Principles and Practice of Engineering Exam (PE) Chemical Engineering (PE Chemical) Syllabus

What is fugacity and why is it used?

Fugacity $f$ is an effective (corrected) pressure that makes the chemical potential expression valid for real fluids: $\mu - \mu^{\circ} = RT \ln(f/f^{\circ})$. For an ideal gas, $f = P$; the deviation is captured by the fugacity coefficient $\phi = f/P$.

Define the compressibility factor $Z$ and its value for an ideal gas.

$$Z = \frac{PV}{nRT}$$ For an ideal gas $Z = 1$; $Z<1$ indicates attractive-dominant behavior, $Z>1$ repulsive-dominant.

State Raoult's law for vapor-liquid equilibrium of an ideal solution.

$$y_i P = x_i P_i^{sat}$$ The partial pressure of component $i$ in the vapor equals its liquid mole fraction times its pure-component saturation (vapor) pressure.

State Henry's law and when it applies.

$$p_i = H_i \, x_i$$ The partial pressure of a dilute solute is proportional to its liquid mole fraction, with Henry's constant $H_i$. It applies to dilute (low-concentration) solutes.

Define the relative volatility $\alpha_{ij}$ and its role in distillation.

$$\alpha_{ij} = \frac{y_i/x_i}{y_j/x_j} = \frac{K_i}{K_j}$$ It measures ease of separation; the larger $\alpha$ is above 1, the easier the distillation. $\alpha = 1$ means no separation by ordinary distillation (azeotrope).

What is an azeotrope, and distinguish minimum- and maximum-boiling types.

An azeotrope is a mixture whose vapor and liquid compositions are identical, so it cannot be separated by simple distillation. A minimum-boiling azeotrope boils below both pure components (positive deviation); a maximum-boiling azeotrope boils above both (negative deviation).

State the Gibbs phase rule.

$$F = C - P + 2$$ where $F$ = degrees of freedom, $C$ = number of components, $P$ = number of phases. The $+2$ accounts for temperature and pressure.

Write the relationship between the equilibrium constant $K$ and standard Gibbs energy of reaction.

$$\Delta G_{rxn}^{\circ} = -RT \ln K$$

State the van't Hoff equation describing temperature dependence of the equilibrium constant.

$$\frac{d \ln K}{dT} = \frac{\Delta H_{rxn}^{\circ}}{RT^2}, \qquad \ln\frac{K_2}{K_1} = -\frac{\Delta H_{rxn}^{\circ}}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)$$

Using Le Chatelier's principle, predict the effect of increasing pressure on a gas-phase reaction that produces more moles of gas.

Equilibrium shifts toward the side with fewer moles of gas (the reactant side), reducing the extent of forward reaction to relieve the increased pressure.

Write the equilibrium expression $K$ in terms of activities for a general reaction $\ce{aA + bB <=> cC + dD}$.

$$K = \frac{a_C^{c}\, a_D^{d}}{a_A^{a}\, a_B^{b}}$$ where $a_i$ are activities (partial pressures for ideal gases, concentrations/mole fractions for ideal solutions).

State the Reynolds number and its laminar/turbulent transition ranges for pipe flow.

$$Re = \frac{\rho v D}{\mu}$$ Laminar for $Re < 2100$, transitional for $2100 < Re < 4000$, turbulent for $Re > 4000$.

Write the mechanical energy (Bernoulli) balance with friction and pump work.

$$\frac{P_1}{\rho} + \frac{v_1^2}{2} + g z_1 + W_p = \frac{P_2}{\rho} + \frac{v_2^2}{2} + g z_2 + h_f$$ where $W_p$ is pump work per unit mass and $h_f$ is friction loss.

State the Darcy-Weisbach equation for frictional head loss in a pipe.

$$h_f = f \frac{L}{D}\frac{v^2}{2g}$$ where $f$ is the Darcy friction factor, $L$ length, $D$ diameter, $v$ velocity.

Give the friction factor for fully developed laminar flow.

$$f = \frac{64}{Re} \quad (\text{Darcy}), \qquad f_{Fanning} = \frac{16}{Re}$$

Write Fourier's law of heat conduction (one-dimensional).

$$q = -k A \frac{dT}{dx}$$ where $k$ is thermal conductivity, $A$ area, and $\dfrac{dT}{dx}$ the temperature gradient.

Write Newton's law of cooling for convective heat transfer.

$$q = h A (T_s - T_\infty)$$ where $h$ is the convective heat transfer coefficient, $T_s$ surface temperature, $T_\infty$ bulk fluid temperature.

Define the log mean temperature difference (LMTD) for a heat exchanger.

$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$ where $\Delta T_1$ and $\Delta T_2$ are the temperature differences at the two ends. Heat duty: $q = U A\,\Delta T_{lm}$.

Define the Nusselt, Prandtl, and overall heat transfer coefficient relationships.

$$Nu = \frac{hL}{k}, \qquad Pr = \frac{C_p \mu}{k}$$ The overall coefficient: $\dfrac{1}{UA} = \dfrac{1}{h_i A_i} + R_{wall} + \dfrac{1}{h_o A_o}$ (sum of resistances in series).

Write Fick's first law of diffusion.

$$J_A = -D_{AB}\frac{dC_A}{dz}$$ where $J_A$ is the molar flux of $A$, $D_{AB}$ the diffusivity, and $\dfrac{dC_A}{dz}$ the concentration gradient.

What does the McCabe-Thiele method determine in distillation, and what are the operating lines?

It graphically determines the number of equilibrium stages. Rectifying line: $y = \frac{R}{R+1}x + \frac{x_D}{R+1}$; stripping line connects the still composition to the feed point. The $q$-line locates the feed condition.

Define the minimum reflux ratio and total reflux limits in distillation.

