How can I calculate pressure change due to reaction?

Question

I'm studying a reactor, that can be imagined simply as a pipe in which a reaction advances.
In this reactor, the inlet and outlet flow are constant as is the temperature.

If that reaction is equi-molar no pressure change happens, otherwise in a reaction like steam reforming, where product moles are greater than reactant ones, I should expect an increase in pressure.

$$CH_4+H_2 O \rightleftharpoons CO+3H_2$$

How can I calculate the pressure change due to change in the number of moles?

PS: In a closed system (like a vessel) I'd use the ideal gas law: $pV=nRT$ ... but this is not the case!

I was thinking about:

steady state grants me $\dot m|_{in} = \dot m|_{out}$
I can express mass flow as $\dot m = \dot n \cdot MM$ where MM is molar mass
and as $\dot m = A \cdot u \cdot \rho$ where A is the cross sectional area (constant) u is mean speed and $\rho$ is density
I express density with ideal gas law $\rho = {p \over {R \cdot T}} \cdot MM$
finally I get $\dot n = A \cdot u \cdot {p \over {R \cdot T}}$
now, since A,R,T are constants taking the reaction abovementioned, if $\dot n$ rise up $u \cdot p$ have to follow this trend.

How, who, what, does make pressure constant while speed rise up ?

Answer 1

The change in pressure depends on the design of the reactor, specifically its boundary conditions like pressure and volumetric flow rate at the inlet and outlet. Without any kind of compressor at the inlet, a reaction like the one you described will generally result in the reactor operating at the same pressure as the feed, but with the exit velocity and volumetric flow rate of the outlet stream being larger/faster than the feed. If the reactants and products are all in the gas phase and are ideal gasses, and the temperature is held constant, you can calculate the change in inlet flow rate to outlet flow rate (volumetric) with this equation:

$$ \Delta \dot V = \xi \left( \nu_p - \nu_r \right) $$

Here, $\xi$ is the extent of reaction, and $\nu_p$, $\nu_r$ are the total stoichiometric numbers of the products and reactants. Note that if $\nu_p=\nu_r$, the reaction is equimolar and the volumetric flow rate entering the reactor is the same as the flow rate exiting the reactor (which is intuitive).

To better explain why the velocity increases while the pressure stays more or less the same, consider that you always need a negative pressure gradient through the reactor (assuming steady-state operation) so that the fluid will always flow from inlet to outlet. That is, the outlet pressure, $P_{out}$, must always be less than the inlet pressure $P_{in}$.

Doing a mass balance on the reactor shows that the mass flow rate coming out must be exactly equal to the inlet flow rate (no gains/losses):

$$ \dot m = \rho_{in} u_{in} A = \rho_{out} u_{out} A $$

where $\rho$ is density, $u$ is velocity, and $A$ is the cross-sectional area of the reactor (constant). We also have the ideal gas equation of state (assuming the reactants and products are in-fact ideal gasses):

$$P=\rho RT = \rho \frac{\bar R}{MW} T$$

where $P$ is pressure, $R$ is specific gas constant, $\bar R$ is universal gas constant, $MW$ is the average molecular weight, and $T$ is temperature. We can solve the equation of state for $\rho$ and plug it in to the mass balance:

$$P_{in} \, MW_{in} \, u_{in} = P_{out} \, MW_{out} \, u_{out} $$

where $T$, $A$, and $R$ have canceled out. If we assume that the pressure gradient is very small, $P_{in} \approx P_{out}$ (just enough to get a little bit of flow), then we can also cancel the pressure terms and rearrange:

$$\frac{MW_{in}}{MW_{out}} = \frac{u_{out}}{u_{in}}$$

Now it should be more apparant that, if the outlet average molecular weight is less than the inlet (i.e. products have a higher stoichiometric number than the reactants), then the outlet velocity must be higher to satisfy our mass balance and equation of state.

Answer 2

I think you're mixing things up. Quoting from this article:

In chemical processes, the most important property to be conserved is the number of atoms of each kind that are present. Unlike nuclear processes, chemical reactions do not create or destroy atoms, or change one kind of atom into another. They only reshuffle the atoms that were originally present into different molecular combinations.

Meaning that mass is conserved whether number of moles is the same or not. For a closed system (combustion reaction inside a piston cylinder), the rise in pressure is not dependent on whether the reaction is equimolar or not.

If we used two different fuels, hydrogen and methane for example, stoichiometric hydrogen reaction is non-equimolar and methane is equimolar, However, both will cause an increase of pressure inside a piston cylinder, the rise in pressure is due to the dramatic increase in temperature of the exothermic reactions (with fixing the volume).

Same thing for an open system, the result of the reaction will affect the temperature (enthalpy) of the stream due to the release of the energy stored in the fuel, while the pressure will stay the same since there is no reason for it to build up.