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 50 mm of gypsum board and 75 mm of brick, with an area of 4 m². The outer faces are at 17 °C and -1 °C. Find the heat flow Q̇ through the wall.
L1 = 50 mm, k1 = 0.17 W/(m·K)L2 = 75 mm, k2 = 0.72 W/(m·K)A = 4 m²T1 = 17 °CT3 = -1 °C
Q̇ = W
2.Heat conductionHard
A wall of 150 mm gypsum board and 25 mm concrete separates a room at 20 °C from outside air at -8 °C. The convection coefficients are 10 W/(m²·K) inside and 20 W/(m²·K) outside, and the wall area is 6 m². Find the temperature Ts of the inside surface of the wall.
L1 = 150 mm, k1 = 0.17 W/(m·K)L2 = 25 mm, k2 = 1.4 W/(m·K)A = 6 m²hi = 10 W/(m²·K)ho = 20 W/(m²·K)Troom = 20 °CTout = -8 °C
Ts = °C
3.Heat exchangers (LMTD)Medium
A counter-flow heat exchanger transfers 90 kW. The hot fluid cools from 80 to 50 °C and the cold fluid heats from 10 to 70 °C. The overall heat transfer coefficient is 300 W/(m²·K). Find the heat transfer area A needed.
Q̇ = 90 kWU = 300 W/(m²·K)Th: 80 → 50 °CTc: 10 → 70 °C
A = m²
4.Heat exchangers (LMTD)Hard
Steam condenses at a constant 140 °C on the outside of the tubes of a condenser with a heat transfer area of 2.5 m² and U = 2000 W/(m²·K). Cooling water (1.5 kg/s, cp = 4.18 kJ/(kg·K)) enters the tubes at 10 °C. Find the water outlet temperature Tc,out.
Ts = 140 °CA = 2.5 m²U = 2000 W/(m²·K)ṁ = 1.5 kg/scp = 4.18 kJ/(kg·K)Tc,in = 10 °C
Tc,out = °C
3Solutions
1.Heat conduction
Answer: Q̇ = 180.8 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) = (50/1000)/(0.17·4) = 0.07353 K/W
R2 = L2/(k2A) = (75/1000)/(0.72·4) = 0.02604 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 18/0.09957 = 180.8 W
Answer
Q̇ = 180.8 W
2.Heat conduction
Answer: Ts = 17.33 °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.
Resistances in series
Ri = 1/(hiA) = 1/(10·6) = 0.01667 K/W
R1 = L1/(k1A) = (150/1000)/(0.17·6) = 0.1471 K/W
R2 = L2/(k2A) = (25/1000)/(1.4·6) = 0.002976 K/W
Ro = 1/(hoA) = 1/(20·6) = 0.008333 K/W
ΣR = 0.175 K/W
Heat loss
Q̇ = (20 − (-8))/0.175 = 160 W
Ts = Troom − Q̇·Ri = 20 − 160·0.01667 = 17.33 °C
Answer
Ts = 17.33 °C
3.Heat exchangers (LMTD)
Answer: A = 13.86 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 = 80 − 70 = 10 K
ΔT₂ = Th,out − Tc,in = 50 − 10 = 40 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (10 − 40)/ln(10/40) = 21.64 K
Area
A = Q̇/(U·ΔTlm) = 90000/(300·21.64) = 13.86 m²
Answer
A = 13.86 m²
4.Heat exchangers (LMTD)
Answer: Tc,out = 81.44 °C
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.
Combine the energy balance and the rate equation
With ΔT₁ = Ts − Tc,in and ΔT₂ = Ts − Tc,out, the difference ΔT₁ − ΔT₂ equals the water temperature rise, so the rise cancels.
ṁcp(ΔT₁ − ΔT₂) = U·A·(ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂)
ln(ΔT₁/ΔT₂) = U·A/(ṁcp) = 2000·2.5/(1.5·4180) = 0.7974
Outlet temperature
ΔT₂ = ΔT₁·e−0.7974 = 130·0.4505 = 58.56 K
Tc,out = 140 − 58.56 = 81.44 °C
Answer
Tc,out = 81.44 °C
Heat transferred: Q̇ = 447.9 kW.
ExamProblems · examproblems.com/courses/heat-transfer