Heat Transfer Lecture 03
Transcript of Heat Transfer Lecture 03
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Lecture Notes forHeat Transfer - 2
ACRi PMDCertified Course
inComputational Fluid Dynamics
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Convection Boundary Condition
The heat transfer from the surface of a body due to convection is q = hA(T - T ) A is the area of the surface from which convectiontakes place
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Convection Boundary Condition
h is the heat transfer coefficientDepends on the fluid propertiesDepends on the nature of the flow Reynoldsnumber etc
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Convection Boundary Condition
One side with convective heat transfer
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Convection Boundary Condition
Convective heat transfer from both sides
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Convection Boundary Condition
Homework problem: Given a specified heatflux on one side and convective heat transfercoefficient on the other side, what is thetemperature distribution in the body?Given: L, q, h, T , k; What is T(x)?
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Radiation Boundary Condition
The heat transfer from the surface of a body due to radiation isEmax = Ts4, =5.67*10-8 w/m 2k 4
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Types of Boundary Conditions - Summary
Dirichlet specified temperatureNeumann specified heat fluxIncludes zero flux case perfectly insulated
Convection Surface specified heat transfercoefficient and medium temperatureRadiation bundary condition
Interface boundary conditionContinuity of temperatureContinuity of heat fux
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Black and Grey body
A black body is an idealised surface that emitsradiation at all frequencies. The Stefan-Boltzmann law is for the perfect blackbody In reality, most bodies are gray they only emitradiation at some frequencies
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Grey body
In reality, most bodies are gray they only emitradiation at some frequenciesSo the amount of radiation emitted is less than thatof a blackbody at the same temperatureEgrey = I Ts4, =5.67*10-8 w/m 2k 4
The fraction I is called emissivity of the surfaceIt is a measure of how efficiently a body emitsradiation compared to a perfect blackbody
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Absorbtion, transmittance
Surfaces generally absorb some incidentradiation, reflect back some radiation and allow some fraction of the radiation to pass throughthem completely
Absorbtion characterised by the absorbtivity EReflectance characterised by the reflectivity
Transmittance characterised by the
transmittivity E + + = 1
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Radiation inside a room
Consider a grey body at temperature T andemissivity I in a room with wall temperature T wall
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Radiation inside a room
Radiation heat energy emitted by body isEbody = A I T4Radiation heat incident on body from wall is
G = A I T w 4
The energy balance equation for the body canbe written as
Qnet = -A I T4 +
AI T w 4
= hA(T - T )
What are the assumptions made in this case?
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Heat Diffusion Cylindrical Coordinates
Consider the conduction of heat through a long cylinderChange in the axial direction is considered to benegligible
Symmetry in the direction
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Heat Diffusion Cylindrical Coordinates
Consider a section along the axisQ(r) Q(r + ( r) + Qg = ( E/ ( t
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Heat Diffusion Cylindrical Coordinates
So the 1st
law for the annular element can ve written asQ(r) Q(r + ( r) + q q ( V = ( ( V)C ( T/ ( t
( V = A* ( r = 2 rL* ( r After simplification and taking limit as ( r 0,
We get
t T
C qr T
kAr A g x
x!
xx
xx V
1
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Heat Diffusion Cylindrical Coordinates
So for a material with variable conductivity
For constant conductivity
t q
r kr
r r g xx
xx
xx1
t T
k
q
r T
r r r g
xx
!
xx
xx
E11
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Heat Diffusion Cylindrical Coordinates
Steady State without heat generationSteady State with heat generation Transient conduction
Similarly for a sphere, we have
t T
C qr T
kr r r g x
x!
xx
xx V22
1
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3D Heat Diffusion Cylindrical Coordinates
In 3D, we have full heat diffusion equation incylindrical coordinates as
t
T C q
z
T k
z
T k
r r
T kr
r r g
x
x!
x
x
x
x
x
x
x
x
x
x
x
x VJ M
2
11
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Ex. 1 - Heat Diffusion in AnnularCylinder Dirichlet BC
Consider an annular cylinder - thermalconductivity k; What is T(r)?
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Ex. 1 - Heat Diffusion in AnnularCylinder Dirichlet BC
For planar wall, area of c/s was constantSo heat flux through each section constantSo temperature gradient between sections was same
For cylinder area of heat transfer increases with increasing radius
Hence heat flux through section of greater radiusless than section of smaller radius
Hence temperature gradient is not a constant
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Ex. 2 - Heat Diffusion in Solid Cylinder Dirichlet BC
Consider a solid cylinder of radius r1 - thermalconductivity k;
Heat generation / unit volume of q q Temperature at outer surface = T1 What is T(r)
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Ex. 2 - Heat Diffusion in Solid Cylinder Dirichlet BC
What are the boundary conditions? At r = r1, T = T1Need one more B.C
Temperature has to be maximum at center of cylinder? Why?
So at r = 0, dT/dr = 0
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Ex. 3 - Wire + insulationConsider a wire of radius r1 and an insulationmaterial with outer radius r2; conductivity k1,k2;
Heat generation / unit volume of q q - electrical Temperature at outer surface = T2 What is T(r)
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Ex. 3 - Wire + insulationSolve in two parts
For the solid wireFor the annular insulation material
Use interface boundary condition
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Summary What is heat transfer coefficient? What are itsunits?
What does convective heat transfer coefficientdepend on?