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An endogenous technological learning formulation for fossil fuel resources
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International Journal of Hydrogen Energy 28 (2003) 1293 – 1298 www.elsevier.com/locate/ijhydene
An endogenous technological learning formulation for fossil fuel resources Walter Seifritz∗ Mulacherstr.44, CH-5212 Hausen, Switzerland Accepted 1 November 2002
Abstract A series of models, based predominantly on the logistic product, is presented. The models contain an endogenous technological learning index which gives rise to an enlarged not yet occupied niche, and thus, to an asymptotically enlarged resource basis. The learning e/ect is formulated in terms of powers of the cumulatively depleted resource. The combination of the resource depletion process and the accompanying technological learning e/ects form an environment where a multiplication of an originally assumed reserve can emerge. Some light is shed on the non-linear mathematics behind this scenario. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.
1. Introduction In most energy system models and in economic studies the technological progress has been treated as ‘manna from heaven’ because an exogenous time-trend fashion is used, i.e., for instance, by improving, exogenously, investment costs [1]. On the other hand, a classical example for an endogenous treatment of technological progress is the so-called ‘learning curves’. One example in energy technology is ‘learning by doing’ in connection with the decreasing speci7c capital costs of a certain technology, SC (in $/kW), as a function of an increasingly installed cumulative capacity, CC (in kW), given by SC(CC) = (SC)o CC−b ;
(1)
where b is the so-called learning by doing index. For b = 0 (no learning e/ect) (SC)o is the constant speci7c cost value of the technology considered. The total costs, TC (in $), for a total cumulative capacity, CCtot , is given by CCtot SC dCC TC = 0
= ∗
(SC)o CC1−b tot : 1−b
Tel.: +41-56-441-42-32.
(2)
Recently, this learning by doing was extended by a ‘learning by searching’ index c taking into account additionally the cumulative R& D expenditures, CRD, by the public hands. Eqs. (1) and (2) are modi7ed to be [1] SC(CC; CRD) = (SC)o CC−b CRD−c and TC =
(SC)o CC1−b CRD−c 1−b
(3)
which reduces to TC = (RC)o CC if b = c = 0, i.e., if there is no accumulation through learning by doing nor by public R& D expenditures. The indices b and c can be considered as an endogenisation because b and c can be determined empirically in a series of industrial product lines. 2. Endogenous formulation for fossil energy resources 2.1. The observation If we consider energy resources in the fossil fuel market as a function of time, we are confronted with the astonishing fact that in the last decades the safe reserves increased steadily although we have been using and burning fossil fuels in large amounts. For example, the oil-reserves of the world increased from 1991 (136:835 Mio t) to
0360-3199/03/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(02)00287-2
1294
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
Nomenclature million metric tons (106 t) billion (109 )
Mio t Bill
Therefore, we modify Eq. (4) by introducing an endogenous learning e/ect by saying that the original assumed reserve mo can be increased by substituting it through t n m( ) ˙ d m o → mo + a 0
n
→ mo + am (t): 151:243 Mio t in 1998, i.e. in 7 years by more than 10%. A similar statistics holds for the world wide natural gas reserves: 1991: 141.561 Bill m3 and 1998: 153.423 Bill m3 , i.e., an increase by more than 8% in 8 years. The most dramatic increase of the reserves of the oil producing countries can be observed in Aizerbejian. Its oil-reserves in 1991 were reported as 157 Mio t and in 1998 they increased to 950 Mio t, i.e., by a factor of 6 in 8 years. All these data are taken from Ref. [2]. Apparently this has to do with the accumulation of knowledge about the deposits, the steadily improved seismological technology to detect new deposits, better extraction and exploiting techniques, etc., all acquired during the working process itself. It is clear that the above trend cannot continue ad in7nitum because, on the one hand, the earth crust contains a limited amount of C-atoms and, on the other hand, the costs will increase from a certain point on considerably due to the higher extraction costs. However, there is no doubt that the acquisition of knowledge together with the accumulation of improved technologies, gained during this exploitation process, will make available much more reserves than originally thought. The question is how to model this adequately in an endogenous manner as in Section 1. In the following we will present some relatively simple models for this. 2.2. The modi/ed logistic model The most simple model for the depletion of a 7xed reservoir with mass mo is the logistic product. This means that the (time-dependent) depletion rate, m, ˙ is proportional to the product of the instantaneously depleted mass m(t) and the actual not yet occupied niche (mo − m(t)), i.e. m m˙ = m 1 − (4) mo yielding m(t) = and m(t) ˙ =
mo 1 + e−(t−to )
(5)
(t − to ) mo sech2 ; 4 2
where to is an integration constant. For t → ∞ the whole reserve mo is depleted (m(∞) = mo ). This model is good only if mo is a 7xed constant from the beginning—which will not be the case in our considerations.
