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Optimisation of use of surplus night electricity in electrochemical processes
Ekcrrochimica
km,
1973 Vol.
18, pp. 897-906.
Pergamon Press. Printed in Great Britain
OPTIMISATION OF USE OF SURPLUS NIGHT ELECTRICITY IN ELECTROCHEMICAL PROCESSES Technisch-Chemisches
N. IBL and P. M. ROBERTSON Laboratorium, Eidg. Technische Hochschule, Zurich, Switzerland
(Received 7 March 1973) Abstract-Cheaper
electric energy is available during certain periods of the day, especially at night. Electrochemical plants could make use of the surplus electricity by applying a variable current density. An analysis of the problem is presented for an electrolysis operated at two different current densities. The optimum values of these current densities were calculated by balancing the investment against the energy (or fuel) costs. Dimensionless groups were introduced in order to reduce the large number of parameters and variables involved. As an extension of previous work the optimization was carried out by two methods: (a) by considering simultaneously the expenses for the electrolytic and the power plant; (b) by assuming that the energy is available at two different but fixed rates. The results depend on a number of parameters such as the specific investment cost for the electrolytic plant, the cell resistance etc. Their influence is shown and discussed in terms of the dimensionless groups used. Numerical values are presented for two examples: chlorine electrolysis and copper refining. In the first case the economically desirable ratio of the day and night current is very small, in the second case it is substantially larger. The optimization technique used is compared to the cash flow method. It is shown that the results obtained by the two procedures are the same.
NOMENCLATURE A a
a’ a” b
c d D E
El3 H
L
number of years during which investment is amortised initially invested capital per m2 electrode area, %/m* total investment cost, $ investment cost of Hg in chlorine electrolysis, $ quantity of electricity needed to make given amount of product in time I, C resistance of cell, 0 m* interest factor = 1 + S’ yearly rate of interest, % time number of time units contained in a year revenue from sales of product, %/yr
n
total electrode area, m’ specific investment cost (more generally, proportionality factor for cost which is proportional to electrode area and thus inversely proportional to current density), $/m’h part of factor a involving both interest and amortization, $/m*h part of factor a involving interest only, $/m2h cost of unit of electric energy, $/kWh specific investment cost for power plant, $jkWh fuel cost, $/kWh replacement cost of Hg in chlorine electrolysis, Slyr applied voltage, V voltage extrapolated to zero current, V cost of electricity to produce a fixed quantity of product, $/yr. current density, A/m2 cost of raw materiaIs, %/yr total cost, $ fixed cost of electrolytic plant, $ fuel cost, $ energy cost, $ investment cost for power plant, $ variable investment cost for electrolytic plant, $ power consumption by consumers other than electrochemical, kW 897
P P*
P
Q R
S S’ t V
W Subscripts period,
I and
2 refer to night
and
day time
respectively. 1.
The demand
INTRODUCTION
for electricity is larger during the day than at night. In general, the capacity of power plants is only partially utilised at night, causing an increase in the investment cost. The electrochemical industry, which is a large consumer of electricity, could in principle contribute to equalise the load of the power plants by operating at a variable current density, designed to counteract the variation in the electricity demand by the other consumers. It is then the investment cost of the electrochemical plant which increases because it is less utilised during the day time. The question is therefore whether it is better
N.
898
IBL AND
P. M. ROBERTSON
to have variable load for the electrochemical or for the power plant. The answer obviously depends on the relative magnitudes of the investment costs of the two types of plant. Typical values are (in s/kW of power used or produced): 200 $ for aluminium electrolysis* ; 200 % for chlorine electrolysis, 150 $ for a thermal and 300 $ for a nuclear power plant[l 1. They are very similar and it is thus of considerable interest to examine more closely from an economic viewpoint the desirability of varying the load of an electrochemical plant. The purpose of this paper is to calculate the optimum current density for an electroIysis, taking into account the surplus electricity available during certain periods, especially at night. Obviously it may turn out that the optimum values thus calculated cannot be realised in practice because of technical reasons, which would lead to additional operational costs. In chlorine electrolysis, for instance, it appears possible to vary the current density over a fairly wide range. However, this would be more difficult in aluminium production, for example. This type of problem will not be considered here. It is a second stage of the complete optimisation procedure, the first one being the calculation of the theoretical optimum current densities without taking complications into account. This calculation is in fact prerequisite to any attempt of achieving a variable current density in an electrolytic pIant. In the following, a simplified but general treatment for the evaluation of the economically desirable current densities will be presented. The method used is similar to that employed in previous papers[2-4]. Let us assume that we want to make by electrolysis a given amount of product A4 in time t. The overall cost can be roughly split into three parts : the fixed and variable investment costs, and the energy cost, The fixed investment K, is that which is virtually independent of the current density. Among others, most of the control instruments belong to this category. The variable (in current density dependent) investment, includes mainly the cells and electrodes, the size of which is inversely proportional to the appIied current density. As a first approximation the variable investment cost Ku is proportional to the total electrode area A needed to make the desired amount of product in the prescribed time. K,=aAt.
(1)
The proportionality factor a has the dimension $/m+h. It represents both the interest on and the amortisation of the invested capital (per unit time). A more detailed discussion of the factor a is given in Appendix I. The energy cost K, is proportional to the energy consumption, which is equal to the product of the * Thanks are due to the Alusuisse AG for providing us the figures necessary for this estimate.
applied voltage and of the quantity of electricity Q needed to make the amount of product M. For the sake of simplicity we will assume Q to be constant, ie that the current efficiency is independent of the current density. The electrode area is related to Q by: A = Q/it. For the voltage we write E=Eo+Ri,
(2)
where E. is the voltage extrapolated to zero current. The linear approximation of (1) and (2) is often fairly good and sufficient for the purpose of this paper (for a more detailed discussion see Ref. 123). If we denote the price of the unit of electricity by b we can thus write for the total cost: K-Kf+K,+Ke=Kf+aAt+bQ(Eo+Ri)
= KY + Q(a/i + bE, + bRi).
(3)
The optimisation is carried out by differentiating K with respect to i and equating the derivative to zero. Lf we take for b a constant (average) value, we obtain for the optimum current density the relationship: iO, = z/(albR)
(4)
which was derived earlier [2]. So far we have tacitly assumed the current density to be constant during time t. We now proceed by taking into account the surplus of night electricity through the use of different current densities during the night and day periods. The calculation of the most economic conditions will be carried out in two ways: (a) by optimising together the electrolysis and the power plant; (b) by assuming fixed costs for the power available at various periods of the day. 2. SIMULTANEOUS ELECTROLYTIC
OPTIMISATION OF THE AND POWER PLANT
In this method of approach we do not regard the price of the electricity as given and consider instead the cost of the investment for the power plant and the fuel consumed in the latter. We will minim& the total cost consisting of the investment cost for the power plant (a), for the electrolytic plant (b) and the cost of fuel used in the electricity production (c). We will assume that the customers other than the electrolytic plant have different power demands L 1 and L2 during two periods t,(night) and tZ (day), respectively (L, > L,). We will further assume that the difference L2 - L1 is larger than the maximum night consumption of the electrolytic plant (which corresponds to the case where the electrolysis is completely cut off during the day), ie that the power plant can supply the electricity needed by the electrolytic plant independently of the chosen ratio of night to day current density in/i*). It was again assumed that the current efficiency, ie Q, is independent of current density.
Optimisation of use of surplus night electricity in electrochemical (a) The investment cost K, for a conventional power plant is proportional to the power produced. Let c be the proportionality factor connecting KP and L. It includes (as in the case of factor a) both the interest and the amortisation (per unit time) and has the dimension $ per kW and per unit time. The size of the plant is determined by the peak demand, ie by the power to be produced during time t2. Since the amortisatibn and the interest must be paid for the whole period tl + t2 we have: K, = 120, + f2)(E0 + Ri2)AiZ f c(t, + f2)L2, (5) (b) The quantity of electricity needed to make the desired amount of product must now be expressed in terms of the two current densities i, and il (corresponding to the periods tl and fL) Q ==Ai,tl + Ai2rz
(6)
from which it follows that: A = The investment
Q/(i~rl + izrd
cost for the electrolytic
KU = aA(rI + t2) := uQ(tl
We thus write:
plant is thus:
+ r2j/(ilrl + i2rZ). (8)
ai,
(11)
The terms with t2/Q and (LItI + L&>/Q do not depend on the current densities il and i2 and thus drop out from the optimisation calculations. The remaining variable (ie cd dependent) total cost still depends on a number of parameters and variables (il, j2, a, c, d, Eo, R, rl , fZ). The same is true of the optimum current densities oil and J2 calculated from the condition (11). A general calculation and representation of the results in terms of all the parameters involved would be extremely intricate. We therefore simplify the problem by introducing dimensionless groups according to the principles of dimensional analysis (Appendix IL). For a complete description of the cost as a function of all the parameters we need six dimensionless groups, which we have selected as Follows:
Equation (10) can be readily rewritten in terms .of these dimensionless groups. The optimum current densities OiI and 0i2 were calculated for rl = r,t with the aid of a CDC 6400/6500 digital computer. They were obtained by computing for several fixed values of c/d and aR/cEfj the minimum of the function
+ (EO + Ri&4i,r, + Lltl
+ Lf21.
