This article introduces the concept of specific impulse, an important value in determining the performance and capabilities of a space propulsion system. It is intended primarily for those people who do not have a firm technical knowledge of space propulsion systems, but would like to gain a working knowledge of some of the parameters used in the field. I have compiled, into one paper, notes and information from several sources, notably space-systems and propulsion courses at the USAF Academy. In some classes, the original reference cited in the coursework is Dr. Glasstone's "Sourcebook on the Space Sciences." Although the original material is now four times removed, it may be safely assumed that credit for everything correct that follows belongs to my USAFA instructors, or, if traced back, to Dr. Glasstone. Conversely, blame for everything incorrect belongs to me. I hope there's not too much of the latter, but occasionally I had to rely on memory. If you find something, please let me know, and I'll correct it.
First, a note on units. To ensure that units' expressions appear the same on all computers, the following conventions are used: all superscripts are indicated by numbers following, e.g. sec2 is read as "seconds squared;" kg-m/sec2 is "kilogram meters per second squared;" the dash between kg and m is present as a separator, and does not indicate subtraction. Similarly, the following abbreviations are used:
N--Newtons (kg-m/sec2), units of force, the metric equivalent to what we normally think of as pounds. Note that kilograms are NOT equivalent to what we normally think of as pounds. A Newton is the force required to accelerate a mass of one kilogram at the rate of one meter per second per second. Quantitatively, a Newton is approximately the same as a Big Mac.
lbf--pounds-force, what we normally think of as pounds; the English qualitative equivalent of Newtons. A lbf is the force required to accelerate a lbm at 32.174 ft/sec2.
lbm--pounds-mass, the English qualitative equivalent of kilograms. 2.2 lbm really equals a kilogram. 1 lbm = 0.4545 kg. We use lbm because it simplifies some equations, as we'll see later. Besides, otherwise we'd be using slugs.
g--one Earth gravity, equal to 32.174 ft/sec2 or 9.78 m/sec2. The "g" is used in converting from mass to (Earth) weight.
In formulas, the qualitative unit we are working with (such as thrust, which is usually denoted F), is followed by an example of corresponding units in parenthesis. For example, thrust: F (N); mass: m (kg). Any word with a suffix "dot" is the equivalent of writing the unit with a dot over it, and represents the rate equivalent of the unit. For example, mass: mdot (kg/sec), is the mass flow RATE (in kg/sec). (That's "em-dot" phonetically.)
While this may seem overly simplistic, I can recall fervently wishing, some years back, at 2 a.m., that the author of some textbook had maybe been a bit more simplistic, and I'll bet I'm not the only one. (I'd also like non-engineers to be able to read and understand it.) Off we go.
The performance of a rocket engine can be primarily judged by two factors: its thrust, and its specific impulse. For the sake of completeness, we'll examine thrust first.
Thrust is understood commonly enough as the resultant of Newton's Third Law, action-reaction. Thrust is expressed in units of force: Newtons or pounds-force (lbf). More specifically, thrust is defined as a "change in momentum." The momentum of a body is the product of its mass and its velocity. Momentum = m*v. This definition of thrust makes sense qualitatively; we have a vehicle of mass m; it is moving at velocity v; we apply thrust, and the velocity changes. Since m*v now equals some new value, momentum has changed.
The value of an engine's thrust depends primarily on the engine's mass flow rate and its exhaust velocity. Mass flow rate (mdot, expressed in kg/sec or lbm/sec) is simply the rate at which propellant is consumed and expelled out the nozzle. It is also called propellant flow rate. The exhaust velocity is often referred to with the letter "c" (we'll stick to the metric m/sec, and note that this is NOT the speed of light). Thus, we have:
F (N) = mdot*c = (kg/sec)(m/sec) = (kg-m/sec2)
We see that the units check out--we expect thrust, F, to be a force (since a force is defined as the change of momentum), and the units work out correctly to Newtons (kg-m/sec2). This quantity is referred to as "momentum thrust," as it results from the change of momentum of the exhaust gas.
Atmospheric pressure can rob an engine of thrust, depending on what altitude the engine was designed for. This contribution to thrust is called "pressure thrust," and is usually negative (i.e. it causes a decrease in overall thrust). I don't like it; it's confusing, so we'll ignore it here except to say that an engine's performance can always be designed highest for a vacuum--there is a performance loss from operating in an atmosphere, or from designing to operate in an atmosphere.
Another important measure of rocket performance, specific impulse, is related to the thrust produced. Specific impulse is written Isp. (Normally, the "sp" would be a subscript.) Two definitions have evolved for the term, and it's important for engineers to know which they're using to solve problems.
First, the technical definitions. NASA's glossaries define specific impulse as the thrust (force, in lbf or N) obtained from each unit mass of propellant (lbm or kg of fuel and oxidizer), consumed in 1 second. The rate at which the propellant is consumed is exactly the same as the rate at which exhaust gas is expelled, because the propellant material is converted into exhaust gas by the operating rocket.
