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 · 45% NaOHStream 2ṁ₂ = ? · 20% NaOHProduct25% NaOH

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 25% NaOH? Find ṁ2.

ṁ1 = 650 kg/hx1 = 45 %x2 = 20 %x3 = 25 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F1500 kg/h · 17% NaOHProduct P55% NaOHVapour W (pure water)W = ? kg/h

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

F = 1500 kg/hxF = 17 %xP = 55 %
W = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 120 °C90 °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: 120 → 90 °Cṁc = 1 kg/scp,c = 4.18 kJ/(kg·K)tin = 15 °C
tout = °C
4.Energy balanceHard
hot sidecold sidelight oil · 160 °C70 °Cwater · 20 °C55 °C · ṁ = ?5% lost

Hot light oil heats water in an exchanger that loses 5% 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: 160 → 70 °Ct: 20 → 55 °Ccp,c = 4.18 kJ/(kg·K)loss = 5 %
ṁc = kg/s
5.Ideal gas lawMedium
430 g of CH₄V = 20 LT = 85 °CP = ?

A tank of 20 L contains 430 g of CH₄ at 85 °C. Treat it as an ideal gas and find the pressure P.

m = 430 gM = 16.04 g/molV = 20 LT = 85 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
He, M = 4.003 g/molP = 700 kPaT = 100 °Cρ = ?

Find the density ρ of He at 700 kPa and 100 °C, treating it as an ideal gas.

M = 4.003 g/molP = 700 kPaT = 100 °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)

20 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(N2​)n(N2​)n(NH3​)m(NH3​)÷ M(N2​)× 2/1× M(NH3​)

In the reaction N2 + 3H2 → 2NH3, 20 g of N2 reacts completely (the other reactants are in excess) and 18.96 g of NH3 is collected. Find the percent yield Y.

M(N2) = 28.02 g/molM(NH3) = 17.03 g/molactual: 18.96 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 2600 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·(45 − 25)/(25 − 20) = 2600 kg/h

Answer

ṁ2 = 2600 kg/h
2.Mass balance

Answer: W = 1036 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 = 1500·17/55 = 463.6 kg/h

Total balance

W = F − P = 1500 − 463.6 = 1036 kg/h

Answer

W = 1036 kg/h
3.Energy balance

Answer: tout = 27.92 °C

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

Heat released by the hot stream

Q = 0.5·3.6·(120 − 90) = 54 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 15 + 54/(1·4.18) = 27.92 °C

Answer

tout = 27.92 °C
4.Energy balance

Answer: ṁc = 1.841 kg/s

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

Heat released

Qh = 1.5·2.1·(160 − 70) = 283.5 kW

Heat that reaches the water

Qc = (1 − 0.05)·283.5 = 269.3 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 269.3/(4.18·35) = 1.841 kg/s

Answer

ṁc = 1.841 kg/s
5.Ideal gas law

Answer: P = 3991 kPa

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

Moles

n = m/M = 430/16.04 = 26.81 mol
T = 85 + 273.15 = 358.1 K

Pressure

P = nRT/V = 26.81·8.314·358.1/0.02 = 3991 kPa

Answer

P = 3991 kPa
6.Ideal gas law

Answer: ρ = 0.9032 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 100 + 273.15 = 373.1 K

Density

ρ = 700·4.003/(8.314·373.1) = 0.9032 kg/m³

Answer

ρ = 0.9032 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) = 20/2.016 = 9.921 mol
n(O2) = 40/32 = 1.25 mol

Limiting reactant

9.921/2 = 4.96, 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 = 77.99 %

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(N2) = 20/28.02 = 0.7138 mol
m = 0.7138·2/1·17.03 = 24.31 g

Percent yield

Y = 18.96/24.31·100 = 77.99%

Answer

Y = 77.99%

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