At total reflux ($R \to \infty$) the number of stages is minimum (Fenske equation). At minimum reflux ($R_{min}$) an infinite number of stages is required (pinch point). Operating reflux is typically $R = 1.2$ to $1.5 \times R_{min}$.

Name the controlling mass-transfer driving force in absorption and the role of HTU/NTU.

The driving force is the concentration (or partial pressure) difference between bulk and interface. Packed-column height $Z = \text{HTU} \times \text{NTU}$, where HTU is the height of a transfer unit and NTU the number of transfer units.

Write the design (mole-balance) equations for the three ideal reactors: batch, CSTR, and PFR.

Batch: $t = N_{A0}\int_0^X \frac{dX}{-r_A V}$. CSTR: $V = \frac{F_{A0} X}{-r_A}$. PFR: $V = F_{A0}\int_0^X \frac{dX}{-r_A}$.

Write the Arrhenius equation for the rate constant.

$$k = A\, e^{-E_a/(RT)}$$ where $A$ is the pre-exponential factor and $E_a$ the activation energy. Linear form: $\ln k = \ln A - \dfrac{E_a}{R}\cdot\dfrac{1}{T}$.

Compare a CSTR and a PFR for a positive-order reaction at the same conversion.

A PFR requires less volume than a CSTR for the same conversion (for normal positive-order kinetics), because the PFR maintains a higher average reactant concentration. A CSTR operates entirely at the low exit concentration.

Define space time $\tau$ and space velocity for a flow reactor.

$$\tau = \frac{V}{v_0}$$ where $V$ is reactor volume and $v_0$ the inlet volumetric flow rate. Space velocity $= 1/\tau$ (volumes of feed processed per reactor volume per time).

Write the integrated rate law for a first-order irreversible reaction.

$$\ln\frac{C_A}{C_{A0}} = -kt \quad\Rightarrow\quad C_A = C_{A0}\, e^{-kt}$$ Half-life: $t_{1/2} = \dfrac{\ln 2}{k}$ (independent of concentration).

Give the closed-loop transfer function of a feedback control system.

$$\frac{Y}{Y_{sp}} = \frac{G_c G_p}{1 + G_c G_p G_m}$$ where $G_c$ is controller, $G_p$ process, and $G_m$ measurement transfer functions.

Write the ideal PID controller equation in the time domain.

$$u(t) = K_c\left[ e(t) + \frac{1}{\tau_I}\int_0^t e(t')\,dt' + \tau_D \frac{de(t)}{dt} \right]$$ where $K_c$ is gain, $\tau_I$ integral time, $\tau_D$ derivative time.

Compare proportional, integral, and derivative control actions.

Proportional reduces error but leaves offset; Integral eliminates steady-state offset but can cause oscillation/windup; Derivative anticipates change to improve stability and reduce overshoot but amplifies noise.

Define the time constant $\tau$ and gain $K$ for a first-order process and its transfer function.

$$G(s) = \frac{K}{\tau s + 1}$$ $K$ is the steady-state output/input ratio; $\tau$ is the time to reach $63.2\%$ of the final response to a step input.

Define LFL and UFL (flammability limits) and the consequence of operating between them.

LFL (lower flammable limit) is the minimum fuel-in-air concentration that supports combustion; UFL (upper flammable limit) is the maximum. Between LFL and UFL the mixture is flammable/explosive; outside this range it is too lean or too rich to ignite.

Define flash point and autoignition temperature.

Flash point is the lowest temperature at which a liquid gives off enough vapor to form an ignitable mixture with air near its surface. Autoignition temperature is the lowest temperature at which a substance spontaneously ignites without an external ignition source.

What is a BLEVE and how does it occur?

A BLEVE (Boiling Liquid Expanding Vapor Explosion) occurs when a vessel holding a pressurized liquid above its atmospheric boiling point fails, causing rapid flashing of the liquid to vapor and a violent explosion, often with a fireball if the material is flammable.

Name the components of the fire triangle and how extinguishment relates to it.

Fuel, oxygen (oxidizer), and heat (ignition energy). Removing any one element extinguishes the fire; many systems add the chemical chain reaction as a fourth element (fire tetrahedron).

What is the purpose of a relief valve / rupture disk, and how do they differ?

Both protect equipment from overpressure. A relief (safety) valve reopens/reseats after relieving and is reusable; a rupture disk is a one-time burst membrane that must be replaced after activation but offers fast, leak-tight protection.

Write the formula for the present worth of a future single payment (discounting).

$$P = \frac{F}{(1+i)^n}$$ where $F$ is the future value, $i$ the interest (discount) rate per period, and $n$ the number of periods.

Define net present value (NPV) and the decision rule for project acceptance.

$$\text{NPV} = \sum_{t=0}^{n} \frac{C_t}{(1+i)^t}$$ where $C_t$ is the net cash flow in period $t$. Accept the project if $\text{NPV} > 0$.

Define the rate of return (IRR / DCFROR) of a project.

The internal rate of return is the discount rate $i^*$ that makes the net present value zero: $\sum_{t=0}^{n}\dfrac{C_t}{(1+i^*)^t} = 0$. A project is attractive when IRR exceeds the minimum acceptable rate of return (hurdle rate).

Write the straight-line depreciation formula.

$$d = \frac{C - S}{n}$$ where $d$ is annual depreciation, $C$ the initial cost, $S$ the salvage value, and $n$ the useful life in years.

State the necessary condition for an unconstrained optimum and how to distinguish a minimum from a maximum.

Necessary: $\dfrac{df}{dx}=0$ (stationary point). For a minimum $\dfrac{d^2 f}{dx^2}>0$; for a maximum $\dfrac{d^2 f}{dx^2}<0$. With multiple variables, set $\nabla f = 0$ and check the Hessian's definiteness.