**3.2 Yield stress fluids**

The NewRocket Green Propellant (NRGP) gelled fuel has been classified as a yield-stress fluid, and this feature has been demonstrated and quantified by tests that included rheological characterization, application of dynamic environment such as acceleration in centrifuge, and real-time storage and handling. The following paragraphs elaborate on that, while being extensively based on Spicer and Gilchrist [53], and are included here in order to make the present chapter quite self-contained.

Yield-stress fluids have the feature of solid-like materials in that they do not flow until a critical stress (**σy**) is exceeded, after which they flow like a liquid. Modeling such behavior often begins with a nonzero value of the yield stress term *σy* in the Herschel-Bulkley-Extended (HBE) equation.

$$
\sigma = \sigma\_y + k \, \dot{\mathbf{y}}^n + \mu\_{\text{so}} \dot{\mathbf{y}} \tag{1}
$$

**13**

**Figure 9.**

*The time interval between sequent pictures is 2 ms [24].*

*Green Comparable Alternatives of Hydrazines-Based Monopropellant and Bipropellant Rocket…*

Eq. (1) is able to describe power law behavior and includes the additional yield stress term **σy**. Yield-stress fluids are typically shear thinning and have an exponent

The expectation that a fluid might have a yield stress comes from an understanding of the fluid microstructure and its relevant length and time scales. Generally, attractive interactions between colloids, physical crowding of larger particles, and cross-links between polymers or micelles can all provide a finite yield stress to a fluid. Concentration is also a key variable in yield-stress fluids. Very dilute suspensions can have a yield stress but only if the particles attract each other strongly such that they stick together upon collision. The rheology of a suspension gel is highly dependent on whether the particles attract one another strongly enough to form a network that resists flow. Gel microstructure is often a unique function of its

*A sequence of high-speed photographs demonstrating hypergolic ignition of hydrogen peroxide with kerosene.* 

*DOI: http://dx.doi.org/10.5772/intechopen.82676*

of **n** < 1.

Here *σ* is the stress applied on the fluid, ̇ is the shear rate, *k* is a proportionality constant termed the consistency coefficient (viscosity at ̇ = 1), **n** is a power law exponent termed the flow index, and **∞** is the constant viscosity in the very high shear rate range.

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

Eq. (1) is able to describe power law behavior and includes the additional yield stress term **σy**. Yield-stress fluids are typically shear thinning and have an exponent of **n** < 1.

The expectation that a fluid might have a yield stress comes from an understanding of the fluid microstructure and its relevant length and time scales. Generally, attractive interactions between colloids, physical crowding of larger particles, and cross-links between polymers or micelles can all provide a finite yield stress to a fluid. Concentration is also a key variable in yield-stress fluids. Very dilute suspensions can have a yield stress but only if the particles attract each other strongly such that they stick together upon collision. The rheology of a suspension gel is highly dependent on whether the particles attract one another strongly enough to form a network that resists flow. Gel microstructure is often a unique function of its

**Figure 9.**

*A sequence of high-speed photographs demonstrating hypergolic ignition of hydrogen peroxide with kerosene. The time interval between sequent pictures is 2 ms [24].*

*Aerospace Engineering*

the proper material.

8% higher average density.

achieved as shown in **Figure 11**.

Herschel-Bulkley-Extended (HBE) equation.

= *<sup>y</sup>* + *k* ̇

Here *σ* is the stress applied on the fluid, ̇

considerations.

self-contained.

shear rate range.

**3.2 Yield stress fluids**

By nature, hydrogen peroxide and kerosene do not ignite upon contact. However, in a gelled fuel, the existence of yield stress assures that particles (reactive or catalytic) can be added without the effect of sedimentation or buoyancy. Gels enable the suspension of reactive or catalyst particles, uniformly distributed in the fuel, without compromising the energetic performance of the system. The use of suspended particles enables a quite large variety of combinations of fuels and oxidizers that can become hypergolic by gelling one of the liquids and adding

Natan et al. [49] came up with the idea of embedding reactive particles with hydrogen peroxide in gelled kerosene. Drop-on-drop tests exhibited that this kind of gelled kerosene is hypergolic with hydrogen peroxide as shown in a sequence of photographs in **Figure 9**. Total ignition delay time was 8 ms. Connell et al. [50–52]

