Chemical Engineering June 2016 - 53

tween 100 and 200°C.
However, the knowledge of properties
over a wide temperature and
pressure range is vital for other fields,
such as drying, to allow accurate
prediction of heat and mass transfer
phenomena during the physical processes
involved.
Prandtl number. The Prandtl number
is a dimensionless number approximating
the ratio of momentum
diffusivity (kinematic viscosity) and
thermal diffusivity, as shown in Equation
(1), with all nomenclature presented
in the box on p. 52):
Pr = =
Cp
k
µ
(1)
Unlike the Reynolds and Grashof
numbers, the Prandtl number contains
no length scale in its definition,
so Pr is dependent only on the fluid
involved and the state of the fluid.
As such, Prandtl numbers are often
found in property tables alongside
other properties such as viscosity
and thermal conductivity.
Over the years, considerable research
efforts have investigated free
and forced convection. However, according
to the authors' knowledge,
there is no simple-to-use correlation
to rapidly estimate Pr of compressed
air at elevated pressures. This is important
because Pr is widely used in
heat transfer and calculations related
to convection [2-5].
With this in mind, two tools are
presented below for predicting Pr
over a wide range of temperatures
and pressures. The first, an empirical
correlation, can be used to
quickly and easily estimate a value
of Pr for air. The second method is
based on a multi-layer perceptron
(MLP) type of artificial neural networks
(ANNs), which is shown to be
able to accurately predict the Pr of
air over a wide range of temperatures
and pressures.
Empirical correlation
The empirical correlation presented
here has been tuned using a widely
accepted methodology described
in the literature. The empirical data
required to develop this correlation
includes the reported data [6] for the
Pr of air as a function of pressure
and temperature (K).
Equation (2) represents the proln(Pr)
= + + +
T
a
Where:
a = +1A B P C P D P+
2
1
= +
2
2
b A B P C P D P+
2
+ 1
+
3
+
4
+
2
3
2
c = +3A B P C P D P+
2
3
d = +4A B P C P D P+
4
4
3
3
2
1
3
3
(3)
(4)
(5)
(6)
These optimized, tuned coefficients
are applicable for calculating the Pr
of air for temperatures up to 1,000K
and pressures up to 10,000 kPa.
If more data become available in
the future, the coefficients in Table
1 can be quickly retuned according
to the approach outlined here.
In order to obtain the values of the
coefficients for the new empirical
method, the following procedure
has been applied:
1. The air Pr is correlated as a function
of temperature for the selected
pressure.
2. Step 1 is repeated for other pressures.
3.
The corresponding polynomial
constants, which were obtained
for various Pr of air versus temperature,
are then correlated.
4. The Pr of air at any temperature
and pressure can then be calculated
from Equation (2).
Equation (2) and its coefficients
provided in Table 1 is believed to be
the first simple-to-use correlation
for quickly predicting Pr of air over
a wide range of temperatures and
pressures. Furthermore, the exponential
function that was selected
ChemiCal engineering www.Chemengonline.Com June 2016
T
b
c
2
d
T
3
(2)
Coefficient
A1
B1
C1
D1
A2
B2
C2
D2
Table 1. Tuned coefficienTs used in equaTions (3-6)
Coefficient
Value
-2.60209064
-4.64712011
-3.17817228
2.88726660
-1.145969902
4.810244580
4.902744144
-4.240021674
posed governing equation in which
four coefficients are used to correlate
the Prandtl numbers of air as a
function of temperature (expressed
in Kelvin) and pressure (kPa), with
the corresponding coefficients listed
in Table 1.
A3
B3
C3
D3
A4
B4
C4
D4
Value
3.5995496368
-1.5920349698
-2.224437123
1.8214824677
-2.930937692
2.32398913631
3.17489914017
-2.428258824
to develop this tool leads to wellbehaved
(that is, smooth and nonoscillatory)
equations enabling fast
and accurate predictions.
Neural-based model
Overview of ANNs. Briefly, ANNs
are constructed of simple processing
elements, known as neurons, in
a parallel computational algorithm.
The neurons are inspired by biological
nervous systems. Figure 1 shows
an artificial neuron. The mathematical
representation of a neuron, n, is
given by Equations (7) and (8).
n
r ()m =
i=1
w x b+ m
mi i
y F r=
m ()m
(7)
(8)
Where x1, x2,...,xn indicate the input
signals; wm1, wm2, ..., wmn denote
the synaptic weights; rm is defined
as the linear combiner output; bm is
the bias term; f is the activation function
used; and ym is the neuron's
output signal.
ANNs are robust types of computational
intelligences with the ability
to express non-linear and complex
relationships. This is done by using
several input-output training patterns
from introduced datasets to the network.
A nonlinear mapping between
inputs and outputs is provided intrinsically
by ANNs [7, 8]. ANNs have
primarily been used by investigators
for pattern recognition, classification,
and prediction [9, 10].
Among the wide variety of ANN
types and architectures, such as
recurrent neural networks (RNNs),
feed-forward neural networks
(FFNNs), and self-organizing maps
(SOMs), most investigators use
FFNNs. Most believe that FFNNs,
including functional link networks
(FLNs), radial basis function networks
(RBFNs) and MLPs, are the
most potent, versatile and trustworthy
nonlinear classifier recognizers
53
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