Question 1

Compare the osmotic power available from freshwater flowing at a rate of $1000 \mathrm {~m} ^ { 3 } / \mathrm { s }$ into the ocean (assuming an efficiency of $40 \%$ ) and the thermal power available from water flowing at $1000 \mathrm {~m} ^ { 3 } / \mathrm { s }$ through a heat exchanger and cooling by $5 ^ { \circ } \mathrm { C }$ (assuming an efficiency of $3 \%$ ).

Accepted Answer

For a total osmotic power of 1 $$\text { MW per } \mathrm { m } ^ { 3 } / \mathrm { s } \text { of flow (at an assumed efficiency of }$$ $40 \%$ ) the total power is
$$\left( 10 ^ { 6 } \mathrm { Ws } / \mathrm { m } ^ { 3 } \right) \times \left( 1000 \mathrm {~m} ^ { 3 } / \mathrm { s } \right) = 1.0 \times 10 ^ { 9 } \mathrm {~W}$$
The thermal power available at an efficiency of $3 \%$ (typical of an OTEC facility) is
$$P = \eta ( d m / d t ) C \Delta T$$
So
$$P = ( 0.03 ) \times \left( 1000 \mathrm {~m} ^ { 3 } / \mathrm { s } \right) \times \left( 1000 \mathrm {~kg} / \mathrm { m } ^ { 3 } \right) \times \left( 4180 \mathrm {~J} / \left( \mathrm { kg } \cdot { } ^ { \circ } \mathrm { C } \right) \right) \times \left( 5 ^ { \circ } \mathrm { C } \right) = 0.627 \times 10 ^ { 9 } \mathrm {~W}$$
Only slightly less than the osmotic power.

Question 2

An offshore OTEC facility operates with a warm reservoir temperature of $22 ^ { \circ } \mathrm { C }$ and a cold reservoir temperature of $5 ^ { \circ } \mathrm { C }$.
(a) If $65 \%$ of the output power is required for operation of the facility (e.g., primarily water pumps), what is the net efficiency?
(b) In order to transport the electricity that has been generated to land, hydrogen is used as an energy storage mechanism. Hydrogen is produced by the electrolysis of water and is converted back to electricity using a fuel cell, with an overall efficiency of $40 \%$ (see Section 20.9). What is the net overall efficiency, and how does this compare with, for example, wind or solar photovoltaics?

Accepted Answer

$$\text { (a) For the temperatures } T _ { h } = 22 ^ { \circ } \mathrm { C } = 295 \mathrm {~K} \text { and } T _ { c } = 5 ^ { \circ } \mathrm { C } = 278 \mathrm {~K} \text { the Carnot }$$ efficiency is $$\eta _ { \text {Carnot } } = 100 \times ( 295 \mathrm {~K} - 278 \mathrm {~K} ) / ( 2 \times 295 \mathrm {~K} ) = 2.88 \%$$
If $65 \%$ of the output power is required for operation then the net efficiency is
$$( 0.35 ) \times ( 2.88 \% ) = 1.0 \%$$
(b) The net efficiency after transporting the energy to shore is
$$( 0.4 ) \times ( 1.0 \% ) = 0.4 \%$$ This may be compared to typical efficiencies for other renewable technologies such as photovoltaics $( \sim 15 \% )$ and wind $( \sim 40 \% )$. OTEC is significantly less efficient that these other technologies.

Question 3

(a) Calculate the energy available when a 106 kg parcel of water is cooled by $4 ^ { \circ } \mathrm { C }$.
(b) If this energy is used (at an efficiency of $100 \%$ ) to lift the parcel of water against gravity, what would be its elevation?
(c) How long would this energy satisfy the average per-capita primary energy needs in Norway?

