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 1800 kg/h · 50% sugarStream 2ṁ₂ = ? · 15% sugarProduct20% sugar

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

ṁ1 = 800 kg/hx1 = 50 %x2 = 15 %x3 = 20 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F1000 kg/h · 20% sugarProduct PP = ? · 40%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 = 1000 kg/hxF = 20 %xP = 40 %
P = kg/h
3.Energy balanceMedium
hot sidecold sidelight oil · 130 °C80 °Cwater · 30 °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 kg/scp,h = 2.1 kJ/(kg·K)Th: 130 → 80 °Cṁc = 1.5 kg/scp,c = 4.18 kJ/(kg·K)tin = 30 °C
tout = °C
4.Energy balanceHard
hot sidecold sideglycol mix · 140 °C80 °Cwater · 20 °C70 °C · ṁ = ?15% lost

Hot glycol mix 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 = 1.5 kg/scp,h = 3.6 kJ/(kg·K)Th: 140 → 80 °Ct: 20 → 70 °Ccp,c = 4.18 kJ/(kg·K)loss = 15 %
ṁc = kg/s
5.Ideal gas lawMedium
140 g of N₂V = 10 LT = 65 °CP = ?

A tank of 10 L contains 140 g of N₂ at 65 °C. Treat it as an ideal gas and find the pressure P.

m = 140 gM = 28.01 g/molV = 10 LT = 65 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
CH₄, M = 16.04 g/molP = 750 kPaT = 180 °Cρ = ?

Find the density ρ of CH₄ at 750 kPa and 180 °C, treating it as an ideal gas.

M = 16.04 g/molP = 750 kPaT = 180 °CR = 8.314 J/(mol·K)
ρ = kg/m³
7.StoichiometryMedium
m(X)n(X)n(NaCl)m(NaCl)÷ M(X)× 2/ν× M(NaCl)

12 g of Na and 80 g of Cl2 react according to 2Na + Cl2 → 2NaCl. What mass m of NaCl can form?

M(Na) = 22.99M(Cl2) = 70.9M(NaCl) = 58.44 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, 30 g of CH4 reacts completely (the other reactants are in excess) and 57.3 g of H2O is collected. Find the percent yield Y.

M(CH4) = 16.04 g/molM(H2O) = 18.02 g/molactual: 57.3 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 4800 kg/h

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

Balances

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

Solve for ṁ2

ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 800·(50 − 20)/(20 − 15) = 4800 kg/h

Answer

ṁ2 = 4800 kg/h
2.Mass balance

Answer: P = 500 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 = 1000·20/40 = 500 kg/h

Answer

P = 500 kg/h
3.Energy balance

Answer: tout = 46.75 °C

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

Heat released by the hot stream

Q = 1·2.1·(130 − 80) = 105 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 30 + 105/(1.5·4.18) = 46.75 °C

Answer

tout = 46.75 °C
4.Energy balance

Answer: ṁc = 1.318 kg/s

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

Heat released

Qh = 1.5·3.6·(140 − 80) = 324 kW

Heat that reaches the water

Qc = (1 − 0.15)·324 = 275.4 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 275.4/(4.18·50) = 1.318 kg/s

Answer

ṁc = 1.318 kg/s
5.Ideal gas law

Answer: P = 1405 kPa

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

Moles

n = m/M = 140/28.01 = 4.998 mol
T = 65 + 273.15 = 338.1 K

Pressure

P = nRT/V = 4.998·8.314·338.1/0.01 = 1405 kPa

Answer

P = 1405 kPa
6.Ideal gas law

Answer: ρ = 3.193 kg/m³

Ideal gas density: ρ = PM/(RT).

Rearrange the gas law

PV = (m/M)RT → ρ = m/V = PM/(RT)
T = 180 + 273.15 = 453.1 K

Density

ρ = 750·16.04/(8.314·453.1) = 3.193 kg/m³

Answer

ρ = 3.193 kg/m³
7.Stoichiometry

Answer: m = 30.5 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(Na) = 12/22.99 = 0.522 mol
n(Cl2) = 80/70.9 = 1.128 mol

Limiting reactant

0.522/2 = 0.261, 1.128/1 = 1.128 → Na

Mass of product

n(NaCl) = 0.261·2 = 0.522 mol
m = 0.522·58.44 = 30.5 g

Answer

m = 30.5 g
8.Stoichiometry

Answer: Y = 85.01 %

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(CH4) = 30/16.04 = 1.87 mol
m = 1.87·2/1·18.02 = 67.41 g

Percent yield

Y = 57.3/67.41·100 = 85.01%

Answer

Y = 85.01%

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