Heat and mass transfer
September 2023
Contents
1 Review 3
1.1 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Heat convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Heat radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Emission and reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 BlackBody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.5 Radiation emission from real surfaces(at uniform temperature) . . . . 12
1.3.6 Net energy irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.7 Surfaces facing eachother . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Surface irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Surface absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Surface reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3 Surface transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Power exchanged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Under which condition absorption is equal to emission? . . . . . . . . 21
1.5.2 Net power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Sun irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Surfaces enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.1 2 surface enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.2 N surface enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.3 3 grey surface enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.4 3 surface enclosure, 1 adiabatic surface . . . . . . . . . . . . . . . . . . 27
1.7.5 3 surface enclosure with 1 black surface . . . . . . . . . . . . . . . . . 27
1.8 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.9 Radiation absorption in partecipating media . . . . . . . . . . . . . . . . . . . 28
1.10 Radiation emission from a partecipant medium . . . . . . . . . . . . . . . . . 31
1.10.1 Isolated isothermal cavity . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.10.2 Hottel-Sarofim method . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
,2 Analythical part (just summary) 35
2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Internal energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Species equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Incompressible fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Diffusion 37
3.1 Energy diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Buckingham theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Steady state conduction in a slender body . . . . . . . . . . . . . . . . . . . . 41
3.2.1 2D Temperature distribution in a fin(2D-case) . . . . . . . . . . . . . 42
3.2.2 1D Temperature distribution . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Power transferred in the fin . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.4 Fin array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.5 Generic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.6 Circular fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Thermal properties dependion on the temperature only . . . . . . . . . . . . 52
3.3.1 Steady state case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Transient heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Lumped parameter approach . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 First term approximation(one term approximation) . . . . . . . . . . . 60
3.4.3 Heat transferred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.4 Limped paramter approach pt2? . . . . . . . . . . . . . . . . . . . . . 68
4 Mass transfer by diffusion 71
4.1 Species equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Evaporation of a pure fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Stationary medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Catalytic chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.2 Homogeneus chemical reaction . . . . . . . . . . . . . . . . . . . . . . 78
5 Convective heat transfer 80
5.1 External forced convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 Velocity boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.2 Thermal boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.3 Concentration boundary layer . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.4 Boundary layers recap . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.5 Navier stokes equation with boundary layer approximation . . . . . . 88
5.2 Internal convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Axial distribution of the bulk temperature . . . . . . . . . . . . . . . . 98
5.3.2 Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Natural convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2
,1 Review
The heat has three type of transfer mechanism:
• Conduction (K)
• Convection (C)
• Radiation
1.1 Heat conduction
The starting point of the analisys is the Fourier’s law
dT W
q = −k [ ] (1)
dr m2
where r indicate a spacial coordinate and k is the thermal conductivity of the material where
the heat flows through. It’s very important the minus in the equation, because it tell us
that the heat flux in the opposite direction to the temperature gradient, which is what the
2nd law of thermodynamics says. To study this we can do some hypotesis:
• 1D
• Steady state(SS) which implies no time dependence
• constant properties
• No power generation which implies passive walls
The absence of generated power tell us that the temperature depends only on its position
in the space T = T (r).We can get the heat power by integrating Fourier’s law
Z Z
dT
Q̇ = qdA = −k dA (2)
A dr
From our hypotesys the value of Q̇ is constant, so we can rewrite everithing
dT
Q̇ = −k A(r) (3)
dr
to solve the equation we must separate the variables:
Z r2 Z T2 Z r2
dr k k k dr
=− dT −→ − (T2 − T1 ) = (T1 − T2 ) = (4)
r1 A(r) T1 Q̇ Q̇ Q̇ | {z } r1 A(r)
∆T
To specify the area it depends:
• planar wall of thickness L, A(r) = A = constant.
We can simply solve the integral and we will have
R2
R2 − R1
Z
k∆T dr 1 L
= = [r]R 2
= = (5)
Q̇ R1 A A R1 A A
3
, L
−→ ∆T = Q̇ (6)
· A}
|k {z
Rk,pw
So we can express the heat flow through an eletrical analogy like
∆T
Q̇ = (7)
Rk,pw
• cylindrical wall of length H, A(r) = 2πrH
R2 R2
ln( R )
Z
k∆T dr 1
= = [ln(r)]R2
R1 = 1
(8)
Q̇ R1 2πrH 2πH 2πH
Expressing through the electrical analogy
R2
ln( R 1
)
∆T = Q̇ (9)
|2πkH
{z }
Rk,cw
• Spherycal wall, A(r) = 4πr2
Z R2
k∆T dr 1 1 1 1 1
= [− ]R2 = ( − ) (10)
Q̇ R1 4πr2 4π r R1 4π R1 R2
Expressing through the electrical analogy
1 1 1
∆T = ( − ) Q̇ (11)
4πk R1 R2
| {z }
Rk,sw
Sometimes we can see different variation of the disposition of the resistance:
• Series of R, in case of conduction through different materials
• Parallel of R, in case of different mechanism of heat transmission
Q̇
Also, sometimes it is indicated A = q ′′ and
Rk Q̇ = Rk q ′′ A (12)
Where Rk · A = Rk′′ which indicates a specific resistance to conduction.
We don’t have only the resistance of the material, when two materials are in contact, it form
a new resistance Rcontact . To reduce Rcontact we can try to fill the gap with oil or water or
apply a force to compress the two materials together.