(6)
This means that the original reserve will be additionally increased by a term proportional to the power n of the hitherto-depleted mass m. a and n are (dimensionless) learning indices. (It should be noted that the learning e/ect works promptly, i.e., without any time delay). For a = 0 Eq. (6) is reduced to Eq. (4). 2.3. The case n = 1 It is reasonable to consider 7rst the case n = 1 because here the learning e/ect is directly proportional to the hitherto depleted mass m. Introducing mo + am instead of mo into Eq. (4) we obtain m (7) m˙ = m 1 − mo + am pointing out that the niche of the bracket is increased by this modi7cation. Unfortunately, the solution of Eq. (7) can be given only in an implicit form as follows: m = e (t − to ): (8) [mo − (1 − a)m]1=(1−a) The plateau-value for t → ∞ is m(∞) = mo =(1 − a) instead of mo for Eq. (4) meaning that the depletable mass here is by a factor 1=(1 − a) larger than in the case without any knowledge accumulation. For instance, if a = 0:5, the e/ective reserve increases by a factor of 2. For a = 1 we obtain from Eq. (7) mo ln(m)+m=mo (t−to ) which reduces to m = mo t for t1. The penetration time to , i.e., the time until half of the plateau value is reached increases in this model from to = 0 for a = 0 to m 1 mo 1 o ln − ln : (9) to = 2(1 − a) 1−a 2 Figs. 1 and 2 visualize Eq. (8) as a function of t with a as a parameter and Eq. (9) as a function of a, respectively. The gain in the reserves, mo =(1 − a), with increasing knowledge accumulation, a, is clearly borne out as well as the growing penetration time in Fig. 2 when the learning index a increases. The case a = 1 in Fig. 1 is a limiting case. It would really mean ‘manna from heaven’: The reserves would grow linearly with time to an unlimited end. We have to exclude this case as well as all values a ¿ 1 because of their unrealistic over-estimation of the learning e/ect. The increase of the reserve from mo to mo =(1 − a) by knowledge accumulation has an analogy in reactor physics.
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
1295
Fig. 1. m(t) of Eq. (8) for various learning indices 0 ¡ a ¡ 1. The ‘subcritical multiplication e/ect’ of the plateaus is clearly borne out.
Fig. 2. The penetration time, i.e., the time where half of the plateau value of Fig. 1 is reached as function of the learning index. The concave shape expresses the increasing time-stretching e/ect when a increases.
There, the multiplication of a neutron source with strength So in a sub-critical multiplying medium is given by So =(1 − k), where k is the multiplication constant. The equilibrium values of m in Eq. (8), m(∞) = mo =(1 − a), can therefore be considered in a 7gurative sense as an endogenously ‘subcritically multiplied original reserves mo ’ whereby the co-operation of the actual depletion process and the accompanying knowledge accumulation form a subcritical environment to ‘enrich e/ectively the endowment of nature’.
2.4. The case n = 1=2 Consider learning e/ects amn for n ¿ 1 makes no sense, since, then, the niche does not shrink to zero, i.e., we lose the second ground state and no plateau value can be observed anymore. Thus, the cases n ¿ 1 makes us believe an in7nite endowment of C-atoms in nature whenever we accumulate knowledge. We reject therefore this regime for practical applications.
