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 1700 kg/h · 30% NaOHStream 2ṁ₂ = ? · 10% 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 = 700 kg/hx1 = 30 %x2 = 10 %x3 = 20 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F2300 kg/h · 16% sugarProduct P30% 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 = 30 %
W = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 160 °C110 °Cwater · 15 °Ct = ? °C

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

ṁh = 0.5 kg/scp,h = 3.6 kJ/(kg·K)Th: 160 → 110 °Cṁc = 1 kg/scp,c = 4.18 kJ/(kg·K)tin = 15 °C
tout = °C
4.Energy balanceHard
hot sidecold sidelight oil · 180 °C140 °Cwater · 10 °C85 °C · ṁ = ?10% lost

Hot light oil heats water in an exchanger that loses 10% of the heat released by the hot stream to the surroundings. Find the water flow rate ṁc needed.

ṁh = 1.5 kg/scp,h = 2.1 kJ/(kg·K)Th: 180 → 140 °Ct: 10 → 85 °Ccp,c = 4.18 kJ/(kg·K)loss = 10 %
ṁc = kg/s
5.Ideal gas lawMedium
220 g of ArP = 200 kPaT = 40 °CV = ?

What volume V does 220 g of Ar occupy at 200 kPa and 40 °C? Treat it as an ideal gas.

m = 220 gM = 39.95 g/molP = 200 kPaT = 40 °CR = 8.314 J/(mol·K)
V = L
6.Ideal gas lawHard
CH₄, M = 16.04 g/molP = 1000 kPaT = 120 °Cρ = ?

Find the density ρ of CH₄ at 1000 kPa and 120 °C, treating it as an ideal gas.

M = 16.04 g/molP = 1000 kPaT = 120 °CR = 8.314 J/(mol·K)
ρ = kg/m³
7.StoichiometryMedium
m(X)n(X)n(H2​O)m(H2​O)÷ M(X)× 2/ν× M(H2​O)

24 g of H2 and 40 g of O2 react according to 2H2 + O2 → 2H2O. What mass m of H2O can form?

M(H2) = 2.016M(O2) = 32M(H2O) = 18.02 g/mol
m = g
8.StoichiometryHard
m(Al)n(Al)n(AlCl3​)m(AlCl3​)÷ M(Al)× 2/2× M(AlCl3​)

In the reaction 2Al + 3Cl2 → 2AlCl3, 10 g of Al reacts completely (the other reactants are in excess) and 40.52 g of AlCl3 is collected. Find the percent yield Y.

M(Al) = 26.98 g/molM(AlCl3) = 133.33 g/molactual: 40.52 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 700 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) = 700·(30 − 20)/(20 − 10) = 700 kg/h

Answer

ṁ2 = 700 kg/h
2.Mass balance

Answer: W = 1073 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 = 2300·16/30 = 1227 kg/h

Total balance

W = F − P = 2300 − 1227 = 1073 kg/h

Answer

W = 1073 kg/h
3.Energy balance

Answer: tout = 36.53 °C

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

Heat released by the hot stream

Q = 0.5·3.6·(160 − 110) = 90 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 15 + 90/(1·4.18) = 36.53 °C

Answer

tout = 36.53 °C
4.Energy balance

Answer: ṁc = 0.3617 kg/s

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

Heat released

Qh = 1.5·2.1·(180 − 140) = 126 kW

Heat that reaches the water

Qc = (1 − 0.1)·126 = 113.4 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 113.4/(4.18·75) = 0.3617 kg/s

Answer

ṁc = 0.3617 kg/s
5.Ideal gas law

Answer: V = 71.69 L

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

Moles

n = 220/39.95 = 5.507 mol
T = 40 + 273.15 = 313.1 K

Volume

V = nRT/P = 5.507·8.314·313.1/200 = 71.69 L

Answer

V = 71.69 L
6.Ideal gas law

Answer: ρ = 4.907 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 120 + 273.15 = 393.1 K

Density

ρ = 1000·16.04/(8.314·393.1) = 4.907 kg/m³

Answer

ρ = 4.907 kg/m³
7.Stoichiometry

Answer: m = 45.05 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(H2) = 24/2.016 = 11.9 mol
n(O2) = 40/32 = 1.25 mol

Limiting reactant

11.9/2 = 5.952, 1.25/1 = 1.25 → O2

Mass of product

n(H2O) = 1.25·2 = 2.5 mol
m = 2.5·18.02 = 45.05 g

Answer

m = 45.05 g
8.Stoichiometry

Answer: Y = 81.99 %

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(Al) = 10/26.98 = 0.3706 mol
m = 0.3706·2/2·133.33 = 49.42 g

Percent yield

Y = 40.52/49.42·100 = 81.99%

Answer

Y = 81.99%

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