Theory
The invention can utilize the power from waves without moving parts. The absolute pressure at sea level when a sea is at rest is one atmosphere. Waves cause the pressure to vary at sea level with an average pressure greater than one atmosphere. It is known that surface waves do not significantly affect the pressure in deep water. A single conduit oriented with the top end at sea level and the bottom end well beneath the waves will have an average net pressure on it and a down-flow within it.
Because the surface of the ocean is warmer than deep water, the warm water within the down-flow conduit has a buoyancy that opposes the flow. If the temperature profile of the ocean is measured and the average temperature of the down-flow is known, the buoyancy can be estimated and taken into account when determining the rate of down-flow. Expressed as head, the buoyancy in a vertical pipe of constant diameter due to a change of temperature is the length of pipe minus the product of the length of pipe and the ratio of the two densities. With given conduit dimensions, pressure from surface waves, the temperatures, and Newton's third law of motion, the down-flow rate can be estimated by those practiced in the art of fluid mechanics.
In the counter flow heat exchanger portion of the present invention the down-flow will lose heat energy through the conduit walls to the adjacent up-flow. The principle of conservation of energy requires that the rate at which the heat energy is lost from the down-flow is equal to the rate at which heat energy is gained in the up-flow. The rate at which heat energy in the up or down-flow is lost or gained is equal to the mass flow rate times the specific heat of seawater times the temperature change. This rate of energy transfer is also equal to the rate of energy flow across the exchange surface. The rate of energy flow across the exchange surface in the counter flow heat exchanger can be estimated if the surface area, the heat transfer coefficients and the log-mean temperature difference are known. The heat transfer coefficients are combined to obtain the overall the heat transfer coefficient. One way to find the heat transfer coefficients is with the Dittus-Boelter equation. The Dittus-Boelter equation is a familiar empirical equation to those practiced in the art of the fluid mechanics of heat exchangers.
The addition of heat in the up-flow causes a buoyancy in the up-flow conduit and an upward flow. If the average temperature of the up-flow is known the upward pressure due to buoyancy can be estimated. With given conduit dimensions, pressure due to buoyancy, the friction of the up-flow, and Newton's third law of motion, the up-flow rate can be estimated.
The performance of the invention is discoverable by those familiar with the art of fluid mechanics of heat exchangers. With a given wave-height at the ocean surface, a given temperature profile of the ocean, and with the dimensions and positions of the two kinds of conduit material, the rate of artificial upwelling can be estimated.
Strategy
The parameters are interdependent. I used a strategy of selecting a known temperature profile for the ocean, a down-flow egress depth, an up-flow ingress depth, an up-flow egress depth, an up-flow egress temperature, an up-flow rate, the down-flow conduit diameters, a down-flow rate and guessing a heat transfer coefficient for the up-flow. The heat transfer coefficient for the up-flow is needed to calculate the parameters of the down-flow. The guess is later compared to the calculated value from the Dittus-Boelter equation.
The total rate of energy exchange is discovered from the selected up-flow rate and selected up-flow egress temperature. The coefficient of heat transfer for the down-flow is calculated from the Dittus-Boelter equation. The down-flow egress temperature is discovered when the rate at which heat energy is lost from the down-flow of one down-flow conduit is equal to the rate at which heat energy is transferred to the up-flow through the exchange area of this one down-flow conduit. With this rate at which heat energy is transferred from one down-flow conduit and the total rate of energy exchange the total number of down-flow conduits is calculated. With the down-flow egress temperature the buoyancy in the down-flow is discovered. The buoyancy in the down-flow and the friction of the down-flow determine the approximate wave height required.
The diameter of the upwelling conduit (and the hydraulic diameter) is discovered when the magnitude of the force from the buoyancy of the up-flow matches the magnitude of the friction force from the up-flow. With the hydraulic diameter of the upwelling conduit and the selected up-flow rate, the velocity of the up-flow is discovered. With the velocity of the up-flow and the hydraulic diameter of the up-flow the coefficient of heat transfer for the up-flow can be calculated. If it does not match the guess, the guess is adjusted and the procedure repeated.
