# HCN reactor design

For my diploma work, I should design a hydrogen cyanide (HCN) producing reactor.

I found that for the BMA process, overall

$$\ce{CH4 + NH3 -> HCN + 3 H2 \quad{} \Delta{}H_r = \pu{251 kJ / mol} }$$ Also, there is side reaction of ammonia decomposition.

the decomposition of ammonia is the rate-limiting step. So I should consider it to find the reactor volume. Besides Langmuir - Hinshelwood model, I also found power law rate to calculate the parameter manually. The expression is like this:

r = 2.27 * 10^23* exp (-21000/RT)*P(NH3)

Previously, I have used plug-flow design (PFR) equation which integrates dx/-r, where r is expressed in terms of x - concentration.

However, I do not know how to work with the previous equation, how to relate with conversion. And I am not sure about how to get the design parameter from it. My question is how this kind of power law rate expressions with pressure instead of concentration should be used in design equations?

Any contribution is highly appreciated. Thanks in advance

• What are the units of $r$ and of $\pu{2.27 10^{23}}$. Is $21000$ in Joules ? $\pu{10^{23}}$ is an extremely high number. Is it expressed in molecules per second and $\pu{m^3}$ ? Feb 15 at 17:31

I will lead you to the general equation. We can relate the partial pressure of a species with conversion by using a simple stoichiometric table.

The reaction is

$$\ce{NH3(g) + CH4(g) -> HCN(g) + 3H2(g)}$$ $$\ce{A(g) + B(g) -> C(g) + 3D(g)} \tag{R}$$ where we defined for convenience new letters for all the species. We concentrate on $$\ce{A}$$.

1. Mole balance

For a PFR the volume needed to achieve a conversion $$X_\ce{A}$$ is $$$$V = \int_0^{X_\mathrm{A}} \frac{F_\mathrm{A0} \; \mathrm{d}X_\ce{A}} {-r_\ce{A}(X_\ce{A})} \tag{1}$$$$ where $$F_\mathrm{A0}$$ is the inlet molar flow rate of species $$\ce{A}$$ in $$\pu{mol s^-1}$$

2. Rate law

We have a power law in the form of $$-r_\ce{A} = kp_\ce{A} \tag{2}$$

3. Stoichiometry

We set up a stoichiometric table for reaction $$\ce{R}$$ in terms of the molar flows. For a general species $$j$$ in a reaction where $$\ce{A}$$ is the limiting reagent, the molar flow rate of species $$j$$ is $$F_{j} = F_\mathrm{A0} (\Theta_j + \nu_j X_\ce{A}) \tag{3}$$ where $$\Theta_j = F_{j0}/F_\mathrm{A0}$$ is the molar relation of species $$j$$ with respect to $$\ce{A}$$ at the entrance of the reactor, and $$\nu_j$$ is the stoichiometric coefficient of species $$j$$.

For the reaction to take place, we need the presence of $$\ce{A}$$ and $$\ce{B}$$. Considering that there are no products at the reactor inlet, application of Eq. (3) to reaction $$\ce{R}$$ gives \begin{align} F_\ce{A} &= F_\mathrm{A0}(1 - X_\ce{A}) \tag{4} \\ F_\ce{B} &= F_\mathrm{A0}(\Theta_\ce{B} - X_\ce{A}) \tag{5} \\ F_\ce{C} &= F_\mathrm{A0} X_\ce{A} \tag{6} \\ F_\ce{D} &= 3F_\mathrm{A0} X_\ce{A} \tag{7} \\ \end{align} where the total molar flow rate is \begin{align} F_\ce{T} &= \sum_j F_j \\ F_\ce{T} &= F_\mathrm{A0}(1 - X_\ce{A}) + F_\mathrm{A0}(\Theta_\ce{B} - X_\ce{A}) + F_\mathrm{A0} X_\ce{A} + 3F_\mathrm{A0} X_\ce{A} \\ F_\ce{T} &= F_\mathrm{A0}(1 - X_\ce{A} + \Theta_\ce{B} - X_\ce{A} + X_\ce{A} + 3X_\ce{A}) \\ F_\ce{T} &= F_\mathrm{A0}(1 + \Theta_\ce{B} + 2X_\ce{A}) \tag{9} \\ \end{align}

Remembering that the partial pressure of species $$j$$ is given by $$p_j = y_j p = \frac{F_j}{F_\ce{T}}p \tag{10}$$ we combine Eqs. (4), (9), and (10) so that \begin{align} \require{cancel} p_\ce{A} &= \frac{\cancel{F_\mathrm{A0}}(1 - X_\ce{A})} {\cancel{F_\mathrm{A0}}(1 + \Theta_\ce{B} + 2X_\ce{A})}p \\ p_\ce{A} &= \left(\frac{1 - X_\ce{A}}{1 + \Theta_B + 2X_\ce{A}}\right)p \tag{11} \end{align}

4. Design equation

Combining Eqs. (1), (2), and (11) gives the final result $$$$\boxed{V = \int_0^{X_\mathrm{A}} \frac{F_\mathrm{A0} \; \mathrm{d}X_\ce{A}} {k\left(\dfrac{1 - X_\ce{A}}{1 + \Theta_\ce{B} + 2X_\ce{A}}\right)p}} \tag{12}$$$$

References

An excellent explanation for stoichiometry for liquid-phase and gas-phase reactive systems can be read at Chapter 3 of:

• Elements of Chemical Reaction Engineering, H. S. Fogler, 5th ed., Prentice Hall (2016).