Mechanical Engineering Exam > Mechanical Engineering Videos > Topper Handwritten Notes & Videos for GATE ME > Derivation of Conduction in Cartesian Coordinates HMT Tutorial- 22
Derivation of Conduction in Cartesian Coordinates HMT Tutorial- 22 Video Lecture | Topper Handwritten Notes & Videos for GATE ME - Mechanical Engineering
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FAQs on Derivation of Conduction in Cartesian Coordinates HMT Tutorial- 22 Video Lecture - Topper Handwritten Notes & Videos for GATE ME - Mechanical Engineering
| 1. What are Caretisian coordinates in the context of conduction? |  |
Ans. Caretisian coordinates, also known as Cartesian coordinates, refer to a system of coordinates used to define the position of a point in three-dimensional space. In the context of conduction, these coordinates are used to describe the spatial distribution of temperature, heat, and mass transfer within a system.
| 2. How is conduction defined in the context of heat and mass transfer? | |
Ans. Conduction is the process of heat or mass transfer through a solid or fluid medium due to molecular diffusion. It occurs when there is a temperature or concentration gradient within the medium, causing the molecules to collide and transfer their energy or mass to neighboring molecules.
| 3. What is the significance of the HMT tutorial-22 in the GATE exam? | |
Ans. The HMT tutorial-22 is relevant to the GATE (Graduate Aptitude Test in Engineering) exam as it covers the topic of heat and mass transfer. This tutorial provides valuable insights and explanations related to conduction in Caretisian coordinates, which is an important concept in the field of heat and mass transfer. Understanding this topic can help candidates perform well in the GATE exam.
| 4. How can one derive the equation for conduction in Caretisian coordinates? | |
Ans. The equation for conduction in Caretisian coordinates can be derived using the Fourier's Law of Heat Conduction. This law states that the heat flux (q) in a solid is proportional to the negative gradient of temperature (dT/dx) and the thermal conductivity (k) of the material. Mathematically, it can be expressed as q = -k(dT/dx), where q is the heat flux, k is the thermal conductivity, and dT/dx is the temperature gradient.
| 5. What are some practical applications of conduction in Caretisian coordinates? | |
Ans. Conduction in Caretisian coordinates has various practical applications. Some examples include heat transfer in building materials, where the temperature distribution within walls and floors can affect energy efficiency; thermal management in electronic devices, where proper heat dissipation is crucial for optimal performance; and heat conduction in cooking utensils, where even heat distribution is important for efficient cooking. Understanding conduction in Caretisian coordinates helps engineers and scientists design and analyze systems involving heat and mass transfer.
Text Transcript from Video
My dear the Sujal Kemal scene and today
we are going to derive an equation the
conduction of I mean it does what is
going on for cut is in coordinates and
how to drive the equation if we get this
question in the university examination
so are you ready to go for it so we are
having a Cartesian coordinates right now
and in this coordinate we are
considering the area of the coordinate
as a and Q X is entering here and
outgoing heat is Q X plus DX right now
what is happening here we are saying
that we wanted to find out its you know
how much heat transfer is going on in
this coordinate system the ultimate aim
is that only right so first of all what
we are doing consider a difference and
element of the slab of length DX as we
have shown in the figure that length is
DX right now we're saying Q X is equal
to heat conducted into the element along
egg by X direction the same feed
conducted into the element along x
direction then minus a we know that t we
X is equal to minus ka DT by DX this is
the first equation we generally studied
that is known as Fourier law of for heat
transfer right now what we are saying Q
X plus DX what is it this is heat
conducted in conducted out of the
element
along X Direction is equal to Q X Carly
why Carly X Q X DX now you can see it
right so we got two equations here one
is for Q X another one is QX plus D X
both the equation is very useful and
very important for Cartesian coordinates
okay now what we have to do we have to
find out what no that equation I have to
show you that's why you can understand
more about it that is heat generated in
the element so what is that it generated
in the element so the equation giving
you a DX this is heat generated in the
element now we have to write another
equation for I mean through that
equation we easily find out the heat
transfer rate okay through these
Cartesian coordinates so the name of
that equation is energy balance equation
so we will write an equation energy
balance equation along X direction so
this equation is known as energy balance
equation along X direction right so what
would be that
I mean how we can write that equation
that is also interesting and that is one
of the interesting part I must say that
so what is that field heat conducted it
conducted into the element
it conducted into the element plus heat
generated in the in the heat conducted
into the element is equal to M si is
equal to it conducted out of the element
