Discipline: Chemical Engineering

4th Semester
Subject: Process Heat Transfer
Unit 3
Module Name: Convection

Dr. Babita Sur

Lecturer in Chemical Engineering
Hooghly Institute of Technology

1 Dr. B. Sur, HIT, Hooghly

 Convection
Basic concept of natural & forced convection —Importance of dimensionless numbers involved
in convective heat transfer process: Reynolds’s number –Prandtl number – Nusselt number –
Grashoff number —Forced convection inside tube — Simple problem

The rate of heat transfer in a solid body or medium can be calculated by Fourier’s law. Moreover,
the Fourier law is applicable to the stagnant fluid also. However, there are hardly a few physical
situations in which the heat transfer in the fluid occurs and the fluid remains stagnant. The heat
transfer in a fluid causes convection (transport of fluid elements) and thus the heat transfer in a
fluid mainly occurs by convection.

Principle of heat flow in fluids and concept of heat transfer coefficient

When a hot metal plate is placed in front of a fan, it will cool faster than when it is exposed to a
stagnant air. In the process, the heat is transfered away with the flowing air, and we call the
process convective heat transfer. The term convective refers to transport of heat (or mass) in a
fluid medium due to the motion of the fluid. Convective heat transfer, thus, associated with the
motion of the fluid.
Convective heat transfer may be of two types – forced convection and free or natural convection.
Both are caused by motion in the medium. If the motion in the medium is generated by the
application of an external force (e.g.,by a pump, a blower, an agitator etc.), heat transfer is said to
occur by forced convection. But if the motion in the medium occurs as a result of density
difference (which may be caused by a temperature difference), the concerned mode of transfer is
called free convection.

Convective heat transfer from a heated wall to a fluid

Consider the heated wall shown in figure. The temperature of the wall and bulk fluid is denoted
by Tw and T∞ respectively. The velocity of the fluid layer at the wall will be zero. Thus the heat

2 Dr. B. Sur, HIT, Hooghly

will be transferred through the stagnant film of the fluid by conduction only. Thus we can
compute the heat transfer using Fourier’s law if the thermal conductivity of the fluid and the
fluid temperature gradient at the wall is known. Why, then, if the heat flows by conduction in
this layer, do we speak of convective heat transfer and need to consider the velocity of the fluid?
The answer is that the temperature gradient is dependent on the rate at which the fluid carries the
heat away; a high velocity produces a large temperature gradient, and so on. However, it must be
remembered that the physical mechanism of heat transfer at the wall is a conduction process.
It is apparent from the above discussion that the prediction of the rates at which heat is convected
away from the solid surface by an ambient fluid involves thorough understanding of the
principles of heat conduction, fluid dynamics, and boundary layer theory. All the complexities
involved in such an analytical approach may be lumped together in terms of a single parameter
by introduction of Newton’s law of cooling,
= ℎ ( − ∞)
where, h is known as the heat transfer coefficient or film coefficient. It is a complex function of
the fluid composition and properties, the geometry of the solid surface, and the hydrodynamics
of the fluid motion.
If k is the thermal conductivity of the fluid, the rate of heat transfer can be written directly by
following the Fourier’s law. Therefore, we have,

( ∞ − )
=− = ( − ∞)

where,
( ∞ − )

is the temperature gradient in the thin film where the temperature gradient is linear.
On comparing above two equations, we have,
ℎ=

It is clear from the above expression that the heat transfer coefficient can be calculated
if k and δ are known. Though the k values are easily available but the δ is not easy to determine.
Therefore, the above equation looks simple but not really easy for the calculation of real
problems due to non-linearity of k and difficulty in determining δ. The heat transfer coefficient is
important to visualize the convection heat transfer phenomenon as discussed before. In fact, δ is
the thickness of a heat transfer resistance as that really exists in the fluid under the given
hydrodynamic conditions. Thus, we have to assume a film of δ thickness on the surface and the
heat transfer coefficient is determined by the properties of the fluid film such as density,
viscosity, specific heat, thermal conductivity etc. The effects of all these parameters are lumped
or clubbed together to define the film thickness. Henceforth, the heat transfer coefficient (h) can

3 Dr. B. Sur, HIT, Hooghly

be found out with a large number of correlations developed over the time by the researchers.
These correlations will be discussed in due course of time as we will proceed through the
modules. Table below shows the typical values of the convective heat transfer coefficient under
different situations.

Typical values of heat transfer coefficient h under different situations

Description Heat transfer coefficient (W/m2°C)

Free convection in air 5-25
forced convection in air 10-500
Free convection in water 500-1,000
forced convection in water 1,000-15,000
Boiling water 2,500-25,000
Condensing water 5000-1,00,000

Some important dimensionless numbers used in forced heat transfer convection

The discussion on heat transfer correlations consists of many dimensionless groups. Therefore,
before we discuss the importance of heat transfer coefficients, it is important to understand the
physical significance of these dimensionless groups, which are frequently used in forced
convection heat transfer. Here are some of the dimensionless numbers used in the forced
convective heat transfer.

