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 75 mm of gypsum board, with an area of 8 m². The outer faces are at 18 °C and 7 °C. Find the temperature Ti at the interface between the two layers.
L1 = 200 mm, k1 = 0.72 W/(m·K)L2 = 75 mm, k2 = 0.17 W/(m·K)A = 8 m²T1 = 18 °CT3 = 7 °C
Ti = °C
2.Heat conductionHard
A wall of 200 mm glass and 150 mm mineral wool separates a room at 20 °C from outside air at -9 °C. The convection coefficients are 6 W/(m²·K) inside and 20 W/(m²·K) outside, and the wall area is 6 m². Find the heat loss Q̇ through the wall.
L1 = 200 mm, k1 = 0.96 W/(m·K)L2 = 150 mm, k2 = 0.04 W/(m·K)A = 6 m²hi = 6 W/(m²·K)ho = 20 W/(m²·K)Troom = 20 °CTout = -9 °C
Q̇ = W
3.Heat exchangers (LMTD)Medium
A counter-flow heat exchanger transfers 40 kW. Its heat transfer area is 5 m². The hot fluid cools from 120 to 40 °C and the cold fluid heats from 15 to 55 °C. Find the overall heat transfer coefficient U.
Q̇ = 40 kWA = 5 m²Th: 120 → 40 °CTc: 15 → 55 °C
U = W/(m²·K)
4.Heat exchangers (LMTD)Hard
Hot oil (0.5 kg/s, cp = 2.3 kJ/(kg·K)) cools from 95 to 55 °C in a counter-flow exchanger. It heats 3 kg/s of water (cp = 4.18 kJ/(kg·K)) that enters at 15 °C. With U = 1000 W/(m²·K), find the heat transfer area A needed.
ṁh = 0.5 kg/scp,h = 2.3 kJ/(kg·K)Th: 95 → 55 °Cṁc = 3 kg/scp,c = 4.18 kJ/(kg·K)Tc,in = 15 °CU = 1000 W/(m²·K)
A = m²
3Solutions
1.Heat conduction
Answer: Ti = 13.75 °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) = (200/1000)/(0.72·8) = 0.03472 K/W
R2 = L2/(k2A) = (75/1000)/(0.17·8) = 0.05515 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 11/0.08987 = 122.4 W
Interface temperature
The temperature drops across the first layer by Q̇·R₁.
Ti = T1 − Q̇·R1 = 18 − 122.4·0.03472 = 13.75 °C
Answer
Ti = 13.75 °C
2.Heat conduction
Answer: Q̇ = 41.68 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.
Resistances in series
Ri = 1/(hiA) = 1/(6·6) = 0.02778 K/W
R1 = L1/(k1A) = (200/1000)/(0.96·6) = 0.03472 K/W
R2 = L2/(k2A) = (150/1000)/(0.04·6) = 0.625 K/W
Ro = 1/(hoA) = 1/(20·6) = 0.008333 K/W
ΣR = 0.6958 K/W
Heat loss
Q̇ = (20 − (-9))/0.6958 = 41.68 W
Answer
Q̇ = 41.68 W
3.Heat exchangers (LMTD)
Answer: U = 191.1 W/(m²·K)
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 = 120 − 55 = 65 K
ΔT₂ = Th,out − Tc,in = 40 − 15 = 25 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (65 − 25)/ln(65/25) = 41.86 K
Overall coefficient
U = Q̇/(A·ΔTlm) = 40000/(5·41.86) = 191.1 W/(m²·K)
Answer
U = 191.1 W/(m²·K)
4.Heat exchangers (LMTD)
Answer: A = 0.8182 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) = 0.5·2.3·40 = 46 kW
Tc,out = 15 + 46/(3·4.18) = 18.67 °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 = 95 − 18.67 = 76.33 K
ΔT₂ = Th,out − Tc,in = 55 − 15 = 40 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (76.33 − 40)/ln(76.33/40) = 56.22 K
Area
A = Q̇/(U·ΔTlm) = 46000/(1000·56.22) = 0.8182 m²
Answer
A = 0.8182 m²
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