Now with a specific arrangement of conduits that function at a specific wave height, the performance with different wave heights can be tested. I performed this test by adjusting the down-flow velocity. Again I assume a value for the coefficient of heat transfer for the up-flow. The coefficient of heat transfer for the down-flow is calculated as before. As before, the down-flow egress temperature is discovered when the rate at which heat energy is lost from the down-flow of one down-flow conduit is equal to the rate at which heat energy is transferred to the up-flow through the exchange area of this one down-flow conduit. With this rate at which heat energy is transferred from one down-flow conduit and the total number of down-flow conduits the total rate of energy exchange is calculated. The required wave height is discovered as before.
The up-flow egress temperature is discovered when the magnitude of the force from the buoyancy of the up-flow is equal to the magnitude of the friction force from the up-flow. The up-flow rate is calculated from the total rate of heat energy exchange and the temperature rise of the up-flow. The up-flow velocity is calculated from the up-flow rate and cross-sectional area of up-flow. With the velocity of the up-flow and the hydraulic diameter of the up-flow the coefficient of heat transfer for the up-flow can be calculated. If it does not match the guess, the guess is adjusted and the procedure repeated.
It is not enough to test the static performance at specific wave heights. Because the coefficient of heat transfer for laminar flow is much less than for turbulent flow it is necessary to test if the transition to turbulent flow occurs as the wave height increases. There is a hysteresis effect where at some wave heights the performance depends on the history of wave heights.
Physical Constants
Seawater Density
Data from the online Water Density Calculator
starting at 5 degrees C for each degree C - salinity at 35000 ppm
1027.701,1027.578,1027.445,1027.300,1027.144,
1026.979,1026.802,1026.616,1026.420,1026.215,1026.000,1025.776,1025.543,1025.300,1025.050,
1024.790,1024.523,1024.247,1023.962,1023.670,1023.370,1023.062,1022.747,1022.424,1022.093,
1021.755,1021.410,1021.058,1020.699,1020.333,1019.960,1019.581,1019.195,1018.802,1018.404
Essential Equations
Example
A substantial performance can be predicted if my invention is installed in the deep waters of the Pacific off the coast of San Diego, California where the water temperature has been measured and is typically about
15.2 degrees Celsius at sea level,
12 degrees at 50 meters depth,
10.2 degrees at 100 meters depth,
9.2 degrees at 150 meters depth,
8.8 degrees at 200 meters depth,
8.2 degrees at 250 meters depth,
and 7.8 degrees at 300 meters depth.
With the program "HeatCalcs" (available on the Reference page) conduit dimensions were adjusted and the performance tested.
Conduit Dimensions:
- The upwelling conduit has a diameter of 4.13 meters and extends between the depths of 50 meters and 300 meters.
- There are 165 down-flow conduits
- The down-flow conduits have a diameter of 0.2 meters within the exchanger.
- The down-flow conduits have a diameter of 0.3 meters between the depth of 50 meters and sea level.
One meter (crest to trough) wave height performance
- The waves provide a head of approximately 0.25 meters.
- The down-flow velocity is approximately 0.3 meters per second (where the diameter is 0.2 meters).
- The down-flow egress temperature is approximately 11.3 degrees Celsius.
- The coefficient of heat transfer for the down-flow is approximately 690 watts per square meter degree Kelvin.
- The up-flow volume rate is approximately 1.4 cubic meters per second.
- The up-flow exit temperature is approximately 12.3 degrees Celsius.
- The coefficient of heat transfer for the up-flow is approximately 521 watts per square meter degree Kelvin.
- The rate of heat energy exchange is approximately 24.5 megawatts.
Two meters (crest to trough) wave height performance:
- The waves provide a head of approximately 0.5 meters.
- The down-flow velocity is approximately 0.51 meters per second.
- The down-flow egress temperature is approximately 12.2 degrees Celsius.
- The coefficient of heat transfer for the down-flow is approximately 1055 watts per square meter degree Kelvin.
- The up-flow volume rate is approximately 1.65 cubic meters per second
- The coefficient of heat transfer for the up-flow is approximately 594 watts per square meter degree Kelvin
- The up-flow exit temperature is approximately 12.8 degrees Celsius
- The rate of heat energy exchange is approximately 32.2 megawatts