1296
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
Fig. 3. m(t) of Eq. (11) for a learning e/ect being proportional to the square root of the hitherto cumulatively depleted resource. The learning index c has here no upper bound.
On the other hand, the cases n ¡ 1 may be of interest because the niche tends always to zero independent of a. Especially, the case n = 12 may be considered here as an example for this regime. √ We introduce Eq. (6) for n= 12 in Eq. (4) and set a= mo c to get also here a dimensionless learning index c and obtain m m˙ = m 1 − (10) √ mo + c m o m resulting in the implicit solution m
A (m) = e(t−to ) B(m)
(11)
with the abbreviations √ √ c2 + 4mo − cmo + 2 mmo A(m) = √ ; √ c2 + 4 mo + cmo − 2 mmo √ B(m) = c mmo − m + mo ; = √
Again, if there are no learning e/ects (c = 0) we obtain m(∞) = mo . Fig. 3 visualizes Eq. (11) for various learning indices c ¿ 0. In Fig. 1 (n=1) as well as in Fig. 3 (n= 12 ) we observe that the rate of depletion, m(t), becomes more asymmetrical with respect to time zero, provided the learning e/ect is higher or in other words the (negative) curvature of the right branch of m(t) is absolutely smaller than the (positive) curvature of the left branch, whereas for a 7xed reserve mo the curve is symmetrical in this sense.
c : c2 + 4
It is clear that for c = 0, Eq. (11) reduces to Eq. (5), the classical logistic result for a 7xed reserve mo . In Eq. (10) c is not bound to be ¡ 1. For any c ¿ 0 the niche in Eq. (10) tends to zero if n ¡ 1. The plateau value in Eq. (11) is given by √ 2 2 c +4+c m(∞) = mo : (12) 2
3. The logarithmic niche In this section we look at another interesting case: The logarithmic niche. Here, the non-linear di/erential equation of the 7rst order for the depletion process reads m o m˙ = m ln (13) m possessing also two ground states m = 0 and mo and having the solution m(t) = mo exp[ − exp[ − (t − to )]]:
(14)
This solution possesses the form of a so-called Gompertz function [3] where the time is nested in two exponential functions.
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
A learning e/ect can also be introduced here in such a way that the niche increases when there is a positive learning e/ect. The substitution m m + am o o ln → ln (15) m m accomplishes this task: if the (dimensionless) learning index a was zero we obtain Eq. (13). The solution for Eq. (15) can only be determined by numerical integration leading to similar skew-shaped m(t)-functions as in Fig. 1 and possessing plateau-values of the asymptotic form m(∞) = mo =(1 − a).
4. Conclusions Of course, a series of other depletion models are theoretically conceivable, too. We have con7ned ourselves to modi7cations of the logistic product—being the obvious thing. The endogenisation of the learning process, pointing to substantially larger fossil fuel reserves as originally thought, leads to an enlarged not yet occupied niche and to a decelerated transition in reaching the plateau value—stretching the depletion process in amplitude and time (see Fig. 2 for the penetration time to ). Depending on the model, the increased asymptotic amplitude can therefore be considered as a ‘subcritically multiplied resource’ mo =(1 − a). The above study should be considered as an answer to the 1972-Club of Rome studies which considered exclusively resource reservoirs with 7xed reserves. This was one of the reasons why they failed completely with their predictions. The modelling presented may be instrumental to predict more reliably the fossil fuel resources. The statistics, available for the author were not detailed enough to apply the above models practically in order to 7nd out empirically a justi7ed learning process with the corresponding learning index. Oil, gas and coal companies have certainly better data to make more reliable 7ts. What is it good for? An improved knowledge of the fossil fuel resources is absolutely necessary for the future because we have to live with them still for a long time. And this is also true if we were switching to a climate neutral hydrogen economy. The reason is that the production costs for hydrogen from fossil fuels are considerably cheaper than hydrogen production costs via a C-free primary energy source. At any rate, this action has to be accompanied by CO2 -removal and CO2 -storage outside the atmosphere. This transitional period can be accomplished either by autothermal or exothermal hydrogen production processes— in the latter case by using, for example, high temperature process heat from high temperature nuclear reactors. Summa summarum, for this ‘bridge to a pure hydrogen economy’ based on water and C-free primary energies, we need a good understanding of the e/ective fossil fuel reserves—including the endogenous technological learning e/ect stretching them in amplitude and time.