K -=f QG c
(9)
Adding (S), (8) and (9) and expressing the electrode area by means of (7) we obtain for the overall cost* :
K = K, + K, -I- Kb =: i,rluf;.2r2 + ctQ (Eo + Rir)ir + 5 i,r, + hr2 Q
aK ai,
a~
(7)
(c) Finally the fuel cost is proportional to the energy produced. Let d be the proportionality factor ($ /kWh). The fuel cost for the total energy ptoduced by the power plant is thus:
Kb = d[(& + Ril)Ai,fI
899
processes
ilR i2R c aR
E,‘-‘-‘-2. E,, d cE,,! c
(13)
The Fibonacci search method [5], which is a very fast procedure, was used. The values of ilR/Eo and i2 R/E,, for which K is a minimum are the optimum values oilR]Eo and ,,i2 R/Eo. They depend on the parameters c/d and aR/cEi . Figure 1 shows ,ilR/Eo
1
+ Qd (Ed + Ril)ilrl + (E. + Ri2F2r2
[
tlrl + br2 (10)
where r = r, + t2. This equation expresses the cost as a function of two independent variables il and iZ. The optimum values of i, and ir are given by the condition that the total cost should be a minimum. + This is the total cost excluding the fixed cost of the electrolytic plant (ie which does not depend on current density). +This is a particular case. remains the same for I, # r2
However,
the
method
-2
0 log.
aR/cE,
I *
2
3
Fig. 1. Optimum night current density for simultaneous optimisation of power plant and electrochemical plant.
900
NIBLAND
-I
P. M. ROBERTSON
0
Fig. 2. Optimum
aR/cEo’ for three values of c/d, Fig. 2 shows the optimum ratios of day to night current densities as a function of aRicE: again for four values of cjd. It is seen that 0i2/0il ie the economically desirable variation of current density, strongly depends on the values of c/d and aRIcE:. It increases with increasing aR/cEo’. It is readily understandable that the optimum ratio of day and night current becomes larger when the investment cost for the electrolytic plant a increases with respect to that for the power plant c. When the investment for the electrolysis is very high it should be fully utilised all the time and it is then preferable to have a variable load on the power plant. However, the ratio a/c is by no means the only relevant parameter. The ratio of the optimum currents also depends-and this is less obvious-on R/E: as well as on c/d. It increases with decreasing c/d, ie when the fuel cost becomes large, compared
* For still smaller values of aR/cEd the most economic way of operation would be to reverse the direction of the current and use the electrolysis cell for power generation during the day. Part of the product made during the night is thus consumed during the day. The system then acts, at least in part, as a storage battery. + The investment cost per kW of energy consumed is to be distinguished from a which is referred to one square metre of electrode area. It is more directly comparable than a to the investment cost for the power plant. Table 1. Data used in the calculation
t * see Appendix
3
oR/cE,2
ratios of day to night current density plant and electrochemical
(which is the dimensionless group representing the night optimum current density) as a function of
Chlorine Copper
2
I
tog.
for simultaneous plant.
optimisation
of power
to the investment cost for the power plant. For a given c/d the ratio 0 i2jo i, tends, at high aRIcE:, towards a constant value, which is larger the smaller c/d is. For c/d = O-1 and aR/cEo’ larger than about 100, 0i,/0 i, is close to 1 (0.8). Under such circumstances it would be hardly worth while to try to operate at different current densities during night and day. However, in the common case of a thermal power plant typical values of c and dare 0+)027 and O-004 $/kWh, respectively [3] which yieIds for c/d ca 0.7. In this case the optimum current ratio is 0.4 at very high values of aR/cEi but when the latter becomes smaller than about 10 the optimum ratio decreases substantially and drops to zero for aR/cEi = 0.32.* Let us consider two examples: chlorine electrolysis and copper refining. The values of the relevant parameters are given in Table 1. With these values one obtains for ,i,/,i, O-15 and 040 for chlorine and copper electrolysis, respectively. In the case of chlorine it would be best to shut down almost completely the electrolysis during the day period, whereas for the copper refinery the economically desirable change in current density is a much less drastic
one.
This
illustrates
well
the
fact that the
ratio a/c is not the only relevant factor. The investment cost and the ratio a/c is much smaller in the case of the copper plant, but the optimum ratio of day to night current is much higher than for the chlorine plant. This is due to the influence of R and particularly of E. (which is very different in the two cases) and which overcompensates that of a. As has been mentioned at the beginning, in chlorine electro-
of the operating
costs of chlorine
and copper refining plants
a $/m2h*
Eo v
R Qm*
c $/kWh
d s]kWh
aRicE:
0.108 0.00493
3.1 0.05
2 x IO-4 6 x IO-“
00l27 0.0027
0.004 0+04
0.83 438.00
I.
Optimisation of use of surplus night electricity in electrochemical processes lysis, the investment cost per kWt is even roughly the same as for a thermal power plant. It may thus seem at first sight surprising that the 0 iJo il is only 0.15. The explanation again appears to lie in the roleof the electrochemical parameters R and E,,, which represent, respectively, the resistance of the cell and the applied voltage extrapolated to zero current. Figure 3 illustrates the savings which can be achieved by operating at a variable current density. The figures shown were calculated for chlorine electrolysis with the numerical values of Q, E,, and R given in Table I. The total variable cost (per coulomb of electricity used) has been plotted in a three dimensional diagram as a function of the night and day current densities i, and i2 (Fig. 3). The curves shown are the projections of the lines of equal cost. The broken line corresponds to the operation at a single current density (iI = i2), the optimum currents being represented by the points A, B, C, which have been calculated from (4) for a power cost b of 0.0067, 0.01 and 0.015 $/kWh, respectively. The corresponding costs calculated from (3) (excluding KJ are.4.483 x 10-5;6*039 x 10-5;8*250 x 10-s$/Ah. The minimum cost for operation at variable current densities (point 0) is 3.80 x 10e5 $/Ah. By operating at two different current densities one can therefore decrease the cost by 15-2 %, 37.1 % and 53.9 %, respectively. The corresponding figures for copper refining are 8-7 %, 25.9 % and 40.0 %, respectively. The decrease in the cost which could be achieved by operating at a variable current density depends on the values of the parameters a, b, c, R and E. Greatest savings are obtained when aR/cEt is low, but more especially when c/d is high. For example, the decreases* in variable operating cost are 43 % and
6000
12000
901
14 % for aR/cEi values of 0.01 and 100 respectively and c/d = 1. However, when c/d is only 0.1 the savings possible are negligible, being only 2.6 y0 and 0.44 % for aR/cEi values of 0.1 and 1000 respectively. The treatment presented in this section, which involves the simultaneous optimisation of electrolytic and power plant may appear at first sight rather unrealistic because the current for the electrolysis is often drawn from a power plant (or a system of power plants) with a much higher capacity than the electrochemical plant. In reality this is not so because as has been already mentioned earlier the values of 0 i, and OiZdo not depend on L1 and L1 provided that the condition mentioned earlier is fulfilled, namely that LZ - L1 is larger than the maximum power demand of the electrolytic plant. In fact it doesn’t even matter if the electrolytic plant draws its energy from a network of interconnected power plants, as is often the case. Nevertheless the above treatment is rather theoretical in its nature. Ln principle the simultaneous consideration of the power production and consumption is the most desirable method of approach since it leads to the most economic overall conditions. However, it presupposes a close cooperation between the power producer and consumer, which has rarely been achieved so far. Nevertheless the simultaneous optimisation of the energy production and consumption is a goal which is worth while to try to approach, even if it may be a distant one. The above treatment is intended to be a first step in that direction. At present, however, in many cases the problem of optimisation of the use of surplus night electricity will rather present itself in a different manner:
l9000
24000
3ooc
600(
Fig. 3. Variable cost K/Q (S/Ah) as a function of il and iz (A/m’) for simultaneous of power plant and electrochemical plant (chlorine electrolysis).
optimisation
N.
902
IBL
AND
P. M. ROBERTSON
The power will be available at different rates at different times of the day. This problem will be treated in the following section. 3. OPTIMISATION
W;AmilVO
FIXED
ENERGY
In this section we assume that the electrolytic plant gets energy at two different rates c, and c1 ($/kWh) corresponding to the night and day time periods t, and f2, respectively. The calculation is similar to that of section 1 except that we have to consider two time periods tl and t2 with two different current densities il and i2 and two costs. Instead of (1) we now have for the variable investment cost: K” = aA(t1 + t2).
(14)
log.
oR/c,Ef
On the other hand the energy cost is: K. = A[(& + Ril)ilcltl
+ (E. + Ridi
cl t21.
