ME7302

HEAT & MASS

TRANSFER

Dr.(Mrs.) K.T.K.M. De silva

Department of Mechanical & Manufacturing Engineering, FOE, UOR
CONDUCTION
Conduction
❖Heat conduction in a medium is three-
dimensional and time dependent.
Temperature, T =T(x, y, z, t)
❖Steady heat conduction - the temperature does
not vary with time
❖Unsteady or transient heat conduction - the
temperature vary with time as well as position
❖Lumped systems - the temperature variation
with time but not with position
❖Unlike temperature, heat transfer is a vector
quantity (has direction as well as magnitude).
Conduction…
❖One dimensional heat conduction - conduction is significant in one
dimension only and negligible in the other two primary dimensions, T(x,t)
❖Three-dimensional heat conduction - conduction in all dimensions is
significant, T(x, y, z, t)
Steady Heat Conduction
Heat conduction in plane walls
❖Consider steady heat conduction through the walls of a
house during a winter day.
- heat transfer normal direction to the wall surface
- wall surface is nearly isothermal
- significant heat transfer from the inner surface to the
outer one
- steady and one-dimensional
❖Energy balance for the wall can be expressed as,
(Rate of heat transfer – (Rate of heat transfer (Rate of change of
=
into the wall) out of the wall) energy of the wall)
Heat conduction in plane walls…
❖dEwall/dt = 0 for steady operation
❖The rate of heat transfer through the wall must
be constant, Q̇cond,wall = constant

❖Q̇cond,wall and the wall area A are constant;

dT/dx = constant
❖The temperature through the wall varies linearly
with x
Thermal Resistance
Rwall is the thermal resistance of the wall against
heat conduction or conduction resistance of the
wall.
Thermal Resistance…
❖Recall Newton’s law of cooling for convection heat transfer rate,

❖Rconv is the thermal resistance of the

surface against heat convection, or simply the
convection resistance of the surface.

❖When the convection heat transfer coefficient is very large, the convection
resistance becomes zero and Ts ≈ Tα, no resistance to convection.
Thermal Resistance…
❖When the wall is surrounded by a gas, the radiation effects. Recall Stefan–
Boltzmann law,

❖Rrad is the thermal resistance of a surface against

radiation, or the radiation resistance, and hrad is the
radiation heat transfer coefficient.
Thermal Resistance Network
❖Consider a plane wall of thickness L, area A, and thermal conductivity k that is
exposed to convection on both sides to fluids at temperatures Tα1 and Tα2 with heat
transfer coefficients h1and h2, respectively.

which can be rearranged as,

Thermal Resistance Network…

❖The rate of steady heat transfer between two surfaces is equal to the ratio
between temperature difference and the total thermal resistance .

❖U is the overall heat transfer coefficient

with the unit W/m2·K
Multilayer Plane Walls
❖Consider a plane wall that consists of
two layers (such as a brick wall with a
layer of insulation). The rate of steady
heat transfer through this two-layer
composite wall can be expressed as,
Thermal contact resistance
➢ Flat surfaces that appear smooth to the
eye turn out to be rather rough when
examined under a microscope -
microscopically rough surface
➢ An interface contains numerous air gaps
act as insulation because of the low
thermal conductivity of air
➢ An interface offers some resistance to
heat transfer, and this resistance for a unit
interface area is called the thermal
contact resistance
Heat conduction in cylinders
➢ Heat transfer through the pipe is in the normal direction
to the pipe surface and no significant heat transfer takes
place in the pipe in other directions.
➢ Temperature of the pipe, T = T(r)
𝒅𝑻
𝑸̇𝒄𝒐𝒏𝒅,𝒄𝒚𝒍 = −𝒌𝑨
𝒅𝒓

