 ### Theory

The down-flow conduits are positioned vertically so that the top ends are at the surface of the ocean. A mass of water contained by a down-flow conduit is subject to three large forces: a force from pressure at the top of the conduit, a force from pressure at the bottom of the conduit, and the force from gravity on the contained water. These three forces are in balance in a quiescent ocean and there is no vertical flow.
Surface waves increase the pressure at the top of the down-flow conduit. The three large forces are no longer in balance resulting in a down-flow. The dynamic pressure field is known for surface waves (Theoretical Hydrodynamics by L. M. Milne-Thompson, 1968, p.433), therefore the magnitude of the wave pressure can be estimated. The maximum theoretical net pressure imposed on the down-flow conduit is one quarter of the wave height (crest to trough) expressed as head. Though wave pressure varies with amplitude over time at the top of the down-flow conduit, it can be treated as constant since the momentum of the water in the down-flow is large.
A consequence of the down-flow is that the weight of the contained water is altered. This change in weight is called the buoyant force and results from of a change of temperature and/or salinity of the contained water.
Another consequence of the down-flow is that friction at the walls of the down-flow conduit must be taken into account when analyzing the flow.

For steady flow in a down-flow conduit, (equation 1)

where is the pressure head from wave pressure, is the buoyancy pressure head, and is the pressure head loss from friction.

The buoyancy pressure head is a function of the density of the contained water. The density of seawater is a known function of salinity and temperature. The salinity of the down-flow is the same as the salinity of the ocean at the surface. The temperature of the down-flow in the header sections of the down-flow conduits is assumed to be constant and equal to the surface temperature of the ocean. The temperature of the down-flow in the heat exchange sections of down-flow conduits is the average of the surface temperature of the ocean and the egress temperature. The temperature of the down-flow in the footer sections of the down-flow conduits is assumed to be constant and equal to the egress temperature. Once the densities are determined, (equation 2)

where is the length of the header sections of down-flow conduits, is the density in the header sections of down-flow conduits, is the average ambient density for the header sections of down-flow conduits, is the length of the heat exchange sections of down-flow conduits, is the average density in the heat exchange sections of down-flow conduits, is the average ambient density for the heat exchange sections of down-flow conduits, is the length of the footer sections of down-flow conduits, is the density in the footer sections of down-flow conduits, and is the average ambient density for the footer sections of down-flow conduits.

To obtain a pressure head loss from friction it is convenient to use the Darcy-Weisbach expression: (equation 3)

where is the pressure head loss from friction, is the Darcy friction factor, is the length of the conduit, is the velocity of flow, is the hydraulic diameter, and is the acceleration due to gravity. Because of differences in velocity and hydraulic diameter, the friction head losses are calculated separately for the header sections of down-flow conduits, the heat exchange sections of down-flow conduits, and footer sections of down-flow conduits.

As the down-flow cools and the up-flow warms, there is a transfer of heat energy from the down-flow to the up-flow. The principle of conservation of energy requires that the rate at which heat energy is lost in the down-flow is equal to the rate at which heat energy is gained in the up-flow. This rate of heat energy transfer is also equal to the rate that heat energy crosses the heat exchange area. (equation 5) (equation 6)

where is the overall heat conductance of the exchanger, is the area of the heat exchange surface, is the log-mean temperature difference across the heat exchanger, is the mass flow rate of one down-flow conduit, is the surface temperature of the ocean, is the down-flow egress temperature, is the number of down-flow conduits of identical dimensions, is the mass flow rate of the up-flow, is the egress temperature of the up-flow, is the temperature of the ocean at the up-flow ingress depth, is the convective heat transfer coefficient for the down-flow, is the convective heat transfer coefficient for the up-flow, and is the conductance of the heat exchange conduit material.

A common method of determining convective heat transfer coefficients is with the Dittus- Boelter equation. For the down-flow, (equation 7)

where is the Nusselt number for the down-flow, is the Reynolds number for the down-flow, is the Prandtl number, is the hydraulic diameter for a single down-flow conduit in the heat exchange section of down-flow conduits, is the heat conductivity of seawater, is the density of seawater, is the velocity of the down-flow in the heat exchange section of down-flow conduits, is the viscosity of sea- water, and is the specific heat of seawater.

For the up-flow, (equation 8)

where is the Nusselt number for the up-flow, is the Reynolds number for the up-flow, is the hydraulic diameter for the up-flow, and is the velocity of the up-flow.

Solving equations 1, 4, 5 and 6 simultaneously yields the upwelling predictions on the results page.

### 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