Short Notes: Dimensionless Parameters in Free and Forced Convective | Short Notes for Mechanical Engineering PDF Download

 Page 1


Dimensionless Parameters in Free and Forced Convective 
The following dimensionless parameters are significant in evaluating the 
convection heat transfer coefficient:
Nusselt Number (Nu):
• It is a dimensionless quantity defined as hL/ k,
° where h = convective heat transfer coefficient 
o L is the characteristic length 
o k is the thermal conductivity of the fluid.
• The Nusselt number could be interpreted physically as the ratio of the 
temperature gradient in the fluid immediately in contact with the surface to a 
reference temperature gradient (Ts - T„) /L.
• The convective heat transfer coefficient can easily be obtained if the Nusselt 
number, the thermal conductivity of the fluid in that temperature range and the 
characteristic dimension of the object is known.
• Let us consider a hot flat plate (temperature Tw ) placed in a free stream 
(temperature T„ < Tw ). The temperature distribution is shown in the following 
figure.
T
A , - Thermal boundary 
layer thickness
Page 2


Dimensionless Parameters in Free and Forced Convective 
The following dimensionless parameters are significant in evaluating the 
convection heat transfer coefficient:
Nusselt Number (Nu):
• It is a dimensionless quantity defined as hL/ k,
° where h = convective heat transfer coefficient 
o L is the characteristic length 
o k is the thermal conductivity of the fluid.
• The Nusselt number could be interpreted physically as the ratio of the 
temperature gradient in the fluid immediately in contact with the surface to a 
reference temperature gradient (Ts - T„) /L.
• The convective heat transfer coefficient can easily be obtained if the Nusselt 
number, the thermal conductivity of the fluid in that temperature range and the 
characteristic dimension of the object is known.
• Let us consider a hot flat plate (temperature Tw ) placed in a free stream 
(temperature T„ < Tw ). The temperature distribution is shown in the following 
figure.
T
A , - Thermal boundary 
layer thickness
Newton's Law of Cooling says that the rate of heat transfer per unit area by 
convection is given by
Q A =h(Tw-T 2e)
hL L 
Nu = — = —
k 5.
N u=----------------------------------------: —
Rate of heat transfer by conduction
Rate of heat transfer bv convection h. A ._T
• Nu measure energy transfer by convection occurring at the surface. Larger the 
value of Nu, larger will be the rate of heat transfer by convection.
Temperature distribution in a boundary layer: Nusselt modulus
• The heat transfer by convection involves conduction and mixing motion of 
fluid particles. At the solid fluid interface (y = 0), the heat flows by conduction 
only, and is given by
• Since the magnitude of the temperature gradient in the fluid will remain the 
same, irrespective of the reference temperature, we can write dT = d(T -T-w) 
and by introducing a characteristic length dimension L to indicate the 
geometry of the object from which the heat flows, we get
A dv
. and
in dimensionless form,
d(Tw-T )/(T w-T 30) * •
d (y /L )
Reynold Number (Re):
• Reynold number is defined as
Re =
Inertia force
Viscous force
Page 3


Dimensionless Parameters in Free and Forced Convective 
The following dimensionless parameters are significant in evaluating the 
convection heat transfer coefficient:
Nusselt Number (Nu):
• It is a dimensionless quantity defined as hL/ k,
° where h = convective heat transfer coefficient 
o L is the characteristic length 
o k is the thermal conductivity of the fluid.
• The Nusselt number could be interpreted physically as the ratio of the 
temperature gradient in the fluid immediately in contact with the surface to a 
reference temperature gradient (Ts - T„) /L.
• The convective heat transfer coefficient can easily be obtained if the Nusselt 
number, the thermal conductivity of the fluid in that temperature range and the 
characteristic dimension of the object is known.
• Let us consider a hot flat plate (temperature Tw ) placed in a free stream 
(temperature T„ < Tw ). The temperature distribution is shown in the following 
figure.
T
A , - Thermal boundary 
layer thickness
Newton's Law of Cooling says that the rate of heat transfer per unit area by 
convection is given by
Q A =h(Tw-T 2e)
hL L 
Nu = — = —
k 5.
N u=----------------------------------------: —
Rate of heat transfer by conduction
Rate of heat transfer bv convection h. A ._T
• Nu measure energy transfer by convection occurring at the surface. Larger the 
value of Nu, larger will be the rate of heat transfer by convection.