𝑻1 − 𝑻2
𝑸̇𝒄𝒐𝒏𝒅,𝒄𝒚𝒍 =
𝑹𝒄𝒚𝒍

𝒓2
𝒍𝒏( ൰ 𝑰𝒏(𝒐𝒖𝒕𝒆𝒓 𝒓𝒂𝒅𝒊𝒖 𝒔Τ𝑰 𝒏𝒏𝒆𝒓 𝒓𝒂𝒅𝒊𝒖𝒔ሻ
𝒓1
𝑹𝒄𝒚𝒍 = =
2𝝅𝑳𝒌 2𝝅 ∗ 𝒍𝒆𝒏𝒈𝒕𝒉 ∗ 𝒕𝒉𝒆𝒓𝒎𝒂𝒍 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒊𝒗𝒊𝒕𝒚
Heat conduction in spheres
➢ Temperature of the sphere, T = T(r)

𝑻1 − 𝑻2
𝑸̇𝒄𝒐𝒏𝒅,𝒔𝒑𝒉 =
𝑹𝒔𝒑𝒚

𝒓2 − 𝒓1 𝒐𝒖𝒕𝒆𝒓 𝒓𝒂𝒅𝒊𝒖𝒔 − 𝑰𝒏𝒏𝒆𝒓 𝒓𝒂𝒅𝒊𝒖𝒔

𝑹𝒔𝒑𝒉 = =
4𝝅𝒓1 𝒓2 𝒌 4𝝅 ∗ 𝒐𝒖𝒕𝒆𝒓 𝒓𝒂𝒅𝒊𝒖𝒔 ∗ 𝒊𝒏𝒏𝒆𝒓 𝒓𝒂𝒅𝒊𝒖𝒔 ∗ 𝒕𝒉𝒆𝒓𝒎𝒂𝒍 𝒄𝒐𝒏𝒅𝒖𝒄𝒕𝒊𝒗𝒊𝒕𝒚
Heat conduction in cylinders and spheres
➢ Now consider steady one-dimensional 𝑻𝛂𝟏 − 𝑻𝛂𝟐
heat transfer through a cylindrical or 𝑸̇ =
𝑹𝒕𝒐𝒕𝒂𝒍
spherical layer that is exposed to
convection on both sides to fluids at
temperatures Tα1 and Tα2 with heat transfer
coefficients h1 and h2, respectively.

For cylinder, For sphere,

𝑹𝒕𝒐𝒕𝒂𝒍 = 𝑹𝒄𝒐𝒏𝒗,𝟏 +𝑹𝒄𝒚𝒍 +𝑹𝒄𝒐𝒏𝒗,𝟐
𝑹𝒕𝒐𝒕𝒂𝒍 = 𝑹𝒄𝒐𝒏𝒗,1 +𝑹𝒔𝒑𝒚 +𝑹𝒄𝒐𝒏𝒗,2

𝒓𝟐 1 𝒓2 − 𝒓1 1
𝟏 𝑰𝒏( ൰ 𝟏
𝒓𝟏 𝑹𝒕𝒐𝒕𝒂𝒍 = + +
𝑹𝒕𝒐𝒕𝒂𝒍 = + + 2
൫4𝝅𝒓1 ሻ𝒉1 4𝝅𝒓 𝒓
1 2 𝒌 ൫4𝝅𝒓22 ሻ𝒉2
൫𝟐𝝅𝒓𝟏 𝑳ሻ𝒉𝟏 𝟐𝝅𝑳𝒌 ൫𝟐𝝅𝒓𝟐 𝑳ሻ𝒉𝟐
Heat transfer from finned surfaces
➢ Recall the Newton’s law of cooling,
𝑄̇𝑐𝑜𝑛𝑣 = ℎ𝐴𝑠 (𝑇𝑠 − 𝑇∞ ሻ
➢ When the temperatures 𝑇𝑠 and 𝑇∞ are fixed by design
considerations, as is often the case, there are two
ways to increase the rate of heat transfer;
❑ increase the convection heat transfer
coefficient h - may or may not be practical
❑ increase the surface area 𝐴𝑠 - by attaching to
the surface extended surfaces called fins
✓ car radiator - closely packed thin metal sheets attached to the
hot-water tubes increase the surface area for convection
Lumped Systems
Lumped system analysis
(a) Consider a small hot copper ball coming out of an
oven. The temperature of the ball remains nearly
uniform at all times, and we can talk about the
temperature of the ball with no reference to a
specific location.