(15)
The electrode area A can be expressed in terms of the current densities il and i2 by means of (7). We thus obtain for the total cost (excluding the fixed cost)
K=ilt,
+Pi2tl
[@I f tz) + 6% + Ril)ctiltt + (I?%+ Riz)cl i2 rJ.
(16)
As in section 2 we simplify the problem by intros ducing six dimensionless groups which were selected as follows : -9-
K
QEocz
aR
Rewriting (16) in groups yields : K -= QEo
cl
Ril
CZEO2’~‘~)~)fi.
terms
of
Ri,
these
tl
dimensionless
Fig. 4. Optimum night current density for electricity at fixeq costs. As a first approximation R. iliE varies according to and CaRlc2 Eo2)o*5. Figure 5 shows the (c,/cJ-“~7 optimum ratio of day to night current as a function of cI/c2 for several values of aR/c2 Eb. As is to be expected, the ratio o i210ir decreases with decreasing ratio of night to day energy costs. It is asymptotic to the value cl/c2 at very large values of aR[c2E02. This behaviour is similar to that observed in section 2 with respect to aRIcE& For c2 =0*02 $/kWh and cI/c2 = O-5 we obtain (with the values of a, R and E. given in Table 1) for chlorine and copper electrolysis optimum current ratios of 0.12 and O-48, respectively. Again, as in the analysis of section 2, the optimum ratio is very small for chlorine electrolysis, whereas for the copper plant a decrease of the current by a factor of only 2 would be the economically desirable one. Because of the smaller E,, and the larger R
cz
[(?)+1](2&)+ (:)[1+(2)] (2)(g) + (g)
lf
(g/[(2)(:)]
[If($)]
-+]+(p)/(g)
(17)
The minimum of this function with respect to RillEo and Ri2/Eo was again calculated for tl = t2 with a digital computer using the Fibonacci search technique. The corresponding values of Ril/Eo and Riz/Eo give the optimum current densities 0 il and 0 iz. The calculation was repeated for various values of aR/c2 Eo2 and c1/c2. Figure 4 shows in dimensionless form (Roil/E,,) the optimum night current as a function of aR/c2 Eo2 for several values of cI/cI .
Fig. 5.
Optimum ratios of day to night current density for electricity at fixed costs.
Optimisation
of use of surplus night electricity
(which overcompensates the influence of the small investment cost) the value of &Z/c2 Eo2 is much larger for the copper(59) than for the chlorine (0,112) case. This is the reason for the very much larger 0 iJo i,. In the range of aR/cz E,,z between 0.05 and 10 the optimum current ratio is quite sensitive to this dimensionless group. However, the uR/c~E~~ value ‘for copper lies already in the range where 0 i2/oi becomes almost independent of aR/cr Eo2. Such high values of aR/c2 EoZ will be encountered mainly in refining processes (where E, is usually small) or when LI and R are very high. The cell resistance is very large in the case of many organic solvents, for instance. However, processes with very large values of a and R are economically inefficient and will in general be of little potential interest. Figure 6 shows the variable cost as a function of il and il for chlorine electrolysis, with c2 =0.02 and
0
6000
in electrochemical processes
903
4. CONCLUSIONS
For the values of the relevant parameters most likely to be encountered in practice it would be definitely desirable to strongly vary the current density to make proper use of the surplus night electricity. Qualitatively one reaches the same conclusion independently of the method used (simultaneous optimisation of the electrolytic and power plant, assumption of two given energy rates). However, the two numerical examples presented (copper and chlorine) show that for otherwise identical conditions the optimum current ratio much depends upon the electrochemical system. In the case of copper only a reduction of the current by one half at day time would be useful (when the ratio of night to day energy price is one half). It would be economically wrong to reduce the current further during
12000
18000
24000
i2
Fig. 6. Variable cost K/Q ($/Ah) for chlorine electrolysis as a function of ii and i2 (A/m2) for electricity
at fixed costs (Q/C, = 2).
c, =O*Ol $/kWh. The representation is similar to that of Fig. 3. Point B corresponds to the optimum conditions when operating at a single current density, b being taken equal to (c, + c2)/2 = 0.015 $/kWh. The saving which could be achieved by operating at variable current, with the optimum current ratio (point A) is 12-S%. For c1 = O-005 and c1 = 0.02 $/kWh the corresponding figure would be 35.9 ‘/& For copper refining the savings would be 6.1 0k and 20.8 0A respectively, for the two foregoing sets of values of c1 and c2. A further decrease of c1/c7 to O-1 would increase the possible saving to 61.8 y0 for chlorine and 43.7 o/0for copper electrolysis.
the day. For chlorine, on the other hand, it would be economically best to virtually interrupt the electrolysis during the day. The savings which can be achieved are illustrated by Figs. 3 and 6. For the examples given they are of the order of 13-35 ‘A (chlorine). However, in many cases operation at a variable current density will involve additional cost. Obviously these should not overcompensate the savings due to the preferential use of cheap night electricity. This aspect is of importance since percentage-wise the saving which can be achieved is not very large if the comparison is made under optimum conditions for both types of operation, constant and variable current.
904
N.16~ AND
P. M.
In the above analyses only the case that involved two periods of equal lengths was considered (t , = r2). However, the calculations would be quite similar for values of 1,/r* different from one. On the other hand, one could also cut the day into more than two periods (or even consider several periods during the year). The calculation of section 3 would remain in principle the same but the computation would then be more cumbersome. However, the general trends observed in the present simplified treatment are believed to be qualitatively representative of the more complicated cases as well.
ROBERTSON
year of operation. capital we have
Since X is used to amortise the initial
i;
I=1
x, = P.
(19)
Since the unamortised capital on which interest has to bc paid progressively decreases X increases and 2 decreases with time. During the nth (last) year the unpaid capital left is X. and the interest due on it is z.=
x.s’=
X,(S-
l),
Y=Z.+Z,/(S-l)orZ.== REFERENCES
Y(1 -l/S).
J. Seetzen, Chemie-lngr-Tech. 41,141(1969). N. Ibl and E. Adam, Chemie-lngr.Tech. 37, 573 (1965). N. Ibl and E. Schalch, Chemie-lrzgr-Tech. 41,208 (1969) N. Ibl, Proc. Symp. on Elecrrochemicaf Engineering, Newcastle-upon-Tyne,(1921) 5. D. J. Wilde and C. S. Beightler, Foundations of Optimisafion, p. 236. Prentice-Hall, Englewood Cliffs, New Jersey, (1967). 6. A. Schmidt, J. ekcrrochem. Sot. 118, 2046 (1971). 7. Assessing Projects. I.C.l.-Methuen London (1970).
Z”_, =(X,
+ X”_,)(S-
1) =z.
+ (X.-,)(S
of the specific
investment
I
Y(I - l/S) + X._,(SCombinmg
(22)
with Y’= X,_, $ Z,_, we obtain: Y(1 -l/S)
Z”_, -(Y-Z,_,)(S-I)= or z,_,
= Y(1 - l/S’).
(23)
For the next term we have:. Y(l -l/S)
Y=X,+Z,=X,+Z2=~~~=Xn+Zn=constant(18) the first, second,
(24)
Z”_, = Y(1 - l/S3)
cost
etc. Summation
yields:
i;z,=nr-Ex, i=l
‘=1 Y(1 -ris+1
=nY-P=
=nY-
efc
* 15% was added to Schmidt’s figures to allow for inflation (rate of exchange = 0.313 $/DM).
-l/s*$~~~l
Y(lISI~l/S~+-*‘+l/S”)
-l/S”) (25)
Or
PS”=
Y(li-s+s2+...$s”-‘).
(26)
Equation (26) can be interpreted as follows. Instead of paying back each year the amount Y, we keep it and cash the interest on it. We thus have at the end of n yr accumulated an amount which is equal to that given by the right hand side of (26). We have now to use this money to return, in a single payment done after n yr, the investment P and the accumulated interest, which is equal to PS”. This is indeed the term on the left hand side of (26). The specific investment a’ is obtained from Y by dividing by the number of time units u contained in a year (for instance, 8760 if a’ is expressed in $/mth). Taking into account that S” - 1 = (S - 1x1 i- S + S* + ’ . . + S”- ‘) we finally obtain from (26) a’ =
1,2... represent
1).
and
The specific investment cost a for the electrolytic plant includes the amount a’ to be paid per unit electrode area per unit time for the amortisation of and the interest on the invested capital. For reasons of convenience we also include into a payments a# which are inversely proportional to the current density but do not involve the amortisation of an invested capital (an example will be given below). The factor a thus consists of two parts a = a’ + a”. We will discuss in some more detail the term a’ which is probably much larger than a” in most cases. The simplest way of evaluating a’ is to divide the initial investment P(%/m’) by the number of time units corresponding to the life time of the cells, and to add to this the interest (per unit time) on the investment. However, this interest has to be an average value owing to the successive amortisation of the invested capital. A more accurate way of evaluating a’ is to take explicitely into account the decrease in the invested capital due to the progressive amortisation. P is the initially invested capital per m* electrode area. We wish to pay it back in n equal yearly amounts Y (where n is the number of years of amortisation which is in principle given by the life time of the cells). Y includes a part X corresponding lo the. amurtisation and a part Z representing the interest. We thus write:
where the indices
- 1) =
.
z,_,-(Y-Z._*)(S-I)=
Evaluafion
(21)
Similarly we have for the (n - 1)“’ year:
1. 2. 3. 4.
