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 1650 kg/h · 15% NaOHStream 2ṁ₂ = ? · 40% NaOHProduct30% NaOH

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 30% NaOH? Find ṁ2.

ṁ1 = 650 kg/hx1 = 15 %x2 = 40 %x3 = 30 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F1900 kg/h · 18% sugarProduct PP = ? · 45%Vapour W (pure water)0% sugar

An evaporator concentrates a sugar solution by boiling off pure water. Find the product flow P in kg/h.

F = 1900 kg/hxF = 18 %xP = 45 %
P = kg/h
3.Energy balanceMedium
hot sidecold sideethanol · 180 °C80 °Cwater · 10 °Ct = ? °C

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

ṁh = 2 kg/scp,h = 2.44 kJ/(kg·K)Th: 180 → 80 °Cṁc = 3 kg/scp,c = 4.18 kJ/(kg·K)tin = 10 °C
tout = °C
4.Energy balanceHard
hot sidecold sideethanol · 140 °C70 °Cwater · 15 °C90 °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 = 3 kg/scp,h = 2.44 kJ/(kg·K)Th: 140 → 70 °Ct: 15 → 90 °Ccp,c = 4.18 kJ/(kg·K)loss = 15 %
ṁc = kg/s
5.Ideal gas lawMedium
230 g of HeV = 30 LT = 30 °CP = ?

A tank of 30 L contains 230 g of He at 30 °C. Treat it as an ideal gas and find the pressure P.

m = 230 gM = 4.003 g/molV = 30 LT = 30 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
N₂, M = 28.01 g/molP = 650 kPaT = 140 °Cρ = ?

Find the density ρ of N₂ at 650 kPa and 140 °C, treating it as an ideal gas.

M = 28.01 g/molP = 650 kPaT = 140 °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)

30 g of H2 and 20 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(CH4​)n(CH4​)n(CO2​)m(CO2​)÷ M(CH4​)× 1/1× M(CO2​)

In the reaction CH4 + 2O2 → CO2 + 2H2O, 10 g of CH4 reacts completely (the other reactants are in excess) and 21.4 g of CO2 is collected. Find the percent yield Y.

M(CH4) = 16.04 g/molM(CO2) = 44.01 g/molactual: 21.4 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 975 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) = 650·(15 − 30)/(30 − 40) = 975 kg/h

Answer

ṁ2 = 975 kg/h
2.Mass balance

Answer: P = 760 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 = 1900·18/45 = 760 kg/h

Answer

P = 760 kg/h
3.Energy balance

Answer: tout = 48.92 °C

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

Heat released by the hot stream

Q = 2·2.44·(180 − 80) = 488 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 10 + 488/(3·4.18) = 48.92 °C

Answer

tout = 48.92 °C
4.Energy balance

Answer: ṁc = 1.389 kg/s

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

Heat released

Qh = 3·2.44·(140 − 70) = 512.4 kW

Heat that reaches the water

Qc = (1 − 0.15)·512.4 = 435.5 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 435.5/(4.18·75) = 1.389 kg/s

Answer

ṁc = 1.389 kg/s
5.Ideal gas law

Answer: P = 4827 kPa

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

Moles

n = m/M = 230/4.003 = 57.46 mol
T = 30 + 273.15 = 303.1 K

Pressure

P = nRT/V = 57.46·8.314·303.1/0.03 = 4827 kPa

Answer

P = 4827 kPa
6.Ideal gas law

Answer: ρ = 5.3 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 140 + 273.15 = 413.1 K

Density

ρ = 650·28.01/(8.314·413.1) = 5.3 kg/m³

Answer

ρ = 5.3 kg/m³
7.Stoichiometry

Answer: m = 22.52 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) = 30/2.016 = 14.88 mol
n(O2) = 20/32 = 0.625 mol

Limiting reactant

14.88/2 = 7.44, 0.625/1 = 0.625 → O2

Mass of product

n(H2O) = 0.625·2 = 1.25 mol
m = 1.25·18.02 = 22.52 g

Answer

m = 22.52 g
8.Stoichiometry

Answer: Y = 78 %

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(CH4) = 10/16.04 = 0.6234 mol
m = 0.6234·1/1·44.01 = 27.44 g

Percent yield

Y = 21.4/27.44·100 = 78%

Answer

Y = 78%

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