1) Heat transfer by conduction occurs due to random molecular motion within a material. The rate of heat transfer by conduction is proportional to the temperature gradient and the thermal conductivity of the material.
2) Fourier's law of heat conduction describes conduction in Cartesian, cylindrical, and spherical coordinate systems. It relates the heat flux to the temperature gradient through the thermal conductivity.
3) The heat equation can be derived by applying the law of conservation of energy combined with Fourier's law. It describes the distribution of temperature as a function of time and space within a body undergoing transient or steady-state heat conduction.
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-I.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Contents: Why Numerical methods? – Advantages – Finite difference formulation from differential eqns – 1D steady state conduction in cartesian coordinates – formulation by energy balance method – different BC’s – Problems
Lectures on Heat Transfer - Introduction - Applications - Fundamentals - Gove...tmuliya
This file contains Introduction to Heat Transfer and Fundamental laws governing heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
Compressible flows in fluid mechanics in chemical engineeringUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process.
Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow.
Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths.
Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference.
Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow.
Determine the friction factor and Nusselt number in fully developed turbulent flow using empirical relations, and calculate the heat transfer rate.
This file contains slides on NUMERICAL METHODS IN STEADY STATE 1D and 2D HEAT CONDUCTION – Part-I.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students.
Contents: Why Numerical methods? – Advantages – Finite difference formulation from differential eqns – 1D steady state conduction in cartesian coordinates – formulation by energy balance method – different BC’s – Problems
Lectures on Heat Transfer - Introduction - Applications - Fundamentals - Gove...tmuliya
This file contains Introduction to Heat Transfer and Fundamental laws governing heat transfer.
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India.
This file contains slides on Transient Heat conduction: Part-I
The slides were prepared while teaching Heat Transfer course to the M.Tech. students in Mechanical Engineering Dept. of St. Joseph Engineering College, Vamanjoor, Mangalore, India, during Sept. – Dec. 2010. Contents: Lumped system analysis – criteria for lumped system analysis – Biot and Fourier Numbers – Response time of a thermocouple - One-dimensional transient conduction in large plane walls, long cylinders and spheres when Bi > 0.1 – one-term approximation - Heisler and Grober charts- Problems
Compressible flows in fluid mechanics in chemical engineeringUsman Shah
This slide will explain you the chemical engineering terms .Al about the basics of this slide are explain in it. The basics of fluid mechanics, heat transfer, chemical engineering thermodynamics, fluid motions, newtonian fluids, are explain in this process.
Automobile Management System Project Report.pdfKamal Acharya
The proposed project is developed to manage the automobile in the automobile dealer company. The main module in this project is login, automobile management, customer management, sales, complaints and reports. The first module is the login. The automobile showroom owner should login to the project for usage. The username and password are verified and if it is correct, next form opens. If the username and password are not correct, it shows the error message.
When a customer search for a automobile, if the automobile is available, they will be taken to a page that shows the details of the automobile including automobile name, automobile ID, quantity, price etc. “Automobile Management System” is useful for maintaining automobiles, customers effectively and hence helps for establishing good relation between customer and automobile organization. It contains various customized modules for effectively maintaining automobiles and stock information accurately and safely.
When the automobile is sold to the customer, stock will be reduced automatically. When a new purchase is made, stock will be increased automatically. While selecting automobiles for sale, the proposed software will automatically check for total number of available stock of that particular item, if the total stock of that particular item is less than 5, software will notify the user to purchase the particular item.
Also when the user tries to sale items which are not in stock, the system will prompt the user that the stock is not enough. Customers of this system can search for a automobile; can purchase a automobile easily by selecting fast. On the other hand the stock of automobiles can be maintained perfectly by the automobile shop manager overcoming the drawbacks of existing system.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Quality defects in TMT Bars, Possible causes and Potential Solutions.PrashantGoswami42
Maintaining high-quality standards in the production of TMT bars is crucial for ensuring structural integrity in construction. Addressing common defects through careful monitoring, standardized processes, and advanced technology can significantly improve the quality of TMT bars. Continuous training and adherence to quality control measures will also play a pivotal role in minimizing these defects.