APPENDIX
(20)
where S’ is the annual rate of interest and S= S’ + 1 is the interest factor. Combination with (18) yields:
Y/v =
PS(S
-
u(S”-1)
1)
.
(27)
This relationship allows us to calculate a’ from the initial investment P, the interest rate S’ (S= s’+ 1) and the number of years n during which the invested capital is to be amortised. The quantity to be used in (4) (or in the equations of sections 2 and 3) is calculated from II = (I’ + a”, the value of a’ being given by (27). In
Optimisation
of use of surplus night electricity
the evaluation of a” no complications arise, since no amortization is involved. We evaluate u for chlorine electrolysis from the data given by Schmidt [6]*. For a mercury cell with 20 mz electrode area the initial investment is 94 000 $ per unit cell or 4700 .$/ml electrode area. The corresponding value of a’ calculated according to (27) is 0.073 $/mlh. For the period of amortisation n we have taken 10 yr and for the rate of interest 6% (S = l-06). The cost of the mercury of the cathodes is not included in the above amount because there is a slow consumption roughly proportional to the current density. We thus have to pay interest on but do not amortise the capital cost of the mercury (see Ref. [2]). For a cell with an electrode area of 20 mz about 2 tons of mercury are needed, the cost of which is 29,000 $. Since there is no amortisation involved we need not use (27) and we simply calculate the interest on the basis of 6 % per yr which yields 1740 %= 0.01 S/m’h. The quantity of mercury needed and therefore the interest to be paid for it is inversely proportional to the current density. This expense is thus a typical example of a contribution to the aforementioned term a”. Finally, according to Schmidt the maintenance cost (labour and material cost = 4300 S/20 m2/yr) is also inversely proportional to the current density?. We therefore lump this expense (which corresponds to 0.025 S/m*h) into a” too. We thus obtain a’ (capital cost of cells) a” (interest on mercury) (maintenance)
0,073 $/m2h 0.01 S/m2h 0,025 $/mlh
a
0,108 $/m’h
This is the value of a which has been used in the calculation of the variable cost for chorine electrolysis as a function of il and i2 (Figs. 3 and 6). An alternative method of approach for the optimisation is the nowadays often used cash-flow procedure [7]. Computations based on this have been applied to electrochemical systems by Schmidt[6]. In the cash flow method the present worth G of a project is maximised.4 In a formal derivation of the optimum current density the cash inflow and cash outflow for each year of operation must be considered. At the end of year 0 we will invest sum P* in electrolysis plant. There is no production in this year; therefore the net cash-flow is -P*. In the rth year of operation, however, the cash outflow is the cost of electricity (H) and raw materials (J) consumed, while the cash-inflow is the revenue from the product
t This is obviously the case if the number of square meters of electrode area fixes the number of cells (k if the size of the cells is given) and if the maintenance cost per cell is independent of current density. In reality one should also optimise the rectifiers. This imposes a restraint on the number of cells, ie one may have to change to some extent the cell size when one modifies the overall electrode area. This additional problem, which is less important than that of the optimisation of the electrode area with respect to current density, will not he discussed here. $ The present worth G of a sum of money U in year r is the amount that must be invested in year 0 such that after r years at ..S% interest it is equal to U.
in electrochemical
905
processes
sales ( W). The net cash-flow in year r is therefore -H--J + W. We disregard inflation and assume that these quantities do not depend on r. The present worth of the project (G) may now be written as G = --P* + e (---H--J+ .=I
W)/S W--l) w+--q
= --P* +(-H--J+
(28)
We now express P* and H in terms of the variables previously. Thus P* = PA = ‘2
and H = %
used
(E” + Ri)
and therefore G/Q=
--go+
-(&+R&$+~
&+
I
I (29)
where Q is now the quantity of electricity needed during the whole duration of the project (I = nu). Let us note that J/Q and W/Q are independent of current density. After multiplying both sides of (29) by nS”(S - l)/(S” - 1) the problem is that of maximising the function GnS(S
- 1)
QW - 1) The optimum
= -_(E,+Ri)b---
p “g:
l).
(30)
current density is found to be ol=
JR??].
(31)
This is identical with (4) if we set a = a’ (a” = 0) and remember that a’is given by (27). Let us now consider a case with a” # 0. In chlorine electrolysis we have to take into account separately from P* the investment P’ due to the mercury. The net cashflow in year 0 has now to be written as -P* -P ‘.The consumed mercury is regularly replaced at a cost of D%/yr. The initial amount of Hg is therefore available at the end of the n yr. of operation; its present worth is P//s”.Taking into account that a” is given by aY_P’(S-l)_P’in(S-l) AU
Q
’
(32)
we thus obtain instead of (29)
G/Q= -&
a"(S- 1) (s- l)Snni
-&;-(E,+Ri);+;
+ [
1
6.
(33)
We may note that in addition to W/Q and J/Q, D/Q is also independent of i, because the consumption of mercury per mz electrode area is roughly proportional to i. The optimisation problem therefore reduces to maximising the function -(&.+Ri)b. ! (34)
N. IBL
906
AND
P. M.ROBERTSON
The optimum current density is now found to be : (35) This is again identical with (4) if we remember that (I = a’+ a” and that a’ is given by (27)§. This remains true if a” incIudes contributions such as the maintenance cost in chlorine electrolysis mentioned earlier. Although in the foregoing treatment we have considered only the case of a constant current during the whole period of operation it is easy to see that the cash flow method and the procedure used in this paper give also
damental units: length; mass; time and $, which are sufficient when the electrical quantities are represented by the “Gaussian” system of units. We have then the dimensional matrix A (Table 2). The rank of a matrix is given by the order of the largest determinant which is contained in it and which is different from zero. It is thus found that the rank of matrix A is 4. It is therefore equal to the number of fundamental units involved, as is usually the case. Let us now consider the second case in which the quantity of electricity is taken as a fundamental unit of which there are now 5 in all. We thus have the dimensional matrix B. Its rank is found to be 4. We have here the much less usual situation where the rank of the matrix
the same results if two different currents at fixed costs are used (see section 3). A slight complication arises in the simultaneous optimisation of the electrolytic and the power plant (section 2) because the two in general have different periods of amortisation. In this case we have to assume that at the end of the amortisation of the shorter lived installation the latter is replaced under the same
conditions as the original one, Under these circumstances the cash flow method yields again the same results as the procedure
of this paper.
APPENDIX
II
Determinatian of the dimensionless groups by means of dimensional analysis In the simultaneous optimisation of the electrochemical and the power plant (section 2) the cost K/Q depends on a, c, d, t,, tl, il, i2, E. and R, ie there are ten variables in all. Their number can be reduced bv combinine them to dimensionless groups which are ihen used as the variables of the problem instead of the initial ones. According to the II-theorem of dimensional analysis the number of dimensionless groups which have tc be introduced in order to completely describe the problem is equal to the number of the initial variables minus the rank of the dimensional matrix. In most cases the rank of this matrix is equal to the number of fundamental units involved. In our case there is an ambiguity with respect to this point. The number of fundamental units is 4 or 5 depending upon whether one considers the electrical quantities as derived solely from the mechanical system of units, or not. In the first case we have as funI In the relationships given by Schmidt (see (17) of Reference [6] the factor taking into account the accrued interest is written as S”+‘(S - I)/(.9 - 1) instead of S”(S - 1)&S” - 1). This is due to the fact that Schmidt assumes the capital investment to have been made one year before the plant actually starts operation.
Table 2. Dimensional matrix A I
mt%
a
3-l
2 Eo I1 i2
-2-l -:
K/Q R t1 t2
B
0 0
Im
0
-1
a jJ +
-1 -2 -2
0 0
21
10 10
1
-2
0 0 0
-2-l -2 2 -2 -2
f
%
Q
O-II
0 0
21 2 1 1 -2 0 O-10 O-IO 0 1 -1 0 0 10 0 10
0
-1
0 -1 1 1 -1 -2 0 0
is different from the number of fundamental units. However, the rank is the same as in the first case. The two methods therefore lead to the same result: We need 10 - 4 = 6 dimensionless groups. There are many possibilities to combine the original variables to dimensionless groups which suffice the requirements of dimensional analysis. We have selected the dimensionless groups described by the definition (12) which appeared adequate for our purposes. In particular, they have been so chosen that the quantities in which we are primarily interested, i,, and il, appear separately with the power 1 in the dimensionless groups. The calculation of the optimum current implies the use of the additional conditions aK/ai, = aKlaiz = 0. thereby eleminating two variables. For r,/t2 = 1 we can thus represent the dimensionless ratio oil/oi2 as a function of two dimensionless groups only, as has been done iii Fig. 2. Similar arguments apply to the introduction of the dimensionless groups used in the case of fixed electricity prices (section 3).
km,
1973 Vol.
18, pp. 897-906.