In English units, if F (lbf) is the rocket thrust and mdot is the rate in lbm/sec at which exhaust gas is expelled (or propellant consumed), then the specific impulse, Isp, can be represented by:
Isp = --------------------- , or
F (lbf sec)
Isp = --------------------
Although this version of specific impulse should be given in units of lbf/lbm/sec or lbf-sec/lbm, it is sometimes, erroneously, expressed in seconds. The error arises when people equate lbf and lbm--mass isn't force! Consistent units are important.
We can see immediately that specific impulse is sort of an efficiency rating--it tells us how much thrust we get for a propellant consumption rate. We'll compare this English-units version to one with metric units shortly.
The second NASA definition does give a result expressed appropriately in seconds. It defines specific impulse as the force of thrust (N or lbf) produced for a propellant consumption rate expressed in N/sec (or lbf/sec). Note the subtle difference in units: this definition gives "correct" values when the propellants' weight flowrate is used. Weight flowrate is calculated using a gravity value of one earth's surface "g" (32.174 ft/sec2 or 9.78 m/sec2). In English units, the lbf is defined by "g", so, letting wdot represent weight flow rate:
Isp = ------------------- = (sec)
In metric units, to convert from mass (kg) to weight at 1g, we include 9.8 m/sec2 and call it g:
F mdot*c c (m/sec)
Isp = ------ = ------------ = --- = ------------- = (sec)
wdot mdot*g g (m/sec2)
Now, let's return to the first NASA definition, and use metric units. (We’re going somewhere here, keep going—this is “thrust obtained from each unit mass of propellant consumed in 1 second”):
F (N) (kg-m/sec2)
Isp = --------------------- = ------------------- = m/sec
mdot (kg/sec) (kg/sec)
Now we have a third set of units for Isp! Or do we? Here's where it can get complicated. Notice the difference between the two metric-unit equations above (sec and m/sec resultants). If we calculate Isp using each of them, the second one will give a value almost 10 times greater! Specifically, it will be 9.8 times greater--and that's because of "g"--9.8 m/sec2.
In English units, lbm and lbf are defined in such a way that "g" (32.2 ft/sec2 in English units) is cancelled out. 1 lbf is the force required to accelerate 1 lbm at 32.2 ft/sec2. (So, in case you were wondering, a lbm is not the same as a slug. One lbm = 0.4545 kg; a slug = 14.5959 kg.) In metric units, the units are nice and neat--a Newton is defined as one kg-ONE m/sec2. As a consequence, ironically, we must include "g" explicitly, or we get results 9.8 times larger than expected. In English units, to get around using slugs and to simplify our calculations, we define our units of mass and force to incorporate "g." (Actually, we already have the lbf = slug ft/sec2. But since this is a pretty useless mass, we define lbm so that lbf = lbm 32.2 ft/sec2, which is a much more convenient mass measure.)
All of which means?
Follow along: we have an engine which produces 1 lbf thrust per 1 lbm/sec mass flow rate. Then, Isp = 1 lbf/lbm/sec. Now, using the same equation, break lbf into its subunits: 1 lbf = 32.2 lbm ft/sec2. The resultant = 32.2 ft/sec. But, we must now divide by "g" in the English system, since we took it out when we broke up 1 lbf. The result is Isp = 1 sec.
Now, convert to metric--1 lbm = 0.4545 kg; 1 ft = 0.3048 m. Then, 1 lbf = 4.45 N. Isp = 4.45 N / 0.4545 kg/sec. The result is 9.8 m/sec, until we divide by the appropriate "g"--9.8 m/sec2. Then, we get Isp = 1 sec. So, whether we use English or metric units, the Isp, in seconds, is equivalent! This makes sense, since seconds are the same in both systems. Numerically, Isp won't change, either, if we use lbf/lbm/sec.
The point is worth repeating: Isp, in seconds or lbf/lbm/sec, is numerically equivalent, regardless of the system used to calculate it!
Now, what about that last equation, where we want to use metric units in NASA's preferred definition:
F (N) (kg-m/sec2)
Isp = ------------------- = -------------------- = m/sec
mdot (kg/s) (kg/sec)
Since "g" isn't built into the metric units, as it is into the lbf/lbm relationship, the Isp will indeed be expressed in m/sec, and will be approximately 10 times larger. An Isp of 1 lbf/lbm/sec, numerically equivalent to an Isp of 1 sec, is the same as an Isp of 9.8 m/sec (just call it about 10 m/sec).
This can lead to difficulties in certain applications of the specific impulse concept, but all three definitions are currently used. You can usually determine which definition to use by checking the units, but the confusion is pervasive. A good check is to be aware of the "normal" Isp values of the class of engine you're working with.
Remember when I said we were going somewhere? An interesting highlight of the metric version of this equation is that the resulting Isp is in terms of the exhaust velocity of the engine! You can easily see, therefore, that a higher exhaust velocity means a more efficient engine--which makes sense, since that means the engine is getting more momentum change out of each particle of exhaust gas.
