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 1850 kg/h · 30% NaOHStream 2ṁ₂ = ? · 10% NaOHProduct15% NaOH

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 15% NaOH? Find ṁ2.

ṁ1 = 850 kg/hx1 = 30 %x2 = 10 %x3 = 15 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F4900 kg/h · 20% saltProduct P30% saltVapour W (pure water)W = ? kg/h

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

F = 4900 kg/hxF = 20 %xP = 30 %
W = kg/h
3.Energy balanceMedium
hot sidecold sidelight oil · 150 °C100 °Cwater · 20 °Ct = ? °C

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

ṁh = 1.5 kg/scp,h = 2.1 kJ/(kg·K)Th: 150 → 100 °Cṁc = 0.5 kg/scp,c = 4.18 kJ/(kg·K)tin = 20 °C
tout = °C
4.Energy balanceHard
hot sidecold sideethanol · 130 °C100 °Cwater · 30 °C70 °C · ṁ = ?5% lost

Hot ethanol 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 kg/scp,h = 2.44 kJ/(kg·K)Th: 130 → 100 °Ct: 30 → 70 °Ccp,c = 4.18 kJ/(kg·K)loss = 5 %
ṁc = kg/s
5.Ideal gas lawMedium
280 g of CO₂P = 150 kPaT = 45 °CV = ?

What volume V does 280 g of CO₂ occupy at 150 kPa and 45 °C? Treat it as an ideal gas.

m = 280 gM = 44.01 g/molP = 150 kPaT = 45 °CR = 8.314 J/(mol·K)
V = L
6.Ideal gas lawHard
O₂, M = 32 g/molP = 350 kPaT = 40 °Cρ = ?

Find the density ρ of O₂ at 350 kPa and 40 °C, treating it as an ideal gas.

M = 32 g/molP = 350 kPaT = 40 °CR = 8.314 J/(mol·K)
ρ = kg/m³
7.StoichiometryMedium
m(X)n(X)n(AlCl3​)m(AlCl3​)÷ M(X)× 2/ν× M(AlCl3​)

30 g of Al and 32 g of Cl2 react according to 2Al + 3Cl2 → 2AlCl3. What mass m of AlCl3 can form?

M(Al) = 26.98M(Cl2) = 70.9M(AlCl3) = 133.33 g/mol
m = g
8.StoichiometryHard
m(CH4​)n(CH4​)n(H2​O)m(H2​O)÷ M(CH4​)× 2/1× M(H2​O)

In the reaction CH4 + 2O2 → CO2 + 2H2O, 6 g of CH4 reacts completely (the other reactants are in excess) and 7.41 g of H2O is collected. Find the percent yield Y.

M(CH4) = 16.04 g/molM(H2O) = 18.02 g/molactual: 7.41 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 2550 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) = 850·(30 − 15)/(15 − 10) = 2550 kg/h

Answer

ṁ2 = 2550 kg/h
2.Mass balance

Answer: W = 1633 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 = 4900·20/30 = 3267 kg/h

Total balance

W = F − P = 4900 − 3267 = 1633 kg/h

Answer

W = 1633 kg/h
3.Energy balance

Answer: tout = 95.36 °C

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

Heat released by the hot stream

Q = 1.5·2.1·(150 − 100) = 157.5 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 20 + 157.5/(0.5·4.18) = 95.36 °C

Answer

tout = 95.36 °C
4.Energy balance

Answer: ṁc = 0.4159 kg/s

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

Heat released

Qh = 1·2.44·(130 − 100) = 73.2 kW

Heat that reaches the water

Qc = (1 − 0.05)·73.2 = 69.54 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 69.54/(4.18·40) = 0.4159 kg/s

Answer

ṁc = 0.4159 kg/s
5.Ideal gas law

Answer: V = 112.2 L

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

Moles

n = 280/44.01 = 6.362 mol
T = 45 + 273.15 = 318.1 K

Volume

V = nRT/P = 6.362·8.314·318.1/150 = 112.2 L

Answer

V = 112.2 L
6.Ideal gas law

Answer: ρ = 4.302 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 40 + 273.15 = 313.1 K

Density

ρ = 350·32/(8.314·313.1) = 4.302 kg/m³

Answer

ρ = 4.302 kg/m³
7.Stoichiometry

Answer: m = 40.12 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(Al) = 30/26.98 = 1.112 mol
n(Cl2) = 32/70.9 = 0.4513 mol

Limiting reactant

1.112/2 = 0.556, 0.4513/3 = 0.1504 → Cl2

Mass of product

n(AlCl3) = 0.1504·2 = 0.3009 mol
m = 0.3009·133.33 = 40.12 g

Answer

m = 40.12 g
8.Stoichiometry

Answer: Y = 54.97 %

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) = 6/16.04 = 0.3741 mol
m = 0.3741·2/1·18.02 = 13.48 g

Percent yield

Y = 7.41/13.48·100 = 54.97%

Answer

Y = 54.97%

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