Convection Heat transfer

Introduction
Convection is a mode of heat transfer where the thermal energy is transported due to the bulk
motion of fluid. The convection heat transfer is governed by Newton's law of cooling. The
analysis of heat transfer by convection therefore involves the evaluation of the convective heat
transfer coefficient. The heat transfer coefficient for convection depends on the bulk fluid
motion. The fluid motion depends not only on the external factors as pressure difference but
also on the fluid properties such as density, and viscosity.

Flow basics
The flow of a fluid (liquid or gas) is broadly divided into the following
According to geometry
1. External flow: The flow is not confined inside a closed space like pipes or tubes
2. Internal flow: Flow in pipes, tubes, ducts are called internal flows
According to the effects of fluid viscosity:
1. Viscous flow: The effect of fluid viscosity is pronounced. Example, boundary layer flow
2. Inviscid flow: The effect of fluid viscosity is negligible. Example, Flow away from boundary
According to density variation
1. Compressible flow: The fluid density varies due to the high speed flow
2. Incompressible flow: The fluid density remains constant at low speed flow
According to fluid mixing due to eddies
1. Laminar flow: No fluid mixing due to flow, example fully developed pipe flow

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2. Turbulent flow: Fluid rapidly mixes due to eddies, example flow over a flat plate initially is
laminar and turns turbulent.
According to the flow initiation force
1. Natural flow: Flow is due to buoyancy effect and is not due to external aide such as a blower
2. Forced flow: Flow is induced by the use of fan or blower
According to the flow rate
1. Steady flow: Flow rate is constant with time
2. Unsteady or transient flow: Flow rate varies with time
All the above classifications of fluid flows are governed by a set of equations called the Navier-
Stokes equations. These include the mass conservation equation called the continuity
equation, the force conservation equation called the momentum equation and the energy
conservation equation or the first law of thermodynamics.
Mechanism of convection
Consider a hot metallic plate kept on the table to cool. A common method to find out if the
plate is safe to touch is by placing the hand just over the plate without touching it. The warm
air rising due to the hot plate is perceived and thus decision is made whether it is safe to touch
or not. This day to day activity is a good example of natural convection. The warm air rising
over a hot plate is due to the buoyancy effect, where chunks of air that come in contact with
the surface of the plate gets heated up and becomes lighter due to the reduced density. The
lighter air moves up due to gravity and the cooler heavier air comes in contact with the hot
plate and process in repeated till the hot plate reaches the ambient temperature. In the case
shown in figure 2, the air is blown over the hot plate with the use of a fan. The fan produces a
bulk fluid motion that is not due to the buoyancy effect or other naturally occurring
phenomenon. Such kind of heat transfer due to conduction that is based on external
influences is called the forced convection. Forced convection is responsible for the
faster cooling of food when we blow air over hot food before placing the hot morsel in our
mouth.

Fig. 1 and 2 : Free and Forced convection

Some dimensionless numbers associated with convection heat transfer:
Reynolds number: It is defined as the ratio of product of velocity times characteristic length
and kinematic viscosity. It gives the ratio of momentum dissipation to the viscous dissipation.
The value of the Reynolds number depends on the flow velocity and the geometry under
consideration.

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Prandtl number: This dimensionless parameter is a measure of viscous dissipation to
the thermal dissipation. The value of Prandtl number ranges from zero to a finite value less
than 20.

Nusselt number: This dimensionless parameter is similar to the Biot number but it is with
respect to the fluid. That is, it provides a ratio of effect of convection to that of conduction in
the fluid. It is a general practice to have correlations between the dimensionless parameters
such that by measuring Reynolds number and Prandtl number, we find out the Nusselt number.
This Nusselt number is used along with the fluid properties to find the heat transfer coefficient
due to convection.

No slip boundary, no temperature jump boundary:

At the physical boundary of the solid that is in contact with the fluid, we have two important
boundary conditions namely, the no-slip fluid boundary condition and the no
temperature jump boundary condition. These boundary conditions are based on the
observations that the fluid that is in contact with the solid at the boundary is stationery for all
practical purposes, and there exists a smooth transition in temperature between the solid surface
and the fluid in contact with it.

Velocity boundary: Due to the no-slip boundary condition the velocity of the fluid in the
leading edge of the flat plate is very close to zero. The fluid elements that are in contact with
the non-moving fluid at the physical boundary move at a very low velocity due to fluid friction
commonly called the viscosity. This viscosity makes the consecutive fluid element layers to
have a much lower velocities compared to the bulk fluid motion. An imaginary boundary where
the fluid velocity inside the boundary layer reaches approximately 99% of bulk fluid

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velocity can be drawn. This boundary is the physical boundary of the viscous flow
near the solid body interface with the fluid. This is called the velocity boundary layer.

Thermal boundary: Similar to the velocity boundary layer that is due to fluid viscosity and
the no-slip boundary condition the no temperature jump boundary condition and the viscosity
lead to thermal boundary layer. The thermal boundary for a heated flat plate with bulk fluid
motion due to a fan is shown in figures 4 and 5. Figure 4, the thermal boundary layer is thicker
and in figure 5 it is thinner than the velocity boundary layer. These conditions can be attributed
to the fact that depending on the fluid properties, the Prandtl number may be lesser than or
greater than unity. If the Prandtl number is lesser than unity the momentum dissipation is
higher, and we have velocity boundary layer thicker. If the Prandtl number is lesser than one,
the thermal dissipation is higher than momentum dissipation and thus the thermal boundary is
much thicker than the velocity boundary layer.

