Temperatures
Reaction
Heat transfer
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This simulation shows how temperature and concentration profiles develop when a jacketed plug flow reactor is started up. The exothermic, liquid-phase, first-order reaction A → B takes place. At time zero the reactor is full of solvent (no A) at the initial temperature Ti, and feed containing A starts to enter at the inlet temperature T0.
Sliders are used to set the inlet temperature, the initial reactor temperature, and the jacket temperature. There are no radial temperature gradients in the reactor. The rate constant and residence time together are set with the Damköhler number (Da) slider. The activation energy for the reaction is also set with a slider. The adiabatic temperature rise, which is set with a slider, is related to the heat of reaction and heat capacities. It is the temperature rise for complete conversion in an adiabatic reactor. The dimensionless heat transfer coefficient (β) is set with a slider, as is the thermal capacitance ratio, which is defined in the Details.
Press Play to run the start-up and Reset to return to the initial state. The simulation stops automatically after the reactor has been at steady state for 3 dimensionless times (time/τ); press Play to continue, or change a slider and press Play to follow the response to that step change. The tube at the top is colored by temperature. The two profile plots show CA/CA0 and temperature versus axial position; dashed curves are the steady-state profiles for the current slider values. The bottom plot tracks outlet conversion, outlet temperature, and the maximum (hot-spot) temperature in the reactor versus dimensionless time t/τ.
Every slider except the initial reactor temperature can be changed while the simulation runs; the change acts as a step change in that variable. For example, let the reactor reach steady state, then lower the inlet temperature. When the thermal capacitance ratio σ is greater than 1, the hot spot temporarily rises well above both the old and new steady states (wrong-way behavior). Set σ = 1 to see this effect disappear.
Details
The liquid-phase reaction A → B is first order and exothermic. Mass and energy balances for a plug flow reactor with a cooling (or heating) jacket, including the heat capacity of the reactor wall or packing, are:
\[ \frac{\partial C_A}{\partial t} + u\,\frac{\partial C_A}{\partial z} = -k(T)\,C_A \] \[ \sigma\,\frac{\partial T}{\partial t} + u\,\frac{\partial T}{\partial z} = \frac{(-\Delta H_R)\,k(T)\,C_A}{\rho c_p} - \frac{U a}{\rho c_p}\,(T - T_a) \] \[ k(T) = k_{\mathrm{ref}}\exp\!\left[\frac{E_a}{R}\left(\frac{1}{T_{\mathrm{ref}}} - \frac{1}{T}\right)\right], \qquad T_{\mathrm{ref}} = 350\ \mathrm{K} \]where u is the fluid velocity, z is axial position, a is the heat-transfer area per reactor volume, and Ta is the jacket temperature. The thermal capacitance ratio is
\[ \sigma = \frac{\rho c_p + \rho_w c_{p,w}\,(V_w/V)}{\rho c_p} \ge 1 \]Because the wall must also be heated, a temperature change travels down the reactor at speed u/σ, while concentration changes travel at u. With σ = 1, every fluid element behaves like a batch reactor moving down the tube.
With dimensionless position ζ = z/L, time θ = t/τ (space time τ = L/u), and f = CA/CA0:
\[ \frac{\partial f}{\partial \theta} + \frac{\partial f}{\partial \zeta} = -\mathrm{Da}\,e^{\gamma}\,f \] \[ \sigma\,\frac{\partial T}{\partial \theta} + \frac{\partial T}{\partial \zeta} = \Delta T_{\mathrm{ad}}\,\mathrm{Da}\,e^{\gamma}\,f - \beta\,(T - T_a) \] \[ \mathrm{Da} = k_{\mathrm{ref}}\,\tau, \quad \gamma = \frac{E_a}{R}\left(\frac{1}{T_{\mathrm{ref}}} - \frac{1}{T}\right), \quad \Delta T_{\mathrm{ad}} = \frac{(-\Delta H_R)\,C_{A0}}{\rho c_p}, \quad \beta = \frac{U a\,\tau}{\rho c_p} \]Initial conditions: f(ζ, 0) = 0 and T(ζ, 0) = Ti. Boundary conditions: f(0, θ) = 1 and T(0, θ) = T0. Conversion is X = 1 − f.
The steady-state profiles (dashed) are found by setting the time derivatives to zero and integrating the resulting ordinary differential equations from ζ = 0 to 1 with a fourth-order Runge–Kutta method:
\[ \frac{df}{d\zeta} = -\mathrm{Da}\,e^{\gamma} f, \qquad \frac{dT}{d\zeta} = \Delta T_{\mathrm{ad}}\,\mathrm{Da}\,e^{\gamma} f - \beta\,(T - T_a) \]The steady state does not depend on σ; only the approach to it does.
The transient equations are solved by the method of lines on 400 axial grid points with operator splitting. Each time step (Δθ = Δζ) first shifts the concentration profile exactly one grid point (first-order upwind with Courant number 1) and advances temperature with first-order upwind differencing (Courant number 1/σ). Reaction and heat exchange are then integrated at each grid point with adaptive substeps. The small numerical dispersion in the temperature equation makes the final transient profiles differ slightly from the steady-state curves.
Symbols
| \(a\) | heat-transfer area per reactor volume (m2/m3) |
| \(C_A,\ C_B\) | concentrations of A and B (mol/m3) |
| \(C_{A0}\) | concentration of A in the feed (mol/m3) |
| \(c_p\) | heat capacity of the reacting fluid (J/(kg K)) |
| \(c_{p,w}\) | heat capacity of the reactor wall or packing (J/(kg K)) |
| \(\mathrm{Da}\) | Damköhler number, krefτ (dimensionless) |
| \(E_a\) | activation energy (kJ/mol) |
| \(f\) | dimensionless concentration of A, CA/CA0 |
| \(k,\ k_{\mathrm{ref}}\) | rate constant at T and at Tref (1/s) |
| \(L\) | reactor length (m) |
| \(R\) | ideal gas constant, 8.314 J/(mol K) |
| \(t\) | time (s) |
| \(T\) | fluid temperature (K) |
| \(T_0\) | feed (inlet) temperature (K) |
| \(T_a\) | jacket temperature (K) |
| \(T_i\) | initial reactor temperature (K) |
| \(T_{\mathrm{ref}}\) | reference temperature for k, 350 K |
| \(u\) | fluid velocity (m/s) |
| \(U\) | overall heat-transfer coefficient (W/(m2 K)) |
| \(V,\ V_w\) | reactor volume and wall (or packing) volume (m3) |
| \(X\) | conversion of A, 1 − f |
| \(z\) | axial position (m) |
| \(\beta\) | dimensionless heat-transfer coefficient, Uaτ/(ρcp) |
| \(\gamma\) | dimensionless Arrhenius exponent |
| \(\Delta H_R\) | heat of reaction (J/mol) |
| \(\Delta T_{\mathrm{ad}}\) | adiabatic temperature rise (K) |
| \(\zeta\) | dimensionless axial position, z/L |
| \(\theta\) | dimensionless time, t/τ |
| \(\rho,\ \rho_w\) | densities of the fluid and the wall or packing (kg/m3) |
| \(\sigma\) | thermal capacitance ratio (dimensionless) |
| \(\tau\) | space time, L/u (s) |
About
This simulation was created in the Department of Chemical and Biological Engineering at University of Colorado Boulder for LearnChemE.com by John L. Falconer using Claude AI. It was prepared with financial support from the National Science Foundation (DUE 2336987 and 2336988). Address any questions or comments to LearnChemE@gmail.com.