CONVECTION HEAT TRANSFER CLASS NOTES FOR MECHANICAL ENGINEERING
CONVECTION HEAT TRANSFER
CLASS NOTES FOR MECHANICAL ENGINEERING
Convection heat transfer is of two types. That is free
and forced convection. Free convection starts in
stationary fluids. Forced convection heat transfer is
in moving fluids. Motion is caused by a pump or a
fan. It takes place in fluids (liquids, vapors and
gases). Convection is due to the bulk motion of the
fluid. This mode is at macro level i.e. visible to the
naked eye. Driving force is temperature difference.
Convection takes place from higher temperature to
lower temperature. Governing law for convection is
Newton’ law of cooling. There are two types of
convection heat transfer.
(i) Free Convection or natural convection heat transfer
Fig. Free Convection Boundary Layer Over a Heated Vertical Plate
In this, bulk motion of the fluid is caused by the density difference or by the buoyancy force. Thermal boundary layer coincide with the hydrodynamic boundary layer. Velocity of fluid is zero at the solid surface. Velocity of fluid is zero at the boundary of the boundary layer and also beyond the boundary layer. Dimensionless numbers used in free convection are Grashoff’s and Prandtl number. There are two types of free Convection.
(i) Laminar free convection
(ii) turbulent free convection.
Examples of free convection are
(a) Thunderstorms
(b) Glider planes
(c) Sea breeze
(d) Land breeze
(e) Cooling of Electric motors, pumps, compressors, transformer, IC Engines, mixers
(f) Hot coffee cooling in a cup
(g) Motion of hot balloons
(ii) FORCED CONVECTION HEAT TRANSFER
There are two cases of forced convection.
(a) Forced Convection Over a Flat Plate
Fig. Hydrodynamic Boundary Layer in Forced Convection Over a Flat Plate
Fig. Hydrodynamic & Thermal Boundary Layers in Forced Convection
(b) Forced Convection Heat Transfer in flow
It is further of two types.
(i) Laminar flow in a pipe
(a) shear stress distribution
τ = (∂p/∂x) (r/2)
τ_{max} = (∂p/∂x) (R/2)
Negative sign shows pressure decreases in the direction of flow
(b) Velocity distribution
u = (1/4μ )(∂p/∂x) (R^{2} –r^{2})
(c) Temperature distribution
ts–t = (u_{max}/α)(∂t/∂x)[3R^{2}/16 r^{2}/4 + r^{4}/16R^{2}]
Fig. Shear stress and velocity distribution during laminar flow in a pipe
Fig. Velocity & Temperature distribution during laminar flow in a pipe
Turbulent Flow in a Pipe
(i) Shear stress distribution
τ_{max} = (f/8)ρU^{2}
u/u_{max} = (y/R)^{1/7} (Power Law)
Boundary layer thickness
δ/x = 0.371/(Re_{x})^{1/5}
Fig. Velocity and temperature profiles during turbulent flow in a pipe
EXAMPLES OF FORCED CONVECTION
Liquid is moved by a pump.
Vapors and gases are moved by a fan/blower.
(There is bulk fluid motion in forced convection.)
Motion of the fluid is at macro level.
Driving force for convection heat transfer is also the temperature difference.
Convection heat transfer takes place from higher temperature to lower temperature.
Reynolds and Prandtl numbers are involved.
It is further of two types.
