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 concrete and 50 mm of gypsum board, with an area of 4 m². The outer faces are at 18 °C and 4 °C. Find the heat flow Q̇ through the wall.
L1 = 50 mm, k1 = 1.4 W/(m·K)L2 = 50 mm, k2 = 0.17 W/(m·K)A = 4 m²T1 = 18 °CT3 = 4 °C
Q̇ = W
2.Heat conductionHard
A wall of 250 mm mineral wool and 25 mm expanded polystyrene separates a room at 22 °C from outside air at -14 °C. The convection coefficients are 7.7 W/(m²·K) inside and 25 W/(m²·K) outside, and the wall area is 8 m². Find the overall heat transfer coefficient U of the wall.
L1 = 250 mm, k1 = 0.04 W/(m·K)L2 = 25 mm, k2 = 0.035 W/(m·K)A = 8 m²hi = 7.7 W/(m²·K)ho = 25 W/(m²·K)Troom = 22 °CTout = -14 °C
U = W/(m²·K)
3.Heat exchangers (LMTD)Medium
A counter-flow heat exchanger transfers 450 kW. The hot fluid cools from 155 to 85 °C and the cold fluid heats from 10 to 115 °C. The overall heat transfer coefficient is 800 W/(m²·K). Find the heat transfer area A needed.
Q̇ = 450 kWU = 800 W/(m²·K)Th: 155 → 85 °CTc: 10 → 115 °C
A = m²
4.Heat exchangers (LMTD)Hard
Steam condenses at a constant 130 °C on the outside of the tubes of a condenser with a heat transfer area of 6 m² and U = 1500 W/(m²·K). Cooling water (2 kg/s, cp = 4.18 kJ/(kg·K)) enters the tubes at 20 °C. Find the water outlet temperature Tc,out.
Ts = 130 °CA = 6 m²U = 1500 W/(m²·K)ṁ = 2 kg/scp = 4.18 kJ/(kg·K)Tc,in = 20 °C
Tc,out = °C
3Solutions
1.Heat conduction
Answer: Q̇ = 169.8 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)/(1.4·4) = 0.008929 K/W
R2 = L2/(k2A) = (50/1000)/(0.17·4) = 0.07353 K/W
Heat flow
Q̇ = (T1 − T3)/(R1 + R2) = 14/0.08246 = 169.8 W
Answer
Q̇ = 169.8 W
2.Heat conduction
Answer: U = 0.1402 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/(7.7·8) = 0.01623 K/W
R1 = L1/(k1A) = (250/1000)/(0.04·8) = 0.7813 K/W
R2 = L2/(k2A) = (25/1000)/(0.035·8) = 0.08929 K/W
Ro = 1/(hoA) = 1/(25·8) = 0.005 K/W
ΣR = 0.8918 K/W
Overall coefficient
U = 1/(A·ΣR) = 1/(8·0.8918) = 0.1402 W/(m²·K)
Answer
U = 0.1402 W/(m²·K)
3.Heat exchangers (LMTD)
Answer: A = 10.1 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 = 155 − 115 = 40 K
ΔT₂ = Th,out − Tc,in = 85 − 10 = 75 K
Log-mean temperature difference
ΔTlm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) = (40 − 75)/ln(40/75) = 55.68 K
Area
A = Q̇/(U·ΔTlm) = 450000/(800·55.68) = 10.1 m²
Answer
A = 10.1 m²
4.Heat exchangers (LMTD)
Answer: Tc,out = 92.52 °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) = 1500·6/(2·4180) = 1.077
Outlet temperature
ΔT₂ = ΔT₁·e−1.077 = 110·0.3408 = 37.48 K
Tc,out = 130 − 37.48 = 92.52 °C
Answer
Tc,out = 92.52 °C
Heat transferred: Q̇ = 606.2 kW.
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