Wednesday, September 21, 2011

How To Build A Spaceship - Part 3

In Part 2 we looked at some of the variables we need to know or calculate to determine our spaceship design. Now, we'll look at the elements of the ship itself.

Just as a Ferrari has specific parts such as engine, chassis, suspension, brakes, body etc., a spaceship can be broken down into necessary, specific components.

We'll need a drive (no warp drives here!) and propellant, hull, and a payload section.

The design I'm going to initially copy is from the NASA Artificial Gravity for Human Exploration Missions report. I must admit, I picked this one because it looks cool and has some rather interesting quirks that I think would make for a great story setting. It uses the "fire baton" concept to create artificial gravity by spinning the entire ship. The power reactors are on one end, and the payload (the habitat or crew module) is balanced on the other end. The ship rotates around the center, where the propellant tanks and thrusters are located. Our thrust will be provided by six Magnetoplasmadynamic (MPD) thrusters, which get their power from six turbo alternators which convert the thermal energy from two nuclear reactors into electrical power. So it will break down like this:
  • Drive - 2 nuclear reactors, 6 turbo alternators, radiators, radiation shield and assorted gizmos.
  • Propellant  - This is the Lithium propellant that our MPD thrusters will use.
  • Hull - this is the central core, the truss that connects the Drive and Habitat modules to the core, and guy wires, etc. In this particular design, our thrusters are also considered to be part of the Hull section, although they are commonly in the Drive section. This will also include the mass of our tanks that hold the propellant. You'll note the hull is nothing like an ocean liner or the Starship Enterprise. It is pretty much just a skeleton that connects the other parts of the ship. Ah, the elegance of space travel!
  • Payload - This consists of our habitat module, which is a self-contained, inflatable TransHab design.
First, we'll need to figure out how much mass our spaceship has...

As anybody who's ever read The Cold Equations knows, mass is serious business for a spaceship. Pretty much everything else depends on what the mass is. The most important numbers are the Total Mass (also called Wet Mass) and the Empty Mass (also called Dry Mass or the mass of everything except the propellant.) For our design, we also need to be sure the masses of our Drive and Payload are roughly equal, since they will be rotating around the central core. Thanks to the lightning slide rules of NASA we already have some good numbers to start out with. That's important, because for "realistic" spaceship design, we really need accurate estimates for the mass of various systems such as life support and reactors and whatnot. It breaks down like this:
  • Drive - 32,200kg - This includes our reactors, turbo alternators, radiation shield, radiators, etc.
  • Propellant - 70,000kg - We might modify this number later to optimize it for the mission.
  • Hull - 9,000kg - just a bunch of exotic metal and composites, but necessary!
  • Payload - 34,200kg - Our payload is us, the astronauts! And everything necessary to keep us alive and cozy for a lengthy voyage to the red planet.
Add it all up and we come up with 145,400kg or about 145 metric tons.

Now, what about these sci-fi sounding Magnetoplasmadynamic MPD thrusters and nuclear reactors and turbo-alternators? As you will see, some of the most important factors in going anywhere in space are power and thrust and specific impulse. You can read all about the MPD thrusters, but for now we'll just use the data in the Nasa Artificial Gravity Report. We'll estimate that each thruster will use 1 Megawatt (MW) of power and that it will generate 25 N of thrust and have a Specific Impulse (Isp) of 5000 seconds. We'll use 6 thrusters for a total of 150N of thrust. The nuclear reactors generate heat energy and that is converted to electricity by the turbo alternators. The reactors actually generate 15MW of heat energy each, for a total of 30MW, of which we are only able to squeeze out 6MW of usable electricity, so if our electricity conversion was more efficient, we'd be able to use more power. But, we're stuck with just 6MW for now.

Also, we're using up all of our electricity output for the thrusters. We really should set aside some power for little things such as life support systems and computers and communications, etc. But let's not worry about that for now; let's crunch some numbers!

