Appendix A — Frequently Used Equations

There are many, many equations spread throughout Reaction Engineering Basics. This appendix collects some of the most frequently used equations in one place. The equations are not numbered, nor explained, so to use this listing one must be able to recognize the equation that is needed in a given situation.

A.1 Stoichiometry, Composition and State Equations

\[ n=\sum_i n_i \qquad \dot{n} = \sum_i \dot{n}_i \qquad PV=nRT \qquad P\dot{V}=\dot nRT \qquad P_iV=n_iRT \] \[ P_i\dot V=\dot n_iRT \qquad P_i = y_iP = C_iRT \qquad C_i = \frac{n_i}{V} \qquad C_i = \frac{\dot n_i}{\dot V} \qquad x_i \text{ or }y_i = \frac{n_i}{\displaystyle \sum _i\left(n_i\right)} \]

\[ x_i \text{ or }y_i = \frac{\dot n_i}{\displaystyle \sum _i\left(\dot n_i\right)} \qquad \sum_i x_i = 1 \qquad \sum_i y_i = 1 \qquad f_i=\frac{n_{i,0}-n_i}{n_{i,0}} \qquad f_i=\frac{\dot{n}_{i,in}-\dot{n}_i}{\dot{n}_{i,in}} \]

\[ P \frac{dV}{dt} + V \frac{dP}{dt} - R \left( T \sum_i \frac{dn_i}{dt} + \left( \sum_i n_i \right) \frac{dT}{dt} \right) = 0 \quad \frac{\dot n_{lr,in}}{\left| \nu_{lr} \right|} < \frac{\dot n_{nlr,in}}{\left| \nu_{nlr} \right|} \] \[ P \frac{d\dot{V}}{dt} + \dot{V} \frac{dP}{dt} - R \left( T \sum_i \frac{d\dot{n}_i}{dt} + \left( \sum_i \dot{n}_i \right) \frac{dT}{dt} \right) = 0 \qquad \frac{n_{lr,0}}{\left| \nu_{lr} \right|} < \frac{n_{nlr,0}}{\left| \nu_{nlr} \right|} \] \[ n_i = n_{i,0} + \sum _j \nu_{i,j} \Xi_j = n_{i,0} + \sum_{j^\prime} \nu_{i,j^\prime} \xi_{j^\prime} \qquad \dot n_i = \dot n_{i,in} + \sum\limits_{j} \nu_{i,j} \dot \Xi_j = \dot n_{i,in} + \sum_{j^\prime=1} \nu_{i,j^\prime} \dot \xi_{j^\prime} \]

A.2 Rates, Rate Expressions and Reaction Mechanisms

\[ r_j = \frac{1}{V} \frac{d\Xi_j}{dt} \qquad r_{i,j} = \nu_{i,j} r_j \qquad \frac{r_{i,j}}{\nu _{i,j}} = \frac{r_{k,j}}{\nu _{k,j}} \qquad k_j = k_{0,j} \exp{\left( \frac{-E_j}{RT} \right)} \]

\[ r_{\text{rev}} = r_{\text{irrev}}\left( 1 - \frac{\prod\limits_i \left[ i \right]^{ \nu_{i,j}}}{K_{j,eq}} \right) \qquad r_j = k_{j,f} \prod_{i^\prime}\left[ i^\prime \right]^{-\nu_{i^\prime,j}} - k_{j,r}\prod_{i^{\prime\prime}}\left[ i^{\prime\prime} \right]^{\nu_{i^{\prime\prime},j}} \]

\[ r_{i,j} =\sum_{e}\nu_{i,e}r_e \qquad r_{e,\text{insig}} = 0 \qquad r_{e,\text{irrev}} = k_{e,f} \prod_{i^\prime} \left[ i^\prime \right]^{-\nu_{i^\prime,e}} \qquad 0=\sum_{e}\nu_{RI,e}r_e \] \[ r_j=r_{e_{rd}} \qquad C_{0,cat} = C_{cat,free} + \sum_c\kappa_cC_c \qquad 1 = \theta_{v} + \sum_i\theta_i \qquad 0=\sum_pC_pq_p + \sum_nC_nq_n \] \[ C_{ma} \gg C_{nma} \qquad \theta_{ma} \gg \theta_{nma} \qquad r_{\text{growth}} = \frac{1}{V} \frac{dm_{cells}}{dt} \qquad \mu = \frac{r_g}{C_{cells}} \] \[ r_{i,net} = \frac{n_i - n_{i,0}}{t_{process} + t_{turn}} \qquad r_{i,net} = \frac{n_i - n_{i,0} - \displaystyle\int_0^{t_{process}}\dot{n}_{i,in}dt}{t_{process} + t_{turn}} \]