(b) Consider a large roast in an oven. The temperature

distribution within the roast is not even close to
being uniform.
Lumped system analysis…
➢ Consider a body of arbitrary shape of mass m,
volume V, surface area As, density ρ, and specific
heat cp initially at a uniform temperature Ti. At time
t = 0, the body is placed into a medium at
temperature T∞ (T∞> Ti).
➢ During a differential time interval dt, the
temperature of the body rises by a differential
amount dT.

(Heat transfer into the body (The increase in the energy of the body
=
during dt) during dt)

𝒉𝑨𝒔 𝑻∞ − 𝑻 𝒅𝒕 = 𝒎𝒄𝒑 𝒅𝑻
Lumped system analysis…
𝑻(𝒕ሻ − 𝑻∞
= 𝒆−𝒃𝒕
𝑻𝒊 − 𝑻 ∞

𝒉𝑨𝒔 b is a positive quantity whose dimension is

𝒃=
𝝆𝒗𝒄𝒑 (time)-1 and called time constant.

➢ The temperature of the body changes rapidly at

the beginning, but rather slowly later on.
➢ A large value of b indicates that the body
approaches the environment temperature in a
short time.
Biot number (Bi)
𝒉𝑳𝒄
𝑩𝒊 =
𝒌

𝒗
𝑳𝒄 = ;Lc is the characteristic length
𝑨𝒔

𝒉 ∆𝑻 𝐶𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦

𝑩𝒊 = =
𝒌ൗ ∆𝑻 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦
𝑳𝒄

𝑳𝒄ൗ
𝒌 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑤𝑖𝑡ℎ𝑖𝑛 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦
𝑩𝒊 = =
1ൗ 𝐶𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑜𝑑𝑦
𝒉
Lumped system analysis…
➢ People from the mainland are to go by boat to
an island whose entire shore is a harbor, and
from the harbor to their destinations on the
island by bus.

➢ The overcrowding of people at the harbor

depends on the boat traffic to the island and the
ground transportation system on the island.
Transient Heat Conduction
Transient heat conduction in objects
➢ The wall temperature at and near
the surfaces starts to drop as a
result of heat transfer from the
wall to the surrounding medium
creating a temperature gradient in
the wall and initiates heat
conduction from the inner parts of
the wall toward its outer surfaces.
➢ Temperature profile within the wall remains symmetric at
all times about the center plane.
➢ The wall reaches thermal equilibrium with its
surroundings at the end.
Transient heat conduction in plane wall
𝑻 𝒙, 𝒕 − 𝑻∞
𝜽 𝑿, 𝝉 = = 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 𝑡𝑒𝑚𝑝𝑒𝑟𝑎𝑡𝑢𝑟𝑒
𝑻𝒊 − 𝑻∞

𝒉𝑳
𝑩𝒊 = = 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑒𝑛𝑡 (𝐵𝑖𝑜𝑡 𝑛𝑢𝑚𝑏𝑒𝑟ቇ
𝒌
𝜶𝒕
𝝉 = 2 = 𝑭𝒐 = 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠 𝑡𝑖𝑚𝑒 ( 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑛𝑢𝑚𝑏𝑒𝑟൰
𝑳

𝜶 = 𝒌/𝝆𝒄𝒑 ;Thermal diffusivity of the material

Lumped systems
𝜽 = 𝒇(𝑿, 𝑩𝒊, 𝑭𝒐ሻ 𝜽 = 𝒇(𝑭𝒐, 𝑩𝒊ሻ
Approximate analytical solution of one-
dimensional transient conduction problem
𝑻 𝒙,𝒕 −𝑻∞ −𝝉𝝀 𝟐 𝛌𝟏 𝒙
𝑷𝒍𝒂𝒏𝒆 𝒘𝒂𝒍𝒍: 𝜽𝒘𝒂𝒍𝒍 = = 𝑨𝟏 𝒆 𝟏 𝒄𝒐𝒔( ሻ, 𝛕 > 𝟎. 𝟐
𝑻𝒊 −𝑻∞ 𝑳

