Table 1 Models to size the diameter of a penstock
From: Design models for small run-of-river hydropower plants: a review
Parameter | Model name | Description | Â |
|---|---|---|---|
Velocity of water | Velocity of water | Model | \(D = \sqrt {\frac{4Q}{{V\pi }}} \quad \quad (5)\) \(V\): Average velocity in the penstock (m/s) \(Q\): Design flow (m3/s) |
Description | This equation is usually used to make sure the velocity of water in the penstock is not lower or higher than some recommended values because otherwise is, it can cause loss in power output (and thus be uneconomical) or an unacceptable surge pressure. | ||
Limitations | With this methodology, the velocity of water in the penstock cannot be less than 1Â m/s or superior to 5Â m/s. This method does not seem to consider enough economic requirements. | ||
Comment | Edeoja et al. (2016) said the velocity in the penstock of a small hydropower system is typically 3 m/s. Obinna et al. (2017) also mentioned that for low heads small hydropower projects, the velocity should not exceed 1–3 m/s to maintain head losses and surge pressure within acceptable limits. In the same way, a first estimation of the diameter of a penstock of a small hydropower plant can be done fixing the maximum velocity flow (2–3 m/s for low-head plants, 3–4 m/s for medium head plants and 4–5 m/s for high head plants) (Ramos et al., 2000). | ||
The thickness, the maximum head and stress of the material | The thickness, the maximum head and stress of the material | Model | \(H = \frac{0.002 + \sigma e}{{D + 0.002e}}\quad \quad (6)\) \(e\): Thickness (m), \(H\): Maximum pressure head (m) \(\sigma\): Stress of the material (ton/m2) |
Description | Â | ||
Limitations | To use this method, the thickness and the stress of the material of the penstock. This method does not seem to consider economic requirements. | ||
Comment | This formula was used for the design of a micro hydro electrical power at Brang Rea river in West Sumbawa of Indonesia (Hoesein & Montarcih, 2011). | ||
Economic requirements | Analytical—only friction losses | Model | \(D^{{^{22/3} }} = \frac{{2.36 \times 10^{6} Q^{3} n^{2} ep_{{\text{f}}} C_{{\text{p}}} }}{{\left[ {1.39C_{{\text{e}}} + 0.6C_{{\text{c}}} + \left( {\frac{{121H_{{\text{g}}} C_{{\text{s}}} \left( {1 + i} \right)}}{{\sigma e_{{\text{j}}} }}} \right)} \right]p}}\quad \quad (7)\) \(\sigma\): Permissible stress in penstock \(C_{{\text{e}}}\): Cost of excavation / cum for laying penstock \(C_{{\text{c}}}\): Unit rate of concrete lining \(C_{{\text{p}}}\): Cost of 1 kWh of energy \(e_{j}\): Joint efficiency of penstock (–) \(e\): Turbine/generator efficiency (–) \(p\): Ratio of annual charges to installation cost of penstock (–) \(i\): Ratio of weight of stiffeners and weight of penstock (–) \(p_{{\text{f}}}\): Annual load factor/Plant load factor (–) \(Q\): Penstock discharge (m3/s) \(H_{{\text{g}}}\): Gross head (m) |
Description | The purpose here is to minimize the total annual expenditure on penstock considering only friction loss. | ||
Limitations | Many parameters whose values are not easily accessible and necessary for the use of these formulae (cost of excavation, unit rate of concrete lining, ratio of annual charges to installation cost of penstock, ratio of weight of stiffeners to weight of penstock, annual load factor/plant load factor). | ||
Comment | Singhal and Kumar (2015) presented these relations in their work on optimum design of penstock for hydro-projects. | ||
Total head loss | Analytical—total head loss | Model | \(D^{7} = \frac{{0.04627 \times 10^{6} Q^{3} \lambda ep_{{\text{f}}} C_{{\text{p}}} \left( {\frac{L}{{H_{{\text{g}}} }}} \right)^{ - 0.19} }}{{\left[ {1.39C_{{\text{e}}} + 0.6C_{{\text{c}}} + \left( {\frac{{121H_{{\text{g}}} C_{{\text{s}}} \left( {1 + i} \right)}}{{\sigma e_{{\text{j}}} }}} \right)} \right]p}}\quad \quad (8)\)See Analytical—only friction losses for the parameter description |
Description | The purpose here is to minimize the total annual expenditure on penstock considering total head loss. | ||
Limitations | Same limitations for Analytical—only friction losses apply | ||
Comment | Singhal and Kumar (2015) presented these relations in their work on optimum design of penstock for hydro-projects. | ||
Penstock diameter | Empirical penstock Diameter | Model | \(D_{{\text{e}}} = C_{{{\text{EC}}}} C_{{{\text{MP}}}} Q^{0.43} H_{{\text{n}}}^{ - 0.24} \quad \quad (9)\) \((\text{Warnick et al }.)\)\(D_{e} = 0.72Q^{0.5} \quad \quad (10)\) \(({\text{Bier}}^{\prime}\text{s equation})\)\(D_{{\text{e}}} = 0.176(P/Q)^{0.466} \quad \quad (11)\) \(({\text{Sarkaria}}^{\prime}\text{s equation})\)\(D_{e} = \frac{{0.71P^{0.43} }}{{H_{n}^{0.65} }}\quad \quad (12)\) \((\text{Moffat et al}.)