Temperature distribution in a boundary layer: Nusselt modulus
• The heat transfer by convection involves conduction and mixing motion of 
fluid particles. At the solid fluid interface (y = 0), the heat flows by conduction 
only, and is given by
• Since the magnitude of the temperature gradient in the fluid will remain the 
same, irrespective of the reference temperature, we can write dT = d(T -T-w) 
and by introducing a characteristic length dimension L to indicate the 
geometry of the object from which the heat flows, we get
A dv
. and
in dimensionless form,
d(Tw-T )/(T w-T 30) * •
d (y /L )
Reynold Number (Re):
• Reynold number is defined as
Re =
Inertia force
Viscous force
vl
Re= —
v
where, p = Density of fluid, v= Velocity of fluid passing through length (I), p = 
Kinematic viscosity, and v = Dynamic viscosity
a
P
Critical Reynold Number:
• It represents the number where the boundary layer changes from laminar to 
turbine flow.
For flat plate
• Re < 5 x 105 (laminar)
• Re > 5 x 105 (turbulent)
For circular pipes
• Re < 2300 (laminar flow)
• 2300 < Re < 4000 (transition to turbulent flow)
• Re > 4000 (turbulent flow)
Stanton Number (St)
• It is defined as
Heat transfer coefficient
St =-
Heat flow per unit temperature rise 
Nu
St =
Re x Pr
Grashof Number (Gr)
• In natural or free convection heat transfer, die motion of fluid particles is 
created due to buoyancy effects. The driving force for fluid motion is the body 
force arising from the temperature gradient.
• If a body with a constant wall temperature Tw is exposed to a quiscent 
ambient fluid at T„, the force per unit volume can be written as:
pgP(tw-t»)
where p = mass density of the fluid, p = volume coefficient of expansion and g is 
the acceleration due to gravity.
• It is used for free convection.
Inertia force x buovancv foce
Gr =
viscous force
where, p = Coefficient of volumetric expansion = 1/7
• The ratio of inertia force x Buoyancy force/(viscous force)2 can be written as
(pV2 L*)xpgp(T0 . -Tr)L3
G r = -
(pVLV
Page 4


Dimensionless Parameters in Free and Forced Convective 
The following dimensionless parameters are significant in evaluating the 
convection heat transfer coefficient:
Nusselt Number (Nu):
• It is a dimensionless quantity defined as hL/ k,
° where h = convective heat transfer coefficient 
o L is the characteristic length 
o k is the thermal conductivity of the fluid.
• The Nusselt number could be interpreted physically as the ratio of the 
temperature gradient in the fluid immediately in contact with the surface to a 
reference temperature gradient (Ts - T„) /L.
• The convective heat transfer coefficient can easily be obtained if the Nusselt 
number, the thermal conductivity of the fluid in that temperature range and the 
characteristic dimension of the object is known.
• Let us consider a hot flat plate (temperature Tw ) placed in a free stream 
(temperature T„ < Tw ). The temperature distribution is shown in the following 
figure.
T
A , - Thermal boundary 
layer thickness
Newton's Law of Cooling says that the rate of heat transfer per unit area by 
convection is given by
Q A =h(Tw-T 2e)
hL L 
Nu = — = —
k 5.
N u=----------------------------------------: —
Rate of heat transfer by conduction
Rate of heat transfer bv convection h. A ._T
• Nu measure energy transfer by convection occurring at the surface. Larger the 
value of Nu, larger will be the rate of heat transfer by convection.
Temperature distribution in a boundary layer: Nusselt modulus
• The heat transfer by convection involves conduction and mixing motion of 
fluid particles. At the solid fluid interface (y = 0), the heat flows by conduction 
only, and is given by
• Since the magnitude of the temperature gradient in the fluid will remain the 
same, irrespective of the reference temperature, we can write dT = d(T -T-w) 
and by introducing a characteristic length dimension L to indicate the 
geometry of the object from which the heat flows, we get
A dv
. and
in dimensionless form,
d(Tw-T )/(T w-T 30) * •
d (y /L )
Reynold Number (Re):
• Reynold number is defined as
Re =
Inertia force
Viscous force
vl
Re= —
v
where, p = Density of fluid, v= Velocity of fluid passing through length (I), p = 
Kinematic viscosity, and v = Dynamic viscosity
a
P
Critical Reynold Number:
• It represents the number where the boundary layer changes from laminar to 
turbine flow.
