# CSTR First-Order Reaction: Step-by-Step Solution Given Data: - Volumetric flow rate, \( v_ = 2. \) m³/h - Inlet concentration of A, \( C_{A} = 2.5 \) kmol/m³ - Reactor volume, V = 5 \ m³ - Rate constant, \( k = 45 \) h¹ - Reaction: \( \text{A} \rightarrow \text{B} \) - Density is constant; system is ideal and at steady-state. --- ## (a) Space Time and Space Velocity ### **Space Time (\(\tau\))** \[ \tau = \frac{V}{v_} \] \[ \tau = \frac{5~\text{m}^3}{2.~\text{m}^3/\text{h}} = 2.5~\text{h} \] ### **Space Velocity (SV)** \[ SV = \frac{v_}{V} \] \[ SV = \frac{2.~\text{m}^3/\text{h}}{5~\text{m}^3} = .4~\text{h}^{-1} \] --- ## (b) Fractional Conversion of A (\(X\)) For a CSTR with a first-order reaction: \[ X = \frac{k\tau}{1 + k\tau} \] \[ k\tau = .45 \times 2.5 = 1.125 \] \[ X = \frac{1.125}{1 + 1.125} = \frac{1.125}{2.125} \approx .529 \] **Fractional conversion, \(X\), is approximately .529 (or 52.9%).** --- ## (c) Outlet Concentrations of A and B ### **Outlet Concentration of A (\(C_A\))** \[ C_A = C_{A}(1 - X) \] \[ C_A = 2.5 \times (1 - .529) = 2.5 \times .471 = 1.1775~\text{kmol/m}^3 \] ### **Outlet Concentration of B (\(C_B\))** \[ C_B = C_{A} - C_A = 2.5 - 1.1775 = 1.3225~\text{kmol/m}^3 \] --- ## (d) Rate of Reaction of A in the Reactor (\(r_A\)) For a first-order reaction: \[ -r_A = kC_A \] \[ -r_A = .45 \times 1.1775 = .530~\text{kmol/m}^3\text{-h} \] --- ## (e) Molar Flow Rate of B Leaving the Reactor \[ \dot{n}_B = v_ \times C_B \] \[ \dot{n}_B = 2.~\text{m}^3/\text{h} \times 1.3225~\text{kmol/m}^3 = 2.645~\text{kmol/h} \] --- ## (f) Reactor Volume for 80% Conversion Given desired conversion, \(X = .80\): For a CSTR with first-order irreversible reaction: \[ X = \frac{k\tau}{1 + k\tau} \] \[ .80 = \frac{k\tau}{1 + k\tau} \] \[ .80(1 + k\tau) = k\tau \] \[ .80 + .80k\tau = k\tau \] \[ .80 = k\tau - .80k\tau = k\tau(1 - .80) = .20k\tau \] \[ k\tau = \frac{.80}{.20} = 4 \] \[ \tau = \frac{4}{k} = \frac{4}{.45} = 8.889~\text{h} \] \[ V = \tau \times v_ = 8.889\times 2. = 17.78~\text{m}^3 \] --- # **Summary of Answers** | Part | Quantity | Value | |------|------------------------ |------------------------------------| | (a) | Space time, \(\tau\) | \(2.5~\text{h}\) | | | Space velocity, SV | \(.4~\text{h}^{-1}\) | | (b) | Fractional conversion | \(.529\) (52.9%) | | (c) | Outlet \(C_A\) | \(1.1775~\text{kmol/m}^3\) | | | Outlet \(C_B\) | \(1.3225~\text{kmol/m}^3\) | | (d) | Rate of reaction | \(.530~\text{kmol/m}^3\text{-h}\) | | (e) | Molar flow rate of B | \(2.645~\text{kmol/h}\) | | (f) | Volume for \(X=.80\) | \(17.78~\text{m}^3\) |