**3.5 Calculation of the sedimentation velocity and distance of a particle through the NRGP gelled fuel**

When the yield stress **σy** = 10 Pa, which is the conservative threshold for movement, has been surpassed by applying acceleration 2400 *g* (in excess of 2352 *g*), the sediment movement velocity of a particle through fluids can be calculated using Eq. (2). Here the acceleration of gravity *g* would be replaced by the applicable acceleration *a*.

*Green Comparable Alternatives of Hydrazines-Based Monopropellant and Bipropellant Rocket… DOI: http://dx.doi.org/10.5772/intechopen.82676*

$$\begin{aligned} \text{It:} & \text{!/dx.do.org/na.ff7/2/int.topen.82676} \\\\ \text{v} &= \frac{(\text{p}\_\text{p} - \rho\_\text{l}) \cdot \text{d}^2 \cdot a}{18\mu} = \frac{650 \frac{\text{kg}}{m^3} \cdot \left(2 \cdot 10^{-6} \, m\right)^2 \cdot (2400 \cdot 9.81) m/s^2}{18 \cdot 100 \, Pa \cdot s} = 3.5 \cdot 10^{-9} \, m/s \quad (6) \end{aligned}$$

This, with a gel viscosity measured in experiments of *μg* = 100 *Pa* <sup>∙</sup> *<sup>s</sup>*. However, the viscosity derived from the Herschel-Bulkley rheological model coefficients in the following paragraph would be taken into account here.

In 10 s under acceleration of 2400 *g*, the resultant sedimentation distance is 3.5 × 10<sup>−</sup><sup>8</sup> m, namely one-thirtieth of a micron sedimentation. In 10 years (10 × 3600 × 24 × 365 = 3.15 × 108 s), this represents sedimentation of 1 m, namely full sedimentation. Therefore, it is important to remember that the extremely high acceleration value, in the order of km/s2 , was assumed here just in order to compare to the actually high yield stress of the fuel, while in reality any considerable accelerations are applied on the fuel for very short periods only, such as experienced by space launch.

For a fuel with similar viscosity, but without a yield stress (not being the case here), for gravitational acceleration of 9.81 m/s2 , in 10 years the sedimentation would merely be 4 mm.

It can be seen that the sedimentation distance of a particle is proportional to its squared diameter, gravity, and particle density and inversely proportional to the viscosity of the gel. Thus, it is possible to reduce the sediment distance by the following ways: reducing particle diameter, reducing particle density, and increasing the viscosity of the gel.

Based on Technion experience, it can be stated that after storage of a couple of years, there is no degradation in terms of phase separation, sedimentation, agglomeration, ignition delays, etc. For quantitative evaluation of these behaviors, both real-time and accelerated tests are relevant. These were obtained by centrifuge tests for assessing the stability of the gel in accelerations.

To simulate the mechanical environmental loads during a typical rocket launch, which might cause concern regarding gel separation in the tank and propellant feed system, centrifuge tests have been conducted. Example of result obtained for a gelled fuel with a suspended particle with a diameter of 250 μm is depicted in **Figure 12**, which shows the gel stability as a function of operating time or degree of acceleration. In this experiment, a gelled fuel sample within a test tube was tested in a centrifuge for assessing the influence of two different conditions: firstly, applying constant acceleration (40 g) while varying the time duration (**Figure 12** left) on the test and secondly, applying constant duration time (2 minutes) while varying the magnitude of the acceleration (**Figure 12** middle). After each centrifuge test, the separated liquid due to the acceleration has been sought in order to be compared with the initial mass to quantify the stability of the investigated gel.

It is important to note that from a visual examination of all samples, no particle sedimentation was observed.

#### **3.6 NRGP fuel rheological characterization**

#### *3.6.1 Measuring system*

For characterization of the rheological behavior of the gelled fuels, a TA Instruments AR 2000 rotational rheometer [56] operated in controlled rate mode is being used. The rotational rheometer imposes strain to the liquid and measures the resulting stress for shear rates up to 1000 1/s. Most common test geometries

*Aerospace Engineering*

**Y** = **2**

suspension of small particles.

for velocity, obtains Eq. (5).

**the NRGP gelled fuel**

applicable acceleration *a*.

whereas viscosity merely slows particle motion.

*<sup>a</sup>* **<sup>=</sup> <sup>3</sup> <sup>y</sup>** \_\_\_\_\_\_\_\_\_

**through the NRGP gelled fuel**

( \_\_ **d 2** ) **2 <sup>y</sup>** \_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_

(**<sup>p</sup>** – **l**)*g*

where **d** is the particle radius and **σy** is the fluid yield stress. It is worthwhile noting that the critical **Y, Ycrit**, bounding the states of suspension and sedimentation, is less than unity because of the finite fluid volume yielded by the particle. This means that the yield stress required to suspend a given particle is actually less than the gravitational stress the particle exerts. Simulations give a value of **Ycrit** = 0*.*14 [54], while experiments produce **Ycrit** values between 0.1 and 0.6 [55]. Since the critical criterion can vary significantly, so can the suspension efficiency of a yield-stress fluid. Eq. (4) can be used to estimate the yield stress required to stably suspend a small solid particle by assuming a worst case of a **Ycrit** = 1. If the worst case application is not satisfying the requirements, then Eq. (4) may be used to remove the extraconservatism by using it as a nondimensional index. This can be experimentally determined for a specific fluid-particle system using a test in which the suspension stability of a range of particle sizes or densities is recorded for a specified yield-stress fluid and the transition from stability to sedimentation is recorded. The approach described above applies to sedimentation of a dilute suspension of particles through a homogeneous yield stress fluid or, equivalently, of a much larger single particle through a homogeneous

It can be demonstrated [53] that yield stress can be a very efficient means of stabilizing particle suspensions because it can entirely prevent any particle motion,

Rearranging Eq. (4) and replacing the gravitational acceleration *g* with the applicable acceleration *a* in order to solve it, and substituting the height to shelf life ratio

**Y ∙ d**(**<sup>p</sup>** – **l**)

Using Eq. (5) for a particle with diameter **d** = 2 μm and density of **ρp** = 1.45 g/ cc, immersed in a gel with density **ρl** = 0.8 g/cc and with a **σy** of 10 Pa (a conservative order of magnitude representative of the 16 Pa measured in the paragraph below), while assuming a worst case of a **Ycrit** = 1, the solid particle will start to move when the acceleration reaches a threshold value of *a* = 23,077 m/s2

**3.5 Calculation of the sedimentation velocity and distance of a particle through** 

When the yield stress **σy** = 10 Pa, which is the conservative threshold for movement, has been surpassed by applying acceleration 2400 *g* (in excess of 2352 *g*), the sediment movement velocity of a particle through fluids can be calculated using Eq. (2). Here the acceleration of gravity *g* would be replaced by the

(5)

or

**3.4 Calculation of the sedimentation threshold acceleration of a particle** 

<sup>=</sup>**<sup>3</sup> <sup>y</sup>** \_\_\_\_\_\_\_\_ **d**(**<sup>p</sup>** – **l**)*g*

(4)

**4 3** ( \_\_ **d 2** ) **3**

**16**

*a* **= 2352** *g*.

**Figure 12.**

*Gel stability as a function of test duration time (left) and acceleration value (middle). On the right is a test tube before centrifuge test.*

for rotational rheometers are the parallel plates (see **Figure 13** right) and the cone and plate. The parallel plates configuration has been used here for gel characterization. A Peltier plate-type temperature regulation system inside the equipment ensures the prescribed controlled fluid temperatures during rheological measurements.
