Heat loss factor of evacuated tubular receivers

Energy Convers. Mgmt Vol. 26, No. 3/4, pp. 313-316, 1986 Printed in Great Britain. All rights reserved

0196-8904/86 $3.00 + 0.00 Copyright ~ 1986 Pergamon Journals Ltd

HEAT LOSS FACTOR OF EVACUATED TUBULAR RECEIVERS R. K. M A Z U M D E R , 1 N. C. B H O W M I K , 1 M. H U S S A I N 2 and M. S. H U Q I tRenewable Energy Research Center and Department of Applied Physics and Electronics, University of Dhaka, Dhaka-2, Bangladesh :Renewable Energy Research Center and Department of Physics, University of Dhaka, Dhaka-2, Bangladesh

(Received 15 October 1985) A~tract--Tubular receivers with an evacuated space between the absorber and concentric glass cover to suppress convection heat loss are employed as absorbers of linear concentrators in the intermediate temperature range. A knowledge of their heat loss factor is important for a study of the thermal performance of such solar concentrating systems. The heat loss factor of a collector can be calculated by solving the governing heat transfer equations or estimated from an empirical equation, if available. The governing equations must be solved simultaneously by iterations, but this is tedious and cumbersome. Although several correlations exist for determining the heat loss factor for flat-plate collectors and non-evacuated tubular absorbers of linear solar collectors, there is no available correlation for predicting the heat loss factor of evacuated receivers. A correlation to calculate the heat loss factor (UL) of evacuated tubular receivers as a function of the variables involved (absorber temperature, emittance, diameter and wind loss coefficient) has been obtained. The correlation developed by a least square regression analysis predicts the heat loss factor to within + 1.5% of the value obtained by exact solution of the simultaneous equations in the following range of variables: wind loss coefficient, 10-60 W/m 2 °C; emittance, 0.I~).95; and absorber temperature, 50-200°C. Evacuated tubular receiver Linear solar concentrators Emissivity Wind loss coefficient

NOMENCLATURE A = D = hc = h, = H, = R = Rax = r = T= V= a = E=

Area (m 2) Diameter (m) Convective heat transfer coefficient (W/m 2 °C) Radiative heat transfer coefficient (W/m 2 °C) Wind loss coefficient (W/m 2 °C) Heat transfer resistance (°C/W) Rayleigh number based on x Radius (m) Temperature (K unless otherwise stated) Wind velocity (m/s) Stefan-Boltzman constant (W/m 2 K -41 Emittance

Subscripts a = Atmosphere 1 = Absorber 2 = Glass cover

INTRODUCTION Efforts have been m a d e by m a n y investigators to reduce optical a n d t h e r m a l losses from a b s o r b e r s o f linear solar concentrators. T u b u l a r receivers with an e v a c u a t e d space between the a b s o r b e r a n d glass cover to suppress convection heat loss are often employed as targets o f linear c o n c e n t r a t o r s in the intermediate t e m p e r a t u r e range. T o predict

Heat loss factor

Operating temperature

with reasonable accuracy the thermal p e r f o r m a n c e o f a solar c o n c e n t r a t i n g system u n d e r various operating conditions, a knowledge o f the heat loss factor of the collector is required. This can be comp u t e d analytically from the governing heat transfer equations. The solutions need numerical iterations which is usually tedious, c u m b e r s o m e a n d timeconsuming. M a n y investigators, such as Hottel a n d Woertz [1], Klein [2] a n d M a l h o t r a et al. [3] worked on heat transfer p r o b l e m s o f fiat plate collectors. Mullick a n d N a n d a [4], a n d afterwards B h o w m i k a n d Mullick [5], suggested correlations for the heat loss factor of n o n - e v a c u a t e d t u b u l a r receivers. However, no correlation for determining the heat loss factor for a n evacuated t u b u l a r receiver has a p p e a r e d in the literature. Figure l(a) is a schematic d i a g r a m o f a t u b u l a r receiver with a n evacuated space between the absorber a n d glass cover. U n d e r steady-state conditions, the total radiative heat loss (convection heat loss is zero in the case o f a n evacuated t u b u l a r receiver) from the a b s o r b e r at t e m p e r a t u r e T~ to the t r a n s p a r e n t glass cover at t e m p e r a t u r e T 2 equals that from the glass cover to a m b i e n t air at t e m p e r a t u r e T o. T h e overall heat loss factor UL of the a b s o r b e r 313

314

MAZUMDER et al.: EVACUATED TUBE RECEIVERS and the rate of heat loss from the glass cover to the atmosphere at temperature To is given by

(o)

(

QL = aA2E2(T~- T4) + A2H,.(T2 - T,).

