The process for manufacturing PVC involves two primary chemical reactions. Initially, ethylene (\(C_2H_4\)) is chlorinated to form 1,2-dichloroethane (\(C_2H_4Cl_2\)), commonly known as ethylene dichloride (EDC). This reaction is the chlorination step. Subsequently, the EDC undergoes a dehydrochlorination process to produce vinyl chloride monomer (VCM), the precursor for PVC polymerization.
Step 1: Chlorination of Ethylene: Ethylene reacts with chlorine gas, forming EDC. \[ C_2H_4 + Cl_2 \rightarrow C_2H_4Cl_2 \]
Step 2: Dehydrochlorination of EDC: EDC is heated and decomposed to yield VCM and hydrogen chloride. \[ C_2H_4Cl_2 \rightarrow CH_2=CHCl + HCl \]
Consider a process with transfer function: \[ G_p = \frac{2e^{-s}}{(5s + 1)^2} \] A first-order plus dead time (FOPDT) model is to be fitted to the unit step process reaction curve (PRC) by applying the maximum slope method. Let \( \tau_m \) and \( \theta_m \) denote the time constant and dead time, respectively, of the fitted FOPDT model. The value of \( \frac{\tau_m}{\theta_m} \) is __________ (rounded off to 2 decimal places).
Given: For \( G = \frac{1}{(\tau s + 1)^2} \), the unit step output response is: \[ y(t) = 1 - \left(1 + \frac{t}{\tau}\right)e^{-t/\tau} \] The first and second derivatives of \( y(t) \) are: \[ \frac{dy(t)}{dt} = \frac{t}{\tau^2} e^{-t/\tau} \] \[ \frac{d^2y(t)}{dt^2} = \frac{1}{\tau^2} \left(1 - \frac{t}{\tau}\right) e^{-t/\tau} \]
Methanol is produced by the reversible, gas-phase hydrogenation of carbon monoxide: \[ {CO} + 2{H}_2 \rightleftharpoons {CH}_3{OH} \] CO and H$_2$ are charged to a reactor, and the reaction proceeds to equilibrium at 453 K and 2 atm. The reaction equilibrium constant, which depends only on the temperature, is 1.68 at the reaction conditions. The mole fraction of H$_2$ in the product is 0.4. Assuming ideal gas behavior, the mole fraction of methanol in the product is ____________ (rounded off to 2 decimal places).