Chapter 9: Convection Processes and
Properties
Basics of Heat and Mass transfer
By D. S. Kumar
Lecture 13
CONTENTS
Significance of Dimensionless Groups
Reynolds Number (Assigned)
Grashof Number
Prandtl Number
Stanton Number
Nusselt Number
Example
2
GRASHOF NUMBER (Gr)
Grashof number indicates the relative strength of the
buoyant to viscous forces.
𝐷3 𝜌2 (∆𝑇 𝛽 𝑔) 3
𝜌
𝐺𝑟 = 2
= 𝜌𝐷 𝛽𝑔 ∆𝑇 2
𝜇 𝜇
2 2
3
𝜌𝑉 𝐷
= 𝜌𝐷 𝛽𝑔 ∆𝑇 ×
(𝜇𝑉𝐷)2
𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑓𝑜𝑟𝑐𝑒
= 𝑏𝑢𝑜𝑦𝑎𝑛𝑡 𝑓𝑜𝑟𝑐𝑒 ×
(𝑣𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒)2
3
GRASHOF NUMBER (Gr)
Obviously the Grashof number represents the ratio of
the product of buoyant and inertial forces to the square
of the viscous forces.
Grashof number has a role in free convection similar to
that played by Reynolds number in forced convection.
4
PRANDTL NUMBER (Pr)
Prandtl number is indicative of the relative ability of the
fluid to diffuse momentum and inertial energy by
molecular mechanisms.
𝜇𝐶𝑝 𝜌𝑣𝐶𝑝 𝑣
𝑃𝑟 = = =
𝑘 𝑘 𝑘 𝜌𝐶𝑝
Recalling that the parameter ( 𝑘/𝜌𝐶𝑝 ) is thermal
diffusivity 𝛼.
𝑣 𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦
𝑃𝑟 = =
𝛼 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 5
PRANDTL NUMBER (Pr)
Apparently Pr is the ratio of the kinematic viscosity to
thermal diffusivity.
The kinematic viscosity indicates the momentum
transport by molecular friction and thermal diffusivity
represents the heat energy transport through
conduction.
Obviously Pr provides a measure of the relative
effectiveness of momentum and energy transport by
diffusion.
6
STANTON NUMBER (St)
Stanton number is the ratio of heat transfer coefficient to the
flow of heat per unit temperature rise due to the velocity of
the fluid.
ℎ ℎ𝐷 𝑘
𝑆𝑡 = =
𝜌𝑉𝐶𝑝 (𝑃𝑉𝐷 𝜇) × (𝜇𝐶𝑝 𝑘)
Thus the Stanton number can be expressed in terms of other
dimensionless numbers as:
𝑁𝑢𝑠𝑠𝑒𝑙𝑡 𝑛𝑢𝑚𝑏𝑒𝑟
𝑆𝑡 =
𝑅𝑒𝑦𝑛𝑜𝑙𝑑𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 × 𝑃𝑟𝑎𝑛𝑑𝑡𝑙 𝑛𝑢𝑚𝑏𝑒𝑟
Stanton number can be used only in correlating forced7
convection.
EXAMPLE 9.7
Air enters a rectangular duct measuring 30 cm x 40 cm
with a velocity of 8.5 m/s and a temperature of 40oC. The
flowing air has a thermal conductivity 0.028 W/m-deg,
kinematic viscosity 16.95 x 10-6 m2/s and from empirical
correlations the Nusselt number has been approximated to
be 425. Work out the equivalent diameter of the flow
passage, the flow Reynolds number and the convective
heat flow coefficient.
8
NUSSELT NUMBER
Consider double pipe exchanger
and assume hot fluid is flowing
inside the pipe and cold fluid is
flowing through annular space.
Also both surfaces of tube are clear
of dirt or scale.
The change in temperature with
distance is shown in diagram.
Temperature gradient is large
through the boundary layer and
small in the turbulent region.
The heat must flow through
boundary layer by conduction but
due to low thermal conductivity of
fluids large temperature gradient
occurs, whereas eddies in core are
effective in equalizing the 9
temperature in turbulent zone.
NUSSELT NUMBER
The mechanism of heat transfer at the wall is conduction,
and the heat flux can be written as:
𝑑𝑞 𝑑𝑇
= −𝑘
𝑑𝐴 𝑑𝑦 𝑤
We know:
𝑑𝑞
= ℎ 𝑇 − 𝑇𝑤
𝑑𝐴
So,
(𝑑𝑇 𝑑𝑦)𝑤
ℎ = −𝑘
𝑇 − 𝑇𝑤
By multiplying both sides of above equation, to make it dimensionless,
with D/k we get:
10
ℎ𝐷 (𝑑𝑇 𝑑𝑦)𝑤
= −𝐷
𝑘 𝑇 − 𝑇𝑤
NUSSELT NUMBER
On the cold fluid side of the tube, denominator becomes Tw – T. The
dimensionless group hD/k is called Nusselt number Nnu.
Nusselt number is the ratio of temperature gradient at the wall,
(dT/dy)w, to the average temperature gradient across the entire
pipe, (T – Tw)/D.
In case, where all the resistance to heat transfer was due to
laminar layer of thickness x and heat transfer mechanism was only
conduction, then:
𝑑𝑞 𝑘 𝑇 − 𝑇𝑤
=
𝑑𝐴 𝑥
𝑘
ℎ=
𝑥
ℎ𝐷 𝑘𝐷 𝐷
= 𝑁𝑁𝑢 = =
𝑘 𝑥𝑘 𝑥
11
In this case, Nusselt no. is the ratio of tube diameter to the thickness
of the laminar layer.
