GATE Chemical Engineering Plant Design and Economics Syllabus

What is the discrete compound interest formula vs. continuous compounding?

Discrete: $F = P(1+i)^{n}$. Continuous: $F = P\,e^{rn}$, where $r$ is the nominal continuous rate. The effective annual rate for continuous compounding is $i_{eff} = e^{r} - 1$.

Define an optimum in process design optimization.

An optimum is the value of a design variable that gives the most favorable (minimum cost or maximum profit) value of the objective function, found where $\dfrac{d(\text{cost})}{dx}=0$ (a stationary point with a minimum/maximum).

In an optimization where total cost = (cost increasing with $x$) + (cost decreasing with $x$), how is the economic optimum located?

Set the derivative of total cost to zero. The optimum occurs where the marginal increasing cost balances the marginal decreasing cost; for many engineering problems this means equating the two opposing cost terms' derivatives.

Give the classic optimum economic pipe diameter trade-off.

Pumping (operating/power) cost decreases with larger diameter while the fixed (capital) cost of pipe increases with diameter. The optimum diameter minimizes the sum: $$\text{Total cost} = \text{Pumping cost} + \text{Fixed cost}$$ found by $\frac{d(\text{total})}{dD}=0$.

State the optimum insulation thickness trade-off.

Cost of heat lost decreases with insulation thickness while the capital cost of insulation increases with thickness. Optimum thickness minimizes their sum, where the marginal saving in heat loss equals the marginal cost of added insulation.

What is the optimum reflux ratio concept in distillation design?

As reflux ratio increases, the number of stages (column fixed cost) falls but vapor/liquid loads and reboiler/condenser (operating) costs rise. The optimum reflux ratio, typically $1.1$–$1.5$ times $R_{min}$, minimizes total annual cost.

In Lagrange-multiplier optimization, what condition defines the constrained optimum?

For objective $f$ subject to constraint $g=0$, the optimum satisfies $\nabla f = \lambda\,\nabla g$, i.e. $$\frac{\partial f}{\partial x_i} = \lambda\frac{\partial g}{\partial x_i}\quad\text{for all }i,$$ together with $g=0$.

What does the cylindrical vessel design trade-off (optimum L/D) balance?

For a fixed volume, the surface area (hence material cost) of a cylindrical vessel varies with the length-to-diameter ratio. The optimum $L/D$ minimizes total surface area / material cost for the required volume.

For a closed cylindrical vessel of fixed volume $V$, what L/D ratio minimizes surface area (ignoring head effects)?

Minimizing total surface area $S = \dfrac{\pi D^2}{2} + \dfrac{4V}{D}$ gives the optimum at $L = D$, i.e. $$\frac{L}{D} = 1.$$ (Height equals diameter for a closed cylinder.)

Write the thin-wall cylinder formula for shell thickness under internal pressure.

$$t = \frac{P\,r}{S\,E - 0.6P} + C_c$$ where $P$ is internal pressure, $r$ inside radius, $S$ allowable stress, $E$ joint efficiency, and $C_c$ corrosion allowance (ASME formula).

What is the hoop (circumferential) stress in a thin-walled cylinder of internal pressure $P$, radius $r$, thickness $t$?

$$\sigma_{hoop} = \frac{P\,r}{t}$$ The hoop stress is twice the longitudinal stress $\sigma_{long} = \dfrac{Pr}{2t}$, so cylinders fail along the axial seam first.

State the basic design equation for sizing a heat exchanger (rate equation).

$$Q = U\,A\,\Delta T_{lm}\,F$$ where $Q$ is duty, $U$ overall heat-transfer coefficient, $A$ heat-transfer area, $\Delta T_{lm}$ the log-mean temperature difference, and $F$ the correction factor for non-countercurrent flow.

Write the Log Mean Temperature Difference (LMTD).

$$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}$$ where $\Delta T_1$ and $\Delta T_2$ are the terminal temperature differences at the two ends of the exchanger.

Write the overall heat-transfer coefficient (based on outside area) including wall and fouling resistances.

$$\frac{1}{U_o A_o} = \frac{1}{h_i A_i} + R_{f,i} + \frac{\ln(r_o/r_i)}{2\pi k L} + R_{f,o} + \frac{1}{h_o A_o}$$ summing inside film, inside fouling, wall conduction, outside fouling, and outside film resistances.

Why is a correction factor $F$ applied to LMTD in a 1-2 shell-and-tube exchanger?

Because flow is neither purely cocurrent nor countercurrent (mixed cross/parallel passes), the true mean temperature difference is less than the countercurrent LMTD. $F$ ($\leq 1$) corrects for this; typically design requires $F > 0.75$.