Other conditions (especially operating pressure) being equal, it can be shown that the velocity of the exhaust gas is proportional to the square root of the temperature of the gas (before ejection), divided by its molecular weight. That is:
c2 = k * ----------
where T is the temperature of the gas and M.W. is its molecular weight. A high exhaust velocity, and consequently, a high specific impulse, results from high temperature and low molecular weight exhaust gases. (Since the exhaust gas is usually a mixture of different substances, the molecular weight is a weighted average value.) This equation explains the desirability of hydrogen as a fuel, and the need for high combustion chamber temperatures. It also highlights another point: the source of the high temperature is irrelevant. In chemical rockets, we get the gas hot by burning a fuel and oxidizer. In a nuclear rocket, a nuclear reactor heats the fuel (which can be pure hydrogen--the lowest molecular weight). In some fusion rocket designs, the fusion reaction generates microwaves which heat hydrogen fuel. The point is to get the "working fluid" as hot as possible, so it will exit out the nozzle as fast as possible.
Another way of looking at the Isp equation is to solve for thrust:
F = mdot*Isp
Systems with higher specific impulse can produce more thrust for a given fuel burn rate, mdot. Alternatively, a particular thrust value can be obtained with a lower propellant consumption rate.
From a mission perspective, specific impulse determines required propellant loading. The total impulse is defined as the product of the thrust multiplied by the operating time, and a given mission can be defined in terms of total impulse required. If we take the specific impulse equation and multiply numerator and denominator by the total operating time:
F*t Total impulse
Isp = ------------ = -------------------------------------
mdot*t Propellant mass consumed
We see the origin of the term specific impulse--it is the impulse per unit mass of propellant consumed. To perform a given mission, the required propellant mass will be less with a higher Isp. This, in turn, increases the thrust-to-weight ratio.
The concept of total impulse points out that we can obtain the same mission result two ways--we can burn with very little thrust for a long time, or we can burn with a lot of thrust for a short time. It is the nature of most very high Isp engines—such as ion engines—that the mass flow rate is very low, and hence so is the thrust. (And conversely, most high thrust engines have low Isp.) But, consider a design where a certain amount of thrust is needed—for example, to achieve a thrust-to-weight ratio greater than one, for liftoff from a planet's surface. We can raise the specific impulse, perhaps by increasing the engine's operating temperature. If we can do this, while maintaining the same mass flow rate, mdot, then our thrust goes up. At the same time, with a higher Isp, the necessary propellant mass goes down. Thus, we have gained improvement in the thrust-to-weight ratio from two areas, instead of one. Alternately, if we have excess thrust, that can be traded for more payload or more velocity. This shows why today's engine designs are being pushed to the limit.
To illustrate this principle with a real-world example, the Atlas rocket had 3 first stage engines, necessary to get the thrust-to-weight ratio greater than one. But, after a few minutes, the rocket was light enough that two “booster” engines were jettisoned, and the rocket continued to orbit with just the “sustainer” engine burning. Dropping the booster engines reduced weight, but also thrust. The rocket didn’t accelerate as fast as it would have with three engines, so the sustainer simply burned longer to reach the necessary velocity.
Specific impulse is a characteristic property of the propellant used. Exact figures for a specified propellant combination will vary depending on the operating conditions and engine design. The SSME's Isp varies from 363 seconds at sea level to 455 seconds in vacuum. Some other values are given below:
|Ethyl alcohol (75%)||280|
|Inhibited Red Fuming-Nitric Acid (IRFNA)
(above are for 1000 psi chamber operating in 1 atm)
(STS Solid Rocket Boosters)
|Hydrogen peroxide (monopropellant)||160|
|Electrostatic ion thrusters (milli-Newtons)||1500-10000|
|Magnetoplasmadynamic thrusters (100 N)||3000-4000|
|Solid core nuclear thermal rocket||950|
|Gas core nuclear thermal rocket||1400|
|Fusion engine (various types)||
|Antimatter engines (various types)||1000-100000|
Note the increase in Isp for the LOX/Liquid hydrogen combination in the RS-25 SSME, which uses 3000 psi chamber pressure, over the 1000 psi baseline: 452 sec vs. 390 sec.
Looking at SpaceX, the Merlin 1C engine, which burned LOX and RP-1 kerosene at 982 psi chamber pressure, had a sea level Isp of 275 sec and a vacuum Isp of 305 sec—which is essentially unchanged from the Isp values from decades ago. The Merlin 1D, with a chamber pressure of 1410 psi, only achieves a slightly better sea level Isp of 282 sec and a vacuum Isp of 311 sec, but has nearly twice the thrust—205,000 lbf versus 110,000 lbf.
Notice if you look these up on Wikipedia, you’ll see both English and metric Isp values—the Merlin 1D has 3050 m/s vacuum Isp, exactly what we’d expect.
There are two types of Raptor engines planned, burning LOX and liquid methane. One has a sea level nozzle and only achieves a vacuum Isp of 356 sec. The other has a larger, vacuum nozzle and achieves a vacuum Isp of 375 sec.
The Merlin and Raptor numbers also suggest why there wasn’t much excitement about SpaceX’s choices of propellant—one wonk even suggesting about Merlin, “The world doesn’t need another LOX/kerosene engine.” Of course, reusability makes these anything but “another” engine.