Pergamon Press. Printed in Great Britain
OPTIMISATION OF USE OF SURPLUS NIGHT ELECTRICITY IN ELECTROCHEMICAL PROCESSES Technisch-Chemisches
N. IBL and P. M. ROBERTSON Laboratorium, Eidg. Technische Hochschule, Zurich, Switzerland
(Received 7 March 1973) Abstract-Cheaper
electric energy is available during certain periods of the day, especially at night. Electrochemical plants could make use of the surplus electricity by applying a variable current density. An analysis of the problem is presented for an electrolysis operated at two different current densities. The optimum values of these current densities were calculated by balancing the investment against the energy (or fuel) costs. Dimensionless groups were introduced in order to reduce the large number of parameters and variables involved. As an extension of previous work the optimization was carried out by two methods: (a) by considering simultaneously the expenses for the electrolytic and the power plant; (b) by assuming that the energy is available at two different but fixed rates. The results depend on a number of parameters such as the specific investment cost for the electrolytic plant, the cell resistance etc. Their influence is shown and discussed in terms of the dimensionless groups used. Numerical values are presented for two examples: chlorine electrolysis and copper refining. In the first case the economically desirable ratio of the day and night current is very small, in the second case it is substantially larger. The optimization technique used is compared to the cash flow method. It is shown that the results obtained by the two procedures are the same.
NOMENCLATURE A a
a’ a” b
c d D E
El3 H
L
number of years during which investment is amortised initially invested capital per m2 electrode area, %/m* total investment cost, $ investment cost of Hg in chlorine electrolysis, $ quantity of electricity needed to make given amount of product in time I, C resistance of cell, 0 m* interest factor = 1 + S’ yearly rate of interest, % time number of time units contained in a year revenue from sales of product, %/yr
n
total electrode area, m’ specific investment cost (more generally, proportionality factor for cost which is proportional to electrode area and thus inversely proportional to current density), $/m’h part of factor a involving both interest and amortization, $/m*h part of factor a involving interest only, $/m2h cost of unit of electric energy, $/kWh specific investment cost for power plant, $jkWh fuel cost, $/kWh replacement cost of Hg in chlorine electrolysis, Slyr applied voltage, V voltage extrapolated to zero current, V cost of electricity to produce a fixed quantity of product, $/yr. current density, A/m2 cost of raw materiaIs, %/yr total cost, $ fixed cost of electrolytic plant, $ fuel cost, $ energy cost, $ investment cost for power plant, $ variable investment cost for electrolytic plant, $ power consumption by consumers other than electrochemical, kW 897
P P*
P
Q R
S S’ t V
W Subscripts period,
I and
2 refer to night
and
day time
respectively. 1.
The demand
INTRODUCTION
for electricity is larger during the day than at night. In general, the capacity of power plants is only partially utilised at night, causing an increase in the investment cost. The electrochemical industry, which is a large consumer of electricity, could in principle contribute to equalise the load of the power plants by operating at a variable current density, designed to counteract the variation in the electricity demand by the other consumers. It is then the investment cost of the electrochemical plant which increases because it is less utilised during the day time. The question is therefore whether it is better
N.
898
IBL AND
P. M. ROBERTSON
to have variable load for the electrochemical or for the power plant. The answer obviously depends on the relative magnitudes of the investment costs of the two types of plant. Typical values are (in s/kW of power used or produced): 200 $ for aluminium electrolysis* ; 200 % for chlorine electrolysis, 150 $ for a thermal and 300 $ for a nuclear power plant[l 1. They are very similar and it is thus of considerable interest to examine more closely from an economic viewpoint the desirability of varying the load of an electrochemical plant. The purpose of this paper is to calculate the optimum current density for an electroIysis, taking into account the surplus electricity available during certain periods, especially at night. Obviously it may turn out that the optimum values thus calculated cannot be realised in practice because of technical reasons, which would lead to additional operational costs. In chlorine electrolysis, for instance, it appears possible to vary the current density over a fairly wide range. However, this would be more difficult in aluminium production, for example. This type of problem will not be considered here. It is a second stage of the complete optimisation procedure, the first one being the calculation of the theoretical optimum current densities without taking complications into account. This calculation is in fact prerequisite to any attempt of achieving a variable current density in an electrolytic pIant. In the following, a simplified but general treatment for the evaluation of the economically desirable current densities will be presented. The method used is similar to that employed in previous papers[2-4]. Let us assume that we want to make by electrolysis a given amount of product A4 in time t. The overall cost can be roughly split into three parts : the fixed and variable investment costs, and the energy cost, The fixed investment K, is that which is virtually independent of the current density. Among others, most of the control instruments belong to this category. The variable (in current density dependent) investment, includes mainly the cells and electrodes, the size of which is inversely proportional to the appIied current density. As a first approximation the variable investment cost Ku is proportional to the total electrode area A needed to make the desired amount of product in the prescribed time. K,=aAt.
(1)
The proportionality factor a has the dimension $/m+h. It represents both the interest on and the amortisation of the invested capital (per unit time). A more detailed discussion of the factor a is given in Appendix I. The energy cost K, is proportional to the energy consumption, which is equal to the product of the * Thanks are due to the Alusuisse AG for providing us the figures necessary for this estimate.
applied voltage and of the quantity of electricity Q needed to make the amount of product M. For the sake of simplicity we will assume Q to be constant, ie that the current efficiency is independent of the current density. The electrode area is related to Q by: A = Q/it. For the voltage we write E=Eo+Ri,
(2)
where E. is the voltage extrapolated to zero current. The linear approximation of (1) and (2) is often fairly good and sufficient for the purpose of this paper (for a more detailed discussion see Ref. 123). If we denote the price of the unit of electricity by b we can thus write for the total cost: K-Kf+K,+Ke=Kf+aAt+bQ(Eo+Ri)
= KY + Q(a/i + bE, + bRi).
(3)
The optimisation is carried out by differentiating K with respect to i and equating the derivative to zero. Lf we take for b a constant (average) value, we obtain for the optimum current density the relationship: iO, = z/(albR)
(4)
which was derived earlier [2]. So far we have tacitly assumed the current density to be constant during time t. We now proceed by taking into account the surplus of night electricity through the use of different current densities during the night and day periods. The calculation of the most economic conditions will be carried out in two ways: (a) by optimising together the electrolysis and the power plant; (b) by assuming fixed costs for the power available at various periods of the day. 2. SIMULTANEOUS ELECTROLYTIC
OPTIMISATION OF THE AND POWER PLANT
In this method of approach we do not regard the price of the electricity as given and consider instead the cost of the investment for the power plant and the fuel consumed in the latter. We will minim& the total cost consisting of the investment cost for the power plant (a), for the electrolytic plant (b) and the cost of fuel used in the electricity production (c). We will assume that the customers other than the electrolytic plant have different power demands L 1 and L2 during two periods t,(night) and tZ (day), respectively (L, > L,). We will further assume that the difference L2 - L1 is larger than the maximum night consumption of the electrolytic plant (which corresponds to the case where the electrolysis is completely cut off during the day), ie that the power plant can supply the electricity needed by the electrolytic plant independently of the chosen ratio of night to day current density in/i*). It was again assumed that the current efficiency, ie Q, is independent of current density.
Optimisation of use of surplus night electricity in electrochemical (a) The investment cost K, for a conventional power plant is proportional to the power produced. Let c be the proportionality factor connecting KP and L. It includes (as in the case of factor a) both the interest and the amortisation (per unit time) and has the dimension $ per kW and per unit time. The size of the plant is determined by the peak demand, ie by the power to be produced during time t2. Since the amortisatibn and the interest must be paid for the whole period tl + t2 we have: K, = 120, + f2)(E0 + Ri2)AiZ f c(t, + f2)L2, (5) (b) The quantity of electricity needed to make the desired amount of product must now be expressed in terms of the two current densities i, and il (corresponding to the periods tl and fL) Q ==Ai,tl + Ai2rz
(6)
from which it follows that: A = The investment
Q/(i~rl + izrd
cost for the electrolytic
KU = aA(rI + t2) := uQ(tl
We thus write:
plant is thus:
+ r2j/(ilrl + i2rZ). (8)
ai,
(11)
The terms with t2/Q and (LItI + L&>/Q do not depend on the current densities il and i2 and thus drop out from the optimisation calculations. The remaining variable (ie cd dependent) total cost still depends on a number of parameters and variables (il, j2, a, c, d, Eo, R, rl , fZ). The same is true of the optimum current densities oil and J2 calculated from the condition (11). A general calculation and representation of the results in terms of all the parameters involved would be extremely intricate. We therefore simplify the problem by introducing dimensionless groups according to the principles of dimensional analysis (Appendix IL). For a complete description of the cost as a function of all the parameters we need six dimensionless groups, which we have selected as Follows:
Equation (10) can be readily rewritten in terms .of these dimensionless groups. The optimum current densities OiI and 0i2 were calculated for rl = r,t with the aid of a CDC 6400/6500 digital computer. They were obtained by computing for several fixed values of c/d and aR/cEfj the minimum of the function
+ (EO + Ri&4i,r, + Lltl
+ Lf21.
