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Constructal design of distributed energy systems: Solar power and

water desalination

S. Lorente a, A. Bejan b,, K. Al-Hinai c, A.Z. Sahin d, B.S. Yilbas d

a University of Toulouse, INSA, LMDC, 135 Avenue de Rangueil, 31077 Toulouse, Franceb Duke University, Department of Mechanical Engineering and Materials Science, Durham, NC 27708-0300, USAc King Fahd University of Petroleum & Minerals, Earth Sciences Department, Dhahran 31261, Saudi Arabiad King Fahd University of Petroleum & Minerals, Department of Mechanical Engineering, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o

Article history:

Received 1 November 2011

Accepted 6 January 2012

Available online 8 February 2012

Keywords:

Constructal

Distributed energy systems

Desalination

Solar

Size effect

Landscape design

Sustainable

a b s t r a c t

Here we show the fundamental tradeoffs that underpin the design of a distributed energy system with

two objectives: the production and distribution of electric power drivenby solar heating, and desalinated

water produced by consuming solar power. We show analytically that larger solar power plants anddesa-

lination plants are more efficient than smaller plants. This phenomenon of economies of scale is coun-

tered by the greater losses associated with larger distribution networks. From this conflict emerges the

proper allocation of nodes of production of power and water on a territory. We show that as the individ-

ual needs of power and water increase in time, the sizes of solar plants and desalination plants increase,

and so does the size of the territory served by each power plant. At the same time, the territory served by

each desalination plant decreases, and this means that the number of desalination plants allocated to one

power plant increases.

2012 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213

2. The effect of size on solar power generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214

3. The size effect on desalination plants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215

4. Desalination for a string of users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215

5. Solar power for a string of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216

6. Solar power and water for a string of users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218

7. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218

Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218

1. Introduction

Current research in constructal design is showing that the opti-

mal distribution of flows and services on a populated area consists

of balances between the sizes of centers of production and the

sizes of the distribution networks that connect the centers with

every inhabitant of the area [13]. The balances result form an

important trade-off, which is fundamental, universally applicable,

and worth exploring.

The trade-off is a consequence of an essential characteristic ofall flow systems: the larger flow systems are more efficient

thermodynamically [4,5]. This accounts for the tendency toward

the centralization of the generation of power, refrigeration, air con-

ditioning and other useful streams that are required by the popu-

lation. At the same time, larger centers produce larger streams that

must be distributed on larger areas. In this direction, the losses

associated with the distribution and collection networks increase

because they are proportional to the length scale of the served

area. The global flow system consists of production, distribution

and collection, and it is most efficient when the size of the produc-

tion center is properly matched to the size of the networks. From

0017-9310/$ - see front matter 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijheatmasstransfer.2012.01.020

Corresponding author. Tel.: +1 919 660 5309; fax: +1 919 660 8963.

E-mail address: [email protected] (A. Bejan).

International Journal of Heat and Mass Transfer 55 (2012) 22132218

Contents lists available at SciVerse ScienceDirect

International Journal of Heat and Mass Transfer

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j h m t
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.01.020mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.01.020http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.01.020mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.01.020

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this balance results the size of the area served by one center, and

the sizes of the production center, and distribution networks.

This principle of constructal design underpins the emergence of

distributed energy systems on the globe, at all scales. These aretapestries of nodes and links on area elements. This is the design

of the civilized landscape, and by exploring it we shed light on

the future of globalization. In engineering science, we construct a

basis for the concepts of globalization, limits to growth, sustain-

ability, and evolution.

So far, the application of this principle was demonstrated by

considering distributed energy systems permeated by a single kind

of flow: hot water for heating [1,2] and refrigeration [3]. In this

paper we extend this design principle to the more complicated

domain where two or more energy systems are distributed and

superimposed on the same populated area. These multiple tapes-

tries are coupled: they feed on each other, and in order to be

efficient together (as an ensemble) their sizes must be in balance.

The features of multiple distributed-system design are fundamen-tal [1]: here we illustrate them with two coupled systems: solar

power generation on an area, and the production and distribution

of desalinated water for the inhabitants of the area (Fig. 1).

