Thermodynamics vs. Heat Transfer Thermodynamics is concerned with the amount of heat transfer as a system undergoes a process from one equilibrium state to another, Thermodynamics gives no indication about how long the process takes, Heat Transfer determines how fast heat can be transferred to or from a system and thus the times of cooling or heating.
Spring 2003 ECE309: Chapter 8 1
Heat Transfer Mechanisms
Conduction: ∆T dT Q& = kA or Q& = −kA (W) cond ∆x cond dx
Convection:
Q&convvection = hA (Ts − T∞ ) (W)
Radiation:
4 4 Q& rad = εσA(Ts −Tsurr ) (W)
Spring 2003 ECE309: Chapter 8 2
1 Example Determine the rate of heat transfer between the plates per unit surface area assuming the gap between the plates is 1. filled with atmospheric air, 2. evacuated, 3. filled with urethane insulation, and 4. filled with superinsulation with k = 0.00002 W/m.°C.
Spring 2003 ECE309: Chapter 8 3
Steady Heat Conduction in Plane Walls Energy Balance: rate of rate of rate of change of the heat transfer − heat transfer = energy content into the wall out of the wall of the wall dE Q& −Q& = wall in out dt dT Q& = −kA (W) cond ,wall dx
T − T Q& = kA 1 2 (W) cond,wall L
Spring 2003 ECE309: Chapter 8 4
T T T − T s − ∞ 1 2 Q&conv = (W) Q&cond ,wall = (W) Rconv Rwall L 1 R = (°C/W) R = (°C/W) wall kA conv hA
4 4 Ts −Tsurr Q& rad = εσA(Ts −Tsurr ) = hrad A(Ts −Tsurr) = (W) Rrad
1 Rrad = (K/W) hrad A
2 2 2 hrad = εσA(Ts −Tsurr )(Ts −Tsurr ) (W/(m .K))
Spring 2003 ECE309: Chapter 8 5
Thermal Resistance Network
T − T Q& = ∞1 ∞2 (W) Rtotal
1 L 1 Rtotal = Rconv,1 + Rwall + Rconv,2 = + + (°C/W) h1A kA h2 A
Spring 2003 ECE309: Chapter 8 6
3 Mutilayer Plane Wall
T − T Q& = ∞1 ∞2 Rtotal
Rtotal = Rconv,1 + Rwall,1 + Rwall,2 + Rconv ,2 1 L L 1 = + 1 + 2 + h1 A k1 A k1 A h2 A
Spring 2003 ECE309: Chapter 8 7
Thermal Contact Resistance
Perfect Thermal Contact Actual Thermal Contact
The thermal resistance due to the roughness at the contact areas is called thermal contact resistance.
Spring 2003 ECE309: Chapter 8 8
4 Example
Given: W = 1.5 m, H = 0.8 m, k = 0.78 W/m.°C, Calculate: 1. Steady rate of heat transfer through this glass window and, 2. Temperature of the inner surface.
Spring 2003 ECE309: Chapter 8 9
Heat Conduction in Cylinders & Spheres
T1 − T2 Q&Cond = Rth
ln(r / r ) r − r 2 1 R 2 1 Rth,cyl = th,sph = 2πkL 4πr1r2k
Spring 2003 ECE309: Chapter 8 10
5 Example Given: k = 15 W/(m.°C),
T∞1 = 0 °C, T∞2 = 22 °C, h1 = 80, h2 = 10W/(m2.°C), Latent heat of fusion = hif = 333.7 kJ/kg. Determine: (a) Rate of heat transfer, (b) Amount of ice at 0 °C that melts during a 24-h period.
Spring 2003 ECE309: Chapter 8 11
Critical Radius of Insulation, rcr d Q& T − T = 0 Q& = 1 ∞ ⇒ d r Rins + Rconv
k 2k r = (m) r = cr,cylinder h cr,sphere h
k: Thermal conductivity of insulation h: Ambient heat transfer coefficient
Spring 2003 ECE309: Chapter 8 12
6 Heat Generation in a Solid
Q& = g&V = hA(Ts −T∞ ) gV T = T + & s ∞ hA g& : Heat generation / unit volume, (W/m3)
Spring 2003 ECE309: Chapter 8 13
Example
Given: 2-kW resistance heater wire, k = 15 W/(m.°C), D = 4 mm, L = 0.5 m,
Ts = 105 °C, Calculate: Centerline temperature.
Spring 2003 ECE309: Chapter 8 14
7 Heat Transfer from Finned Surfaces
Ab = w× t
Q& conv = h(T −T∞ )dA 2 ∫ Arec. fin = 2 × w× L + w×t Apin fin = πDL + πD / 4 L
Spring 2003 ECE309: Chapter 8 15
Fin Efficiency, ηfin
Q& fin Actual heat transfer rate from the fin η fin = = Q& fin,max Ideal (maximum) heat transfer rate from the fin
Spring 2003 ECE309: Chapter 8 16
8 Fin Effectiveness, εfin Thermal conductivity of the fin material should be as high as possible.The most widely used fins are made of aluminum.
The ratio of the perimeter to the cross- sectional area of the fin, p/Ac, should be as high as possible.
The use of fins is most effective in Q& applications that involve low ε = fin convection heat transfer coefficient fin Q&no fin (gases not liquids).
Spring 2003 ECE309: Chapter 8 17
Example
Given Case-to-ambient thermal resistance = 20 °C/W, Maximum power rating = 10W, Maximum allowable temperature = 85 °C, Ambient temperature = 25 °C.
Determine: Power at which this transistor can be operated safely.
Spring 2003 ECE309: Chapter 8 18
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