ENSC 14a Engineering Thermodynamics and Heat Transfer
Chapter 10 Conduction Heat Transfer
Department of Engineering Science
BSAE / MSMSE
University of the Philippines – Los Los Banos
Conductive Heat Transfer Introduction
Conduction
Transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions interactions between the particles. 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 Conducting solids: energy transport by free electrons Non-conducting Non-conducting solids: via lattice vibration
Conductive Heat Transfer Introduction
Location of a point in different coordinate system
Conductive Heat Transfer Introduction
Steady-State vs. Transient Heat Conduction
Conductive Heat Transfer Fourier’s Law of Heat Conduction
Fourier’s Law of Heat Conduction
Based on experimental observation Basic equation for the analysis of heat conduction Can be expresses as,
′′
Q = −k Q ′′ k
T n
Heat transfer rate in the n direction per unit area in W/m 2 Thermal conductivity in n direction, in W/m-K Temperature gradient in n direction;
Conductive Heat Transfer Fourier’s Law of Heat Conduction
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
Conductive Heat Transfer Fourier’s Law of Heat Conduction
Thermal Conductivity
Measure of the ability of the material to conduct heat In general, = (, ) Type of material based on k Isotropic – k is the same in all directions Anisotropic – k has strong directional dependence
Conductive Heat Transfer Fourier’s Law of Heat Conduction
In rectangular coordinates, coordina tes, heat conduction vector can be expressed in terms of its components as,
= = −
= − = −
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
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,
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
Q − Q +∆ G eee =
∆Eeee ∆t
The change in energy content of the element is
∆Eeee = E+∆ − E = mC m C T+∆ − T = ρCA∆x T+∆ − T
The rate of heat generation within the element is G eee = gV = gA∆ A∆x
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
Q − Q +∆ g A∆x = ρCA∆x
∆t
Dividing by A∆x −
T+∆ − T
1 Q +∆ − Q A
∆x
g = ρC
T+∆ − T ∆t
Applying the definition of derivative and Fourier’s Law lim
∆→
Q +∆ − Q ∆x
=
Q x
=
x
−kA
T x
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
Taking the limit as → 0
and
Δ → 0 1 A x
kA
T x
g = ρC
T t
The one-dimensional, transient heat conduction equation in plane wall with variable thermal conductivity is
T T k g = ρC x x t
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
The one-dimensional, transient heat conduction equation in plane wall with constant thermal conductivity is, T x
where α =
p
g k
=
1 T α t
is called the thermal
diffusivity or the measure of how fast heat propagates through a material
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Plane Wall
The one-dimensional, transient heat conduction equation in plane wall with constant thermal conductivity and
Steady-state:→
=0
=0
Transient, no heat generation
=
1
Steady-state, no heat generation
=0
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
One-Dimensional Heat Conduction in Cylinders The area normal to the direction of heat transfer is = 2 The energy balance for the thin element shown during small time interval is,
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
Q − Q +∆ G eee =
∆Eeee ∆t
The change in energy content of the element is
∆Eeee = E+∆ − E = mC T+∆ − T = ρCA∆r T+∆ − T
The rate of heat generation within the element is G eee = gV = gA∆ A∆r
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
Q − Q +∆ g A∆r = ρCA∆r
∆t
Dividing by A∆r (where A=2) −
T+∆ − T
1 Q +∆ − Q A
∆r
g = ρC
T+∆ − T ∆t
Applying the definition derivative and Fourier’s Law
lim
∆→
Q +∆ − Q ∆r
=
Q r
=
r
−kA
of T r
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
Taking the limit as ∆ → 0
and
Δ → 0 1 A r
kA
T r
g = ρC
T t
The one-dimensional, transient heat conduction equation in cylinders with variable thermal conductivity conductivity is, 1
T T rk g = ρC r r r t
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
The one-dimensional, transient heat conduction equation in cylinders with constant thermal conductivity conductivity is, 1 r r
r
T r
g k
=
1 T α t
where α = is called the thermal p
diffusivity or the measure of how fast heat propagates through a material
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Cylinders
The one-dimensional, transient heat conduction equation in cylinders with constant thermal conductivity and
Steady-state:→
=0
1
=0
Transient, no heat generation 1 r r
r
T r
=
1 T α t
Steady-state, no heat generation
= 0
=0
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Spheres
One-Dimensional Heat Conduction in Spheres The area normal to the direction of heat transfer is = 4 The one-dimensional, one-dimensional, transient heat conduction equation in spheres with variable thermal conductivity conductivity is,
1
T T r k g = ρC r r r t
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Spheres
The one-dimensional, transient heat conduction equation in spheres with constant thermal conductivity is, 1
r
r r
T r
g k
=
1 T α t
where α = is called the thermal p
diffusivity or the measure of how fast heat propagates through a material
Conductive Heat Transfer One-Dimensional Heat Conduction Equations – Spheres
The one-dimensional, transient heat conduction equation in spheres with constant thermal conductivity and
Steady-state:→
=0
1 d
r
dr
dT dr
g k
=0
Transient, no heat generation 1 r r
r
r
T r
=
1 T α t
Steady-state, no heat generation d dr
r
dT dr
= 0 or r
d T dr
2
dT dr
=0
Conductive Heat Transfer One-Dimensional Heat Conduction Equations 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.
Conductive Heat Transfer One-Dimensional Heat Conduction Equations 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 ∞ = 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.
