- Email: [email protected]
An analytical model for a biomass system
Energy Printed
Vol. IO, No. 9. PP. 1023-1028, m Great Britain.
AN
0360-5442/85 $3.00 + MJ 0 1985 Pewmm~ F’rcss Ltd.
1985
ANALYTICAL
MODEL
FOR A BIOMASS
SYSTEM
M. J. MWANDOSYA~ and M. L. LUHANGA$ Department of Electrical Engineering, University of Dar es Salaam, P.O. Box 35 131, Dar es Salaam, Tanzania (Received 10 August 1984) Abstract-More than half of the world’s population depends on biomass as a source of energy. The rapid population growth of rural areas has, however, placed severe strains on the biomass resource, which has led to deforestation and desertification of some areas. We present a linear, dynamic, deterministic model that captures the interaction between the supply and demand for biomass from both natural forests and man-made forests. Tanzania has been used as an example to derive parameters of interest, but we believe the model will be applicable to other developing countries.
1. INTRODUCTION
To the vast majority of people in the rural areas of Africa, Asia, and Latin America, biomass in the form of fuelwood or charcoal is the main source of energy for household, agricultural, and cottage industrial uses. The pressures of rapidly increasing rural populations have led to deforestation and, for areas such as the Sahel region of Africa, to desertification. It is therefore important to understand interactions between population growth and biomass supply and demand. We model the dynamics of a biomass system by assuming the demand for biomass to be linearly dependent on the size of the population, whereas the supply of biomass is assumed to depend on the regeneration rate and size of natural forests and the biomass available from afforestation efforts. The model yields results that show the impacts of policy alternatives (such as limitation of the rate of population growth or reduction in the per capita demand for biomass) on the area deforested. 2. THE
MODEL
We assume the time axis is divided into equal-length segments whose duration is an integral number of 7 years. We shall refer to each segment of 7 years as a slot. We define Vd(?z),n = 0, 1) . . . ) as the volume associated with deforestation at the beginning of the nth slot [i.e. V4(n) is the cut over and above the potential for annual cut from a nation’s woodlands at the beginning of the nth slot]. A simple equation governs the relationship between V4(n) and demand and supply potentials, viz. V4/4(n)
=
v,(n)
-
V,(n)
-
V3/3(n),
(1)
where VI(n) is the total demanded wood volume in steres (cubic meters of wood), V,(n) is the potential for the annual cut in steres from the woodlands, and ‘C/3(n)is the expected volume of cut in steres from afforestation efforts, all referring to the beginning of the nth slot. The potential for annual cut from woodlands is essentially the volume increment of woodlands from natural regeneration. Thus, Vz(n) = bh(~
- I),
(2)
where A,(n - 1) is the area in hectares remaining at the beginning of the nth slot, and X, is the regeneration rate, in steres per hectare, of the woodlands. t Currently a Visiting Research Scientist, Center for Energy and Environmental Studies, Princeton University, Princeton, NJ 08544, U.S.A. $ Currently with the Department of Electrical Engineering, Columbia University, New York, NY 10027, U.S.A. 1023
1024
M. J. MWANDOSYA and M. L. LUHANGA
The area of woodland remaining at the end of the nth slot is
(3)
where A&) is the area, in hectares, cleared over and above the regeneration of the woodlands during the kth slot, and A0 is the area of all woodlands at the beginning of the zeroth slot. An area is considered deforested when it has been cleared of all vegetation. Assuming that such clearing of vegetation yields X, steres per hectare of biomass, V,(n)
= ukf(n>.