(i) Laminar forced convection
(ii) Turbulent forced convection
Examples are
(a) Fluid flow in boilers
(b) Refrigerant flow in refrigeration and air conditioning plants
(c) Flow over condensers
(d) Cooling of Internal combustion engines with fan in a radiator
(e) Nuclear reactors cooling
(f) Heat exchangers
Comparison of Hydrodynamic & Thermal Boundary Layers for different fluids
Fig. HBL and TBL for Air & Gases (δ_{TBL} = δ_{HBL})
Fig. HBL and TBL for Oils (δ_{TBL} < δ_{HBL})
Fig. HBL and TBL for Liquid Metals (δ_{TBL} > δ_{HBL})
METHODS FOR THE ANALYSIS OF FORCED CONVECTION HEAT TRANSFER
(i) Empirical Correlations
(ii) Hydrodynamic and Thermal Boundary Layers
(iii) Dimensional Analysis
(iv) Reynolds Analogy
CRITERIA & EMPIRICAL CORRELATIONS FOR LAMINAR FLOW & TURBULENT FLOW IN FREE CONVECTION HEAT TRANSFER
Free Convection 
Laminar condition & Empirical Relation 
Turbulent condition & empirical relation 
Over a horizontal flat plate with hot surface upwards 
Gr Pr < 10^{9 }Nu=0.54 (Gr Pr)^{0.25} 
Gr Pr > 10^{9 }Nu=0.14(Gr Pr) ^{0.33} 
Over a horizontal flat plate with cold surface upwards 
Gr Pr < 10^{9 }^{ } Nu=0.27 (Gr Pr)^{0.25} 
Gr Pr> 10^{9 } ^{ }Nu=0.1o7(Gr Pr) ^{0.33} 
Horizontal long cylinder L/D > 60 
Gr Pr <10^{9 } ^{ }Nu=0.53 (Gr Pr)^{0.25} 
Gr Pr >10^{9 } ^{ }Nu=0.13(Gr Pr) ^{0.33} 
Vertical Plates/Vertical cylinder 
Gr Pr < 10^{9 } Nu=0.59 (Gr Pr)^{0.25} 
Gr Pr > 10^{9 } Nu=0.13(Gr Pr) ^{0.33} 
Grashoff’s number= Gr = Buoyant force x inertia force/ (viscous force)^{2}
Gr= L^{3}g β ΔT/ν^{2}
Where L is the length of the plate g is acceleration due to gravity
β = 1/T_{av }
_{where Tav is in K=(T(high in C) +T(low in C))/2 + 273}
ΔT is the temperature difference
ν is the kinematic viscosity
PRANDTL NUMBER
Pr = momentum diffusivity/Thermal diffusivity
Pr =µc_{p}/k_{f}
Take Pr=0.7 for gases if not given
Take Pr = 10 for water if not given
GOVERNING EQUATIONS IN FORCED CONVECTION
It is an empirical equation of Nusselt number in terms of Reynolds and Prandtl numbers.
TABLE: Boundary layer parameters for different velocity profiles
Sr.No. 
Velocity Profile 
Boundary ConditionsAt y=0 
BoundaryconditionsAt y=δ 
Boundarylayer thicknessδ 
Average Friction CoefficientC^{–}_{f} 
1. 
u/U =y/δ 
u=0 
u=U 
3.46 x/(Re_{x})^{0.5} 
1.155 x/(Re_{L})^{0.5} 
2. 
u/U =2(y/δ) –(y/δ)^{3} 
u=0 
u =U∂u/∂y =0 
5.48 x/(Re_{x})^{0.5} 
1.46 x/(Re_{L})^{0.5} 
3. 
u/U = (3/2) (y/δ) –(1/2) (y/δ)^{3} 
u=0∂^{2}u/∂^{2}y =0 
u =U∂u/∂y =0 
4.64 x/(Re_{x})^{0.5} 
1.292 x/(Re_{L})^{0.5} 
4. 
u/U = Sin (π/2) (y/δ) 
u=0 
u =U 
4.795 x/(Re_{x})^{0.5} 
1.31 x/(Re_{L})^{0.5} 
5. 
Blasius Exact Solution 
— 
—– 
5x/(Re_{x})^{0.5} 
1.328 x/(Re_{L})^{0.5} 
The most commonly used velocity profile is u/U =2(y/δ) –(y/δ)^{3}.
TABLE: Reynolds number Criteria for laminar and turbulent flow in forced convection
and the Empirical Correlations
Forced convection 
Condition for laminar flow & empirical relation 
Condition for turbulent flow & empirical relation 
over a flat plate 
Re < 5×10^{5}Nu_{x}=0.332 Re_{x}^{1/2}Pr^{1/3 (Local)}Nu_{av}=0.664 Re_{L}^{1/2}Pr^{1/3 }Nu=h x/k_{f} 
Re > 7×10^{7 }Nu_{x}=00296 Re_{x}^{0.8}Pr^{1/3 (local)}Nu_{av}=0.037( Re_{L}^{0.8}—871)Pr^{1/3}Nu=h L/k_{f} 
Through a Rough pipe 
Re<2100Nu=1.86(Re Pr (L/D))^{1/3 (}μ_{f }/μ_{w})^{0.14 }^{ }Nu=h D/k_{f} 
Re > 4000 Nu=0.023 Re_{x}^{0.8}Pr^{n }^{ }n=0.4 for fluid being heatedn=0.3 for fluid being cooledNu=hD/k_{f} 
Through a Smooth pipe 
Re<10000Nu=1.86(Re Pr(L/D))^{1/3}(μ_{f}/μ_{w})^{0.14 }Nu=h D/k_{f} 
Re > 20000Nu=0.023 Re_{x}^{0.8}Pr^{n }n=0.4 for fluid being heatedn=0.3 for fluid being cooled Nu=h D/k_{f} 
DERIVATION OF NUSSELT NUMBER EXPRESSION FOR BOTH FREE AND FORCED CONVECTION
The equation is obtained by equating diffusion in stationary fluid in contact with the hot solid surface to the convection by the bulk motion of the fluid.
