Certainly! I'll walk you through each part with clear explanations. To establish the independent reaction matrix for the pyrolysis network, you need to write the stoichiometric matrix \( \nu \) for the species **A, B, C, D, E**. The matrix provided is: \[ \begin{bmatrix} -1 & 1 & & -2 & \\ & -2 & 1 & 1 & \end{bmatrix} \begin{bmatrix} n_A \\ n_B \end{bmatrix} = \] To determine the rank, use Gaussian elimination. If the matrix has rank 2, it means you need at least 2 independent reactions to describe the system. For the selectivity profile \( S_{P/Q} \) versus conversion of A in a parallel reaction, plot selectivity as \( S_{P/Q} = (k_1/k_2) \frac{[A]}{[A]_} \), where \( k_1 \) and \( k_2 \) are the rate constants for P and Q formation. High \( k_1/k_2 \) and a strategy to keep \( C_A \) high (such as high feed or low residence time) will maximize selectivity towards P. The maximum yield of P is reached when the conversion is optimized such that the formation of P is maximized compared to Q. For parallel reactions, this happens at low conversions if \( k_2 \) is significant, so you want to operate at a point where the rate of P formation is much higher than Q. For the internal effectiveness factor \( \eta \) of a non-isothermal pellet, plot \( \eta \) versus the Thiele modulus \( \phi \), taking into account the coupled effects of high activation energy \( \gamma \) and heat of reaction \( \beta \). The mathematical model is given by the effectiveness factor integral and the non-isothermal steady-state equation. When plotting, expect regions of multiple steady states when the pellet temperature exceeds the bulk temperature due to reaction heat. To calculate the mean residence time \( \bar{t} \) and variance \( \sigma^2 \) from the pulse tracer data, first normalize the concentration \( C(t) \) to get the E-curve, \( E(t) = C(t)/\int C(t)dt \). Using the data, calculate the area under \( C(t) \) for normalization, then compute: \[ \bar{t} = \int t E(t) dt \] \[ \sigma^2 = \int (t - \bar{t})^2 E(t) dt \] Sum the products as per the table to get the values numerically. The dispersion number \( D/uL \) is then found from the variance using the relationship for axial dispersion in plug flow: \[ \frac{D}{uL} = \frac{\sigma^2}{2\bar{t}^2} \] For the adiabatic packed bed problem, solve for the catalyst weight \( W \) required to achieve 40% conversion by integrating the mole balance equation numerically. The equation includes the exponential Arrhenius rate with the temperature increasing as the reaction proceeds, given by the adiabatic temperature rise \( \Delta T_{ad} \). Substitute all given terms and integrate from \( X= \) to \( X=.4 \). For the hysteresis loop in an autothermal reactor, sketch a plot of reactor temperature versus conversion with product recycle. The S-shaped curve \( G(T) \) (heat generation) and the linear \( T_{in} + \Delta T_{ad} X \) (heat removal) should be drawn. The ignition and extinction points are where the slopes of these curves are equal, indicating the onset and loss of multiple steady states. To calculate the minimum ignition temperature, set the heat generated equal to the heat removed and solve for the lowest temperature at which this equality is satisfied, typically at the tangent point between the \( G(T) \) and heat removal curves. Let me know if you need any of the calculations or plots explicitly worked out!