(4)
If A,(n) represents the area planted due to afforestation or reforestation in the nth slot, and if we assume tha.t this area will be ready for harvesting after an integral number of 7 years, then the area harvested by the end of the nth slot will be c&(n - [T/T]), where (Y is the survival rate of the seedlings, and [x] is the smallest integer greater than x. Thus, V3(n) = aX3A,(n
-
(9
[T/71),
where X3 is the yield in m3/ha from the afforested areas. The average volume of wood demanded at the beginning of the nth slot, V,(n), will depend on the size of the population and the per capita demand for wood. Thus, V,(n) = POP”&, where p = (1 + at the beginning beginning of the for the shortfall
(6)
y)T, y is the annual rate of increase of populations, p. is the population of the zeroth slot, and P,, is the per capita wood demand in steres at the nth slot. Using eqns (2)-(6) in eqn (l), we obtain the following expression between supply and demand:
(7) 3. DISAGGREGATION
OF
THE
SUPPLY
AND
Equation (7) describes the interrelationships between and can further be disaggregated once detailed knowledge structure, the population dynamics, and the types and available. We pursue here further disaggregation of the more parameters.
DEMAND
FACTORS
supply and demand of biomass and information on the demand potentials of the woodlands are model through introduction of
3. I Disaggregation of the supply Although, for the purpose of eqn (7), AoX, appears as a single constant, it is actually the sum of a number of constants, which have to be evaluated depending on the classification of forest types in a country. If we let A, be the area and Xi the regeneration volume (in m3/ha), respectively, for forest type i, then AoX,y=
5 A;&; I=
I
we have here assumed that there are a total of p forest types. For example, if we let A, represent the area of officially gazetted forests, A, the area of woodland, and A, the area covered by bushland and thicket, and if X,, X,, and X, are their respective yield volumes (in m3/ha), then AoX, can be replaced by A&, + A,X, + A&,. A similar argument can be applied to disaggregating the yields of afforestation efforts if the spatial variation by type of plantation, survival rates, and their yields per hectare are known.
An analyticalmodel for a biomass system
1025
3.2 Disaggregation of the demand
The terms on the right side of eqn (6) represent a high degree of aggregation. The population term assumes homogeneity of character, and the utilization factor is an average for the entire population. The demand term may depend on the age composition of the population, i.e. the per capita wood demand for people of different ages is hypothesized to be different. Here, we use a cohort population model to disaggregate the demand term. We assume that the female population is divided into N + 1 cohorts, each of duration r years, and that the male population is divided into M + 1 cohorts, each of duration 7 years. This is the usual graphical representation of the population pyramid. We now define the following quantities based on our original assumption that the time axis is divided into equal intervals of duration T years: _@(n) = [&(n), P{(n), . . . , P&n)]‘, a 1 X (N + 1) matrix, with p{(n) being the per capita biomass demand at the beginning of the nth slot in steres for the female cohort i; /3”(n) = [@f(n), /3?(n), . . . , P%(n)]‘, a 1 X (A4 + 1 matrix, with /3?(n) being the per capita biomass demand at the beginning of the nth slot for the male cohort i. E/(n) [E”(n)] is a C X (N + 1) [C X (M + l)] matrix whose elements E$(n) [E,fl(n)] represent the per capita useful energy needed for an activity (e.g. cooking, heating, etc.) at the beginning of the nth slot for female (male) cohort j. There are assumed to be a total of C activities. Also, n = (17;‘) 7;‘) . . , VP’), where vi is the efficiency of energy use in activity i, y is the energy in one stere of fuelwood, x$(n) is the female population of the ith cohort at the beginning of the nth slot, and x?(n) is the male population of the ith cohort at the beginning of the nth slot. @X0i)is the average number of living female (male) offspring per female from cohort i in one slot of duration T years. di = 1 - ef
and
x/o(n+ 1) . . . x/&n -__---- + 1) Jan
+ 1) . . .