Equations (4) and (5) must be solved simultaneously by iterations to find the value of T2. Then, substituting the value of T2, one can obtain UL, the heat loss factor as

Rrzo

LJhC2o h¢i2=o

UL = QL/A,(T, -- T,). (b)

7"I

Rri2 ~

To

/?cao

R¢l 2

T,

R,

T2

T

R2

(c)

/•f2o

r~

(6)

However, the radiative coefficients are functions of T2. To find T2 by iterative solution of the simultaneous equations is tedious. Mullick and Nanda [4] proposed a procedure to estimate the heat loss factor of tubular receivers without the requirement of iterations. They found that the value of T2 can be easily determined if the ratio of the outside to inside heat transfer resistances ( f ) is known. The ratiofmust be considered as a function of wind velocity, absorber temperature, emittance of black coating and ratio of the absorber to glass cover diameters. From Fig. 1b, we get

R f 2o

~

(5)

%7.> RC2o

R2 T2 -- T o _ Dlhl2 f = R~l = T , - T 2 D2h2~

(7)

T 2 = (fT~ + T~)/(1 + f )

(8)

and R,

T2

Ra

To

Fig. 1. (a) Evacuated receiver, (b) electrical analogy for non-evacuated receiver and (c) electrical analogy for evacuated receiver. per unit target area for a non-evacuated receiver is given by [4]

where f is a solution of the variables V,,, Ti, El, DI/D2 and Ta. After a detailed study of the variation o f f with the above variables, they recommended a correlation for f. Afterwards, Bhowmik and Mullick [5] obtained a modified form: f =/)1 (a + bEl )H~. c exp [d(Ti - 273)], D2

UL=[~---~2+ Dl

and the overall heat loss factor for an evacuated receiver is [1

UL--- ~

+D l

1

]-',

~'h,~+hc~a

aAj(T~- T~)

e,= 1 o , ( ! --+ El

", ~

D2 k,~2

-1

)

A,hcl~(T,- r2).

[°lT(-1.5×lO-3+g.5,l)mo.9 L~J x exp[3.6 x 10-3(Tl - 273)].

(10)

The procedures to calculate the heat loss factor of evacuated and non-evacuated tubular receivers are shown in Appendices A and B, respectively.

RESULTS AND DISCUSSION

a A , ( r ; - r;) D~ { 1

f =

(3)

For evacuated receivers the expression reduces to QL = 1

where a = 1.22, b = 2.6, c = 0.785 and d = 0.00325. In the present work, a simple correlation for f for the evacuated tubular receiver has been developed using a similar form. The correlation developed by least square regression analysis is based on the basic procedure of Mullick and Nanda [4]. The developed correlation is

(2)

where hi2 and h2a are the inside and outside heat transfer coefficients, respectively. Under steady-state conditions, the rate of heat loss from the absorber (non-evacuated) to the glass cover is given by [4],

(9)

"~'

)

(4)

Figures 2-4 show the comparison of the estimated value of evacuated receiver heat loss factor UL obtained using the correlations for f with exact calculations. It is found that the correlation developed for the evacuated tubular receiver predicts the value

MAZUMDER et al.: /

Ot • 0.045m /

0 2 -0 065m

t6

EVACUATED TUBE RECEIVERS

/

C1 -O.Sm

J

/

x/

H W ,20 W/m2 K

To =50OK

%

////x /

x

/ / /

/

o

x/

/

×7 xJ

o

x./

x/

x~

x ~ X~

"6 "1-

U L - N0¢I- Evocuoted ( from correlation )

U L - Evocuoted (exocl) UL-EVacuoted (con'elotion developed)

~o

xxxxx

I

I

I

I

I

420

520

620

720

820

Averoge

obsorber temperoture,

T1 ( K )

Fig. 2. O 1 • 0.045m

D 2 -0.065m 46--

T I • 473 K H. -20W/m

x¢

% x.

