14.2 Equipment Selection, Sizing, and Cost Estimation

Equipment Selection, Sizing, and Cost Estimation

Process equipment sizing and cost estimation are critical stages in design that bridge engineering calculations and economic feasibility. The FE Chemical exam requires candidates to apply fluid dynamics, thermodynamics, heat transfer, and economic formulas from the NCEES Reference Handbook to size vessels, pumps, and heat exchangers, and to estimate capital costs using exponents and cost indexes.

Process Vessel Sizing (Liquid-Vapor Separators)

Vapor-liquid separators (or knock-out drums) are designed to separate liquid droplets from a gas stream. Sizing a vertical separator is governed by the maximum allowable vapor velocity ($v_{max}$) to prevent liquid droplets from being entrained back into the gas stream. This velocity is calculated using the Souders-Brown equation:

$v_{max} = C \sqrt{\frac{\rho_L - \rho_V}{\rho_V}}$

where:

• $v_{max}$ is the maximum allowable vapor velocity (ft/s or m/s).
• $\rho_L$ is the liquid density at operating conditions (lb/ft³ or kg/m³).
• $\rho_V$ is the vapor density at operating conditions (lb/ft³ or kg/m³).
• $C$ is an empirical sizing coefficient (typically ranging from 0.1 to 0.35 ft/s, depending on tray spacing or presence of a mesh demister pad).

To design a vessel, the actual vapor velocity is typically set between 75% and 85% of $v_{max}$. The minimum cross-sectional area ($A$) of the vessel is determined from the volumetric vapor flow rate ($Q_V$) under operating conditions:

$A = \frac{Q_V}{v_{actual}}$

From the cross-sectional area, the separator diameter ($D$) is computed as $D = \sqrt{4A / \pi}$. The height of the separator must accommodate liquid holdup (typically sized for 5–10 minutes of liquid surge volume) and sufficient vapor disengagement height above the feed nozzle. The aspect ratio (height-to-diameter, $H/D$) of vertical vessels is typically designed to be between 3.0 and 4.0.

Pump Selection and Sizing

Pump sizing requires calculating the total dynamic head ($H_p$) that the pump must provide to move a liquid from a source vessel to a destination vessel:

$H_p = (z_2 - z_1) + \frac{P_2 - P_1}{\rho g} + h_f + \frac{v_2^2 - v_1^2}{2g}$

where $z$ is elevation, $P$ is pressure, $h_f$ is the sum of friction losses in the piping and fittings, and $v$ is fluid velocity. The hydraulic power (or fluid power) transferred to the fluid is:

$P_{hyd} = \dot{m} g H_p = Q \rho g H_p = Q \Delta P$

The brake horsepower ($P_{brake}$), which is the shaft power required from the motor, accounts for pump efficiency ($\eta_p$):

$P_{brake} = \frac{P_{hyd}}{\eta_p}$

To prevent cavitation (the formation and violent collapse of vapor bubbles at low-pressure regions inside the pump), the Net Positive Suction Head Available ($NPSHA$) at the pump inlet must exceed the Net Positive Suction Head Required ($NPSHR$) specified by the pump manufacturer:

$NPSHA = H_{sp} + H_s - H_f - H_{vp}$

where:

• $H_{sp}$ is the absolute pressure head at the feed reservoir surface.
• $H_s$ is the static suction head (positive if the reservoir liquid level is above the pump centerline, negative if below).
• $H_f$ is the friction loss head in the suction line piping and fittings.
• $H_{vp}$ is the vapor pressure head of the liquid at the inlet temperature.

To avoid cavitation, the safety margin $NPSHA - NPSHR \ge 2 \text{ to } 3 \text{ feet}$ (or $\sim 0.6 \text{ to } 1.0 \text{ meter}$) is maintained.

Heat Exchanger Sizing and Fouling

Heat exchangers are sized using the design equation:

$Q = U A F \Delta T_{lm}$

where:

• $Q$ is the heat transfer duty (W or Btu/hr).
• $U$ is the overall heat heat transfer coefficient (W/m²·K or Btu/hr·ft²·°F).
• $A$ is the heat transfer area (m² or ft²).
• $F$ is the configuration correction factor (equal to 1.0 for true countercurrent flow, and less than 1.0 for multi-pass shell-and-tube exchangers).
• $\Delta T_{lm}$ is the log-mean temperature difference:

$\Delta T_{lm} = \frac{(T_{h,in} - T_{c,out}) - (T_{h,out} - T_{c,in})}{\ln \left( \frac{T_{h,in} - T_{c,out}}{T_{h,out} - T_{c,in}} \right)}$

The overall heat transfer coefficient ($U$) changes over time due to fouling (deposition of dirt, scale, or chemical products on tube walls). The fouling factor (or fouling resistance, $R_f$) is incorporated into the calculation of $U$ based on the initial clean coefficient ($U_{clean}$):

$\frac{1}{U_{dirty}} = \frac{1}{U_{clean}} + R_{fo} + R_{fi}$

where $R_{fo}$ and $R_{fi}$ are the outer and inner fouling resistances.

Equipment Cost Sizing and Capacity Scaling

When sizing a new piece of equipment, historical cost data can be scaled to estimate the purchased cost of a different capacity using the cost exponent method:

$C_B = C_A \left( \frac{S_B}{S_A} \right)^n$

where $C_A$ is the cost of equipment $A$ with capacity $S_A$, $C_B$ is the cost of equipment $B$ with capacity $S_B$, and $n$ is the cost exponent. If $n$ is not specified, $n = 0.6$ is used (representing the six-tenths rule for economy of scale).

Inflation Scaling using Cost Indexes

To account for changes in the value of money over time, cost indexes are applied to adjust historical costs to current costs:

$C_{current} = C_{historical} \left( \frac{I_{current}}{I_{historical}} \right)$

where $I$ represents a cost index, most commonly the Chemical Engineering Plant Cost Index (CEPCI).

Bare Module Cost Estimation

The bare module cost ($C_{BM}$) represents the total installed cost of a piece of equipment, including piping, instrumentation, structure, electrical, and installation labor. It is calculated from the base purchased cost ($C_{p}^0$ under standard conditions: carbon steel and ambient pressure) using the bare module factor ($F_{BM}$):

$C_{BM} = C_{p}^0 F_{BM}$

If the equipment operates under high pressure or requires specialized materials (e.g., stainless steel), pressure ($F_P$) and material ($F_M$) correction factors are applied to find the actual purchased cost ($C_p$):

$C_p = C_{p}^0 F_P F_M$