The idea was adopted by a start-up company, NewRocket that proceeded with the development of a prototype motor using gelled kerosene with reactive particles and hydrogen peroxide [24]. **Table 3** shows the characteristics of their NRGP

to comparing MMH/N2O4 and the kerosene and hydrogen peroxide-based green bipropellant NRGP. The specific impulse *Isp* of NRGP, as shown in **Table 4**, is some 4% lower than that of MMH/N2O4, in accordance with its considerably lower chamber temperature; but it is noteworthy that its *ρIsp* is higher by 4% thanks to its

Experiments have been conducted in a lab-scale motor to verify the feasibility of the idea. The main problems were the atomizers because the particles initially caused plugging of the exit. The problem was solved by changing the type of reactive particles and by increasing the atomizer diameter. The system (**Figure 10**) was found to operate properly, and by using adequate valves, operation in pulses was

In the next sections, the stability of the fuel for no phase separation or sedimentation throughout its life cycle is demonstrated by theoretical and experimental

The NewRocket Green Propellant (NRGP) gelled fuel has been classified as a yield-stress fluid, and this feature has been demonstrated and quantified by tests that included rheological characterization, application of dynamic environment such as acceleration in centrifuge, and real-time storage and handling. The following paragraphs elaborate on that, while being extensively based on Spicer and Gilchrist [53], and are included here in order to make the present chapter quite

Yield-stress fluids have the feature of solid-like materials in that they do not flow until a critical stress (**σy**) is exceeded, after which they flow like a liquid. Modeling such behavior often begins with a nonzero value of the yield stress term *σy* in the

ity constant termed the consistency coefficient (viscosity at ̇ = 1), **n** is a power law exponent termed the flow index, and **∞** is the constant viscosity in the very high

\_\_\_\_\_\_

*<sup>n</sup>* + ∞ ̇ (1)

is the shear rate, *k* is a proportional-

*Tc*/*M*, is applicable

also investigated the issue and showed that it is feasible.

propellant in comparison to other candidate propellants.

Here again the specific impulse relation to temperature, *Isp*~√

**12**


#### **Table 3.**

*Characteristics of candidate propellants [24].*


## **Table 4.**

*Comparison of the properties of the bipropellant composition MMH/N2O4 vs. NRGP with kerosene-based fuel/ H2O2.*

**15**

flow.

**Figure 11.**

*NewRocket engine operation in pulses.*

*Green Comparable Alternatives of Hydrazines-Based Monopropellant and Bipropellant Rocket…*

processing history because the particle networks can grow, break, and reform under

The ability of the NRGP fuel, as a yield-stress suspension, to suspend solid particles in the gelled kerosene without any displacement occurring until a shear stress of **σy** or above is applied, is of key importance. For that, an estimate is made of the magnitude of yield stress required to suspend a given particle. It is important to contrast this treatment with the following Stokes law description of particle settling in a Newtonian fluid, derived via a force balance between the buoyant and

v = (**<sup>p</sup>** – **l**) **d<sup>2</sup>** *<sup>g</sup>* \_\_\_\_\_\_\_\_\_ **<sup>18</sup>**

where the sedimentation velocity at low Reynolds numbers, **v**, is a function of the particle **ρp** and liquid **ρl** densities, the particle diameter **d**, gravitational acceleration *g*, and the fluid viscosity **μ**. For other than gravitational accelerations, *g* would

Rearranging Eq. (2) to solve for viscosity and substituting the height to shelf life ratio for velocity obtains Eq. (3), with sedimentation length and time, **h** and **t**,

The performance of viscosity with that of a yield stress for the same application can numerically be contrasted by day by day examples [53]. A fluid with a yield stress not exceeded by the acceleration stress of a particle is not described by Eq. (2) because it essentially possesses an infinite viscosity at low stresses and no flow can occur. By taking the ratio of the particle gravitational stress to the fluid yield stress and assuming a hemispherical characteristic area of the yield surface formed, a dimensionless parameter, **Y**, is obtained to be used to calculate whether a particle

(2)

**18h** (3)

**3.3 Yield-stress suspension of particles vs. Stokes law settling**

drag forces acting on a suspended particle:

be replaced by the applicable acceleration *a*.