_.$f(n + 1)
I=
We now rewrite eqn (9) as
_-4(n)
I . (9)
(10) The parameters of eqn (9) are usually listed in actuarial tables. Using eqns (8)-( lo), we find
(11) When the expression for AoX, obtained in Section 4.1 and eqn (2) are used in eqn (7) we obtain a very general equation describing the dynamics of a biomass system. The values of the parameters in our model can be easily obtained for any one country from the country’s gross data on population and forestry. The generalizations of our model generally introduce more parameters into the model. The utility of these more EGY IO:‘)-c
1026
M. J. MWANDOSYA and M. L. LUHANGA
general models thus depends on whether values of all of the model parameters can be obtained easily and at reasonable cost. These more general models thus require the definition and solution of a parameter estimation problem. Disaggregation of the population term via the cohort population model is not unique. The population can be disaggregated using criteria other than age groups: e.g. (i) a broad division into rural and urban populations; (ii) the income per household, thus identifying the urban rich, the urban poor, the rural rich, and the rural poor; and (iii) broad division according to cottage industry activities. 4. APPLICATION
TO
TANZANIA
The total forested area of Tanzania is about 28 X lo6 ha, which is about 32% of the total land area. The main part of the forest is savanna and intermediate type forest, covered by what is generically referred to as Miombo Woodlands. Miombo refers to an open canopy type of woodland characteristic of the drier areas of Eastern Africa between 300 and 1200 m above sea level. The predominant species are Brachstegia, Isoberlinia, Julbernadia, and Pterocarpus.2 These natural forests, from which 99% of the total fuelwood requirement in Tanzania is harvested, have rather low annual volume increments per ha, which range between 0.5 and 2.0 steres. 3,4An average value of natural regeneration of 1 stere/ha per annum has been used in the examples. It has been estimated that the potential harvest of wood from the natural forest in Tanzania, when clear felling is practiced, is of the order of 30 to 50 m3 of solid wood per ha.3,4 In the examples we adopt the lower figure of 30 m3 of solid wood per ha, and the same figure is used for yields arising from clear felling from afforestation plantations. Based on the output from nurseries, a total of 49,000 ha was estimated to have been planted in afforestation efforts between 1975 and 1981. 4.5 With an average survival rate of seedlings of 50%, we find that it is prudent to assume a base-year value of 10,000 ha for afforestation in 1985. On the demand side, several studies have been done in Tanzania. Figures for the per capita demand per annum vary from 2.6, 2.2, 2.0 to 0.7 m3.2,5 These variations seem to reflect geographical locations of survey sites, as well as availability of alternative fuels and types of foods cooked in the various survey areas. An average figure of 2.0 m3 per capita per annum is presently used in accounting for biomass in the country4 and is also used in the model as the per capita demand in the base year. We use eqn (7) to calculate V,(n) and the gross standing volume of biomass at the beginning of each slot for Tanzania, taking 1985 as the base year. We assume T = 1 and use an annual population growth rate of 3.3%, starting with a population of 2 1.7 million in 1985. We also assume that the annual rate of increase of afforested area (pa) is 0.2, starting with an area of 10,000 ha in 1985. Other values used are X, = 1, X, = 30, CY= 0.5, T = 8, and A0 = 2.8 X 10’ ha. Figures l-2 exemplify our results when using eqn (7). By varying the parameters in eqn (7) we have been able to conclude that, for Tanzania, the uses of short rotation tree species (i.e. trees with small values of T) or trees with very high survival rates (i.e. trees with high values of LY)have minimal short-term impact on both the deficit fuelwood supply and the gross standing volume of wood. The population growth rate and the afforestation rate (&, however, have considerable impact on the deficit fuelwood supply and the gross standing volume of wood. Furthermore, we have been able to show, through Figs. 1-2, that the management of fuelwood demand would have dramatic and positive consequences on the availability of fuelwood in Tanzania, provided it is coupled with an appropriate annual growth rate in afforested area. Since over 90% of wood use is accounted for by domestic fuel use, it is logical to assume that the utilization factor p, is primarily determined by the efficiency of use and, in the present case, the efficiency of cookstoves. Thus, for the nth year after the base year, the demand relation can be associated with the total number of people using more efficient stoves. If we assume that stoves leading to doubling of efficiency, compared to the base year efficiency, are gradually introduced, then
1027
An analytical model for a biomass system
I
I
I996
1991
b86
20(
t (YEAR) Fig. 1. Deficit fuelwood demand.
