Heat conduction
- Fourier’s law
- Q̇ = kAΔT/L
- Conduction resistance
- R = L/(kA)
- Convection resistance
- R = 1/(hA)
- Resistances in series
- Q̇ = ΔTtotal/ΣR
- U-value
- U = 1/(ΣR·A)
Heat Transfer: cram pack
Formula sheets, the key method steps for every topic and 4 practice problems, with full solutions at the back.
1Formulas and methods
Heat exchangers (LMTD)
- Heat duty
- Q = U·A·ΔTlm
- Log mean temperature difference
- ΔTlm = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2)
- Stream balance
- Q = ṁcp(Tin − Tout)
2Practice problems
1.Heat conductionMedium
A wall is made of 200 mm of brick and 150 mm of concrete, with an area of 20 m². The outer faces are at 18 °C and -7 °C. Find the heat flow Q̇ through the wall.
L1 = 200 mm, k1 = 0.72 W/(m·K)L2 = 150 mm, k2 = 1.4 W/(m·K)A = 20 m²T1 = 18 °CT3 = -7 °C
Q̇ = W
2.Heat conductionHard
A wall of 200 mm wood and 150 mm expanded polystyrene separates a room at 21 °C from outside air at -9 °C. The convection coefficients are 10 W/(m²·K) inside and 20 W/(m²·K) outside, and the wall area is 10 m². Find the overall heat transfer coefficient U of the wall.
L1 = 200 mm, k1 = 0.17 W/(m·K)L2 = 150 mm, k2 = 0.035 W/(m·K)A = 10 m²hi = 10 W/(m²·K)ho = 20 W/(m²·K)Troom = 21 °CTout = -9 °C
U = W/(m²·K)
3.Heat exchangers (LMTD)Medium
A counter-flow heat exchanger transfers 170 kW. The hot fluid cools from 180 to 120 °C and the cold fluid heats from 10 to 105 °C. The overall heat transfer coefficient is 1000 W/(m²·K). Find the heat transfer area A needed.
Q̇ = 170 kWU = 1000 W/(m²·K)Th: 180 → 120 °CTc: 10 → 105 °C
A = m²
4.Heat exchangers (LMTD)Hard
Steam condenses at a constant 105 °C on the outside of the tubes of a condenser with a heat transfer area of 1.5 m² and U = 2000 W/(m²·K). Cooling water (1.5 kg/s, cp = 4.18 kJ/(kg·K)) enters the tubes at 30 °C. Find the water outlet temperature Tc,out.
Ts = 105 °CA = 1.5 m²U = 2000 W/(m²·K)ṁ = 1.5 kg/scp = 4.18 kJ/(kg·K)Tc,in = 30 °C
Tc,out = °C
3Solutions
1.Heat conduction
Answer: Q̇ = 1299 W
Thermal resistance: steady heat flow through layers in series behaves like current through resistors in series: Q̇ = ΔT/ΣR, with R = L/(kA) for a conducting layer and R = 1/(hA) for a convecting surface. The temperature drop across each layer is proportional to its resistance.
Layer resistances
R1 = L1/(k1A) = (200/1000)/(0.72·20) = 0.01389 K/W
R2 = L2/(k2A) = (150/1000)/(1.4·20) = 0.005357 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 25/0.01925 = 1299 W
Answer
Q̇ = 1299 W
2.Heat conduction
Answer: U = 0.1782 W/(m²·K)
Thermal resistance: steady heat flow through layers in series behaves like current through resistors in series: Q̇ = ΔT/ΣR, with R = L/(kA) for a conducting layer and R = 1/(hA) for a convecting surface. The temperature drop across each layer is proportional to its resistance.
Resistances in series
Ri = 1/(hiA) = 1/(10·10) = 0.01 K/W
R1 = L1/(k1A) = (200/1000)/(0.17·10) = 0.1176 K/W
R2 = L2/(k2A) = (150/1000)/(0.035·10) = 0.4286 K/W
Ro = 1/(hoA) = 1/(20·10) = 0.005 K/W
ΣR = 0.5612 K/W
Overall coefficient
U = 1/(A·ΣR) = 1/(10·0.5612) = 0.1782 W/(m²·K)
Answer
U = 0.1782 W/(m²·K)
3.Heat exchangers (LMTD)
Answer: A = 1.86 m²
LMTD method: the temperature difference between the streams changes along the exchanger, so the rate equation uses its logarithmic mean: Q̇ = U·A·ΔTlm, with ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), where ΔT₁ and ΔT₂ are the differences at the two ends. Counter-flow gives a larger ΔTlm than parallel flow for the same inlet and outlet temperatures.
End temperature differences
In counter-flow the hot inlet faces the cold outlet, and the hot outlet faces the cold inlet.
ΔT₁ = Th,in − Tc,out = 180 − 105 = 75 K
ΔT₂ = Th,out − Tc,in = 120 − 10 = 110 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (75 − 110)/ln(75/110) = 91.39 K
Area
A = Q̇/(U·ΔTlm) = 170000/(1000·91.39) = 1.86 m²
Answer
A = 1.86 m²
4.Heat exchangers (LMTD)
Answer: Tc,out = 58.52 °C
LMTD method: the temperature difference between the streams changes along the exchanger, so the rate equation uses its logarithmic mean: Q̇ = U·A·ΔTlm, with ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), where ΔT₁ and ΔT₂ are the differences at the two ends. Counter-flow gives a larger ΔTlm than parallel flow for the same inlet and outlet temperatures.
Combine the energy balance and the rate equation
With ΔT₁ = Ts − Tc,in and ΔT₂ = Ts − Tc,out, the difference ΔT₁ − ΔT₂ equals the water temperature rise, so the rise cancels.
ṁcp(ΔT₁ − ΔT₂) = U·A·(ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂)
ln(ΔT₁/ΔT₂) = U·A/(ṁcp) = 2000·1.5/(1.5·4180) = 0.4785
Outlet temperature
ΔT₂ = ΔT₁·e−0.4785 = 75·0.6197 = 46.48 K
Tc,out = 105 − 46.48 = 58.52 °C
Answer
Tc,out = 58.52 °C
Heat transferred: Q̇ = 178.8 kW.
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