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Economic aspects of low-temperature multi-effect desalination plants
Desalination, 71 (1989) 177-201 Elsevier Science Publishers B.V., Amsterdam -
177 Printed in The Netherlands
Economic Aspects of Low-Temperature MultiEffect Desalination Plants Y.G. CAOURIS, E.T. KANTSOS and N.G. ZAGOURAS* University of Patras, Mechanical Engineering Department, 265 00 Patras (Greece), Telex: 312447 UNPA GR (Received May 20,1987; in revised form August 30,1988) SUMMARY
The technical parameters of multi-effect low-temperature distillation plants are determined and the production cost of every input is calculated separately for the successive levels of production under various loads of annual operation, by using a computational model devised for this purpose. The study of the results obtained leads to the development of the respective production map diagrams. Observation of the iso-cost and iso-product curves in these diagrams allows the determination of the optimal number of effects for each combination of production level and annual operation load. The sensitivity of the optimal number of effects relatively to the price system is also examined. Further, the average total cost is formulated as a function of the size of the plant under full load operation. This cost is also formulated as a function of the annual operation load for each plant size. The present work is obviously relevant to the design of desalination plants on Greek islands and other arid zones of the littoral countryside. Keywords: desalination optimization, solar low-temperature multi-effect desalination, economics.
SYMBOLS
AC
-
B
-
Ch
-
c them
-
c them
-
c me”
-
C me”
-
c heat
-
Cheat
-
Average unit cost of the desalination process Number of effects Daily heating cost Annual cost of chemical treatment Average unit cost of chemical treatment Annual cost of evaporators maintenance Average unit cost of evaporators maintenance Annual cost of heating Average unit cost of heating
*In alphabetical order. OOll-9164/89/$03.50
0 1989 Elsevier Science Publishers B.V.
(h/m3)
178
CPump CPumP
-
C wages
-
C wages
-
CW
-
c
-
csi
-
C 0”” & (BE)
-
4
(BE)
-
& (UE)
-
4
-
(UE)
EL
-
ET ei
-
GY
-
G’i
-
g Hi h
-
Ji
-
Li (BE) Li (UE) kl
-
M NW ipvi (UE)
-
AP,t
-
Annual cost of pumping requirements Average unit cost of pumping requirements Annual cost of labour Average unit cost of labour Daily wage, other fees and social contributions of employers Specific heat of pure water Specific heat of saline water of the ith effect Specific heat of seawater Temperature drop between successive effects External diameter of boiling section of the ith effect Internal diameter of boiling section of the ith effect External diameter of condenser-type feed heater of the ith effect Internal diameter of condenser-type feed heater of the ith effect Installed electrical power for the pumps operation Price of heating energy Quantity of pure water obtained by condensation in the condenser-type feed heater of the ith effect Specific enthalpy of water vapour at the jth effect Specific enthalpy of liquid phase at the ith effect Acceleration of gravity Latent heat of vaporization at the ith effect Plant’s altitude Local pressure loss coefficient Length of boiling section of the jth effect Length of feed heater section of the ith effect Mean length of a heat-exchanger tube between two successive bends Saline water flow rate through feed heaters Total number of evaporators Lifetime of evaporators Pressure drop through the feed heater of an evaporator of the ith effect Inlet-outlet static pressure difference
(dr) (h/m3) (h) (dr/m31 (dr)
&J/kg K) &J/kg K) (kJ/kg K) WI (4 (ml b) (m) (kw) (dr/kWhh)
(t/d) &J/k) &J/k) (m/s”) &J/k)
b-d
(t/d) (Y)
(Pa) (Pa)
179
APvi(BE) -
PET” S : Q" 4i
-
2
-
SP
-
STAC
-
ri
-
W x
-
Yi
-
4 u2
Subscripts h Y
-
Pressure drop through boiling section of an evaporator of the ith effect Capital cost of each evaporator Price of electrical energy Effective annual production Effective daily production Nominal production capacity Quantity of saline water evaporated at the ith effect Nominal interest rate Inflation rate Average salary Required rate of thermal energy for the plant’s operation Short term average cost of desalination process Duration of the period of operation Temperature of the available hot water from the heat source Temperature of flashing brine at the ith effect Total heat exchange coefficient of condensing-type feed heaters Total heat exchange coefficient of boiling section Velocity of distillate Velocity of saline water Flowrate of recycled hot water (heat carrier) Ratio of the capital and maintenance cost of the pumps, as a proportion of their operating cost, supposing a full year operation Number of parallel tubes forming a plane of the i-th heat exchanger
Heat as economic input Evaporators plus maintenance plus pump as economic input
(Pa)
(h) (WkWh,)
(m3/y) b3/d) (m3/d) (t/d)
(dr/employer, month) (kwt,)
(h/m3) (h/y) (K) (K)
(W/m2K) (W/m2K)
(dr) (dr)
Greek symbols E - Inflation adjusted interest rate [ t = (ri- rj ) / (l+rj)l
Ii
,uandv
-
-
Friction factor at the ith effect Duration of yearly operation
(months)
180
P @
-
Water density Coefficient of utilization
(kg/m3)
INTRODUCTION
A computational model has been developed for the calculation of the technical and economic factors, engaged in the process of multi-effect low-temperature desalination, as a function of the number of effects, B, and the production capacity, Q. The most important technical factors are the required thermal energy SP, the number of the effects and the number of evaporators in each effect, the heat-transfer surface area, the pumping requirements, EL, and the seawater feed mass flow rate. Besides, the most important economic factors are the capital cost of the evaporators and their maintenance, CeVapand C,,,, the cost of thermal energy Cheat, the pumping cost Cpump,the cost of chemicals C them and labour Cwages. The above technical factors are not influenced by the duration of the plant operation and are determined only by the production capacity. On the contrary, the economic factors are strongly influenced by the duration of the plant operation. Thus the economic analysis requires the consideration of different loads of the plant operation on annual basis (full load and different partial loads ) . Substitutional relationships are observed between the heat cost and the cost of all other technical inputs. These relationships are revealed by considering different numbers of effects in each level of production capacity, while keeping the load of annual operation constant. This can be described for each level of annual operation load by a production function diagram from which we can derive the optimal number of effects. Production functions with substitutional factors of production are often used in applied economics (e.g., [ 1,9] ). On the other hand, observation of the production costs for plants of different capacity operating under full load leads to the formulation of a function expressing the average cost in relation to the size of the plant (function of the long-term average cost under full load operation). Finally, observation of the production costs of the plants with varying load during the year allows the formulation of another cost function expressing the average cost in relation to the annual operation load with given capacity of plant (short-term average cost). In this paper the process analysis is shortly presented, in order to support the economic data. We do not present the technical results. The work is divided in three sections: (i) process analysis; (ii) modelling; and (iii) treatment of the economic results. The costs are computed on the basis of the prices observed in the Greek market at the end of 1985 and the beginning of 1986 (in drachmas). During this period the exchange rate of U.S.$/dr was l/150.
181
Note that combined technical-economic zation of water production [ 2,5,8].
models are useful in cost minimi-
PROCESS ANALYSIS
Plant description Fig. 1 illustrates the arrangement of a desalination plant of N effects, as considered for the present analysis. Before entering the last effect, seawater is preheated in condenser-type feed heaters mounted in the upper part of the evaporators and by using part of the generated vapour as heating medium. Prior to that, the seawater is preheated in a condenser in which the vapour from the last effect is condensed. The external heat carrier is water at an approximate temperature of 70’ C. This heat is produced by solar energy and is transferred to the plant via heat exchangers at the bottom of the first effect evaporators. All the evaporators are of similar dimensions and their number in each effect depends on the required heat-transfer surface (see Note). The energy balance In order to facilitate the analysis and estimation of the construction cost, the following assumptions have been made: (a) Equal temperature drop between successive effects, i.e.
(b) Constant specific heats for the feed distillate and brine. (c) The salt concentration of flashing brine must not exceed the value of 7% pwt (part per weight ) . (d) No heat loss occurs between the evaporators and the surroundings.
Fig. 1. Multi-effect desalination plant arrangement.
