# Nozzles and Flow Velocity

• Alexrey
In summary, the addition of a convergent-divergent nozzle to a simple model rocket motor with no nozzle would increase the exhaust velocity, but the mass flow rate (dm/dt) would still be dependent on the design of the rocket and the area ratios of the nozzle. The nozzle must be started before the throat is choked, which requires a higher pressure ratio, but if the areas of the throat and pipe are equal, then equal reservoir conditions will produce the same mass flow rate through the nozzle or the straight pipe. Conservation of mass cannot be broken and the mass flow rate out of the combustion chamber may decrease, but the mass flow through the nozzle will be dependent on various factors.

#### Alexrey

Hi guys,

I'm no engineer, but I have a question regarding engine thrust and nozzles:

Consider a simple model rocket motor with no nozzle (i.e. just a straight cylindrical pipe with its mass flow coming out the back) that has a mass flow rate of 2kg/s. Since F = v dm/dt we would have dm/dt = 2kg/s and the exhaust velocity would most likely be quite low. Now consider the addition of a convergent-divergent nozzle. My question is this: The effect of the nozzle would be to increase the exhaust velocity, BUT due to the conservation of mass, would the mass flow rate (dm/dt) still be dm/dt = 2kg/s or close to 2kg/s?

That's why c-d nozzles are useful. In order to maintain the continuity of the mass flow thru the nozzle, the flow must accelerate thru the throat, so Vout > Vin.

if you have a supersonic nozzle (and I'm assuming you do since there is no other reason to have a divergent portion), then you will choke the flow in the throat. Whether you are close to 2kg/s depends on the design of the rocket, though. It is much more likely that if a straight tube was ejecting 2kg/s, that you will have significantly less mass flow

Wouldn't conservation of mass be broken if the mass flow rate decreases? If the mass flow is decreased with the addition of a nozzle, does that mean that there will be a pressure increase within the motor? I'm guessing that the increase in exhaust velocity more than makes up for the decrease in mass flow.

It depends on your constraints on your question. If the throat area of the nozzle is the same as the cross-section of the hypothetical straight tube, then for a given total temperature and pressure, the mass flow will be the same. Of course with the nozzle, the velocity will be a great deal higher so you'll get more thrust from it.

Conservation of mass can't be broken. Your mass flow rate out of the combustion chamber will go down. The mass flow through the nozzle will be dependent on the fuel, operating temperatures and pressures in the combustion chamber, and the area ratios of the nozzle.

No, the mass flow will not go down in the situation I described. If the nozzle is started, then the throat is choked and it's mass flow is dependent on the total pressure, total temperature and the throat area. Starting a converging-diverging nozzle requires more pressure ratio than choking a straight pipe, so if the pressure ratio is enough to start the nozzle, then the pipe will be choked and it's mass flow equation is identical to the nozzle. In that case, if the areas of the throat and the pipe are equal, then equal reservoir conditions will produce the same mass flow rate through the nozzle or the straight pipe.