K -=f QG c
(9)
Adding (S), (8) and (9) and expressing the electrode area by means of (7) we obtain for the overall cost* :
K = K, + K, -I- Kb =: i,rluf;.2r2 + ctQ (Eo + Rir)ir + 5 i,r, + hr2 Q
aK ai,
a~
(7)
(c) Finally the fuel cost is proportional to the energy produced. Let d be the proportionality factor ($ /kWh). The fuel cost for the total energy ptoduced by the power plant is thus:
Kb = d[(& + Ril)Ai,fI
899
processes
ilR i2R c aR
E,‘-‘-‘-2. E,, d cE,,! c
(13)
The Fibonacci search method [5], which is a very fast procedure, was used. The values of ilR/Eo and i2 R/E,, for which K is a minimum are the optimum values oilR]Eo and ,,i2 R/Eo. They depend on the parameters c/d and aR/cEi . Figure 1 shows ,ilR/Eo
1
+ Qd (Ed + Ril)ilrl + (E. + Ri2F2r2
[
tlrl + br2 (10)
where r = r, + t2. This equation expresses the cost as a function of two independent variables il and iZ. The optimum values of i, and ir are given by the condition that the total cost should be a minimum. + This is the total cost excluding the fixed cost of the electrolytic plant (ie which does not depend on current density). +This is a particular case. remains the same for I, # r2
However,
the
method
-2
0 log.
aR/cE,
I *
2
3
Fig. 1. Optimum night current density for simultaneous optimisation of power plant and electrochemical plant.
900
NIBLAND
-I
P. M. ROBERTSON
0
Fig. 2. Optimum
aR/cEo’ for three values of c/d, Fig. 2 shows the optimum ratios of day to night current densities as a function of aRicE: again for four values of cjd. It is seen that 0i2/0il ie the economically desirable variation of current density, strongly depends on the values of c/d and aRIcE:. It increases with increasing aR/cEo’. It is readily understandable that the optimum ratio of day and night current becomes larger when the investment cost for the electrolytic plant a increases with respect to that for the power plant c. When the investment for the electrolysis is very high it should be fully utilised all the time and it is then preferable to have a variable load on the power plant. However, the ratio a/c is by no means the only relevant parameter. The ratio of the optimum currents also depends-and this is less obvious-on R/E: as well as on c/d. It increases with decreasing c/d, ie when the fuel cost becomes large, compared
* For still smaller values of aR/cEd the most economic way of operation would be to reverse the direction of the current and use the electrolysis cell for power generation during the day. Part of the product made during the night is thus consumed during the day. The system then acts, at least in part, as a storage battery. + The investment cost per kW of energy consumed is to be distinguished from a which is referred to one square metre of electrode area. It is more directly comparable than a to the investment cost for the power plant. Table 1. Data used in the calculation
t * see Appendix
3
oR/cE,2
ratios of day to night current density plant and electrochemical
(which is the dimensionless group representing the night optimum current density) as a function of
Chlorine Copper
2
I
tog.
for simultaneous plant.
optimisation
of power
to the investment cost for the power plant. For a given c/d the ratio 0 i2jo i, tends, at high aRIcE:, towards a constant value, which is larger the smaller c/d is. For c/d = O-1 and aR/cEo’ larger than about 100, 0i,/0 i, is close to 1 (0.8). Under such circumstances it would be hardly worth while to try to operate at different current densities during night and day. However, in the common case of a thermal power plant typical values of c and dare 0+)027 and O-004 $/kWh, respectively [3] which yieIds for c/d ca 0.7. In this case the optimum current ratio is 0.4 at very high values of aR/cEi but when the latter becomes smaller than about 10 the optimum ratio decreases substantially and drops to zero for aR/cEi = 0.32.* Let us consider two examples: chlorine electrolysis and copper refining. The values of the relevant parameters are given in Table 1. With these values one obtains for ,i,/,i, O-15 and 040 for chlorine and copper electrolysis, respectively. In the case of chlorine it would be best to shut down almost completely the electrolysis during the day period, whereas for the copper refinery the economically desirable change in current density is a much less drastic
one.
This
illustrates
well
the
fact that the
ratio a/c is not the only relevant factor. The investment cost and the ratio a/c is much smaller in the case of the copper plant, but the optimum ratio of day to night current is much higher than for the chlorine plant. This is due to the influence of R and particularly of E. (which is very different in the two cases) and which overcompensates that of a. As has been mentioned at the beginning, in chlorine electro-
of the operating
costs of chlorine
and copper refining plants
a $/m2h*
Eo v
R Qm*
c $/kWh
d s]kWh
aRicE:
0.108 0.00493
3.1 0.05
2 x IO-4 6 x IO-“
00l27 0.0027
0.004 0+04
0.83 438.00
I.
Optimisation of use of surplus night electricity in electrochemical processes lysis, the investment cost per kWt is even roughly the same as for a thermal power plant. It may thus seem at first sight surprising that the 0 iJo il is only 0.15. The explanation again appears to lie in the roleof the electrochemical parameters R and E,,, which represent, respectively, the resistance of the cell and the applied voltage extrapolated to zero current. Figure 3 illustrates the savings which can be achieved by operating at a variable current density. The figures shown were calculated for chlorine electrolysis with the numerical values of Q, E,, and R given in Table I. The total variable cost (per coulomb of electricity used) has been plotted in a three dimensional diagram as a function of the night and day current densities i, and i2 (Fig. 3). The curves shown are the projections of the lines of equal cost. The broken line corresponds to the operation at a single current density (iI = i2), the optimum currents being represented by the points A, B, C, which have been calculated from (4) for a power cost b of 0.0067, 0.01 and 0.015 $/kWh, respectively. The corresponding costs calculated from (3) (excluding KJ are.4.483 x 10-5;6*039 x 10-5;8*250 x 10-s$/Ah. The minimum cost for operation at variable current densities (point 0) is 3.80 x 10e5 $/Ah. By operating at two different current densities one can therefore decrease the cost by 15-2 %, 37.1 % and 53.9 %, respectively. The corresponding figures for copper refining are 8-7 %, 25.9 % and 40.0 %, respectively. The decrease in the cost which could be achieved by operating at a variable current density depends on the values of the parameters a, b, c, R and E. Greatest savings are obtained when aR/cEt is low, but more especially when c/d is high. For example, the decreases* in variable operating cost are 43 % and
6000
12000
901
14 % for aR/cEi values of 0.01 and 100 respectively and c/d = 1. However, when c/d is only 0.1 the savings possible are negligible, being only 2.6 y0 and 0.44 % for aR/cEi values of 0.1 and 1000 respectively. The treatment presented in this section, which involves the simultaneous optimisation of electrolytic and power plant may appear at first sight rather unrealistic because the current for the electrolysis is often drawn from a power plant (or a system of power plants) with a much higher capacity than the electrochemical plant. In reality this is not so because as has been already mentioned earlier the values of 0 i, and OiZdo not depend on L1 and L1 provided that the condition mentioned earlier is fulfilled, namely that LZ - L1 is larger than the maximum power demand of the electrolytic plant. In fact it doesn’t even matter if the electrolytic plant draws its energy from a network of interconnected power plants, as is often the case. Nevertheless the above treatment is rather theoretical in its nature. Ln principle the simultaneous consideration of the power production and consumption is the most desirable method of approach since it leads to the most economic overall conditions. However, it presupposes a close cooperation between the power producer and consumer, which has rarely been achieved so far. Nevertheless the simultaneous optimisation of the energy production and consumption is a goal which is worth while to try to approach, even if it may be a distant one. The above treatment is intended to be a first step in that direction. At present, however, in many cases the problem of optimisation of the use of surplus night electricity will rather present itself in a different manner:
l9000
24000
3ooc
600(
Fig. 3. Variable cost K/Q (S/Ah) as a function of il and iz (A/m’) for simultaneous of power plant and electrochemical plant (chlorine electrolysis).
optimisation
N.
902
IBL
AND
P. M. ROBERTSON
The power will be available at different rates at different times of the day. This problem will be treated in the following section. 3. OPTIMISATION
W;AmilVO
FIXED
ENERGY
In this section we assume that the electrolytic plant gets energy at two different rates c, and c1 ($/kWh) corresponding to the night and day time periods t, and f2, respectively. The calculation is similar to that of section 1 except that we have to consider two time periods tl and t2 with two different current densities il and i2 and two costs. Instead of (1) we now have for the variable investment cost: K” = aA(t1 + t2).
(14)
log.
oR/c,Ef
On the other hand the energy cost is: K. = A[(& + Ril)ilcltl
+ (E. + Ridi
cl t21.
(15)
The electrode area A can be expressed in terms of the current densities il and i2 by means of (7). We thus obtain for the total cost (excluding the fixed cost)
K=ilt,
+Pi2tl
[@I f tz) + 6% + Ril)ctiltt + (I?%+ Riz)cl i2 rJ.