2. The effect of size on solar power generation

Efficiency data of existing power plants show that larger power

plants are more efficient [5]. Additional data are reproduced in

Figs. 2 [6] and 3 [7]. Here we show that the same size effect rules

the efficiencies of power plants driven by solar energy. We can

demonstrate this with simple models and analyses of solar-ther-

mal power plants for example, the model shown in Fig. 4. The solarheat current qs is collected by a reflector, and it is absorbed in a

water heating tank of temperature T. Heat leaks from the tank to

the ambient (T0) at the rate C(T T0). The net heat input available

for driving the power plant is qs C(T T0). For simplicity, we as-

sume that the power plant operates reversibly between T and T0,

therefore the power output is

_W qs CT T0 1 T0T

1

The power output reaches its maximum when the water heater has

the temperature

Topt T0 1 qs

CT0

1=2

2

This temperature is situated between the two possible limits of T,

namely, the lowest (T= T0) and the highest (T= T0 + qs/C). The max-

imum power output per collected heat current is

_Wmaxqs

1 qs

CT01=2 1

1 qsCT0

1=2 1

3

Nomenclature

A area (m2)c1, . . . , c5 constant factorsC thermal conductance (W/K)D length scale (m)h heat transfer coefficient (W/m2K)

hfg latent heat (J/kg)k exponentL length

_m mass flow rate (kg/s)_m1 mass flow rate for one user (kg/s)

q heat current (W)S solar power plantT temperature (K)W water desalination plant

_W power (W)

Greek symbolsa exponentDT temperature difference (K)

Subscriptsb boilerc condensermax maximumdiss dissipateds solar0 ambient

Fig. 1. Two energy systems superimposed on the same area: solar powergeneration and the delivery of desalinated water. Fig. 2. The effect of size on the efficiency of power plants [6].

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The size of the solar power plant is represented by the dimension-

less group qs/(CT0). The collected heat current (qs) increases with the

size of the concentrating reflector. The conductance (C) for the heat

leak from the central heater to the ambient also increases with the

length sale of the heater (D). In the most conservative limit (that is,

if we assume that the heater is not insulated), the heat leak in-

creases in proportion with the external surface of the heater (D2),

whereas the volume of the heater (D3) increases in proportion with

the heat input, qs. Consequently, the group qs/(CT0) is proportional

to D, or q1=3s , and this means that it increases monotonically with

the size of the power plant.

In sum, the group qs/(CT0) is a measure of the size of the power

plant. In the limit of large sizes [qs/(CT0)) 1], the energy conver-

sion efficiency ratio _Wmax=qs approaches 1. In the limit of small

sizes [qs/(CT0)( 1], the efficiency ratio approaches qs/(4CT0), which

means that the efficiency is proportional to the size parameter qs/

(CT0).

This behavior does not change if we modify Eq. (1) to account

for the fact that that power plant does not operate reversibly. This

means that on the right side of Eq. (1) the second law efficiency

(gII) will appear as a factor, and that gII increases monotonically

with the size of the power plant [4,5]. This factor will not change

the conclusion that an optimal operating temperature exists, and

that the maximum power output increases with the size of the

power plant.

3. The size effect on desalination plants

Larger desalination installations produce more distilled water

per unit of fuel used than smaller installations. This is indicated

by the data reproduced in Figs. 5 [8] and 6 [9], and it can also be

demonstrated analytically based on the model of Fig. 7. Sea water

of flow rate _m is heated at the rate q and transformed into steam.

During this process, heat is leaking to the ambient at a rate of order

hbAbDT, where DT is the order of the temperature difference be-

tween boiler and ambient. In sum, we have

q $ _mhfg hbAbDT 4

where Ab is the area of contact between boiler and ambient.

Further downstream, the steam _m is condensed by cooling in

contact with sea water, across the thermal resistance (hcAc)1,

where Ac is the size of the condenser surface. By writing

_mhfg $ hcAcDT 5

and combining with Eq. (4), we construct the ratio _m=q, which rep-

resents the efficiency of the desalination process,

_mhfgq

$1

1 hbAbhcAc

6

The condenser size determines the overall size of the installation.

Assuming that the group (hbAb/hc) does not change, the ratio _m=q

increases monotonically as the size of the condenser (Ac or _m) in-creases. It also increases as the boiler insulation improves, i.e. as

hb decreases.

4. Desalination for a string of users

Consider the production of desalinated water of flow rate _m at

one location, and the delivery of this water stream to a string of

length L populated equidistantly by N users of desalinated water,

N= c1L, where c1 is a constant (see Fig. 8a). Each user receives

the water flow rate _m1, which is fixed. The total flow rate produced

by the desalination plant is

_m c1L _m1 7

Fig. 3. The effect of size on the efficiencies of thermoelectric (TE) power generators,

relative to engines [7].

Fig. 4. Model of power plant driven by solar heating.

Fig. 5. The effect of size on the efficiency of high-pressure pumps for desalination

[8].

