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INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2004; 28:12691291Published online 18 August 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/er.1029
Modelling and performance analysis of an electric heater
Ashraf M. Bassily1,n,y and Gerald M. Colver2
1Gharian University, P.O. Box 64735, Gharian, Libya2Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, U.S.A.
SUMMARY
Electric heaters are used extensively in many industrial applications. There are several interactingparameters that affect heater performance and contribute to its cost. Such parameters are: coil length, coildiameter, helix diameter, coil pitch, number of turns, coil emissivity, heater wall emissivity, applied voltage,air flow rate, air temperature at the heater outlet, insulation thickness, and the heater dimensions. Three
conical heater configurations were selected for preliminary optimization. A conical heater configurationwith outer ring coils was found to give the highest heater efficiency, the easiest and least expensive tomanufacture, and was selected for detailed modelling. In the simulation model, the heater wall was dividedinto four annular sections and the continuous heater coil was divided into four segments of four ring coils.Energy and heat transfer equations were written for each ring coil, each section of the wall, and the air pasteach ring coil. Equations for coil resistance and power, air properties, heater geometry, and configurationfactors are added to form a system of 220 nonlinear equations. Engineering equation solver (EES) was usedto solve the system of equations. The results were checked by comparing the heater efficiency based on theaverage inlet and outlet air temperatures, and the heater efficiency based on the heater losses. Bothefficiencies matched well in all calculations. The effects of varying the identified heater parameters on theheater performance were studied and discussed. The results indicate that increasing the coil length andairflow rate, while reducing coil emissivity, wall emissivity, and wire diameter could improve heaterperformance. Copyright # 2004 John Wiley & Sons, Ltd.
KEY WORDS: coiled wires; a conical heater; energy efficiency; an efficient heater; an insulated heater
1. INTRODUCTION
Electric heaters have a wide variety of industrial applications such as drying, air conditioning,
and water heating. Electric heaters involve many interacting parameters such as coil length, wire
diameter, helix diameter, coil pitch, the number of turns, coil emissivity, wall emissivity, applied
voltage, insulation thickness, air flow rate, and the heater dimensions. Increasing coil diameter
increases its heat transfer area, but also reduces coil resistivity; thus, increasing coil power for
the same coil length. Increasing coil length increases its heat transfer area and coil resistivity;
thus, reducing coil power for the same coil diameter. Varying the coil dimensions also influences
its temperature, which affects the temperature and properties of the surrounding air. That affectsthe heat transfer coefficient of the coil, which influences its temperature even further. Reducing
Received 22 May 2003Accepted 4 September 2003Copyright# 2004 John Wiley & Sons, Ltd.
yE-mail: bassily2000@yahoo.com
nCorrespondence to: A. M. Bassily, 31 Ahmed El Barrad Street, 11241 Cairo, Egypt.
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coil emissivity reduces the radiation loss to the surrounding and increases heater efficiency.
Detrich et al. (1990) showed that increasing coil pitch increases the heat transfer coefficient of
the coil. Literature review has shown that no model for an electric heater has yet considered the
interaction of all the above-mentioned parameters. Modelling of the electric heater is an
essential step for optimization and performance analysis. An optimum design of the electricheater minimizes extreme coil temperatures and the associated thermal stresses, leading to
longer operating cycles, longer life for the heating element, and saving in energy usage. This
paper gives a detailed model comprised of 220 nonlinear equations for a conical electric heater.
The model determines the energy losses from the heater, the temperature distribution along the
coil (simulated by separate ring coils), along the heater wall, and of air past each ring coil. In
addition, the model determines the effect of changing the above-mentioned heater parameters
on heater performance.
2. PRELIMINARY OPTIMIZATION OF THREE CONFIGURATIONS
OF THE HEATER
Figure 1 shows the three conical heater configurations that were selected for optimization. The
inner rows of ring coils are in contact with increasingly hotter air. The main idea of the heater
configurations is to reduce the temperature of the inner coils (the second, third, and fourth rows)
by increasing the convection heat transfer coefficient of ring coils as air passes through the
heater; thus, reducing the temperature of the inner coils and achieving a relatively uniform
temperature distribution. Energy balance and heat transfer equations were written for each of
the three heater configurations, which included balances on the coil, heater wall, and the air flow
passing each coil. In that model, a single value of temperature represented the temperature
distribution for the coil and a single value represented the temperature distribution for the
heater wall. Interactive Thermodynamics (IT) software, Intelli Pro, Inc. (1996) was used to solve
the resulting 30 nonlinear equations.The optimization was based on maximizing the heater efficiency in terms of the dimensions of
the coils, emissivity of the coil, and of the wall subject to a constant mass flow rate for air and an
Outer Coil Inner Coil
1st
row
2nd
row
3rd
row
4th
row
Air
(a) (b) (c)
Figure 1. Three different configurations for the electric heater. (a) a heater with outer coils; (b) a heaterwith inner coils; (c) a heater with inner and outer coils.
Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Energy Res. 2004; 28:12691291
A. M. BASSILY AND G. M. COLVER1270
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outlet air temperature of 400 K. The univariate search was carried out. The procedure involved
holding all but one of the parameters constant while optimizing the heater efficiency relative to
the remaining parameter. This maximizing process was then repeated for each parameter. By
this method, the heater configuration of Figure 1(a) was found to give the highest heater
efficiency and also the least sensitivity of efficiency to the parameters (coil dimensions, etc.)variation. In addition, the same heater configuration was also found to be the easiest and least
expensive to manufacture.
3. DETAILED MODELLING FOR ONE CONFIGURATION OF THE HEATER
Detailed modelling for the best heater configuration (Figure 1(a)) will be illustrated. The heater
wall was divided into four sections of equal area, as shown in Figure 2. The heater coil was
divided into four ring coils that simulate the continuous coiled helix of an actual heater. The
four sections of the heater wall are labelled wall 1, wall 2, wall 3, and wall 4, respectively. The
four ring coils are labelled coil 1, coil 2, coil 3, and coil 4, respectively. In the analysis, energy
equations were written for each section of the heater wall, each ring coil, and for the air flowpast each ring coil. The equations of heat transfer were written to include convection and
radiation at each section of the heater wall and each ring coil. Equations for air properties,
configuration factors, and heater geometry made up the additional equations necessary to
complete the model.
Section 4
of the heater wallHw4
Hw3
Hw2
Hw1
Section 3
of the heater wall
Section 2
of the heater wall
Section 1
of the heater wall
Coil 3
Coil 1
Coil 2
Coil 4
0.1524m
0.05m
0.05m
0.05m
Hh
Air flow
Doh
Hs
Figure 2. Illustrated diagram of the electric heater.
Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Energy Res. 2004; 28:12691291
MODELLING AND PERFORMANCE ANALYSIS 1271
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3.1. Calculation assumptions
1. Steady state conditions.
2. The four ring coils are located at 0.1524, 0.2032, 0.254, and 0.3048 m from the heater
inlet, respectively.3. The heater outlet is connected to a heater duct that is 0.1016 m in diameter so that the
outlet diameter of the heater is constant at 0.1016 m.
4. The length of each coil is proportional to the diameter of the heater cone at the point of
location of that ring coil.
5. The materials of the heater wall and of the ring coils are homogenous.
6. A single value of temperature represents the temperature distribution on each ring
coil.
7. A single value of temperature represents the temperature distribution on each section of
the heater wall.
8. The inlet air temperature to the heater is constant and equal to the surrounding
temperature of 305 K.
9. The hemispherical emissivity of the heater wall is not a function of the direction ofradiation or its wavelength.
10. The hemispherical emissivity of the surroundings is taken as a constant of 0.9 and that it
is not a function of the radiation direction or its wavelength.
11. The hemispherical emissivity of the ring coils is not a function of the direction of
radiation or its wavelength.
12. Radiation heat that reaches any ring coil from a neighbouring ring coil is very small
compared with the radiation heat emitted from the neighbouring ring coil so that
radiation exchange between ring coils is neglected.
13. The four ring coils simulate a continuous coiled helix with a constant applied voltage at
its coil terminals.
14. The flow through the heater is strongly non-uniform with higher velocity and cool
temperature in the empty centre and lower velocity and hot temperature in the coilregions; both due to wall friction and the resistance of the coils. Therefore, the flow is
assumed to be in two zones. Cool zone through the empty centre and hot zone near the
ring coils. The two zones mix as the flow passes the fourth ring coil, as shown in
Figure 3. The thickness of the hot zone is assumed to be 3.3cm measured
perpendicularly from the heater walls, as shown in Figure 3. Such a value is determined
assuming that the centre line of the hot zone passes through the centre of the insulators
that support and centre the ring coils. The flow area of the hot zone is determined by
subtracting the cross-section areas of the ring coils and insulators from the cross-section
area of the hot zone. For the purpose of determining the temperature difference of
convection heat transfer, it is assumed that the temperature of air in the region occupied
by each ring coil in the hot zone is constant and equal to the average temperature of the
inlet and outlet for that ring coil, as shown in Figure 3.15. For the purpose of determining a correction factor for the Nusselt number due to
ring coil arrangement, each ring coil is modelled as two adjacent rows of tubes.
The diameter of each tube equals the coil diameter. The transverse distance between
tubes equals the pitch of the ring coil and the longitudinal distance equals the helix
diameter.
Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Energy Res. 2004; 28:12691291
A. M. BASSILY AND G. M. COLVER1272
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16. The heat loss occurs by convection from the external wall to the surroundings at a
constant convection heat transfer coefficient of 6.5 W m2 K1.