1297
A last comment has to be made on the new book ‘The Hydrogen Economy’ by Jeremy Rifkin [4]. His analysis of the fossil fuel reserves is exclusively based on ‘Hubbert’s Bell Curve’ for the depletion rate—established by the geophysisist and Shell Oil Company coworker M. King Hubbert back in 1956 for oil production. His main thesis is that, despite new discoveries and progress, the bell shape remains essentially sti/ (due to the statistical foundation of the curve) and that exactly half of the ultimatively estimated oil is produced if the maximum of that classic bell-shaped curve is reached. Rifkin adopts this view uncritically and projects a soon and deterministic end of the oil-era only by observing the crossing of the maximum of the depletion rate. However, as we have shown through the formalisation of the problem in this article, the depletion rate curves can get extremely asymmetrical due to the learning e/ects and more than half of the reserves may be hidden behind the maximum of the derivatives in Fig. 1. The author doubts therefore that Hubbert’s symmetrical bell curve is the last resort in predicting the end of the fossil fuel era. Rifkin’s view resembles therefore very much the perception of the public, that the depletion of a resource is like eating a cake: there is a certain amount of a resource (cake), and after it was eaten, it is gone; there is no reserve anywhere else. It was the purpose of this article to contradict this mental attitude. De/nitions: ‘Exogenous’ treatment means to assess a speci7c technological process from outside. In Economy, for instance, when the temporal evolution of the costs of a speci7c product is judged by means of the general productivity development of the corresponding industrial sector. In contrast to this, ‘endogenous’ treatment means to assess the future of a speci7c technological process from inside. For instance, endogenous learning starts with the consideration of its proper inherent dynamical parameters and how they can be inQuenced by learning and technological progress. The simplest approach is, for example, to postulate that the endogenous learning e/ect is proportional to a power of the number of the speci7c pieces produced as it came true by many examples so far. Acknowledgements O autor hRa tenido a ideia para este trabalho durante a primeira reuni˜ao de nosso ‘Instituto Mundial dos Cientistas Aposentados’ (World Institute for Retired Scientists, WIRS) em Canela, Rio Grande do Sul/Brasil. O ambiente exclusivo dos cientistas reunidos constituiu o fundamento intelectual para as reQex˜oes neste artigo. O autor agradece ao Prof. Dr.F. Se7dvash da Universidade Federal de Rio Grande do Sul em Porto Alegre pela organizaTca˜ o deste evento. The idea of this work stems from the 7rst conference of our ‘World Institute for Retired Scientists (WIRS)’ in Canela, RS, Brazil. The intellectual climate of that exclusive meeting stimulated this article. The author thanks Prof. Dr. F. Se7dvash
1298
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
of the Federal University in Porto Alegre for the organization of the meeting. References [1] Bahn O, Kypreos S. MERGE-ETL: an optimisation equilibrium model with two di/erent endogenous technological learning
formulations. PSI Bericht No. 02–16, ISSN 1019-0643, July, Villigen, Switzerland: Paul Scherrer Institut, 2002. [2] Energie Daten 2000. Bundesministerium fVur Wirtschaft und Technologie, Bonn, Berlin, July, 2000. [3] Any BW, Ng TT. The use of growth curves in energy studies. Energy 1992;17:25. [4] Rifkin J. In: Tarcher JP, editor. The hydrogen economy. New York: Putnam, 2002. ISBN 1-58542-193-6.