182
After these assumptions, the governing equations can be written: Bottom heat exchangers (boiling section of the evaporator): - (M-q,)C,lD+q,H1
(for the lSteffect)
- WCD=O
Upper heat exchangers (condenser-type feed heaters): ci(G’i+l -G;)
+MC,D=O
(Ll,.....,
N)
Apart from the unknowns ci and qi, the quantities M and W also remain undetermined. That is, a system of 2N equations has to be solved with 2N+ 2 unknowns. Yet it is evident that O
under the above constraints. Heat transfer surface and the evaporators’size The size and the number of evaporators in each effect are dependent on the appropriate size of heat exchangers required in order to achieve the necessary amounts of ei and qi. After a number of calculations it has been shown that it is preferable to have a number of parallel evaporators in each effect instead of a single big one for each effect. As an upper limit for the size of each evaporator has been considered a height of 3 m with a diameter approximately one half of the height. For heat exchangers we selected the type of serpentine shape in parallel series. They are made from an Cu-Ni alloy. The total heat transfer coefficient for the condensation heat exchangers (feed heaters) is considered to be UoI= 3500 W/m” K while for the bottom heat exchanger a value of UoZ= 2500 W/m2 K is considered. As optimum velocities (in order to obtain the values of U,,, and U,,,) we take 2.5 m/s for the feed and 1 m/s for the distillate. The length of the feed heater heat exchanger of an evaporator of the itheffect is Li (UE) =MC,,D/
[ Uo, 0.5 Dx d,i (UE)] ~8.812~
10m3M/do, (UE)
where M is the flow rate of brine through the feed heater heat exchanger of an evaporator of the ith effect (t/d). The internal tube diameter can be derived by the flow rats equation
183
Therefore dii(UE)=2.43~10-~JM The length of the boiling section feed exchanger of the evaporator of the first effect is L,(BE)=WCD/[U,,D~~,,(BE)]=~.~~~X~O-~
W/d,,(BE)
where W is the flow rate of hot water (heat carrier) from heat source through the heat exchanger (t/d). The internal tube diameter dii (BE) is di, (BE) = 2.43 x 10m3Jw The length of the boiling section of the ith effect is Li(BE)=((qi_l--i_1)Hi/[D [D&i
=[(Qi-I-
(BE)
1.6~d,i(BE)
u,,]}+{(qi_,-eei_,)CD/
&I}
ei_l)/d,,i (BE)]
[ (6.578x10S4Hi/D)
+6.168~10-~]
where Hi (kJ/kg) is the latent heat of vaporization at the ith effect. dii(BE) can be derived from the flow rate equation
Therefore dii (BE) =3.838~10-~
dz
The size of the evaporators is limited by the size of upper and bottom heat exchangers, assuming that the distance between them is 0.7 m. If the size of the evaporator exceeds the predetermined upper limit, then the evaporator is splitted into two or more equal parts, operating in parallel. Inox steel is chosen as the construction material while the thickness and weldings are computed according to DIN standards and depend on the pressure under which each effect operates. Pumping requirements (i) Pumping of brine through the feed heaters The pressure drop through the feed heater of an evaporator of the ith effect is
184
dP”i (UE) =
C ji +A,L, (UE)/dii (UE) {0.28($
1
u$P/~=
+dyi) +li[Li(UE)/dii
(UE)]}3125
whereji is the local factor of losses and iii is the frictional factor. In the case of turbulent flow, the latter can be obtained from the equation l/J&
-2 log[2.5/(Re
fi)+K/(3.71
d,)]
yf +dyi expresses the number of bends of the serpentine-type tube and yi is the number of parallel tubes of each plane. In general, there must also exist a number yi of parallel planes, in order for the length Li (UE ) ( = L) to be equal to l,yf. The distance between two successive parallel tubes is equal to 2doi (UE) ( = 2d,). However, the length L cannot be in fact exactly equal to y: 1,) because commercially available tubes are of certain diameters. Consequently, L must be equal to y:Z, +dyi. According to the geometry of the heat exchanger, the values yi and dyi are obtained from the following equations: 3d,ya-1.287
d:yi-L=O
dyi= (L-3doya
+1.287 d~y~)/(1+0.7123
do)
where yi is the greatest integer less than the real positive solution. The heat H can be expressed as
H=
F
i=l
[dPvi(UE)l
/@I
(ii) Pumping of brine through the bottom heat exchangers The heat H can be expressed as
where dP”i (BE) is the pressure drop through the bottom heat exchangers of the ith effect and AP,, is the difference between static pressures of inlet and outlet, taken equal to 7 x lo5 Pa, a realistic value when the site of consumption is near the production plant. The flow through the bottom heat exchangers is separated into two parts, one for the vapor phase and the other for the liquid. SO the length L,(BE) is calculated as Li(BE) = Liv + Lif where Liv= [ (qi-1-ei_l)/doi(BE)]6.578X10-4 L,=6*168X
10d3 (qi_1 -ei_l
)/d,i (BE)
The pressure drop across the part Liv is
(Hi/D)
185
dP”i (v) = [0*28(yP +&i) +A,& (BE)/& (BE)]
(/Q&/2)Liv/Li(BEI
the pressure drop across the part Lif is dP”i(f)=[0.28(y~+dyi)+~iLi(BE)/dii(BE)]FiOOL,f/Li(BE) and the total pressure drop is dP”i (BE) =dP,i (f) +dPvi
(V)
MODELLING
A computer program has been developed which simulates the operation of the desalination plant and calculates both the individual and total costs, related to the level of production and the number of effects B (3
e/11-
(l--E)-nl
(ii) The annual cost for the maintenance of the evaporators, (estimated as 3% of the owning value of the evaporators) [5] C,,, =P,,N,,O.O3 defined as the sum of of the cost of electrical (iii) The pumping cost CpUmp power for the pump operation and the capital plus maintenance cost of pumps. This is inserted in the model by the expression Cp,,=EL~~+P,[l+
(x/100)(12/~)]
where x/100 is the percentage coefficient which expresses the capital and maintenance cost of pumps, x = 13.45 - (EL/14.57), for 100 m3/day < & < 1000
186
m3/day and x = 0.003 for Q> 1000 m3/day (as a proportion of the functioning cost, supposing a full year operation) and p is the duration of operation expressed in months. (iv) The annual cost of heating requirements expressed as c heat= SP- T-ET The cost of heating is considered constant and equal to 0.75 dr/kWhth, and is independent of any seasonal fluctuation in potable water demand, because of the possibility of using the excess heat for other purposes. (v) The annual total cost of chemical treatment. From the fact that the plant operates at low temperature levels the treatment must be made with phosphates [ 61. The cost per m3 is estimated to 11.1 dr, thus the total cost is expressed as Cthem=Q*11.1.30 (vi) The labour total cost. An assumption has been made that for a nominal production capacity of less than 200 m3/d, four employees are necessary for the plant’s operation. One additional employee is necessary for each additional 200 m3, up to a total of 1000 m3/d. For a nominal production capacity greater than 1000 m3/d, one additional employee is necessary for each additional 1000 m3. Therefore the annual labour cost is considered to depend on the level of annual production and is expressed as c wages = Sp4
if Q < 200 m3/d
C,,,,=S~u(4+INT[(Q-200)/20~]+1}if200m3/d~Q~1OOOm3/d C,,~=S~{8+1NT[(&-1000)/1000]+1)~~Q~>1000m~/d The annual total cost is considered as the sum of all the preceding particular costs, without taking account of the land cost. This is a typical characteristic of the waterless islands of Greece, which we had in mind from the beginning of this work. On these islands, there are plenty of arid communal lands unsuitable for any other use. This total cost concerns only the production process and the pumping of potable water to a height of approximately 70 m and does not include the distribution cost. The average unit cost AC (per m3 of distillate) is calculated by dividing the particular terms of annual cost by the annual production Q,,= &*~*@30.
187
The results discussed in the next sections are obtained from the following standard values of parameters: Pev= 698,000 dr T= 8,760 h, n= 15 T= 6,570 h, n= 18.333 T = 4,380 h, n= 21.666 T=2,190 h, n=25 t= (ri-rj)/(l+rj)
when when when when
the plant the plant the plant the plant
operates operates operates operates
12 months yearly 9 months yearly 6 months yearly 3 months yearly
= (0.27-0.21)/(1+0.21)=0.0496
$=0.85, P1 =8 dr/kWh, ETz0.75
dr/kWh,,[7],
and S=125,000 dr/
employer, month.
TREATMENT OF THE ECONOMIC RESULTS
Maximum load operation In the first part of this study, the desalination plant is considered to operate under maximum load continuously, all over the year, except for the necessary maintenance periods which are supposed to be 15% of the total time. Under this condition, data from the model are obtained and evaluated. Classification of factors ofproduction and inputs involved: substitutional and complementary factors The factors of production and inputs involved in the process of desalination costing can be classified into three classes, according to their variation with the production level or to the technical procedures. (i) The first class is composed of the factors of production or the inputs of which the quantities per product unit of the desalination plant (per m3 desalted water) are dependent exclusively (or almost exclusively) on the number of effects and not on the level of production. When the capacity of the plant increases without any change in the number of effects, the proportion of these factors of production or inputs per product unit is not modified. In other words, the total quantities of these factors increase proportionally with the quantity of the product. On the contrary, when the number of effects is modified in any equal product level, the quantities of these factors or inputs per product unit is modified too. This class refers to the following factors or inputs: - the capital invested in evaporators; - the materials and labour spent on the maintenance of these evaporators;
188
- the capital invested in pumping machinery and the energy consumed for its operation; - the heat consumed for the evaporation in the first effect. The average unit cost of each of the first three factors (per m3 desalted water) increases generally as the number of effects increases, that is to say when passing from 3 effects to 4, or from 4 to 5, or from 8 effects to 9, as shown in Fig. 2. On the contrary, the average unit cost of heat (per m3 desalted water) decreases generally with increasing number of effects (see Fig. 2 ) . It is important to observe in Fig. 2 the curve representing the sum of the previous costs. The lowest point on this curve corresponds to the technical combination at B = 5. Indeed, the average cost of substitutional factors as well as the average total cost is minimum at 5 effects, as shown in Table I. However, the cost at 4 effects and those at 6 or 7 effects are only a little higher than the cost of the optimal combination, as shown in the same table. (ii) The second class corresponds to the labour cost. As expected, the labour cost per unit of product depends mainly on the level of production. Especially, the average cost of labour decreases when the capacity of production increases. From the model expression of the total annual cost of labour, the average cost of labour (per m3 desalted water) was derived as
1 3
4
5
6
7
8
0
9
Fig. 2. Average unit cost (dr) of substitutional factors per m3 of desalted water as a function of the number of effects B. Exchange rate during the elaboration of this study: 1 US $ = 150 dr.
189 TABLE I Average cost of multi-effect water desalination by alternative technical combinations and alternative production capacities (full annual load operation, the cost in drachmas) Technical combinations B=7
B=8
B=9
Nominal production capacity: 100 ma/d (Q,=85 m3/d) ICib 136 119.5 117 118.5 AC 335.4 319.1 316.8 318.3
121 320.5
125 323.6
129.5 329.3
Nominalproduction capacity: 250 m3/d (Qd=212.5 m3/d) CCi 132.4 117 114.2 115.7 AC 238 222.8 219.8 222.7
118.1 224.2
125.4 231.2
126.7 232.5
Nominal production capacity: 400 m3/d (Qd= 340 m3fd) XCi 133.8 118.7 116.2 117 AC 215.7 200.8 198.2 200
119.2 201.5
125 207.2
126.2 208.4
Nominal production capacity: 550 m3/d (Q,=467.5 ZCi 132.5 116.4 114 AC 195.2 179.8 176.7
m3/d) 117.6 181
119.4 182.3
124.2 186.9
125.5 188.3
Nominalproduction capacity: 700 m3fd (Q,=595 m3/d) CCi 133 117.7 114.3 117.8 AC 191.4 176.1 173.1 176.9
119.5 178
123.5 182
127.5 185.9
Nominal production capacity: 850 m3/d (Qd= 722.5 m3/d) I&i 132.4 116.7 113.5 115.9 AC 188 172.4 169.1 172.2
119.3 175.2
122.8 178.5
126 181.7
Nominal production capacity: 1000 m3/d (Q,=850 m3/d) XCi 132.4 116.7 113.6 115.6 AC 186.1 170.5 167.1 169.9
118.8 172.7
122 175.7
124.8 178.2
B=3
B=4
B=5
B=6
“The cost was evaluated for small production capacities, e.g. Qd=85 m3/d, without any restriction on the height of the evaporators. For higher production capacities, e.g. Qd> 200 m3/d, the cost was computed considering evaporators with a height of 3 m. bi = evap, mev, pump and heat.
