Multiple effect evaporators are used extensively in industries that require the concentration of solutions by evaporating the solvent, typically water. They utilize steam efficiently by cascading it through several effects where the temperature and pressure decrease sequentially.
Step 1: Analysis of Backward Feed for Cold Feed:
Backward feed is preferred when the feed is cold, as it minimizes thermal shock and avoids excessive heating of the cold feed. In this configuration, the cold feed enters the last effect (with the lowest temperature), reducing the chances of thermal degradation or fouling.
Step 2: Backward Feed for Highly Viscous Products:
Backward feed is advantageous for processing highly viscous solutions because the feed enters the last effect where the temperature is lowest and the viscosity is consequently reduced, making it easier to pump and handle. As the solution moves to hotter effects (where temperature is higher), its viscosity further decreases, improving heat transfer and reducing the risk of fouling.

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).