heat conducted out of the element plus
rate of change of internal energy rate
of change of internal energy of element
with expert to time rate of change of
internal energy energy of the element
with respect to time okay so these
things we are going to write in
mathematical form that's why you can
understand it much better way let me use
another color pen that's why it would be
quite easier for you to understand okay
so let's go for it
so what would be that cube X cube X plus
Q Jennsen
okay generation is equal to Q X plus DX
plus W by curly toe internal energy of
element so internal energy of elements
now what next Q X
q generation q x plus curly white ally X
cube X DX
curly bike a little m CP T where what is
M M is mass of the element and we know
density is equal to mass upon volume
therefore mass is equal to density into
fee-simple
okay now after writing this what we can
write here that Q X and P versus
cancelling out so Q generation is equal
to l QX q dash DX is equal to curly
Wycherley x minus ka DT by DX into DX
plus calibike a little row is the X C DT
into CP so what we are doing we are
writing a DX is volume this volume is
equal to a B X okay no no what next work
nice that is most important what we are
going to do we are going to send all
these men we just cancel out this ATX
ATX and ATX so we can cancel out right
now what will be remaining Q generation
curly by curly X minus a DT by DT X plus
Kalia PI Carlito Rho CPT right after you
know after calculation we can get this
also I mean with the sending this one
this side so curly by curly X K DT by DX
plus Q generation curly bike a little CP
T okay that's it now we have to
differentiate this so k d square I mean
college party Cali square
new generation curly Baikal Utah Rho CP
and this must say okay we can say this
CPT okay now what we got here
this is only for one axis X exists or
x-direction we must say that but if we
wanted to find out what we wanted to get
energy BLM situation for all the axes
for all the axes then what we should do
now all the axis means I'm talking about
and I have already shared with you then
we are going to find out the energy
balance I mean we are going to find out
the equation energy balance equation for
x axis y axis and z axis now we are
having we can already find out the
energy balance equation for x direction
now we have to find out y axis as well
as Z exists once we'll come to know the
energy balance equation for x axis we
can easily write that an energy balance
equation for y axis as well as for J
axis so we don't need to find out
separately to for for writing the energy
balance equation right so let's go for
it how to write I mean how to do that
and that's why you will not face any
kind of problem in it so what we can do
we just need to write the axis
accordingly so first of all killing
Kali's vertical yep square plus K Khalif
Square t can dy square plus scale Cali
Square t curly ja d square plus heat
generation is equal to so not inhalation
asleep we have the new trick denoting
this in in the way that so you can
understand it CP Cali T upon colleague
Tom so stone right now what exactly
doing we are sending this thermal
conductivity to this side and we are
dividing here I mean just taking common
from it and we are dividing here let me
share with you Callie squaty Callie X
square plus Callie's purty Kelly Y
square plus del square t plus curly
Jasper
you lord it's scale is equal to Rho CP
by give curly D by Carlito so this is
the equation this is the equation for
Cartesian coordinates so most of our
time we have been asked for all the XS
and maybe sometime we have been asked
for only X direction so this is how you
can derive the equation for what is in
coordinates in case of 1/2 conduction
only so most of the time you know there
are three types of coordinate system
which gently which we find out that is
any spherical as well as this
cylindrical so this one is Cartesian in
my next lectures I will definitely try
to cover all other types of coordinate
system there so I will not go if not go
to watch any other videos you will get
each and everything from here only ok so
thanks for watching see you soon in my
next video bye
we are going to derive an equation the
conduction of I mean it does what is
going on for cut is in coordinates and
how to drive the equation if we get this
question in the university examination
so are you ready to go for it so we are
having a Cartesian coordinates right now
and in this coordinate we are
considering the area of the coordinate
as a and Q X is entering here and
outgoing heat is Q X plus DX right now
what is happening here we are saying
that we wanted to find out its you know
how much heat transfer is going on in
this coordinate system the ultimate aim
is that only right so first of all what
we are doing consider a difference and
element of the slab of length DX as we
have shown in the figure that length is
DX right now we're saying Q X is equal
to heat conducted into the element along
egg by X direction the same feed
conducted into the element along x
direction then minus a we know that t we
X is equal to minus ka DT by DX this is
the first equation we generally studied
that is known as Fourier law of for heat
transfer right now what we are saying Q
X plus DX what is it this is heat
conducted in conducted out of the
element
along X Direction is equal to Q X Carly
why Carly X Q X DX now you can see it
right so we got two equations here one
