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 1350 kg/h · 5% sugarStream 2ṁ₂ = ? · 35% sugarProduct25% sugar

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

ṁ1 = 350 kg/hx1 = 5 %x2 = 35 %x3 = 25 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F3800 kg/h · 9% sugarProduct PP = ? · 60%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 = 3800 kg/hxF = 9 %xP = 60 %
P = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 180 °C140 °Cwater · 25 °Ct = ? °C

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

ṁh = 2.5 kg/scp,h = 3.6 kJ/(kg·K)Th: 180 → 140 °Cṁc = 1 kg/scp,c = 4.18 kJ/(kg·K)tin = 25 °C
tout = °C
4.Energy balanceHard
hot sidecold sidelight oil · 140 °C80 °Cwater · 10 °C45 °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 = 3 kg/scp,h = 2.1 kJ/(kg·K)Th: 140 → 80 °Ct: 10 → 45 °Ccp,c = 4.18 kJ/(kg·K)loss = 5 %
ṁc = kg/s
5.Ideal gas lawMedium
500 g of ArV = 10 LT = 60 °CP = ?

A tank of 10 L contains 500 g of Ar at 60 °C. Treat it as an ideal gas and find the pressure P.

m = 500 gM = 39.95 g/molV = 10 LT = 60 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
State 1State 2

A fixed amount of CO₂ goes from 300 kPa, 20 L, 10 °C to 450 kPa and 60 °C. Find the new volume V2.

P1 = 300 kPaV1 = 20 LT1 = 10 °CP2 = 450 kPaT2 = 60 °C
V2 = L
7.StoichiometryMedium
m(X)n(X)n(NaCl)m(NaCl)÷ M(X)× 2/ν× M(NaCl)

20 g of Na and 10 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(CaCO3​)n(CaCO3​)n(CaO)m(CaO)÷ M(CaCO3​)× 1/1× M(CaO)

In the reaction CaCO3 → CaO + CO2, 16 g of CaCO3 reacts completely (the other reactants are in excess) and 4.93 g of CaO is collected. Find the percent yield Y.

M(CaCO3) = 100.09 g/molM(CaO) = 56.08 g/molactual: 4.93 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
sugar: ṁ1x1 + ṁ2x2 = (ṁ1 + ṁ2)x3

Solve for ṁ2

ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 350·(5 − 25)/(25 − 35) = 700 kg/h

Answer

ṁ2 = 700 kg/h
2.Mass balance

Answer: P = 570 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 = 3800·9/60 = 570 kg/h

Answer

P = 570 kg/h
3.Energy balance

Answer: tout = 111.1 °C

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

Heat released by the hot stream

Q = 2.5·3.6·(180 − 140) = 360 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 25 + 360/(1·4.18) = 111.1 °C

Answer

tout = 111.1 °C
4.Energy balance

Answer: ṁc = 2.455 kg/s

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

Heat released

Qh = 3·2.1·(140 − 80) = 378 kW

Heat that reaches the water

Qc = (1 − 0.05)·378 = 359.1 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 359.1/(4.18·35) = 2.455 kg/s

Answer

ṁc = 2.455 kg/s
5.Ideal gas law

Answer: P = 3467 kPa

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

Moles

n = m/M = 500/39.95 = 12.52 mol
T = 60 + 273.15 = 333.1 K

Pressure

P = nRT/V = 12.52·8.314·333.1/0.01 = 3467 kPa

Answer

P = 3467 kPa
6.Ideal gas law

Answer: V2 = 15.69 L

Combined gas law: for a fixed amount of ideal gas, P1V1/T1 = P2V2/T2.

Absolute temperatures

T1 = 283.1 K, T2 = 333.1 K

Combined gas law

V2 = V1·(P1/P2)·(T2/T1) = 20·(300/450)·(333.1/283.1) = 15.69 L

Answer

V2 = 15.69 L
7.Stoichiometry

Answer: m = 16.49 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) = 20/22.99 = 0.8699 mol
n(Cl2) = 10/70.9 = 0.141 mol

Limiting reactant

0.8699/2 = 0.435, 0.141/1 = 0.141 → Cl2

Mass of product

n(NaCl) = 0.141·2 = 0.2821 mol
m = 0.2821·58.44 = 16.49 g

Answer

m = 16.49 g
8.Stoichiometry

Answer: Y = 54.99 %

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(CaCO3) = 16/100.09 = 0.1599 mol
m = 0.1599·1/1·56.08 = 8.965 g

Percent yield

Y = 4.93/8.965·100 = 54.99%

Answer

Y = 54.99%

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