Question 1

has been suggested that tidal turbines could be used to generate power
from ocean currents. The Gulf Stream, for example, has a constant velocity of about 1.5 m/s. Discuss the advantages or disadvantages of this approach compared to the use of turbines in a tidal current that varies from $$- 3.0 \mathrm {~m} / \mathrm { s }$$ (in one direction) to +3.0 m/s (in the other direction).

Accepted Answer

For the Gulf Stream, the velocity is constant so the power output will be constant. This is advantageous compared to the tidal stream current which varies between 0 and a maximum velocity. On the other hand, the total energy produced by the tidal stream generator is greater than that produced by the Gulf Stream generator. This is because the power is proportional to the cube of the velocity and the mean value of v3 for (e.g.) a linearly varying velocity is greater than half the maximum value. This can be shown quantitatively as follows:

For a constant velocity of $v _ { 0 } / 2$, the total power over half a tidal cycle (time $\left. = T / 2 \right)$ will be proportional to
$$E \sim \left( v _ { 0 } / 2 \right) ^ { 3 } \times ( T / 2 ) = v _ { 0 } T / 16$$
For a velocity that varies linearly between 0 and $v _ { 0 }$ over half a tidal cycle then the velocity may be written as
$$v ( t ) = 2 v _ { 0 } t / T$$
So the energy will be proportional to
$$E \sim \frac { 8 v _ { 0 } ^ { 3 } } { T ^ { 3 } } \int _ { 0 } ^ { T / 2 } t ^ { 3 } d t = \left[ \frac { 8 v _ { 0 } { } ^ { 3 } } { T ^ { 3 } } \frac { t ^ { 4 } } { 4 } \right] _ { 0 } ^ { T / 2 } = \frac { v _ { 0 } { } ^ { 3 } T } { 8 }$$
Or twice the energy for the constant velocity case.

Question 2

Calculate the average current velocity that is necessary for an 8-m diameter
water turbine with a coefficient of performance of 0.3 to generate 1 MW.

Accepted Answer

From equation (13.5)
$$P = \frac { 1 } { 2 } C A \rho v ^ { 3 }$$ Solving for velocity gives
$$v = \left[ \frac { 2 P } { \pi r ^ { 2 } \rho C } \right] ^ { 1 / 3 }$$
Substituting values from the problem
$$v = \left[ \frac { 2 \times 10 ^ { 6 } \mathrm {~W} } { ( 3.14 ) \times ( 4 \mathrm {~m} ) ^ { 2 } \times \left( 1025 \mathrm {~kg} / \mathrm { m } ^ { 3 } \right) \times ( 0.3 ) } \right] ^ { 1 / 3 } = 5.06 \mathrm {~m} / \mathrm { s }$$

Question 3

A 10 m diameter tidal turbine is placed in an inlet and operates with a coefficient of performance of $0.3$. During a tidal period of $12.4$ hours, the tidal current in the inlet varies from $3.5 \mathrm {~m} / \mathrm { s }$ in one direction to $3.5 \mathrm {~m} / \mathrm { s }$ in the opposite direction. If it is assumed that the change in tidal current velocity is linear in time, what is the total energy output of the turbine during one tidal period.

