16
Conduction & Convection

Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Embed Size (px)

Citation preview

Page 1: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Conduction & Convection

Page 2: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Quiz 8 – 2014.01.24

Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies linearly with temperature according to the equation

TIME IS UP!!!

20 0 5k T . T

1 in

350F 250F

2 ft

3 ft

4 ft

o2 o

Btu F

h ft Fk , T

Page 3: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Outline

2. Conduction Heat Transfer

2.1. Series/Parallel Resistances

2.2. Geometric Considerations

Page 4: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Outline

3. Convection Heat Transfer

3.1. Heat Transfer Coefficient

3.2. Dimensionless Groups for HTC

Estimation

Page 5: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Thermal Resistance Circuits

Recall: Fourier’s Law of Heat Conduction

𝑄𝐴

=−𝑘𝑑𝑇𝑑𝑥

ONE-DIMENSIONAL ONLY

For a constant cross-section and isotropic thermal conductivity (e.g. a flat slab).

𝑄=−∆𝑇

(∆ 𝑥𝑘𝐴 )

Driving force*

Thermal Resistanceǂ

* where T1> T2ǂ where x2> x1

Page 6: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

For series resistances (flat slab):

𝑅𝐴=∆𝑥𝐴

𝑘𝐴 𝐴𝑅𝐵=

∆ 𝑥𝐵

𝑘𝐵 𝐴

Driving Force at A:

Driving Force at B:

Overall driving force:

(−∆𝑇 𝐴 )=𝑄𝑡𝑜𝑡𝑅𝐴

(−∆𝑇 𝐵 )=𝑄𝑡𝑜𝑡𝑅𝐵

(−∆𝑇 𝑡𝑜𝑡 )=𝑄𝑡𝑜𝑡 (𝑅𝐴+𝑅𝐵 )

Page 7: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

For series resistances (flat slab):

𝑄𝑡𝑜𝑡=(−∆𝑇 𝑡𝑜𝑡 )∆ 𝑥𝐴

𝑘𝐴 𝐴+∆ 𝑥𝐵

𝑘𝐵 𝐴

𝑅𝑡𝑜𝑡=𝑅𝐴+𝑅𝐵

𝑄𝑡𝑜𝑡=𝑄𝐴=𝑄𝐵

(−∆𝑇 𝑡𝑜𝑡 )=(−∆𝑇 𝐴 )+(−∆𝑇 𝐵 )𝑅𝐴=

∆𝑥𝐴

𝑘𝐴 𝐴𝑅𝐵=

∆ 𝑥𝐵

𝑘𝐵 𝐴

Page 8: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

For parallel resistances (flat slab):

𝑅𝐴=∆𝑥𝐴

𝑘𝐴 𝐴𝑅𝐵=

∆ 𝑥𝐵

𝑘𝐵 𝐴

Driving Force at A:

Driving Force at B:

Overall driving force:

(−∆𝑇 𝑡𝑜𝑡 )=𝑄𝐴𝑅𝐴

(−∆𝑇 𝑡𝑜𝑡 )=𝑄𝐵𝑅𝐵

(−∆𝑇 𝑡𝑜𝑡 )𝑅𝑡𝑜𝑡

=𝑄𝐴+𝑄𝐵

Page 9: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

For parallel resistances (flat slab):

1𝑅𝑡𝑜𝑡

=1𝑅𝐴

+1𝑅𝐵

𝑄𝑡𝑜𝑡=𝑄𝐴+𝑄𝐵

(−∆𝑇 𝑡𝑜𝑡 )=(−∆𝑇 𝐴 )=(−∆𝑇 𝐵 )𝑅𝐴=

∆𝑥𝐴

𝑘𝐴 𝐴𝑅𝐵=

∆ 𝑥𝐵

𝑘𝐵 𝐴

(−∆𝑇 𝑡𝑜𝑡 )𝑅𝑡𝑜𝑡

=(−∆𝑇 𝑡𝑜𝑡 )

𝑅𝐴

+(−∆𝑇 𝑡𝑜𝑡 )

𝑅𝐵

Substituting the individual Q:

Page 10: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

Exercise! (PIChE Quiz Bowl Nationals 2009, Easy Round, 2 min)

A composite wall consists of 2-in. corkboard (inner), 6-in. concrete, and 3-in. wood (outer). The thermal conductivities of the materials are 0.025, 0.8, and 0.065 Btu/hr/ft/°F, respectively. The temperature of the inner surface of the wall is 55°F while the outer surface is at 90°F. What are the temperatures in °F:

a) Between the cork and concrete?b) Between the concrete and wood?