Dimensinless number Physical significance Expression

Wall temperature gradient hl

= , Nu =
Temperature gradient across the fluid in the pipe

Inertia force
Reynolds number = , Re =
Viscous force

Momentum diffusivity
Prandtl number = , Pr = = =
Thermal diffusivity

Heat flux by convection ℎ Nu

Stanton number = , St = = =
Heat flux by bulk flow uρ

Heat flux by bulk flow

Peclet number = , = =
Heat flux by conduction

4 Dr. B. Sur, HIT, Hooghly

Notations
h = heat transfer coefficient
l = charateristic length
k = thermal conductivity
ν = momentum diffusivity
u = velocity
ρ = density
µ = viscosity
cp = specific heat capacity

Flow through a pipe or tube

Turbulent flow
A classical expression for calculating heat transfer in fully developed turbulent flow in smooth
tubes/pipes of diameter (d) and length (L) is given by Dittus and Boelter

.8
Nu = 0.023
where,
n = 0.4, for heating of the fluid
n = 0.3, for cooling of the fluid
The properties in this equation are evaluated at the average fluid bulk temperature. Therefore, the
temperature difference between bulk fluid and the wall should not be significantly high.
Application of this equation lies in the following limits

0.7 ≤ Pr ≤ 160; (d/L) < 0.1; Re ≥ 10,000

When the temperature difference between bulk fluid and wall is very high, the viscosity of the
fluid and thus the fluid properties changes substantially. Therefore, the viscosity correction must
be accounted using Sieder – Tate equation given below

∙
∙ ∙
= 0 ∙ 027

Where the conditions for this equation are

0∙7≤ ≤ 16700; ≥ 10,000; ≤0.1

5 Dr. B. Sur, HIT, Hooghly

However, the fluid properties have to be evaluated at the mean bulk temperature of the fluid
except μw which should be evaluated at the wall temperature.
The earlier relations were applicable for fully developed flow when entrance length was
negligible. Nusselt recommended the following relation for the entrance region when the flow is
not fully developed.

∙
∙ ∙
= 0 ∙ 036

where, L is the tube length and d is the tube diameter.

Applicability condition of this equation is 10 < (L/d)
The fluid properties in this equation should be evaluated at mean bulk temperature of the fluid.

Different temperature terms used in this chapter

Bulk temperature/mixing cup temperature: Average temperature in a cross-section.
Average bulk temperature: Arithmetic average temperature of inlet and outlet bulk temperatures.
Wall temperature: Temperature of the wall.
Film temperature: Arithmetic average temperature of the wall and free stream temperature.
Free stream temperature: Temperature free from the effect of wall.

Laminar flow
Hausen presents the following empirical relations for fully developed laminar flow in tubes at
constant wall temperature.

0 ∙ 0668
= 3 ∙ 66 + ∙
1 + 0 ∙ 04

The heat transfer coefficient calculated from the above equation is the average value over the
entire length (including entrance length) of tube

ℎ
=

Sieder and Tate suggested a simple relation for laminar heat transfer in tubes.

0.33
ePr 0.14
= .86

6 Dr. B. Sur, HIT, Hooghly

The conditions for applicability of the equation are
0.48<Pr<16700; 0.0044< < 9.75

where, μ is the viscosity of the fluid at the bulk temperature and μw is that at the wall
temperature Tw . The other fluid properties are at mean bulk temperature of the fluid. Here also
the heat transfer coefficient calculated from the equation is the average value over the entire
length (including entrance length) of tube .
All the above empirical relations are for smooth pipe. However, it case of rough pipes, it is
sometimes appropriate that the Reynolds analogy between fluid friction and heat transfer be used
to effect a solution under these conditions and can be expressed in terms of Stanton number.
In order to account the variation of the thermal properties of different fluids the following
equations may be used (i.e. Stanton number multiplied by Pr2/3 ),

⁄
=
8

Where f is, Fanning friction factor defined by

̅
∆ =
2

where, ̅ is the mean free velocity. The friction factor can be evaluated from Moody’s chart.

Flow through non-circular ducts–Equivalent diameter

The same co-relations can be used for the non-circular ducts. However, the diameter of the tube
has to be replaced by the hydraulic diameter or equivalent diameter for the non-circular ducts.
The hydraulic diameter is defined as

cross-sectional area of low

=4× =4×
wetted perimeter

Where rH is hydraulic radius.

