OverviewThere are several different approaches that can be taken when modelling tidal flows and hydraulics, of differing depth and complexity, and consequently time-spans. Whereas a CFD model may provide an expert with the most accurate data over a long enough time period, it was decided that a simple Excel parametric tool would facilitate sufficient insight for the purposes of ascertaining the main characteristics of a particular tidal flow regime, for a specific site, and within a more reasonable time frame. In mathematical terms, channel hydraulics can be calculated for anything from frictionless, steady-flow conditions to unsteady flow with boundary layer interaction, salinity mixing and sediment transport. In this regard, the tidal model calculation tool used was midway between the two. The motivation for this was that some of the previous efforts were simplistic and perhaps likely to overestimate the resource, or too involved for early phase feasibility studies, and more disposed to errors from inexperienced users. This then left several options for calculations:
- Bernoulli’s energy equation – easy to use, though difficult to account for frictional losses (2).
- Chézy, Kutter, Manning all share similarities and are well established for open channel flow, accounting for roughness factors through various coefficients (3)(4)(5).
- Newton’s gravitational equations (6).
Rather than single out one method, it was decided that it may be beneficial to integrate aspects of all the approaches, to possibly derive further conclusions about the channel characteristics, error compensation, and validation. Of Chézy, Kutter and Manning’s equations, it is Manning's simple equation that seems most favoured in hydraulic circles, with much research and experimentally derived coefficient data available for general consumption:
“It is absolutely remarkable that such a simple formula gives such good results”
J. B. Calvert (2003)
Thus, provision was made in the model for the simultaneous solution of both Manning’s and Bernoulli’s equations. Although as expected, Bernoulli tends to give higher velocities, and this is also site specific, having applied the model to three different west coast sites. This does however enable the calculation of a frictional coefficient for Bernoulli’s equation, which can be useful for further study. As for Newton’s gravitational equations, this theory was partially used for simple flow rejuvenation after the Significant Impact Factors (SIF) had been calculated.
A factor that arose from considering the complexity of the tide and the various governing parameters, such as:
- Interactions between the sun, moon and earth
- Electromagnetic field changes
- Centrifugal forces
- Surface resultant forces
- Land/sea interactions
- Harmonics/resonance effects
- Thermal current interactions
- Corriolis effects
These factors among others are difficult to take into these into account with any degree of accuracy. Thus, the input data for the model was from a validated, continually evolving tidal model courtesy of the UKHO tidal prediction service, which is free upon registration; negating the inherent difficulties and imbuing the tidal calculation model with realistic data.
Having located a good input data resource, and a suitable site fitting the criteria listed in the site selection part of the methodology, the tidal calculation model was then set up as follows:
- Since the tidal data was in a different format, that could
not be exported readily, Excel was set up so that sinusoidal
functions replicated the tidal height data at each port. One
draw back however was that the tides were normalised over the
period of study, effectively inducing slight error in the ebb
and flood tidal flow differences. It was deemed that this averaging
was acceptable due to its minimal effect. A second sine function
was applied to the first to allow for spring and neap variations
over the period, which in turn can be set as desired. The period
shown is for 7 days, from a new moon spring tide to a neap tide
at port Rubha A’mhnail on Jura. Purple cells represent
user input areas, and spawn changes throughout the worksheet.
This process is repeated for the second port, in this case,
port Askaig on Jura, on a separate worksheet.
- After tidal height data has been entered for both ports, Excel
will calculate the resultant phase difference of the tides,
as sufficient distance separates the two locations such that
one site will lag or lead the other during flood and ebb tides.
The resultant phase graph below shows the magnitude and effective
polarity of the head difference at the ports, which in turn
yields the energy gradient in the Manning equation, or the gravity
head in Bernoulli’s. Also, note that a little effort has
to be exerted to input all key channel dimensions and parameters
when done manually. Alternatively, these parameters can be exported
from CAD using vector software and profile mapping, assuming
the use of digital hydrographic charts (available from UKHO).
Using the the model the average bulk flow rate is calculated for the channel. A trait however of the Sound of Islay (in the Casestudy), is that the cross sectional area at the point of fastest flow (Port Askaig) is significantly smaller than at the channel entrance, which must be accounted for.
By using continuity and the venturi effect equations, the true average flow rate can be arrived at for Askaig. Using linear interpolation, and the frictional loss factor governed by the simultaneous solution of both Benoulli and Manning, a velocity of less than 2.57m/s (5 knots) should be arrived at an average spring tide. With an n value of 0.031027, the model predicts a velocity of 2.27 m/s at Askaig (Figure 3), which when applied to the velocity profile model of the channel is accurate to within 10% of the Admiralty chart quoted value. However, even using the selection guides given (7) it is a difficult process to accurately select values, and, if done incorrectly can yield high inaccuracies.
