On optimized solar-pond-driven irreversible heat engines

R e n e w a b l e Ener#y, Vol. 7, No. 1, pp. 6% 69, 1996

~ ) Pergamon

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O N O P T I M I Z E D S O L A R - P O N D - D R I V E N IRREVERSIBLE HEAT ENGINES SELAHATTiN GOKTUN ]stanbul Technical University Maritime Faculty, 81716 Tuzla-]stanbul, Turkey

(Received 30 January 1995 ; accepted28 August 1995) Abstraet--Optimum operating temperature and design parameters are defined for a salt gradient solar pond in combination with an irreversible Carnot heat engine. The influence of the design limits of the heat engine and the design parameter of the pond on the operating temperature are considered. It was found that when a solar-pond-driven heat engine is designed to operate close to a maximum power point, the operating temperature must be increased. Apart from this, the heat engine will operate close to the maximum efficiency limit.

irreversible Carnot heat engine has been determined.

INTRODUCTION The salt gradient solar pond (SGSP) is a shallow water-body which acts as a trap for solar radiation. The solar radiation is converted into thermal energy, which is accumulated in the bottom layers of the water-body. Conventional SGSPs are usually characterized by a surface layer at temperature Ts, which is completely mixed due to the effect of the atmosphere ; an insulating layer, which is comprised of a stagnant fluid subject to a large stabilizing gradient, and a homogeneous thermal layer at temperature TH, which is completely mixed due to thermal convection. It is known that SGSP efficiency decreases when TH increases. When an SGSP as a hot reservoir is integrated with a heat engine for power generation, the efficiency of the heat engine increases with TH. This incompatibility in the temperature--efficiency relationship indicates that a solar p o n d ~ e a t engine combination will have an optimum operating temperature. K a m i u t o [1,2] has studied this problem by utilizing the C u r z o n - A h l b o r n [3] model. However heat engines are usually designed between two limiting cases of practical interest. One limiting case is the irreversible Carnot limit which represents the efficiency below the Carnot limit. The other limiting case is that of maximum power operation of an irreversible Carnot engine, which represents the efficiency below the C u r z o n - A h l b o r n efficiency. In this study, assuming that heat engine designs fall between these two limits, the optimal operating temperature of a solar pond in combination with an

ANALYSIS F o r steady-state conditions, the thermal efficiency of an SGSP can be written as [4] : r/p = ( z ~ ) - UL T . -- Ts,

(1)

is where (z~) and UL are the transmittance-absorbtance product and bottom overall heat loss coefficient, respectively. Ts is the surface temperature and Is is the solar radiation just below the surface of the pond. The maximum efficiency and the efficiency at maximum power output for an irreversible Carnot heat engine are given by [5]: /']max

=

1 - Tt,/RTH,

R < 1,

(2)

and

~m = 1--(TL/RTH) j/2, R < 1,

(3)

where R is the irreversibility parameter and TL is the cold reservoir temperature of the heat engine. It is possible to calculate the optimum T . value, TH(opt) for the solar pond-irreversible Carnot heat engine system in two ways : (i) using q . . . . eq. (2), and (ii) using qm, eq. (3), for heat engine efficiency, q. The overall system efficiency is : 67

68

S. G O K T U N

~/s = qP" q.

(4)

Here TH(opt) is obtained by equating d~/s/dTH to zero. In the maximum efficiency limit, from eqs (1), (2) and (4) the overall system efficiency can be written as:

Since the SGSP is also the cold reservoir of the heat engine for practical purposes, by taking TL = Ts and defining : 0op t =

TH(opt)/TL,

(9)

and

.s =

@ = (~O01s/(ULTL),

j

¢ >~ O,

(10)

eqs (6) and (8) can then be written, respectively, as : and

0opt = [(~t -{- 1)/R]

Tu(opt) = L R \ UL + Ts

:

(0opt) 3/2 = 0.5(0opt-~- I/]-]-

(6)

In the maximum power limit of eq. (3), ~/s becomes

[1_ (,je .jl' 21j

(7)

1/2, 1)/R '/2.