For the annular space of a double pipe heat exchanger the equivalent diameter can be calculated
as follows:
If the outer diameter of the inner pipe = do and
inner diameter of the outer pipe = Di,

Cross sectional area of low = −

4

7 Dr. B. Sur, HIT, Hooghly

And,
The wetted perimeter = π(Di+do)

So, the equivalent diameter will be

−
=4× =4×4 = −
π( + )

Natural convection
In the previous chapter, we have discussed about the forced convective heat transfer when the
fluid motion relative to the solid surface was caused by an external input of work by means of
pump, fan, blower, stirrer, etc. However, in this chapter we will discuss about the natural or free
convection. In natural convection, the fluid velocity far from the solid body will be zero.
However, near the solid body there will be some fluid motion if the body is at a temperature
different from that of the free fluid. In this situation there will be a density difference between
the fluid near the solid surface and that far away from the system. There will be a positive or
negative buoyancy force due to this density difference. Hot surface will create positive buoyancy
force whereas the cold surface will create the negative buoyancy force. Therefore, buoyancy
force will be the driving force which produce and maintain the free convective process.

In case of perfect natural-convection and in absence of heat dissipation,

Nu = f(Gr,Pr)
The dimensionless numbers involved are evaluated at the average film temperature.
It can be easily found that in case of the forced convection and in absence of heat dissipation the
function for average heat transfer will be,
Nu = f(Re,Pr)
On comparing these two relations, one can see that the Grashof number will perform for free
convection in a same way as the Reynolds number for forced convection.
The Grashof number is defined as

( − ) Buoyancy force
= =
Viscous force

Where, g = acceleration due to gravity

β = coefficient of volume expansion = −
Ts = surface temperature
Tb = bulk fluid temperature
L = characteristic length
ν = kinematic viscosity

8 Dr. B. Sur, HIT, Hooghly

Air flows through a 10 cm internal diameter tube at a rate of 75 kg/hr. Measurements indicate
that at a particular point in the tube, the pressure and temperature of air are 1.5 bar and 325 K
respectively whilst the tube wall temperature is 375 K. Make calculations for the heat transfer
rate from 1 m length in the region of this point.
For air at 325 K and 1 atm, the thermophysical properties are:
µ = 1.967×10–5 kg/ms, k = 0.02792 W/mK and cp = 1.038 kJ/kgK

Solution
A general non-dimensional correlation for calculating heat transfer in fully developed turbulent
flow in smooth tubes/pipes of diameter (d) and length (L) is given by Dittus and Boelter

.8
Nu = 0.023
where,
n = 0.4, for heating of the fluid
n = 0.3, for cooling of the fluid
The properties in this equation are evaluated at the average fluid bulk temperature. Application
of this equation lies in the following limits

0.7 ≤ Pr ≤ 160; (d/L) < 0.1; Re ≥ 10,000

Here, air is being heated, so the equation for this case will be

.8 .
Nu = 0.023

̇ ̇ 4 ̇ 4 × (75⁄3600)
= = = = = = 13492 > 10,000
× 0.1 × 1.967 × 10–
4

1.967 × 10– × 1.038 × 10

= = = 0 ∙ 73128 > 0 ∙ 7
0.02792

Now from
. ∙ ∙
Nu = 0.023Re .8 Pr = 0.023(13492) (0 ∙ 73128) = 40 ∙ 87

This gives the convective heat transfer coefficient as

0.02792
ℎ= × = 40 ∙ 87 × = 11 ∙ 41 /
0∙1

Rate of heat transfer, Q = hA∆T = 11.41×(π×0.1×1)×(375–325) = 179.23 W

9 Dr. B. Sur, HIT, Hooghly

Within a condenser shell, water flows through one hundred thin walled circular tubes (diameter
22.5 mm and length 5 m) which have been arranged in parallel. The mass flow rate of water is
65 kg/s and its inlet and outlet temperatures are known to be 22°C and 28°C respectively.
Predict the average convective heat transfer coefficient associated with water flow.
For water at 25 °C, the thermophysical properties are:
ρ = 996.65kg/m3, µ = 903.01×10–6 kg/ms, k = 2.1893 kJ/mhrK and cp = 4.1776 kJ/kgK

Solution
Mean bulk temperature = (22+28)/2 = 25°C
So, the thermophysical properties of water are given at the mean bulk temperature.

(903.01 × 10– × 3600) × 4.1776

= = =6∙2>0∙7
2.1893

65
Mass low rate of water through each tube = = 0 ∙ 65 kg/s
100

̇ ̇ 4 ̇ 4 × 0 ∙ 65
= = = = = = 40749 ∙ 78 > 10,000
× 0.0225 × 903.01 × 10–
4

So, here for calculating heat transfer Dittus and Boelter equation is used, as water is being heated,
the equation for this case will be

.8 .
Nu = 0.023

∙ (6 ∙ 2) ∙
= 0.023(40749 ∙ 78) = 232 ∙ 69

This gives the convective heat transfer coefficient as

2.1893
ℎ= × = 232 ∙ 69 × = /
0.0225

Source:
1. B. K. Dutta: Heat Transfer- Principles and Applications.
2. D.Q. Kern: Process Heat Transfer
3. Dr. D. S. Kumar: Heat and Mass Transfer
4. NPTEL and other Online Courses.

10 Dr. B. Sur, HIT, Hooghly