)

Fig. 5. Variations of the heat flux going into the air-conditioned space and the enclosed air temperature with time, keeping the width of the air gap and L 1 constant and varying

L2•

I

ROOFRADIATION TRAPSYSTEMWITH AIRGAP = 0.30 m

L1 L2 II

030 m = 0.30 m

=

SINGLEGLAZEDROOFOF THICKNESS0.30m

10 0

x

._J u_ -

~

~

~

1'1

1'3

l's

~

1~

2~

A

2~

TIME {HF) 10

-20 -30 -l.C

Fig. 6. Time variations of the heat flux going into the air-conditioned space through the roof radiation trap with L 1 = 0 . 3 m , L 2 = 0 . 3 m and width of the air gap = 0 . 3 m , and through a single glazed roof of thickness 0-3 m.

Periodic analysis of roof radiation trap system

181

space (which is maintained at 20 °C). Numerical'predictions corresponding to the meterological data for a typical harsh winter day, i.e. 11 January at Boulder, Colorado (USA) show that: (i)

in a r o o f radiation trap the air gap should have a width of 1.22 m for the average flux coming into the air-conditioned space (assumed to be at 20°C) to be positive; (ii) a r o o f radiation trap performs better than a Single Glazed R o o f with an air gap of width 0.30 m; (iii) the thickness o f the concrete roof, L 1, has a significant effect on the load levelling; and (iv) the thickness of the insulation layer does not affect the load levelling but influences the performance marginally.

REFERENCES 1. B. Givoni, Solar heating and night radiation cooling by a roof radiation trap, Energy and Building, Netherlands (In Press). 2. J. A. Duffle and W. A. Beckman, Solar energy thermal processes, John Wiley and Sons, New York, 1970. 3. A. M. Zarem and D. D. Erway (Eds), Introduction to the utilization of solar energy, McGraw-Hill Book Company, New York, 1963. 4. W. H. Giedt, Principles of engineering heat transfer, D. Van Nostrand Company, Inc., New York, 1964. 5. M. Necati Ozisik, Basic heat transfer, McGraw-Hill Book Company, New York, 1977.