Sachpazis:Terzaghi Bearing Capacity Estimation in simple terms with Calculati...Dr.Costas Sachpazis
Terzaghi's soil bearing capacity theory, developed by Karl Terzaghi, is a fundamental principle in geotechnical engineering used to determine the bearing capacity of shallow foundations. This theory provides a method to calculate the ultimate bearing capacity of soil, which is the maximum load per unit area that the soil can support without undergoing shear failure. The Calculation HTML Code included.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Event Management System Vb Net Project Report.pdfKamal Acharya
In present era, the scopes of information technology growing with a very fast .We do not see any are untouched from this industry. The scope of information technology has become wider includes: Business and industry. Household Business, Communication, Education, Entertainment, Science, Medicine, Engineering, Distance Learning, Weather Forecasting. Carrier Searching and so on.
My project named “Event Management System” is software that store and maintained all events coordinated in college. It also helpful to print related reports. My project will help to record the events coordinated by faculties with their Name, Event subject, date & details in an efficient & effective ways.
In my system we have to make a system by which a user can record all events coordinated by a particular faculty. In our proposed system some more featured are added which differs it from the existing system such as security.
Courier management system project report.pdfKamal Acharya
It is now-a-days very important for the people to send or receive articles like imported furniture, electronic items, gifts, business goods and the like. People depend vastly on different transport systems which mostly use the manual way of receiving and delivering the articles. There is no way to track the articles till they are received and there is no way to let the customer know what happened in transit, once he booked some articles. In such a situation, we need a system which completely computerizes the cargo activities including time to time tracking of the articles sent. This need is fulfilled by Courier Management System software which is online software for the cargo management people that enables them to receive the goods from a source and send them to a required destination and track their status from time to time.
4. Conduction:
The heat flux (W/m2), q’’ is the heat transfer rate in the x direction per unit area
perpendicular to the direction of transfer, and it is proportional to the temperature gradient, dT/dx, in
this direction. The parameter k is a transport property known as the thermal conductivity (W/m K) and is
a characteristic of the wall material. The minus sign is a consequence of the fact that heat is transferred
in the direction of decreasing temperature.
Under the steady-state conditions
and the heat flux
or
5. Problem.1 The wall of an industrial furnace is constructed from 0.15-m-thick fireclay brick having a
thermal conductivity of 1.7 W/m K. Measurements made during steadystate operation reveal
temperatures of 1400 and 1150 K at the inner and outer surfaces, respectively. What is the rate of heat
loss through a wall that is 0.5 m by 1.2 m on a side? (Ans: 1700W).
Heat Diffusion Equation:
The conduction heat rates
perpendicular to each of
the control surfaces at the
x, y, and z coordinate
locations are indicated by
the terms qx, qy, and qz,
respectively. The conduction
heat rates at the opposite
surfaces can then be expressed
as a Taylor series expansion
where, neglecting higher order terms,
6. Within the medium there may also be an energy source term associated with the rate of thermal energy
generation. This term is represented as
and the energy storage term may be expressed as
the general form of the conservation of energy requirement is
7. substitute all inthe conservation of energy equation
and therefore
The conduction heat rates may be evaluated from Fourier’s law
is the general form, in Cartesian coordinates, of the heat diffusion equation. This equation, often referred to as
the heat equation
8. if the thermal conductivity is constant, the heat equation is
there can be no change in the amount of energy storage
Cylidrical Co ordinates
9. dt
dr
r
T
dz
rd
k
qr .
)
(
r
r q
r
q
)
(
)
( r
r
r
r q
r
q
r
q
q
heat is transfered in the cylindrical coordinates are radial, angular and axial directions
apply the fourier law of conduction for this coordinates
heat transfer radial (dr) direction:
heat inflow for time rate ‘dt’
heat out flow is =
net heat accumulate or efflux in radial direction is = heat in - heat out =
11. dt
r
T
dz
rd
dr
k
dt
rd
r
T
drdz
k
q .