P,*P(n)
where the population,
= &W)
+ PoM4
when using doubled-efficiency N(n) = 2P(Ml
-
-
N~)l,
(12)
cookstoves, is
PmlPol,
(13)
and p(n) is the total population for the nth year after the base year. If we assume an average household of five people, then the number of more efficient stoves n years after the base year is N,(n) = 0.4P@r)[l -
PmlPol,
(14)
and the number of new stoves introduced in that year is NJn) - NJn -
1).
Using Figs. 1 and 2, we estimate that the number of efficient stoves should be 1.7 million by 1990 and 7 million by 2000. Assuming that an improved stove will cost $20, these represent total investments of $34 million and $140 million, respectively.
1028
M. J. MWANDOSYAand M. L. LUHANGA
80-
60 -
1986
1991
1996
201
t(YEAR) Fig. 2. Gross standing volume of fuelwood
The forestry division in Tanzania2 has estimated that an investment of $600/ha is required in village afforestation efforts. At a growth rate of pcrr= 0.2 ha/yr., an investment of $54 million will be required by the year 1990. 5. CONCLUSIONS
We have presented a linear, dynamic, deterministic model of a biomass system. Our model is also suitable for handling stochastic problems.(j Acknowledgemenf-The
authors wish to acknowledge the support of IDRC and the University of Dar es Salaam.
REFERENCES 1. M. J. Mwandosya and M. L. P. Luhanga, Energy Resources, Flows and End-Uses in Tanzania, Dar es Salaam. University Press, Dar es Salaam, Tanzania (1983). 2. E. M. Mnzava, Village Afforestation: Lessons of Experience in Tanzania. FAO Report TF-INT 271 (SWE), Rome, Italy (1980). 3. E. M. Mnzava, Unasylva 33, 24 (1981). 4. B. K. Kaale, Tanzania Five-Year National Village Afforestation Plan 1982/83-1986187. Ministry of Natural Resources and Tourism, Dar es Salaam, Tanzania (1983). 5. 1. Rikula, Report to Support Village Ajbrestation in Tanzania. Institute of Resource Assessment, University of Dar es Salaam, Tanzania (1983). 6. R. G. Leibundgut, IEEE Trans. Auto. Control AC-28, 427 (1983).
Vol. IO, No. 9. PP. 1023-1028, m Great Britain.
AN
0360-5442/85 $3.00 + MJ 0 1985 Pewmm~ F’rcss Ltd.
1985
ANALYTICAL
MODEL
FOR A BIOMASS
SYSTEM
M. J. MWANDOSYA~ and M. L. LUHANGA$ Department of Electrical Engineering, University of Dar es Salaam, P.O. Box 35 131, Dar es Salaam, Tanzania (Received 10 August 1984) Abstract-More than half of the world’s population depends on biomass as a source of energy. The rapid population growth of rural areas has, however, placed severe strains on the biomass resource, which has led to deforestation and desertification of some areas. We present a linear, dynamic, deterministic model that captures the interaction between the supply and demand for biomass from both natural forests and man-made forests. Tanzania has been used as an example to derive parameters of interest, but we believe the model will be applicable to other developing countries.