3.2. Defining the stretching factors
From the geometry of Figure 2, the length of ring coil 1 is given by the following equation:
Lc1 pDhxnc1 1
whereDhxis the helix diameter andnc1 is the number of turns of ring coil 1. Similarly, the helix
diameter to the pitch ratio for ring coil 1pc1can be determined from the coil geometry (please
see Figure 3) and is given by the following equation:
pc1 Dhxnc1
2pHh Dho=2 tana 0:1524 tan a Hs
Lc1
2p2Hh Dho=2 tana 0:1524 tan a Hs
fc1Lc
pLch2
whereHhis the total height of the heater, Dhois its outlet diameter,Hsis height of the insulators
that support the ring coil,Lchis a chosen length,fc1 is the stretching factor for ring coil 1, and a
is the cone half angle. The stretching factor is introduced to determine the effect of varying the
transverse distance and is defined as a ring coil length ratio multiplied by a dimensionless
number. The ring coil length ratio is the ratio of the ring coil length to the length of the
peripheral of the ring coil path. The dimensionless number is chosen to give the stretchingfactors values between 0.25 and 1.0 as feasible values that can be manufactured and installed.
The corresponding values for pciare 0.8 and 2.55, respectively. The dimensionless number is
chosen by picking a value for the length Lch. The stretching factor affects the length of the ring
coil and its convection heat transfer coefficient by changing the helix diameter to pitch ratio.
Equations for other ring coils dimensions can be written in a similar manner.
Coil 1
Coil 2
Coil 3
Coil 4
Ti
Tav1
Tav2Tav3
Tav4
To
0.0508 m
0.1778 m
0.127 m
0.0508 m
0.127 m
0.0254 m
Tav1
Tav2Tav3
Tav4
ToToTi Cool Zone
0.033 m
0.033 m
Hh
Hs
Dho
Figure 3. The temperature distribution of air in the heater.
Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Energy Res. 2004; 28:12691291
MODELLING AND PERFORMANCE ANALYSIS 1273
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3.3. Energy balance for the heater ring coils
Applying the energy equation for ring coil 1, gives
QQec1 QQrc1w1 QQrc1w2 QQrc1w3 QQrc1w4 QQrc1w5 QQrc1ehc1Ac1Tc1 Tav1 3
The energy equation can be applied for ring coils 24 in a similar manner.
3.4. Energy balance for air past the ring coils
Applying the energy equation for a control volume of air in the hot zone as it passes from the
heater inlet across ring coil 1, gives,
hw1Aw1Tw1Tinl hw2Aw20:127Hw1
Hw2Tw2Tinl hw2Aw2
Hw1Hw2 0:127
Hw2Tw2Tav1
hw3Aw30:1778Hw1 Hw2
Hw3Tw3 Tav1 hc1Ac1Tc1 Tav1 2 mmahzcpa Tav1Tinl 4
Similarly, the energy equation can be applied for a control volume of air in the hot zone as itpasses across ring coils 24.
3.5. Energy balance for the heater wall
Applying the energy equation for section 1 of the heater wall gives
QQrc1w1 QQrc2w1 QQrc3w1 QQrc4w1 QQrw2w1 QQrw3w1 QQrw4w1 QQcw2w1
hw1Aw1Tw1 Tinl hw1oAw1oTw1 Tinl QQrw1e 5
Similarly, the energy equation can be applied for sections 24 of the heater wall.
3.6. Evaluating the heat transfer coefficient from the heater ring coils
The angle between the flow direction and the ring coil surface is not 908as in the case of the flow
past a cylinder. Dutrich et al. (1990) showed that the average Nusselt number for flow past
coiled wires is less than that for cross flow past a single cylinder. The data of Dutrich et al.
(1990) indicate that for a helix diameter-to-pitch ratio of less than 1.0, and a Reynolds number
of more than 20, the ratio of the Nusselt number for flow past a coiled wire to that for flow past
a single cylinder (Nur) approaches 0.95 and increases as the Reynolds number increases. Dutrich
et al. (1990) also found that the wire diameter does not influence the ratio of the Nusselt number
(Nur). Extrapolating the data given by Dutrich et al. (1990), we get the following expression
for the correction factor for the Nusselt number as a function of the helix diameter to the pitch
ratio (p):
Nur 1 1:15 103 Re8:0555106 Re2p 6
Equation (6) is valid for a Reynolds number of 140 or less. For a Reynolds number of greater
than 140, Nuris assumed to be 1.0. The heat transfer coefficient of coil 1, hc1 can be found using
the Churchill-Bernsteins equation for flow over a single cylinder (Hodge, 1990) and the
Copyright # 2004 John Wiley & Sons, Ltd. Int. J. Energy Res. 2004; 28:12691291
A. M. BASSILY AND G. M. COLVER1274
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correction factor of Equation (6) so that,
NuD Nur 0:3 0:62Re
1=2D Pr
1=3
1 0:4=Pr2=31=4 1
ReD
282 000
5=8" #4=58