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 250 mm of concrete and 150 mm of gypsum board, with an area of 12 m². The outer faces are at 16 °C and -12 °C. Find the temperature Ti at the interface between the two layers.
L1 = 250 mm, k1 = 1.4 W/(m·K)L2 = 150 mm, k2 = 0.17 W/(m·K)A = 12 m²T1 = 16 °CT3 = -12 °C
Ti = °C
2.Heat conductionHard
A wall of 250 mm expanded polystyrene and 150 mm gypsum board separates a room at 20 °C from outside air at 1 °C. The convection coefficients are 10 W/(m²·K) inside and 30 W/(m²·K) outside, and the wall area is 10 m². Find the overall heat transfer coefficient U of the wall.
L1 = 250 mm, k1 = 0.035 W/(m·K)L2 = 150 mm, k2 = 0.17 W/(m·K)A = 10 m²hi = 10 W/(m²·K)ho = 30 W/(m²·K)Troom = 20 °CTout = 1 °C
U = W/(m²·K)
3.Heat exchangers (LMTD)Medium
A counter-flow heat exchanger transfers 120 kW. The hot fluid cools from 135 to 45 °C and the cold fluid heats from 20 to 80 °C. The overall heat transfer coefficient is 500 W/(m²·K). Find the heat transfer area A needed.
Q̇ = 120 kWU = 500 W/(m²·K)Th: 135 → 45 °CTc: 20 → 80 °C
A = m²
4.Heat exchangers (LMTD)Hard
Hot oil (0.5 kg/s, cp = 2.1 kJ/(kg·K)) cools from 125 to 80 °C in a counter-flow exchanger. It heats 0.5 kg/s of water (cp = 4.18 kJ/(kg·K)) that enters at 25 °C. With U = 800 W/(m²·K), find the heat transfer area A needed.
ṁh = 0.5 kg/scp,h = 2.1 kJ/(kg·K)Th: 125 → 80 °Cṁc = 0.5 kg/scp,c = 4.18 kJ/(kg·K)Tc,in = 25 °CU = 800 W/(m²·K)
A = m²
3Solutions
1.Heat conduction
Answer: Ti = 11.29 °C
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) = (250/1000)/(1.4·12) = 0.01488 K/W
R2 = L2/(k2A) = (150/1000)/(0.17·12) = 0.07353 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 28/0.08841 = 316.7 W
Interface temperature
The temperature drops across the first layer by Q̇·R₁.
Ti = T1 − Q̇·R1 = 16 − 316.7·0.01488 = 11.29 °C
Answer
Ti = 11.29 °C
2.Heat conduction
Answer: U = 0.1226 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) = (250/1000)/(0.035·10) = 0.7143 K/W
R2 = L2/(k2A) = (150/1000)/(0.17·10) = 0.08824 K/W
Ro = 1/(hoA) = 1/(30·10) = 0.003333 K/W
ΣR = 0.8159 K/W
Overall coefficient
U = 1/(A·ΣR) = 1/(10·0.8159) = 0.1226 W/(m²·K)
Answer
U = 0.1226 W/(m²·K)
3.Heat exchangers (LMTD)
Answer: A = 6.308 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 = 135 − 80 = 55 K
ΔT₂ = Th,out − Tc,in = 45 − 20 = 25 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (55 − 25)/ln(55/25) = 38.05 K
Area
A = Q̇/(U·ΔTlm) = 120000/(500·38.05) = 6.308 m²
Answer
A = 6.308 m²
4.Heat exchangers (LMTD)
Answer: A = 0.9009 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.
Heat duty and water outlet temperature
Q̇ = ṁhcp,h(Th,in − Th,out) = 0.5·2.1·45 = 47.25 kW
Tc,out = 25 + 47.25/(0.5·4.18) = 47.61 °C
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 = 125 − 47.61 = 77.39 K
ΔT₂ = Th,out − Tc,in = 80 − 25 = 55 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (77.39 − 55)/ln(77.39/55) = 65.56 K
Area
A = Q̇/(U·ΔTlm) = 47250/(800·65.56) = 0.9009 m²
Answer
A = 0.9009 m²
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