The APEX 180ml Buoyancy Engine.

The energy consumed by the high pressure pump motor is given by

$$E = \int i V dt$$ (2)

where i is the electrical current, V is the voltage, and t is time. For subsequent computations, it will be convenient to express this integral in terms of the volume pumped rather than time. The following model of the pump motor applies,

$$V = i R + \frac{dv}{dt} K~.$$ (3)

where $R=\ensuremath{4.6\Omega}$ is the internal resistance of the motor, $K=\ensuremath{73.7\frac{\textrm{V}\cdot\textrm{s}}{\textrm{ml}}}$ is a generation factor for back-EMF, and $\frac{dv}{dt}$ is the volume pump-rate (see Appendix A). Using (3) to effect a change-of-variables in (2) yields,

$$E = \int \frac{V i K}{V - i R} dv~,$$ (4)

Due to active ascent-rate control, the energy consumed by the buoyancy engine occurs at unpredictable depths while measuring the hydrographic profile. Figure 1 shows a profile of in situ density for the upper 800 decibars of the JES. A reasonable estimate is obtained by considering the buoyancy generation to have occurred at only 4 points in the water column (marked with blue circles in Figure 1). Accordingly, (2) can be approximated as

$$E \doteq \sum_{n=1}^4 \frac{V_n i_n K \delta v_n}{V_n - i_n R}~.$$ (5)

  

Figure 1: This is a plot of the in situ density of the JES.

1.

The initial piston extension at the parking depth of 800 dbars increases the buoyancy by 31ml. This buoyancy is considered to be a marginal force above and beyond that required for neutral buoyancy at any given depth and its function is to maintain the float's ascent rate. The energy consumed by pumping 31ml against 800 dbars of pressure is

$$\delta E_1 = \frac{\ensuremath{12.5\textrm{V}}\cdot 0.42\en... ...\ensuremath{4.6\Omega} } = 1.135~\ensuremath{\textrm{kJ}} ~.$$ (6)

2.

The buoyancy difference between the parking depth and the base of the pycnocline ( $\sim 50~\textrm{dbar}$) is $\delta v = 41\textrm{g}$ (see Appendix B). The energy consumed by pumping 44ml against 450 dbar of pressure is

$$\delta E_2 = \frac{\ensuremath{12.5\textrm{V}}\cdot 0.34\en... ...\ensuremath{4.6\Omega} } = 1.260~\ensuremath{\textrm{kJ}} ~.$$ (7)

3.

The buoyancy difference between the base of the pycnocline ( $\sim 50~\textrm{dbar}$) and the surface is $\delta v = 63\textrm{g}$ (see Appendix B). The energy consumed by pumping 66ml against 50 dbar of pressure is

$$\delta E_3 = \frac{\ensuremath{12.5\textrm{V}}\cdot 0.25\en... ...\ensuremath{4.6\Omega} } = 1.339~\ensuremath{\textrm{kJ}} ~.$$ (8)

4.

This model of APEX has 180ml of gross displacement capacity but there is only 157ml of displacement available between the ballast piston position (24 A/D counts) and full piston extension (248 A/D counts). The displacements in items 1-3 have consumed 138ml leaving 16ml to be pumped at the surface in order for the piston to reach full extension. The energy consumed by pumping 16ml against atmospheric pressure is

$$\delta E_4 = \frac{\ensuremath{12.5\textrm{V}}\cdot 0.24\en... ...\ensuremath{4.6\Omega} } = 0.310~\ensuremath{\textrm{kJ}} ~.$$ (9)

According to (5), the total energy consumed by the APEX buoyancy engine during each profile is

$$E_{APEX} = \sum_{n=1}^4 \delta E_n = 4.04~\ensuremath{\textrm{kJ}} ~.$$ (10)