1. INTRODUCTION
To the vast majority of people in the rural areas of Africa, Asia, and Latin America, biomass in the form of fuelwood or charcoal is the main source of energy for household, agricultural, and cottage industrial uses. The pressures of rapidly increasing rural populations have led to deforestation and, for areas such as the Sahel region of Africa, to desertification. It is therefore important to understand interactions between population growth and biomass supply and demand. We model the dynamics of a biomass system by assuming the demand for biomass to be linearly dependent on the size of the population, whereas the supply of biomass is assumed to depend on the regeneration rate and size of natural forests and the biomass available from afforestation efforts. The model yields results that show the impacts of policy alternatives (such as limitation of the rate of population growth or reduction in the per capita demand for biomass) on the area deforested. 2. THE
MODEL
We assume the time axis is divided into equal-length segments whose duration is an integral number of 7 years. We shall refer to each segment of 7 years as a slot. We define Vd(?z),n = 0, 1) . . . ) as the volume associated with deforestation at the beginning of the nth slot [i.e. V4(n) is the cut over and above the potential for annual cut from a nation’s woodlands at the beginning of the nth slot]. A simple equation governs the relationship between V4(n) and demand and supply potentials, viz. V4/4(n)
=
v,(n)
-
V,(n)
-
V3/3(n),
(1)
where VI(n) is the total demanded wood volume in steres (cubic meters of wood), V,(n) is the potential for the annual cut in steres from the woodlands, and ‘C/3(n)is the expected volume of cut in steres from afforestation efforts, all referring to the beginning of the nth slot. The potential for annual cut from woodlands is essentially the volume increment of woodlands from natural regeneration. Thus, Vz(n) = bh(~
- I),
(2)
where A,(n - 1) is the area in hectares remaining at the beginning of the nth slot, and X, is the regeneration rate, in steres per hectare, of the woodlands. t Currently a Visiting Research Scientist, Center for Energy and Environmental Studies, Princeton University, Princeton, NJ 08544, U.S.A. $ Currently with the Department of Electrical Engineering, Columbia University, New York, NY 10027, U.S.A. 1023
1024
M. J. MWANDOSYA and M. L. LUHANGA
The area of woodland remaining at the end of the nth slot is
(3)
where A&) is the area, in hectares, cleared over and above the regeneration of the woodlands during the kth slot, and A0 is the area of all woodlands at the beginning of the zeroth slot. An area is considered deforested when it has been cleared of all vegetation. Assuming that such clearing of vegetation yields X, steres per hectare of biomass, V,(n)
= ukf(n>.
(4)
If A,(n) represents the area planted due to afforestation or reforestation in the nth slot, and if we assume tha.t this area will be ready for harvesting after an integral number of 7 years, then the area harvested by the end of the nth slot will be c&(n - [T/T]), where (Y is the survival rate of the seedlings, and [x] is the smallest integer greater than x. Thus, V3(n) = aX3A,(n
-
(9
[T/71),
where X3 is the yield in m3/ha from the afforested areas. The average volume of wood demanded at the beginning of the nth slot, V,(n), will depend on the size of the population and the per capita demand for wood. Thus, V,(n) = POP”&, where p = (1 + at the beginning beginning of the for the shortfall
(6)
y)T, y is the annual rate of increase of populations, p. is the population of the zeroth slot, and P,, is the per capita wood demand in steres at the nth slot. Using eqns (2)-(6) in eqn (l), we obtain the following expression between supply and demand:
(7) 3. DISAGGREGATION
OF
THE
SUPPLY
AND
Equation (7) describes the interrelationships between and can further be disaggregated once detailed knowledge structure, the population dynamics, and the types and available. We pursue here further disaggregation of the more parameters.
DEMAND
FACTORS
supply and demand of biomass and information on the demand potentials of the woodlands are model through introduction of
3. I Disaggregation of the supply Although, for the purpose of eqn (7), AoX, appears as a single constant, it is actually the sum of a number of constants, which have to be evaluated depending on the classification of forest types in a country. If we let A, be the area and Xi the regeneration volume (in m3/ha), respectively, for forest type i, then AoX,y=
5 A;&; I=
I
we have here assumed that there are a total of p forest types. For example, if we let A, represent the area of officially gazetted forests, A, the area of woodland, and A, the area covered by bushland and thicket, and if X,, X,, and X, are their respective yield volumes (in m3/ha), then AoX, can be replaced by A&, + A,X, + A&,. A similar argument can be applied to disaggregating the yields of afforestation efforts if the spatial variation by type of plantation, survival rates, and their yields per hectare are known.
