2a heat conduction equations (ensc 14a)

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;

5

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

6

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

7

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

8

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,

9

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

10

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

11

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

12

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

13

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

14

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,

15

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

16

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

17

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

18

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

19

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

20

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

21

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

22

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

23

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.

24

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.

25

One-Dimensional Heat Conduction Equations

Conductive Heat Transfer