The graphical presentation of the average unit cost of labour is given in Fig. 3. (iii) Finally, we have the chemical material, of which the quantities per product unit are the same, irrespective of the production level and the number of effects. Therefore, the average cost of the chemical treatment can be considered the same for every level of production and number of effects: cthem
=
11.1 dr/m3 of distillate
Fig. 3. Average unit cost (dr) of complementary factors per m3 of desalted water as a function of the effective daily production Qd. C”i
50
25 (drx103)
l5
ch
Fig. 4. Iso-product curves in the case of full annual operation. C,=daily cost (drx 103) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr x 103) of heating.
191
Production map diagram
The modification of the technical process of desalination, in the sense of increasing the number of effects, implies modification of composition of the average unit cost of the desalination only for the following inputs and factors of production: - capital cost of evaporators; - inputs and labour for maintenance of the evaporators; - capital cost of the pumping system and energy consumed for its operation; - heat for the desalination process. In particular, the increase in the number of effects, on the same level of production capacity in full operation, implies an increase in the first three elements of the average unit cost of desalted water (per m3) and, inversely, a decrease in the last element; in other words, it implies partial substitution of the fourth element (heat) by the first three elements. Fig. 4 gives the graphical presentation of the preceding combination of C, and C,.,for six successive levels of production, between 212.5 m3/d and 850 m”/ d. The vertical axis C, represents the daily cost of inputs and factors of production: evaporators, maintenance, and pumping; the horizontal axis C, represents the cost of heat per day; the iso-product curves Qd= 225 m3/d to Qd= 850 m3/d represent the successive levels of product, obtained by alternative combinations of C, and C,. The points B= 3 to B=9 on each iso-product curve represent the alternative technical procedures, by which the level of production represented by the corresponding iso-product curve can be obtained. Each technical procedure B = 3-9 is characterized by the number of effects. Using this diagram, the substitutional relationships between C, and C, in different levels of production is clearly shown. It is clear that this substitution is due to the substitution of one technical procedure by another. For example, passing from the technical procedure B = 5 to B = 6, for a certain level of production, implies the substitution of the input h by the group y. The measure of this substitution is given by the rate of technical substitution, i.e. the ratio between the increase in C,,, (C, - C,,), and the decrease in Ch, (C,, - C,,), while maintaining the same production. Graphical representation of the rate of technical substitution when passing from the technical procedure B = 5 to the B = 6 in a certain level of production is given by the inclination of the line which is simultaneously tangent to the corresponding iso-product curves at the points B = 5 and B = 6. Numerical values of the rate of technical substitution (passing from B = 5 to B = 6 and from B = 5 to B = 4 ) in different levels of production are given in Table A.1, in the Appendix. Optimal number of effects
For each level of production the optimal technical procedure is defined by such a point on the corresponding iso-product curve, where an iso-cost line (geometrical space of distribution of a quantity of expenses) is tangent to this
192
iso-product curve. Every other distribution of the same expense leads to an inferior theoretical iso-product curve. Note that in every iso-product curve the point where an iso-cost line is tangent to it represents the technical procedure B = 5. Thus, the same result derived elsewhere by observing the curve of the average cost of substitutional factors is now derived by observing the iso-cost and iso-product curves. This optimum is defined here according to the prices of the inputs and factors of production of the group y and of the input h, valid in the Greek market at the end of 1985. The answer to the question how much a modification of the price system influences the optimal technical procedure B = 5 is founded on the rate of technical substitution, as given in the Appendix. In general, a great change in the relative prices of the production factors is necessary in order to achieve a change in the optimal number of effects. For example, if the real (inflation adjusted) interest rate is doubled concerning the entire investment (e.g. 9.92% instead of the value 4.96% considered for the derivation of the present results), this increase does not influence the optimal technical solution B = 5 (see Appendix). Similarly, when the real interest rate is tripled (14.88% ) , the optimal technical solution B = 5 is influenced only in two cases, viz. when Qd= 212.5 m3/d or Qd= 340 m3/d, in which cases the technical combination B = 4 becomes optimal (see Appendix). Inversely, when the real interest rate becomes zero, the optimal technical solution B = 5 is not influenced at all (see Appendix). Average total cost as function of the size of the plant (long-term average total cost) The principle of decreasing cost is valid in water desalination, in so far as increasing capacity of production results in decreasing average total cost. For increasing levels of production capacity up to 212.5 m3/d, the observed decrease in the average total cost is due to the decrease in the cost of technical inputs (since the increased production level allows improved organization of the technical terms of production) as well as to the decrease in the cost of labour. Above this level of production, the decrease in the total average cost is due to the decrease in the labour cost exclusively or almost exclusively. On the basis of the data concerning the average total cost obtained by the optimal technical combination in each level of production, this cost may be written approximately as a function of the size of production capacity AC= (118.1-v (1)
QP)+ll.l+{4(C,/Q)+ (2)
[INT(Q-200)/200+1]C,/Q} (3)
We take C, = 3778 dr as mean daily wage plus social contributions. We also take v=O.1764 (constant) and p=O.496 (constant). Given these values for the constants v and p, the first part (1) of the above expression describes the impact of the size of production capacity on the average cost of
193 AC
A
320-
240.
160.
80.
I I
I I
‘3l-v
I I I I
‘%mt <-
I I
‘%mp 200
400
600
800
Qd (mYday>
Fig. 5. Long-term average unit cost (dr) under full load operation per m3 of desalted water as a function of the effective daily production Qti
technical inputs except chemicals (with a standard deviation equal to 0.90). The second part (2) of the expression is the average cost of chemicals (taken as constant ) . The third part (3) describes the impact of the size of production capacity on the average cost of labour. Partial annual load operation Desalination plants designed to cover seasonal water needs operate only for some periods during the year. In this case the following questions arise: - Which is the optimal number of effects, for each intended period of operation (for example, 3 or 6 or 9 months per year)? - How much does the average cost of desalination increase for each such case of limited operation? ho-product curves under different load Figs. 6-8 illustrate the position and form of the iso-product curves of different plants with a real production capacity of 850,595 and 340 m3/d of desalted water (considering only 85% of the nominal capacity, the remaining 15% being ineffective due to interruptions of production for maintenance), for an annual duration of production of 9 months (Fig. 6), 6 months (Fig. 7), and 3 months (Fig. 8). When comparing these figures, it can be seen that for decreasing annual duration of operation the iso-product curves move to the right and
Fig. 6. Iso-product curves in the case of 9 months operation annuaIly. C,=d&y cost (drx103) of capital invested in evaporators and pumping machineries and their maintenance; C, = daily cost (drx 103) of heating. Fig. 7. Iso-product curves in the case of 6 months operation annually. C, =daily cost (dr X lo3 ) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr x 103) of heating.
Fig. 8. Iso-product curves in the case of 3 months operation annually. C,=daily cost (drx 103) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr X 103) of heating.
195 TABLE II Short-term average total cost in drachmas” per m3 of distilled water for alternative technical combinations Technical combination
Production capacity: 340 m3/d B=4 B=5
Production capacity: 595 m3/d B=4 B=5
Production capacity: 850 m3/d B=4 B=5
Duration of operation (months per year) 3
6
9
12
250.3 (1.669) 258.1 (1.721)
216 (1.44) 216.6 (1.444)
205.2 (1.368) 203.6 (1.357)
200.8 (1.339) 198.2 (1.321)
222.2 (1.481) 230.2 (1.535)
189.5 (1.263) 190.3 (1.269)
179.2 (1.195) 178.1 (1.187)
176.1 (1.174) 173.1 (1.154)
216.3 (1.442) 222.5 (1.483)
184.5 (1.23) 184 (1.227)
174.5 (1.163) 172.6 (1.151)
170.5 (1.137) 167.1 (1.114)
“Valuesin parentheses are in US $; exchange rate during the elaboration of this study: 1 US $ = 150 dr.
upwards, and become steeper. This evolution of the iso-product curves occurs because of the increase in capital cost and in pumping cost. This also involves a removal of the points B = 6-B = 9 from the iso-cost line, while the points B = 3 and B = 4 approach the iso-cost line. This is clear even in the case of an annual operation of 9 months instead of 12 (Fig. 6 compared with Fig. 4). However, in this case the technical combination B = 5 is the optimal one as is the case for the full year load operation. In the case of an annual operation of 6 months (Fig. 7)) the point B =4 approaches the iso-cost line which sometimes ceases to be tangent to the isoproduct curve at the point B = 5. Finally, in the case of an annual operation of 3 months (Fig. 8), the point B= 5 is removed from the iso-cost line, which is now tangent to the iso-product curve at the point B = 4. The technical combination B = 5 ceases to be optimal when the annual operation period is reduced to below 6 months. In this case, the technical combination B=4 is the only optimal one. However, from Table II it is evident that for 9 and 12 months operation the average total desalination cost at 4 effects is only 1.1-3.6 dr/m3 higher than the corresponding cost at 5 effects. On the contrary, in all cases except one
196
(occurring in the 6 months operation period), the average total desalination cost at 5 effects is significantly higher than the corresponding cost at 4 effects. Therefore in general, the technical combination at 4 effects can be preferred as more advantages for a plant destined to operate under partial load operation, based on the fact that it is optimal in all cases except one, when operation is less than or equal to 6 months annually, while it is nearly optimal in the cases of operation of 9 or 12 months annually. Average total cost as function of the utilization factor of the plant on annual basis (short- term average cost)
The costs obtained by the technical combinations at 4 and 5 effects under different operation loads, for a production of more than 340 m3/d of desalted water can be modeled by a polynomial least squares expression (R2 N 1) for the short-term average cost (STAC) as follows: STAC=c,,~(a+j&~+0.02178v~-0.00073v~) where cl2 is the average total cost obtained in full yearly load operation, v the number of months of annual operation and a! and p are constants taking the values: cy= 1.624, /?= - 0.169 for 4 effects, o!= 1.821, p= - 0.225 for 5 effects. The short-term average cost of desalination is a decreasing function of the productive duration per annum, with a decreasing slope, as shown in Fig. 9. In spite of the decreasing nature of the cost function, it would however be erroPd=212.5
-__-__-_--_-_Qd=340
m3/day
--_--_----
Qd=467.5
-_--_--_-
ad=595
--------04=722.5
160 I/I,,II,,,,,I,,I,,,,,II~,II,I,,I,,I O.OE+OOO 1 .OE+005
ANNUAL
m3/day
2.OE+005
PRODUCTION
m3/day mJ/day mJ/day
3.OE+005 m3/year
Fig. 9. Short-term average cost (STAC) for technical combination B =5 and for various plant sizes ae a function of the annual production.