International Journal of Hydrogen Energy 28 (2003) 1293 – 1298 www.elsevier.com/locate/ijhydene
An endogenous technological learning formulation for fossil fuel resources Walter Seifritz∗ Mulacherstr.44, CH-5212 Hausen, Switzerland Accepted 1 November 2002
Abstract A series of models, based predominantly on the logistic product, is presented. The models contain an endogenous technological learning index which gives rise to an enlarged not yet occupied niche, and thus, to an asymptotically enlarged resource basis. The learning e/ect is formulated in terms of powers of the cumulatively depleted resource. The combination of the resource depletion process and the accompanying technological learning e/ects form an environment where a multiplication of an originally assumed reserve can emerge. Some light is shed on the non-linear mathematics behind this scenario. ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved.
1. Introduction In most energy system models and in economic studies the technological progress has been treated as ‘manna from heaven’ because an exogenous time-trend fashion is used, i.e., for instance, by improving, exogenously, investment costs [1]. On the other hand, a classical example for an endogenous treatment of technological progress is the so-called ‘learning curves’. One example in energy technology is ‘learning by doing’ in connection with the decreasing speci7c capital costs of a certain technology, SC (in $/kW), as a function of an increasingly installed cumulative capacity, CC (in kW), given by SC(CC) = (SC)o CC−b ;
(1)
where b is the so-called learning by doing index. For b = 0 (no learning e/ect) (SC)o is the constant speci7c cost value of the technology considered. The total costs, TC (in $), for a total cumulative capacity, CCtot , is given by CCtot SC dCC TC = 0
= ∗
(SC)o CC1−b tot : 1−b
Tel.: +41-56-441-42-32.
(2)
Recently, this learning by doing was extended by a ‘learning by searching’ index c taking into account additionally the cumulative R& D expenditures, CRD, by the public hands. Eqs. (1) and (2) are modi7ed to be [1] SC(CC; CRD) = (SC)o CC−b CRD−c and TC =
(SC)o CC1−b CRD−c 1−b
(3)
which reduces to TC = (RC)o CC if b = c = 0, i.e., if there is no accumulation through learning by doing nor by public R& D expenditures. The indices b and c can be considered as an endogenisation because b and c can be determined empirically in a series of industrial product lines. 2. Endogenous formulation for fossil energy resources 2.1. The observation If we consider energy resources in the fossil fuel market as a function of time, we are confronted with the astonishing fact that in the last decades the safe reserves increased steadily although we have been using and burning fossil fuels in large amounts. For example, the oil-reserves of the world increased from 1991 (136:835 Mio t) to
0360-3199/03/$ 30.00 ? 2003 International Association for Hydrogen Energy. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S0360-3199(02)00287-2
1294
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
Nomenclature million metric tons (106 t) billion (109 )
Mio t Bill
Therefore, we modify Eq. (4) by introducing an endogenous learning e/ect by saying that the original assumed reserve mo can be increased by substituting it through t n m( ) ˙ d m o → mo + a 0
n
→ mo + am (t): 151:243 Mio t in 1998, i.e. in 7 years by more than 10%. A similar statistics holds for the world wide natural gas reserves: 1991: 141.561 Bill m3 and 1998: 153.423 Bill m3 , i.e., an increase by more than 8% in 8 years. The most dramatic increase of the reserves of the oil producing countries can be observed in Aizerbejian. Its oil-reserves in 1991 were reported as 157 Mio t and in 1998 they increased to 950 Mio t, i.e., by a factor of 6 in 8 years. All these data are taken from Ref. [2]. Apparently this has to do with the accumulation of knowledge about the deposits, the steadily improved seismological technology to detect new deposits, better extraction and exploiting techniques, etc., all acquired during the working process itself. It is clear that the above trend cannot continue ad in7nitum because, on the one hand, the earth crust contains a limited amount of C-atoms and, on the other hand, the costs will increase from a certain point on considerably due to the higher extraction costs. However, there is no doubt that the acquisition of knowledge together with the accumulation of improved technologies, gained during this exploitation process, will make available much more reserves than originally thought. The question is how to model this adequately in an endogenous manner as in Section 1. In the following we will present some relatively simple models for this. 2.2. The modi/ed logistic model The most simple model for the depletion of a 7xed reservoir with mass mo is the logistic product. This means that the (time-dependent) depletion rate, m, ˙ is proportional to the product of the instantaneously depleted mass m(t) and the actual not yet occupied niche (mo − m(t)), i.e. m m˙ = m 1 − (4) mo yielding m(t) = and m(t) ˙ =
mo 1 + e−(t−to )
(5)
(t − to ) mo sech2 ; 4 2
where to is an integration constant. For t → ∞ the whole reserve mo is depleted (m(∞) = mo ). This model is good only if mo is a 7xed constant from the beginning—which will not be the case in our considerations.