When we consider active walls, the formulation of before are not valid anymore. We have a
power generation per unit volume these can happen trough
• nuclear reactions
• chemical reactions
4
September 2023
Contents
1 Review 3
1.1 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Heat convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Heat radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.1 Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3.2 Irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.3 Emission and reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.4 BlackBody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.5 Radiation emission from real surfaces(at uniform temperature) . . . . 12
1.3.6 Net energy irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.7 Surfaces facing eachother . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Surface irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Surface absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4.2 Surface reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4.3 Surface transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5 Power exchanged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Under which condition absorption is equal to emission? . . . . . . . . 21
1.5.2 Net power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.6 Sun irradiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.7 Surfaces enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.1 2 surface enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.2 N surface enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.7.3 3 grey surface enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.7.4 3 surface enclosure, 1 adiabatic surface . . . . . . . . . . . . . . . . . . 27
1.7.5 3 surface enclosure with 1 black surface . . . . . . . . . . . . . . . . . 27
1.8 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.9 Radiation absorption in partecipating media . . . . . . . . . . . . . . . . . . . 28
1.10 Radiation emission from a partecipant medium . . . . . . . . . . . . . . . . . 31
1.10.1 Isolated isothermal cavity . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.10.2 Hottel-Sarofim method . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1
,2 Analythical part (just summary) 35
2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Linear momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Internal energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Species equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Incompressible fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3 Diffusion 37
3.1 Energy diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1 Buckingham theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Steady state conduction in a slender body . . . . . . . . . . . . . . . . . . . . 41
3.2.1 2D Temperature distribution in a fin(2D-case) . . . . . . . . . . . . . 42
3.2.2 1D Temperature distribution . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Power transferred in the fin . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.4 Fin array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.5 Generic solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.6 Circular fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3 Thermal properties dependion on the temperature only . . . . . . . . . . . . 52
3.3.1 Steady state case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Transient heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Lumped parameter approach . . . . . . . . . . . . . . . . . . . . . . . 59
3.4.2 First term approximation(one term approximation) . . . . . . . . . . . 60
3.4.3 Heat transferred . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.4 Limped paramter approach pt2? . . . . . . . . . . . . . . . . . . . . . 68
4 Mass transfer by diffusion 71
4.1 Species equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.3 Evaporation of a pure fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Stationary medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Catalytic chemical reactions . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.2 Homogeneus chemical reaction . . . . . . . . . . . . . . . . . . . . . . 78
5 Convective heat transfer 80
5.1 External forced convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.1 Velocity boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.1.2 Thermal boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1.3 Concentration boundary layer . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.4 Boundary layers recap . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.1.5 Navier stokes equation with boundary layer approximation . . . . . . 88
5.2 Internal convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Axial distribution of the bulk temperature . . . . . . . . . . . . . . . . 98
5.3.2 Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Natural convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2
,1 Review
The heat has three type of transfer mechanism:
• Conduction (K)
• Convection (C)
• Radiation
1.1 Heat conduction
The starting point of the analisys is the Fourier’s law
dT W
q = −k [ ] (1)
dr m2
where r indicate a spacial coordinate and k is the thermal conductivity of the material where
the heat flows through. It’s very important the minus in the equation, because it tell us
that the heat flux in the opposite direction to the temperature gradient, which is what the
2nd law of thermodynamics says. To study this we can do some hypotesis:
• 1D
• Steady state(SS) which implies no time dependence
• constant properties
• No power generation which implies passive walls
The absence of generated power tell us that the temperature depends only on its position
in the space T = T (r).We can get the heat power by integrating Fourier’s law
Z Z
dT
Q̇ = qdA = −k dA (2)
A dr
From our hypotesys the value of Q̇ is constant, so we can rewrite everithing
dT
Q̇ = −k A(r) (3)
dr
to solve the equation we must separate the variables:
Z r2 Z T2 Z r2
dr k k k dr
=− dT −→ − (T2 − T1 ) = (T1 − T2 ) = (4)
r1 A(r) T1 Q̇ Q̇ Q̇ | {z } r1 A(r)
∆T
To specify the area it depends:
• planar wall of thickness L, A(r) = A = constant.
We can simply solve the integral and we will have
R2
R2 − R1
Z
k∆T dr 1 L
= = [r]R 2
= = (5)
Q̇ R1 A A R1 A A
3
, L
−→ ∆T = Q̇ (6)
· A}
|k {z
Rk,pw
So we can express the heat flow through an eletrical analogy like
∆T
Q̇ = (7)
Rk,pw
• cylindrical wall of length H, A(r) = 2πrH
R2 R2
ln( R )
Z
k∆T dr 1
= = [ln(r)]R2
R1 = 1
(8)
Q̇ R1 2πrH 2πH 2πH
Expressing through the electrical analogy
R2
ln( R 1
)
∆T = Q̇ (9)
|2πkH
{z }
Rk,cw
• Spherycal wall, A(r) = 4πr2
Z R2
k∆T dr 1 1 1 1 1
= [− ]R2 = ( − ) (10)
Q̇ R1 4πr2 4π r R1 4π R1 R2
Expressing through the electrical analogy
1 1 1
∆T = ( − ) Q̇ (11)
4πk R1 R2
| {z }
Rk,sw
Sometimes we can see different variation of the disposition of the resistance:
• Series of R, in case of conduction through different materials
• Parallel of R, in case of different mechanism of heat transmission
Q̇
Also, sometimes it is indicated A = q ′′ and
Rk Q̇ = Rk q ′′ A (12)
Where Rk · A = Rk′′ which indicates a specific resistance to conduction.
We don’t have only the resistance of the material, when two materials are in contact, it form
a new resistance Rcontact . To reduce Rcontact we can try to fill the gap with oil or water or
apply a force to compress the two materials together.
When we consider active walls, the formulation of before are not valid anymore. We have a
power generation per unit volume these can happen trough
• nuclear reactions
• chemical reactions
4