197
neous to propose a 12-month operation period of a smaller unit accompanied by a large water storage to cover the demand in peak periods, because of the prohibitive cost of storage (50-150 dr/m3). Important therefore is the decreasing-slope property of the cost function which implies a minor increase in the cost for small reductions of the 12-month production period. It is feasible to reduce the duration of the operation significantly before reaching a level of average total cost equivalent to the sum of production and storage costs of the alternative solution. CONCLUSIONS
(1) The long-term average total cost of low-temperature desalination under full load annual operation of the plant is determined as a function depending on the capacity of the plant. Under the values adopted for the interest rate (4.96% real interest rate) and for the price of heating (0.75 dr/kWh,,), the average total cost is found to vary between a maximum value of 316.8 dr/m3 (2.112 $/m3) an d a minimum value of 167.1 dr/m3 (1.114 $/m”). (2 ) The decrease in .the annual period of the plant operation entails a strong increase in the average production cost. The relation between the average cost increase and the annual decrease in the period of operation, observed for different plants having the same number of effects but different production capacities and for the two optimal numbers of effects, leads to the formulation of the corresponding function, here called the short-term average cost function. (3) The search for the optimal number of effects, by examination of the properties of the iso-product and iso-cost curves, has shown the following: (a) In the case of full annual load operation, the optimal number of effects is 5 for a production level below 850 m3/d. This optimal solution is remarkably stable under changes in the system of prices, even if technical solutions of one effect more or less than the optimal number (i.e. 4 or 6 effects instead of 5) are under initial prices nearly optimal as it is shown by cost analysis. (b) In the case of partial annual load operation of more than about 6 months, the same optimum solution (5 effects) applies. However, in the case of partial annual load operation of less than about 6 months, the optimum number is 4 effects rather than 5. This case is relevant to most Greek islands, where the population increases drastically (sometimes ten fold) during the summer months due to the tourist surge. (4) As a general observation concerning Greece we note that the cost of low temperature desalination is never prohibitive. Even in the case of annual operation of 3 months only, to cover strong seasonal needs, the average production cost is between 250.3 and 216.1 dr/m3, for a production rate between 340 and 850 m3/d, respectively (under the assumption adopted for the source and price of heating). These values are impressive in view of the price of the water
198
transported to the Greek islands (sometimes 2500 dr/m3) or even to some arid zones of the littoral country (1200 dr/m3). REFERENCES 1 E.O. Heady and J.L. Dillon, Agricultural Production Functions, Iowa State University Press, Ames, IA, 1961. 2 L.T. Fan, C.Y. Cheng, C.L. Hwang, L.E. Erickson and K.D. Kiang, Analysis and optimization of a multieffect multistage flash distillation system - Part I. Process analysis - Part II. Optimization, Desalination, 4 (1968) 336-388. 3 C.E. Ferguson, Microeconomic Theory, Irwin Homewood, Illinois, IL, 3rd ed., 1972. 4 B. Franquelin, F. Murat and C. Tern&et, Application of the multi-effect process at high temperature for large seawater desalination plants, Desalination, 45 (1983) 81-92. 5 N. Koumoutsos and I. Makatsoris, Sea water desalination by using surface geothermal fields Examination of the application to the Milos island, Proc. Second Greek Congress on Renewal sources of energy, November 1985, pp. 603-609 (translated from Greek). 6 K.S. Spiegler and A.D.K. Laird, Principles of desalination, Academic Press, New York, NY, 2nd ed., 1980. 7 H. Tabor, Solar ponds as heat source for low-temperature distillation plants, Desalination, 17 (1975) 289-302. 8 W.B. Tleimat, Optimal water cost from solar-powered multi-effect distillation dual purpose plant, Desalination, 44 (1983) 153-165. 9 J. Wei, T.W.F. Russel and M.W. Swartzlander, The structure of the chemical processing industries, Function and Economics, McGraw-Hill, New York, NY, 1979. NOTE
All the evaporators are of similar dimensions. Each effect consists of more than one evaporator, their number depending mainly on the required heattransfer surface. According to the model developed, a representative evaporator has a shell made of stainless steel, a height of 272 cm, a diameter of 156 cm, a shell thickness of 2 mm, and a cover thickness of 20 mm. Heat exchangers are of the serpentine type in parallel, and they are made of Cu-Ni alloy 90/10. The total heat-transfer surface of the feed heater is 12.305 m2 and the total heat-transfer surface of the heat exchanger of the boiling section is 2.43 m2. For a plant whose average daily production is 476 m3, the optimal number of effects if 5, while the first effect consists of 21 evaporators and each of the other 4 effects consists of 13 evaporators of the above basic specifications. The cost of each evaporator was estimated to be 698,000 dr. APPENDIX
Sensitivity of the optimum under price changes
A given sum of money (ET) destined to cover the cost of the technical factors in the distillation process can be distributed between the total cost of evapo-
199
rators plus maintenance plus pumping (C,) and (or) the total cost of heat (C!,), according to the alternative combinations represented by the appropriate iso-cost line. ET=Cy+Ch=Qy+‘,,+Qh.P~
(1)
where Q, is the quantities of the assortment (of factors) y, Qh is the quantity of the factor h, P,, is the price of the assortment y during a certain period, and Ph is the price of factor h during a certain period. If Qh= 0 then
ET=EyT=QyT*Py
(2)
Similarly, if Q,,= 0 then (3)
ET =&T =QhT.Ph
Eqns. (2) and (3) determine the end points of the iso-cost line on the C, and Ch axes, respectively. This is a straight line with slope &/DC,
=&r/&T
= (Q,WPy)/(QhT’Ph)
Further, from Eqns. (2) and (3 ) , (4)
&y-r*PY=&wr%
or Qyr/Q,T = phlpy
(4a)
which is the well-known principle of economic analysis that the ratio of technical substitution of the factors y and h (expressed in physical terms) is equal to the inverse ratio of the respective prices (see Ref. 3, paragraph 6.3). Similarly, for new prices PS, and Ph we have (5)
Q;r.Py.(P~IP,,)=Qh~.Ph*(PhIPh)
or (Q;VP,)/(QhT%)=
(ph/p,)/(p;/&)
(5a)
that is to say that if the costs are calculated as the quantities bought with the modified prices multiplied by the old prices P, and Pi.,,then the slope of the corresponding iso-cost line is equal to the inverse ratio of the new process expressed in units of the old prices. Consequently, this ratio expresses also the ratio of technical substitution of C, by Ch. How much does a modification of the price system influence the optimal technical procedure B=5? For example, which modification of the prices of the assortment y and the factor h would make the technical combination B = 6 equivalent to B=5? And which modification of the same prices would make B = 4 equivalent to B = 5?
200
In order for the technical procedure B = 6 to be equivalent to that of B = 5, the iso-cost line corresponding to the modified prices must be tangent to the iso-product curve at the points B = 6 and B = 5 simultaneously, while the isocost line corresponding to the initial prices must be tangent to the point B = 5. The new iso-cost line has the inclination (Qb *P,) / ( QhT *Ph) which is also expressed by the ratio of technical substitution of the assortment C, by the factor Ch:
(Q~.Py)I(QI~.Ph)=(Cy6-Cys)/(Ch5-Chs)
(6)
where C,, is the quantity of C,, with technical combination B = 6 prices), C,, is the quantity of C,, with technical combination B = 5 prices), C,, is the quantity of C, with technical combination B = 5 prices), and C,, is the quantity of Ch with technical combination initial prices). From Eqns. (5a) and (6) it can be written that
(in initial (in initial (in initial B =6 (in
- Cl&-)= uwJul(q$J
(7)
Thus, the ratio of the new prices to the initial prices required to make B = 6 equivalent to B = 5 can be found from the rate of technical substitution between C, and Ch when passing from B = 5 to 23= 6. This is shown in Table A.1 for the successive production levels. Similarly, the ratio of the new prices to the initial prices required to make B =4 equivalent to B = 5 is found in Table A.1 as the corresponding rate of technical substitution (C,, - C,,) / ( Ch4- C,,) . TABLE A.1 Rate of technical substitution of C, by C,,, and the required ratio of the new prices to the initial ones in order for the technical combinations B = 5 and B = 6 or B = 4 to become equivalent
212.5 340 467.5 595 722.5 850
Passing from B = 5 to B = 6
Passing from B = 5 to B = 4
1.229 1.122 1.573 1.555 1.382 1.334
0.791 0.811 0.782 0.763 0.762 0.760
0.814 0.891 0.636 0.643 0.724 0.750
1.264 1.233 1.279 1.311 1.312 1.316
201
Applications
If the real (inflation adjusted) interest rate is doubled, than, concerning the entire investment (e.g. 9.92% instead of the value 4.96% considered for the derivation of present results), the cost of product is increasing by 8.07 dr/m3 which represents a 13.8% increase in the value of C,,. This increase does not influence the optimal technical solution B =5 (see Table A.1, 5th column). Similarly, when the real interest rate is tripled (14.88%) the value of C,, is increasing by 27.61% and the optimal technical solution B=5 is influenced only in two cases, viz. when Qd-- 212.5 m3/d or when Qd= 340 m3/d, in which cases the technical combination B = 4 becomes optimal again (see Table A.1, 5th column). Inversely, when the real interest rate becomes zero, the value of C, is decreasing by 13.8% and the optimal technical solution B = 5 is not influenced at all (see Table AI, 3rd column). The use of oil instead of solar pond as primary heat source, has as result a quadruplication of the cost of kWh,, (3.56 dr/kWhth including taxes, instead of 0.75 dr/kWh,,) and leads to the search for the optimal solution far away from the region around B = 5, towards the region B = 9.