Accepted Answer

Since the velocity and hence the power varies over the tidal cycle, it is necessary to write the energy as an integral over the tidal period. As the magnitude of the velocity is the same during the raising tide and the falling tide we write the energy as twice the integral over half a cycle as
$$E = 2 \int _ { 0 } ^ { T / 2 } P ( t ) d t$$
where $T$ is the total tidal period. The power as a function of time is
$$P ( t ) = \frac { 1 } { 2 } A C \rho v ( t ) ^ { 3 }$$
where $C$ is the coefficient of performance. Since the velocity varies linearly, we write
$$v ( t ) = 2 v _ { 0 } t / T$$
Converting time to seconds (so the answer will be in joules) gives the integral
$$E = \frac { 8 A C \rho v _ { 0 } ^ { 3 } } { ( T / 2 ) ^ { 3 } } \int _ { 0 } ^ { T / 2 } t ^ { 3 } d t = \frac { A C \rho v _ { 0 } ^ { 3 } } { 8 } T$$
Using the values from the problem
$$\begin{array} { l } 
A = ( 3.14 ) \times ( 5 \mathrm {~m} ) ^ { 2 } = 78.5 \mathrm {~m} ^ { 2 } \
C = 0.3
\end{array}$$ $$\begin{array} { l } 
\rho = 1025 \mathrm {~kg} / \mathrm { m } ^ { 3 } \
v _ { 0 } = 3.5 \mathrm {~m} / \mathrm { s } \
T / 2 = 6.2 \mathrm {~h} = 22320 \mathrm {~s}
\end{array}$$
gives the energy as
$$E = ( 1 / 4 ) \times \left( 78.5 \mathrm {~m} ^ { 2 } \right) \times ( 0.3 ) \times \left( 1025 \mathrm {~kg} / \mathrm { m } ^ { 3 } \right) \times ( 3.5 \mathrm {~m} / \mathrm { s } ) ^ { 3 } \times ( 22320 \mathrm {~s} ) = 5.78 \times 10 ^ { 9 } \mathrm {~J}$$
This may be expressed in $\mathrm { kWh }$ as
$$\left( 5.78 \times 10 ^ { 9 } \mathrm {~J} \right) / \left( 3.6 \times 10 ^ { 6 } \mathrm {~J} / \mathrm { kWh } \right) = 1.60 \times 10 ^ { 3 } \mathrm { kWh }$$

Question 4

What is the total tidal energy available (at 100% efficiency) during the falling tide from a basin of area 100 km² with a tidal range of 8 m?

Accepted Answer

The answer of What is the total tidal energy available...

Question 5

velocity variation of a tidal current through a tidal cycle is more
accurately described as sinusoidal (rather than linear as assumed in Problem 13.9). Reconsider this previous problem using a sinusoidal variation. How does this new assumption affect the result?

Accepted Answer

The answer of velocity variation of a tidal current through...

Question 6

(a) A tidal turbine has an efficiency of 40%. In a tidal stream with a velocity of 7 m/s, what is the diameter of a turbine that is required to produce 1 MWe?
(b) A wind turbine has an efficiency of 40%. In a wind with a velocity of 7 m/s, what is the diameter of a turbine that is required to produce 1 MWe?
(c) From the answers to parts (a) and (b), calculate the relative mass of water to air passing through the rotor per second.

Accepted Answer

The answer of (a) A tidal turbine has an efficiency...

Question 7

Estimate the average velocity of water leaving the Gulf of Khambhat
during the falling tide cycle.

Accepted Answer

The answer of Estimate the average velocity of water leaving...

Question 8

Using the tidal range and basin area for Penzhinsk inlet given in the chapter,
estimate the monetary value of the electrical energy that can be generated (at 25% efficiency) during one falling tide cycle. Assume electricity has a value of $0.10/kWh.

Accepted Answer

The answer of Using the tidal range and basin area...

Question 9

Although the Gulf Stream is probably the best-known ocean current, the
Antarctic Circumpolar Current is the largest ocean current. This current circulates around the continent of Antarctica and represents the transport of about $1.5 \times 10 ^ { 8 } \mathrm {~m} ^ { 3 } / \mathrm { s }$ with a velocity of up to $1.1 \mathrm {~m} / \mathrm { s }$. Calculate the total power available from this current, and compare this result to the current world power use.

Accepted Answer

The answer of Although the Gulf Stream is probably the...

Question 10

What is the power available from an underwater turbine with a coefficient
of performance of 0.4 that is 10 m in diameter in a current of seawater traveling at a velocity of 2.0 m/s?

Accepted Answer

The answer of What is the power available from an...

