CHAPTER 94 THE ECONOMICAL ECONOMICAL PENSTOCK PENSTOCK DIAMETER
Different diameters d may be considered for a penstock required to carry a given discharge Q. Although the weight and thus the first cost of the penstock increase parallel with increasing diameter, the output in electrical energy is also increased owing to the reduction in frictional head loss. The economic diameter may be determined on the basis of the following considerations: the economically justified diameter for a penstock required to carry a given flow Q is the one at which annual costs due to the greater investment do not exceed the annual value of the resulting increment energy output. The governing criterion is thus to regain economically the last increment kilowatt-hour made available by reducing the head loss through using a larger diameter. Mathematically this criterion of economical nature may be expressed by the relation
where C1 is the annual cost due to the investment for a pipe of any diameter d, and C 2 is the value of energy that can be produced at the same diameter d. With H metres, denoting the design head for the penstock section under consideration, the necessary shell thickness is from Eq. (9/92)
and the weight of the penstock section having a length / cm is
where γ1 is the specific weight of steel: 7.85 kg/dm 3. Considering that the installed weight of the penstock is higher by about 20 per cent due to the weight of joints, rivets, etc., we may write
Denoting the specific average first cost for the penstock Co [$/kg] and the annual operating charges including the depreciation, maintenance by a per cent the average annual specific cost of the penstock is
The annual cost of the penstock having a diameter d is thus:
Let us now consider the influence of the changes in the diameter d on the annual energy output and its value. Using Eqs (10/90) and (11/90) the resistance of the pipe having a diameter d cm and a length I cm, and delivering Q m3/sec may be written as
(Factor 1005 is introduced since d has been substituted in metres into the original expression). The power output may be considered as the difference of an initial No and the power loss due to frictional resistance and written into the form:
(Reference is made to Eq. (1/11) derived in Volume I according to which the power output is 3 obtained from the expression 9.8η Q∆h in kW, if the discharge is introduced in m /sec and the head in m units). Assuming the overall efficiency of the development as 0.77 and denoting the average annual duration of operation by / hours, the annual energy output is, after reductions and rearrangements,
which, when substituting, the expression of ∆h assumes the final form If the value of power at the generator terminals is c 2 $/kWh, the annual energy produced may be represented by the sum:
Differentiating Eqs (3/94) and (4/94) with respect to d and substituting into Eq. (1/94), we obtain
Rearranging terms and rounding off
and
or
wherein λ = the friction coefficient, values of which have been given in Chapter 90 (may be taken as 0.02 for preliminary estimates} 2 σ a= the allowable stress for steel in kg/cm t = the annual duration of operation in hours, c2 = the value of one kWh at the generator terminals, 3 Q = the discharge conveyed by the penstock in m /sec, c1 = the annual cost of 1 kg weight of the penstock in the same currency units as C2, and H = the design head in m. lt should be pointed out that the investigation is more involved if the utilized discharge is subject to considerable fluctuations, in as much as under such circumstances both Q and / are to be determined by auxiliary computations before being entered into Eq. (5/94). As can be seen from Eq. (5/94) the economic diameter depends upon the head thus upon the elevation of the pipe section under consideration. Theoretically, the relationship indicates the necessity of gradually reducing the penstock diameter towards the lower end but this, and even the use of small decrements, is not feasible for practical reasons. When applying therefore Eq. (5/94) to any proposed installation, the head pertaining to the central section of the investigated pipe length, i.e. the mean head should be introduced. Reductions in diameter are practicably carried out at anchorages. Decrements of at least 50 to 100 mm are commonly used, the shell thickness is specified in mm. Neglecting slight variations in λ with the diameter, each step in the latter can be computed in keeping with Eq.(5/94) from the relationship
where H, respectively Hi are the mean heads pertaining to the compared sections. Constituents of the annual charges are: Depreciation for a useful life 50 to 33 years………………………………………………………. 2 to 3% Annual maintenance…………………………………………………………………………………..3 to 5% Other charges (cost of money, etc.)………………………………………………………………... 2 to 4% Thus coefficient "a"…………………………………………………………………………………..7 to 12% In estimating the cost co it should be considered that the cost of shop-welded pipe sections is about 1.5 to 2.0 times that of steel plates delivered from the mill. The higher coefficient applies to longer sections designed for higher heads. Erection may be taken as from 20 to 25 per cent of the cost of fabrication, but may be considerably higher. (E.g. if longitudinal joints must be prepared at the site, i.e. if half cylinders only can be transported). The method of investigation described above and referred to on several occasions in the literature should, of course, not be accepted without criticism, in as much as the validity of the basic considerations may justly be questioned. It is not settled yet whether the diameter at which an infinitely small additional investment is balanced by the increment return, should actually be accepted as the most economical. The cost of the penstock represents an item of varying significance relative to the total cost of the development and it is the mean unit cost of energy produced that is eventually of decisive importance. In some instances the unit cost of energy production may be very low and in order to increase energy output a relative investment higher than the limit defined in the foregoing may be justified for the penstock without raising the mean unit cost of energy above the permissible value. This consideration will become clear if it is remembered that, under exceptional conditions, the investment required for the penstock (or penstocks) may be as low as from 3 to 5 per cent of that of the entire development. Under average conditions the cost of the penstock amounts to about 5 to 10 per cent of the total investment, yet in some cases the share of the penstock was from 15 to 20 per cent. As demonstrated by A W. K. Billings, the total cost of the penstock is at high-head developments in many cases equal to the aggregate cost of all mechanical and electrical equipment (including the switchyard and the transformer station). In connection with the above deduction the following should be remembered: the treatment of the head H as a constant value is an approximation, since changes in the pipe diameter involve changes in the flow velocity as well as in the dynamic pressure component caused by water hammer. For determining the economic diameter the graph shown in Fig . 1/94 has been proposed by G. Ferrand. The graph represents a function of two variables d = f (Q, H)
in the form of a family of curves d = φ (Q) for different H = const. values between 150 and 2000 m. The graph provides information as to the mean diameter. In connection with the economic analysis of the penstock, it should be noted that by dividing the discharge to 2, 3, 4, 5 penstocks of equal diameter, their cost is increased to 1.10, 1.17, 1.22, 1.26 times that of a single pipe, respectively. (After W. Bauersfeld and A. Schoklitsch). It should be emphasized in conclusion that any diameter determined by the method described above, or on the basis of other considerations of an economic character, is not feasible unless requirements of fabrication, handling, transport and installation as well as limitations as regards the velocity of flow are complied with. For a more accurate analysis of the economical penstock diameter the reader is referred to the exact mathematical transaction of J. O. de Mello Flores Owe bibliography).