Step-1: Find the distance (D) we need to travel. We're going to Mars, so the distance is .5 Astronomical Units, or about 74.8 million kilometers. To make things simpler, we'll multiply that by 1000 to get the result in meters.
D = 74,800,000,000m

Step-2: Figure our Total Mass (M), which is 145,000kg.
M = 145,000kg

Step-3: Find our Mass Ratio (R.) Our Total Mass (Wet Mass) is 145,400kg. Our Propellant is 70,000kg, so our Dry Mass (Mass without Propellant) is 145,400 - 70,000 = 75,400. Mass Ratio = Total Mass / Dry Mass, so 145,400 / 75,400 = 1.92.
R = 1.92

Step-4: Find our Exhaust Velocity (Ve) which is Isp * 9.81. So it is 5000 * 9.81 = 49,050 meters per second.
Ve = 49,050m/s

Step-5: Let's see what our Ship Delta-V is. We use the Rocket Equation, which is Δv = Ve * ln[R] Hah, yeah right. Okay, the triangle stands for "change" and "v" is our velocity, so Delta-V really means "change in velocity."  ln is the Natural Logarithim of R, which is our Mass Ratio. Just switch your Mac or PC built-in calculators to scientific mode and type 1.92 then press the ln key then multiply that by our Ve, which is 49,050. So, Δv = 49,050 * ln[1.92] = 31,996m/s.
Δv = 31,996m/s

Step-6: Now let's see what our acceleration is. Acceleration (A) = Thrust (N) / Mass (M). So we have A = 150N / 145,000kg = .001 m/s.
A = .001 m/s

Step-7: Okay, let's put some of those numbers to work! Using the Brachistochrone Equation, we can find out how long it will take to get to Mars with amount of thrust we found in Step-6. This means we will accelerate constantly till we get to the halfway point, then flip around and decelerate until we come to a stop at our destination. T (time in seconds) = 2 * Square Root [Distance / Acceleration] so it will be T = 2 * Sqrt[74,800,000,000 / .001] = 17,297,398 seconds. Well divide that by 86,400 to convert from seconds to days, which gives us: 200 days.
T = 200 days

Step-8: We need to determine our Mission Delta-V. That will be Mission Delta-V = 2 * Square Root[D*A] so Mission Delta-V = 2 * Sqrt[74,800,000,000 * .001] = 17,297m/s
Mission Δv = 17,297m/s

So once we put all of that together, we can get to Mars in about 200 days or 6.5 months. We have enough Delta-V to get there, but wait! Our Ship Delta-V is 31,996 and our Mission Delta-V is 17,297. But we need to multiply that by 2 for a round trip. So our Mission Delta-V is really 34,594. So we'll be about 2,500 m/s short on the way back home.

Here's a summery of all the eye-glaizing mathematical goodness of our ship:
D = 74,800,000,000m
D is distance in meters

M = 145,000kg
M is mass in kg

R = Mw / Md
R is Mass Ratio
Mw is Wet Mass in kg
Md is Dry Mass kg
R = 1.92

Thrust = 150N (Newtons)
Specific Impulse = 5000 sec.

Ve = Isp * 9.81
Ve is Exhaust Velocity in m/s
Isp is Specific Impulse in seconds
Ve = 49,050m/s

Ship Δv = Ve * ln[R]
Ve is Exhaust Velocity is m/s
R is Mass Ratio
Ship Δv = 31,996m/s

A = Thrust / Mass
A is Acceleration in m/s
A = .001 m/s

T = 2 * Sqrt[D / A]
T is Transit Time in seconds
D is Distance in meters
A is Acceleration in m/s
Divide by 86,400 to convert seconds to days
T = 200 days (6.7 months)

Mission Δv = 2 * Sqrt[D * A]
D is Distance in meters
A is Acceleration in m/s
Mission Δv = 17,297m/s

In Part 4 we'll look at what all those numbers mean and whether they really are realistic or not. And just for suffering through all the math, you will also get a prize!

Other parts of this series:
How To Build A Spaceship

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