An analyticalmodel for a biomass system
1025
3.2 Disaggregation of the demand
The terms on the right side of eqn (6) represent a high degree of aggregation. The population term assumes homogeneity of character, and the utilization factor is an average for the entire population. The demand term may depend on the age composition of the population, i.e. the per capita wood demand for people of different ages is hypothesized to be different. Here, we use a cohort population model to disaggregate the demand term. We assume that the female population is divided into N + 1 cohorts, each of duration r years, and that the male population is divided into M + 1 cohorts, each of duration 7 years. This is the usual graphical representation of the population pyramid. We now define the following quantities based on our original assumption that the time axis is divided into equal intervals of duration T years: _@(n) = [&(n), P{(n), . . . , P&n)]‘, a 1 X (N + 1) matrix, with p{(n) being the per capita biomass demand at the beginning of the nth slot in steres for the female cohort i; /3”(n) = [@f(n), /3?(n), . . . , P%(n)]‘, a 1 X (A4 + 1 matrix, with /3?(n) being the per capita biomass demand at the beginning of the nth slot for the male cohort i. E/(n) [E”(n)] is a C X (N + 1) [C X (M + l)] matrix whose elements E$(n) [E,fl(n)] represent the per capita useful energy needed for an activity (e.g. cooking, heating, etc.) at the beginning of the nth slot for female (male) cohort j. There are assumed to be a total of C activities. Also, n = (17;‘) 7;‘) . . , VP’), where vi is the efficiency of energy use in activity i, y is the energy in one stere of fuelwood, x$(n) is the female population of the ith cohort at the beginning of the nth slot, and x?(n) is the male population of the ith cohort at the beginning of the nth slot. @X0i)is the average number of living female (male) offspring per female from cohort i in one slot of duration T years. di = 1 - ef
and
x/o(n+ 1) . . . x/&n -__---- + 1) Jan
+ 1) . . .
_.$f(n + 1)
I=
We now rewrite eqn (9) as
_-4(n)
I . (9)
(10) The parameters of eqn (9) are usually listed in actuarial tables. Using eqns (8)-( lo), we find
(11) When the expression for AoX, obtained in Section 4.1 and eqn (2) are used in eqn (7) we obtain a very general equation describing the dynamics of a biomass system. The values of the parameters in our model can be easily obtained for any one country from the country’s gross data on population and forestry. The generalizations of our model generally introduce more parameters into the model. The utility of these more EGY IO:‘)-c
1026
M. J. MWANDOSYA and M. L. LUHANGA
general models thus depends on whether values of all of the model parameters can be obtained easily and at reasonable cost. These more general models thus require the definition and solution of a parameter estimation problem. Disaggregation of the population term via the cohort population model is not unique. The population can be disaggregated using criteria other than age groups: e.g. (i) a broad division into rural and urban populations; (ii) the income per household, thus identifying the urban rich, the urban poor, the rural rich, and the rural poor; and (iii) broad division according to cottage industry activities. 4. APPLICATION
TO
TANZANIA
The total forested area of Tanzania is about 28 X lo6 ha, which is about 32% of the total land area. The main part of the forest is savanna and intermediate type forest, covered by what is generically referred to as Miombo Woodlands. Miombo refers to an open canopy type of woodland characteristic of the drier areas of Eastern Africa between 300 and 1200 m above sea level. The predominant species are Brachstegia, Isoberlinia, Julbernadia, and Pterocarpus.2 These natural forests, from which 99% of the total fuelwood requirement in Tanzania is harvested, have rather low annual volume increments per ha, which range between 0.5 and 2.0 steres. 3,4An average value of natural regeneration of 1 stere/ha per annum has been used in the examples. It has been estimated that the potential harvest of wood from the natural forest in Tanzania, when clear felling is practiced, is of the order of 30 to 50 m3 of solid wood per ha.3,4 In the examples we adopt the lower figure of 30 m3 of solid wood per ha, and the same figure is used for yields arising from clear felling from afforestation plantations. Based on the output from nurseries, a total of 49,000 ha was estimated to have been planted in afforestation efforts between 1975 and 1981. 