Fluid Loop Attitude Actuation

In this section, the PPs that influence attitude actuation are discussed for the fluid loop specifically: mass, angular momentum, and torque. They are calculated based on the fundamental equations from Section 2.1, and written to apply specifically to the various configurations of the fluid loop, expressed in the most fundamental design parameters (Requirement SM-5; Figure 4- 1). These are then adopted in the sizing model.

4.1.1 Mass

The total mass of a fluid loop comprises the mass of the pump, electronics, channel and fluid mass. The fluid mass can be generically determined for all fluid loops by Equations 4.1 and 4.2, where it is assumed that the loop radius is much larger than the corner radius. Various fluid densities can be found in Table 4-1 in Section 4.2.1. To obtain the total fluid loop mass, the masses of the other components can be estimated with data from pumps, channels and electronics boards.

( ) ( ∑ )

4.1 4.2 The cross-sectional area of a circular and rectangular channel are given by the equations below.

4.3 4.4 The channel length can be substituted with one of the configuration-specific channel

lengths, given by the equations below.

( ) √( ) 4.5 4.6 4.7 4.8

4.1.2 Angular Momentum

The angular momentum of a fluid loop can also be generically determined, based on the traditional angular momentum equation for a collection of point masses (Eq. 4.9). Because the channel diameter is much smaller than the loop diameter, an infinitesimally thin slice of fluid can be considered a point mass. The sum of the angular momentums of these slices is the total angular momentum of the fluid loop.

| | ∑

∫ 4.9

Circular Fluid Loop

For a circular fluid loop the simplified Equation 4.10 can be used, because it represents all point masses at a radius , moving at a velocity perpendicular to the radius (see Figure 4-3). Their integration can be done by multiplication with the total mass , because these variables are the same for all point masses along the length of the fluid loop.

4.10

Next, this can be expressed in terms of the fluid loop design parameters. The mass of the inertial fluid can be obtained by the configuration-specific Equations 4.1 to 4.8, and depends on the fluid density, the channel length and the channel cross-sectional area. The fluid velocity is dictated by the system‟s continuity, or the volumetric flow rate of the pump over the cross- sectional area . The radius of the loop is a design parameter itself, so this can stay in place. The two cross-sectional areas cancel each other out, which results in the general notation for a fluid loop‟s angular momentum [Eq. 4.11], and that specifically for a circular fluid loop [Eq. 4.12].

It can be seen that the angular momentum is proportional to the density of the inertial fluid, the radius of the loop squared, and the flow rate of the pump. Put differently, a heavy fluid pumped through a loop as large as possible at a high flow rate produces a high angular momentum.

4.11 4.12

This relation has some far-reaching consequences for the sizing process of the fluid loop. Traditionally, the angular momentum capacity of a reaction wheel is increased by increasing the inertial mass of the rotating wheel. The most intuitive way of doing this in the case of the fluid loop is to increase the channel diameter, so there is more fluid mass that can be rotated.

However, due to continuity this will not lead to a higher angular momentum of the fluid loop. At a given pump flow rate, increasing the cross-sectional area results in a mass increase, but the fluid velocity decreases by the same amount (see Figure 4-2). The mass flow, and thus the angular momentum stay the same in the end.

This also means that, ceteris paribus, the channel diameter of a fluid loop can be made arbitrarily small, while retaining its angular momentum. This allows for optimizing the actuator‟s mass. Decreasing the diameter does increase the viscous friction, so where the optimum diameter lies must be weighed off with the theory developed in this dissertation and assessed on a case per case basis.

Figure 4-2 Two fluid loops with different masses and channel diameters, but equal angular momentums.

Square Fluid Loop

In theory, a square fluid loop should work according to the same principles as a circular fluid loop. However, as opposed to a circular fluid loop, the point masses along the perimeter of a square fluid loop have constantly changing perpendicular velocities and radii to the axis of rotation as they move along the channel, so at first sight the simple Equation 4.10 cannot be used here. However, a similar simple angular momentum equation is possible here, too.

Figure 4-3 The radius and velocity of an arbitrary fluid slice point mass in a circular and square fluid loop. These variables dictate the point mass‟ angular momentum.

Figure 4-3 shows that the radius of the point mass is a factor higher than the perpendicular radius (which does remain the same throughout the loop). Conversely, the perpendicular velocity of that point mass is a factor lower than the fluid velocity (which also remains the same throughout the loop). Hence, the two cosine factors cancel each other out which results in a similar equation that applies to all point masses along the square perimeter [Eq. 4.13]. Because the variables are the same for all point masses, the equation can be used for all point masses, resulting in a very similar equation to that of a circular fluid loop, where the only difference is the channel length ( versus ) [Eq. 4.14].

4.13 4.14

In reality, however, the square loop will most likely have (slightly) rounded corners to reduce fluid friction. Unlike in a square, the radius and perpendicular velocity of a point mass in a rounded corner is not a simple function of (see image 4.4). Although Equation 4.15 is a good approximation and computationally easier, it is not strictly correct.

Hence, integration of the point masses‟ angular momentums in the four corners is required. As usual, the angular momentum of a point mass around the fluid loop‟s central axis is given by:

4.16

Where, in this case:

Where: √ ( ) 4.19 4.20 4.21 4.22

The above formulation is verified with multiple numerical examples whereby the fluid mass of all corners was split into 36,000 slits of (See Appendix 3) and integrated for angular momentum. It was found that the deviation of the approximate Equation 4.15 from the exact method above depends on the ratio of and , and always has a maximum deviation of –5.8% at a corner ratio of

. Depending on the required accuracy, either the simple

approximation or the more complex exact method can be used.

Helical Fluid Loop

As opposed to increasing the channel diameter, each extra coil should increase the effective flow rate, as illustrated in Figure 4-5. This makes coiling a key insight to scaling up the fluid loop. A helical fluid loop can be seen as multiple small fluid loops acting in parallel, but with the benefit of being pumped by only one pump. With each winding, the angular momentum capacity of the fluid loop is multiplied, within approximately the same constraints of the satellite envelope and pump flow rate. The extra coils do come at the expense of extra mass and viscous friction. This can be weighed off using the theory developed in this dissertation.

√

4.17 4.18

Figure 4-4 The geometry of the angular

momentum of a point mass in a rounded corner of a FLIA.

Figure 4-5 The angular momentum of the fluid loop can be scaled up by winding it in coils, thus

increasing the effective flow rate. It can be seen as multiple small fluid loops working in parallel, actuated by a single pump.

4.1.3 Torque

In general, the torque that a mass exerts on its environment depends on the rate of change of its angular momentum. Hence, the actuator‟s control torque depends on the same variables as the angular momentum, but instead of the flow rate it is multiplied by the acceleration of the flow rate ̇ [Eq. 4.25]. How strongly the flow rate is accelerated depends on many factors, such as the pump characteristics, pressure differential and fluid speed. Hence, a generic approach is difficult, if at all possible. With Equation 4.25, some ̇ values of MHD-pumped fluid loops are obtained from previous work and are shown in Table 4-3 in Section 4.2.3 (derivation in Appendix 4). These can be used for assuming realistic values for ̇ in the sizing process. This is verified as described in Section 5.3.2. ̇ ̇ 4.25 √( ) 4.23 4.24