177 Printed in The Netherlands
Economic Aspects of Low-Temperature MultiEffect Desalination Plants Y.G. CAOURIS, E.T. KANTSOS and N.G. ZAGOURAS* University of Patras, Mechanical Engineering Department, 265 00 Patras (Greece), Telex: 312447 UNPA GR (Received May 20,1987; in revised form August 30,1988) SUMMARY
The technical parameters of multi-effect low-temperature distillation plants are determined and the production cost of every input is calculated separately for the successive levels of production under various loads of annual operation, by using a computational model devised for this purpose. The study of the results obtained leads to the development of the respective production map diagrams. Observation of the iso-cost and iso-product curves in these diagrams allows the determination of the optimal number of effects for each combination of production level and annual operation load. The sensitivity of the optimal number of effects relatively to the price system is also examined. Further, the average total cost is formulated as a function of the size of the plant under full load operation. This cost is also formulated as a function of the annual operation load for each plant size. The present work is obviously relevant to the design of desalination plants on Greek islands and other arid zones of the littoral countryside. Keywords: desalination optimization, solar low-temperature multi-effect desalination, economics.
SYMBOLS
AC
-
B
-
Ch
-
c them
-
c them
-
c me”
-
C me”
-
c heat
-
Cheat
-
Average unit cost of the desalination process Number of effects Daily heating cost Annual cost of chemical treatment Average unit cost of chemical treatment Annual cost of evaporators maintenance Average unit cost of evaporators maintenance Annual cost of heating Average unit cost of heating
*In alphabetical order. OOll-9164/89/$03.50
0 1989 Elsevier Science Publishers B.V.
(h/m3)
178
CPump CPumP
-
C wages
-
C wages
-
CW
-
c
-
csi
-
C 0”” & (BE)
-
4
(BE)
-
& (UE)
-
4
-
(UE)
EL
-
ET ei
-
GY
-
G’i
-
g Hi h
-
Ji
-
Li (BE) Li (UE) kl
-
M NW ipvi (UE)
-
AP,t
-
Annual cost of pumping requirements Average unit cost of pumping requirements Annual cost of labour Average unit cost of labour Daily wage, other fees and social contributions of employers Specific heat of pure water Specific heat of saline water of the ith effect Specific heat of seawater Temperature drop between successive effects External diameter of boiling section of the ith effect Internal diameter of boiling section of the ith effect External diameter of condenser-type feed heater of the ith effect Internal diameter of condenser-type feed heater of the ith effect Installed electrical power for the pumps operation Price of heating energy Quantity of pure water obtained by condensation in the condenser-type feed heater of the ith effect Specific enthalpy of water vapour at the jth effect Specific enthalpy of liquid phase at the ith effect Acceleration of gravity Latent heat of vaporization at the ith effect Plant’s altitude Local pressure loss coefficient Length of boiling section of the jth effect Length of feed heater section of the ith effect Mean length of a heat-exchanger tube between two successive bends Saline water flow rate through feed heaters Total number of evaporators Lifetime of evaporators Pressure drop through the feed heater of an evaporator of the ith effect Inlet-outlet static pressure difference
(dr) (h/m3) (h) (dr/m31 (dr)
&J/kg K) &J/kg K) (kJ/kg K) WI (4 (ml b) (m) (kw) (dr/kWhh)
(t/d) &J/k) &J/k) (m/s”) &J/k)
b-d
(t/d) (Y)
(Pa) (Pa)
179
APvi(BE) -
PET” S : Q" 4i
-
2
-
SP
-
STAC
-
ri
-
W x
-
Yi
-
4 u2
Subscripts h Y
-
Pressure drop through boiling section of an evaporator of the ith effect Capital cost of each evaporator Price of electrical energy Effective annual production Effective daily production Nominal production capacity Quantity of saline water evaporated at the ith effect Nominal interest rate Inflation rate Average salary Required rate of thermal energy for the plant’s operation Short term average cost of desalination process Duration of the period of operation Temperature of the available hot water from the heat source Temperature of flashing brine at the ith effect Total heat exchange coefficient of condensing-type feed heaters Total heat exchange coefficient of boiling section Velocity of distillate Velocity of saline water Flowrate of recycled hot water (heat carrier) Ratio of the capital and maintenance cost of the pumps, as a proportion of their operating cost, supposing a full year operation Number of parallel tubes forming a plane of the i-th heat exchanger
Heat as economic input Evaporators plus maintenance plus pump as economic input
(Pa)
(h) (WkWh,)
(m3/y) b3/d) (m3/d) (t/d)
(dr/employer, month) (kwt,)
(h/m3) (h/y) (K) (K)
(W/m2K) (W/m2K)
(dr) (dr)
Greek symbols E - Inflation adjusted interest rate [ t = (ri- rj ) / (l+rj)l
Ii
,uandv
-
-
Friction factor at the ith effect Duration of yearly operation
(months)
180
P @
-
Water density Coefficient of utilization
(kg/m3)
INTRODUCTION
A computational model has been developed for the calculation of the technical and economic factors, engaged in the process of multi-effect low-temperature desalination, as a function of the number of effects, B, and the production capacity, Q. The most important technical factors are the required thermal energy SP, the number of the effects and the number of evaporators in each effect, the heat-transfer surface area, the pumping requirements, EL, and the seawater feed mass flow rate. Besides, the most important economic factors are the capital cost of the evaporators and their maintenance, CeVapand C,,,, the cost of thermal energy Cheat, the pumping cost Cpump,the cost of chemicals C them and labour Cwages. The above technical factors are not influenced by the duration of the plant operation and are determined only by the production capacity. On the contrary, the economic factors are strongly influenced by the duration of the plant operation. Thus the economic analysis requires the consideration of different loads of the plant operation on annual basis (full load and different partial loads ) . Substitutional relationships are observed between the heat cost and the cost of all other technical inputs. These relationships are revealed by considering different numbers of effects in each level of production capacity, while keeping the load of annual operation constant. This can be described for each level of annual operation load by a production function diagram from which we can derive the optimal number of effects. Production functions with substitutional factors of production are often used in applied economics (e.g., [ 1,9] ). On the other hand, observation of the production costs for plants of different capacity operating under full load leads to the formulation of a function expressing the average cost in relation to the size of the plant (function of the long-term average cost under full load operation). Finally, observation of the production costs of the plants with varying load during the year allows the formulation of another cost function expressing the average cost in relation to the annual operation load with given capacity of plant (short-term average cost). In this paper the process analysis is shortly presented, in order to support the economic data. We do not present the technical results. The work is divided in three sections: (i) process analysis; (ii) modelling; and (iii) treatment of the economic results. The costs are computed on the basis of the prices observed in the Greek market at the end of 1985 and the beginning of 1986 (in drachmas). During this period the exchange rate of U.S.$/dr was l/150.
181
Note that combined technical-economic zation of water production [ 2,5,8].
models are useful in cost minimi-
PROCESS ANALYSIS
Plant description Fig. 1 illustrates the arrangement of a desalination plant of N effects, as considered for the present analysis. Before entering the last effect, seawater is preheated in condenser-type feed heaters mounted in the upper part of the evaporators and by using part of the generated vapour as heating medium. Prior to that, the seawater is preheated in a condenser in which the vapour from the last effect is condensed. The external heat carrier is water at an approximate temperature of 70’ C. This heat is produced by solar energy and is transferred to the plant via heat exchangers at the bottom of the first effect evaporators. All the evaporators are of similar dimensions and their number in each effect depends on the required heat-transfer surface (see Note). The energy balance In order to facilitate the analysis and estimation of the construction cost, the following assumptions have been made: (a) Equal temperature drop between successive effects, i.e.
(b) Constant specific heats for the feed distillate and brine. (c) The salt concentration of flashing brine must not exceed the value of 7% pwt (part per weight ) . (d) No heat loss occurs between the evaporators and the surroundings.
Fig. 1. Multi-effect desalination plant arrangement.