4.5 With an average survival rate of seedlings of 50%, we find that it is prudent to assume a base-year value of 10,000 ha for afforestation in 1985. On the demand side, several studies have been done in Tanzania. Figures for the per capita demand per annum vary from 2.6, 2.2, 2.0 to 0.7 m3.2,5 These variations seem to reflect geographical locations of survey sites, as well as availability of alternative fuels and types of foods cooked in the various survey areas. An average figure of 2.0 m3 per capita per annum is presently used in accounting for biomass in the country4 and is also used in the model as the per capita demand in the base year. We use eqn (7) to calculate V,(n) and the gross standing volume of biomass at the beginning of each slot for Tanzania, taking 1985 as the base year. We assume T = 1 and use an annual population growth rate of 3.3%, starting with a population of 2 1.7 million in 1985. We also assume that the annual rate of increase of afforested area (pa) is 0.2, starting with an area of 10,000 ha in 1985. Other values used are X, = 1, X, = 30, CY= 0.5, T = 8, and A0 = 2.8 X 10’ ha. Figures l-2 exemplify our results when using eqn (7). By varying the parameters in eqn (7) we have been able to conclude that, for Tanzania, the uses of short rotation tree species (i.e. trees with small values of T) or trees with very high survival rates (i.e. trees with high values of LY)have minimal short-term impact on both the deficit fuelwood supply and the gross standing volume of wood. The population growth rate and the afforestation rate (&, however, have considerable impact on the deficit fuelwood supply and the gross standing volume of wood. Furthermore, we have been able to show, through Figs. 1-2, that the management of fuelwood demand would have dramatic and positive consequences on the availability of fuelwood in Tanzania, provided it is coupled with an appropriate annual growth rate in afforested area. Since over 90% of wood use is accounted for by domestic fuel use, it is logical to assume that the utilization factor p, is primarily determined by the efficiency of use and, in the present case, the efficiency of cookstoves. Thus, for the nth year after the base year, the demand relation can be associated with the total number of people using more efficient stoves. If we assume that stoves leading to doubling of efficiency, compared to the base year efficiency, are gradually introduced, then
1027
An analytical model for a biomass system
I
I
I996
1991
b86
20(
t (YEAR) Fig. 1. Deficit fuelwood demand.
P,*P(n)
where the population,
= &W)
+ PoM4
when using doubled-efficiency N(n) = 2P(Ml
-
-
N~)l,
(12)
cookstoves, is
PmlPol,
(13)
and p(n) is the total population for the nth year after the base year. If we assume an average household of five people, then the number of more efficient stoves n years after the base year is N,(n) = 0.4P@r)[l -
PmlPol,
(14)
and the number of new stoves introduced in that year is NJn) - NJn -
1).
Using Figs. 1 and 2, we estimate that the number of efficient stoves should be 1.7 million by 1990 and 7 million by 2000. Assuming that an improved stove will cost $20, these represent total investments of $34 million and $140 million, respectively.
1028
M. J. MWANDOSYAand M. L. LUHANGA
80-
60 -
1986
1991
1996
201
t(YEAR) Fig. 2. Gross standing volume of fuelwood
The forestry division in Tanzania2 has estimated that an investment of $600/ha is required in village afforestation efforts. At a growth rate of pcrr= 0.2 ha/yr., an investment of $54 million will be required by the year 1990. 5. CONCLUSIONS
We have presented a linear, dynamic, deterministic model of a biomass system. Our model is also suitable for handling stochastic problems.(j Acknowledgemenf-The
authors wish to acknowledge the support of IDRC and the University of Dar es Salaam.
REFERENCES 1. M. J. Mwandosya and M. L. P. Luhanga, Energy Resources, Flows and End-Uses in Tanzania, Dar es Salaam. University Press, Dar es Salaam, Tanzania (1983). 2. E. M. Mnzava, Village Afforestation: Lessons of Experience in Tanzania. FAO Report TF-INT 271 (SWE), Rome, Italy (1980). 3. E. M. Mnzava, Unasylva 33, 24 (1981). 4. B. K. Kaale, Tanzania Five-Year National Village Afforestation Plan 1982/83-1986187. Ministry of Natural Resources and Tourism, Dar es Salaam, Tanzania (1983). 5. 1. Rikula, Report to Support Village Ajbrestation in Tanzania. Institute of Resource Assessment, University of Dar es Salaam, Tanzania (1983). 6. R. G. Leibundgut, IEEE Trans. Auto. Control AC-28, 427 (1983).