- Email: [email protected]
Effect of perforation shape and porosity on effective thermal conductivity of matrix heat exchanger plates
Accepted Manuscript Effect of Perforation Shape and Porosity on Effective Thermal Conductivity of Matrix Heat Exchanger Plates P. Navaneethakrishnan, K. Krishnakumar, Anish K John PII: DOI: Reference:
S1359-4311(16)30927-9 http://dx.doi.org/10.1016/j.applthermaleng.2016.06.024 ATE 8435
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
28 February 2016 3 June 2016 4 June 2016
Please cite this article as: P. Navaneethakrishnan, K. Krishnakumar, A.K. John, Effect of Perforation Shape and Porosity on Effective Thermal Conductivity of Matrix Heat Exchanger Plates, Applied Thermal Engineering (2016), doi: http://dx.doi.org/10.1016/j.applthermaleng.2016.06.024
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of Perforation Shape and Porosity on Effective Thermal Conductivity of Matrix Heat Exchanger Plates P Navaneethakrishnana1, K Krishnakumarb Anish K Johnc a
b
Student, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
Professor, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
c
Asst.Professor, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
Abstract
The effective thermal conductivity of a matrix heat exchanger plate is used to determine the heat transfer characteristics of the matrix exchanger in transient testing methods. In this paper, the influence of two parameters namely porosity, which is an indication of the void space and perforation shape, which is the geometry of the void, on effective thermal conductivity is analysed numerically and experimentally. The perforated copper plates of square, rhomboidal, rectangular and circular shapes of different porosities are used. A new parameter called perforation coefficient which generalises the different perforation shapes is coined. Correlations are developed using regression analysis to understand the effect of these parameters on effective thermal conductivity.
Keywords: Effective thermal conductivity; Matrix heat exchanger plates; Correlations; Perforation coefficient; Porosity; Numerical analysis; Experimental Analysis; Design and Manufacturing.
1. Introduction
The matrix heat exchangers [1] are high effectiveness heat exchangers used in cryogenic applications. Its increased effectiveness is due to the increased surface area of perforations and decrease in axial conduction due to alternate stacking of low thermal conductivity stainless steel spacers. Heat transfer characteristics of matrix heat exchangers are obtained by conducting single blow transient test [2]. The result of the test along with the effective thermal conductivity is used to calculate the effectiveness of the heat exchanger. Thus determination of effective thermal conductivity of the perforated plates is important. The plates can be made with different shapes of perforation and with different porosities. Thus the perforation shape and porosity become important influencing parameters. Porosity is the ratio of volume of the void spaces to the total volume of the plate.
1
Corresponding author. Tel.: +918547374786; E-mail address: [email protected]
1
The paper describes in detail the design and manufacturing of the plates, numerical and experimental methodologies and setups used to determine effective thermal conductivity (keff) of the plates, the study also develops a generalised parameter for the perforation shapes based on observations and arrives at correlations which quantifies the effect of the parameters.
2. Design and Manufacturing
The plates are made of copper with 5cm diameter and 0.3mm thickness, i.e. the material and thickness of the plates are considered to be constant parameters. The following set of analysis is done numerically and experimentally as provided in Table 1. Experiment
Parameters fixed
Parameter varied
Remarks
1
Shape: Rhomboidal
Porosity
(0.22, 0.30, 0.33, 0.37)
2
Shape: Circular
Porosity
(0.22, 0.30, 0.33, 0.37)
3
Porosity: 0.30
Shape
Circle, Rhomboid, Square, Rectangle
Table 1: List of Experiments To study the effect of porosity, plates of same shape but with different porosities are chosen. The porosity values are shown in brackets. To study the effect of perforation geometry, plates of same porosity (0.30) but of different shape of perforation are taken. The dimensions of individual perforations are shown in Fig 1(a). Four geometries namely circular, square, rectangular and rhomboidal are used. A drawing of the plate with porosity 0.30 and of rhomboidal shape is shown in Fig 1(b). Nomenclature keff
Effective thermal conductivity (W/mK)
kreff
Effective thermal conductivity ratio
p
Porosity
dpc
Perforation coefficient
D
Diameter of the perforated plate (mm)
T1
Temperature at the left end of the circular plate (⁰ C)
T2
Temperature at the right end of the circular plate (⁰ C)
w
Uncertainty in measurement
q
Heat flux (W/cm2)
dx
Gradient length (mm)
2
Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30. The manufacturing method used for the production of plates is photochemical milling [3]. In this process the pattern showing the desired profile is drawn to a convenient scale and photographed, to make a mask of exact size, on transparent plastic. Copper sheets coated with photo resist are exposed from an appropriate light source through the transparent mask. Exposed sheets are then developed and etched in chemical baths to get the desired plates. This method is used as the plate thickness is less than 0.5mm.
