Process Heat Transfer Lec 3: Heat Conduction Equation

Chemical Engineering Department | University of Jordan | Amman 11942, Jordan

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Steady-State Conduction without Thermal Energy Generation

Process Heat Transfer

Lec 3: Heat Conduction Equation

Dr. Zayed Al-Hamamre


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1-

2

One-Dimensional, Steady-State Conduction without Thermal Energy Generation

o Plane Wall

o Tube and cylinders

o Spherical shell

Content


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• Specify appropriate form of the heat equation.

• Solve for the temperature distribution.

• Apply Fourier’s law to determine the heat flux.

Simplest Case: One-Dimensional, Steady-State Conduction with No Thermal Energy

Generation.

• Common Geometries:

– The Plane Wall: Described in rectangular (x) coordinate. Area

perpendicular to direction of heat transfer is constant (independent of x).

– The Tube Wall: Radial conduction through tube wall.

– The Spherical Shell: Radial conduction through shell wall.

Methodology of a Conduction Analysis


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Consider a plane wall between two fluids of different temperature:

The Plane Wall

• Implications:

Heat Equation:

• Boundary Conditions: ,1 ,20 , s sT T T L T

• Temperature Distribution for Constant k :k

For steady-state conditions with no distributed

source or sink of energy within the wall

the heat flux is a constant, independent of x


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• Heat Flux and Heat Rate:

Thermal Resistances and Thermal Circuits:

o Conduction in a plane wall:

o Convection

:o Thermal circuit for plane wall with adjoining fluids:

The Plane Wall


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Composite Wall with Negligible Contact Resistance:

The Plane Wall


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Overall Heat Transfer Coefficient (U) :

A modified form of Newton’s Law of Cooling to encompass multiple resistances

to heat transfer.

Where

The Plane Wall


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Series – Parallel Composite Wall:

The Plane Wall


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The Plane Wall


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• Note departure from one-dimensional conditions

for .

• Circuits based on assumption of isothermal

surfaces normal to x direction or adiabatic

surfaces parallel to x direction provide

approximations for qx .


The Plane Wall

∦kF kG


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The Plane Wall

Series – Parallel Resistances :

Thermal Resistance for Unit Surface Area:

,t condL

Rk

,1

t convRh

Units: W/KtR 2m K/WtR

,1

t radr

Rh


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The Plane Wall

Contact Resistance:

In composite systems, the temperature drop

across the interface between materials may

be appreciable.

This temperature change is attributed to what

is known as the thermal contact resistance

,,

t ct c

c

RR

A

,

A Btc

x

T TR

q


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Contact Resistance:


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To reduce the heat loss rate, a person wears special sporting gear (snow suit and wet suit) made

from a nanostructured silica aerogel insulation with an extremely low thermal conductivity of

0.014 W/m K. Consider surroundings conditions (air or water) are at 10oC. The emissivity of the

outer surface of the snow and wet suits is 0.95. What thickness of aerogel insulation is needed to

reduce the heat loss rate to 100 W (a typical metabolic heat generation rate) in air and water?

What are the resulting skin temperatures?

Example

For air


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Example cont.


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Example cont.


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Example cont.


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• Heat Equation:

Cylindrical tube and shell

• Temperature Distribution for Constant k :

Given that


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• Heat Flux and Heat Rate:

Conduction Resistance:

Why is it inappropriate to base the thermal resistance on a unit surface area?

K/W



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Composite Wall with Negligible Contact Resistance



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This definition is arbitrary, and the overall coefficient may also be defined in terms

of A4 or any of the intermediate areas.


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1-

22Process Heat Transfer



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Critical Thickness of Insulation


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Example

Insulation thickness is


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Example cont.


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• Heat Equation

Spherical Shell

• Temperature Distribution for Constant k :

= 0.0

But

1/,1 ,1 ,2

1 2

1

1 /s s s

r rT r T T T

r r


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• For Composite Shell:

and

and

Spherical Shell

for steady-state conditions with no heat generation and no heat loss

from the sides


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A 3-m internal diameter spherical tank made of 2-cm-thick stainless steel (k 15 W/m · °C) is

used to store iced water at T1 = 0°C. The tank is located in a room whose temperature is T2 =

22°C. The walls of the room are also at 22°C. The outer surface of the tank is black and heat

transfer between the outer surface of the tank and the surroundings is by natural convection and

radiation. The convection heat transfer coefficients at the inner and the outer surfaces of the tank

are h1 80 W/m2 · °C and h2 10 W/m2 · °C, respectively. Determine (a) the rate of heat transfer

to the iced water in the tank and (b) the amount of ice at 0°C that melts during a 24-h period.

Example


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Example cont.


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Example cont.


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Example cont.


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the amount of ice that will melt during a 24-h period is

Example cont.


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Example


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Assessment of thermal barrier coating (TBC) for protection of turbine blades. Determine

maximum blade temperature with and without TBC.

Schematic:

Example cont.


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With a heat flux of

ANALYSIS: For a unit area, the total thermal resistance with the TBC is

Example cont.


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Example cont.

Process Heat Transfer Lec 3: Heat Conduction Equation

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