Idealized models for power potential
4.2 Balance between time derivative of veloc- veloc-ity, pressure gradient and turbine friction
Back to 3.28 where only the advection term is neglected this time. From the solutions obtained in this section it will be possible to decide the importance of the time derivate of the velocity.
4.2 Balance between time derivative of velocity, pressure gradient and
turbine friction 37
Again, as in Section 4.1, it is assumed that the channel length is short com-pared to the wave length of the tide, so that the volume flux is constant through Rystraumen and is given by Q(t)) = Au, where A is the cross sec-tional area of the channel anduis the current velocity. Substituting this into the time derivative term in 3.28, the following equation is obtained:
1 A
Q.
dt +g∂ζ
∂x =Fx (4.12)
where the balance now is between the time derivative, the pressure gradient and the resistance due to turbine friction.
The equation has been integrated along the channel length to give the fol-lowing: the surface elevation in the basin outside the channel, and ζB is the surface elevation on the east side of the channel. As for Section 4.1 the surface elevationζ0 in the large basin is unaffected by energy extraction and is given as the M2 tidal constituent whereζ0 =acos(ωt). The surface elevationζB at the other end of the channel is not unaffected by energy extraction from the current. The relationship between the surface elevation ζB and the volume flux through Rystraumen is given by 4.4.
The natural friction along the bottom and channel walls is neglected for both the linear case and the quadratic case following and the term Fx represents only the turbine friction.
Linear Friction
As a start the turbine friction is assumed to be linearly proportional with the current velocity and is given by 4.5 in Section 4.1. and is inserted into 4.13 and together with 4.4 forms the following system of differential equations:
cdQ
dt +g(ζB−acos(ωt)) =−λLQ (4.14)
and
AB∂ζB
∂t =Q (4.15)
where c is given as
c= Z L
0
1
Adx (4.16)
As in Garrett & Cummins (2005), the integral in 4.16 is evaluated at the narrowest part of the channel with the cross sectional area A and length L. The constants are given in Table 4.1. and the differential equations are solved in matlab with the solver ode23 for non stiff differential equations. The decrease in average volume flux asλL increases is presented in Figure 4.6.
The power extracted from the current when the turbine friction is linear, is given as in 4.7 in Section 4.1. The result is given in Figure 4.7. As the time derivative term is of an order one ten times smaller than the pressure gradient, it is expected that the result obtained here will be very similar to the result in Section 4.1. The maximum power potential is estimated to be 89 MW and is achieved when the average volume flux is reduced to 16 490 m3/s. The reduction from the estiamte given in Section 4.1 is larger than what expected and indicates that the time derivative might be of higher order, or the pressure gradient of lower. It should also be noted that the solution of the set of equation given in Figure 4.6 appears to be unstable for very smallλL. This can be seen from the graph as it is not completely smooth and close toλL= 0 the solution makes a jump. But as the graph smoothens out for largerλL the solution is assumed to be correct and therefore the time derivative of greater importance than predicted.
Quadratic Friction
In Section 4.1 it is shown that the maximum power potential for quadratic friction is reduced to 98% of the maximum power potential for linear friction.
Hence it is also assumed that introducing quadratic friction to 4.13, will reduce the estimated maximum power potential. The friction integrated over the length of the channel is given as 4.9 in 4.1. This gives the following system of differential equations:
4.2 Balance between time derivative of velocity, pressure gradient and
turbine friction 39
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−3 0.5
1 1.5 2 2.5
3x 104 Average volume flux with increasing friction parameter
Lambda Average Volume Flux (m3/s)
Figure 4.6: Average volume flux calculated from 4.14 and 4.15 with a balance between the time derivative of the volume flux, the pressure gradient and the re-sistance due to linear friction, for an increasing λL. The red square indicates the average volume flux when maximum power potential is reached, this is given in Figure 4.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 10−3 0
1 2 3 4 5 6 7 8
9x 107 Power calculated from avereage volume flux
Lambda
Power (watt)
Figure 4.7: Power calculated from the 4.7 where the volume flux is calculated from 4.14 and 4.15 where the turbine friction is linear proportional to the velocity.
cdQ
dt +g(ζB−acos(ωt)) =−λS|Q|Q (4.17) and
A∂ζB
∂t =Q (4.18)
where c is given as in 4.16. The constants are given in Table 4.1, and the system of differential euqations is solved numerically in matlab with the same solver as for the linear case. The result is presented in Figure 4.8.
The power is calculated from 4.11 and from the results presented in Fig-ure 4.9. The maximum power potential is 72 MW and is achieved when the average volume flux is reduced to 14 492m3/s, which is represented as a red square in Figure 4.8.
Similar to the solution of the linear case the the graph in Figure 4.8 is not smooth and indicates that the solution is not stable. The quadratic repre-sentation of the turbine friction reduces the power potential to 80 % of the
4.2 Balance between time derivative of velocity, pressure gradient and
turbine friction 41
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x 10−7 0.6
0.8 1 1.2 1.4 1.6 1.8 2 2.2
2.4x 104 Average volume flux with increasing friction parameter
Lambda Average Volume Flux (m3/s)
Figure 4.8: Average volume flux calculated from 4.17 and 4.18 with a balance between the time derivative of the volume flux, the pressure gradient and the re-sistance due to quadratic friction, for an increasing λS. The red square indicates the average volume flux when maximum power potential is reached. This is given in Figure 4.9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
8x 107 Power calculated from avereage volume flux
Lambda
Power (watt)
Figure 4.9: Power calculated from the 4.11. The volume flux is calculated from 4.17 and 4.18. The turbine friction is quadratic proportional to the velocity.
power potential given for the linear case.