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 1250 kg/h · 30% saltStream 2ṁ₂ = ? · 15% saltProduct20% salt

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

ṁ1 = 250 kg/hx1 = 30 %x2 = 15 %x3 = 20 %
ṁ2 = kg/h
2.Mass balanceHard
EvaporatorFeed F2300 kg/h · 19% saltProduct P40% 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 = 2300 kg/hxF = 19 %xP = 40 %
W = kg/h
3.Energy balanceMedium
hot sidecold sideglycol mix · 180 °C90 °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 = 3 kg/scp,h = 3.6 kJ/(kg·K)Th: 180 → 90 °Cṁc = 3 kg/scp,c = 4.18 kJ/(kg·K)tin = 25 °C
tout = °C
4.Energy balanceHard
hot sidecold sideglycol mix · 160 °C60 °Cwater · 20 °C80 °C · ṁ = ?5% lost

Hot glycol mix 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 = 2 kg/scp,h = 3.6 kJ/(kg·K)Th: 160 → 60 °Ct: 20 → 80 °Ccp,c = 4.18 kJ/(kg·K)loss = 5 %
ṁc = kg/s
5.Ideal gas lawMedium
100 g of N₂V = 50 LT = 75 °CP = ?

A tank of 50 L contains 100 g of N₂ at 75 °C. Treat it as an ideal gas and find the pressure P.

m = 100 gM = 28.01 g/molV = 50 LT = 75 °CR = 8.314 J/(mol·K)
P = kPa
6.Ideal gas lawHard
Ar, M = 39.95 g/molP = 950 kPaT = 180 °Cρ = ?

Find the density ρ of Ar at 950 kPa and 180 °C, treating it as an ideal gas.

M = 39.95 g/molP = 950 kPaT = 180 °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 CH4 and 48 g of O2 react according to CH4 + 2O2 → CO2 + 2H2O. What mass m of H2O can form?

M(CH4) = 16.04M(O2) = 32M(H2O) = 18.02 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 59.32 g of H2O is collected. Find the percent yield Y.

M(CH4) = 16.04 g/molM(H2O) = 18.02 g/molactual: 59.32 g
Y = %

3Solutions

1.Mass balance

Answer: ṁ2 = 500 kg/h

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

Balances

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

Solve for ṁ2

ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 250·(30 − 20)/(20 − 15) = 500 kg/h

Answer

ṁ2 = 500 kg/h
2.Mass balance

Answer: W = 1208 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 = 2300·19/40 = 1093 kg/h

Total balance

W = F − P = 2300 − 1093 = 1208 kg/h

Answer

W = 1208 kg/h
3.Energy balance

Answer: tout = 102.5 °C

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

Heat released by the hot stream

Q = 3·3.6·(180 − 90) = 972 kW

Heat picked up by the water

tout = tin + Q/(ṁccp,c) = 25 + 972/(3·4.18) = 102.5 °C

Answer

tout = 102.5 °C
4.Energy balance

Answer: ṁc = 2.727 kg/s

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

Heat released

Qh = 2·3.6·(160 − 60) = 720 kW

Heat that reaches the water

Qc = (1 − 0.05)·720 = 684 kW

Water flow rate

ṁc = Qc/(cp,c·Δt) = 684/(4.18·60) = 2.727 kg/s

Answer

ṁc = 2.727 kg/s
5.Ideal gas law

Answer: P = 206.7 kPa

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

Moles

n = m/M = 100/28.01 = 3.57 mol
T = 75 + 273.15 = 348.1 K

Pressure

P = nRT/V = 3.57·8.314·348.1/0.05 = 206.7 kPa

Answer

P = 206.7 kPa
6.Ideal gas law

Answer: ρ = 10.07 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 = 950·39.95/(8.314·453.1) = 10.07 kg/m³

Answer

ρ = 10.07 kg/m³
7.Stoichiometry

Answer: m = 27.03 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(CH4) = 30/16.04 = 1.87 mol
n(O2) = 48/32 = 1.5 mol

Limiting reactant

1.87/1 = 1.87, 1.5/2 = 0.75 → O2

Mass of product

n(H2O) = 0.75·2 = 1.5 mol
m = 1.5·18.02 = 27.03 g

Answer

m = 27.03 g
8.Stoichiometry

Answer: Y = 88 %

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 = 59.32/67.41·100 = 88%

Answer

Y = 88%

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