Review:Conduction

CH EN 3453 – Heat Transfer

Reminders…

• No homework assignment this week

• Midterm Wednesday– Covers chapters 1, 2, 3, 4, 5– Closed book– One 8.5x11 sheet of notes (both sides) allowed– Kevin will provide tables (see “misc” of website)– Bring calculator and paper. Do not write answers on test– No cell phones allowed

• Review session today at 4:00 pm in WEB 1230– Come with questions

• Lecture overheads posted online

Chapter 1:Introduction to Heat Transfer

• Heat vs. heat flux vs. heat per length

• Conduction– Fourier’s Law– Ranges of k (Table 2.4 + appendices)

• Convection– Newton’s Law of cooling– Ranges of h (Table 1.1)

• Radiation– Stefan-Boltzmann Law– Emissivity, absorptivity

Chapter 2:Introduction to Conduction

• Thermal properties of matter

• Heat diffusion equation:

∂∂x

k∂T∂x

⎛⎝⎜

⎞⎠⎟

+∂∂y

k∂T∂y

⎛⎝⎜

⎞⎠⎟

+∂∂z

k∂T∂z

⎛⎝⎜

⎞⎠⎟

+ q = ρcp

∂T∂t

Range of Thermal Conductivities

Figure 2.4 Range of thermal conductivity for various states of matter at normal temperatures and pressure.

Thermal Conductivity of Gases

Figure 2.8 The temperature dependence of the thermal conductivity of selected gases at normal pressures. The molecular weight of the gases is also shown.

Chapter 3:Steady-State Conduction (1-D)

• The plane wall

• Radial systems

• Energy generation

• Extended surfaces

Heat Transfer through a Wall

R1 R2 R3

Complex Heat Transfer

This type assumes lateral heat transfer (in the y direction) at interfaces, which is often not a good assumption.

This is a “safer” way to structure the thermal circuit. Assumes parallel heat transfer through all layers.

Contact Resistance

A Cylinder

Review of Conduction…

Page 126

Extended Surfaces (fins)

Figure 3.12 Use of fins to enhance heat transfer from a plane wall.(a) Bare surface. (b) Finned surface.

Types of Fins

Four Scenarios for Treating Pin Tips

Fin Effectiveness vs Efficiency• Fin Effectiveness (ε)

– A measure of how much more heat one can transfer to a given base (attachment) area by adding a fin.

– Greater than 1.0– qwith fin = ε·qwithout fin

• Fin Efficiency (η)– Ranges from 0 to 1.0– Relative heat transfer from fin compared to case

where all of fin surface is at base temperature

Fin Efficiencies

Fin Efficiencies

Fin Efficiencies

Modified Bessel function of the first kind(Appendix B.5)

Modified Bessel function of the second kind(Appendix B.5)

Fin Efficiencies, continued

Chapter 4:2-D Steady-State Conduction

• Graphical methods

• Shape factors

• Finite-difference equations

2-D Heat Flow

• Heat flow lines ("adiabats") represent how heat "flows." • There is no heat transfer in a direction perpendicular to

heat flow lines• Isotherms – constant temperature• Adiabats and isotherms are perpendicular to one another

Shape Factors q = Sk(T1-T2)

Shape Factors, Cont.

2-D Solid

Conduction between Nodes

Figure 1.5

Chapter 5:Unsteady-State Conduction

• Lumped analysis and the Biot number

• Spatial effects

• Semi-infinite solids

• Constant surface temp. and const. heat flux

Review: The Biot Number

• If Bi < 0.1 then the lumped capacitance approach can be used– Eq. 5.5 to find time to reach a given T– Eq. 5.6 to find T after a given time– Eq. 5.8a to find total heat gain (loss) for given time

• L depends on geometry– General approach is L = V/As

• L/2 for wall with both sides exposed• ro/2 for long cylinder• ro/3 for sphere• Use L = V/As when geometry is neither plate, cylinder nor sphere

– Conservative approach (preferred) is to use the maximum length• L for wall (note that L is measured from midpoint

when heated from both sides)• ro for cylinder or sphere

Bi = hL

k

Lumped Capacitance Equations

• Time as a function of temperature

• Temperature as a function of time

Spatial Effects(When lumped analysis cannot be used)

Dimensionless Variables

Temperature: θ*  ≡  θθi = T − T∞

Ti − T∞

Position: x* ≡xL

Time: t* ≡ αtLc2 =

ktρcLc

2

Table 5.1 – ζ1 and C1 vs. Bi