The Tidal Velocity tool can be accessed by clicking on the tool icon.
The tool consists of several sheets of an Excel spreadsheet and instructions are contained within the file.
- Inputs: Tidal heights for two ports at either end of the channel of interest. Cross sectional areas at these points, a characterisation of bottom roughness in terms of Manning coefficients.
- Outputs: Tidal velocity (average bulk flow) for a period of interest in the channel. Energy capture, kinetic energy of the body of water, gravitational effective attraction, rejuvenation, SIF and net flow alteration.
As a stand-alone tool, the tidal velocity calculation model is useful as an initial scoping study, but it is perhaps better used when validated against measured values for flow, for specific technology and SIF comparisons, as shall be discussed.
Power and Energy Capture
Further to tidal flow calculations, the Excel tool can be used to calculate power extraction and the SIF for different combinations of energy extraction devices. For more details on the parametric technology study please see the technology section. By applying the function velocity sine function over small increments of time for a tidal cycle, a sensitivity analysis of energy capture and deployment can be carried out. Also, because of the nature of the Manning’s approach, the installation of energy extraction devices in the channel stream can be iteratively modelled in terms of the blockage effects they display (5)(8), as a proportion of flow through the device and their swept area, at set efficiencies, rated power and cut in speeds.
As an initial comparison, it was decided to model a linearly increasing array of axial turbines, such that the effective portion of the whole CSA occupied by the turbines increased from 5, to 10, and then to 15, which yielded some interesting results. Since a portion of the kinetic energy flux of the channel is removed by the turbines, the flow velocity could be expected to reduce slightly, as it would by increasing the blockage of the channel. As such, it was found that after a certain point, subsequent sequential addition of turbines adversely affected the energy capture (Figure 4). It must be remembered though that the level of channel blockage and extraction at which this phenomenon occurs is substantially outwith the advocated SIF of 10%.
Figure 4. Graph of extrapolated yearly energy capture against turbine deployment.
Figure 5. Energy extraction section of the model, showing primary inputs calculation steps. Notice in cell K, M and O10 the potential SIF of each deployment ratio.
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It is potentially very useful for a developer to have tools at their disposal to quickly assess the feasibility of a site before embarking on expensive surveying (Acoustic Doppler Profiling for instance). On the same note, the tool described herein offers a reasonable level of insight into tidal flow regimes, though it must be remembered that in any modelling there is scope for error. A major assumption of this tool is that the flow does not differ too much in either direction, whereas it often can, and by a large amount depending on surrounding tidal flux. Similarly, it can be time consuming and awkward to assess the impacts of different technologies on energy capture, and almost impossible to investigate more complicated matters without the use of computing power. In this regard it is hoped that the model described might be useful as a stepping stone to more in-depth and focused investigations.
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|1||GARRAD HASSAN & PARTNERS, et al., DTI: Uk Atlas of Offshore Renewable Energy. 2004 [cited 02 May 2006]; Government renewable energy policy; Available from http://www.dti.gov.uk/renewables/renew_atlaspages.htm|
|2||«ENGEL, Y. and BOLES, M., Thermodynamics: An Engineering Approach. Mcgraw Hill Higher Education. 1989, New York, London: McGraw Hill. pp. 867. ISBN 0071250840.|
|3||CALVERT, J., Open Channel Flow. 2003 [cited 02 May 2006]; Available from http://www.du.edu/~jcalvert/tech/fluids/opench.htm|
|4||GIERKE, Michigan Tech ENG5300 Engineering Applications in the Earth Sciences: River Velocity. [PDF] 2002 [cited 02 May 2006]; Available from http://www.cee.mtu.edu/peacecorps/resources/use_of_manning_equation_for_measuring_river_velocity.pdf|
|5||CHOW, V.T., Open-Channel Hydraulics. Mcgraw-Hill Civil Engineering Series. 1959, New York, London: McGraw-Hill Education. pp. 692. ISBN 007085906X.|
|6||BOWDITCH, N., The American Practical Navigator Chapter 9: Tides and Tidal Currents, and Nautical Charts. [cited 02 May 2006]; IRBS Available from http://www.irbs.com/bowditch/pdf/chapt09.pdf|
|7||ARCEMENT, G. and SCHNEIDER, V., USGS Water-supply Paper 2339: Guide for Selecting Manningís Roughness Coefficients for Natural Channels and Flood Plains, 1989 United States Geological Survey|
|8||CHARBENEAU, R., et al., US Dept. of Transportation: Backwater Effects of Piers in Subcritical Flow. [PDF] 2001 [cited 02 May 2006]; Available from http://www.utexas.edu/research/ctr/pdf_reports/1805_1.pdf|
- Bournemouth University: Exponentially Changing Sinusoids
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