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Engr. FRANCIS M. MULIMBAYAN
BSAE / MSMSE INSTRUCTOR 4
ENSC 14a
Engineering Thermodynamics and Heat Transfer
Department of Engineering Science University of the Philippines –Los Banos
College, Los Banos, Philippines
Chapter 10
Conduction Heat Transfer
Conduction
Transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles.
Requires an intervening medium
Can take place in solids, liquids or gas.
Stationary fluids: results of interaction between molecules of different energy level
Conducting solids: energy transport by free electrons
Non-conducting solids: via lattice vibration
2
Introduction
Conductive Heat Transfer
Location of a point in different coordinate system
3
Introduction
Conductive Heat Transfer
Steady-State vs. Transient Heat Conduction
4
Introduction
Conductive Heat Transfer
Fourier’s Law of Heat Conduction
Based on experimental observation
Basic equation for the analysis of heat conduction
Can be expresses as,
Q ′′ = −kn𝜕T
𝜕n
Q ′′ Heat transfer rate in the n direction per unit area in W/m2
kn Thermal conductivity in n direction, in W/m-K
𝜕T
𝜕n Temperature gradient in n direction;
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Fourier’s Law of Heat Conduction
Conductive Heat Transfer
Temperature gradient
Slope on the T-x diagram at a given point in the medium
Direction of heat
Always from higher to lower temperature
Negative sign indicate that heat transfer in the positive x direction is a positive quantity
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Fourier’s Law of Heat Conduction
Conductive Heat Transfer
Thermal Conductivity
Measure of the ability of the material to conduct heat
In general, 𝑘 = 𝑓(𝑇, 𝑛)
Type of material based on k
Isotropic – k is the same in all directions
Anisotropic – k has strong directional dependence
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Fourier’s Law of Heat Conduction
Conductive Heat Transfer
In rectangular coordinates, heat conduction vector can be expressed in terms of its components as,
𝑸 𝒏 = 𝑸 𝒙𝐢 + 𝑸 𝒚𝐣 + 𝑸 𝒛𝐤
𝑸 𝒙 = −𝒌𝒙𝑨𝒙𝝏𝑻
𝝏𝒙
𝑸 𝒚 = −𝒌𝒚𝑨𝒚𝝏𝑻
𝝏𝒚
𝑸 𝒛 = −𝒌𝒛𝑨𝒛𝝏𝑻
𝝏𝒛
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Fourier’s Law of Heat Conduction
Conductive Heat Transfer
One-Dimensional Heat Conduction in Plane Wall
Heat conduction is dominant in one direction and negligible in other directions
The energy balance for the thin element shown during small time interval is,
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
Q x − Q x+∆x + G element =∆Eelement
∆t
The change in energy content of the element is ∆Eelement = Et+∆t − Et = mC Tt+∆t − Tt
= ρCA∆x Tt+∆t − Tt
The rate of heat generation within the element is
G element = g V = g A∆x
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
Q x − Q x+∆x + g A∆x = ρCA∆xTt+∆t − Tt
∆t
Dividing by A∆x
−1
A
Q x+∆x − Q x∆x
+ g = ρCTt+∆t − Tt
∆t
Applying the definition of derivative and Fourier’s Law
lim∆x→0
Q x+∆x − Q x∆x
=𝜕Q
𝜕x=
𝜕
𝜕x−kA
𝜕T
𝜕x
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
Taking the limit as 𝑥 → 0 and Δ𝑡 → 0
1
A
𝜕
𝜕xkA
𝜕T
𝜕x+ g = ρC
𝜕T
𝜕t
The one-dimensional, transient heat conduction equation in plane wall with variable thermal conductivity is
𝜕
𝜕xk𝜕T
𝜕x+ g = ρC
𝜕T
𝜕t
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in plane wall with constant thermal conductivity is,
𝜕2T
𝜕x2+g
k=1
α
𝜕T
𝜕t
where α =k
ρCp is called the thermal
diffusivity or the measure of how fast heat propagates through a material
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in plane wall with constant thermal conductivity and
Steady-state:→𝜕T
𝜕t= 0
𝑑2𝑇
𝑑𝑥2+𝑔
𝑘= 0
Transient, no heat generation 𝜕2𝑇
𝜕𝑥2=1
𝛼
𝜕𝑇
𝜕𝑡
Steady-state, no heat generation 𝑑2𝑇
𝑑𝑥2= 0
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One-Dimensional Heat Conduction Equations – Plane Wall
Conductive Heat Transfer
One-Dimensional Heat Conduction in Cylinders
The area normal to the direction of heat transfer is 𝐴 = 2𝜋𝑟𝐿
