2.1 Units, Bases, and Process Variables

Material and energy balances dominate Domain A of the UPDA/MMUP Chemical exam, the largest single strand at about 20% of a weight-faithful study plan. Almost every balance problem starts the same way: identify the process units, fix a calculation basis, convert compositions into consistent flows, and only then write equations. This section trains that front end so later chapters on balances with reaction, recycle, and energy can assume you already handle units and stream variables cleanly.

Why units matter on a timed MCQ exam

Qatar plants and textbooks mix SI with common process engineering units. A feed may be given in kg/h, a gas rate in Nm³/h, pressure in bar(g) or kPa(a), and temperature in °C. The UPDA format is computer-based multiple choice with roughly 25 questions in 60 minutes, so you rarely have time to redo a long solution after a unit slip. Treat unit conversion as a mandatory first step, not a cleanup at the end.

Gauge vs absolute pressure: $P_{\mathrm{abs}} = P_{\mathrm{gauge}} + P_{\mathrm{atm}}$. At sea level, $P_{\mathrm{atm}} \approx 101.325,\mathrm{kPa} \approx 1.013,\mathrm{bar}$. Many Gulf plant gauges report barg; ideal-gas and vapor–liquid equilibrium calculations need absolute pressure.

Standard gas volume (quick recall): At standard conditions often taken as 0 °C and 1 atm, 1 kmol of ideal gas occupies about $22.4,\mathrm{m}^3$. At 25 °C and 1 atm the molar volume is about $24.5,\mathrm{m}^3/\mathrm{kmol}$. Always check which “standard” a problem states before converting Nm³ to kmol.

Composition: mole fraction, mass fraction, and average MW

For a mixture of $n$ species, mole fraction of $i$ is $y_i = \frac{n_i}{\sum_j n_j}, \qquad \sum_i y_i = 1$ and mass fraction is $w_i = \frac{m_i}{\sum_j m_j}, \qquad \sum_i w_i = 1.$ Mole and mass fractions are related through molecular weights $M_i$: $w_i = \frac{y_i M_i}{\bar{M}}, \qquad y_i = \frac{w_i / M_i}{\sum_j (w_j / M_j)},$ where the average molecular weight is $\bar{M} = \sum_i y_i M_i = \frac{1}{\sum_i (w_i / M_i)}.$

Flows connect the same way:

• Mole flow of $i$: $\dot{n}i = y_i \dot{n}{\mathrm{tot}}$
• Mass flow of $i$: $\dot{m}i = w_i \dot{m}{\mathrm{tot}} = \dot{n}_i M_i$
• Total mass flow: $\dot{m}{\mathrm{tot}} = \dot{n}{\mathrm{tot}} \bar{M}$

Density $\rho$ links mass and volume: $\dot{m} = \rho \dot{V}$. For liquids, density is often nearly constant; for gases, use the equation of state (ideal gas first unless the problem flags non-ideality).

Choosing a calculation basis

A basis is a fixed quantity you invent (when the problem does not fix it) so all other streams scale to it. Good habits:

1. If composition is mole %, choose 100 mol or 100 kmol (or 100 mol/h for continuous flow).
2. If composition is mass % or wt%, choose 100 kg or 100 kg/h.
3. If a flow is already given, use that flow as the basis (e.g., 5000 kg/h feed).
4. If only ratios are given, pick any convenient basis and scale at the end if a real plant rate is needed.
5. Never change basis mid-solution without explicitly rescaling every stream.

For continuous plants, thinking in rate units (kmol/h, kg/h) is natural. For batch problems, use amounts (kmol, kg) charged or produced per batch. The algebra of balances is the same; only the interpretation of the numbers changes.

Stream tables

A stream table is a spreadsheet-like layout of every important stream variable. Typical columns: stream ID, total mole flow, component mole flows (or mole fractions), total mass flow, temperature, pressure, phase, and density if known. Rows are components plus a total row that must equal the sum of components.

Filling stream tables before writing equations reduces algebra mistakes and makes degree-of-freedom checks visible: unknown boxes become the variables you must solve for.

Worked example 1 — Composition conversion

A process gas contains 40 mol% CO₂, 50 mol% H₂, and 10 mol% N₂. Molecular weights: CO₂ = 44, H₂ = 2, N₂ = 28. Find mass fractions and average MW.

Solution. Basis: 100 mol of gas.

• CO₂: 40 mol → $40 \times 44 = 1760,\mathrm{g}$
• H₂: 50 mol → $50 \times 2 = 100,\mathrm{g}$
• N₂: 10 mol → $10 \times 28 = 280,\mathrm{g}$
• Total mass = $2140,\mathrm{g}$
• $\bar{M} = 2140/100 = 21.4,\mathrm{g/mol}$
• Mass fractions: CO₂ $1760/2140 = 0.822$, H₂ $100/2140 = 0.0467$, N₂ $280/2140 = 0.131$

Check: $0.822 + 0.047 + 0.131 = 1.000$. Hydrogen looks “small” on a mass basis even though it is half the moles—classic exam trap if someone treats mol% as wt%.

Worked example 2 — Flow from volume and density

A liquid hydrocarbon stream has SG = 0.78 (relative to water at 1000 kg/m³) and volumetric flow 25 m³/h. What is the mass flow in kg/h?

Solution. $\rho = 0.78 \times 1000 = 780,\mathrm{kg/m}^3$. $\dot{m} = \rho \dot{V} = 780 \times 25 = 19{,}500\,\mathrm{kg/h}.$

If the same stream had average MW 78 g/mol and you needed mole flow: $\dot{n} = \frac{19{,}500}{78} = 250\,\mathrm{kmol/h}.$

Worked example 3 — Unit conversion discipline

A gas feed is 1200 Nm³/h of pure methane (assume 0 °C, 1 atm standard; 22.4 m³/kmol). Convert to kmol/h and kg/h ($M = 16$).

Solution. $\dot{n} = \frac{1200}{22.4} = 53.57\,\mathrm{kmol/h}, \qquad \dot{m} = 53.57 \times 16 = 857\,\mathrm{kg/h}.$ If someone wrongly used 24.5 m³/kmol (25 °C standard), $\dot{n}$ would be 49.0 kmol/h—about 9% low. Always match the standard stated in the problem.

Exam habits for UPDA process-variable items

• Write units next to every intermediate number.
• Convert gauge pressures before using $PV = nRT$.
• Confirm whether % means mol% or wt%; if unspecified, the surrounding data (MW given vs density given) often reveals intent.
• Sum fractions and component flows as a sanity check before selecting the MCQ option nearest your answer.
• For multi-stream problems, label each stream (F, V, L, R, P) and never reuse the same symbol for two different quantities.

These habits transfer directly into Chapter 3 material balances: once units, basis, and stream tables are automatic, the balance equations themselves become the only remaining challenge.