Define the temperature-effectiveness parameters $P$ and $R$ used to find the LMTD correction factor $F$.

$$P = \frac{t_2 - t_1}{T_1 - t_1}, \qquad R = \frac{T_1 - T_2}{t_2 - t_1}$$ where $T$ = shell-side (hot) and $t$ = tube-side (cold) inlet (1) and outlet (2) temperatures.

State the NTU and effectiveness definitions in the effectiveness-NTU method.

$$\text{NTU} = \frac{U A}{C_{min}}, \qquad \varepsilon = \frac{Q}{Q_{max}} = \frac{Q}{C_{min}(T_{h,in} - T_{c,in})}$$ where $C_{min}$ is the smaller of the two fluid heat-capacity rates $\dot m c_p$.

For a countercurrent exchanger with capacity ratio $C_r = C_{min}/C_{max}$, write the effectiveness $\varepsilon$.

$$\varepsilon = \frac{1 - \exp[-\text{NTU}(1 - C_r)]}{1 - C_r\,\exp[-\text{NTU}(1 - C_r)]}$$ For $C_r = 1$: $\varepsilon = \dfrac{\text{NTU}}{1+\text{NTU}}$.

Compare cocurrent and countercurrent heat exchanger arrangements.

Countercurrent gives a larger mean temperature difference, higher effectiveness for the same area, and can raise the cold outlet above the hot outlet; cocurrent cannot exceed an outlet temperature crossover and is thermally less efficient but limits thermal stress at the inlet.

Define LMTD correction and the special case when both fluids change temperature with $C_r=1$.

When $C_r = 1$ (equal heat-capacity rates) in countercurrent flow, $\Delta T_1 = \Delta T_2$, the temperature profiles are parallel lines, and $\Delta T_{lm}$ equals the constant terminal difference $\Delta T$.

What is the purpose of a multistage contactor in mass transfer?

A multistage (cascade) contactor brings phases into repeated contact across several equilibrium stages so that the overall separation greatly exceeds what one stage achieves, used in distillation, absorption, and extraction columns.

Define an ideal (theoretical/equilibrium) stage.

An ideal stage is one in which the two leaving streams are in thermodynamic equilibrium with each other, so the exiting vapor and liquid (or two liquid phases) compositions lie on the equilibrium curve.

Define Murphree stage (plate) efficiency.

$$E_{MV} = \frac{y_n - y_{n+1}}{y_n^{*} - y_{n+1}}$$ It compares the actual change in vapor composition across a tray to the change if the leaving vapor were in equilibrium with the leaving liquid ($y_n^{*}$).

Relate the number of actual plates to ideal plates via overall efficiency.

$$E_O = \frac{N_{ideal}}{N_{actual}}$$ so $N_{actual} = \dfrac{N_{ideal}}{E_O}$, where $E_O$ is the overall column (plate) efficiency.

State the operating-line equation for the rectifying section in distillation.

$$y_{n+1} = \frac{R}{R+1}\,x_n + \frac{x_D}{R+1}$$ where $R$ is the reflux ratio and $x_D$ the distillate composition. Slope $=\dfrac{R}{R+1}$, intercept $=\dfrac{x_D}{R+1}$.

What is the McCabe-Thiele method used for?

It graphically determines the number of theoretical stages for binary distillation by stepping off between the equilibrium curve and the operating lines (rectifying and stripping), joined by the $q$-line.

Define the $q$-line and the meaning of $q$ in McCabe-Thiele analysis.

$q$ is the fraction of feed that is liquid (mole-basis heat to vaporize feed / molar latent heat). The $q$-line is $$y = \frac{q}{q-1}x - \frac{x_F}{q-1}$$ with slope $\dfrac{q}{q-1}$; it locates the intersection of the two operating lines.

State the Kremser equation for the number of ideal stages in countercurrent absorption/stripping.

For an absorption factor $A = L/(mV)$, $$N = \frac{\ln\!\left[\frac{x_{in}-y_{in}/m}{x_{out}-y_{in}/m}(1-1/A)+1/A\right]}{\ln A}$$ The Kremser (Kremser-Brown-Souders) equation gives stages for dilute, constant-$A$ systems.

Define the absorption factor $A$ and stripping factor $S$ and their significance.

$$A = \frac{L}{mV}, \qquad S = \frac{mV}{L} = \frac{1}{A}$$ where $L,V$ are liquid/gas flow rates and $m$ the equilibrium slope. $A>1$ favors absorption; $S>1$ favors stripping.

In a countercurrent multistage extraction, what defines the minimum solvent rate, and what is the trade-off with number of stages?

The minimum solvent rate corresponds to an infinite number of stages (operating line touches the equilibrium curve, a pinch). Increasing solvent above the minimum reduces the required stages but raises solvent recovery cost—an economic optimum (typically $1.2$–$1.5$ times minimum) balances the two.