).
.
.
.(
.
)
(
Heat transfer in angular (rdθ) direction:
heat inflow , qθ,
heat out flow =
Net heat efflux in angular direction is =
heat transfer in axial direction:
heat inflow , qz,
heat out flow is =
q
r
q
)
(
)
(
q
r
q
r
q
q
dt
r
T
dz
rd
dr
k
dt
rd
r
T
drdz
k
r
.
.
.
.
.
.
.
2
2
2
dt
dz
z
T
rd
dr
k
qz .
)
.
.(
z
z q
z
q
12. net heat efflux along axial (z) direction is =
Heat generated within the volume = qv =
net heat rate stored within the volume =
)
(
)
( z
z
z
z q
z
q
z
q
q
dt
z
T
dz
rd
dr
k
dt
z
T
z
dz
rd
dr
k
dt
dz
z
T
rd
dr
k
z
.
)
.
.
.(
)
.
.
.(
.
.
(
2
2
dt
dz
rd
dr
q ).
.
.
.(
dt
t
T
C
dz
rd
dr
dt
t
T
C
V
dt
t
T
C
m p
p
p
.
).
.
.
.(
.
.
.
.
.
13. therefore the energy balance of the control volume
net Heat efflux within the volume + = rate of heat stored in the volume
Cancel the volume (dr.rdθ.dz) , on both sides and divide the thermal conductivity ‘k’ , then we get
and finally,
Where α is the thermal diffusivity =
dt
t
T
C
dz
rd
dr
dt
dz
rd
dr
q
dt
z
T
dz
rd
dr
k
dt
r
T
dz
rd
dr
k
dt
r
T
r
r
T
dz
rd
dr
k p
.
).
.
.
.(
).
.
.
.(
.
)
.
.
.(
.
.
.
.
.
1
)
.
.
( 2
2
2
2
2
2
2
t
T
k
C
k
q
z
T
r
T
r
T
r
r
T p
2
2
2
2
2
2
2
1
t
T
k
q
z
T
r
T
r
T
r
r
T
1
1
2
2
2
2
2
2
2
p
C
k
15. net heat efflux in radial, angular and peripharal directions are
apply the energy balance
the above equation is divided by volume
take
now
q
q
q
q
q
q d
d
r
dr
r
t
T
C
d
r
rd
dr
d
r
rd
dr
q
q
q
q
q
q
q p
d
d
r
dr
r
.
.
sin
.
.
.
sin
.
.
t
T
C
q
d
r
rd
dr
q
q
d
r
rd
dr
q
q
d
r
rd
dr
q
q
p
d
d
r
dr
r
.
sin
.
.
.
sin
.
.
.
sin
.
.
d
dq
d
q
q
d
dq
d
q
q
dr
dq
dr
q
q d
d
r
r
dr
r
;
;
t
T
C
q
d
dq
r
rd
dr
d
dq
d
r
r
dr
dr
dq
d
r
rd
p
r
.
.
sin
.
.
1
.
.
sin
.
.
1
.
.
.
sin
.
1
17. Boundary conditions
• To determine the
in a medium, it is necessary to solve the
appropriate form of the heat equation.
• Such a solution depends on the physical
conditions existing at the boundaries of the
medium and, if the situation is time
dependent, on conditions existing in the
medium at some initial time.
• The first condition corresponds to a
situation for which the surface is
maintained at a fixed temperature Ts. It is
commonly termed a , or
a boundary condition of the first kind.