(16)
As in section 2 we simplify the problem by intros ducing six dimensionless groups which were selected as follows : -9-
K
QEocz
aR
Rewriting (16) in groups yields : K -= QEo
cl
Ril
CZEO2’~‘~)~)fi.
terms
of
Ri,
these
tl
dimensionless
Fig. 4. Optimum night current density for electricity at fixeq costs. As a first approximation R. iliE varies according to and CaRlc2 Eo2)o*5. Figure 5 shows the (c,/cJ-“~7 optimum ratio of day to night current as a function of cI/c2 for several values of aR/c2 Eb. As is to be expected, the ratio o i210ir decreases with decreasing ratio of night to day energy costs. It is asymptotic to the value cl/c2 at very large values of aR[c2E02. This behaviour is similar to that observed in section 2 with respect to aRIcE& For c2 =0*02 $/kWh and cI/c2 = O-5 we obtain (with the values of a, R and E. given in Table 1) for chlorine and copper electrolysis optimum current ratios of 0.12 and O-48, respectively. Again, as in the analysis of section 2, the optimum ratio is very small for chlorine electrolysis, whereas for the copper plant a decrease of the current by a factor of only 2 would be the economically desirable one. Because of the smaller E,, and the larger R
cz
[(?)+1](2&)+ (:)[1+(2)] (2)(g) + (g)
lf
(g/[(2)(:)]
[If($)]
-+]+(p)/(g)
(17)
The minimum of this function with respect to RillEo and Ri2/Eo was again calculated for tl = t2 with a digital computer using the Fibonacci search technique. The corresponding values of Ril/Eo and Riz/Eo give the optimum current densities 0 il and 0 iz. The calculation was repeated for various values of aR/c2 Eo2 and c1/c2. Figure 4 shows in dimensionless form (Roil/E,,) the optimum night current as a function of aR/c2 Eo2 for several values of cI/cI .
Fig. 5.
Optimum ratios of day to night current density for electricity at fixed costs.
Optimisation
of use of surplus night electricity
(which overcompensates the influence of the small investment cost) the value of &Z/c2 Eo2 is much larger for the copper(59) than for the chlorine (0,112) case. This is the reason for the very much larger 0 iJo i,. In the range of aR/cz E,,z between 0.05 and 10 the optimum current ratio is quite sensitive to this dimensionless group. However, the uR/c~E~~ value ‘for copper lies already in the range where 0 i2/oi becomes almost independent of aR/cr Eo2. Such high values of aR/c2 EoZ will be encountered mainly in refining processes (where E, is usually small) or when LI and R are very high. The cell resistance is very large in the case of many organic solvents, for instance. However, processes with very large values of a and R are economically inefficient and will in general be of little potential interest. Figure 6 shows the variable cost as a function of il and il for chlorine electrolysis, with c2 =0.02 and
0
6000
in electrochemical processes
903
4. CONCLUSIONS
For the values of the relevant parameters most likely to be encountered in practice it would be definitely desirable to strongly vary the current density to make proper use of the surplus night electricity. Qualitatively one reaches the same conclusion independently of the method used (simultaneous optimisation of the electrolytic and power plant, assumption of two given energy rates). However, the two numerical examples presented (copper and chlorine) show that for otherwise identical conditions the optimum current ratio much depends upon the electrochemical system. In the case of copper only a reduction of the current by one half at day time would be useful (when the ratio of night to day energy price is one half). It would be economically wrong to reduce the current further during
12000
18000
24000
i2
Fig. 6. Variable cost K/Q ($/Ah) for chlorine electrolysis as a function of ii and i2 (A/m2) for electricity
at fixed costs (Q/C, = 2).
c, =O*Ol $/kWh. The representation is similar to that of Fig. 3. Point B corresponds to the optimum conditions when operating at a single current density, b being taken equal to (c, + c2)/2 = 0.015 $/kWh. The saving which could be achieved by operating at variable current, with the optimum current ratio (point A) is 12-S%. For c1 = O-005 and c1 = 0.02 $/kWh the corresponding figure would be 35.9 ‘/& For copper refining the savings would be 6.1 0k and 20.8 0A respectively, for the two foregoing sets of values of c1 and c2. A further decrease of c1/c7 to O-1 would increase the possible saving to 61.8 y0 for chlorine and 43.7 o/0for copper electrolysis.
the day. For chlorine, on the other hand, it would be economically best to virtually interrupt the electrolysis during the day. The savings which can be achieved are illustrated by Figs. 3 and 6. For the examples given they are of the order of 13-35 ‘A (chlorine). However, in many cases operation at a variable current density will involve additional cost. Obviously these should not overcompensate the savings due to the preferential use of cheap night electricity. This aspect is of importance since percentage-wise the saving which can be achieved is not very large if the comparison is made under optimum conditions for both types of operation, constant and variable current.
904
N.16~ AND
P. M.
In the above analyses only the case that involved two periods of equal lengths was considered (t , = r2). However, the calculations would be quite similar for values of 1,/r* different from one. On the other hand, one could also cut the day into more than two periods (or even consider several periods during the year). The calculation of section 3 would remain in principle the same but the computation would then be more cumbersome. However, the general trends observed in the present simplified treatment are believed to be qualitatively representative of the more complicated cases as well.
ROBERTSON
year of operation. capital we have
Since X is used to amortise the initial
i;
I=1
x, = P.
(19)
Since the unamortised capital on which interest has to bc paid progressively decreases X increases and 2 decreases with time. During the nth (last) year the unpaid capital left is X. and the interest due on it is z.=
x.s’=
X,(S-
l),
Y=Z.+Z,/(S-l)orZ.== REFERENCES
Y(1 -l/S).
J. Seetzen, Chemie-lngr-Tech. 41,141(1969). N. Ibl and E. Adam, Chemie-lngr.Tech. 37, 573 (1965). N. Ibl and E. Schalch, Chemie-lrzgr-Tech. 41,208 (1969) N. Ibl, Proc. Symp. on Elecrrochemicaf Engineering, Newcastle-upon-Tyne,(1921) 5. D. J. Wilde and C. S. Beightler, Foundations of Optimisafion, p. 236. Prentice-Hall, Englewood Cliffs, New Jersey, (1967). 6. A. Schmidt, J. ekcrrochem. Sot. 118, 2046 (1971). 7. Assessing Projects. I.C.l.-Methuen London (1970).
Z”_, =(X,
+ X”_,)(S-
1) =z.
+ (X.-,)(S
of the specific
investment
I
Y(I - l/S) + X._,(SCombinmg
(22)
with Y’= X,_, $ Z,_, we obtain: Y(1 -l/S)
Z”_, -(Y-Z,_,)(S-I)= or z,_,
= Y(1 - l/S’).
(23)
For the next term we have:. Y(l -l/S)
Y=X,+Z,=X,+Z2=~~~=Xn+Zn=constant(18) the first, second,
(24)
Z”_, = Y(1 - l/S3)
cost
etc. Summation
yields:
i;z,=nr-Ex, i=l
‘=1 Y(1 -ris+1
=nY-P=
=nY-
efc
* 15% was added to Schmidt’s figures to allow for inflation (rate of exchange = 0.313 $/DM).
-l/s*$~~~l
Y(lISI~l/S~+-*‘+l/S”)
-l/S”) (25)
Or
PS”=
Y(li-s+s2+...$s”-‘).
(26)
Equation (26) can be interpreted as follows. Instead of paying back each year the amount Y, we keep it and cash the interest on it. We thus have at the end of n yr accumulated an amount which is equal to that given by the right hand side of (26). We have now to use this money to return, in a single payment done after n yr, the investment P and the accumulated interest, which is equal to PS”. This is indeed the term on the left hand side of (26). The specific investment a’ is obtained from Y by dividing by the number of time units u contained in a year (for instance, 8760 if a’ is expressed in $/mth). Taking into account that S” - 1 = (S - 1x1 i- S + S* + ’ . . + S”- ‘) we finally obtain from (26) a’ =
1,2... represent
1).
and
The specific investment cost a for the electrolytic plant includes the amount a’ to be paid per unit electrode area per unit time for the amortisation of and the interest on the invested capital. For reasons of convenience we also include into a payments a# which are inversely proportional to the current density but do not involve the amortisation of an invested capital (an example will be given below). The factor a thus consists of two parts a = a’ + a”. We will discuss in some more detail the term a’ which is probably much larger than a” in most cases. The simplest way of evaluating a’ is to divide the initial investment P(%/m’) by the number of time units corresponding to the life time of the cells, and to add to this the interest (per unit time) on the investment. However, this interest has to be an average value owing to the successive amortisation of the invested capital. A more accurate way of evaluating a’ is to take explicitely into account the decrease in the invested capital due to the progressive amortisation. P is the initially invested capital per m* electrode area. We wish to pay it back in n equal yearly amounts Y (where n is the number of years of amortisation which is in principle given by the life time of the cells). Y includes a part X corresponding lo the. amurtisation and a part Z representing the interest. We thus write:
where the indices
- 1) =
.
z,_,-(Y-Z._*)(S-I)=
Evaluafion
(21)
Similarly we have for the (n - 1)“’ year:
1. 2. 3. 4.