182
After these assumptions, the governing equations can be written: Bottom heat exchangers (boiling section of the evaporator): - (M-q,)C,lD+q,H1
(for the lSteffect)
- WCD=O
Upper heat exchangers (condenser-type feed heaters): ci(G’i+l -G;)
+MC,D=O
(Ll,.....,
N)
Apart from the unknowns ci and qi, the quantities M and W also remain undetermined. That is, a system of 2N equations has to be solved with 2N+ 2 unknowns. Yet it is evident that O
under the above constraints. Heat transfer surface and the evaporators’size The size and the number of evaporators in each effect are dependent on the appropriate size of heat exchangers required in order to achieve the necessary amounts of ei and qi. After a number of calculations it has been shown that it is preferable to have a number of parallel evaporators in each effect instead of a single big one for each effect. As an upper limit for the size of each evaporator has been considered a height of 3 m with a diameter approximately one half of the height. For heat exchangers we selected the type of serpentine shape in parallel series. They are made from an Cu-Ni alloy. The total heat transfer coefficient for the condensation heat exchangers (feed heaters) is considered to be UoI= 3500 W/m” K while for the bottom heat exchanger a value of UoZ= 2500 W/m2 K is considered. As optimum velocities (in order to obtain the values of U,,, and U,,,) we take 2.5 m/s for the feed and 1 m/s for the distillate. The length of the feed heater heat exchanger of an evaporator of the itheffect is Li (UE) =MC,,D/
[ Uo, 0.5 Dx d,i (UE)] ~8.812~
10m3M/do, (UE)
where M is the flow rate of brine through the feed heater heat exchanger of an evaporator of the ith effect (t/d). The internal tube diameter can be derived by the flow rats equation
183
Therefore dii(UE)=2.43~10-~JM The length of the boiling section feed exchanger of the evaporator of the first effect is L,(BE)=WCD/[U,,D~~,,(BE)]=~.~~~X~O-~
W/d,,(BE)
where W is the flow rate of hot water (heat carrier) from heat source through the heat exchanger (t/d). The internal tube diameter dii (BE) is di, (BE) = 2.43 x 10m3Jw The length of the boiling section of the ith effect is Li(BE)=((qi_l--i_1)Hi/[D [D&i
=[(Qi-I-
(BE)
1.6~d,i(BE)
u,,]}+{(qi_,-eei_,)CD/
&I}
ei_l)/d,,i (BE)]
[ (6.578x10S4Hi/D)
+6.168~10-~]
where Hi (kJ/kg) is the latent heat of vaporization at the ith effect. dii(BE) can be derived from the flow rate equation
Therefore dii (BE) =3.838~10-~
dz
The size of the evaporators is limited by the size of upper and bottom heat exchangers, assuming that the distance between them is 0.7 m. If the size of the evaporator exceeds the predetermined upper limit, then the evaporator is splitted into two or more equal parts, operating in parallel. Inox steel is chosen as the construction material while the thickness and weldings are computed according to DIN standards and depend on the pressure under which each effect operates. Pumping requirements (i) Pumping of brine through the feed heaters The pressure drop through the feed heater of an evaporator of the ith effect is
184
dP”i (UE) =
C ji +A,L, (UE)/dii (UE) {0.28($
1
u$P/~=
+dyi) +li[Li(UE)/dii
(UE)]}3125
whereji is the local factor of losses and iii is the frictional factor. In the case of turbulent flow, the latter can be obtained from the equation l/J&
-2 log[2.5/(Re
fi)+K/(3.71
d,)]
yf +dyi expresses the number of bends of the serpentine-type tube and yi is the number of parallel tubes of each plane. In general, there must also exist a number yi of parallel planes, in order for the length Li (UE ) ( = L) to be equal to l,yf. The distance between two successive parallel tubes is equal to 2doi (UE) ( = 2d,). However, the length L cannot be in fact exactly equal to y: 1,) because commercially available tubes are of certain diameters. Consequently, L must be equal to y:Z, +dyi. According to the geometry of the heat exchanger, the values yi and dyi are obtained from the following equations: 3d,ya-1.287
d:yi-L=O
dyi= (L-3doya
+1.287 d~y~)/(1+0.7123
do)
where yi is the greatest integer less than the real positive solution. The heat H can be expressed as
H=
F
i=l
[dPvi(UE)l
/@I
(ii) Pumping of brine through the bottom heat exchangers The heat H can be expressed as
where dP”i (BE) is the pressure drop through the bottom heat exchangers of the ith effect and AP,, is the difference between static pressures of inlet and outlet, taken equal to 7 x lo5 Pa, a realistic value when the site of consumption is near the production plant. The flow through the bottom heat exchangers is separated into two parts, one for the vapor phase and the other for the liquid. SO the length L,(BE) is calculated as Li(BE) = Liv + Lif where Liv= [ (qi-1-ei_l)/doi(BE)]6.578X10-4 L,=6*168X
10d3 (qi_1 -ei_l
)/d,i (BE)
The pressure drop across the part Liv is
(Hi/D)
185
dP”i (v) = [0*28(yP +&i) +A,& (BE)/& (BE)]
(/Q&/2)Liv/Li(BEI
the pressure drop across the part Lif is dP”i(f)=[0.28(y~+dyi)+~iLi(BE)/dii(BE)]FiOOL,f/Li(BE) and the total pressure drop is dP”i (BE) =dP,i (f) +dPvi
(V)
MODELLING
A computer program has been developed which simulates the operation of the desalination plant and calculates both the individual and total costs, related to the level of production and the number of effects B (3
e/11-
(l--E)-nl
(ii) The annual cost for the maintenance of the evaporators, (estimated as 3% of the owning value of the evaporators) [5] C,,, =P,,N,,O.O3 defined as the sum of of the cost of electrical (iii) The pumping cost CpUmp power for the pump operation and the capital plus maintenance cost of pumps. This is inserted in the model by the expression Cp,,=EL~~+P,[l+
(x/100)(12/~)]
where x/100 is the percentage coefficient which expresses the capital and maintenance cost of pumps, x = 13.45 - (EL/14.57), for 100 m3/day < & < 1000
186
m3/day and x = 0.003 for Q> 1000 m3/day (as a proportion of the functioning cost, supposing a full year operation) and p is the duration of operation expressed in months. (iv) The annual cost of heating requirements expressed as c heat= SP- T-ET The cost of heating is considered constant and equal to 0.75 dr/kWhth, and is independent of any seasonal fluctuation in potable water demand, because of the possibility of using the excess heat for other purposes. (v) The annual total cost of chemical treatment. From the fact that the plant operates at low temperature levels the treatment must be made with phosphates [ 61. The cost per m3 is estimated to 11.1 dr, thus the total cost is expressed as Cthem=Q*11.1.30 (vi) The labour total cost. An assumption has been made that for a nominal production capacity of less than 200 m3/d, four employees are necessary for the plant’s operation. One additional employee is necessary for each additional 200 m3, up to a total of 1000 m3/d. For a nominal production capacity greater than 1000 m3/d, one additional employee is necessary for each additional 1000 m3. Therefore the annual labour cost is considered to depend on the level of annual production and is expressed as c wages = Sp4
if Q < 200 m3/d
C,,,,=S~u(4+INT[(Q-200)/20~]+1}if200m3/d~Q~1OOOm3/d C,,~=S~{8+1NT[(&-1000)/1000]+1)~~Q~>1000m~/d The annual total cost is considered as the sum of all the preceding particular costs, without taking account of the land cost. This is a typical characteristic of the waterless islands of Greece, which we had in mind from the beginning of this work. On these islands, there are plenty of arid communal lands unsuitable for any other use. This total cost concerns only the production process and the pumping of potable water to a height of approximately 70 m and does not include the distribution cost. The average unit cost AC (per m3 of distillate) is calculated by dividing the particular terms of annual cost by the annual production Q,,= &*~*@30.
187
The results discussed in the next sections are obtained from the following standard values of parameters: Pev= 698,000 dr T= 8,760 h, n= 15 T= 6,570 h, n= 18.333 T = 4,380 h, n= 21.666 T=2,190 h, n=25 t= (ri-rj)/(l+rj)
when when when when
the plant the plant the plant the plant
operates operates operates operates
12 months yearly 9 months yearly 6 months yearly 3 months yearly
= (0.27-0.21)/(1+0.21)=0.0496
$=0.85, P1 =8 dr/kWh, ETz0.75
dr/kWh,,[7],
and S=125,000 dr/
employer, month.
TREATMENT OF THE ECONOMIC RESULTS
Maximum load operation In the first part of this study, the desalination plant is considered to operate under maximum load continuously, all over the year, except for the necessary maintenance periods which are supposed to be 15% of the total time. Under this condition, data from the model are obtained and evaluated. Classification of factors ofproduction and inputs involved: substitutional and complementary factors The factors of production and inputs involved in the process of desalination costing can be classified into three classes, according to their variation with the production level or to the technical procedures. (i) The first class is composed of the factors of production or the inputs of which the quantities per product unit of the desalination plant (per m3 desalted water) are dependent exclusively (or almost exclusively) on the number of effects and not on the level of production. When the capacity of the plant increases without any change in the number of effects, the proportion of these factors of production or inputs per product unit is not modified. In other words, the total quantities of these factors increase proportionally with the quantity of the product. On the contrary, when the number of effects is modified in any equal product level, the quantities of these factors or inputs per product unit is modified too. This class refers to the following factors or inputs: - the capital invested in evaporators; - the materials and labour spent on the maintenance of these evaporators;
188
- the capital invested in pumping machinery and the energy consumed for its operation; - the heat consumed for the evaporation in the first effect. The average unit cost of each of the first three factors (per m3 desalted water) increases generally as the number of effects increases, that is to say when passing from 3 effects to 4, or from 4 to 5, or from 8 effects to 9, as shown in Fig. 2. On the contrary, the average unit cost of heat (per m3 desalted water) decreases generally with increasing number of effects (see Fig. 2 ) . It is important to observe in Fig. 2 the curve representing the sum of the previous costs. The lowest point on this curve corresponds to the technical combination at B = 5. Indeed, the average cost of substitutional factors as well as the average total cost is minimum at 5 effects, as shown in Table I. However, the cost at 4 effects and those at 6 or 7 effects are only a little higher than the cost of the optimal combination, as shown in the same table. (ii) The second class corresponds to the labour cost. As expected, the labour cost per unit of product depends mainly on the level of production. Especially, the average cost of labour decreases when the capacity of production increases. From the model expression of the total annual cost of labour, the average cost of labour (per m3 desalted water) was derived as
1 3
4
5
6
7
8
0
9
Fig. 2. Average unit cost (dr) of substitutional factors per m3 of desalted water as a function of the number of effects B. Exchange rate during the elaboration of this study: 1 US $ = 150 dr.
189 TABLE I Average cost of multi-effect water desalination by alternative technical combinations and alternative production capacities (full annual load operation, the cost in drachmas) Technical combinations B=7
B=8
B=9
Nominal production capacity: 100 ma/d (Q,=85 m3/d) ICib 136 119.5 117 118.5 AC 335.4 319.1 316.8 318.3
121 320.5
125 323.6
129.5 329.3
Nominalproduction capacity: 250 m3/d (Qd=212.5 m3/d) CCi 132.4 117 114.2 115.7 AC 238 222.8 219.8 222.7
118.1 224.2
125.4 231.2
126.7 232.5
Nominal production capacity: 400 m3/d (Qd= 340 m3fd) XCi 133.8 118.7 116.2 117 AC 215.7 200.8 198.2 200
119.2 201.5
125 207.2
126.2 208.4
Nominal production capacity: 550 m3/d (Q,=467.5 ZCi 132.5 116.4 114 AC 195.2 179.8 176.7
m3/d) 117.6 181
119.4 182.3
124.2 186.9
125.5 188.3
Nominalproduction capacity: 700 m3fd (Q,=595 m3/d) CCi 133 117.7 114.3 117.8 AC 191.4 176.1 173.1 176.9
119.5 178
123.5 182
127.5 185.9
Nominal production capacity: 850 m3/d (Qd= 722.5 m3/d) I&i 132.4 116.7 113.5 115.9 AC 188 172.4 169.1 172.2
119.3 175.2
122.8 178.5
126 181.7
Nominal production capacity: 1000 m3/d (Q,=850 m3/d) XCi 132.4 116.7 113.6 115.6 AC 186.1 170.5 167.1 169.9
118.8 172.7
122 175.7
124.8 178.2
B=3
B=4
B=5
B=6
“The cost was evaluated for small production capacities, e.g. Qd=85 m3/d, without any restriction on the height of the evaporators. For higher production capacities, e.g. Qd> 200 m3/d, the cost was computed considering evaporators with a height of 3 m. bi = evap, mev, pump and heat.
