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Thanks in advance for any help on this. I'm kind of stumped by this problem I've come across in the lab.

We have a tube reactor with a nozzle, and we want to measure the combustion/pyrolysis products coming out the model using GC/MS and MBMS. Right now, we are just running pure nitrogen and taking pressure measurements to make sure we understand well how the system works.

Here is a description of our experimental setup:

We have a ~4 mm diameter quartz tube going through a 20 in (~500 mm) heating section which maintains a temperature of around 1000K. The tube has a converging nozzle at the downstream end (just past the heated section) which is approximately 70 microns. Due to the small nozzle size, we can get the tube up to relatively high pressure with relatively low volume flow rates. We have a pressure transducer downstream of the mass flowmeter, but upstream of the heated section.

The equation we are using to relate mass flow rate to pressure in the tube is:

mdot=(A*)[itex]\sqrt{\frac{γ}{R}(\frac{2}{γ+1})^{\frac{γ+1}{γ-1}}}\frac{p_{0}}{\sqrt{T_{0}}}[/itex],

where A* is the area of the nozzle. We know that as long as the pressure inside the tube is greater than ~1.8 times the pressure outside the orifice, then the flow is choked, and we have reached Mach 1 at the throat.

We have found that this equation works well for choked AND cold conditions, and accurately predicts what pressure we will observe when we flow a particular mass flowrate from the flowmeter without heating.

However, when we turn the heaters on, we no longer get the results we expect. For example, the above equation predicts that pressure should increase as the √T for a set mass flow rate. This is the procedure we followed:

1.) Picked a flowrate, measured the pressure and compared to the formula predicted pressure at 300K. We had good agreement for this step.

2.) Increased temperature from 300K to 1000K inside the tube reactor, and recorded the new pressure.

Because the temperature ratio is 3.33, we expected to see √3.33=1.826 ratio increase in pressure. So something which was at 2 atm cold should increase to almost 4 atm. Instead, we saw an increase of only about 12 psi!

Now, here are the possible issues which we have considered could be causing our miscalculation:

- The above equation from compressible flow is assuming isentropic and adiabatic. We have, until now, assumed the heat transfer problem is seperate from the compressible flow problem. i.e. The tube heats up, reaches 1000K, AND THEN reaches the nozzle and experiences isentropic expansion. Personally, I feel this assumption is ok. The end of the quartz tube is well insulated, and the nozzle is outside of the heated section. Finally, I should note that the low mass flow rates and the small nozzle size lead to very low velocities, on the order of cm/s. I do not think there are any compressible effects upstream of the nozzle. Am I wrong? Is there some coupling between the heat transfer/compressible flow problem I am not considering?

- There is some sort of viscous effect not being accounted for. I think this is unlikely to be the reaosn as well. The problem is our model currently OVERpredicts the pressure rise from increasing temperature. Viscosity of N2 gas INCREASES with temperature increase, which should make the boundary layer larger, which should in turn make effective nozzle diameter smaller. With smaller diameter nozzle, we should, if anything, be UNDERpredicting the pressure measured inside the tube.

- The flow inside the tube is not reaching the target temperature. To me, this is the easiest option and jives well with the experimental results. However, a quick back-of-envelope calculation suggests to me the N2 at this low of a flow rate (on the order of .1 SLMP) quickly takes the temperature of the wall, and should easily reach 1000K before reaching the nozzle.

- The transducer measured pressure is significantly different than the pressure in the hot section. This can't be possible because the fluid velocities are much less than Mach 1 until we get near the nozzle, right?

Are there some issues I am not considering? Any help would be much appreciated. As a reference, I attached the matlab script I used to check the heat transfer calculation, as well as the script to calculate mdot for a certain pressure. I know no one wants to debug my code, but if you see any glaring errors, I would be so happy to have this problem solved because something stupid in my code. Sorry this is so long. I hope I've given enough detail without making the whole problem unreadable.

Thanks so much!

----------------------------Heatx.m------------------------------

%Takes range of mass flow rates, .1 SLPM to 1 SLPM, plots temperature vs. x in mm. The plot will show that N2 gas takes wall temperature far before the 500 mm heating length is reached

To=300;

syms('T');

CpN2=(0.03298677E+02 + 0.01408240E-01*T - 0.03963222E-04*T^2 + 0.05641515E-07*T^3 - 0.02444855E-10*T^4)*8.314/.028013;

SLPM=[.101, .123, .165, .208, .250, .293, .336, .378, .421, .485, .5, .527, .611, .656, .739, .823, .868, .996];

Twall=1000;

length=500; %mm

for i=1:18

mdot=SLPM(i)*1.25/1000/60; %kg/s

Nu=3.66;

D=3.98e-3; %m

KN2=.026;

hN2=Nu*KN2/D;

TempN2(1)=To;

for n=1:length

Twall(n+1)=1000;

TempN2(1+n)=TempN2(n)+hN2*pi*D*1e-3/(mdot*subs(CpN2,T,TempN2(n)))*(Twall(n)-TempN2(n));

l(n)=n;

end

l(n+1)=n+1;

plot(l,Twall, l, TempN2)

hold on

end

----------------------------mdotcalc.m---------------------

%takes as user input desired pressure, target temperature, and outputs set mass flowrate in kg/s and scc/min

atm=101325; %pascal

P=2*atm;

T=3;

D_tube=.00398;

% nozzle

D_noz=70E-6;

A_noz=pi*D_noz^2/4;

P_noz=P*0.528

T_noz=T*0.833

gamma=1.4;

Mass=29;

densityatSTP=1.205; %N2 kg/m^3

Rspec=8.314/Mass*1000

const=sqrt((gamma/Rspec)*(2/(gamma+1))^((gamma+1)/(gamma-1)))

mdot=const*A_noz*P/sqrt(T)

% kg/s to scc/m for gas

mdotsccm=mdot*60/densityatSTP*100^3

--------------------------------------------------------------