A.3 Thermodynamics

\[ \Delta H_j^0\Bigr\rvert_{T=298K} = \sum_i \left( \nu_{i,j} \Delta H_{f,i}^0\Bigr\rvert_{T=298K} \right) \qquad \Delta H_j^0\Bigr\rvert_{T=298K} = \sum_i \left( -\nu_{i,j} \Delta H_{c,i}^0\Bigr\rvert_{T=298K} \right) \] \[ \Delta H_j^0 = \Delta H_j^0\Bigr\rvert_{T=298K} + \sum_i \left( \nu_{i,j} \int_{298\text{ K}}^T \hat C_{p,i}dT \right) \qquad \Delta G_j^0\Bigr\rvert_{T=298K} = \sum_i \left( \nu_{i,j} \Delta G_{f,i}^0\Bigr\rvert_{T=298K} \right) \] \[ K_j \Bigr\rvert_{T=298K} = \exp{ \left( \frac{-\Delta G_j^0\Bigr\rvert_{T=298K}}{R(298\text{ K})} \right)} \qquad K_j = K_j \Bigr\rvert_{T=298K} \exp{\left\{ \frac{1}{R} \int_{298K}^T\frac{\Delta H_j^0}{T^2}dT \right\}} \] \[ K_{j} = \prod_i a_i^{\nu_{i,j}} \qquad a_i = x_i \quad \text{(ideal liquid solution)} \qquad a_i = \frac{P_i}{1 \text{ atm}} = \frac{y_iP}{1 \text{ atm}} \quad \text{(ideal gas)} \] \[ \left(\sum_i n_i \hat C_{p,i} \right) \frac{dT}{dt}\ \Leftrightarrow\ \rho V \tilde C_p \frac{dT}{dt}\ \Leftrightarrow\ V \breve C_p \frac{dT}{dt} \qquad \]

\[ \left(\sum_i \dot{n}_i \hat C_{p,i} \right) \frac{dT}{dz}\ \Leftrightarrow\ \rho \dot{V} \tilde C_p \frac{dT}{dz}\ \Leftrightarrow\ \dot{V} \breve C_p \frac{dT}{dz} \]

\[ \sum_i \dot n_{i,in} \int_{T_{in}}^T \hat C_{p,i}dT\ \Leftrightarrow\ \rho \dot V_{in} \int_{T_{in}}^T \tilde C_pdT\ \Leftrightarrow\ \dot V_{in} \int_{T_{in}}^T \breve C_pdT \]

\[ \frac{V}{\dot V}\sum_i \left( \dot n_i \hat C_{p,i} \right) \frac{dT}{dt}\ \Leftrightarrow\ \rho V \tilde C_p \frac{dT}{dt}\ \Leftrightarrow\ V \breve C_p \frac{dT}{dt} \]

A.4 Ideal Reactors

\[ \frac{dn_i}{dt} = V \sum_j \nu_{i,j}r_j \qquad \left(\sum_i n_i \hat C_{p,i} \right) \frac{dT}{dt} - V\frac{dP}{dt} - P \frac{dV}{dt} = \dot Q - \dot W - V \sum_j \left(r_j \Delta H_j \right) \] \[ \frac{dn_i}{dt} = \dot n_{i,in} + V \sum_j \nu_{i,j}r_j \qquad \frac{V}{\dot V}\frac{d \dot n_i}{dt} + \frac{\dot n_i}{\dot V}\frac{dV}{dt} - \frac{\dot n_iV}{\dot V^2}\frac{d \dot V}{dt} = \dot n_{i,in} - \dot n_i + V \sum_j \nu_{i,j}r_j \] \[ \sum_i \left( n_i \hat C_{p,i}\right) \frac{dT}{dt} -V\frac{dP}{dt} - P\frac{dV}{dt} = \dot Q - \dot W - \sum_i \dot n_{i,in} \int_{T_{in}}^T \hat C_{p,i}dT - V\sum_j r_j \Delta H_j \] \[ \frac{V}{\dot V}\sum_i \left( \dot n_i \hat C_{p,i} \right) \frac{dT}{dt} - V \frac{dP}{dt} - P\frac{dV}{dt} = \dot Q - \dot W - \sum_i\dot n_{i,in} \int_{T_{in}}^T \hat C_{p,i}dT - V\sum_j r_j \Delta H_j \] \[ \frac{\partial \dot n_i}{\partial z} + \frac{\pi D^2}{4\dot V} \frac{\partial\dot n_i}{\partial t} - \frac{\pi D^2\dot n_i}{4\dot V^2} \frac{\partial \dot V}{\partial t} =\frac{\pi D^2}{4}\sum_j \nu_{i,j}r_j \qquad \tau = \frac{V}{\dot V_{in}} \qquad SV = \frac{1}{\tau} \] \[ \left(\sum_i \dot n_i \hat C_{p,i} \right) \frac{\partial T}{\partial z} + \frac{\pi D^2}{4\dot V} \sum_i \left(\dot n_i \hat C_{p,i} \right) \frac{\partial T}{\partial t} - \frac{\pi D^2}{4} \frac{\partial P}{\partial t} = \pi D U\left( T_{ex} - T \right) - \frac{\pi D^2}{4}\sum_j r_j \Delta H_j \] \[ \frac{dP}{dz} = - \frac{4G}{\pi D^2} \frac{d \dot V}{dz} - \frac{fG^2}{2D \rho} \qquad \rho_{ex} V_{ex} \tilde C_{p,ex}\frac{dT_{ex}}{dt} = -\dot Q - \dot m_{ex} \int_{T_{ex,in}}^{T_{ex}} \tilde C_{p,ex}dT \] \[ \frac{\rho_{ex} V_{ex} \Delta H_{\text{latent},ex}^0}{M_{ex}} \frac{d \gamma}{dt} = - \dot Q - \gamma \dot m_{ex} \frac{\Delta H_{\text{latent},ex}^0}{M_{ex}} \qquad \dot Q = UA\left( T_{ex} - T \right) \]

