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 wood and 50 mm of concrete, with an area of 15 m². The outer faces are at 20 °C and -11 °C. Find the heat flow Q̇ through the wall.
L1 = 250 mm, k1 = 0.17 W/(m·K)L2 = 50 mm, k2 = 1.4 W/(m·K)A = 15 m²T1 = 20 °CT3 = -11 °C
Q̇ = W
2.Heat conductionHard
A wall of 50 mm gypsum board and 150 mm wood separates a room at 20 °C from outside air at 0 °C. The convection coefficients are 7.7 W/(m²·K) inside and 30 W/(m²·K) outside, and the wall area is 12 m². Find the overall heat transfer coefficient U of the wall.
L1 = 50 mm, k1 = 0.17 W/(m·K)L2 = 150 mm, k2 = 0.17 W/(m·K)A = 12 m²hi = 7.7 W/(m²·K)ho = 30 W/(m²·K)Troom = 20 °CTout = 0 °C
U = W/(m²·K)
3.Heat exchangers (LMTD)Medium
A parallel-flow heat exchanger transfers 410 kW. Its heat transfer area is 6 m². The hot fluid cools from 165 to 130 °C and the cold fluid heats from 20 to 70 °C. Find the overall heat transfer coefficient U.
Q̇ = 410 kWA = 6 m²Th: 165 → 130 °CTc: 20 → 70 °C
U = W/(m²·K)
4.Heat exchangers (LMTD)Hard
In a counter-flow double-pipe heat exchanger, hot oil (0.2 kg/s, cp = 2.3 kJ/(kg·K)) flows in the inner tube and cools from 115 to 65 °C. Water (1.5 kg/s, cp = 4.18 kJ/(kg·K)) in the annulus enters at 25 °C. The inner tube has a diameter of 20 mm, and U = 200 W/(m²·K) is based on this diameter. Find the tube length L needed.
ṁh = 0.2 kg/scp,h = 2.3 kJ/(kg·K)Th: 115 → 65 °Cṁc = 1.5 kg/sTc,in = 25 °CD = 20 mmU = 200 W/(m²·K)
L = m
3Solutions
1.Heat conduction
Answer: Q̇ = 308.7 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) = (250/1000)/(0.17·15) = 0.09804 K/W
R2 = L2/(k2A) = (50/1000)/(1.4·15) = 0.002381 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 31/0.1004 = 308.7 W
Answer
Q̇ = 308.7 W
2.Heat conduction
Answer: U = 0.7465 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/(7.7·12) = 0.01082 K/W
R1 = L1/(k1A) = (50/1000)/(0.17·12) = 0.02451 K/W
R2 = L2/(k2A) = (150/1000)/(0.17·12) = 0.07353 K/W
Ro = 1/(hoA) = 1/(30·12) = 0.002778 K/W
ΣR = 0.1116 K/W
Overall coefficient
U = 1/(A·ΣR) = 1/(12·0.1116) = 0.7465 W/(m²·K)
Answer
U = 0.7465 W/(m²·K)
3.Heat exchangers (LMTD)
Answer: U = 709.4 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 = 165 − 20 = 145 K
ΔT₂ = Th,out − Tc,out = 130 − 70 = 60 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (145 − 60)/ln(145/60) = 96.33 K
Overall coefficient
U = Q̇/(A·ΔTlm) = 410000/(6·96.33) = 709.4 W/(m²·K)
Answer
U = 709.4 W/(m²·K)
4.Heat exchangers (LMTD)
Answer: L = 30.39 m
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 = 0.2·2.3·(115 − 65) = 23 kW
Tc,out = 25 + 23/(1.5·4.18) = 28.67 °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 = 115 − 28.67 = 86.33 K
ΔT₂ = Th,out − Tc,in = 65 − 25 = 40 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (86.33 − 40)/ln(86.33/40) = 60.22 K
Area and length
A = 23000/(200·60.22) = 1.91 m²
L = A/(πD) = 1.91/(π·0.02) = 30.39 m
Answer
L = 30.39 m
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