• The second condition corresponds to the
existence of a fixed or constant heat flux
qs’’ at the surface.It is termed a
is the
existence of convection heating or cooling
at the surface
18. Problems
2. Assume steady-state, one-dimensional heat conduction through the symmetric shape shown.
Assuming that there is no internal heat generation, derive an expression for the
thermal conductivity k(x) for these conditions: A(x)= (1 - x), T(x) = 300 (1 - 2x - x3),
and q = 6000 W, where A is in square meters, T in kelvins, and x in meters. Ans: k=
3. A one-dimensional plane wall of thickness 2L=100 mm experiences uniform thermal energy generation
of q=1000 W/m3and is convectively cooled at x= 50 mm by an ambient fluid characterized by Tα= 20oC. If
the steady-state temperature distribution within the wall is T(x) = a(L2-x2)+b where a= 10 C/m2 and b = 30
C, what is the of the wall? What is the value of the convection
?
4. In the two-dimensional body illustrated, the gradient at surface A is found
to be δT/δy= 30 K/m. What are δT/δy and δT/δx at surface B? Ans:60 K/m.
b. Consider the above problem , for the case where the thermal conductivity
varies with temperature as k= (ko+aT), where ko= 10W/m.K,
a=-10-3 W/m.K2, and T is in kelvins. The gradient at surface B is δT/δx=30 K/m.
What is δT/δy at surface A? Ans=14.85 K/m
19. 5. At a given instant of time, the temperature distribution within an infinite homogeneous body is given
by the function
Assuming constant properties and no internal heat generation, determine the regions where the
temperature changes with time.[Ans: temp distribution is independant with time]
6.A cylindrical rod of stainless steel is insulated on its exterior surface except for the ends. The steady-
state temperature distribution is T(x)= a - bx/L, where a=305 K and b=10 K. The diameter and length of
the rod are D=20 mm and L=100 mm, respectively. Determine the heat flux along the rod, The mass of
the rod is M= 0.248 kg.[Ans=1490 W/m2]
7.Uniform internal heat generation at q=5x107W/m3 is occurring in a cylindrical nuclear reactor fuel rod
of 50 mm diameter, and under steady-state conditions the temperature distribution is of the form
T(r)=a+ br2, where T is in degrees Celsius and r is in meters, while a= 800oC and b= - 4.167 x105 oC/m2.
The fuel rod properties are k=30 W/m K, ρ=1100 kg/m3, and Cp= 800 J/kg K.
(a) What is the rate of heat transfer per unit length of the rod at r= 0 (the centerline) and at
r=25 mm (the surface)?
(b) If the reactor power level is suddenly increased to q2= 108 W/m3, what is the initial time
rate of temperature change at r= 0 and r=25 mm?
20. • For 1DConduction in a plane wall, temperature is a
function of the x-coordinate only and heat is transferred
exclusively in this direction.
• In Figure , a plane wall separates two fluids of different
temperatures. Heat transfer occurs by convection from
the hot fluid at Tα,1 to one surface of the wall at Ts,1, by
conduction through the wall, and by convection from the
other surface of the wall at Ts,2 to the cold fluid at Tα,2.
• We begin by considering conditions within the wall. We
first determine the temperature distribution, from which
we can then obtain the conduction heat transfer rate.
Temperature Distribution
• For steady-state conditions with no distributed source or
sink of energy within the wall, the appropriate form of the
heat equation is
Conduction- Plane wall
21. 2
1
)
( C
x
C
x
T
• equation may be integrated twice to obtain the general solution is
• To obtain the constants of integration, C1 and C2, boundary conditions must be introduced. We choose
to apply conditions of the first kind at x = 0 and x = L, in which case
• T(0) = Ts,1 and T(L) = Ts,2. therfore Ts,1 = C2 then the equation is re write T(x) = C1x +Ts,1, now apply
second boundary condition T(L) = Ts,2 ch gives Ts,2 = C1L+Ts,1; (Ts,2- Ts,1)/L = C2 substitute C2 value in
the comman equation we get
• From this result it is evident that, for one-dimensional, steady-state conduction in a plane wall with no
heat generation and constant thermal conductivity, the temperature varies linearly with x.
