Hybrid Rocket Modelling
3.3 Mass and Sizing
A key factor for any flight vehicle is the vehicle’s total mass, for its direct relation to the propulsion subsystem. Create a lighter structure and optimizing the mass distribution in the rocket are key factors to achieve the optimal design and minimize the costs to build the vehicle. The rocket size is a factor as important as the mass since the rocket stability is directly related to its length.
In this section, the models used to calculate and estimate both mass and sizing of the rocket are presented.
3.3.1 Mass Estimation
In a sounding rocket the mass can vary according to the material of the components and the respective dimensions. But since the mass of each compartment is necessary for the model, a multivariate linear regression was done by Klammer [13]. At this linear regression, twelve rockets mass data [3, 20, 21, 29–
36, 59] was used to create a linear fit to estimate the structural massmstructuralin function of the rocket external diameterDex and propellant mass mprop, calculated as the sum of the fuel mass mf uel and oxidizer massmox, as shown in Equation 3.52. The linear fit is shown in Figure 3.9 and the multivariate linear regression is shown in Equation 3.53.
Figure 3.9: Linear Regression for Mass Estimation[13]
mprop=mf uel+mox (3.52)
mstructural= 0.4349mprop+ 259.015Dex−13.262 (3.53) The structural mass calculated represents the sum of the masses of all the rockets components.
However, since the masses of the rocket external structurembody, the nose conemcone, the finsmf ins, combustion chambermCC and oxidizer tankmOT are calculated, the function can be modified to es- timate only the missing values, which are the recovery system massmrec, avionics system massmav and nozzle massmnozzle. The expression for the estimated massmestimatedis shown in Equation 3.54.
To distribute the estimated mass to the three remaining components the average value of the masses of each of them were taken from the reference models resulting in a distribution of 60%designated for the recovery bay, since this compartment is composed of two parachutes, the main and drogue, and the parachute launch system, meaning that between the three its mass and volume vary the most,25%des- ignated for the avionics system, since the electronic components usually are lighter than the recovery components, and15%for the nozzle mass, since the design and material selection of this part is usually focused to minimize the mass for stability purposes.
mestimated=mrec+mav+mnozzle=mstructural−mbody−mcone−mf ins−mCC−mOT (3.54)
Another mass that is necessary to estimate is the payload mass. At the Spaceport America Cup rules [60] it is stated that the payload mass must be equal or higher to 8.8 lb, or approximately 4 kg. To take this rule into account, the payload mass was seated as 4 kg.
3.3.2 Mass Calculation
Having determined a model to estimate the masses of the nozzle, recovery and avionic system, it is also necessary to calculate the masses of the remaining components. These components can be separated into three groups: the cylindrical components, the nose cone and fins.
To calculate the mass of the cylindrical components, such as the combustion chamber, the oxidizer tank and the rocket body, a model of a thin wall cylinder is adopted. The masses of these components are calculated as a function of its density ρcylinder and volume Vcylinder, that can be calculated using the cylinder’s lengthLcylinder, internal rcylinder,inand external radius rcylinder,ex, as in Equation 3.55.
To calculate the internal radius of the cylinder, a value for the wall thickness is necessary. Using the data from six different rockets launched at the 2018 Spaceport America Cup [3, 17, 20, 21, 29, 30], an average value for the wall thickness was created. The thickness selected for the combustion chamber was tCC = 3mm, for the oxidizer tank tOT = 6mmand for the rocket body tbody = 3mm. Since the external radius is a design variable and therefore is considered a known value, the internal radius can be calculated by subtracting the cylinder’s thicknesstcylinderas in Equation 3.56.
mcylinder=ρcylinderVcylinder=ρcylinderLcylinderπ(r2cylinder,ex−r2cylinder,int) (3.55)
rcylinder,int=rcylinder,ex−tcylinder (3.56) To define the density of each component it is necessary to define the most commonly used materials.
For a sounding rocket, it is a common practice to use carbon fiber-epoxy as the rocket body material, since it presents a lower density in comparison to other lightweight materials (such as aluminum), a high tensile strength, toughness, corrosion and impact resistance, having a high strength to weight ratio [61].
Knowing this, the properties of a carbon fiber-epoxy composite were considered for the rocket body. As
for the oxidizer tank and combustion chamber, using the data from the previously mentioned six rockets, the most commonly used material is the Aluminum 6061-T6 for its low price, for being easily machinable and to attend the resistances requisites during operation[3]. Therefore it was selected as the material for the combustion chamber and oxidizer tank.
The nose cone mass calculation has a similar procedure to the cylindrical parts calculations. The simplification of a thin-walled nose cone is applied to the model. Since the nose cone shape can vary according to each team’s project, a conical shape was adopted as a reference, with a correction factor ζcone. The nose cone mass can be calculated based on its material densityρcone, superficial areaAcone, thicknesstconeand length Lcone, as shown in Equation 3.57. The material selection for the nose cone is very similar to the one done for the rocket body. However, carbon fiber can generate radio frequency shielding [62] and since the nose cone is commonly used to accommodate electronic sensors, the carbon fiber is not commonly used as the nose cone material. With similar mechanical properties as the carbon fiber, fiberglass is an alternative material that does not generate radio frequency shielding.
