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) (R² –r²)

(c) Temperature distribution

ts–t = (u_max/α)(∂t/∂x)[3R²/16 -r²/4 + r⁴/16R²]

 

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)ρU²

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 < 109   Nu=0.54 (Gr Pr)0.25	Gr Pr > 109   Nu=0.14(Gr Pr) 0.33
Over a horizontal flat plate with cold surface upwards	Gr Pr < 109    Nu=0.27 (Gr Pr)0.25	Gr Pr> 109     Nu=0.1o7(Gr Pr) 0.33
Horizontal long cylinder   L/D > 60	Gr Pr <109        Nu=0.53 (Gr Pr)0.25	Gr Pr >109       Nu=0.13(Gr Pr) 0.33
Vertical Plates/Vertical cylinder	Gr Pr <  109        Nu=0.59 (Gr Pr)0.25	Gr Pr >  109       Nu=0.13(Gr Pr) 0.33

Grashoff’s number= Gr = Buoyant force x inertia force/ (viscous force)²

Gr= L³g β ΔT/ν²

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 Conditions At y=0	Boundary conditions At y=δ	Boundary layer thickness δ	Average Friction Coefficient C–f
1.	u/U =y/δ	u=0	u=U	3.46 x/(Rex)0.5	1.155 x/(ReL)0.5
2.	u/U =2(y/δ) –(y/δ)3	u=0	u =U ∂u/∂y =0	5.48 x/(Rex)0.5	1.46 x/(ReL)0.5
3.	u/U = (3/2) (y/δ) –(1/2) (y/δ)3	u=0 ∂2u/∂2y =0	u =U ∂u/∂y =0	4.64 x/(Rex)0.5	1.292 x/(ReL)0.5
4.	u/U = Sin (π/2) (y/δ)	u=0	u =U	4.795 x/(Rex)0.5	1.31 x/(ReL)0.5
5.	Blasius Exact Solution	—	—–	5x/(Rex)0.5	1.328 x/(ReL)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 < 5×105Nux=0.332 Rex1/2Pr1/3  (Local) Nuav=0.664 ReL1/2Pr1/3     Nu=h x/kf	Re > 7×107       Nux=00296 Rex0.8Pr1/3 (local) Nuav=0.037( ReL0.8—871)Pr1/3 Nu=h L/kf
Through a Rough pipe	Re<2100 Nu=1.86(Re Pr (L/D))1/3 (μf /μw)0.14    Nu=h D/kf	Re > 4000     Nu=0.023 Rex0.8Prn    n=0.4 for fluid being heated n=0.3 for fluid being cooled Nu=hD/kf
Through a Smooth pipe	Re<10000 Nu=1.86(Re Pr(L/D))1/3(μf/μw)0.14     Nu=h D/kf	Re > 20000Nu=0.023 Rex0.8Prn     n=0.4 for fluid being heated n=0.3 for fluid being cooled Nu=h D/kf

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

1. Q^.= h A dT        

Where Q^.= Rate of heat transfer, W

 h = convective heat transfer coefficient (W/m²K or W/m^{2 o}C)

A = Surface area, m²

 dT = temperature difference between the surface and the bulk fluid (^oC)

1. Q^.= h with A =1m²and dT=1^oC

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/m2K
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 20⁰C, then mean film temperature is (100+20)/2=60⁰C. 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 non-dimensional 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  = L³gβΔT/ν²

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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