Heat exchange by concurrent radiation and regular convection through an optically thick fluid over a hot vertical plate has been contemplated with first-order momentum and hot non continuum boundary conditions. The radiant heat flux was dealt with utilizing the Rosseland diffusion approximation. By solving the local non-similarity two equation model, numerical arrangements were acquired to inspect the slip consequences for the association amongst radiation and regular convection for a scope of thin conditions and radiation impacts. Results including slip speed, temperature hop, skin grinding, and warmth exchange rate are exhibited graphically and talked about. Likewise, a fundamental connection is introduced for the normal Nusselt number as an element of the non-continuum conditions, radiation– conduction parameter, and stream properties.

1. Effects of Surface Radiation on
Natural Convective Non Continuum
flow behavior through a
Heated Solid Cylindrical object
Ravi Kumar 217ME5252, M. Tech Cryogenic & Vacuum Technology, NIT Rourkela
Submitted to Prof. S. Murugan sir
M E 6 8 8A work by R. Li, M. Bousetta, Eric Chénier, Guy Lauriat (2015)

2. 2

3. Boundary layer Diagram
 Boundary layer diagram for
non continuum case is
same like that developed in
continuum type of regime.
 Changes are seen in
velocity profile,
Temperature profile,
amount of heat transfer
rate etc..
3

4.  Continuum means fluid flow is all continuous (homogenous) or
without any void within the concerned control volume
 Usually we assume flow to be continuum despite being fact
that at molecular level it is not continuous
 We use this assumption to validate well known differential
calculus to solve the physical problems
Continuum Vs Non Continuum
4

5.  The degree of the Continuum for fluid flow is usually characterized by
Knudsen number
𝐾𝑛 =
𝜆
𝐿
=
𝑀𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ
𝐿𝑒𝑛𝑔𝑡ℎ
 Categorization of flow regimes
Knudsen Number
0.01 0.1Continuum flow Slip flow Non continuum
flow
 For a fluid to be considered as a continuum its Knudsen number
should be approaching zero
5

6.  Combined radiation and natural convection problems are found in a wide
variety of engineering applications, including high temperature material
processing such as glass production, high temperature heat exchangers etc
 Arpaci et al.[2014] investigated the radiation effects on the boundary-layer
flow of optically thick medium using the Rosseland approximation
 They considered the impact of mass transfer studying the radiation–natural
convection interaction, and transformed the boundary-layer equations to
integral forms for optically thick gases
 Bousetta et al.[2015] examined the radiation effects on natural convection past
isothermal horizontal plate and a thin vertical cylinder
 As the mean free path of the flow approaches the characteristic length scale of
the problem, flows will demonstrate a non-continuum behavior due to fewer
molecular collisions within the dimension of interest
Introduction
6

7. η = Pseudo Similarity position
F = Reduced Stream Function
Θ = Dimensionless Temperature
Θw = Dimensionless wall surface temperature
Rd = Radiation conduction Parameter
ξ = Non continuum variable
Gr = Grasshoff’s Number
Pr = Prandtl’s number
β = Volumetric thermal expansion coefficient
Kn = Knudsen number
Nu = Nusselt Number
G = ξ derivative of F
φ = ξ derivative of theta
Nomenclature
7

8.  As the flow approaches the continuum limit, the conventional no-slip
wall boundary condition fails to model the surface interaction between
the fluid and the wall boundary due to the low collision frequency
 Slip models were proposed to improve the prediction of the non-
continuum phenomenon near wall boundaries within the framework
of the continuum assumption.
 Maxwell slip model relates the slip velocity at the wall to the local
velocity gradient based on the gas kinetic theory, given by
f = skin friction coefficient
8

9.  First-order temperature relationship is given by
 Results suggested that the flow structure, velocity profile, and
boundary layer thickness are changed by the non-continuum
conditions
 It is assumed that viscous dissipation is negligible
 Variations of fluid properties are limited to density variations
appearing in the gravitational body force only
( Boussinesq approximation )
α = thermal diffusivity
Pr = Prandtl number
9

10.  Equations of motion and heat transfer for two-dimensional steady
state, viscous incompressible natural convection in participating
medium has the form
Velocity along
length
Velocity normal to length
Thermal expansion
coefficient
Kinematic Viscosity
Radiation heat flux
Thermal conductivityThermal Diffusivity
Momentum Eq.
for HBL
Energy Eq.
for TBL
10

11.  Total heat transfer at the wall consists of the convective and
radiative heat flux
 Total nusselt value can be given as
 Convective part Radiative Part
 Average Nusselt number
11

12.  Pseudo similarity position
η =
𝑦
𝑥
𝐺𝑟(𝑥)
4
1
4
 Dimensionless Stream
Function
F(η) =
𝜑(𝑥,𝑦)
4 𝑉
𝐺𝑟(𝑥)
4
1
4
 Dimensionless Temperature
θ =
𝑇 − 𝑇(∞)
𝑇 𝑤 −𝑇(∞)
 Radiation Conduction
Parameter
Rd =
(4 𝜎 𝑇(∞)3)
𝛽 𝐾
Included Formulae
12

13.  Valied for medium having Optical thickness greater than 3
 If on average a photon cannot pass through the medium without absorption,
so, if absorption is higher Optically thick medium (Opaque)
if absorption is lower Optically thin medium (Transparent)
 The heat flux can be given as
qr = - 𝛤 ∇ G
where Γ =
1
{3 𝑎+ 𝜎 −𝐶𝜎}
a= absorptivity, σ = stefan constant ,
G = 4 𝜎 T4 G = Incident radiation
qr = - 16 𝜎 T3 𝛤∇ T
Rosseland Approximation Model
13

14. 14

15. 15

16. ξ
 Maximum value of
F ’’(0) is at no slip
condition and decreases
as the flow becomes more
rarefied
ξ = Non continuum variable
Skin friction, Non Continuum, Rd
16

17. 0 1 2 3 4 5
0.2
0.4
0.6
0.8
1
Rd = 0
Rd = 1.0
Rd = 5.0
Rd = 10.0
ξ
Θ(0)
 Temperature drop
increases with the
degree of
rarefaction.
 For a particular non
continuum condition,
magnitude of
temperature slip will
be reduced as the
radiation effect
becomes more
dominant
Wall temp., Non continuum, Rd
17

18.  At ξ = 0 , velocity
profile formed is
unaltered due to
no slip condition
 The peak value
decreases with
increasing
rarefaction
η = Pseudo Similarity position
Dimensionless x- velocity profile
18

19.  Heat Transfer rate decreases on increasing non – continuum
condition
 Wall Slip Velocity initially increases, attains maxima and starts
decreasing
 Skin Friction & Heat Transfer rate increase with increasing
radiation – conduction parameter (Rd )
 Average Nusselt No. as a function of non- continuum and
radiation – conduction parameter (Rd ) increases with Rd
Conclusion
19

20. Reference