Therefore, a fiberglass-epoxy composite was selected as the material for the nose cone.
mcone=ρconeζconeAconetcone=ρconeζconetcone(πrex
pL2cone+r2ex) (3.57) The fins of a rocket are responsible to stabilize the rocket during flight, maintaining its orientation and trajectory [63]. To reduce the structural mass and guarantee higher stability, the fins are usually made of a composite material, like carbon fiber-epoxy. In this model, a trapezoidal carbon fiber-epoxy fin is considered, resulting in a mass expression as in the Equation 3.58. In this equation the mass of the fins mf inis calculated as a function of its densityρf in, spanSf in, root chordBf in, tip chordbf in and thicknesstf in.
mf in=ρf intf in(Bf in+bf in)Sf in
2 (3.58)
3.3.3 Size Estimation
In the project of a rocket, some variables such as the dimensions of the drogue and main parachutes, the avionic components and the fins may vary according to each team decision making, resources and priorities making it difficult to predict or calculate. However, since this model requires the length of each of the rockets compartments, an approximation needed to be done.
Using the data from the same rockets used on mass calculation (Subsection 3.3.2)[3, 17, 20, 21, 29, 30], a linear regression was done relating the length of the recovery system bay to the length of the rocket and the avionics bay to the recovery system bay. The linear regression comparison with the actual rocket recovery and avionics length is shown in Figure 3.10.
Using this fits, the length of recovery bayLrecis calculated as a function of the rocket’s lengthLrocket
and the avionics bayLav is calculated as a function ofLrecas expressed in Equations 3.59 and 3.60.
Lrec= 0.2404Lrocket−0.4193 (3.59)
(a) Recovery System Length in Function of Rocket Length (b) Avionic System Length in Function of Recovery System Length
Figure 3.10: Linear Regression for Size Estimation
Lav = 0.6891Lrec−0.0784 (3.60)
The payloads are separated into two categories: a ”boiler-plate” payload, being a non-functional pay- load, and a scientific experiment or technology demonstration payload. For the boiler-plate payload, a minimum requirement for its length is to fit at least a 3U Cubesat, with 30cmx10cmx10cm. However, the scientific experiment payload may be constructed in any form factor. [60]. To attend the non-functional payload rule, a length ofLpay= 0.4 m was considered for the payload, taking a tolerance into account.
The length of the nose cone and the length of the nozzle are two independent project variables that can vary according to each group’s manufacturing costs and team decisions, therefore are hard to associate with other known parameters. To stipulate these two dimensions, an average value between the six rockets was used leading to a length ofLcone= 0.6 m for the nose cone and a length ofLnozzle= 0.135 m for the nozzle. The assumed lengths are shown in Table 3.1.
Table 3.1: Assumed Lengths
To determine the rocket external diameter, a geometrical constraint was considered. The combustion chamber was considered as a straight fit inside the fuselage, meaning that the external diameter of the combustion chamber is equal to the internal diameter of the rocket external tube and that the oxidizer tank diameter is lower than the combustion chambers. Normally the teams apply a thermal protection coating inside or outside the combustion chamber to minimize the heat transfer from the combustion to the chamber and external structure.
3.3.4 Size Calculation
Since the size of the rocket is directly related to its mass and stability, the calculation of the length of each compartment in a precise way is crucial to the model. The components whose lengths can be calculated are the oxidizer tank and the rocket external body.
To calculate the oxidizer tank’s lengthLOT, a cylindrical shape is considered. Combining two design variables, the tank volume VOT and tank radius rOT, the length of this component is expressed in Equation 3.61.
LOT = VOT
πrOT2 (3.61)
The rocket external body is a carbon fiber-epoxy tube, which needs to contain the recovery, payload, avionics, oxidizer tank, combustion chamber and part of the nozzle. To take into account the nozzle part that is allocated inside the composite tube, a factorNnozzleis created. Since the combustion chamber lengthLCCis a design variable, the rocket external bodyLbody and total lengthLrocketcan be calculated as shown in Equations 3.62 and 3.63.
Lbody=Lrec+Lpay+Lav+LOT +LCC+LnozzleNnozzle (3.62)
Lrocket=Lcone+Lbody+Lnozzle(1−Nnozzle) (3.63) The initial estimation of the rocket’s lengthLrocket,0 is performed by substituting the recovery and avionics lengths expressions in Equation 3.62, rearranging it as a function of the known lengths, as shown in Equation 3.64.
Lrocket,0= Lcone+Lpay+LOT +LCC+Lnozzle
0.5939 −1.324 (3.64)