Mechanical Engineering Exam > Mechanical Engineering Videos > Topper Handwritten Notes & Videos for GATE ME > Radial Conduction Through Hollow and Composite Sphere Coordinates HMT Tutorial- 23, Heat and Mass Transfer
Radial Conduction Through Hollow and Composite Sphere Coordinates HMT Tutorial- 23, Heat and Mass Transfer Video Lecture | Topper Handwritten Notes & Videos for GATE ME - Mechanical Engineering
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FAQs on Radial Conduction Through Hollow and Composite Sphere Coordinates HMT Tutorial- 23, Heat and Mass Transfer Video Lecture - Topper Handwritten Notes & Videos for GATE ME - Mechanical Engineering
| 1. What is radial conduction? |  |
Ans. Radial conduction refers to the transfer of heat or mass through a medium in a radial direction, from the center to the outer surface or vice versa.
| 2. What is the significance of hollow and composite spheres in heat and mass transfer? | |
Ans. Hollow and composite spheres are important in heat and mass transfer as they represent commonly encountered geometries in various engineering applications. Studying conduction through these spheres helps in understanding the behavior of heat and mass transfer in practical scenarios.
| 3. How can the radial conduction through a hollow sphere be calculated? | |
Ans. The radial conduction through a hollow sphere can be calculated using the formula:
q = (4πk(T1 - T2))/(1/r2 - 1/r1)
where q is the rate of heat transfer, k is the thermal conductivity of the material, T1 and T2 are the temperatures at the inner and outer surfaces of the sphere, r1 and r2 are the radii of the inner and outer surfaces of the sphere.
| 4. What is the formula for calculating the radial conduction through a composite sphere? | |
Ans. The radial conduction through a composite sphere can be calculated using the formula:
q = (4πk(T1 - T2))/(1/r2 - 1/r1 + ∑(δ1/k1) + ∑(δ2/k2))
where q is the rate of heat transfer, k is the thermal conductivity of the materials, T1 and T2 are the temperatures at the inner and outer surfaces of the sphere, r1 and r2 are the radii of the inner and outer surfaces of the sphere, δ1 and δ2 are the thicknesses of the respective layers, and k1 and k2 are the thermal conductivities of the respective layers.
| 5. What are the applications of radial conduction through hollow and composite spheres? | |
Ans. The understanding of radial conduction through hollow and composite spheres is widely applicable in various engineering fields. It is used in the design and analysis of heat exchangers, insulation materials, and thermal protection systems. It also finds application in the study of geothermal energy extraction and underground heat transfer.
Text Transcript from Video
so this is little bit probably this
video will be little bit lengthy because
today in this video taught well I'm
going to share with you the radial
conduction heat transfer through hollow
sphere as well as composite spherical
coordinate well both the topic topics
are most important and hopefully we'll
enjoy the total so let's go for it
so first of all first order element
first half of the total I am going to
share with you always fear
I mean how radial conduction of heat
transfer we can find out because for the
case of hollow sphere so first of all we
have two door oh oh we have to draw a
hollow sphere right and suppose that
this is r1 and this is r2 this is
internal ready and this is outer radial
of the sphere okay it's a spherical
cotton suppose that okay now now I'm not
going to share with you what to write
what to do next I'm going to share with
you what exactly you need to do next for
finding out the radial heat transfer
okay now most of the time what happen if
you have be begging for complete
examination then what do you have to do
we have to just remember this form this
diagram and you need to explain it what
you were doing exactly I mean
construction of this diagram right so
for derivation point of view what you
have to do we need to know if you are
deriving anything for conduction then
you need to known for years law for
years slow what this law tells you is
equal to minus K DT by DX this law is
all about this okay and this Q is in
joules per second or what most important
thing is that for years law is this law
is only for conduction so you will not
confuse will not get confused okay you
don't need to be confused if you wanted
to drive a radial conduction heat
transfer through always now we know that
we know this equation right so what
happened we know what is K thermal
conductivity we know DT we do DT DX and
of area you know what is the formula for
finding
area of sphere that is 4 PI R square 4
PI R square right if R is the radius now
we are having a hollow sphere so there
are two radius one is R 1 and 100 is R
two and again suppose that here
temperature is T 1 and temperature T 2
so we are having two temperature T 1 and
K 2 so best way to utilize these two
things is by doing integration so we are
going to integrate these steps while we
can get this one also and we can get
this one also when we can get the exact
value of it
so here DX is not the X here DX is dr
because if conduction is taking place
radially that's why we have to get here
the topic itself what we have to do okay
now let me use another DEATH okay so
what exactly we're going to do here
we're going to integrate this equation
suppose that we are having Q and R 1 to
R 2 so dr and 4 PI R square let's say R
square + 4 5 ok now t 1 to t 2 and here
DT this is it this is it there is
nothing left I think so right now we
have to until this one and this one
integration of this one is quite simple
because we know K 4 PI e t1 to t2 here R
1 to R 2 minus 2 because it gives
monoxide so minus 2 plus 1 minus 2 plus
1 R 1 2 here minus T 4 PI T 2 minus T 1
