Heat Conduction and Transfer in Thermal Systems
Fourier's l
a
w of heat conduction
Heat conduction
•
Steady state heat conduction
•
Steady state
means that temperature in an
object may vary by location but it does not
change with time.
Fourier's law describes the rate of heat conduction in a solid
Boundary Conditions
•
x = x
1,
T = T
1
•
x = x
2
, T = T
2
•
Separating the variables,
•
Integrating from x
1
to x (some interior location within the slab)
Where L= thickness of the slab
Thermal Resistance concept:
•
In the flow of electricity, the electrical resistance is
determined from the ratio of electric potential divided by
the electric current. Similarly, we may consider heat
resistance as a ratio of the driving potential (temperature
difference) divided by the rate of heat transfer.
•
Thus,
HEAT CONDUCTION IN MULTILAYERED SYSTEMS
COMPOSITE RECTANGULAR WALL (IN SERIES)
In a single layer, the rate of heat transfer is:
•
Using the Resistance Concept:
Conduction in cylindrical objects
Boundary Conditions
T =T
i
at r = r
i
T=T
o
at r = r
0
Separating variables:
Integrating :
OVERALL HEAT TRANSFER COEFFICIENT
Heat transfer to the internal wall
Heat transfer through the wall
Where A
lm
•
Outside the wall
•
By summing the three equation 1,2, and 3 the
result is :
•
CRITICAL THICKNESS FOR INSULATION
COMPOSITE CYLINDRICAL TUBE
for a single-layer pipe:
and resistance due to conduction
Thermal resistance diagram:
Heat Conduction in a Spherical Shell
Fourier's Law in radial coordinates
Substituting the area of a sphere
Integrating, between r = r
1
and r
2
, and T
1
and T
2
The thermal resistance is expressed as
Conduction with internal heat
generation
•
In some cases it energy may be produced in an
identical shape inside the system like electrical
heaters or in respiration in fresh food products
such as fruits and vegetables.
Rate of
heat in
+ rate of heat
generation
= rate of
heat out
+ rate of heat
accumulation
By taking the limit
And for a pure conductive material
Note that if the heat generation zero then :
By integration
Then we get
•
At x=-L, T=T
w
•
And at x=L, T=T
w
•
And by solving 1 and 2
At x=0, T
o
=center temperature:
Total heat at both sides =total heat generated at
steady state case.
Home work #2
Temperature change with change of distance from the center of generation
Draw the relationship between temperature and distance x from the center of
the wall knowing that:
Home work should be submitted after 6 days from the lecture.
Explore the fundamental concepts of heat conduction and transfer, including Fourier's law, steady-state conduction, thermal resistance, multi-layered systems, cylindrical objects, and overall heat transfer coefficients. Discover how to analyze boundary conditions, calculate critical insulation thickness, and apply the resistance concept to solve heat transfer problems effectively.
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Presentation Transcript
Heat conduction Fourier's law of heat conduction
Steady state heat conduction Steady state means that temperature in an object may vary by location but it does not change with time.
Fourier's law describes the rate of heat conduction in a solid Boundary Conditions x = x1, T = T1 x = x2, T = T2 Separating the variables, Integrating from x1to x (some interior location within the slab)
Where L= thickness of the slab Thermal Resistance concept: In the flow of electricity, the electrical resistance is determined from the ratio of electric potential divided by the electric current. Similarly, we may consider heat resistance as a ratio of the driving potential (temperature difference) divided by the rate of heat transfer. Thus,
HEAT CONDUCTION IN MULTILAYERED SYSTEMS COMPOSITE RECTANGULAR WALL (IN SERIES) In a single layer, the rate of heat transfer is:
Conduction in cylindrical objects Fourier's Law in cylindrical coordinates Boundary Conditions T =Ti at r = ri T=To at r = r0 ??= ?(2???)?? ??
Separating variables: Integrating :
OVERALL HEAT TRANSFER COEFFICIENT Heat transfer to the internal wall Heat transfer through the wall Where Alm
Outside the wall By summing the three equation 1,2, and 3 the result is :
COMPOSITE CYLINDRICAL TUBE for a single-layer pipe: and resistance due to conduction Thermal resistance diagram:
Heat Conduction in a Spherical Shell Fourier's Law in radial coordinates Substituting the area of a sphere Integrating, between r = r1and r2, and T1and T2
Conduction with internal heat generation In some cases it energy may be produced in an identical shape inside the system like electrical heaters or in respiration in fresh food products such as fruits and vegetables.
Rate of heat in + rate of heat generation = rate of heat out + rate of heat accumulation By taking the limit And for a pure conductive material
Note that if the heat generation zero then : By integration Then we get
At x=-L, T=Tw And at x=L, T=Tw And by solving 1 and 2 At x=0, To=center temperature:
Total heat at both sides =total heat generated at steady state case.
Home work #2 Temperature change with change of distance from the center of generation Draw the relationship between temperature and distance x from the center of the wall knowing that: Ta q' L H k 25 5 1 10 2 C w/m2. k w/m. k w/m3 m Home work should be submitted after 6 days from the lecture.