• this temperature distribution may use Fourier’s law
• then the heat flux is
1
,
1
,
2
,
)
( Ts
x
L
Ts
Ts
x
T
22. • Conduction and convection resistances are in series and may be summed, and
The Composite Wall
Alternatively
interms ovf overall heat transfer co efficient
in general
23. • Composite walls may also be characterized by
series–parallel configurations
Contact Resistance
• normally it is neglected, but, 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, Rt,c
24.
25. Conduction : Porous Media
• In many applications, heat transfer occurs within porous media that are combinations of a
stationary solid and a fluid.
• When the either a , the resulting to be saturated.
in an unsaturated porous medium.
• A saturated porous medium that consists of a stationary solid phase through which a fluid flows is
referred to as a packed bed.
• heat rate may be expressed where keffis an effective thermal conductivity.
• effective thermal conductivity of a saturated porous medium consisting of an interconnected solid
phase within which a dilute distribution of spherical fluid regions exists, resulting in an expression
of the form
• Equation is valid for relatively small porosities (ɛ ≤ 0.25).
26. • Problem1. A thin silicon chip and an 8-mm-thick aluminum substrate are separated by a 0.02-mm-
thick epoxy joint. The chip and substrate are each 10 mm on a side, and their exposed surfaces are
cooled by air, which is at a temperature of 25oC and provides a convection coefficient of 100 W/m2 K.
If the chip dissipates 104 W/m2 under normal conditions, will it operate below a maximum allowable
temperature of 85oC? (take the epoxy conductivity resistance 0.9x10-4 m2K/W and the thermal
conducity of aluminium is 273W/mK).[Ans:Tg=75.3oC, it will work]
• Problem 2. A conical section fabricated from pyroceram ( K=3.46W/mK at 500K). It is of circular cross
section with the diameter D= ax, where a= 0.25. The small end is at x1 = 50 mm and the large end at
x2=250 mm. The end temperatures are T1=400 K and T2= 600 K, while the lateral surface is well
insulated.
1. Derive an expression for the temperature distribution T(x) in symbolic form, assuming one-
dimensional conditions. Sketch the temperature distribution.
2. Calculate the heat rate qx through the cone.
27. • Temperature distribution without and with heat geration: (derivation please refer class notes)
• without heat generation
• with heat gereation
and in dimensionless form
• Problem 3. A current of 200 A is passed through a stainless-steel wire [k =19 W/m· ◦C] 3 mm in
diameter. The resistivity of the steel may be taken as 70 μΩ· cm, and the length of the wire is 1 m. The
wire is submerged in a liquid at 110 ◦C and experiences a convection heat-transfer coefficient of 4
kW/m2 · ◦C. Calculate the center temperature of the wire
Conduction - Cylinder
2
2
2
1
2
1
ln
ln
)
( T
r
r
r
r
T
T
r
T
w
T
r
R
k
q
r
T
2
2
4
)
(
2
0
1
R
r
T
T
T
T
w
w
28. • Adding insulation will always increase the conduction resistance.
• But in the case of a cylinder and sphere ,when the total resistance is made up of both conduction
resistance and convection resistance, the addition of insulation in some cases may reduce the
convection resistance due to the increase in surface area and the total resistance may actually
decrease resulting in increased heat flow.
• It may be shown that the heat flow actually first increases and then decreases in certain cases.
• The thickness upto which heat flow increases and after which heat flow decreases is termed as
critical thickness
• In the case of cylinders and spheres it is called critical radius.
• Cylinder:
• Total resistance, R, for radius r =
Cancelling the common terms
Critical thickness of insulation
rL
h
kL
r
r
2
1
2
1
ln
t
xcons
r
h
r
r
k
R tan
1
1
ln
1
1
29. 2
1
1
1
1
r
h
r
k
dr
dR
h
k
rcr
Equating to zero then we get
For sphere
Problem 3: Calculate the critical radius of insulation for asbestos [k =0.17 W/m◦C] surrounding a pipe
and exposed to room air at 20◦C with h=3.0 W/m2 ◦C. Calculate the heat loss from a 200◦C, 5.0-cm-
diameter pipe when covered with the critical radius of insulation and without insulation.