is for Q X another one is QX plus D X
both the equation is very useful and
very important for Cartesian coordinates
okay now what we have to do we have to
find out what no that equation I have to
show you that's why you can understand
more about it that is heat generated in
the element so what is that it generated
in the element so the equation giving
you a DX this is heat generated in the
element now we have to write another
equation for I mean through that
equation we easily find out the heat
transfer rate okay through these
Cartesian coordinates so the name of
that equation is energy balance equation
so we will write an equation energy
balance equation along X direction so
this equation is known as energy balance
equation along X direction right so what
would be that
I mean how we can write that equation
that is also interesting and that is one
of the interesting part I must say that
so what is that field heat conducted it
conducted into the element
it conducted into the element plus heat
generated in the in the heat conducted
into the element is equal to M si is
equal to it conducted out of the element
heat conducted out of the element plus
rate of change of internal energy rate
of change of internal energy of element
with expert to time rate of change of
internal energy energy of the element
with respect to time okay so these
things we are going to write in
mathematical form that's why you can
understand it much better way let me use
another color pen that's why it would be
quite easier for you to understand okay
so let's go for it
so what would be that cube X cube X plus
Q Jennsen
okay generation is equal to Q X plus DX
plus W by curly toe internal energy of
element so internal energy of elements
now what next Q X
q generation q x plus curly white ally X
cube X DX
curly bike a little m CP T where what is
M M is mass of the element and we know
density is equal to mass upon volume
therefore mass is equal to density into
fee-simple
okay now after writing this what we can
write here that Q X and P versus
cancelling out so Q generation is equal
to l QX q dash DX is equal to curly
Wycherley x minus ka DT by DX into DX
plus calibike a little row is the X C DT
into CP so what we are doing we are
writing a DX is volume this volume is
equal to a B X okay no no what next work
nice that is most important what we are
going to do we are going to send all
these men we just cancel out this ATX
ATX and ATX so we can cancel out right
now what will be remaining Q generation
curly by curly X minus a DT by DT X plus
Kalia PI Carlito Rho CPT right after you
know after calculation we can get this
also I mean with the sending this one
this side so curly by curly X K DT by DX
plus Q generation curly bike a little CP
T okay that's it now we have to
differentiate this so k d square I mean
college party Cali square
new generation curly Baikal Utah Rho CP
and this must say okay we can say this
CPT okay now what we got here
this is only for one axis X exists or
x-direction we must say that but if we
wanted to find out what we wanted to get
energy BLM situation for all the axes
for all the axes then what we should do
now all the axis means I'm talking about
and I have already shared with you then
we are going to find out the energy
balance I mean we are going to find out
the equation energy balance equation for
x axis y axis and z axis now we are
having we can already find out the
energy balance equation for x direction
now we have to find out y axis as well
as Z exists once we'll come to know the
energy balance equation for x axis we
can easily write that an energy balance
equation for y axis as well as for J
axis so we don't need to find out
separately to for for writing the energy
balance equation right so let's go for
it how to write I mean how to do that
and that's why you will not face any
kind of problem in it so what we can do
we just need to write the axis
accordingly so first of all killing
Kali's vertical yep square plus K Khalif
Square t can dy square plus scale Cali
Square t curly ja d square plus heat
generation is equal to so not inhalation
asleep we have the new trick denoting
this in in the way that so you can
understand it CP Cali T upon colleague
Tom so stone right now what exactly
doing we are sending this thermal
conductivity to this side and we are
dividing here I mean just taking common
from it and we are dividing here let me
share with you Callie squaty Callie X
square plus Callie's purty Kelly Y
square plus del square t plus curly
Jasper
you lord it's scale is equal to Rho CP
by give curly D by Carlito so this is
the equation this is the equation for
Cartesian coordinates so most of our
time we have been asked for all the XS
and maybe sometime we have been asked
for only X direction so this is how you
can derive the equation for what is in
coordinates in case of 1/2 conduction
only so most of the time you know there
are three types of coordinate system
which gently which we find out that is
any spherical as well as this
cylindrical so this one is Cartesian in
my next lectures I will definitely try
to cover all other types of coordinate
system there so I will not go if not go
to watch any other videos you will get
each and everything from here only ok so
thanks for watching see you soon in my
next video bye
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Oct 30, 2024 Last updated |
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