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The power needed to produce and distribute this stream is being

used in two ways. First, a portion is used to produce the stream _m

that is delivered by the desalination plant. This power represents

the rate at which the plant destroys useful energy, and it is propor-

tional to the heating rate q, or the rate at which fuel (useful energy,_W1) is being consumed during the desalination process. Because

according to Eq. (6) the ratio _m=q increases as the size of the plant

increases, its inverse ( _W1= _m) decreases as _m increases. If the de-

crease of _W1= _m is approximated locally (in the vicinity of the design

of size _m) as being proportional to _mk, where k is positive, it fol-

lows that _W1 behaves as

_W1 c2 _ma 8

where a < 1, and c2 is a constant. The actual value of a can be de-

duced by comparing Eq. (8) with Eq. (6) in which Ac varies as _m.

The conclusion at this point is that larger plants are more efficient

because they use less_

W1 per unit of_

m as_

m increases.The opposite trend is exhibited by the power ( _W2) needed to

deliver _m to all the users along L. This power scales as _m DP/q,

where for turbulent flow DP varies in proportion with _m2L. From

this follows the scaling relation

_W2 c3 _m3L 9

for which c3 is determined from the fluid mechanics of the linearly

decreasing _m along the duct of length L (e.g., Ref. [10]). Eq. (9) shows

that the power spent on distributing water per unit of _m increases

with the size of the installation and the community of those served

by the installation.

Together, Eqs. (8) and (9) account for all the power spent on

providing water to all the users,

_W c2c1 _m1La c3c1 _m1

3L4 10

The total power spent per user is

_W

N c2ca1

1_

ma

1L

a1

c2

1c3_

m3

1L

3

11

The first term decreases as L(1a) as L increases, while the second

term increases as L3. The total power spent per user is minimal

when the two terms are of the same size, and from this follows

the size L (or N) served most efficiently with desalinated water,

L $ ca31c2c3

_ma31

1=4a12

We learn that the optimal L decreases as _m3a=4a1 as the individ-

ual need _m1 increases. By combining Eqs. (11) and (12) we find that

the power spent per user increases as _m3=4a1 as _m1 increases.

In summary, from the tradeoff between production losses ( _W1)

and distribution losses ( _W2) emerges the length scale of the inhab-

ited territory served by one desalination plant. It is reasonable to

expect that the standard of living evolves such that the individual

use ( _m1) will increase in time. From this follows the prediction that

L will decrease, and that the density of desalination plants on the

landscape will increase.

5. Solar power for a string of users

Next, we consider the problem of allocating a number (N) of

users of solar power to one solar power plant (Fig. 8b). The model

of Section 2 showed that the efficiency of the plant ( _Wmax=qs) in-

creases with the physical size (qs), such that when the size is small

the ratio _Wmax=qs is proportional to qs, and when the size is large_

Wmax=qs approaches 1. More simply, this means that locally (inthe vicinity of the design of size qs) the ratio _Wmax=qs varies as

Fig. 6. The effect of size on the efficiency of nuclear reactors for power and desalination [9].

Fig. 7. Model of desalination by boiling the sea water and condensing the steam.


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qks , where k is positive and smaller than 1. In sum, we expect_Wmax

to increase with size, approximately as q1ks .

The physical size of the solar power plant on the inhabited ter-

ritory (A) is represented by the area occupied by the reflectorcol-

lector installation (Ac), or the field of such installations. The

economic and political reality of approving, funding and building

the installation means that the size Ac scales with the size of the

territory, or population, A. Because the length scale of the territory

is L =A1/2, the conclusion of the preceding paragraph means that

the power output of the solar plant increases with size as A1+k, or

L2(1+k),

_Wmax c4L21k 13

A fraction of this power stream ( _Wdiss) is destroyed along the trans-

mission lines that connect the plant with the recipients of electric

power. This fraction is proportional to the product _WmaxL, where L

is the length of the transmission line,

_Wdiss c5L32k 14

If the users are connected (equidistantly, cf. Fig. 4b) by a transmis-

sion line of length L, then the net power delivered per user

_Wnet _Wmax _Wdiss is proportional to

_WnetL

c4L12k c5L

21k 15

When L is small and increases, the first term increases faster than

the second term, and the net power per user increases. When L is

sufficiently large, the trend reverses and eventually the net power

per user drops to zero (that happens when L reaches c4

/c5

). The

net power per user is maximum when the length scale of the terri-

tory is

L 1 2kc421 kc5

16

This length is larger when the solar plant is more efficient (with a

larger c4) and the transmission line is less dissipative (with a smal-

ler c5). At this scale, the maximized net power per user varies as

c21k4 =c

12k5 , and increases qualitatively in the same direction and

for the same reasons that L increases. As technology advances, we

Fig. 9. Network of solar thermal power plants and water desalination plants on the sea shore.