2 K

ro - 3ooK 12~

315

of the independent variables: wind loss coefficient H,., 10-60 W/m 2 °C; absorber temperature T~, 50-200°C; emittance of black paint q , 0.1-0.95; absorber diameter, 12-76.5mm; gap between absorber and glass cover, 5-15 mm; ambient air temperature, 0-40°C. The correlation also permits evaluation of UL at higher temperatures to within +4.0% of the value obtained by exact solutions for temperatures up to 450°C. It is evident from Figs 2 and 3 that, with the increase in absorber temperature and emissivity, the value of the heat loss factor increases rapidly, but the increase in heat loss factor for increase in wind loss coefficient is not so prominent as shown in Fig. 4. Figure 2 also indicates that the lowering of heat loss factor in comparison with that of the non-evacuated absorber decreases as the temperature T~ of the absorber increases. This is because the convection loss between the absorber and the glass cover is dominant in the medium temperature range, but at higher temperatures the radiation loss dominates. It indicates that evacuation in the medium temperature range is important.

J ~

~

×/

x~ x

REFERENCES 1. H. C. Hottel and B. B. Woertz, Trans. A S M E 64, 91

o

(1972). (1973).

_o

2. S. A. Klein, M.S. thesis, University of Wisconsin

"6 4 ~o

U L - Non- E v o c u o ~ d ( from cocwlotion )

x ~

"I-

U L- Evocuoted ( exoct ) /JL- Evocuofed ( ¢¢m'ekltion developed ) x xxxx 01

I

I

i

I

03

0.5

0.7

O, 9

Emissivity

of obsorber (E 1 l

3. A. Malhotra, H. P. Garg and A. Palit, J. Thermal Engng 2, 59 (1981). 4. S. C. Mullick and S. K. Nanda, Appl. Energy 11, 1 (1982). 5. N. C. Bhowmik and S. C. Mullick, Solar Energy 35, 219 (1985).

Fig. 3. APPENDIX

D 1 "0.045m D 2 =0.O65m ~6

Step 1

T z • 30OK

% --.

In case h,. is known, U L can be computed as follows:

C t =0. S m T I "473K

A

A

Calculate D 2 f = [~-~1 ( - 1'5 x 10-3 + 3'5")H~°9

12

I

/

I

f

x exp[0.0036(T~ - 273)1.

o ~ x _ _ x ~ X _ _ X ~ X ~ X - - X

$

To find T2 substitute the value o f f in

- X - X - X -

o

T 2 = (fT~ + T~)/(i + f ) .

"6

o) T

U L - NON- Evocuoted ( from correlotion )

Step 2

U L - Evocuoted {exact) U L - Evocuoted [ correlorion 0

t0

I 20

] 30

I 40

developed) I 50

x xxxx I 60

W i n d loss coefficient, H w ( W / m 2 K )

Fig. 4.

Find h,~2 = o ( T ~ + T~)(T~ + Tz)/[1/q + (OJO2)(I/~2 -

and h,2o= o~(r~ + r,:,)(r2 + To). Step 3

of heat loss factor to within _+ 1.5% of the value obtained by exact solutions of the governing heat transfer equations in the following range of values

Calculate

UL=[ (hrl2)-I+D2D'(h,2. + h..)-']-'.

1)]

M A Z U M D E R et al.:

316

E V A C U A T E D TUBE RECEIVERS

APPENDIX B

h.~

0.0003064 × T ~ 2 x R°c~5 -

In case H~. is known U, can be calculated as follows:

\D,)

Step 1 Calculate

h~,2 = a(T t + T2)(T ` + T2)/

f = ~ (1.22 + 2.6~ I )n~. °'785exp[0.0325(T I - 273)]

and %~o = c~(T~ + T2.)(T~ +

T 2 = (fT, + T,)/(1 + f )

+ \D2]\% - 1

T,,).

Step 3 Obtain

T,,,,2 = (T t + T2)/2 and

D i UL=[(hrt2 +h,,2)-' +~2(h,2,,+ H~.)-' 1 •

T,,,~, = ( T-, + T,,)/2. Step 2 Find

l-~n(D2~] ' R,o: = 0.0707 x 1020 T, - T2

L \D~JJ

Tm'2"4 [(DI~) 0.6 + (D~) 0.6Is