How is combustion speed regulated in solid propellants?
(I apologize for being too lazy to find all of the citations for my answer.  However, I have a fair amount of experience as an organic chemist, at one point worked in an underground mine with explosives and have built (for entertainment) a solid fuel composite propellant)


Why does propellant "burn" instead of detonating like similar explosives?


A typical solid composite propellant looks like this:  (1) mainly ammonium nitrate (oxidizer) + "fuel" (typically aluminum flake or magnesium) + binder (polymer + perhaps plasticizer) + (maybe) catalysts OR substitute ammonium perchlorate for ammonium nitrate.

The most similar mining explosive is ANFO or variants thereof:  ammonium nitrate + diesel fuel (possible sensitizing substitutions are... some ammonium perchlorate for the ammonium nitrate, some aluminum in addition to fuel, partially oxidized "fuels" such as hexamine nitrate, nitromethane, possibly a small percentage of a secondary explosive such as TNT or RDX, or addition of plastic "bubbles" to simulate the bubbles that sensitize pure nitroglycerine.

The key point is that the latter mining explosive is called a blasting agent.  The taxonomy of high explosives looks like this:  primary explosives can be easily DETONATED (as opposed to deflagration, which is a type of "burning", detonation involves a supersonic thin region of reaction that moves through the explosive, giving it brisance that can shatter rock).  Primary explosives are used in primers, and can be detonated with sparks, physical shocks, etc.  Secondary explosives (dynamites, TNT, PETN) need to be detonate with a primary explosive.  They are relatively insensitive to other means of initiation (one measure of this is the deflagration to detonation distance, which really is the amount of explosive that can burn before detonation takes place.  It is usually quite large).  You can often shoot or burn secondary explosives without them detonating.

Finally, blasting agents are quite hard to detonate.  They require a booster that is a secondary explosive.  Blasting agents are very insensitive and can be burned without detonating, especially if not completely confined.

This answers your first question...  we should not expect solid propellants to detonate without a booster (actually I'd guess that solid propellants are even harder to initiate, as ANFO variants are usually oxygen balanced so that all of the fuel is oxidized, whereas propellants appear to be a bit carbon rich).

Note that blasting agents and solid propellants share the property that the oxidizer and fuel form distinct particles, as opposed to explosives such as nitroglycerine, which are self-oxidizing within the molecule.  The low explosive gunpowder shares the property of having a mixture of discrete particles of oxidizer and fuel, and it merely deflagrates.

TLDR:  The explosives that are most similar to solid composite propellants are difficult to detonate, so it isn't a surprise that they burn instead.

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How is the burn rate controlled?  (note, I use burn as burn typically is decomposition at a rate of a few cm per second, while deflagration is at a rate of many meters per second.  Detonation is thousands of meters per second).


A:

All propellant mixtures have (approximately) a burn rate that is a function of pressure.  This approximation is called Vieille's Law.  The formula is 

$rate = {rate}_0 + aP^\beta$

where $P$ is the pressure.  Thus, as pressure increases in the rocket motor, burn rate increases.  Ideal values of $\beta$ are between 0.5 and 0.8, otherwise initiation can be difficult or else a slight overpressure could be amplified to cause the pressure to increase dramatically and destroy the rocket.  What seems to affect the burn rate?  I'll list a few:

(1)  Ammonium perchlorate burns much faster than ammonium nitrate.

(2)  Generally, the rate limiting property of a propellant is the particle size of the oxidizer, so more finely divided oxidizer means faster burn rate.

(3)  The geometry of the propellant is important.  The burning tends to occur at the exposed surface, so wide and not very long propellant castings will produce more thrust.

(4)  Metallic catalysts increase burn rate, such as iron oxides and chromates.

(5)  I've heard that a small amount of carbon black increases burn rate by increasing infrared absorption in the casting.  It also makes it look cooler.

(6)  Not so good:  cracks and voids in the casting will increase burn speed, as they lead to more exposed area.  That's why typical binders are rubbers, often including plasticizers, as these are less likely to crack under pressure.

Here's a couple of citations:

A amateur rocketry site

metallic wires as a burn rate catalyst

Rocket engine mathematics