Distribution of the Hydraulic Potential in a Hydrographic Basin
Omar Campos Ferreira.
In the nº 36 issue of the periodical Economy & Energy we have presented a model for the morphological study of hydrographic basins  using the entropy concept in the version formulated by Boltzmann for Statistical Mechanics. The model was developed by Yang  from the expression of specific entropy (by unit mass of water) for a sub-basin:
S = k [Σu (Δhu/H)],
where hu is the average fall of the sub-basin streams of order u, H is the total fall of the basin (level difference between the highest stream headwaters and the lowest stream mouth) and k is a proportionality constant. This expression is similar to the Classical Thermodynamic relative to phenomenon of heat exchange between bodies with different temperatures:
S ≥ Σ(Q/T),
where the heat exchanged is substituted by the potential energy variation (proportional to Δhu) and the absolute temperature, by the fall H. An important result of Yang’s work is that the average fall per sub-basin tends to be constant as the system comes closer to the dynamic equilibrium. Therefore, the fall constancy would be an indicator of maturity of the system. Applied to 14 American basins, the model confirmed this property with a relative deviation of about 0.5%.
The proposed analogy is an interesting starting point for extending the entropy concept to phenomena whose interpretation does not necessarily requires heat exchange but involves relevant conceptual difficulties such as:
- The classical entropy concept refers to closed systems (without mass exchange) and its extension to open systems, like the hydrographic basin that gets rainwater, needs a differentiated treatment;
- The entropy of a thermodynamic system is an extensive variable, that is, it is proportional to the mass of the body that receives or gives heat; the entropy used by Yang refers to the unit of water mass, what seems an artifice to conciliate the treatment of open systems with the methodology developed for closed systems, since the unit mass is constant by definition.
The present work aims at circumvent these difficulties using the more general concept of energy dissipation or of irreversibility applicable to any system and conserving the logical sequence of Yang’s work.
This new approach permits to evaluate the distribution of the hydraulic potential per sub-basins and would be useful in the preliminary estimation of a large basin in order to build small power plants (SPP). Hydraulic generation is still the best option for Brazil from the economic and environmental points of view and the small power plants can be well justified in a scenario of high prices of fossil fuels .
The system considered is a water stream limited by input and output sections, by channel and by the water-air separation surface. The system exchanges energy (potential energy of the water that flows from the soil and the drainage work against passive resistances)  and matter (water and material resulting from soil abrasion) with the surroundings through the border.
The method for treating an open system considers two terms concerning entropy variation: the entropy generated by irreversible phenomena inside the system (drainage against passive resistances) and convective flows of the entropy associated with energy and matter exchanges between the system and the surroundings. The generated entropy always grows as in the isolated system and therefore tends to an absolute maximum but the total entropy variation can be positive, negative or null, depending on the surroundings conditions. If the net entropy flux (input less output) compensates the internal generation, the system reaches one or more equilibrium states in its path to the final static equilibrium, each one corresponding to a relative maximum of the entropy and a given condition of the surroundings .
Adaptation of Yang’s model to describe the hydraulic potential distribution.
In the present work we have expressed the entropy generation in terms of the mechanical irreversibility, understood as the loss of work generation capacity due to irreversible phenomena.
The basic equation of the model for open system (infinitesimal part of the stream) is:
dSgenerated + dSconvection ≥ 0
For a sub-basin , the equation above is integrated and becomes:
ΔSger. + ΔSconv. ≥ 0
In a steady state,
ΔSger. + ΔSconv.= 0
which means that the entropy may remain constant, in which case the system would not reach equilibrium, while the entropy flow compensates the generation.
The entropy generation is due, in case of drainage, to the resistance presented by the surroundings to the water movement (friction and viscosity) and by the soil erosion work and transport of suspended material. In the absence of these effects the kinetic energy gain (ΔKrev) in a part of the stream is equal to the potential energy decrease (ΔP):
ΔKrev + ΔP = Δ(Krev + P) = 0
In the real case,
Δ(Kirrev.+ P) < 0,
The mechanical irreversibility would then be measured by the difference:
I = ΔKrev - ΔKirrev. = |ΔP - ΔKirrev| | = |ΔP|. (1 - α ) (1)
In the thermodynamic analog the irreversibility associated with the spontaneous heat exchange between two hot sources with temperatures T1 and T 2, is:
I = Q [(1 -T0/T1) – (1 - T0/T2)] = Q (1/T1 - 1/T2),
That is the expression of loss of the capacity to do work in a Carnot cycle where T0 is the temperature of the cold source. In the proposed analogy, Q is substituted by ΔP and the absolute temperature corresponds to the fall of the sub-basin.
Entropy according to Boltzmann.
In order to use Boltzmann’s entropy concept it is necessary to introduce the thermodynamic probability concept. The state of a thermodynamic system can be represented by a global variable (macroscopic) or by a set of variables that describe the behavior of the elements of the system (microscopic variable). If the system contains a large number of elements (in the case, water and soil particles) that interact through elementary mechanisms (collision, gravitational attraction, etc.) the microscopic description is virtually impossible. So, in economic studies we have macroscopic variables such as gross domestic product, internal saving, commercial balance, etc., that represent observable global values. From the macroscopic description of the system evolution, it is possible to speculate about values of the microscopic variables (income distribution by social classes, participation of a specific production sector in the GDP, etc.).
