Viorel Badescu 4

8/12/2019 Viorel Badescu 4

http://slidepdf.com/reader/full/viorel-badescu-4 1/26

Optimal control of solar

energy systems

Viorel Badescu

Candida Oancea Institute

Polytechnic University of Bucharest



Contents

1. Optimal operation - systems with water

storage tanks

2. Sizing solar collectors

3. Optimal operation - maximum exergyextraction

4. Sizing solar collection area

5. Conclusions






The optimization depends on the way the investoruses the thermal energy obtained from solar energyconversion

Two objectives:

First, to develop a sizing procedure for collectionsurface area, with input variables:

the working fluid mass flow rate and

the inlet and outlet fluid temperatures

Second, propose a procedure to find the best local design solution;

It may be implemented by using various objectivefunctions




Some economical indices, including

net present value and

internal return rate,

are examples of objective functions. V Badescu, Optimum size and structure for solar energy

collection systems, Energy 31 (2006) 1483-1499



Model

The user

may need

heat or

work fluxes

The

classical

system

may

provideheat or

work fluxes



The optimization problem

A (primary) conventional energy transfer system

A (secondary) system based on solar energyconversion.

cT - total energy transfer cost per unit time,

c1 - cost of one energy unit received/removed by using the primary system

c2 - investment and operation costs of the secondary system The optimization problem:

find the surface area A which minimizes the costs

and the optimal structure of the collection system.

Ac F F c Ac unecT 21



Model

The mass flow rate is

fixed

The fluid exits the area

A at temperature T

Adding area dA

increases the

temperature by delta_T



Solar collector model

“Absorbed” heat flux

Lost heat flux

Useful heat flux = “absorbed” - lost



Model

Integration of Hottel-Whillier-Bliss eq. (in J):

The time averaged form is (in W):

The time-averaged efficiency

dt dAT T F U F Gdt dT cmt

a fi R L R

t

p

0

*****

0

**

0

*

dAT U GdT c F ~~

0

GT U GdAdT c F /~~/ 0



Applications

(a)

The energy transferred is a heat rate received by a body and

the primary energy transfer system is a conventional heater.

(b)

The energy transferred is a heat rate received by a body attemperature Ta+T and

the primary energy transfer system is a vapor compressionheat pump.

(c)

The energy transferred is a heat rate extracted from a bodyat temperature

and the primary energy transfer system is an absorptionrefrigerator.

The difference consists in the factors Fnec and Fu

0 T T T T avap



Case (a) as an example

All energy

fluxes

involved are

heat fluxes



Case (a)

Fnec and Fu are heat fluxes

The increase of the heat rate supplied by the

solar energy conversion system,

associated to the increase of collection area dAis:

Then, the economical benefit is

dT cdF F

a

u

a

udF

dAGcdT ccdF cd a

F

aa

u

aa 111$



Economical indicators

The so called “revenue” factor R

cost of saved primary energy over cost of

surface area

The cost C_A per unit time of the solarenergy collection surface area A:

2

1

c

Gc R

a

a

2

1

2

02

T

T

F A

A dT G

ccdAcC




the net present value (NPV)

the present value of cash inflows is

subtracted by the present value of cash

outflows.

2

1

22

1

21

21

,,

T

T tot

a

tot F

aa

red dT Y G

cY

G

c

t

t c

t c

T T NPV T T NPV




the internal rate of return (IRR) is the interest rate that makesNPV equal zero.

It is the return that a company would earn if they expanded orinvested in themselves, rather than investing that money abroad

The

may be found by solving numerically the associated equation

cbaiT T IRR i,,, 21

cbaiT T NPV i

,,0, 21



Examples



Results

The revenue factor R exceeds unity in casethe inlet working fluid temperatureexceeds a certain “threshold value”,depending on solar collector design (Fig.c).

The four threshold temperatures arelower than 50 degrees.

The temperature threshold values in

case of are around 60 degrees forcollectors I and II.

The other two collectors have pooreconomical performance as theassociated NPV is negative for alloperation temperatures (Fig. a).

The IRR values of Fig. b show thecollector I may be used economically forT between 55 and 70 degrees while

collector II is recommended for operationat more than 60 degrees.

Collectors III and IV are notrecommended as the associated IRRvalues do not exceed the interest rate forall T values.



Results

Different economical

indicators induce

different hierarchies

over the set of solar

collectors.



Results

Let us consider a part of thecollection surface consistingof a single type of collector.Integration of the efficiencydefinition yields thenecessary surface area

The necessary collectionarea is slightly smaller forcollector I than for collectorII.

Therefore, if a single type ofcollector must be used,collector I should beselected.

In case both types ofcollectors are available, a

better solution exists.

2

1

/, 21

T

T F

G

dT cT T A



Solar collectors with optimal non-

uniformly distributed parameters

It was proved that systems consisting in

combinations of different collector types may

be a better solution

than systems consisting of a single collectortype.

One could imagine the extreme case of a

collection system with continuously space

variable parameters.

Such a system may be optimized from the

point of view of a given economical indicator.



Optimisation

The cost

is optimised if:

One finds

2

1

~,~

~,~

~,~

0

02

0

T

T

F

A dT GU

cU cU C

0~

/~/ 0 U C C A A

0

2

20

~1

~

~

~1

c

c U

c

cU ~1

~

~

~1 2

2



Theorem

The following condition should be fulfilled bythe optimum parameters distribution:

Theorem. The modified optical efficiencyand the modified overall heat loss coefficient

in an optimal collection system are distributedin such a way that the gradient of

in the bi-dimensional parametric spacevanishes

0

~

ln 2

~,~0

cU

2/~ln c

U ~

,~0



Results

For very small values of Tthe unglazed solarcollector is the besteconomical solution forboth applications (Fig a).

When increases T asingle transparent layercollector should be used.

The thresholdtemperature for which N jumps from 0 to 1 is

smaller for the coldseason application.

A collector without bottomthermal insulation is thebest solution at very smalltemperatures (Fig. b).



Conclusions

The general theorem proposed here showshow the modified optical efficiency and heatloss coefficient should be distributed for costminimization.

One finds that unglazed, single-glazed anddouble-glazed collectors should be used onthe same collection area in order to obtain thebest performance.

Also, the bottom insulation thickness shouldbe changed accordingly.



End of part 4/4

Thank you!

Viorel Badescu 4

Documents