(6)
This means that the original reserve will be additionally increased by a term proportional to the power n of the hitherto-depleted mass m. a and n are (dimensionless) learning indices. (It should be noted that the learning e/ect works promptly, i.e., without any time delay). For a = 0 Eq. (6) is reduced to Eq. (4). 2.3. The case n = 1 It is reasonable to consider 7rst the case n = 1 because here the learning e/ect is directly proportional to the hitherto depleted mass m. Introducing mo + am instead of mo into Eq. (4) we obtain m (7) m˙ = m 1 − mo + am pointing out that the niche of the bracket is increased by this modi7cation. Unfortunately, the solution of Eq. (7) can be given only in an implicit form as follows: m = e (t − to ): (8) [mo − (1 − a)m]1=(1−a) The plateau-value for t → ∞ is m(∞) = mo =(1 − a) instead of mo for Eq. (4) meaning that the depletable mass here is by a factor 1=(1 − a) larger than in the case without any knowledge accumulation. For instance, if a = 0:5, the e/ective reserve increases by a factor of 2. For a = 1 we obtain from Eq. (7) mo ln(m)+m=mo (t−to ) which reduces to m = mo t for t1. The penetration time to , i.e., the time until half of the plateau value is reached increases in this model from to = 0 for a = 0 to m 1 mo 1 o ln − ln : (9) to = 2(1 − a) 1−a 2 Figs. 1 and 2 visualize Eq. (8) as a function of t with a as a parameter and Eq. (9) as a function of a, respectively. The gain in the reserves, mo =(1 − a), with increasing knowledge accumulation, a, is clearly borne out as well as the growing penetration time in Fig. 2 when the learning index a increases. The case a = 1 in Fig. 1 is a limiting case. It would really mean ‘manna from heaven’: The reserves would grow linearly with time to an unlimited end. We have to exclude this case as well as all values a ¿ 1 because of their unrealistic over-estimation of the learning e/ect. The increase of the reserve from mo to mo =(1 − a) by knowledge accumulation has an analogy in reactor physics.
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
1295
Fig. 1. m(t) of Eq. (8) for various learning indices 0 ¡ a ¡ 1. The ‘subcritical multiplication e/ect’ of the plateaus is clearly borne out.
Fig. 2. The penetration time, i.e., the time where half of the plateau value of Fig. 1 is reached as function of the learning index. The concave shape expresses the increasing time-stretching e/ect when a increases.
There, the multiplication of a neutron source with strength So in a sub-critical multiplying medium is given by So =(1 − k), where k is the multiplication constant. The equilibrium values of m in Eq. (8), m(∞) = mo =(1 − a), can therefore be considered in a 7gurative sense as an endogenously ‘subcritically multiplied original reserves mo ’ whereby the co-operation of the actual depletion process and the accompanying knowledge accumulation form a subcritical environment to ‘enrich e/ectively the endowment of nature’.
2.4. The case n = 1=2 Consider learning e/ects amn for n ¿ 1 makes no sense, since, then, the niche does not shrink to zero, i.e., we lose the second ground state and no plateau value can be observed anymore. Thus, the cases n ¿ 1 makes us believe an in7nite endowment of C-atoms in nature whenever we accumulate knowledge. We reject therefore this regime for practical applications.