Accepted Answer

(a) The thermal energy is
$$E = m C \Delta T$$ Using the values $$\begin{array} { l } 
m = 10 ^ { 6 } \mathrm {~kg} \
C = 4180 \mathrm {~J} / \left( \mathrm { kg } \cdot { } ^ { \circ } \mathrm { C } \right) \
\Delta T = 4 ^ { \circ } \mathrm { C }
\end{array}$$
gives
$$E = \left( 10 ^ { 6 } \mathrm {~kg} \right) \times \left( 4180 \mathrm {~J} / \left( \mathrm { kg } \cdot { } ^ { \circ } \mathrm { C } \right) \right) \times \left( 4 ^ { \circ } \mathrm { C } \right) = 1.672 \times 10 ^ { 10 } \mathrm {~J}$$
(b) The gravitational potential energy is
$$E = m g h$$
Solving for h gives
$$\left. h = E / ( m g ) = \left( 1.672 \times 10 ^ { 10 } \mathrm {~J} \right) / \left( 10 ^ { 6 } \mathrm {~kg} \right) \times \left( 9.8 \mathrm {~m} / \mathrm { s } ^ { 2 } \right) \right) = 1.71 \times 10 ^ { 3 } \mathrm {~m}$$
(c) From Figure $2.4$ for Norway the per capita energy use is about $440 \mathrm { GJ } / \mathrm { y } = 4.4 \times 10 ^ { 11 } \mathrm {~J} / \mathrm { y }$. From part (a) the energy would last for
$$\left( 1.672 \times 10 ^ { 10 } \mathrm {~J} \right) / \left( 4.4 \times 10 ^ { 11 } \mathrm {~J} / \mathrm { y } \right) = 0.038 \mathrm { y }$$
or about 2 weeks.

Question 4

$$\text { A } 1 \mathrm {~m} ^ { 3 } \text { parcel of seawater is cooled from } 24 ^ { \circ } \mathrm { C } \text { to } 8 ^ { \circ } \mathrm { C } \text { in } 1.2 \text { seconds. }$$ Calculate the average thermal power that is available during that time.

Accepted Answer

The answer of \[\text { A } 1 \mathrm {~m}...

Question 5

A 20 MWe OTEC facility operates at a thermal efficiency of 2.4%. The warm water temperature is $22 ^ { \circ } \mathrm { C }$. Calculate the flow rate of warm water in cubic meters per second $\left( \mathrm { m } ^ { 3 } / \mathrm { s } \right)$. Assume that the conversion from mechanical to electrical energy is $86 \%$ efficient.

Accepted Answer

The answer of A 20 MWe OTEC facility operates at...

Question 6

It has been speculated that an OTEC facility could use wave energy to offset its low efficiency by providing electrical energy for operating the plant. Where would the most advantageous location(s) be for such a facility? If the facility could intercept 200 m of wave front for your chosen location and had a total output of 100 MWe, what fraction of this total output would be from waves? Assume the efficiency of wave-generated electricity to be 35%.

Accepted Answer

The answer of It has been speculated that an OTEC...

Question 7

The Amazon River has a flow rate of $$2.09 \times 10 ^ { 5 } \mathrm {~m} ^ { 3 } / \mathrm { s } \text { at an average }$$ velocity of $0.7 \mathrm {~m} / \mathrm { s }$. Compare the total potential for osmotic energy and the total potential for run-of-the river hydroelectric energy (at $40 \%$ capacity factor) for the Amazon.

Accepted Answer

The answer of The Amazon River has a flow rate...

Question 8

Consider the possible design of an OTEC facility that has an electrical output comparable to an average nuclear power plant $\left( \mathrm { e } . g . , 1 \mathrm { GW } _ { \mathrm { e } } \right)$. Assume a turbine/generator efficiency of $90 \%$.
(a) If the surface temperature is $23 ^ { \circ } \mathrm { C }$ and the cold reservoir is at $4 ^ { \circ } \mathrm { C }$, what is the efficiency?
(b) What is the total flow rate of water needed?
(c) Compare the result of part (b) to the flow rate needed for a $1 \mathrm { GW } _ { \mathrm { e } }$ hydroelectric facility with a head of $200 \mathrm {~m}$.

Accepted Answer

The answer of Consider the possible design of an OTEC...

Question 9

An OTEC facility has a total flow rate of $$500 \mathrm {~m} ^ { 3 } / \mathrm { s } \text { of water. The cold }$$ water from the deep ocean is at 5°C. During the year the surface water temperature varies from 19°C to 22°C. Plot the output as a function of warm water temperature over this range.

Accepted Answer

The answer of An OTEC facility has a total flow...

Question 10

$$\text { An OTEC system operates with a warm reservoir of } 23 ^ { \circ } \mathrm { C } \text { and a cold }$$ reservoir of $5 ^ { \circ } \mathrm { C }$. Calculate the mass of water (vapor) flowing through the evaporator needed to generate $1 \mathrm { MWh }$ of electricity. Assume a $90 \%$ conversion from mechanical to electrical energy.

Accepted Answer

The answer of \[\text { An OTEC system operates with...

Question 11

The salinity (mostly NaCl) in some parts of the Great Salt Lake (Utah) is $27 \%$; this is sometimes expressed as 270 parts per thousand, meaning $270 \mathrm {~g}$ of salt per liter of solution. Estimate the height of a column of water that can be supported by the osmotic pressure between Great Salt Lake water and freshwater. The density of a 27\% salt solution is $\approx 1200 \mathrm {~kg} / \mathrm { m } ^ { 3 }$.