For flat plate
• Re < 5 x 105 (laminar)
• Re > 5 x 105 (turbulent)
For circular pipes
• Re < 2300 (laminar flow)
• 2300 < Re < 4000 (transition to turbulent flow)
• Re > 4000 (turbulent flow)
Stanton Number (St)
• It is defined as
Heat transfer coefficient
St =-
Heat flow per unit temperature rise 
Nu
St =
Re x Pr
Grashof Number (Gr)
• In natural or free convection heat transfer, die motion of fluid particles is 
created due to buoyancy effects. The driving force for fluid motion is the body 
force arising from the temperature gradient.
• If a body with a constant wall temperature Tw is exposed to a quiscent 
ambient fluid at T„, the force per unit volume can be written as:
pgP(tw-t»)
where p = mass density of the fluid, p = volume coefficient of expansion and g is 
the acceleration due to gravity.
• It is used for free convection.
Inertia force x buovancv foce
Gr =
viscous force
where, p = Coefficient of volumetric expansion = 1/7
• The ratio of inertia force x Buoyancy force/(viscous force)2 can be written as
(pV2 L*)xpgp(T0 . -Tr)L3
G r = -
(pVLV
• The magnitude of Grashof number indicates whether the flow is laminar or 
turbulent.
• If the Grashof number is greater than 109 , the flow is turbulent and
• If Grashof number less than 108 , the flow is laminar.
• For 108 < Gr < 109 , It is the transition range.
Prandtl Number (Pr)
• It is a dimensionless parameter defined as:
„ Momentum diffusivitv throueh the fluid 
Thermal diffusivitv through the fluid
Pr = pC_/ k = v/o.
V P
k 1 pc,
where p is the dynamic viscosity of the fluid, v = kinematic viscosity and a = 
thermal diffusivity.
• For liquid metal, Pr < 0.01
• For air and gases, Pr «1
• For water, Pr «10
• For heavy oil and grease, Pr > 105
• It provides a measure of relative effective of momentum and energy transport 
by diffusion in velocity and thermal boundary layers respectively. Higher Pr 
means higher Nu and it shows higher heat transfer.
• This number assumes significance when both momentum and energy are 
propagated through the system.
• It is a physical parameter depending upon the properties of the medium.
• It is a measure of the relative magnitudes of momentum and thermal 
diffusion in the fluid:
• For Pr = 1, the r ate of diffusion of momentum and energy are equal which 
means that the calculated temperature and velocity fields will be Similar, the 
thickness of the momentum and thermal boundary layers will be equal.
• For Pr « 1 (in case of liquid metals), the thickness of the thermal boundary 
layer will be much more than the thickness of the momentum boundary layer 
and vice versa.
• The product of Grashof and Prandtl number is called Rayleigh number. Or, R a 
= Gr x Pr.
Rayleigh Number (Ra)
• It is the product of Grashof number and Prandtl number. It is used for free 
convection.
Ra = GrTr:Ra =
g 3 P i t
V .Q
where,
Page 5


Dimensionless Parameters in Free and Forced Convective 
The following dimensionless parameters are significant in evaluating the 
convection heat transfer coefficient:
Nusselt Number (Nu):
• It is a dimensionless quantity defined as hL/ k,
° where h = convective heat transfer coefficient 
o L is the characteristic length 
o k is the thermal conductivity of the fluid.
• The Nusselt number could be interpreted physically as the ratio of the 
temperature gradient in the fluid immediately in contact with the surface to a 
reference temperature gradient (Ts - T„) /L.
• The convective heat transfer coefficient can easily be obtained if the Nusselt 
number, the thermal conductivity of the fluid in that temperature range and the 
characteristic dimension of the object is known.
• Let us consider a hot flat plate (temperature Tw ) placed in a free stream 
(temperature T„ < Tw ). The temperature distribution is shown in the following 
figure.
T
A , - Thermal boundary 
layer thickness
Newton's Law of Cooling says that the rate of heat transfer per unit area by 
convection is given by
Q A =h(Tw-T 2e)
hL L 
Nu = — = —
k 5.
N u=----------------------------------------: —
Rate of heat transfer by conduction
Rate of heat transfer bv convection h. A ._T
• Nu measure energy transfer by convection occurring at the surface. Larger the 
value of Nu, larger will be the rate of heat transfer by convection.
