Forced Convection

2.3 Forced Convection

2.3.1. Introduction
In its simplest form, the study of Forced Convection is concerned with the transfer of heat
between a moving fluid and a solid surface. The movement is initiated by some force e.g. that of
an electrical fan.

In order to apply Newton’s Law of Cooling, it is important to find the value of h.

( ) ( )

where,

Q = heat transferred, kJ/kg

h = heat transfer coefficient

A = cross-sectional area, m2

tw = temperature of the wall (surface), K or 0C

t = temperature of the fluid, K or 0C

The fluid flowing is also governed by the Reynold’s number, Re.

( )

where,

ρ = fluid density, kg/m3

u = flow velocity, m/s

μ = fluid dynamic viscosity, kg/ms

l = characteristic linear dimension, m

ϑ = fluid kinematic viscosity, m2/s

As mentioned earlier, the solution of convective heat transfer is possible when empirical
information is available. This empirical information can be generalized by using Dimensional
Analysis.

2.3.2 Dimensional Analysis

For forced convection, it is satisfactory to assume that the buoyance effects are negligible. In
such cases, the coefficient of heat transfer, h depends on:

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 Fluid dynamic viscosity, μ

 Fluid density, ρ
 Thermal conductivity, k
 Specific heat capacity, c
 Temperature difference between the surface and the fluid, ϴ
 Fluid velocity, u
 Characteristic linear dimension, l

Mathematically,

( ) ( )

Equation (1) can generally be expressed as:

[ ] [ ] ( )

where,

A and B are Constants

a, b, c, …. are Arbitrary indices

If equation (2) is correct, then each term on the Right Hand Side (RHS) must have the same
dimensions as those of h. Consider the first term on the RHS, and ignore the subscripts of the
indices, and the arbitrary constant, A.

( )

But each of the properties in equation (3) can be expressed in terms of fundamental dimensions:
Mass = M, Length = L, Time = T, Temperature = t, and Heat = Q. The following table can be
developed.

Property Units Dimensions

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 Forced Convection

ϴ K t

l m L

Substituting the dimensions in equation (3) results into the following:

( ) ( ) ( ) ( ) ( ) ( ) ( )

The powers of the dimensions on both sides of equation (4) can be equated. But first it is
required that the dimensions be grouped together with their respective powers as follows:

( )

Hence the powers in equation (5) can be equated.

()

( )

( )

( )

( )

It can be noted that there are five equations (i – v) and seven unknowns. A solution can only be
obtained if a set of unknowns are expressed in terms of others. Usually it is by experience that
one decides which of the unknowns to be expressed by others. In this case, it is most useful to
express a, b, c, e, and g in terms of d and f since they are powers of two unrelated quantities.

()

( )

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 Forced Convection

( )

( )

( )

( )

( )

( ) ( )

Substituting these values in equation (2) result into,

[ ] [ ] ( )

Putting the terms with common powers,

[( ) ( ) ( )] [( ) ( ) ( )]

( ) { [( ) ( ) ] [( ) ( ) ] }

( )( ) ( )

The dimensionless groups are defined as follows:

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 Forced Convection

( )

( )

( )

Hence for forced convection, it can be shown that the Nu no. is a function of the Pr no. and the
Re no., that is:

[( )( )] ( )

Empirical correlations are then used to determine the constants KF in equation (2.6).

In simple cases, evaluation of Nu, Pr, and Re require taking the properties of the fluid at a
suitable mean temperature.

For cases in which the temperature of the bulk of the fluid is not very different from the
temperature of the solid surface, the fluid properties are evaluated at the mean bulk fluid
temperature.

When the temperature difference is large, errors may occur due to using mean bulk temperature,
and hence a mean film temperature is preferred. This is defined as follows:

( )

where,

tf = mean film temperature

tb = mean bulk temperature

tw = surface/wall temperature

2.3.3 Reynold’s Analogy

According to Reynold, the heat transfer from a solid surface is similar to the transfer of fluid
momentum from the surface. It is, therefore, possible to express the heat transfer in terms of
frictional resistance due to shear.

Consider turbulent flow; it can be assumed that particles of mass, m transfer heat and momentum
to and from the surface – moving perpendicular to the surface. On the average, heat transfer per
unit area, q (i.e. Q/A) is given by:

( )

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where,

ṁ = mass flow rate of the fluid

c = specific heat capacity of the fluid

∆t = temperature difference between the surface and the bulk of the fluid

The rate of change of moment across the stream is given by:

( ) ( )

where,

u = velocity of the bulk of fluid

uw = velocity of the fluid at the wall/ surface

But force per unit area is given by:

( )

where,

w = shear stress in the fluid at the wall

From equations (1) and (3),

( )

For turbulent flow, in practice, there is always a thin layer of fluid on the surface on which
viscous effects pre-dorminate. This is called the laminar sub-layer. In this layer, heat is
transferred purely by conduction. Therefore by applying Fourier’s Law of Conduction,

( ) ( )

where,

k = thermal conductivity of the fluid

y = distance from the surface perpendicular to the surface

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For viscous flow, it can be established from Fluid Mechanics that:

( )

Hence shear stress at the wall can be given as,

( ) ( )

where,

μ = fluid dynamic viscosity

Since the laminar sub-layer is very thin, it may be assumed that the temperature and velocity
vary linearly with the distance from the wall i.e.

where,

ẟb = thickness of the laminar sub-layer

If the above expressions are mathematically manipulated,

Neglecting the minus sign, since the distance cannot be negative, the above expression yields,

( )

Equations (4) and (7) are identical. Consequently,

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Therefore for fluids whose Pr ≈ Unity, the Reynold’s analogy can be applied since the heat
transfer across the laminar sub-layer can be considered in a similar way to the heat transferred
from the sub-layer to the bulk of the fluid.

For most gases, dry vapours, and superheated vapours, Pr lies between 0.65 – 1.2 and hence
Reynold’s analogy can safely be applied to them.

Consider a unit area,

Therefore by equating to equation (4),

( )

Dividing equation (8) by ρu gives,

It can be established that both sides of this equation are dimensionless.

The term on the left hand side (LHS) is called the Stanton number, St.

( )

A dimensionless term called friction factor, f is defined by,

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( )

( )

By mathematical manipulation, St can be expressed as:

( )

The friction factor, f can be derived mathematically for some cases. In other cases, a practical
determination is necessary.

For turbulent flow in a pipe, a simple measurement of a pressure drop gives f. Using the equation
for f, the approximate heat flow can be found.

Resistance to flow per unit length in a pipe that has a diameter d is given by,

( )

An important factor in Heat Design is the pumping power, W required. W is the rate at which
work is done in overcoming frictional resistance. Therefore for a flow in a pipe, W is given by:

( )

( )( )( ) ( ) ( )

But from equation (4),

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( )

The ratio of the pumping power, W to the rate of heat flow, Q can be expressed as,

( )

For a heat exchanger, ∆tIn is the logarithmic mean temperature difference (LMTD).

( )
( )

2.3.4 Worked Examples

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