The energy balance for the thin element shown during small time interval is,
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
Q r − Q r+∆r + G element =∆Eelement
∆t
The change in energy content of the element is ∆Eelement = Et+∆t − Et = mC Tt+∆t − Tt
= ρCA∆r Tt+∆t − Tt
The rate of heat generation within the element is
G element = g V = g A∆r
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
Q r − Q r+∆r + g A∆r = ρCA∆rTt+∆t − Tt
∆t
Dividing by A∆r (where A=2𝜋𝑟𝐿)
−1
A
Q r+∆r − Q r∆r
+ g = ρCTt+∆t − Tt
∆t
Applying the definition of derivative and Fourier’s Law
lim∆r→0
Q r+∆r − Q r∆r
=𝜕Q
𝜕r=
𝜕
𝜕r−kA
𝜕T
𝜕r
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
Taking the limit as ∆𝑟 → 0 and Δ𝑡 → 0
1
A
𝜕
𝜕rkA
𝜕T
𝜕r+ g = ρC
𝜕T
𝜕t
The one-dimensional, transient heat conduction equation in cylinders with variable thermal conductivity is,
1
r
𝜕
𝜕rrk𝜕T
𝜕r+ g = ρC
𝜕T
𝜕t
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in cylinders with constant thermal conductivity is,
1
r
𝜕
𝜕rr𝜕T
𝜕r+g
k=1
α
𝜕T
𝜕t
where α =k
ρCp is called the thermal
diffusivity or the measure of how fast heat propagates through a material
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in cylinders with constant thermal conductivity and
Steady-state:→𝜕T
𝜕t= 0
1
𝑟
𝑑
𝑑𝑟𝑟𝑑𝑇
𝑑𝑟+𝑔
𝑘= 0
Transient, no heat generation 1
r
𝜕
𝜕rr𝜕T
𝜕r=1
α
𝜕T
𝜕t
Steady-state, no heat generation 𝑑
𝑑𝑟𝑟𝑑𝑇
𝑑𝑟= 0 𝑜𝑟 𝑟
𝑑2𝑇
𝑑𝑟2+𝑑𝑇
𝑑𝑟= 0
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One-Dimensional Heat Conduction Equations – Cylinders
Conductive Heat Transfer
One-Dimensional Heat Conduction in Spheres
The area normal to the direction of heat transfer is 𝐴 = 4𝜋𝑟2
The one-dimensional, transient heat conduction equation in spheres with variable thermal conductivity is,
1
r2𝜕
𝜕rr2k
𝜕T
𝜕r+ g = ρC
𝜕T
𝜕t
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One-Dimensional Heat Conduction Equations – Spheres
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in spheres with constant thermal conductivity is,
1
r2𝜕
𝜕rr2𝜕T
𝜕r+g
k=1
α
𝜕T
𝜕t
where α =k
ρCp is called the thermal
diffusivity or the measure of how fast heat propagates through a material
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One-Dimensional Heat Conduction Equations – Spheres
Conductive Heat Transfer
The one-dimensional, transient heat conduction equation in spheres with constant thermal conductivity and
Steady-state:→𝜕T
𝜕t= 0
1
r2d
drr2dT
dr+g
k= 0
Transient, no heat generation 1
r2𝜕
𝜕rr2𝜕T
𝜕r=1
α
𝜕T
𝜕t
Steady-state, no heat generation d
drr2dT
dr= 0 or r
d2T
dr2+ 2
dT
dr= 0
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One-Dimensional Heat Conduction Equations – Spheres
Conductive Heat Transfer
Sample Problems: 1. Consider a steel pan placed on top of an
electric range to cook spaghetti.. Assuming a constant thermal conductivity, write the differential equation that describes the variation of the temperature in the bottom section of the pan during steady operation.
2. A 2-kW resistance heater wire is used to boil water by immersing it in water. Assuming the variation of the thermal conductivity of the wire with temperature to be negligible, obtain the differential equation that describes the variation of the temperature in the wire during steady operation.
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One-Dimensional Heat Conduction Equations
Conductive Heat Transfer
Sample Problems:
3. A spherical metal ball of radius R is heated in an oven to a temperature of 600°F throughout and is then taken out of the oven and allowed to cool in ambient air at 𝑇∞ = 75℉ by convection and radiation. The thermal conductivity of the ball material is known to vary linearly with temperature. Assuming the ball is cooled uniformly from the entire surface; obtain the differential equation that describes the variation of the temperature in the ball during cooling.
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One-Dimensional Heat Conduction Equations
Conductive Heat Transfer