Temperature distribution in a boundary layer: Nusselt modulus
• The heat transfer by convection involves conduction and mixing motion of 
fluid particles. At the solid fluid interface (y = 0), the heat flows by conduction 
only, and is given by
• Since the magnitude of the temperature gradient in the fluid will remain the 
same, irrespective of the reference temperature, we can write dT = d(T -T-w) 
and by introducing a characteristic length dimension L to indicate the 
geometry of the object from which the heat flows, we get
A dv
. and
in dimensionless form,
d(Tw-T )/(T w-T 30) * •
d (y /L )
Reynold Number (Re):
• Reynold number is defined as
Re =
Inertia force
Viscous force
vl
Re= —
v
where, p = Density of fluid, v= Velocity of fluid passing through length (I), p = 
Kinematic viscosity, and v = Dynamic viscosity
a
P
Critical Reynold Number:
• It represents the number where the boundary layer changes from laminar to 
turbine flow.
For flat plate
• Re < 5 x 105 (laminar)
• Re > 5 x 105 (turbulent)
For circular pipes
• Re < 2300 (laminar flow)
• 2300 < Re < 4000 (transition to turbulent flow)
• Re > 4000 (turbulent flow)
Stanton Number (St)
• It is defined as
Heat transfer coefficient
St =-
Heat flow per unit temperature rise 
Nu
St =
Re x Pr
Grashof Number (Gr)
• In natural or free convection heat transfer, die motion of fluid particles is 
created due to buoyancy effects. The driving force for fluid motion is the body 
force arising from the temperature gradient.
• If a body with a constant wall temperature Tw is exposed to a quiscent 
ambient fluid at T„, the force per unit volume can be written as:
pgP(tw-t»)
where p = mass density of the fluid, p = volume coefficient of expansion and g is 
the acceleration due to gravity.
• It is used for free convection.
Inertia force x buovancv foce
Gr =
viscous force
where, p = Coefficient of volumetric expansion = 1/7
• The ratio of inertia force x Buoyancy force/(viscous force)2 can be written as
(pV2 L*)xpgp(T0 . -Tr)L3
G r = -
(pVLV
• The magnitude of Grashof number indicates whether the flow is laminar or 
turbulent.
• If the Grashof number is greater than 109 , the flow is turbulent and
• If Grashof number less than 108 , the flow is laminar.
• For 108 < Gr < 109 , It is the transition range.
Prandtl Number (Pr)
• It is a dimensionless parameter defined as:
„ Momentum diffusivitv throueh the fluid 
Thermal diffusivitv through the fluid
Pr = pC_/ k = v/o.
V P
k 1 pc,
where p is the dynamic viscosity of the fluid, v = kinematic viscosity and a = 
thermal diffusivity.
• For liquid metal, Pr < 0.01
• For air and gases, Pr «1
• For water, Pr «10
• For heavy oil and grease, Pr > 105
• It provides a measure of relative effective of momentum and energy transport 
by diffusion in velocity and thermal boundary layers respectively. Higher Pr 
means higher Nu and it shows higher heat transfer.
• This number assumes significance when both momentum and energy are 
propagated through the system.
• It is a physical parameter depending upon the properties of the medium.
• It is a measure of the relative magnitudes of momentum and thermal 
diffusion in the fluid:
• For Pr = 1, the r ate of diffusion of momentum and energy are equal which 
means that the calculated temperature and velocity fields will be Similar, the 
thickness of the momentum and thermal boundary layers will be equal.
• For Pr « 1 (in case of liquid metals), the thickness of the thermal boundary 
layer will be much more than the thickness of the momentum boundary layer 
and vice versa.
• The product of Grashof and Prandtl number is called Rayleigh number. Or, R a 
= Gr x Pr.
Rayleigh Number (Ra)
• It is the product of Grashof number and Prandtl number. It is used for free 
convection.
Ra = GrTr:Ra =
g 3 P i t
V .Q
where,
• g = Acceleration due to gravity
• P = Thermal expansion coefficient
• v = Kinematic viscosity
• a = Thermal diffusivity
• Pr = Prandtl number
• Gr = Grashof number
If free or natural convection
• 104 < Ra < 109 (laminar flow)
• Ra > 109 (turbulent flow).