Diffusion in a fluid
OR
(conduction in a stationary fluid)
q^{.}= k_{f } A бT/бy _{y=0}
Rate of convective heat transfer
q^{. }= hA (T_{sur} –T_{fluid})
Under steady state
k_{f} A бT/бy_{y=0} = hA (T_{sur}–T_{fluid})
k_{f} A бT/бy_{y=0} = hA (T_{sur}–T_{fluid})
h = k_{f} A бT/бy_{ y=0} / A (T_{sur}–T_{fluid})
h = — k_{f} бT/бy_{ y=0} / (T_{sur}–T_{fluid})
hL /k_{f} = — бT/бy_{ y=0} / (T_{sur}–T_{fluid})/L = Nu
Nu = hL /k_{f} =–бT/бy_{ y=0} / (T_{sur}–T_{fluid})/L
Free convection is governed by Grashoff’s and Prandtl Numbers.
Forced convection is governed by Reynold and Prandtl numbers.
Convective heat transfer coefficient.
From Newton’s Law of Cooling

Q^{.}= h A dT
Where Q^{.}= Rate of heat transfer, W
h = convective heat transfer coefficient (W/m^{2}K or W/m^{2 o}C)
A = Surface area, m^{2}
dT = temperature difference between the surface and the bulk fluid (^{o}C)

Q^{.}= h with A =1m^{2}and dT=1^{o}C
Therefore convective heat transfer coefficient is the rate of convective heat transfer through a unit area and unit temperature difference.
Numerical data on convective heat transfer coefficient.
Type of convection 
Fluid used 
Experimental data on ‘h’, W/m^{2}K 
Free Convection 
Air 
5 – 25 
Free Convection 
Water 
20 – 100 
Forced Convection 
Air 
10 – 200 
Forced Convection 
Water 
50 – 10000 
Boiling 
Water 
3000 – 100,000 
Condensation 
Water vapor 
5,000 – 100,000 
It is highest during condensation.
Mean film temperature.
It is arithmetic mean of two temperatures involved in heat transfer. For example the two temperatures are 100 and 20^{0}C, then mean film temperature is (100+20)/2=60^{0}C. At this temperature the properties of the fluid are considered.
Difference between natural and forced convection
Sr.No. 
Natural convection 
Forced convection 
1. 
The motion of the fluid is only due to density difference. It is due to buoyancy force. No external pump or blower is used in this. 
Motion of the liquid is caused by a pump. Motion of the vapor/gas is caused by a blower. 
2. 
It is a slow process of heat transfer 
It is fast process 
3. 
It involves Grashoff’s number to decide laminar/ turbulent flow 
It involves Reynolds number to decide laminar/ turbulent flow 
4., 
Nu = f( Gr, Pr) 
Nu = f( Re, Pr) 
Q. Differentiate Nusselt number and Biot number.
It is a nondimensional number.
Nu = q^{.}_{conv in fluid}/q^{.}_{cond in static fluid} = hAΔT/–k_{f} AΔT/dx
Nu= hL_{c}/k_{f}
Where k_{f} is the thermal conductivity of the fluid
Biot number = Bi= hL_{c}/ k_{s}
Where k_{s} is the thermal conductivity of the solid
It is also a Non dimensional number
Q. Write the formula for Grashoff’s number and discuss its importance.
Gr = Buoyant force x Inertia force/ (viscous force)^{2}
Gr = L^{3}gβΔT/ν^{2}
Where β=1/ (T_{mean}+273)
and ν=µ/ρ
Grashoff’s number used in the analysis of free convection (laminar flow).
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