> Hello list!
>
> I have descided to take the step from reading and admiering to actually
> trying to do soma calculations on a Hybrid. But after reading Sutton's
> book I have come to the conclusion that there are a lot of calculations
> that depend on measured data.
Great to see some one new having a go. Don't be put off by the apparent
>
> Hybrid Rocket with HTPB and gaseous N2O:
>
> Isp = 250s -Estimated
> Chamber pressure = 6.895MPa (=1000Psi)
> Initial thrust = 1000N
> Burn Time = 5s
> Heat coefficient (forgot the exact name), k = 1.3 -Estimated
> Expansion Ratio, e = 10
OK, lets go through these:
HTPB is an excellent fuel for hybrid use, but you can reasonably expect an
ISP of about 200-210 seconds in your first motor, Unless HTPB is convenient, you might want to consider PVC, a good all round material, you can buy it as bar stock and machine any grain you want in much less time than it takes to make a jig to cast HTPB. PVC will give an ISP between 175-200s in a practical motor. You may even get lucky and find some hose or pipe that will do the job!It's unlikely that you'd run a 1000N motor on gaseous NOX. The reason is that NOX is stored as a liquid. As such you would be boiling off the gas to run the motor. At 1000N this would quickly freeze the tank solid.
As others have commented Pc=1000Psi requires an external pressurant. Not hard to do, but it's much harder than saying Pc=300Psi and using the 750ish psi vapour pressure of the NOX to run the show. As the liquid is squirted out of the tank, some NOX boils off to keep the pressure up, this cools the tank and the pressure drops by about 100Psi, so work on a 650-700Psi feed pressure.
OK so our spec now looks like:
Hybrid rocket with PVC and liquid N20
ISP=180s
Pc=300Psi
Average thrust=1000N
Burn Time=5s
Expansion ratio=4
Note I don't go into expansion ratios much. When your starting out, picking close enough expansion ratio saves you a bunch of maths..
Let's design the motor. My apologies if I stuff this up, all my texts are at home so I'll be doing this out of my head.
First up we need a tank to hold the NOX:
We need 5000Ns of impulse from a 180s propellant.
That means we need:
5000/180=27.78N of propellant.
at a realistic O/F ratio of 4:1 that gives 22.2N, or
2.27kg of NOX
at a density of 0.77 that means we need a 2.95L tank.
You MUST leave some space in your tank to allow for over filling, pressure
spikes at ignition etc.. Let's make it 3.5L
But we need the tank to be pressurised when the last bit of
liquid is being squirted out! It takes:
Pt*V/Pa/22.4*M g of NOX to pressurise our tank where:
Pt=Tank Pressure (650Psi)
V=Tank Volume (3.5L)
Pa=Ambient Pressure (14.5Psi)
M=Molar Mass of gas (44g) (It's ok to mix Psi and L here)
Putting our numbers in we get:
650*3.5/14.5/22.4*44=308g which takes up another 308/0.77=400cc (a 3.5L tank
works out about right)
To summarise, our motor will have a 3.5l tank that will hold 2.58Kg of NOX
Working downwards we come to the injector.
We need to flow 2.27Kg of NOX in 5 seconds that's a MOXdot of 0.454Kg/s
The formula for flow across a simple injector is:
MOXdot=Cfl*Ai*SQRT(2*Pox*DeltaP)
Where:
MOXdot=Oxidiser flow rate (Kg/s)
Cfl=Coefficient of flow for that injector(0.5 is a good guess for biphase
flow)
Ai = Area of the injector (m^2)
Pox=The density of NOX in Kg/M^3 (770)
DeltaP =The pressure drop across the injector ie 650-300Psi(2420000Pa)
Rearranging for Ai
Ai=Moxdot/(Cfl*sqrt(2*Pox*DeltaP)))
Plugging in our numbers:
Ai=0.454/(0.5*sqrt(2*770*2420000)))
=0.0000149m^2
or a diameter of 4.35mm
Now we come to the fuel grain.
So far the concepts have all been pretty straight forward, standard mech eng
type calcs. Here's where we get into the "Hybrid stuff"
To keep things simple, we'll stick with a single port grain, A bit like a
big core burning solid. We need to boil off enough fuel to mix
with our 454g/s of oxidiser and give our 4:1 O/F ratio.
The two factors that determine this (Mfdot) flow rate are:
The burning surface area of the fuel grain.
The rate at which the fuel regresses.
The first is straight forward.
The latter can be calculated as r=aGo^n
Where:
r= regression rate.
a= Burn rate coefficient in mm/s
n= Burn rate exponent (dimensionless)
Go= The oxidiser Flux rate (N/m^2/s)
Go is the thing most people have trouble with.
Imagine a length of 75mm pipe laid horizontal on a bench.
Place small candles every 10cm down the length of the pipe and light them.
Now start blowing air down the pip, gently at first.
Initially the candles splutter about in the gentle breeze. This is what happens
in hybrid when Gox is too low ( < about 1500N/m^2/s). The flame front
is unstable and your motor may not light.
As you increase the velocity of the air down the pipe, the candles burn stronger
and more steadily, this is where we want our motor to be at.
At some point the velocity of the air down the pipe gets to a level where it
blows the candles out! This is what happens when we run Gox too
high ( >6500N/m^2/s). Note that for a given volume of air travelling down the
pipe, we can decrease the velocity by increasing the
diameter of the pipe.
hope that helped.
Anyhow, lets pick a sensible Gox of 4500N/m^2/s
This gives a port area of:
4.5/4500=0.001m^2
(4.5 is Moxdot in N)
or a port diameter of 3.6cm
a for PVC is about 0.0052, n is about 0.65 (this will vary with grade
andcolour)
Thus r=0.0052*4500^0.65=1.23mm/s
So, we have a 3.6cm diam port, regressing at 0.123cm/sec.
Given the density of PVC is 1.7, This will produce:
3.6*pi()*0.123*1.7=2.36g of fuel for every centimetre of fuel grain we have.
We know Moxdot is 454g/s and O/F=4:1
so we work out that we need 454/4=114g/s of Fuel
From which we derive a grain length of 114/2.36=48.3cm
Assuming a constant regression rate, (for bonus points work out the
regression rate over the duration of the burn), we need a web thickness of
5*0.123=0.615cm
Which gives a minimum grain diameter of 3.6+2*0.615=4.83cm.
So if you managed to find some 2"OD, 1/4" wall PVC hose, you'd be
laughing!
Lastly we have our nozzle.
Using the ubiquitous rocket equation.
F=Cf*At*Pc
F=Thrust (N)
Cf=Coefficient of thrust worked out from all those nasty Thermodynamic equations
Pc=Chamber Pressure in Pa
Rearranging for At, and working off a Cf of about 1.4 we get
At=1000/(1.4*2070000)
=0.000345m^2
Which gives a throat diam of 21mm and (using Ae/At of 4) an exit diam of 42mm
(looks like the combustion chamber is going to be made from 2" pipe!)
There you have it!
Seems complicated at first. But when you've done a few (and made a spreadsheet:-)
it's all pretty logical.
BTW 5000Ns @1000N is a big motor to learn on. You might want to try something
like http://www.comcen.com.au/~pkelly/40design.zip
While your there, have a look at some of the other designs there. All sorts of ways to put them together...
PK