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 concrete and 50 mm of gypsum board, with an area of 10 m². The outer faces are at 19 °C and -3 °C. Find the heat flow Q̇ through the wall.
L1 = 200 mm, k1 = 1.4 W/(m·K)L2 = 50 mm, k2 = 0.17 W/(m·K)A = 10 m²T1 = 19 °CT3 = -3 °C
Q̇ = W
2.Heat conductionHard
A wall of 50 mm mineral wool and 50 mm concrete separates a room at 22 °C from outside air at -10 °C. The convection coefficients are 7.7 W/(m²·K) inside and 20 W/(m²·K) outside, and the wall area is 20 m². Find the heat loss Q̇ through the wall.
L1 = 50 mm, k1 = 0.04 W/(m·K)L2 = 50 mm, k2 = 1.4 W/(m·K)A = 20 m²hi = 7.7 W/(m²·K)ho = 20 W/(m²·K)Troom = 22 °CTout = -10 °C
Q̇ = W
3.Heat exchangers (LMTD)Medium
A shell-and-tube heat exchanger (one shell pass, two tube passes) must transfer 180 kW. The hot fluid cools from 150 to 50 °C and the cold fluid heats from 10 to 20 °C. The LMTD correction factor for this arrangement is F = 0.8, and U = 500 W/(m²·K). Find the heat transfer area A needed.
Q̇ = 180 kWF = 0.8U = 500 W/(m²·K)Th: 150 → 50 °CTc: 10 → 20 °C
A = m²
4.Heat exchangers (LMTD)Hard
Hot oil (1.5 kg/s, cp = 2.3 kJ/(kg·K)) cools from 140 to 100 °C in a counter-flow exchanger. It heats 0.5 kg/s of water (cp = 4.18 kJ/(kg·K)) that enters at 15 °C. With U = 300 W/(m²·K), find the heat transfer area A needed.
ṁh = 1.5 kg/scp,h = 2.3 kJ/(kg·K)Th: 140 → 100 °Cṁc = 0.5 kg/scp,c = 4.18 kJ/(kg·K)Tc,in = 15 °CU = 300 W/(m²·K)
A = m²
3Solutions
1.Heat conduction
Answer: Q̇ = 503.5 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)/(1.4·10) = 0.01429 K/W
R2 = L2/(k2A) = (50/1000)/(0.17·10) = 0.02941 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 22/0.0437 = 503.5 W
Answer
Q̇ = 503.5 W
2.Heat conduction
Answer: Q̇ = 436.7 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.
Resistances in series
Ri = 1/(hiA) = 1/(7.7·20) = 0.006494 K/W
R1 = L1/(k1A) = (50/1000)/(0.04·20) = 0.0625 K/W
R2 = L2/(k2A) = (50/1000)/(1.4·20) = 0.001786 K/W
Ro = 1/(hoA) = 1/(20·20) = 0.0025 K/W
ΣR = 0.07328 K/W
Heat loss
Q̇ = (22 − (-10))/0.07328 = 436.7 W
Answer
Q̇ = 436.7 W
3.Heat exchangers (LMTD)
Answer: A = 5.893 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 (counter-flow)
In counter-flow the hot inlet faces the cold outlet, and the hot outlet faces the cold inlet.
ΔT₁ = Th,in − Tc,out = 150 − 20 = 130 K
ΔT₂ = Th,out − Tc,in = 50 − 10 = 40 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (130 − 40)/ln(130/40) = 76.36 K
Area
A = Q̇/(U·F·ΔTlm,cf) = 180000/(500·0.8·76.36) = 5.893 m²
Answer
A = 5.893 m²
4.Heat exchangers (LMTD)
Answer: A = 6.461 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) = 1.5·2.3·40 = 138 kW
Tc,out = 15 + 138/(0.5·4.18) = 81.03 °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 = 140 − 81.03 = 58.97 K
ΔT₂ = Th,out − Tc,in = 100 − 15 = 85 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (58.97 − 85)/ln(58.97/85) = 71.19 K
Area
A = Q̇/(U·ΔTlm) = 138000/(300·71.19) = 6.461 m²
Answer
A = 6.461 m²
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