Page 11: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Series/Parallel Resistances

Exercise! (Combined series & parallel)Four different materials were joined together as a block of constant width shown below. The cross-sectional area of B is equal to that of C. Materials A, B, C, and D have k = 0.1, 0.5, 0.4, and 0.1 W/mK, respectively. The thickness of blocks A and D is 1” and the thickness of blocks B and C is 6”. If the left of A is exposed to 110°C and the right of D is exposed to 60°C, calculate (a) the temperature right after material A and (b) the temperature right before material D.

Page 12: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Geometric Considerations

Heat Conduction Through Concentric Cylinders

𝐴=2𝜋𝑟𝐿

𝑄𝐴

=−𝑘𝑑𝑇𝑑𝑟

∫ 𝑄2𝜋𝑟𝐿

𝑑𝑟=−𝑘∫𝑑𝑇

Integrating both sides:

ln ( 𝑟2

𝑟1)=−2𝜋 𝑘𝐿

𝑄 (𝑇 2−𝑇 1 )

Page 13: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Geometric Considerations

Heat Conduction Through Concentric Cylinders

𝐴=2𝜋𝑟𝐿

𝑄𝐴

=−𝑘𝑑𝑇𝑑𝑟

Rearranging: 𝑄=−𝑘2𝜋 𝐿

ln ¿¿¿¿¿

∫ 𝑄2𝜋𝑟𝐿

𝑑𝑟=−𝑘∫𝑑𝑇

Page 14: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Geometric Considerations

Heat Conduction Through Concentric Cylinders

Define a logarithmic mean area:

𝐴𝐿𝑀=2𝜋 𝐿(𝑟 2−𝑟1)ln ¿¿ ¿¿¿

…and a logarithmic mean radius:

𝑟 𝐿𝑀=(𝑟2−𝑟1)ln¿ ¿¿¿¿

𝑄=−𝑘 𝐴𝐿𝑀 (𝑇 2−𝑇1

𝑟2−𝑟1)

𝐴𝐿𝑀=2𝜋 𝐿𝑟 𝐿𝑀 *Final form

𝑄=−𝑘2𝜋 𝐿

ln ¿¿¿¿¿

Page 15: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Geometric Considerations

Heat Conduction Through Concentric Cylinders

When to use logarithmic mean area and when to use arithmetic mean area?

𝐴𝐿𝑀=2𝜋 𝐿(𝑟 2−𝑟1)ln ¿¿ ¿¿¿

𝐴𝑀=2𝜋 𝐿(𝑟 2+𝑟 1)

2

𝒓 𝑳𝑴

𝒓𝑴

r2/r1 rLM rM

1 #DIV/0! 11.05 1.024797 1.0251.1 1.049206 1.05

1.15 1.073254 1.0751.2 1.096963 1.1

1.25 1.120355 1.1251.3 1.143448 1.15

1.35 1.16626 1.1751.4 1.188805 1.2

Page 16: Conduction & Convection Quiz 8 – 2014.01.24 Determine the heat transfer across the 1 inch-thick slab shown at the right, whose thermal conductivity varies

Geometric Considerations

Heat Conduction Through Concentric Cylinders

A tube of 60-mm (2.36-in.) outer diameter is insulated with a 50-mm (1.97-in.) layer of silica foam, for which k = 0.032 Btu/hr-ft-°F, followed by a 40-mm (1.57-in.) layer of cork with k = 0.03 Btu/hr/ft/°F. If the temperature of the outer surface of the pipe is 150°C, and the temperature of the outer surface of the cork is 30°C, calculate the heat loss in W/(m of pipe).