\[ \frac{{dP}}{{dz}} = - \frac{{1 - \varepsilon }}{{{\varepsilon ^3}}}\frac{{{G^2}}}{{\rho {\Phi _s}{D_p}}}\left[ {\frac{{150\left( {1 - \varepsilon } \right)\mu }}{{{\Phi _s}{D_p}G}} + 1.75} \right] \]

A.5 Processes

\[ \Delta {T_{\text{cold}}} = {T_{\substack{\text{hot}\\ \text{out}}}} - {T_{\substack{\text{cold}\\ \text{in}}}} \qquad {R_R} = \frac{\dot V_{\text{recycle}}}{\dot V_{\text{product}}} = \frac{\dot n_{i,\text{recycle}}}{\dot n_{i,\text{product}}} \qquad {\dot n_{i,\text{in}}} = {\dot n_{i,\text{feed}}} + \frac{{{R_R}{{\dot n}_{i,\text{out}}}}}{{1 + {R_R}}} \] \[ 0 = \mathop \sum_i \left( {\dot n_{i,\text{feed}}}\int \limits_{{T_{\text{feed}}}}^{{T_{\text{in}}}} {\hat C_{p,i}}dT \right) + \mathop \sum_i \left( {\dot n_{i,\text{recycle}}}\int \limits_{{T_{\text{out}}}}^{{T_{\text{in}}}} {\hat C_{p,i}}dT \right) \] \[ \sum_i \left( {\dot n_{i,1}}\int \limits_{{T_{in,1}}}^{{T_{out,1}}} {\hat C_{p,i}}dT \right) = - \sum_i \left( {\dot n_{i,2}}\int \limits_{{T_{in,2}}}^{{T_{out,2}}} {\hat C_{p,i}}dT \right) = UA \left( \frac{T_{in,1} - T_{out,2} - T_{out,1} + T_{in,2}}{\ln\left({\frac{\displaystyle{T_{in,1} - T_{out,2}}}{\displaystyle{T_{out,1} - T_{in,2}}}} \right)} \right) \]

A.6 Non-Ideal Reactors

\[ dF\left( \lambda \right) = F\left( \lambda + d\lambda \right) - F\left( \lambda \right) \qquad F\left( \lambda \right) = \int_{y=0}^{y=\lambda}dF\left( y \right) \qquad F_{\text{CSTR}}\left( \lambda \right) = 1 - \exp{\left( \frac{-\lambda}{\tau} \right)} \]

\[ \begin{aligned} F_{\text{PFR}}\left( \lambda \right) &= 0: \lambda < \tau \\&=1: \lambda \ge \tau \qquad \end{aligned} \qquad F\left(\lambda = t^\prime \right) = \frac{\dot{M} \int_{t_0}^{t^\prime}\left(w_{out}-w_0\right)dt}{\dot{M} \int_{t_0}^{\infty}\left(w_{out}-w_0\right)dt} = \frac{\dot{M} \int_{t_0}^{t^\prime}\left(w_{out}-w_0\right)dt}{m_{tot}} \]

\[ F\left(\lambda = t \right) = \frac{w_\lambda - w_0}{w_f - w_0} \qquad dF=\frac{dF}{d\lambda}d\lambda \qquad \bar{f}_i = \int_0^\infty f_i \frac{dF}{d\lambda}\Big\vert_{\lambda = t^\prime}dt^\prime \qquad \frac{dC_i}{dz}\Bigr\rvert_{z=L} = 0 \]

\[ -D_{ax} \frac{d^2C_i}{dz^2} + \frac{d}{dz}\left( u_s C_i \right) = \sum_j \nu_{i,j} r_j \qquad u_sC_i\Bigr\rvert_{z=0} - D_{ax}\frac{dC_i}{dz}\Bigr\rvert_{z=0} = u_sC_{i,feed} \]

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In the equations above \(i\) indexes all reagents in the system, \(i^{\prime}\) indexes all reactants in a reaction.
\(i^{\prime \prime}\) indexes all products in a reaction, \(j\) indexes all reactions occurring in the system, \(e\) indexes all elementary reaction steps in a mechanism, \(c\) indexes all catalyst complexes, \(p\) indexes all positively charged species, and \(n\) indexes all negatively charged species.