Fig. 8. The balance between production and distribution losses pinpoints the size of the energy flow system: (a) desalinated water; (b) solar power; (c) water and power.

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expect that c4 increases and c5 decreases, therefore, the evolution of

L and _Wnet=L is toward greater values in time.

6. Solar power and water for a string of users

Consider finally the combined design of water desalination dri-

ven by solar power on the same territory L, Fig. 8c. The users who

reside on L receive two streams at the same time, water _

m1) andelectric power ( _Wnet=L). One solar power plant (S) drives both

streams, the generation of electric power (as in Section 5) and

the desalination of water in plants of type W (Section 4).

The new feature that emerges is due to the simultaneous tech-

nological evolution of the designs of Fig. 8a and b. In water desali-

nation the number of users allocated to one plant decreases, while

for power generation the number of users allocated to one solar

plant increases. This means that in time, as the solar plant serves

a greater territory, it also serves a larger number of water desalina-

tion plants. The territory served by one desalination plant (LW,

Fig. 8c) decreases while the territory served by one solar plant (L,

Fig. 8c) increases. From this follows the design in which the indus-

try of solar-driven desalination becomes distributed, with many

W plants serving short territories LW

distributed on the territory L

of the solar plant. In this limit, the design approaches the distrib-

uted-power design of Fig. 8b, in which the users distributed along

L are of two kinds, users of electric power for personal needs, and

users of power for desalinating water.

7. Concluding remarks

The distributed energy designs determined in this paper apply

generally to the production of desalinated water and solar power

on an area. We unveiled the fundamental tradeoffs that govern

the distributed design by considering a string of users (Sections 4

and 5), which led to the discovery that in time a string of desalina-

tion plants will be allocated to one power plant (Section 6). These

findings also apply to users that occupy an area A (e.g., Fig. 9), as

opposed to a line (Fig. 8). The reason is that the length scale of

the area (A1/2) is the same as the scale of the length of the network

(L) that might connect all the users onA, provided that the network

has modest (finite) complexity, for example, a tree-shaped design

with a finite number of bifurcation levels [11].

The tradeoffs identified in this paper are based on optimizing

the distribution and use of power. This approach can be taken

to a next level of modeling by considering the cost of the distrib-

uted power & water flow design. In simple terms, the cost of an

installation increases monotonically with its size. This means that

the total cost constraint is represented qualitatively by the total

size constraint, which for the systems considered in Fig. 8 would

represent the sum of the sizes ofS

,W

and the distribution system

L. These three sizes increase in time and so does the number of

desalination plants allocated to one power plant. To be deter-

mined are the balances (the ratios) between these sizes, and

how these balances change in time as the individual sizes

increase.

Acknowledgement

The authors wish to thank King Fahd University of Petroleum &

Minerals (KFUPM) for the support received through the Project #

IN100025.

References

[1] A. Bejan, S. Lorente, Design with Constructal Theory, Wiley, Hoboken, 2008,Section 11.3.

[2] L.A.O. Rocha, S. Lorente, A. Bejan, Distributed energy tapestry for heating the

landscape, J. Appl. Phys. 108 (2010) 124904.

[3] L. Xia, S. Lorente, A. Bejan, Constructal design of distributed cooling on the

landscape, Int. J. Energy Res. 35 (2011).

[4] S. Lorente, A. Bejan, Few large and many small: hierarchy in movement on

earth, Int. J. Des. Nature Ecodyn. 5 (3) (2010) 254267.

[5] A. Bejan, S. Lorente, B.S. Yilbas, A.Z. Sahin, The effect of size on efficiency:

power plants and vascular designs, Int. J. Heat Mass Transfer 54 (2011)

14751481.

[6] N. Greene, R. Hammerschlag, Small and clean is beautiful: exploring the

emissions of distributed generation and pollution prevention practices, The

Electricity J. (2000) 5060.

[7] C.B. Vining, An inconvenient truth about thermoelectrics, Nat. Mater. 8 (2009)

8385.

[8] S. Baumgarten, Hydraulic comparison of high-pressure pumps for RO

desalination systems, World Pumps (2005) 5255.

[9] B.M. Ma, Large reactors for power and water desalting, Nucl. Eng. Des. 12

(1970) 917.

[10] E. Cetkin, S. Lorente, A. Bejan, Natural constructal emergence of vascular

design with turbulent flow, J. Appl. Phys. 107 (2010) 114901.

[11] A. Bejan, S. Lorente, The constructal law origin of the logistics S curve, J. Appl.

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