\)\(D_{{\text{e}}} = \frac{{0.52P^{0.48} }}{{H_{n}^{0.6} }}\quad \quad (13)\) \((\text{Voetsch and Fresen})\)\(D_{{\text{e}}} = B \times D^{\prime}\quad \quad (14)\) For a discharge of more than 0.56 m3/s and with \(B\) and Dʹ selected through \(K\) from appropriate graph. Where \(K = \frac{{k_{s} e\lambda Se_{j} b}}{{ar\left( {1 + n_{s} } \right)}}\quad \quad (15)\) (USBR’s equation) \(D_{{\text{e}}} = \frac{{1.517Q^{0.5} }}{{H_{n}^{0.25} }}\quad \quad (16)\) (Fahlbusch's equation) \(D_{e} = \frac{{1.12Q^{0.45} }}{{H_{n}^{0.13} }}\quad \quad (17)\) \(D_{e} = \frac{{0.05(SMh\lambda eQ^{3} P_{wf} )}}{{WCH_{n} }}\quad \quad (18)\) (ASCE’s equation) with \(P_{{{\text{wf}}}} = \frac{{((1 + {\text{int}})^{{{\text{nr}}}} - 1)}}{{({\text{int}}(1 + {\text{int}})^{{{\text{nr}}}} }}\quad \quad (19)\) \(C_{{{\text{EC}}}}\): Coefficient of energy cost (zones where the energy cost is low = 1.2, medium = 1.3 and high or no alternative source = 1) \(C_{{{\text{MP}}}}\): Coefficient for the material of the penstock (for steel = 1; for wood = 1.05 – 1.1; or plastics = 0.35 – 0.4) \(Q\): Design discharge (m3/s) \(P\): Installed capacity (kW) \(H_{n}\): Rated head (m) \(a\): Cost of pipe in $ per lb \(b\): Cost of 1 kWh of energy ($) \(e\): Turbine/generator efficiency (–) \(e_{j}\): Joint efficiency of penstock (–) \(r\): Ratio of annual charges to installation cost of penstock (–) \(n_{s}\): Ratio of weight of stiffeners and weight of penstock (–) \(S\): Allowable stress (psi) \(k_{s}\): Scobey friction factor (–) \(\lambda\): Friction factor (–) \(h\): Annual hours of operation (hours) \(P_{{{\text{wf}}}}\): Present worth factor (–) \(W\): Specific weight of steel \(C\): Capital cost of penstock installed per unit weight \(M\): Composite value of power \({\text{int}}\): Interest rate (%) \({\text{nr}}\): Repayment period |
| Â | Â | Description | These relations are developed by analyzing and correlating statistical data of existing/installed projects designed as per past practice because of data from existing penstocks of specific places. |
| Â | Â | Limitations | The calibration of the parameters for different context may be complex to perform or not available. For these models, the physical phenomena involved in the process are not described because they come from the systematization and the generalization of experience. It is better to use this type of model while knowing the specific contexts and situations of their elaboration. Unfortunately, this is not the case here because these equations have been presented without information on the contexts and situations of their elaboration. |
|  |  | Comment | Ramos et al. (2000) mentioned Eq. (5) which is used by Obinna et al. (2017) to calculate a first value of the diameter. This diameter will be optimized by adding or removing from it through iteration to get the best diameter that will limit a 4% power loss. Singhal and Kumar (2015) mentioned the rest of the equations in their study on the Optimum Design of Penstock for Hydro Projects. |
Minimum head loss | Minimum head loss | Model | \(D_{{\text{e}}} = 2.69(\frac{{n^{2} Q^{2} L}}{{H_{{\text{g}}} }})^{0.1875} \quad \quad (20)\) \(n\): Manning’s coefficient (–) \(L\): Length of penstock (m) \(Q\): Optimum discharge(m.3/s) \(H_{{\text{g}}}\): Gross head (m) \({\Delta H}_{\text{S}}\): Frction loss (m) |
Description | The purpose here is to limit the friction loss to 4% of the gross head (\({\Delta H}_{\text{S}}\)Â at 4 \({H}_{\text{g}}\) /100). | ||
Limitations | To use this method, the length of the penstock, should be known. This method does not seem to take economic requirements. Into proper consideration | ||
Comment | This method, which consists on limiting the head loss to 4% while using Manning equation, is mentioned in the guide of ESHA (European Small Hydropower Association) (Penche, 2004) and by Nasir (2014) in his work on the Design considerations of Micro-Hydro-Electric Power Plant. This method was also used for feasibility studies of Micro hydro-power plant projects in Cameroon (Kengne Signe et al., 2017a; Kengne Signe et al., 2017b). | ||
The cost and the slope of the penstock | The cost and the slope of the penstock | Model | \(D_{{{\text{opt}}}} = \left[ {\frac{{\lambda Q^{2} }}{{2g\left( {\frac{\pi }{4}} \right)^{2} \left( {\frac{h}{{H_{{\text{g}}} }}} \right)S}}} \right]^{1/5} \quad \quad (21)\) λ Friction factor determined by the surface roughness of the penstock material (–), \(H_{{\text{g}}}\): Gross head (m) L: Length of penstock (m) \(S\): Penstock slope = \({H}_{\text{g}}\)/L (–) Q: Optimum discharge (m3/s) \(h\): Head loss (m) |
Description | The purpose here is to achieve the maximum power for the minimum capital investment. Differentiating the power per unit cost with respect to the flow rate Q, and equating to zero shows that the maximum power per unit cost occurs when the head loss h = \({H}_{\text{g}}\)/3. | ||
Limitations | To use this method, the slope of the penstock (gross head divided by the length) as well as the friction factor usually calculate with the diameter | ||
Comment | Alexander and Giddens (2008) presents this method in their analysis for penstock optimization. Edeoja et al. (2015) also used this method in their work on the Conceptual Design of a Simplified Decentralized Pico. Hydropower with Provision for Recycling Water and by Edeoja et al. (2016) in their work on the Investigation of the Effect of Penstock Configuration on the Performance of a Simplified Pico-hydro System. |