Question 11

In some tidal areas, such as the Bay of Fundy, some smaller basins could be used for tidal generation using barrage systems, while more open areas could be used for in-stream tidal turbines. Discuss how the variations in power output from these two approaches during the tidal cycle would (or would not) complement each other.

Accepted Answer

The answer of In some tidal areas, such as the...

Question 12

A water turbine 15 m in diameter is placed in a channel with a tidal current
moving with a velocity of 3.5 m/s. Estimate the power produced by the turbine.

Accepted Answer

The answer of A water turbine 15 m in diameter...

Question 13

Consider the four proposed tidal barrage generating facilities for Argentina
as described in Table 13.1. Calculate the average power available and compare with the capacity as given in the table. Comment on any differences between these quantities.

Accepted Answer

The answer of Consider the four proposed tidal barrage generating...

Question 14

Based on a calculation of the total energy available and an assumed
capacity factor of 0.24, calculate the total annual energy production for tidal generatingNstations at the three Canadian sites in Table 13.1. Estimate the number of homes that could be supplied with electricity from these three sites.

Accepted Answer

The answer of Based on a calculation of the total...

Question 15

Consider the possibility of implementing either tidal or wave energy in India.
(a) From the maximum capacity for tidal generation in the Gulf of Khambhat as given in Table $13.1$ and an estimated capacity factor of $0.24$, calculate the total annual energy generated.
(b) For wave energy at a capacity factor of $0.25$, how many km of coastline on the south coast of India would have to be developed generate the same annual energy as tidal energy in the Gulf of Khambhat as calculated in part (a)?

Accepted Answer

The answer of Consider the possibility of implementing either tidal...

Question 16

Calculate the total kinetic energy $$\text { (in MJ and in } \mathrm { kWh } \text { ) of a } 1 - \mathrm { m } ^ { 3 } \text { parcel of }$$ seawater moving with a velocity of 1 m/s.

Accepted Answer

The answer of Calculate the total kinetic energy \[\text {...

Question 17

A tidal inlet has an area of 32 $$320 \mathrm {~km} ^ { 2 } \text { and a tidal range of } 6.7 \mathrm {~m}$$ (a) Calculate the total energy available during one tidal cycle.
(b) Calculate the average power available over one tidal cycle.
(c) If the capacity factor is $0.27$, what number of homes (on average) could be supplied with electricity if each home, on average, requires $1.3 \mathrm {~kW}$ ?

Accepted Answer

The answer of A tidal inlet has an area of...

Question 18

For the proposed tidal lagoon facility in Swansea, calculate the average
tidal power available during the tidal cycle. Are the proposed turbines appropriate for this facility?

Accepted Answer

The answer of For the proposed tidal lagoon facility in...

Question 19

tidal barrage system generates 15 MWe from 85% efficient
turbines/generators at the beginning of the power generation cycle. The mean head of water above the turbines in 5.7 m and the water flows through the turbines with a velocity of 6 m/s. What is the total turbine area?

Accepted Answer

The answer of tidal barrage system generates 15 MWe from...

Question 20

For a tidal current with a velocity of 1.5 m/s in a channel that is 0.5 km wide
and 28 m deep (on average), calculate the mass of water per unit time moving through the channel.

Accepted Answer

The answer of For a tidal current with a velocity...

Quiz 13: Tidal Energy

Calculate the average current velocity that is necessary for an 8-m diameter water turbine with a coefficient of performance of 0.3 to generate 1 MW.

What is the total tidal energy available (at 100% efficiency) during the falling tide from a basin of area 100 km² with a tidal range of 8 m?

velocity variation of a tidal current through a tidal cycle is more accurately described as sinusoidal (rather than linear as assumed in Problem 13.9). Reconsider this previous problem using a sinusoidal variation. How does this new assumption affect the result?

Estimate the average velocity of water leaving the Gulf of Khambhat during the falling tide cycle.