= (**<sup>p</sup>** – **l**) **d<sup>2</sup>** *<sup>g</sup>***<sup>t</sup>** \_\_\_\_\_\_\_\_\_

respectively instead of velocity **v**.

will sediment in a yield-stress fluid:

*DOI: http://dx.doi.org/10.5772/intechopen.82676*

**Figure 10.** *NewRocket lab-scale experimental system.*

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

**Figure 11.** *NewRocket engine operation in pulses.*

*Aerospace Engineering*

Ionic propellants ADN-based (HPGP-LMP-103S\*)

Bipropellants NTO/

MMH\*

**Table 3.**

Electric ion thrusters arcjet

Hybrid Low–

*Characteristics of candidate propellants [24].*

none

Low Medium– high

*expansion ratio ε = 50.*

**Table 4.**

*H2O2.*

**Bipropellant MMH/N2O4 NRGP** Average density *ρ* kg/L 1.2 1.3 Specific impulse based on mass flow *Isp* s 341 328 Specific impulse based on volume flow *ρIsp* s kg/L 409 426 Chamber temperature *Tc* °C 3125 2580 *All properties at 25°C, Isp calculated for reaction chamber pressure Pc = 2.0 MPa, ambient pressure Pa = 0.0 MPa,* 

**Propellant Toxicity Storability Cost Safety In flight** 

Solid Low High Low Medium No No

Hydrazine High High High Low Yes Yes

NRGP None High Low High Yes Yes

High Medium Medium–

Medium– high

High High High Medium Yes Yes

None High High Medium Yes \_

high

**control**

Medium Yes Yes

Yes No

**Hypergolic ignition**

*Comparison of the properties of the bipropellant composition MMH/N2O4 vs. NRGP with kerosene-based fuel/*

**14**

**Figure 10.**

*NewRocket lab-scale experimental system.*

processing history because the particle networks can grow, break, and reform under flow.

## **3.3 Yield-stress suspension of particles vs. Stokes law settling**

The ability of the NRGP fuel, as a yield-stress suspension, to suspend solid particles in the gelled kerosene without any displacement occurring until a shear stress of **σy** or above is applied, is of key importance. For that, an estimate is made of the magnitude of yield stress required to suspend a given particle. It is important to contrast this treatment with the following Stokes law description of particle settling in a Newtonian fluid, derived via a force balance between the buoyant and drag forces acting on a suspended particle:

$$\mathbf{v} = \frac{(\rho\_\mathrm{P} - \rho\_\mathrm{I})\,\mathrm{d}^2\mathrm{g}}{\mathbf{1}\,\mathrm{8\mu}} \tag{2}$$

where the sedimentation velocity at low Reynolds numbers, **v**, is a function of the particle **ρp** and liquid **ρl** densities, the particle diameter **d**, gravitational acceleration *g*, and the fluid viscosity **μ**. For other than gravitational accelerations, *g* would be replaced by the applicable acceleration *a*.

Rearranging Eq. (2) to solve for viscosity and substituting the height to shelf life ratio for velocity obtains Eq. (3), with sedimentation length and time, **h** and **t**, respectively instead of velocity **v**.

$$
\mu = \frac{\left(\rho\_p - \rho\_l\right) \mathbf{d}^2 \mathbf{g} \mathbf{t}}{\mathbf{18h}} \tag{3}
$$

The performance of viscosity with that of a yield stress for the same application can numerically be contrasted by day by day examples [53]. A fluid with a yield stress not exceeded by the acceleration stress of a particle is not described by Eq. (2) because it essentially possesses an infinite viscosity at low stresses and no flow can occur. By taking the ratio of the particle gravitational stress to the fluid yield stress and assuming a hemispherical characteristic area of the yield surface formed, a dimensionless parameter, **Y**, is obtained to be used to calculate whether a particle will sediment in a yield-stress fluid:

$$\mathbf{Y} = \frac{2\pi \left(\frac{\mathbf{d}}{2}\right)^2 \sigma\_\mathbf{y}}{\frac{4}{3}\pi \left(\frac{\mathbf{d}}{2}\right)^3 (\rho\_\mathbf{p} - \rho\mathbf{v})\mathbf{g}} = \frac{3\sigma\_\mathbf{y}}{\mathbf{d}(\rho\_\mathbf{p} - \rho\mathbf{v})\mathbf{g}}\tag{4}$$

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 suspension of small particles.

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, whereas viscosity merely slows particle motion.
