AME 60634 Int. Heat Trans.

D. B. Go 1

Heat Transfer Rates

heat flux[W/m2]

thermal conductivity[W/m-K]

temperature gradient[K/m]

heat flux[W/m2]

heat transfer coefficient[W/m2-K]

surface temperature[K]

fluid temperature[K]

emissive power[W/m2]

surface emissivity[ ]

Stefan-Boltzmann constant[5.67×10-8 W/m2-K4]

surface temperature[K]

Conduction: Fourier’s Law

Convection: Newton’s Law of Cooling

Radiation: Stefan-Boltzmann Law (modified)

AME 60634 Int. Heat Trans.

D. B. Go 2

Transient Conduction: Lumped Capacitance • General Transient Problem: Special Case negligible radiation,

heat flux & heat generation

Define: thermal time constant

t cVhAs,c

cV 1hAs,c

lumped capacitance thermal resistance

We can plot the normalized solution to the general problem

Notes:• The change in thermal energy storage due to the transient process is:

AME 60634 Int. Heat Trans.

D. B. Go 3

1-D Steady Conduction: Plane WallGoverning Equation:

Dirichlet Boundary Conditions:

qx kA dTdx

kAL

Ts,1 Ts,2

Solution:

Heat Flux:

Heat Flow:

temperature is not a function of k

T(x) Ts,1 Ts,2 Ts,1 xL

q x k dTdx

kL

Ts,1 Ts,2

Notes:• A is the cross-sectional area of the wall perpendicular to the heat flow• both heat flux and heat flow are uniform independent of position (x)• temperature distribution is governed by boundary conditions and length

of domain independent of thermal conductivity (k)

heat flux/flow are a function of k

AME 60634 Int. Heat Trans.

D. B. Go 4

1-D Steady Conduction: Cylinder WallGoverning Equation:

Dirichlet Boundary Conditions:

T(r1) Ts,1 ; T(r2) Ts,2

Notes:• heat flux is not uniform function of position (r)• both heat flow and heat flow per unit length are uniform independent of

position (r)

Solution:

Heat Flux:

Heat Flow:

T(r) Ts,1 Ts,2

ln r1 r2 ln r

r2

Ts,2

q r k dTdr

k Ts,1 Ts,2 r ln r2 r1

qr kA dTdr

2rL q r 2Lk Ts,1 Ts,2

ln r2 r1

q r qr

L

2k Ts,1 Ts,2 ln r2 r1 heat flow per unit length

heat flux is non-uniform

heat flow is uniform

AME 60634 Int. Heat Trans.

D. B. Go 5

1-D Steady Conduction: Spherical ShellGoverning Equation:

Dirichlet Boundary Conditions:

T(r1) Ts,1 ; T(r2) Ts,2

Solution:

Heat Flux:

Heat Flow:

T(r) Ts,1 Ts,1 Ts,2 1 r1 r 1 r1 r2

q r k dTdr

k Ts,1 Ts,2

r2 1 r1 1 r2

qr kA dTdr

4r2 q r 4k Ts,1 Ts,2

1 r1 1 r2

Notes:• heat flux is not uniform function of position (r)• heat flow is uniform independent of position (r)

heat flux is non-uniform

heat flow is uniform

AME 60634 Int. Heat Trans.

D. B. Go 6

Thermal Resistance

AME 60634 Int. Heat Trans.

D. B. Go 7

Thermal Circuits: Composite Plane Wall

Circuits based on assumption of (a) isothermal surfaces normal to x direction

or (b) adiabatic surfaces parallel to x direction

Rtot LE

kE A

kF A2LF

kG A2LG

1

LH

kH A

Rtot

2LE

kE A

2LF

kF A

2LH

kH A

1

2LE

kE A

2LG

kG A

2LH

kH A

1

1

Actual solution for the heat rate q is bracketed by these two approximations

AME 60634 Int. Heat Trans.

D. B. Go 8

Thermal Circuits: Contact ResistanceIn the real world, two surfaces in contact do not transfer heat perfectly

R t,c TA TB

q x Rt ,c

R t,cAc

Contact Resistance: values depend on materials (A and B), surface roughness, interstitial conditions, and contact pressure typically calculated or looked up

Equivalent total thermal resistance:

Rtot LA

kA Ac

R t,c

Ac

LB

kB Ac