APPENDIX
(20)
where S’ is the annual rate of interest and S= S’ + 1 is the interest factor. Combination with (18) yields:
Y/v =
PS(S
-
u(S”-1)
1)
.
(27)
This relationship allows us to calculate a’ from the initial investment P, the interest rate S’ (S= s’+ 1) and the number of years n during which the invested capital is to be amortised. The quantity to be used in (4) (or in the equations of sections 2 and 3) is calculated from II = (I’ + a”, the value of a’ being given by (27). In
Optimisation
of use of surplus night electricity
the evaluation of a” no complications arise, since no amortization is involved. We evaluate u for chlorine electrolysis from the data given by Schmidt [6]*. For a mercury cell with 20 mz electrode area the initial investment is 94 000 $ per unit cell or 4700 .$/ml electrode area. The corresponding value of a’ calculated according to (27) is 0.073 $/mlh. For the period of amortisation n we have taken 10 yr and for the rate of interest 6% (S = l-06). The cost of the mercury of the cathodes is not included in the above amount because there is a slow consumption roughly proportional to the current density. We thus have to pay interest on but do not amortise the capital cost of the mercury (see Ref. [2]). For a cell with an electrode area of 20 mz about 2 tons of mercury are needed, the cost of which is 29,000 $. Since there is no amortisation involved we need not use (27) and we simply calculate the interest on the basis of 6 % per yr which yields 1740 %= 0.01 S/m’h. The quantity of mercury needed and therefore the interest to be paid for it is inversely proportional to the current density. This expense is thus a typical example of a contribution to the aforementioned term a”. Finally, according to Schmidt the maintenance cost (labour and material cost = 4300 S/20 m2/yr) is also inversely proportional to the current density?. We therefore lump this expense (which corresponds to 0.025 S/m*h) into a” too. We thus obtain a’ (capital cost of cells) a” (interest on mercury) (maintenance)
0,073 $/m2h 0.01 S/m2h 0,025 $/mlh
a
0,108 $/m’h
This is the value of a which has been used in the calculation of the variable cost for chorine electrolysis as a function of il and i2 (Figs. 3 and 6). An alternative method of approach for the optimisation is the nowadays often used cash-flow procedure [7]. Computations based on this have been applied to electrochemical systems by Schmidt[6]. In the cash flow method the present worth G of a project is maximised.4 In a formal derivation of the optimum current density the cash inflow and cash outflow for each year of operation must be considered. At the end of year 0 we will invest sum P* in electrolysis plant. There is no production in this year; therefore the net cash-flow is -P*. In the rth year of operation, however, the cash outflow is the cost of electricity (H) and raw materials (J) consumed, while the cash-inflow is the revenue from the product
t This is obviously the case if the number of square meters of electrode area fixes the number of cells (k if the size of the cells is given) and if the maintenance cost per cell is independent of current density. In reality one should also optimise the rectifiers. This imposes a restraint on the number of cells, ie one may have to change to some extent the cell size when one modifies the overall electrode area. This additional problem, which is less important than that of the optimisation of the electrode area with respect to current density, will not he discussed here. $ The present worth G of a sum of money U in year r is the amount that must be invested in year 0 such that after r years at ..S% interest it is equal to U.
in electrochemical
905
processes
sales ( W). The net cash-flow in year r is therefore -H--J + W. We disregard inflation and assume that these quantities do not depend on r. The present worth of the project (G) may now be written as G = --P* + e (---H--J+ .=I
W)/S W--l) w+--q
= --P* +(-H--J+
(28)
We now express P* and H in terms of the variables previously. Thus P* = PA = ‘2
and H = %
used
(E” + Ri)
and therefore G/Q=
--go+
-(&+R&$+~
&+
I
I (29)
where Q is now the quantity of electricity needed during the whole duration of the project (I = nu). Let us note that J/Q and W/Q are independent of current density. After multiplying both sides of (29) by nS”(S - l)/(S” - 1) the problem is that of maximising the function GnS(S
- 1)
QW - 1) The optimum
= -_(E,+Ri)b---
p “g:
l).
(30)
current density is found to be ol=
JR??].
(31)
This is identical with (4) if we set a = a’ (a” = 0) and remember that a’is given by (27). Let us now consider a case with a” # 0. In chlorine electrolysis we have to take into account separately from P* the investment P’ due to the mercury. The net cashflow in year 0 has now to be written as -P* -P ‘.The consumed mercury is regularly replaced at a cost of D%/yr. The initial amount of Hg is therefore available at the end of the n yr. of operation; its present worth is P//s”.Taking into account that a” is given by aY_P’(S-l)_P’in(S-l) AU
Q
’
(32)
we thus obtain instead of (29)
G/Q= -&
a"(S- 1) (s- l)Snni
-&;-(E,+Ri);+;
+ [
1
6.
(33)
We may note that in addition to W/Q and J/Q, D/Q is also independent of i, because the consumption of mercury per mz electrode area is roughly proportional to i. The optimisation problem therefore reduces to maximising the function -(&.+Ri)b. ! (34)
N. IBL
906
AND
P. M.ROBERTSON
The optimum current density is now found to be : (35) This is again identical with (4) if we remember that (I = a’+ a” and that a’ is given by (27)§. This remains true if a” incIudes contributions such as the maintenance cost in chlorine electrolysis mentioned earlier. Although in the foregoing treatment we have considered only the case of a constant current during the whole period of operation it is easy to see that the cash flow method and the procedure used in this paper give also
damental units: length; mass; time and $, which are sufficient when the electrical quantities are represented by the “Gaussian” system of units. We have then the dimensional matrix A (Table 2). The rank of a matrix is given by the order of the largest determinant which is contained in it and which is different from zero. It is thus found that the rank of matrix A is 4. It is therefore equal to the number of fundamental units involved, as is usually the case. Let us now consider the second case in which the quantity of electricity is taken as a fundamental unit of which there are now 5 in all. We thus have the dimensional matrix B. Its rank is found to be 4. We have here the much less usual situation where the rank of the matrix
the same results if two different currents at fixed costs are used (see section 3). A slight complication arises in the simultaneous optimisation of the electrolytic and the power plant (section 2) because the two in general have different periods of amortisation. In this case we have to assume that at the end of the amortisation of the shorter lived installation the latter is replaced under the same
conditions as the original one, Under these circumstances the cash flow method yields again the same results as the procedure
of this paper.
APPENDIX
II
Determinatian of the dimensionless groups by means of dimensional analysis In the simultaneous optimisation of the electrochemical and the power plant (section 2) the cost K/Q depends on a, c, d, t,, tl, il, i2, E. and R, ie there are ten variables in all. Their number can be reduced bv combinine them to dimensionless groups which are ihen used as the variables of the problem instead of the initial ones. According to the II-theorem of dimensional analysis the number of dimensionless groups which have tc be introduced in order to completely describe the problem is equal to the number of the initial variables minus the rank of the dimensional matrix. In most cases the rank of this matrix is equal to the number of fundamental units involved. In our case there is an ambiguity with respect to this point. The number of fundamental units is 4 or 5 depending upon whether one considers the electrical quantities as derived solely from the mechanical system of units, or not. In the first case we have as funI In the relationships given by Schmidt (see (17) of Reference [6] the factor taking into account the accrued interest is written as S”+‘(S - I)/(.9 - 1) instead of S”(S - 1)&S” - 1). This is due to the fact that Schmidt assumes the capital investment to have been made one year before the plant actually starts operation.
Table 2. Dimensional matrix A I
mt%
a
3-l
2 Eo I1 i2
-2-l -:
K/Q R t1 t2
B
0 0
Im
0
-1
a jJ +
-1 -2 -2
0 0
21
10 10
1
-2
0 0 0
-2-l -2 2 -2 -2
f
%
Q
O-II
0 0
21 2 1 1 -2 0 O-10 O-IO 0 1 -1 0 0 10 0 10
0
-1
0 -1 1 1 -1 -2 0 0
is different from the number of fundamental units. However, the rank is the same as in the first case. The two methods therefore lead to the same result: We need 10 - 4 = 6 dimensionless groups. There are many possibilities to combine the original variables to dimensionless groups which suffice the requirements of dimensional analysis. We have selected the dimensionless groups described by the definition (12) which appeared adequate for our purposes. In particular, they have been so chosen that the quantities in which we are primarily interested, i,, and il, appear separately with the power 1 in the dimensionless groups. The calculation of the optimum current implies the use of the additional conditions aK/ai, = aKlaiz = 0. thereby eleminating two variables. For r,/t2 = 1 we can thus represent the dimensionless ratio oil/oi2 as a function of two dimensionless groups only, as has been done iii Fig. 2. Similar arguments apply to the introduction of the dimensionless groups used in the case of fixed electricity prices (section 3).