Accepted Answer

The answer of The salinity (mostly NaCl) in some parts...

Question 12

(a) Calculate the gravitational potential energy associated with $$1 \mathrm {~m} ^ { 3 } \text { of sea }$$ water that is lifted through a vertical distance of $1 \mathrm {~km}$.
(b) Compare the result in part (a) with the total thermal energy difference between $1 \mathrm {~m} ^ { 3 }$ of seawater at $4 ^ { \circ } \mathrm { C }$ and $1 \mathrm {~m} ^ { 3 }$ of seawater at $22 ^ { \circ } \mathrm { C }$.

Accepted Answer

The answer of (a) Calculate the gravitational potential energy associated...

Question 13

If the temperature drops across the evaporator and condenser of an OTEC
system are equal, prove that the relative temperature drops as given in equation (14.4) will maximize the power output.

Accepted Answer

The answer of If the temperature drops across the evaporator...

Question 14

Estimate the salinity of the Dead Sea from the information in this chapter.

Accepted Answer

The answer of Estimate the salinity of the Dead Sea...

Question 15

Consider the experiment of Jean-Antoine Nollet as described in the text,
which illustrates the osmotic pressure between water and wine. Assume wine is 12% (by volume) of ethanol in water, and estimate, as in Example 14.4, the height of a column of water that the osmotic pressure between water and wine can support. Note that the van 't Hoff factor depends on the nature of the solvent and that a value of i ≈ 1.0 is appropriate for ethanol in water.

Accepted Answer

The answer of Consider the experiment of Jean-Antoine Nollet as...

Question 16

In the Arctic, winter air temperatures can average around $$- 35 ^ { \circ } \mathrm { C } , \text { while sea }$$ temperatures (under the ice) can typically be around $- 2 ^ { \circ } \mathrm { C }$. Consider the possibility of constructing an OTEC facility that uses the air as the cold reservoir and pumps water from the ocean to serve as the "warm" reservoir.
(a) Calculate the ideal efficiency of such a facility.
(b) Would there be constraints on the design of such a system (i.e., open system or closed system) and the working fluid?

Accepted Answer

The answer of In the Arctic, winter air temperatures can...

Question 17

Make a plot of the height of a column of water (in meters) that the osmotic pressure between freshwater and saltwater can support as a function of salinity between 0 and $10 \%$ (weight of salt per weight of solution) of $\mathrm { NaCl }$ in water. Assume a constant density of $1025 \mathrm {~kg} / \mathrm { m } ^ { 3 }$.

Accepted Answer

The answer of Make a plot of the height of...

Question 18

The Mississippi River, the largest of all rivers in the United States, has a flow rate of $1.8 \times 10 ^ { 4 } \mathrm {~m} ^ { 3 } / \mathrm { s }$. If $10 \%$ of the osmotic energy potential of the Mississippi could be harvested, estimate the number of households that could be supplied with electricity.

Accepted Answer

The answer of The Mississippi River, the largest of all...

Question 19

An ideal OTEC system has a flow $$\text { rate of } 100 \mathrm {~m} ^ { 3 } / \mathrm { s } \text { of warm seawater. The }$$ warm reservoir is at $26 ^ { \circ } \mathrm { C }$, and a cold reservoir is at $6 ^ { \circ } \mathrm { C }$. For a conversion efficiency for mechanical to electrical energy of $90 \%$, what is the facility's output in $\mathrm { MW } _ { \mathrm { e } }$ ?

Accepted Answer

The answer of An ideal OTEC system has a flow...

Quiz 14: Ocean Thermal Energy Conversion and Ocean Salinity Gradient Energy

$\text { A } 1 \mathrm {~m} ^ { 3 } \text { parcel of seawater is cooled from } 24 ^ { \circ } \mathrm { C } \text { to } 8 ^ { \circ } \mathrm { C } \text { in } 1.2 \text { seconds. }$ Calculate the average thermal power that is available during that time.

An OTEC facility has a total flow rate of $500 \mathrm {~m} ^ { 3 } / \mathrm { s } \text { of water. The cold }$ water from the deep ocean is at 5°C. During the year the surface water temperature varies from 19°C to 22°C. Plot the output as a function of warm water temperature over this range.

If the temperature drops across the evaporator and condenser of an OTEC system are equal, prove that the relative temperature drops as given in equation (14.4) will maximize the power output.

Estimate the salinity of the Dead Sea from the information in this chapter.