University of Strathclyde Small Scale Hydro: Novel Approaches to Generation & Transmission

Hillside

Individual Systems

This section contains an overview of the main calculations that were utilised in the Excel hydraulic performance calculator. The aim was to develop an Excel spreadsheet that would have the capability of accurately calculating the energy yield of hydro schemes given that they used input flows which are susceptible to seasonal variations. As the economics and performance of a common pressure header are found to be highly dependent on the geographical layout of the scheme, it proved difficult to develop a tool that would model this for general conditions. It was decided therefore that it would be of more benefit to model the economics and performance of individual systems and, through accurate and thorough assumptions, the effect of joining them in a combined pressure header configuration. This section details the calculations that are of significance to this process. Figure 1 outlines the process the group followed during the development of the mathematical modelling tool. This method was developed in order to better assess the initial assumptions that any calculations were based

The spreadsheet was developed in order to calculate the resulting hydraulic performance from the given site parameters of head height, hydraulic gradient and monthly flow exceedance statistics – which were evaluated for a range of pipe diameters from 0.1m< D <1m. Figure 2 shows how the characteristic site parameters are held in the spreadsheet whilst Figure 3 indicates

Initially, the Colebrook-White equation (1.1) was used to evaluate the potential flow that could be taken under gravity for a given pipe diameter and head height.

Where:

  • U= velocity (m/s)
  • g = acceleration due to gravity (m/s2)
  • e/D = relative roughness height (mm)
  • D = pipe diameter (m)
  • ν = kinematic viscosity of fluid (m2/s)
  • S = hydraulic gradient from intake to turbine

Using this form of the Colebrook-White equation, the maximum discharge rate of a pipe of diameter D and hydraulic gradient S could be calculated. This maximum discharge rate is the assumed to be the maximum that the pipe is capable of carrying, and is therefore used as the upper limit for the subsequent calculations. The head loss due to friction (hf) for the pipe diameter, D, and length, L, is then calculated using the Darcy Weisbach equation (1.2)

In order to calculate the head loss, one must initially evaluate the Darcy friction factor, f, using an implicit form of the Colebrook-White equation (1.3). This is calculated in Excel and is found to regularly converge after 4 iterations.

The head loss for the given pipe/flow parameters can then be calculated for each statistical flow rate given in the monthly exceedance data. Figure 4 is indicative of the calculations and results obtained in this section of analysis.

This process is carried out across data for the entire year to evaluate the statistically probable flow that would be available to drive a turbine upon exit from the pipe system. The spreadsheet evaluates that if the net head = 0, i.e. the head= head loss, then pipe must be choked due to its narrow diameter. The user is required to input a turbine rating, which is used by the spreadsheet to calculate the amount of electrical power such a turbine/pipe/flow configuration can potentially generate over a year

The calculations are evaluated for numerous pipe sizes in order to ascertain at which diameter head loss due to friction has little effect on power generated. The spreadsheet outputs graphical data to indicate this to the user, as shown in Figure 5.


Having evaluated the performance of individual systems, the analysis could now begin to consider a pipe network leading to a common pressure header. Details of the pipe network analysis can be found here.