𝑻 𝒓, 𝒕 − 𝑻∞ −𝝉𝝀𝟐𝟏
𝛌𝟏 𝒓
𝑪𝒚𝒍𝒊𝒏𝒅𝒆𝒓: 𝜽𝒄𝒚𝒍 = = 𝑨𝟏 𝒆 𝑱𝟎 , 𝛕 > 𝟎. 𝟐
𝑻𝒊 − 𝑻∞ 𝒓𝟎

𝛌𝟏 𝒓
𝑻 𝒓, 𝒕 − 𝑻∞ 𝒔𝒊𝒏( ሻ
−𝝉𝝀𝟐𝟏 𝒓𝟎
𝑺𝒑𝒉𝒆𝒓𝒆: 𝜽𝒔𝒑𝒉 = = 𝑨𝟏 𝒆 , 𝛕 > 𝟎. 𝟐
𝑻𝒊 − 𝑻∞ 𝛌𝟏 𝒓
𝒓𝟎
Approximate analytical solution….
𝑻 𝟎 −𝑻∞ 𝟐
𝑪𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒑𝒍𝒂𝒏𝒆 𝒘𝒂𝒍𝒍(𝒙 = 𝟎ሻ: 𝜽𝟎,𝒘𝒂𝒍𝒍 = = 𝑨𝟏 𝒆−𝝉𝝀𝟏 , τ>0.2 𝜽𝒘𝒂𝒍𝒍 𝝀𝟏 𝒙
𝑻𝒊 −𝑻∞ = 𝒄𝒐𝒔( ሻ
𝜽𝟎,𝒘𝒂𝒍𝒍 𝑳
𝑻 𝟎 −𝑻∞ 𝟐
𝑪𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒄𝒚𝒍𝒊𝒏𝒅𝒆𝒓(𝒓 = 𝟎ሻ: 𝜽𝟎,𝒄𝒚𝒍 = = 𝑨𝟏 𝒆−𝝉𝝀𝟏 , τ>0.2
𝑻𝒊 −𝑻∞ 𝜽𝒄𝒚𝒍 𝝀𝟏 𝒓
= 𝑱𝟎 ( ሻ
𝜽𝟎,𝒄𝒚𝒍 𝒓𝟎
𝑻 𝟎 −𝑻∞ 𝟐
𝑪𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒔𝒑𝒉𝒆𝒓𝒆(𝒓 = 𝟎ሻ: 𝜽𝟎,𝒔𝒑𝒉 = = 𝑨𝟏 𝒆−𝝉𝝀𝟏 , τ>0.2
𝑻𝒊 −𝑻∞
𝝀 𝒓
𝜽𝒔𝒑𝒉 𝒔𝒊𝒏( 𝟏 ൗ𝒓𝟎 ሻ
=
𝜽𝟎,𝒔𝒑𝒉 𝝀𝟏 𝒓ൗ
➢ Coefficients used in the one-term approximate solution of 𝒓𝟎
transient one-dimensional heat conduction in plane walls,
cylinders, and spheres (Bi=hL/k for a plane wall of thickness
2L, and Bi=hr0/k for a cylinder or sphere radius r0
➢ The constants A1 and λ1 are functions of the Bi number only.
The function J0 is the zeroth-order Bessel function of the first
kind.
Coefficients used in the one-term
approximate solution of transient one
dimensional heat conduction in plane walls,
cylinders, and spheres (Bi = hL/k for a
plane wall of thickness 2L, and Bi = hro/k
for a cylinder or sphere of radius ro).