Material and Energy Balances: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

Mass balance

Steady state
in = out
Total balance
ṁ1 + ṁ2 = ṁ3
Component balance
ṁ1x1 + ṁ2x2 = ṁ3x3
Evaporator
F·xF = P·xP, W = F − P

Energy balance

Sensible heat
Q = ṁ·cp·ΔT
Latent heat
Q = ṁ·hfg
Mixing
ΣṁcpTin = ΣṁcpTout
Water
cp ≈ 4.18 kJ/(kg·K)

Ideal gas law

Ideal gas law
pV = nRT
Gas constant
R = 8.314 J/(mol·K)
Combined gas law
p1V1/T1 = p2V2/T2
Moles
n = m/M
Temperature
T[K] = T[°C] + 273.15

Stoichiometry

Moles
n = m/M
Concentration
c = n/V
Mole ratio
use the coefficients of the balanced equation
Limiting reactant
the one that runs out first
Percent yield
actual / theoretical × 100%

Method

  1. Write and balance the chemical equation.
  2. Convert the given mass to moles: n = m/M.
  3. Use the coefficients as the mole ratio to find the moles of the substance you want.
  4. Convert back to mass with m = n·M. With two reactants, find the limiting reactant first.

2Practice problems

1.Mass balanceMedium
MixerStream 1750 kg/h · 40% NaOHStream 2ṁ₂ = ? · 15% NaOHProduct20% NaOH

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 20% NaOH? Find ṁ2.

ṁ1 = 750 kg/hx1 = 40 %x2 = 15 %x3 = 20 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F2300 kg/h · 16% sugarProduct P65% sugarVapour W (pure water)W = ? kg/h

An evaporator concentrates a sugar solution by boiling off pure water. Find the water evaporated W in kg/h.

F = 2300 kg/hxF = 16 %xP = 65 %
W = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 130 °C80 °Cwater · 10 °Ct = ? °C

Hot glycol mix cools water in a counter-flow heat exchanger with no heat loss. Find the water outlet temperature tout.

ṁh = 3 kg/scp,h = 3.6 kJ/(kg·K)Th: 130 → 80 °Cṁc = 2 kg/scp,c = 4.18 kJ/(kg·K)tin = 10 °C
tout = °C
4.Energy balanceHard
hot sidecold sideethanol · 180 °C90 °Cwater · 15 °C85 °C · ṁ = ?15% lost

Hot ethanol heats water in an exchanger that loses 15% of the heat released by the hot stream to the surroundings. Find the water flow rate ṁc needed.

ṁh = 2.5 kg/scp,h = 2.44 kJ/(kg·K)Th: 180 → 90 °Ct: 15 → 85 °Ccp,c = 4.18 kJ/(kg·K)loss = 15 %
ṁc = kg/s
5.Ideal gas lawMedium
400 g of O₂P = 350 kPaT = 90 °CV = ?

What volume V does 400 g of O₂ occupy at 350 kPa and 90 °C? Treat it as an ideal gas.

m = 400 gM = 32 g/molP = 350 kPaT = 90 °CR = 8.314 J/(mol·K)
V = L
6.Ideal gas lawHard
CO₂, M = 44.01 g/molP = 250 kPaT = 30 °Cρ = ?

Find the density ρ of CO₂ at 250 kPa and 30 °C, treating it as an ideal gas.

M = 44.01 g/molP = 250 kPaT = 30 °CR = 8.314 J/(mol·K)
ρ = kg/m³
7.StoichiometryMedium
m(X)n(X)n(MgO)m(MgO)÷ M(X)× 2/ν× M(MgO)

12 g of Mg and 40 g of O2 react according to 2Mg + O2 → 2MgO. What mass m of MgO can form?

M(Mg) = 24.31M(O2) = 32M(MgO) = 40.3 g/mol
m = g
8.StoichiometryHard
m(Na)n(Na)n(NaCl)m(NaCl)÷ M(Na)× 2/2× M(NaCl)

In the reaction 2Na + Cl2 → 2NaCl, 12 g of Na reacts completely (the other reactants are in excess) and 22.88 g of NaCl is collected. Find the percent yield Y.

M(Na) = 22.99 g/molM(NaCl) = 58.44 g/molactual: 22.88 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 3000 kg/h

Steady-state mass balance: in = out for the total and for each component. Blending problems combine both.

Balances

Total: ṁ3 = ṁ1 + ṁ2
NaOH: ṁ1x1 + ṁ2x2 = (ṁ1 + ṁ2)x3

Solve for ṁ2

ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 750·(40 − 20)/(20 − 15) = 3000 kg/h

Answer

ṁ2 = 3000 kg/h
2.Mass balance

Answer: W = 1734 kg/h

Tie component: a component that passes through unchanged in one stream (here the solute) links the flows directly. Then the total balance gives the rest.

Solute balance

F·xF = P·xP → P = 2300·16/65 = 566.2 kg/h

Total balance

W = F − P = 2300 − 566.2 = 1734 kg/h

Answer

W = 1734 kg/h
3.Energy balance

Answer: tout = 74.59 °C

Exchanger energy balance: with no losses, ṁhcp,h(Th,in − Th,out) = ṁccp,c(tout − tin).

Heat released by the hot stream

Q = 3·3.6·(130 − 80) = 540 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 10 + 540/(2·4.18) = 74.59 °C

Answer

tout = 74.59 °C
4.Energy balance

Answer: ṁc = 1.595 kg/s

Energy balance with losses: heat released = heat absorbed + heat lost.

Heat released

Qh = 2.5·2.44·(180 − 90) = 549 kW

Heat that reaches the water

Qc = (1 − 0.15)·549 = 466.6 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 466.6/(4.18·70) = 1.595 kg/s

Answer

ṁc = 1.595 kg/s
5.Ideal gas law

Answer: V = 107.8 L

Ideal gas law: PV = nRT, with n = m/M.

Moles

n = 400/32 = 12.5 mol
T = 90 + 273.15 = 363.1 K

Volume

V = nRT/P = 12.5·8.314·363.1/350 = 107.8 L

Answer

V = 107.8 L
6.Ideal gas law

Answer: ρ = 4.365 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 30 + 273.15 = 303.1 K

Density

ρ = 250·44.01/(8.314·303.1) = 4.365 kg/m³

Answer

ρ = 4.365 kg/m³
7.Stoichiometry

Answer: m = 19.89 g

Stoichiometry: a balanced equation counts particles, not grams. Convert mass to moles with n = m/M, use the coefficients of the equation as the mole ratio, then convert back with m = n·M. With two reactants, the limiting reactant is the one that runs out first and fixes how much product forms.

Moles of each reactant

n(Mg) = 12/24.31 = 0.4936 mol
n(O2) = 40/32 = 1.25 mol

Limiting reactant

0.4936/2 = 0.2468, 1.25/1 = 1.25 → Mg

Mass of product

n(MgO) = 0.2468·2 = 0.4936 mol
m = 0.4936·40.3 = 19.89 g

Answer

m = 19.89 g
8.Stoichiometry

Answer: Y = 75.01 %

Stoichiometry: a balanced equation counts particles, not grams. Convert mass to moles with n = m/M, use the coefficients of the equation as the mole ratio, then convert back with m = n·M. With two reactants, the limiting reactant is the one that runs out first and fixes how much product forms.

Theoretical yield

n(Na) = 12/22.99 = 0.522 mol
m = 0.522·2/2·58.44 = 30.5 g

Percent yield

Y = 22.88/30.5·100 = 75.01%

Answer

Y = 75.01%

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