For each value of the macroscopic variable there are different combinations of microscopic variables values that satisfy given conditions (number of elements, internal energy, mass, etc.). The thermodynamic probability of the macro-state is defined as the ratio between the number of microstates that satisfy the given conditions, specific for each type of problem, and the total possible microstates. The entropy of the system as a function of the thermodynamic probability p of the macro-state in the Statistical Mechanics is:
S = k ln p
For the purpose of the present study, the state of the basin is represented by the total potential ΔP, considered as the macroscopic variable of the state. For each value of ΔP there are various sets of potential values per sub-basin (Δpu) that satisfy the equation:
ΔP = Σu ΔPu. (2)
Therefore, the thermodynamic probability p of the macro-state will be the ratio between the number of the sets of Δpu values that satisfy equation (2) and the number of all sets of possible values.
The direct calculation of the dissipated energy would require the knowledge of the water kinetic energy, the local declivity, the soil properties, etc., what renders the desired study very difficult since these properties vary at each part of each stream of the basin. In principle, it would suffice to assume the proportionality suggested by equation (1) with constant α and use the dissipation probability distribution as the potential distribution representation. Assuming that the elementary probabilities pu (of irreversibility occurrence) are independent one another, the probability of a given distribution per sub-basin is equal to the product of the elementary probabilities:
p = p1. p2. p3...pm (m is the largest u)
The entropy of the system is therefore,
S = k ln (p1. p2. p3 …pm) = ∑1m ln pu.
The maximum entropy corresponds to the maximum thermodynamic probability since the logarithm is a crescent function of the argument. Considering the probability of irreversibility occurrence as a continuous function of the problem’s variable, given the large number of possible interactions, the condition of maximum entropy occurrence can be written as:
dp = 0 (3)
subject to the condition
(Σu pu) = 1 (definition of probability). (4)
Equation (3) is equivalent to:
Σu (∂p/∂pu) dpu = 0
Σu (dpu /pu) = 0 (5)
For pu finite, the solution of (5) is:
dpu = 0 or pu = constant.
Therefore, the elementary probability is independent of the considered sub-basin order. Using condition (4), one calculates:
pu = 1/m and ΔPu = ΔP/m.
So, one verifies that in the equilibrium states, the basin potential is uniformly distributed among the sub-basins . In the vicinities of the equilibrium state the uniformity of the potential distribution is more or less approximate, depending on the “entropic distance” to this state. This fact could be corroborated by the cartographic comparison of a set of basins in different evolution stages, which requires a laborious but simple study.
For the practical objectives of the present study, it seems sufficient to examine the potential distribution in basins already well used, considering the installed hydroelectric power distribution as a sample of the former, since we don’t have appropriate morphological data. An interesting example concerning the Brazilian basins is that of the Paraná river with about 65% of its potential in operation and about 86% of the potential inventoried. It is expected that the power distribution of the plants in operation by sub-basin can be used for an approximate verification of the results obtained.
Table: Power plants of the Paraná River.
Data from the table show that the installed power in the largest stream (Rio Paraná) – 19,391 MW – is approximately equal to the sum of the power installed in the sub-basins of the next inferior order – 23,920 MW, with 23% approximation, the same order of magnitude obtained by Yang for some Horton’s parameters. As we have shown in the previous article, the empirical laws are presented in a logarithmic scale, so that a good correlation coefficient is not necessarily translated into small deviations when the graphic is converted to a metric scale .
The evolution of hydrographic basins implies periods in a geological scale, whereas the waterfall use for electricity generation is a relatively recent process. Horton’s empiric laws represent an approximation for long term processes and should be used with the necessary reservations. With these considerations, it is clear that the estimation of the potential concerning small plants does not substitute field inventory but it gives preliminary information relative to the capacity of these plants as a complement of the large power plants for satisfying the regional electricity demand.
The validity of the potential distribution for all sub-basins should be verified through a study of the morphology, similar to the one made for the Santo Antonio river but it is out of the scope of the present work since it requires a careful examination regarding the ordering of the streams. If the potential constancy per sub-basin is confirmed, the generation capacity of the SPP largely exceeds the current estimates.
 “Aplicação da Termodinâmica no Estudo da Morfologia da Bacia do Rio Santo Antonio”, Moreira, C.V.R., Martins Jr., P.P, Ferreira, O.C., 2002.
 “Potential Energy and Stream Morphology”, Yang, C.T, Water Research Study, 1971.
 The original notation has been changed to ease the text typing. The integral presented in the previous article and which has been substituted by the sum defines the entropy variation between a reference state and the generic state of the system. By assuming that the entropy in the reference state is zero, it is irrelevant to use entropy or entropy variation.
 In Europe, the average power of small power plants is about 0.6 MW (apud “Geração de Energia Elétrica no Brasil: Histórico e Perspectivas”. Amaral, C.A, MSc Thesis in Energy Planning, UFMG, 1998).
 As a first approximation, the heat exchanges are considered as minor contributions.
 The dynamic equilibrium is equivalent to a permanent regimen used in the engineering text books. In this state the energy and entropy flows are considered as time-independent.
 The ordering criterion designates the singular streams (without effluents) as streams of order 1. The junction of 2 flows of order 1 forms a flow of order 2 and so forth. Sub-basin of order u is the set of flows of this order.
 Other restrictions, if known, could modify the distribution. In all cases where there are no restrictions, besides that expressed by the probability definition, the less biased inference is that of uniform distribution (Principle of Insufficient Reason). It should be observed that changing the total potential value by pluvial modification along some hydrologic cycles or by antropic intervention results in distribution evolution until a new equilibrium state is reached.
 It should be noted that the standard deviation is: (σln x)2 = (d/dx lnx σx)2 = (1/x σx)2 . For x>1, σln x< σx , that is, a logarithmic graphic attenuates the sum of the standard deviations.
Graphic Edition/Edição Gráfica:
Tuesday, 11 November 2008.