if I will multiply this minus sign with
this then what will happen
k 4 K PI T 1 minus T 2 okay now here
what we have to do - 1 by
our one that means okay that's okay
and R 1 to R 2 so minus 1 let's take
this minus sign outside - I'm sorry R 2
minus R 1
okay is equal to the same - is equal to
the same thing to do okay don't write
this do in the examination because if
you write this thing in the examination
the public Simon is not going to be B
marks but if you are be painting for
companies I mentioned don't require to
do derivation if they'll ask I mean if
you have people employed in subjective
examination subject accompanies
dimensions of like UPSC then this is
this is required but don't again don't
repeat this okay this is for
understanding ology now so what we got
here this one we have to do something
with this one so what we got here R 1 R
2 R 1 minus R 1 R 2 so R 2 minus R 1
then what will happen
R 1 minus R 2
right so we got it now because minus n
is 2 this side this way this will become
R 2 minus R 1 R 1 R 2 okay so what
you've got here this side we've got this
side so so what will be the ultimate one
I mean Q is also here we have left some
calculate Q is also here is also here so
ultimately after doing all these what we
have been to get Q is equal to 2 T 1
minus T 2 R 2 minus R 1 4 by killing R 1
and this R 2 minus R 1 this one is known
as RTS thermal resistance now most
important thing well we have to write
the equation for hallway sphere how to
find out Q how to find out you we caught
the formula know what we have to do we
have to find out find out what we have
to find out and
everything that is thermal circuit what
is thermal circuit we need to know with
the out knowing that so the thermal
circuit for this follow event would be
like this so suppose that this is yeah
and d1 d2 there is heat transfer in here
and this is our D H so our D s is equal
to what r2 minus r1 4 pi e r1 alpha this
is the rth thermal resistance for this
hollow sphere so we got this formula
this formula is most important role for
finding out conduction radially
conduction in case of hollow sphere no
no now in the next part and this one is
for followers right I'm going to just I
am going to share with you this one also
composite vehicle coordinate because
most of the time you feel if your third
be bearing for complete examination and
objective type then you are going to get
question from here maybe but for
preparing subjective type then most of
the time you are going to get question
from here because from here question
will be little bit lengthier okay so
let's go for next one vehicle you have
to get another base yes so this
derivation will be for spherical
spherical compose each container and
mercy okay so here suppose that we are
having a hollow sphere then this is a
computer this is a different material
this is made of and this one is made of
different material so this is the
composite one suppose that this one is R
1 this one is R 2 and this one is R 3 T
1 T 2 and T 3 so I am NOT going to rule
the whole details as I have gone through
for follow
they'll coordinate I'm here to share
with you exactly what you have to do
okay now first draw a thermal circuit
diagram this is for t1 this is t2 and
this is t3
so we have to get our t1 and our th2 if
any how we can manage to get these two
then we can easily write or easily find
out heat conduction in this in this case
right so how we can able to get that we
know that
RT s is equal to RT R 2 minus R 1 for 5k
r 1 r 2 for this case and for this case
what will happen R 3 minus R so 4 PI K R
3 R 2
now we know that Q is equal to T 1 minus
T 3 T 1 minus degrees according to the
circuit and writing and summation of RT
s total thermal resistance because of
this components a little coordinate or
container so T 1 minus T 3 summation R 2
minus R 1 we just need to write please
for 5k R 1 R 2 plus R 3 minus R 2 for 5
k r3 r2 what do you think I'm very fast
I am not exactly because this is
repeating I don't want to waste your
time I want a youtube lifetime every
second I hope you understand the topic
right how to find out transfer of heat
and this is Joule per second or what so
this is all about this lecture thanks
for watching see you soon in my next
video bye bye
video will be little bit lengthy because
today in this video taught well I'm
going to share with you the radial
conduction heat transfer through hollow
sphere as well as composite spherical
coordinate well both the topic topics
are most important and hopefully we'll
enjoy the total so let's go for it
so first of all first order element
first half of the total I am going to
share with you always fear
I mean how radial conduction of heat
transfer we can find out because for the
case of hollow sphere so first of all we
have two door oh oh we have to draw a
hollow sphere right and suppose that
this is r1 and this is r2 this is
internal ready and this is outer radial
of the sphere okay it's a spherical
cotton suppose that okay now now I'm not
going to share with you what to write
what to do next I'm going to share with
you what exactly you need to do next for
finding out the radial heat transfer
okay now most of the time what happen if
you have be begging for complete
examination then what do you have to do
we have to just remember this form this
diagram and you need to explain it what
you were doing exactly I mean
construction of this diagram right so
for derivation point of view what you
have to do we need to know if you are
deriving anything for conduction then
you need to known for years law for
years slow what this law tells you is
equal to minus K DT by DX this law is
all about this okay and this Q is in
joules per second or what most important
thing is that for years law is this law
is only for conduction so you will not