1296
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
Fig. 3. m(t) of Eq. (11) for a learning e/ect being proportional to the square root of the hitherto cumulatively depleted resource. The learning index c has here no upper bound.
On the other hand, the cases n ¡ 1 may be of interest because the niche tends always to zero independent of a. Especially, the case n = 12 may be considered here as an example for this regime. √ We introduce Eq. (6) for n= 12 in Eq. (4) and set a= mo c to get also here a dimensionless learning index c and obtain m m˙ = m 1 − (10) √ mo + c m o m resulting in the implicit solution m
A (m) = e(t−to ) B(m)
(11)
with the abbreviations √ √ c2 + 4mo − cmo + 2 mmo A(m) = √ ; √ c2 + 4 mo + cmo − 2 mmo √ B(m) = c mmo − m + mo ; = √
Again, if there are no learning e/ects (c = 0) we obtain m(∞) = mo . Fig. 3 visualizes Eq. (11) for various learning indices c ¿ 0. In Fig. 1 (n=1) as well as in Fig. 3 (n= 12 ) we observe that the rate of depletion, m(t), becomes more asymmetrical with respect to time zero, provided the learning e/ect is higher or in other words the (negative) curvature of the right branch of m(t) is absolutely smaller than the (positive) curvature of the left branch, whereas for a 7xed reserve mo the curve is symmetrical in this sense.
c : c2 + 4
It is clear that for c = 0, Eq. (11) reduces to Eq. (5), the classical logistic result for a 7xed reserve mo . In Eq. (10) c is not bound to be ¡ 1. For any c ¿ 0 the niche in Eq. (10) tends to zero if n ¡ 1. The plateau value in Eq. (11) is given by √ 2 2 c +4+c m(∞) = mo : (12) 2
3. The logarithmic niche In this section we look at another interesting case: The logarithmic niche. Here, the non-linear di/erential equation of the 7rst order for the depletion process reads m o m˙ = m ln (13) m possessing also two ground states m = 0 and mo and having the solution m(t) = mo exp[ − exp[ − (t − to )]]:
(14)
This solution possesses the form of a so-called Gompertz function [3] where the time is nested in two exponential functions.
W. Seifritz / International Journal of Hydrogen Energy 28 (2003) 1293 – 1298
A learning e/ect can also be introduced here in such a way that the niche increases when there is a positive learning e/ect. The substitution m m + am o o ln → ln (15) m m accomplishes this task: if the (dimensionless) learning index a was zero we obtain Eq. (13). The solution for Eq. (15) can only be determined by numerical integration leading to similar skew-shaped m(t)-functions as in Fig. 1 and possessing plateau-values of the asymptotic form m(∞) = mo =(1 − a).
4. Conclusions Of course, a series of other depletion models are theoretically conceivable, too. We have con7ned ourselves to modi7cations of the logistic product—being the obvious thing. The endogenisation of the learning process, pointing to substantially larger fossil fuel reserves as originally thought, leads to an enlarged not yet occupied niche and to a decelerated transition in reaching the plateau value—stretching the depletion process in amplitude and time (see Fig. 2 for the penetration time to ). Depending on the model, the increased asymptotic amplitude can therefore be considered as a ‘subcritically multiplied resource’ mo =(1 − a). The above study should be considered as an answer to the 1972-Club of Rome studies which considered exclusively resource reservoirs with 7xed reserves. This was one of the reasons why they failed completely with their predictions. The modelling presented may be instrumental to predict more reliably the fossil fuel resources. The statistics, available for the author were not detailed enough to apply the above models practically in order to 7nd out empirically a justi7ed learning process with the corresponding learning index. Oil, gas and coal companies have certainly better data to make more reliable 7ts. What is it good for? An improved knowledge of the fossil fuel resources is absolutely necessary for the future because we have to live with them still for a long time. And this is also true if we were switching to a climate neutral hydrogen economy. The reason is that the production costs for hydrogen from fossil fuels are considerably cheaper than hydrogen production costs via a C-free primary energy source. At any rate, this action has to be accompanied by CO2 -removal and CO2 -storage outside the atmosphere. This transitional period can be accomplished either by autothermal or exothermal hydrogen production processes— in the latter case by using, for example, high temperature process heat from high temperature nuclear reactors. Summa summarum, for this ‘bridge to a pure hydrogen economy’ based on water and C-free primary energies, we need a good understanding of the e/ective fossil fuel reserves—including the endogenous technological learning e/ect stretching them in amplitude and time.