(11) (12)

Equation (10) is the design parameter of an SGSP. The effects of the design parameter o n 0op t for the maximum power and maximum efficiency limits of the heat engine are shown in Fig. 1.

and Tu(opt) is the solution to the equation RESULTS AND DISCUSSION

2(Tu(opt)) 3/2- (TL/R) l'2(Tu(opt) + Ts + (zCOIs/(Uc)) = O.

(8)

T o d a y the S G S P is considered to be one of the most attractive m e a n s o f collecting a n d storing solar energy

12.0 - -

-

Pmo× -'r/mox.

f

*-*** ooooo

~ R=0.2 R=0.4

~ * ,*,~ ~ ooooo

8.0

"

4¢"

J J

R=0.6 R-0.8

;-" ;< "," "-" )-" R = 1 . 0 f f

ro J

UP

J

EY-

CY

f

40

~

f

X

j X ~ I i X f x

X f

/

/

0.0 0.0

5.0

10.0

t5.0

20.0

Fig. 1. The effect of the design parameter on the optimum temperature ratio for m a x i m u m power output and m a x i m u m efficiency.

Optimized solar heat engines on a large scale and is currently being utilized for power production. The most important problem of practical importance is how to determine the operating temperature of the SGSP with a secondary system for power production. In this study we determined the optimum pond temperature (0op,) so as to maximize the efficiency or maximize power production in the secondary system, because of different views on design of the heat engine. We have analysed the influence of the design parameter (~) on 0opt for maximum efficiency and for maximum power production in an irreversible Carnot heat engine as a secondary system (Fig. 1). As seen from Fig. 1, the irreversibility parameter (R) of the secondary system has a strong effect on 0opt for a given value of ~p. When R is increased, which implies a decreasing loss mechanism in the heat engine, 0oot decreases and the efficiency of the pond increases, showing that optimization of R is very important for a solar-pond-powered heat engine design. It is well known however, that for a typical power plant heat engine (such as a Rankine cycle engine) it may take values between 0.8 and 0.9. Also Fig. 1 shows that when a solar-pond-powered heat engine is designed to operate closer to the maximum power point, the optimum operating temperature of the pond must be increased. If the operating temperature of the pond is not high enough due to climatological effects, the optimal operation of the heat engine will lie closer to the maximum efficiency limit (the irreversible Carnot efficiency).

69 NOMENCLATURE

& radiation intensity at the pond surface P power R cycle irreversibility parameter bottom zone temperature of the pond, source temperature of the heat engine TH(opt) optimum operating temperature of the pond T~ sink temperature of the heat engine T~ surface zone temperature of the pond UL bottom overall heat loss coefficient of the pond Greek letters re transmittance absorptance product q,, efficiency of a heat engine at maximum power output ?/max maximum efficiency of a heat engine ~/, overall system efficiency design parameter of the pond (eq. 10) 0opt ratio of/'.(opt) to TL. REFERENCES

1. K. Kamiuto, Determination of the optimum pond temperature for maximizing power production of a convecting solar-pond thermal-energy conversion system. Applied Energy 28, 47-57 (1987). 2. K. Kamiuto, Optimum pond temperature for maximizing power production of a non-convecting solar-pond heat conversion system. Applied Energy 30, 305 316 (1988). 3. F. L. Curzon and B. Ahlborn, Efficiency of a Carnot engine at maximum power output. Am. J. Phys. 43, 22 24 (1975). 4. C. F. Kooi, The steady-state salt gradient solar pond. Solar Energy 23, 37 (1979). 5. J. Chen, The maximum power output and maximum efficiency of an irreversible Carnot heat engine. J. Phys. D : Appl. Phys. 27, 114~1149 (1994).