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 150 mm of gypsum board and 150 mm of glass, with an area of 6 m². The outer faces are at 19 °C and 6 °C. Find the temperature Ti at the interface between the two layers.
L1 = 150 mm, k1 = 0.17 W/(m·K)L2 = 150 mm, k2 = 0.96 W/(m·K)A = 6 m²T1 = 19 °CT3 = 6 °C
Ti = °C
2.Heat conductionHard
A wall of 100 mm mineral wool and 50 mm brick separates a room at 20 °C from outside air at -5 °C. The convection coefficients are 10 W/(m²·K) inside and 20 W/(m²·K) outside, and the wall area is 12 m². Find the overall heat transfer coefficient U of the wall.
L1 = 100 mm, k1 = 0.04 W/(m·K)L2 = 50 mm, k2 = 0.72 W/(m·K)A = 12 m²hi = 10 W/(m²·K)ho = 20 W/(m²·K)Troom = 20 °CTout = -5 °C
U = W/(m²·K)
3.Heat exchangers (LMTD)Medium
A parallel-flow heat exchanger transfers 110 kW. Its heat transfer area is 3 m². The hot fluid cools from 135 to 105 °C and the cold fluid heats from 30 to 85 °C. Find the overall heat transfer coefficient U.
Q̇ = 110 kWA = 3 m²Th: 135 → 105 °CTc: 30 → 85 °C
U = W/(m²·K)
4.Heat exchangers (LMTD)Hard
Hot oil (1.5 kg/s, cp = 2.1 kJ/(kg·K)) cools from 170 to 35 °C in a counter-flow exchanger. It heats 1 kg/s of water (cp = 4.18 kJ/(kg·K)) that enters at 10 °C. With U = 800 W/(m²·K), find the heat transfer area A needed.
ṁh = 1.5 kg/scp,h = 2.1 kJ/(kg·K)Th: 170 → 35 °Cṁc = 1 kg/scp,c = 4.18 kJ/(kg·K)Tc,in = 10 °CU = 800 W/(m²·K)
A = m²
3Solutions
1.Heat conduction
Answer: Ti = 7.956 °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) = (150/1000)/(0.17·6) = 0.1471 K/W
R2 = L2/(k2A) = (150/1000)/(0.96·6) = 0.02604 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 13/0.1731 = 75.1 W
Interface temperature
The temperature drops across the first layer by Q̇·R₁.
Ti = T1 − Q̇·R1 = 19 − 75.1·0.1471 = 7.956 °C
Answer
Ti = 7.956 °C
2.Heat conduction
Answer: U = 0.3677 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/(10·12) = 0.008333 K/W
R1 = L1/(k1A) = (100/1000)/(0.04·12) = 0.2083 K/W
R2 = L2/(k2A) = (50/1000)/(0.72·12) = 0.005787 K/W
Ro = 1/(hoA) = 1/(20·12) = 0.004167 K/W
ΣR = 0.2266 K/W
Overall coefficient
U = 1/(A·ΣR) = 1/(12·0.2266) = 0.3677 W/(m²·K)
Answer
U = 0.3677 W/(m²·K)
3.Heat exchangers (LMTD)
Answer: U = 715.3 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 parallel flow both inlets are at the same end, and both outlets at the other.
ΔT₁ = Th,in − Tc,in = 135 − 30 = 105 K
ΔT₂ = Th,out − Tc,out = 105 − 85 = 20 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (105 − 20)/ln(105/20) = 51.26 K
Overall coefficient
U = Q̇/(A·ΔTlm) = 110000/(3·51.26) = 715.3 W/(m²·K)
Answer
U = 715.3 W/(m²·K)
4.Heat exchangers (LMTD)
Answer: A = 13.52 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) = 1.5·2.1·135 = 425.3 kW
Tc,out = 10 + 425.3/(1·4.18) = 111.7 °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 = 170 − 111.7 = 58.27 K
ΔT₂ = Th,out − Tc,in = 35 − 10 = 25 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (58.27 − 25)/ln(58.27/25) = 39.31 K
Area
A = Q̇/(U·ΔTlm) = 425250/(800·39.31) = 13.52 m²
Answer
A = 13.52 m²
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