The graphical presentation of the average unit cost of labour is given in Fig. 3. (iii) Finally, we have the chemical material, of which the quantities per product unit are the same, irrespective of the production level and the number of effects. Therefore, the average cost of the chemical treatment can be considered the same for every level of production and number of effects: cthem
=
11.1 dr/m3 of distillate
Fig. 3. Average unit cost (dr) of complementary factors per m3 of desalted water as a function of the effective daily production Qd. C”i
50
25 (drx103)
l5
ch
Fig. 4. Iso-product curves in the case of full annual operation. C,=daily cost (drx 103) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr x 103) of heating.
191
Production map diagram
The modification of the technical process of desalination, in the sense of increasing the number of effects, implies modification of composition of the average unit cost of the desalination only for the following inputs and factors of production: - capital cost of evaporators; - inputs and labour for maintenance of the evaporators; - capital cost of the pumping system and energy consumed for its operation; - heat for the desalination process. In particular, the increase in the number of effects, on the same level of production capacity in full operation, implies an increase in the first three elements of the average unit cost of desalted water (per m3) and, inversely, a decrease in the last element; in other words, it implies partial substitution of the fourth element (heat) by the first three elements. Fig. 4 gives the graphical presentation of the preceding combination of C, and C,.,for six successive levels of production, between 212.5 m3/d and 850 m”/ d. The vertical axis C, represents the daily cost of inputs and factors of production: evaporators, maintenance, and pumping; the horizontal axis C, represents the cost of heat per day; the iso-product curves Qd= 225 m3/d to Qd= 850 m3/d represent the successive levels of product, obtained by alternative combinations of C, and C,. The points B= 3 to B=9 on each iso-product curve represent the alternative technical procedures, by which the level of production represented by the corresponding iso-product curve can be obtained. Each technical procedure B = 3-9 is characterized by the number of effects. Using this diagram, the substitutional relationships between C, and C, in different levels of production is clearly shown. It is clear that this substitution is due to the substitution of one technical procedure by another. For example, passing from the technical procedure B = 5 to B = 6, for a certain level of production, implies the substitution of the input h by the group y. The measure of this substitution is given by the rate of technical substitution, i.e. the ratio between the increase in C,,, (C, - C,,), and the decrease in Ch, (C,, - C,,), while maintaining the same production. Graphical representation of the rate of technical substitution when passing from the technical procedure B = 5 to the B = 6 in a certain level of production is given by the inclination of the line which is simultaneously tangent to the corresponding iso-product curves at the points B = 5 and B = 6. Numerical values of the rate of technical substitution (passing from B = 5 to B = 6 and from B = 5 to B = 4 ) in different levels of production are given in Table A.1, in the Appendix. Optimal number of effects
For each level of production the optimal technical procedure is defined by such a point on the corresponding iso-product curve, where an iso-cost line (geometrical space of distribution of a quantity of expenses) is tangent to this
192
iso-product curve. Every other distribution of the same expense leads to an inferior theoretical iso-product curve. Note that in every iso-product curve the point where an iso-cost line is tangent to it represents the technical procedure B = 5. Thus, the same result derived elsewhere by observing the curve of the average cost of substitutional factors is now derived by observing the iso-cost and iso-product curves. This optimum is defined here according to the prices of the inputs and factors of production of the group y and of the input h, valid in the Greek market at the end of 1985. The answer to the question how much a modification of the price system influences the optimal technical procedure B = 5 is founded on the rate of technical substitution, as given in the Appendix. In general, a great change in the relative prices of the production factors is necessary in order to achieve a change in the optimal number of effects. For example, if the real (inflation adjusted) interest rate is doubled concerning the entire investment (e.g. 9.92% instead of the value 4.96% considered for the derivation of the present results), this increase does not influence the optimal technical solution B = 5 (see Appendix). Similarly, when the real interest rate is tripled (14.88% ) , the optimal technical solution B = 5 is influenced only in two cases, viz. when Qd= 212.5 m3/d or Qd= 340 m3/d, in which cases the technical combination B = 4 becomes optimal (see Appendix). Inversely, when the real interest rate becomes zero, the optimal technical solution B = 5 is not influenced at all (see Appendix). Average total cost as function of the size of the plant (long-term average total cost) The principle of decreasing cost is valid in water desalination, in so far as increasing capacity of production results in decreasing average total cost. For increasing levels of production capacity up to 212.5 m3/d, the observed decrease in the average total cost is due to the decrease in the cost of technical inputs (since the increased production level allows improved organization of the technical terms of production) as well as to the decrease in the cost of labour. Above this level of production, the decrease in the total average cost is due to the decrease in the labour cost exclusively or almost exclusively. On the basis of the data concerning the average total cost obtained by the optimal technical combination in each level of production, this cost may be written approximately as a function of the size of production capacity AC= (118.1-v (1)
QP)+ll.l+{4(C,/Q)+ (2)
[INT(Q-200)/200+1]C,/Q} (3)
We take C, = 3778 dr as mean daily wage plus social contributions. We also take v=O.1764 (constant) and p=O.496 (constant). Given these values for the constants v and p, the first part (1) of the above expression describes the impact of the size of production capacity on the average cost of
193 AC
A
320-
240.
160.
80.
I I
I I
‘3l-v
I I I I
‘%mt <-
I I
‘%mp 200
400
600
800
Qd (mYday>
Fig. 5. Long-term average unit cost (dr) under full load operation per m3 of desalted water as a function of the effective daily production Qti
technical inputs except chemicals (with a standard deviation equal to 0.90). The second part (2) of the expression is the average cost of chemicals (taken as constant ) . The third part (3) describes the impact of the size of production capacity on the average cost of labour. Partial annual load operation Desalination plants designed to cover seasonal water needs operate only for some periods during the year. In this case the following questions arise: - Which is the optimal number of effects, for each intended period of operation (for example, 3 or 6 or 9 months per year)? - How much does the average cost of desalination increase for each such case of limited operation? ho-product curves under different load Figs. 6-8 illustrate the position and form of the iso-product curves of different plants with a real production capacity of 850,595 and 340 m3/d of desalted water (considering only 85% of the nominal capacity, the remaining 15% being ineffective due to interruptions of production for maintenance), for an annual duration of production of 9 months (Fig. 6), 6 months (Fig. 7), and 3 months (Fig. 8). When comparing these figures, it can be seen that for decreasing annual duration of operation the iso-product curves move to the right and
Fig. 6. Iso-product curves in the case of 9 months operation annuaIly. C,=d&y cost (drx103) of capital invested in evaporators and pumping machineries and their maintenance; C, = daily cost (drx 103) of heating. Fig. 7. Iso-product curves in the case of 6 months operation annually. C, =daily cost (dr X lo3 ) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr x 103) of heating.
Fig. 8. Iso-product curves in the case of 3 months operation annually. C,=daily cost (drx 103) of capital invested in evaporators and pumping machineries and their maintenance; C,,= daily cost (dr X 103) of heating.
195 TABLE II Short-term average total cost in drachmas” per m3 of distilled water for alternative technical combinations Technical combination
Production capacity: 340 m3/d B=4 B=5
Production capacity: 595 m3/d B=4 B=5
Production capacity: 850 m3/d B=4 B=5
Duration of operation (months per year) 3
6
9
12
250.3 (1.669) 258.1 (1.721)
216 (1.44) 216.6 (1.444)
205.2 (1.368) 203.6 (1.357)
200.8 (1.339) 198.2 (1.321)
222.2 (1.481) 230.2 (1.535)
189.5 (1.263) 190.3 (1.269)
179.2 (1.195) 178.1 (1.187)
176.1 (1.174) 173.1 (1.154)
216.3 (1.442) 222.5 (1.483)
184.5 (1.23) 184 (1.227)
174.5 (1.163) 172.6 (1.151)
170.5 (1.137) 167.1 (1.114)
“Valuesin parentheses are in US $; exchange rate during the elaboration of this study: 1 US $ = 150 dr.
upwards, and become steeper. This evolution of the iso-product curves occurs because of the increase in capital cost and in pumping cost. This also involves a removal of the points B = 6-B = 9 from the iso-cost line, while the points B = 3 and B = 4 approach the iso-cost line. This is clear even in the case of an annual operation of 9 months instead of 12 (Fig. 6 compared with Fig. 4). However, in this case the technical combination B = 5 is the optimal one as is the case for the full year load operation. In the case of an annual operation of 6 months (Fig. 7)) the point B =4 approaches the iso-cost line which sometimes ceases to be tangent to the isoproduct curve at the point B = 5. Finally, in the case of an annual operation of 3 months (Fig. 8), the point B= 5 is removed from the iso-cost line, which is now tangent to the iso-product curve at the point B = 4. The technical combination B = 5 ceases to be optimal when the annual operation period is reduced to below 6 months. In this case, the technical combination B=4 is the only optimal one. However, from Table II it is evident that for 9 and 12 months operation the average total desalination cost at 4 effects is only 1.1-3.6 dr/m3 higher than the corresponding cost at 5 effects. On the contrary, in all cases except one
196
(occurring in the 6 months operation period), the average total desalination cost at 5 effects is significantly higher than the corresponding cost at 4 effects. Therefore in general, the technical combination at 4 effects can be preferred as more advantages for a plant destined to operate under partial load operation, based on the fact that it is optimal in all cases except one, when operation is less than or equal to 6 months annually, while it is nearly optimal in the cases of operation of 9 or 12 months annually. Average total cost as function of the utilization factor of the plant on annual basis (short- term average cost)
The costs obtained by the technical combinations at 4 and 5 effects under different operation loads, for a production of more than 340 m3/d of desalted water can be modeled by a polynomial least squares expression (R2 N 1) for the short-term average cost (STAC) as follows: STAC=c,,~(a+j&~+0.02178v~-0.00073v~) where cl2 is the average total cost obtained in full yearly load operation, v the number of months of annual operation and a! and p are constants taking the values: cy= 1.624, /?= - 0.169 for 4 effects, o!= 1.821, p= - 0.225 for 5 effects. The short-term average cost of desalination is a decreasing function of the productive duration per annum, with a decreasing slope, as shown in Fig. 9. In spite of the decreasing nature of the cost function, it would however be erroPd=212.5
-__-__-_--_-_Qd=340
m3/day
--_--_----
Qd=467.5
-_--_--_-
ad=595
--------04=722.5
160 I/I,,II,,,,,I,,I,,,,,II~,II,I,,I,,I O.OE+OOO 1 .OE+005
ANNUAL
m3/day
2.OE+005
PRODUCTION
m3/day mJ/day mJ/day
3.OE+005 m3/year
Fig. 9. Short-term average cost (STAC) for technical combination B =5 and for various plant sizes ae a function of the annual production.