confuse will not get confused okay you
don't need to be confused if you wanted
to drive a radial conduction heat
transfer through always now we know that
we know this equation right so what
happened we know what is K thermal
conductivity we know DT we do DT DX and
of area you know what is the formula for
finding
area of sphere that is 4 PI R square 4
PI R square right if R is the radius now
we are having a hollow sphere so there
are two radius one is R 1 and 100 is R
two and again suppose that here
temperature is T 1 and temperature T 2
so we are having two temperature T 1 and
K 2 so best way to utilize these two
things is by doing integration so we are
going to integrate these steps while we
can get this one also and we can get
this one also when we can get the exact
value of it
so here DX is not the X here DX is dr
because if conduction is taking place
radially that's why we have to get here
the topic itself what we have to do okay
now let me use another DEATH okay so
what exactly we're going to do here
we're going to integrate this equation
suppose that we are having Q and R 1 to
R 2 so dr and 4 PI R square let's say R
square + 4 5 ok now t 1 to t 2 and here
DT this is it this is it there is
nothing left I think so right now we
have to until this one and this one
integration of this one is quite simple
because we know K 4 PI e t1 to t2 here R
1 to R 2 minus 2 because it gives
monoxide so minus 2 plus 1 minus 2 plus
1 R 1 2 here minus T 4 PI T 2 minus T 1
if I will multiply this minus sign with
this then what will happen
k 4 K PI T 1 minus T 2 okay now here
what we have to do - 1 by
our one that means okay that's okay
and R 1 to R 2 so minus 1 let's take
this minus sign outside - I'm sorry R 2
minus R 1
okay is equal to the same - is equal to
the same thing to do okay don't write
this do in the examination because if
you write this thing in the examination
the public Simon is not going to be B
marks but if you are be painting for
companies I mentioned don't require to
do derivation if they'll ask I mean if
you have people employed in subjective
examination subject accompanies
dimensions of like UPSC then this is
this is required but don't again don't
repeat this okay this is for
understanding ology now so what we got
here this one we have to do something
with this one so what we got here R 1 R
2 R 1 minus R 1 R 2 so R 2 minus R 1
then what will happen
R 1 minus R 2
right so we got it now because minus n
is 2 this side this way this will become
R 2 minus R 1 R 1 R 2 okay so what
you've got here this side we've got this
side so so what will be the ultimate one
I mean Q is also here we have left some
calculate Q is also here is also here so
ultimately after doing all these what we
have been to get Q is equal to 2 T 1
minus T 2 R 2 minus R 1 4 by killing R 1
and this R 2 minus R 1 this one is known
as RTS thermal resistance now most
important thing well we have to write
the equation for hallway sphere how to
find out Q how to find out you we caught
the formula know what we have to do we
have to find out find out what we have
to find out and
everything that is thermal circuit what
is thermal circuit we need to know with
the out knowing that so the thermal
circuit for this follow event would be
like this so suppose that this is yeah
and d1 d2 there is heat transfer in here
and this is our D H so our D s is equal
to what r2 minus r1 4 pi e r1 alpha this
is the rth thermal resistance for this
hollow sphere so we got this formula
this formula is most important role for
finding out conduction radially
conduction in case of hollow sphere no
no now in the next part and this one is
for followers right I'm going to just I
am going to share with you this one also
composite vehicle coordinate because
most of the time you feel if your third
be bearing for complete examination and
objective type then you are going to get
question from here maybe but for
preparing subjective type then most of
the time you are going to get question
from here because from here question
will be little bit lengthier okay so
let's go for next one vehicle you have
to get another base yes so this
derivation will be for spherical
spherical compose each container and
mercy okay so here suppose that we are
having a hollow sphere then this is a
computer this is a different material
this is made of and this one is made of
different material so this is the
composite one suppose that this one is R
1 this one is R 2 and this one is R 3 T
1 T 2 and T 3 so I am NOT going to rule
the whole details as I have gone through
for follow
they'll coordinate I'm here to share
with you exactly what you have to do
okay now first draw a thermal circuit
diagram this is for t1 this is t2 and
this is t3
so we have to get our t1 and our th2 if
any how we can manage to get these two
then we can easily write or easily find
out heat conduction in this in this case
right so how we can able to get that we
know that
RT s is equal to RT R 2 minus R 1 for 5k
r 1 r 2 for this case and for this case
what will happen R 3 minus R so 4 PI K R
3 R 2
now we know that Q is equal to T 1 minus
T 3 T 1 minus degrees according to the
circuit and writing and summation of RT
s total thermal resistance because of
this components a little coordinate or
container so T 1 minus T 3 summation R 2
minus R 1 we just need to write please
for 5k R 1 R 2 plus R 3 minus R 2 for 5
k r3 r2 what do you think I'm very fast
I am not exactly because this is
repeating I don't want to waste your
time I want a youtube lifetime every
second I hope you understand the topic
right how to find out transfer of heat
and this is Joule per second or what so
this is all about this lecture thanks
for watching see you soon in my next
video bye bye
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