Using the tidal range and basin area for Penzhinsk inlet given in the chapter, estimate the monetary value of the electrical energy that can be generated (at 25% efficiency) during one falling tide cycle. Assume electricity has a value of $0.10/kWh.

What is the power available from an underwater turbine with a coefficient of performance of 0.4 that is 10 m in diameter in a current of seawater traveling at a velocity of 2.0 m/s?

A water turbine 15 m in diameter is placed in a channel with a tidal current moving with a velocity of 3.5 m/s. Estimate the power produced by the turbine.

Consider the four proposed tidal barrage generating facilities for Argentina as described in Table 13.1. Calculate the average power available and compare with the capacity as given in the table. Comment on any differences between these quantities.

Calculate the total kinetic energy $\text { (in MJ and in } \mathrm { kWh } \text { ) of a } 1 - \mathrm { m } ^ { 3 } \text { parcel of }$ seawater moving with a velocity of 1 m/s.

For the proposed tidal lagoon facility in Swansea, calculate the average tidal power available during the tidal cycle. Are the proposed turbines appropriate for this facility?

tidal barrage system generates 15 MWe from 85% efficient turbines/generators at the beginning of the power generation cycle. The mean head of water above the turbines in 5.7 m and the water flows through the turbines with a velocity of 6 m/s. What is the total turbine area?

For a tidal current with a velocity of 1.5 m/s in a channel that is 0.5 km wide and 28 m deep (on average), calculate the mass of water per unit time moving through the channel.

Quiz 13: Tidal Energy

Calculate the average current velocity that is necessary for an 8-m diameter water turbine with a coefficient of performance of 0.3 to generate 1 MW.

What is the total tidal energy available (at 100% efficiency) during the falling tide from a basin of area 100 km2 with a tidal range of 8 m?

velocity variation of a tidal current through a tidal cycle is more accurately described as sinusoidal (rather than linear as assumed in Problem 13.9). Reconsider this previous problem using a sinusoidal variation. How does this new assumption affect the result?

Estimate the average velocity of water leaving the Gulf of Khambhat during the falling tide cycle.

Using the tidal range and basin area for Penzhinsk inlet given in the chapter, estimate the monetary value of the electrical energy that can be generated (at 25% efficiency) during one falling tide cycle. Assume electricity has a value of $0.10/kWh.

What is the power available from an underwater turbine with a coefficient of performance of 0.4 that is 10 m in diameter in a current of seawater traveling at a velocity of 2.0 m/s?

A water turbine 15 m in diameter is placed in a channel with a tidal current moving with a velocity of 3.5 m/s. Estimate the power produced by the turbine.

Consider the four proposed tidal barrage generating facilities for Argentina as described in Table 13.1. Calculate the average power available and compare with the capacity as given in the table. Comment on any differences between these quantities.

Calculate the total kinetic energy (in MJ and in kWh ) of a 1−m3 parcel of \text { (in MJ and in } \mathrm { kWh } \text { ) of a } 1 - \mathrm { m } ^ { 3 } \text { parcel of } (in MJ and in kWh ) of a 1−m3 parcel of seawater moving with a velocity of 1 m/s.

For the proposed tidal lagoon facility in Swansea, calculate the average tidal power available during the tidal cycle. Are the proposed turbines appropriate for this facility?

tidal barrage system generates 15 MWe from 85% efficient turbines/generators at the beginning of the power generation cycle. The mean head of water above the turbines in 5.7 m and the water flows through the turbines with a velocity of 6 m/s. What is the total turbine area?

For a tidal current with a velocity of 1.5 m/s in a channel that is 0.5 km wide and 28 m deep (on average), calculate the mass of water per unit time moving through the channel.

What is the total tidal energy available (at 100% efficiency) during the falling tide from a basin of area 100 km² with a tidal range of 8 m?

Calculate the total kinetic energy $\text { (in MJ and in } \mathrm { kWh } \text { ) of a } 1 - \mathrm { m } ^ { 3 } \text { parcel of }$ seawater moving with a velocity of 1 m/s.