Forced convection:

Convection calculation of flat plate in parallel flow with laminar; Mixed and turbulent
flow conditions.
The parallel flow over flat plate is shown in the figure above. The no-slip boundary condition
ensures that the initial flow over flat plate is laminar at the plate boundary. The laminar flow

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is eventually changes into turbulent flow characterized by eddies. This transition takes place at
critical length of the flat plate corresponding to a critical Reynolds number as shown below:

Correlations based on experimental investigation are available that relate Reynolds number,
Prandtl number with the Nusselt number. The correlations are given below

The average Nusselt number over the entire plate is got by integrating the above equation over
the entire length of the flat plate and dividing the integral by the length of the plate. This
average Nusselt number correlations are given below:

Internal flow and its mean velocity, velocity profile and the mean temperature
Flow through pipes, tubes and ducts are very common in the engineering applications. It is
therefore important to understand the flow characteristics in these situations. This further helps
in the heat transfer calculations through convection from these geometries. If the cross section
of the tube, pipe or duct is circular we directly use the diameter of the cross section for all the
calculations and correlations. However, if the cross section is non-circular we use the
hydraulic diameter. The hydraulic diameter of a cross section is given below:

The internal flow through a pipe has viscous boundary layer similar to the parallel flow over
flat plate. Unlike flow over flat plate where viscous boundary layer becomes a turbulent flow

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after the critical length, the flow through pipes become completely laminar after the
development length. This is explained with the figure below:

Figure 9: Schematic of development length of pipe flow

The entrance length for laminar flow is given by

Like the velocity boundary layer, temperature boundary layer exists and the entry length is
similar to the velocity entry length.
Newton's law of cooling is used to calculate the heat transfer due to convection, and the
temperature for the calculation is taken to be the idealized mean temperature.
Free convection:
Physical considerations, the governing equations
The free convection is a process of heat transfer where the initiation force is the buoyancy
force. The buoyancy force can be understood by looking at a floating ship. The ship floats
because the bulk density of the ship and its mass is less than the mass of water it displaces. The
buoyancy force is given by:

The other common examples include the smoke moving up the chimney, characteristic shape
of candle flame, smoke from rising the incense stick. The primary reason for the reduced
density of fluid is the volumetric expansion coefficient β. The governing equation for
the free convection is derived from the Navier – Stokes equation and is shown below

Grashof number: The above governing equation can be non dimensionalized and among the
other dimensionless parameters we get a new parameter called the Grashof number.
This number is similar to the Reynolds number that governs the flow regime in the natural
convection. Grashof number is the ratio of buoyancy force to that of viscous force.

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Raleigh number: Often in the analysis of natural convection, we come across the product of
Grashof number and Prandtl number. This product is called the Raleigh number

Convection on a vertical plate

Heating and cooling of plates forms an important topic of natural convection, due to its
applications in food engineering, thermal solar collector applications, automobile engineering,
civil engineering, etc., In general, the plates can be either vertical, horizontal or inclined, and
the plate may be at higher or lower temperature compared to the ambient temperature. The
hotter plate produces low density air that is lifted upwards due to buoyancy force, while the
colder plate produces cold denser air that tends to flow downwards.

Dimensionless numbers:

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Q.1 Water heated to 80C flows through a 2.54 cm I.D. and 2.88 cm O.D steel tube (k = 50
W/mK). The tube is exposed to an environment which is known to provide an average
convection coefficient of ho = 30800 W/m2 K on the outside of the tube. The water velocity is
50 cm/s. Calculate the overall heat transfer coefficient based on the outer area of the pipe.
Properties of water at bulk mean temperature of 80 C: =974 kg/m3,  = 0.364 x 10- 6 m2 /sec,
k= 668.7 x 10- 3 W/m K, Pr = 2.20. Use Dittus Boelter equation:

Q. 2 Air at 2 atm and 200C is heated as it flows at a velocity of 12m/s through a tube with a
diameter of 3cm.The tube wall temperature is 20C above the air temperature all along the
length of the tube. Calculate the rate of heat transfer per unit length of the tube. The properties
of air at bulk mean temperature are; Pr = 0.681;  = 2.57 x 10-5 kg/ms; k = 0.0386 W/mK and
Cp = 1.025 kJ/kg K.
Q. 3 A horizontal cylinder, 3.0 cm in diameter and 0.8 m length is suspended in water at 200C.
Calculate the rate of heat transfer if the cylinder surface is at 550C.Given Nu = 0.53 (Gr x Pr)
0.25. The properties of water at average temperature are as follows: Density = 990 kg/m3,
Viscosity = 2.47 kg/h.m, k = 0.534 kcal/hr.m.0C, Cp = 1 kcal/kg C.
Q. 4 Liquid sodium is to be heated from 120 C to 149 C at a rate of 2.3 kg/sec in a 2.5 cm
diameter electrically heated tube (constant heat flux). Calculate the heat transfer coefficient.
The properties of sodium at 134.5C are: =916 kg/m3, Cp= 1.3565 kJ/kg K, =0.594 x 10-6
m2 /sec, k=84.90 W/m K, Pr = 0.0087. Given NNu =4.82 + 0.0185 NPe 0.827.

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