Collection and analysis of rate data

Chemical Reaction Engineering

Rate data analysis

  • Isothermal reactor design algorithm

Isothermal reactor design algorithm

  1. Mole balance

    F_{A0} - F_A = \int^V r_A dV = \frac{dN_A}{dt}

  2. Rate law

    If -r_A is given as f(X) \rightarrow directly solve the design equations

  3. Stoichiometry

    If -r_A = g(C) \rightarrow use stoichiometry to write -r_A = f(X)

  4. Combine

    Gather all equations to obtain a system of equations that must be solved.

  5. Evaluate

    The system of equation scan be solved analytically, graphically, numerically, or using software

Isothermal reactor design: molar flow rates

  • In many instances it is easier to work with molar flow rates/ no. of moles than conversion

    • Reactors with external mass transfer (such as membrane reactors)
    • Multiple reactions in gas phase

  • We must write a mole balance for each and every species as opposed to just one species

  • Usually this leads to a system of (simultaneous or) ordinary differential equations

  • Solve the combined set of equations using ODE solver (such as scipy.integrate.solve_ivp)

Isothermal reactions in PBR: molar flow rates

Second order reaction \ce{aA + bB -> cC + dD}

  1. Mole balance: Write balance for each species i = 1 \ to \ N

    \frac{dF_i}{dW} = r'_i

  2. Rate law

    -r'_A = kC_A^\alpha C_B^\beta; \qquad \frac{r'_A}{-a} = \frac{r'_B}{-b} = \frac{r'_C}{c} = \frac{r'_D}{d}

  3. Stoichiometry C_i = \frac{C_{A0} (\Theta_i + \nu_i X)} {(1 + \epsilon X)} \left( \frac{P}{P_0} \right) \left( \frac{T_0}{T} \right)

    Pressure: \frac{dP}{dW} = -\frac{\alpha}{2p} \left( \frac{T}{T_0} \right) \frac{F_T}{F_{T0}}

    Total molar flow rate: F_T = \sum_{i=1}^N F_i

  4. Combine:

    Collate all equations from steps 1 to 3 to yield a system of equations

  5. Evaluate:

    Use ODE solver to solve the system of equations obtained in step 4.

Reactor design problem

  • In practice, collection and analysis of rate data is the most time consuming task in reactor design

  • Data collection is done in the lab, where we can simplify mole balance, stoichiometry, and fluid dynamic considerations

Rate law

  • Algebraic equation that relates -r_A to species concentration

    -r_A = k_A(T) f(C_A, C_B, ...)

  • Power law model

    • Most common general forms of the rate law

    -r_A = k_A C_A^\alpha C_B^\beta

  • Order of reaction: the powers to which the concentrations are raised

    • The reaction is \alpha order with respect to A and \beta order with respect to B.
    • Overall order of reaction n = \alpha + \beta.

  • k_A: Specific reaction rate

    • Depends on the temperature

  • Need to determine k, \alpha, and \beta

Collection and analysis of rate data

  • Goal

    • Determine reaction order, \alpha, and specific reaction rate constant, k, in the rate law
    • Want ideal conditions \rightarrow well-mixed (data is easiest to interpret) \rightarrow Select a simple reactor

  • Constant-volume batch reactor

    • For homogenous reactions
    • Concentration vs. time measurements
    • Measurement during the unsteady-state operation

  • Differential reactor

    • For solid-fluid reactions
    • Measurement during steady state operation
    • Product concentration is usually monitored for different feed conditions

Algorithm for rate data analysis

Determination of rate law for homogenous reactions

  • Most often batch reactors are used

    Type of reactor chosen will not affect rate of reaction

  • Batch reactor

    • Simple operations, low cost
    • Ease of sampling, easy clean up, limited waste
    • Uniform concentration can be obtained

  • Mole balance: constant volume -\frac{dC_A}{dt} = -r_A

  • Typical measurements
    • Concentration Pressure
    • Temperature: Many times batch reactions are carried out isothermally.
    • Development of heat during reaction: Reaction calorimetry

Method of excess

  • Given concentration vs. time profile in a batch experiment, determine the reaction order and rate constant.

  • \ce{A + B -> C + D}

    • Rate law: -r_A = k C_A^\alpha C_B^\beta
    • Need to determine: k, \alpha, and \beta

  • Determining reaction orders: \alpha, and \beta

    • Common simplification: One of the reactants is in excess

    • Two separarate experiments

      • Excess B \Rightarrow C_B \gg C_A, C_B can be assumed constant. \Rightarrow determine \alpha.

      • Excess A \Rightarrow C_A \gg C_B, C_A can be assumed constant. \Rightarrow determine \beta.

  • Determining k: Measuring rate at known concentrations of A, and B.

Analysis methods

  • Differential analysis
  • Integral analysis
  • Nonlinear regression
  • Method of half life
  • Initial rates method
  • Differential reactor

Differential analysis

  • Irreversible reaction

  • The rate is essentially a function of the concentration of only one reactant.

    \ce{A -> products}; -r_A = k C_A^\alpha

  • Isothermal, constant volume batch reactor

  • Mole balance: constant volume -\frac{dC_A}{dt} = -r_A; -r_A = k C_A^\alpha

  • Taking natural logarithm

\ln \left( \frac{-dC_A}{dt} \right) = \ln k + \alpha \ln C_A

  • Slope of plot of \ln[ -dC_A/dt ] \ \text{vs.} \ln C_A is the reaction order

  • Specific reaction rate can be determined using a specific point p: k = \frac{(-dC_A/dt)_p}{(C_A)_p^\alpha}

Evaluating \frac{-dC_A}{dt}

  • Graphical diffrentiation

    • Very old method
    • Equal area graphical diffrentiation
    • Disparities in the data are easily seen

  • Numerical diffrentiation

    • Finite difference
    • Independent variable (time) is equally spaced

  • Diffrentiation of a polynomial fit to the data

    • Fit a polynomial to C_A \ \text{vs.} \ t data

    • e.g. C_A = f(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4

    • Analytical derivative

      \frac{dC_A}{dt} = f'(t) = a_1 + a_2 t + a_3 t^2 + a_4 t^3

  • Determine reaction order and specific rate from plot of \ln[ -dC_A/dt ] \ \text{vs.} \ln C_A

Integral analysis

  • Quickest method to determine if the order is 0, 1, or 2.