1297
A last comment has to be made on the new book ‘The Hydrogen Economy’ by Jeremy Rifkin [4]. His analysis of the fossil fuel reserves is exclusively based on ‘Hubbert’s Bell Curve’ for the depletion rate—established by the geophysisist and Shell Oil Company coworker M. King Hubbert back in 1956 for oil production. His main thesis is that, despite new discoveries and progress, the bell shape remains essentially sti/ (due to the statistical foundation of the curve) and that exactly half of the ultimatively estimated oil is produced if the maximum of that classic bell-shaped curve is reached. Rifkin adopts this view uncritically and projects a soon and deterministic end of the oil-era only by observing the crossing of the maximum of the depletion rate. However, as we have shown through the formalisation of the problem in this article, the depletion rate curves can get extremely asymmetrical due to the learning e/ects and more than half of the reserves may be hidden behind the maximum of the derivatives in Fig. 1. The author doubts therefore that Hubbert’s symmetrical bell curve is the last resort in predicting the end of the fossil fuel era. Rifkin’s view resembles therefore very much the perception of the public, that the depletion of a resource is like eating a cake: there is a certain amount of a resource (cake), and after it was eaten, it is gone; there is no reserve anywhere else. It was the purpose of this article to contradict this mental attitude. De/nitions: ‘Exogenous’ treatment means to assess a speci7c technological process from outside. In Economy, for instance, when the temporal evolution of the costs of a speci7c product is judged by means of the general productivity development of the corresponding industrial sector. In contrast to this, ‘endogenous’ treatment means to assess the future of a speci7c technological process from inside. For instance, endogenous learning starts with the consideration of its proper inherent dynamical parameters and how they can be inQuenced by learning and technological progress. The simplest approach is, for example, to postulate that the endogenous learning e/ect is proportional to a power of the number of the speci7c pieces produced as it came true by many examples so far. Acknowledgements O autor hRa tenido a ideia para este trabalho durante a primeira reuni˜ao de nosso ‘Instituto Mundial dos Cientistas Aposentados’ (World Institute for Retired Scientists, WIRS) em Canela, Rio Grande do Sul/Brasil. O ambiente exclusivo dos cientistas reunidos constituiu o fundamento intelectual para as reQex˜oes neste artigo. O autor agradece ao Prof. Dr.F. Se7dvash da Universidade Federal de Rio Grande do Sul em Porto Alegre pela organizaTca˜ o deste evento. The idea of this work stems from the 7rst conference of our ‘World Institute for Retired Scientists (WIRS)’ in Canela, RS, Brazil. The intellectual climate of that exclusive meeting stimulated this article. The author thanks Prof. Dr. F. Se7dvash
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of the Federal University in Porto Alegre for the organization of the meeting. References [1] Bahn O, Kypreos S. MERGE-ETL: an optimisation equilibrium model with two di/erent endogenous technological learning
formulations. PSI Bericht No. 02–16, ISSN 1019-0643, July, Villigen, Switzerland: Paul Scherrer Institut, 2002. [2] Energie Daten 2000. Bundesministerium fVur Wirtschaft und Technologie, Bonn, Berlin, July, 2000. [3] Any BW, Ng TT. The use of growth curves in energy studies. Energy 1992;17:25. [4] Rifkin J. In: Tarcher JP, editor. The hydrogen economy. New York: Putnam, 2002. ISBN 1-58542-193-6.