197
neous to propose a 12-month operation period of a smaller unit accompanied by a large water storage to cover the demand in peak periods, because of the prohibitive cost of storage (50-150 dr/m3). Important therefore is the decreasing-slope property of the cost function which implies a minor increase in the cost for small reductions of the 12-month production period. It is feasible to reduce the duration of the operation significantly before reaching a level of average total cost equivalent to the sum of production and storage costs of the alternative solution. CONCLUSIONS
(1) The long-term average total cost of low-temperature desalination under full load annual operation of the plant is determined as a function depending on the capacity of the plant. Under the values adopted for the interest rate (4.96% real interest rate) and for the price of heating (0.75 dr/kWh,,), the average total cost is found to vary between a maximum value of 316.8 dr/m3 (2.112 $/m3) an d a minimum value of 167.1 dr/m3 (1.114 $/m”). (2 ) The decrease in .the annual period of the plant operation entails a strong increase in the average production cost. The relation between the average cost increase and the annual decrease in the period of operation, observed for different plants having the same number of effects but different production capacities and for the two optimal numbers of effects, leads to the formulation of the corresponding function, here called the short-term average cost function. (3) The search for the optimal number of effects, by examination of the properties of the iso-product and iso-cost curves, has shown the following: (a) In the case of full annual load operation, the optimal number of effects is 5 for a production level below 850 m3/d. This optimal solution is remarkably stable under changes in the system of prices, even if technical solutions of one effect more or less than the optimal number (i.e. 4 or 6 effects instead of 5) are under initial prices nearly optimal as it is shown by cost analysis. (b) In the case of partial annual load operation of more than about 6 months, the same optimum solution (5 effects) applies. However, in the case of partial annual load operation of less than about 6 months, the optimum number is 4 effects rather than 5. This case is relevant to most Greek islands, where the population increases drastically (sometimes ten fold) during the summer months due to the tourist surge. (4) As a general observation concerning Greece we note that the cost of low temperature desalination is never prohibitive. Even in the case of annual operation of 3 months only, to cover strong seasonal needs, the average production cost is between 250.3 and 216.1 dr/m3, for a production rate between 340 and 850 m3/d, respectively (under the assumption adopted for the source and price of heating). These values are impressive in view of the price of the water
198
transported to the Greek islands (sometimes 2500 dr/m3) or even to some arid zones of the littoral country (1200 dr/m3). REFERENCES 1 E.O. Heady and J.L. Dillon, Agricultural Production Functions, Iowa State University Press, Ames, IA, 1961. 2 L.T. Fan, C.Y. Cheng, C.L. Hwang, L.E. Erickson and K.D. Kiang, Analysis and optimization of a multieffect multistage flash distillation system - Part I. Process analysis - Part II. Optimization, Desalination, 4 (1968) 336-388. 3 C.E. Ferguson, Microeconomic Theory, Irwin Homewood, Illinois, IL, 3rd ed., 1972. 4 B. Franquelin, F. Murat and C. Tern&et, Application of the multi-effect process at high temperature for large seawater desalination plants, Desalination, 45 (1983) 81-92. 5 N. Koumoutsos and I. Makatsoris, Sea water desalination by using surface geothermal fields Examination of the application to the Milos island, Proc. Second Greek Congress on Renewal sources of energy, November 1985, pp. 603-609 (translated from Greek). 6 K.S. Spiegler and A.D.K. Laird, Principles of desalination, Academic Press, New York, NY, 2nd ed., 1980. 7 H. Tabor, Solar ponds as heat source for low-temperature distillation plants, Desalination, 17 (1975) 289-302. 8 W.B. Tleimat, Optimal water cost from solar-powered multi-effect distillation dual purpose plant, Desalination, 44 (1983) 153-165. 9 J. Wei, T.W.F. Russel and M.W. Swartzlander, The structure of the chemical processing industries, Function and Economics, McGraw-Hill, New York, NY, 1979. NOTE
All the evaporators are of similar dimensions. Each effect consists of more than one evaporator, their number depending mainly on the required heattransfer surface. According to the model developed, a representative evaporator has a shell made of stainless steel, a height of 272 cm, a diameter of 156 cm, a shell thickness of 2 mm, and a cover thickness of 20 mm. Heat exchangers are of the serpentine type in parallel, and they are made of Cu-Ni alloy 90/10. The total heat-transfer surface of the feed heater is 12.305 m2 and the total heat-transfer surface of the heat exchanger of the boiling section is 2.43 m2. For a plant whose average daily production is 476 m3, the optimal number of effects if 5, while the first effect consists of 21 evaporators and each of the other 4 effects consists of 13 evaporators of the above basic specifications. The cost of each evaporator was estimated to be 698,000 dr. APPENDIX
Sensitivity of the optimum under price changes
A given sum of money (ET) destined to cover the cost of the technical factors in the distillation process can be distributed between the total cost of evapo-
199
rators plus maintenance plus pumping (C,) and (or) the total cost of heat (C!,), according to the alternative combinations represented by the appropriate iso-cost line. ET=Cy+Ch=Qy+‘,,+Qh.P~
(1)
where Q, is the quantities of the assortment (of factors) y, Qh is the quantity of the factor h, P,, is the price of the assortment y during a certain period, and Ph is the price of factor h during a certain period. If Qh= 0 then
ET=EyT=QyT*Py
(2)
Similarly, if Q,,= 0 then (3)
ET =&T =QhT.Ph
Eqns. (2) and (3) determine the end points of the iso-cost line on the C, and Ch axes, respectively. This is a straight line with slope &/DC,
=&r/&T
= (Q,WPy)/(QhT’Ph)
Further, from Eqns. (2) and (3 ) , (4)
&y-r*PY=&wr%
or Qyr/Q,T = phlpy
(4a)
which is the well-known principle of economic analysis that the ratio of technical substitution of the factors y and h (expressed in physical terms) is equal to the inverse ratio of the respective prices (see Ref. 3, paragraph 6.3). Similarly, for new prices PS, and Ph we have (5)
Q;r.Py.(P~IP,,)=Qh~.Ph*(PhIPh)
or (Q;VP,)/(QhT%)=
(ph/p,)/(p;/&)
(5a)
that is to say that if the costs are calculated as the quantities bought with the modified prices multiplied by the old prices P, and Pi.,,then the slope of the corresponding iso-cost line is equal to the inverse ratio of the new process expressed in units of the old prices. Consequently, this ratio expresses also the ratio of technical substitution of C, by Ch. How much does a modification of the price system influence the optimal technical procedure B=5? For example, which modification of the prices of the assortment y and the factor h would make the technical combination B = 6 equivalent to B=5? And which modification of the same prices would make B = 4 equivalent to B = 5?
200
In order for the technical procedure B = 6 to be equivalent to that of B = 5, the iso-cost line corresponding to the modified prices must be tangent to the iso-product curve at the points B = 6 and B = 5 simultaneously, while the isocost line corresponding to the initial prices must be tangent to the point B = 5. The new iso-cost line has the inclination (Qb *P,) / ( QhT *Ph) which is also expressed by the ratio of technical substitution of the assortment C, by the factor Ch:
(Q~.Py)I(QI~.Ph)=(Cy6-Cys)/(Ch5-Chs)
(6)
where C,, is the quantity of C,, with technical combination B = 6 prices), C,, is the quantity of C,, with technical combination B = 5 prices), C,, is the quantity of C, with technical combination B = 5 prices), and C,, is the quantity of Ch with technical combination initial prices). From Eqns. (5a) and (6) it can be written that
(in initial (in initial (in initial B =6 (in
- Cl&-)= uwJul(q$J
(7)
Thus, the ratio of the new prices to the initial prices required to make B = 6 equivalent to B = 5 can be found from the rate of technical substitution between C, and Ch when passing from B = 5 to 23= 6. This is shown in Table A.1 for the successive production levels. Similarly, the ratio of the new prices to the initial prices required to make B =4 equivalent to B = 5 is found in Table A.1 as the corresponding rate of technical substitution (C,, - C,,) / ( Ch4- C,,) . TABLE A.1 Rate of technical substitution of C, by C,,, and the required ratio of the new prices to the initial ones in order for the technical combinations B = 5 and B = 6 or B = 4 to become equivalent
212.5 340 467.5 595 722.5 850
Passing from B = 5 to B = 6
Passing from B = 5 to B = 4
1.229 1.122 1.573 1.555 1.382 1.334
0.791 0.811 0.782 0.763 0.762 0.760
0.814 0.891 0.636 0.643 0.724 0.750
1.264 1.233 1.279 1.311 1.312 1.316
201
Applications
If the real (inflation adjusted) interest rate is doubled, than, concerning the entire investment (e.g. 9.92% instead of the value 4.96% considered for the derivation of present results), the cost of product is increasing by 8.07 dr/m3 which represents a 13.8% increase in the value of C,,. This increase does not influence the optimal technical solution B =5 (see Table A.1, 5th column). Similarly, when the real interest rate is tripled (14.88%) the value of C,, is increasing by 27.61% and the optimal technical solution B=5 is influenced only in two cases, viz. when Qd-- 212.5 m3/d or when Qd= 340 m3/d, in which cases the technical combination B = 4 becomes optimal again (see Table A.1, 5th column). Inversely, when the real interest rate becomes zero, the value of C, is decreasing by 13.8% and the optimal technical solution B = 5 is not influenced at all (see Table AI, 3rd column). The use of oil instead of solar pond as primary heat source, has as result a quadruplication of the cost of kWh,, (3.56 dr/kWhth including taxes, instead of 0.75 dr/kWh,,) and leads to the search for the optimal solution far away from the region around B = 5, towards the region B = 9.