    \Rightarrow Used when reaction order can be guessed or is known.

  1. Guess reaction order

  2. Integate the differential equation to obtain concentration as a function of time

  3. If the guessed order is correct, appropriate plot (determined from integration) should be linear.

Mole balance: -dC_A/dt = -r_A

  • Zero order reaction

    -r_A = k

C_A = C_{A0} - kt

  • First order reaction

    -r_A = k C_A

\ln \frac{C_{A0}}{C_A} = kt

  • Second order reaction

    -r_A = k C_A^2

\frac{1}{C_A} = \frac{1}{C_{A0}} + kt

Need to know appropriate function of concentration corresponding to a rate law that is linear with time.

Integral analysis

Nonlinear regression

  • Search for parameter values that minimize the sum of squares of the difference between the measured values and calculated values for all data points.

  • Best estimate of parameter values

  • Discriminate between different rate law models

  • Godness of fit

    • Root Mean Squared Error (RMSE) and Mean Absolute Error (MAE)

      Lower values indicate a better fit. RMSE is sensitive to outliers, while MAE provides a more robust error metric.

    • P-value of the F-test in ANOVA (Analysis of Variance)

      A p-value smaller than the significance level (commonly 0.05) indicates that there is a statistically significant relationship.

    • Residual Plots

      Ideally, the residuals should be randomly scattered around 0 across the range of fitted values. Patterns in the residual plot can indicate problems with the model

Method of half life

  • Half life (t_{1/2}): Time taken for the concentration of reactant to fall to half of its initial value.

  • Determine half life as a function of initial concentration

  • Requires several experiments

Reaction: \ce{A -> products}

Rate law: -r_A = k C_A^\alpha

Mole balance: -\frac{dC_A}{dt} = k C_A^\alpha

  • Integration: C_A = C_{A0} at t = 0

t = \frac{1}{k C_{A0}^{\alpha -1} (\alpha -1)} \left[ \left( \frac{C_{A0}}{C_A} \right)^{\alpha -1} -1 \right]

  • Integration: C_A = \frac{1}{2} C_{A0} at t = t_{1/2}

t_{1/2} = \frac{1}{k C_{A0}^{\alpha -1} (\alpha -1)} \left[ 2^{\alpha -1} -1 \right]

  • In general:

t_{1/n} = \frac{n^{\alpha -1} -1 }{k (\alpha -1)} \left[ \frac{1}{C_{A0}^{\alpha -1}}\right]

Method of half life

t_{1/2} = \frac{1}{k C_{A0}^{\alpha -1} (\alpha -1)} \left[ 2^{\alpha -1} -1 \right]

Taking log

\ln t_{1/2} = \ln \frac{2^{\alpha - 1} - 1}{(\alpha -1) k} + (1-\alpha) \ln C_{A0}

  • Multiple experiments are performed varying initial concentration and t_{1/2} is recorded.

  • The plot of \ln t_{1/2} vs. \ln C_{A0} is linear with a slope of (1 - \alpha)

Method of initial rates

  • Perform a series of experiments at different initial concentrations C_{A0}

  • Determine initial rate of reaction -r_{A0}

  • Determine rate law parameters by relating -r_{A0} to C_{A0}

Reaction: \ce{A -> products}

Rate law: -r_A = k C_A^\alpha

Mole balance: -\frac{dC_A}{dt} = k C_A^\alpha

Initial rate: -r_{A0} = \left( \frac{-dC_A}{dt}\right)_0 = k C_{A0}^\alpha

  • Taking log

\ln \left( \frac{-dC_A}{dt}\right)_0 = \ln k + \alpha \ln C_{A0}

Rate data from differential reactors

  • For heterogeneous reactions mostly packed bed reactors (PBRs) are used.

  • Differential reactor: The conversion of the reactants in the bed is extremely small, as is the change in reactant concentration through the bed

  • Reactant concentration through the reactor is essentially constant (i.e. the reactor is considered to be gradient-less)

  • Can treat the mole balance like a CSTR

  • Rate of reaction determined for a specified number of pre-determined initial or entering reactant concentrations

  • Determine rate of reaction as a function concentration or partial pressure

  • Operate isothermally

Differential reactor

  • Assumptions

    • No concentration and temperature gradient (gradientless reactor)
    • High volumetric flow rate
    • Small catalyst particles (No mass transfer limitation)
    • Very low conversion
    • Low/ negligible heat release (isothermal)
    • No bypassing/ channeling (uniform flow across catalyst layer)

  • Mole balance:

    in - out + generation = accumulation

F_{A0} - F_{Ae} + r'_A \Delta W = 0

-r'_A = \frac{F_{A0} - F_{Ae}}{\Delta W} = \frac{\upsilon_0 C_{A0} - \upsilon C_{Ae}}{\Delta W}

  • For constant flow rate \upsilon = \upsilon_0

-r'_A = \frac{\upsilon_0 C_{A0} X}{\Delta W} = \frac{\upsilon_0 C_{P}}{\Delta W}

\rightarrow For small conversion -r'_A can be expressed as a function of C_{A0}

Chemical Reaction Engineering