3. Methodology of investigation.
The methodology is to apply a known constant heat flux at the left end (Fig 2) and a constant temperature at the right end of the plate, so that the heat flux passes radially. All the other surfaces except the interior of the pores are insulated. Interior of the pores are assumed to have natural convection due to still air. It could then be solved for the end temperatures. Knowing the temperature difference across the plate ends (dT) and the length dx, the effective thermal conductivity (keff) is found using the Fourier conduction equation.
4. Numerical Investigation
The numerical investigation [4] is done on a finite element scheme on Ansys workbench 15.0. The perforated plates are modelled in Autodesk Inventor 2015 and a steady state thermal analysis is done on the imported model.
3
4.1 Geometry Geometry of one of the perforated plate used for numerical analysis is shown in Fig 2. The plate is inscribed in a rectangular block of dimensions 100mm x 50mm x 0.3mm. A similar model is created for all the other plates. The model is then imported to Ansys for meshing.
Figure 2: Geometry of perforated square plate taken for numerical analysis 4.2 Meshing The geometry is meshed with solid 90 element. It is a 20 node element having temperature degree of freedom at its nodes. The element has compatible temperature shapes and is well suited to model curved boundaries. A grid independence test is conducted for all the meshes. The mesh profile is shown in Fig 3.
Figure 3: Mesh profile 4.3 Boundary conditions The following boundary conditions are given to the meshed geometry. 1.
A constant heat flux of 2.64 W/cm2 (
2.
A constant temperature of 30oC given to the right face (face CD).
3.
Perfectly insulated condition given to the top, bottom, front and back faces.
4.
Convective condition of 5 W/m2K at mean temperature 30oC given to the inner surfaces of the pores.
) given to the left face (face AB).
4
5. Experimental Investigation 5.1 Experimental setup A schematic of the experimental setup is given Fig 4. The constant heat flux to the system is provided using a Nichrome heating coil and is regulated by an auto transformer. K type thermocouples are used to take the temperature at the ends of the plate. Constant temperature condition on the right face is achieved by circulating cooling water in a small water reservoir which takes away the heat. Agilent 34970 bench link data logger is used as the data acquisition system (DAQ) for recording the steady state temperature in every 10 seconds.
Figure 4: Experimental setup
5.2 Procedure The experimental procedure is as follows: 1.
Prepare the plates, wire and thermocouple connections as shown in Fig.4.
2.
Pour water to submerge the plate and insert thermometer as shown.
3.
Switch on the Autotransformer and turn the knob to about 100V.
4.
Switch on the DAQ and set to take the readings in every 10 seconds.
5.
After reaching a steady state note down the temperatures displayed on its monitor.
6.
Repeat the above steps for other plates.
The following initial readings are taken and the heat flux is calculated. Voltage Applied (V) = 104.6V. Current (I) = 38.9 mA. Area of cross section of the support plate (A) = 50mm * 1mm = 50 * 10-6m2. Heat flux =
83400W/m2.
5
6. Result and Discussion 6.1 Numerical Results No.
Shape
Porosity
T1
T2
ΔT
Δx
keff
(⁰ C)
(⁰ C)
(⁰ C)
(mm)
(W/mK)
1
Rhomboidal
0.22
113.4596
62.465
50.994
50
258.851
2
Rhomboidal
0.30
117.712
61.3311
56.3808
50
236.48
3
Rhomboidal
0.33
123.662
61.252
62.4101
50
211.504
4
Rhomboidal
0.37
132.267
60.5048
71.7622
50
185.798
5
Circular
0.22
100.573
62.1003
38.4724
50
346.56
6
Circular
0.30
102.286
61.6446
40.6414
50
328.07
7
Circular
0.33
103.138
61.3643
41.7734
50
319.182
8
Circular
0.37
113.5048
61.187
52.3178
50
252.304
9
Rectangle
0.30
111.640
61.4901
50.1499
50
265.869
10
Square
0.30
116.201
61.3652
54.8358
50
243.150
Table 2: Numerical values of effective thermal conductivity 6.2 Experimental Results No.
Shape
Porosity
T1
T2
ΔT
Δx
keff
(⁰ C)
(⁰ C)
(⁰ C)
(mm)
(W/mK)
1
Rhomboidal
0.22
56.2586
37.4875
18.771
53
237.173
2
Rhomboidal
0.30
60.1850
37.8023
22.3827
53
197.482
3
Rhomboidal
0.33
61.3127
38.2571
23.0556
53
191.719
4
Rhomboidal
0.37
64.6877
39.2542
25.4335
53
173.794
5
Circular
0.22
51.5315
37.8755
13.6560
52
317.573
6
Circular
0.30
51.6405
37.2721
13.9684
52
301.472
7
Circular
0.33
54.1988
38.9823
15.2165
52
290.486
8
Circular
0.37
57.0753
39.0210
19.0543
52
227.601
9
Rectangle
0.30
56.5180
38.3262
18.1918
52
238.393
10
Square
0.30
60.2911
38.1247
22.1663
53
199.410
Table 3: Experimental values of effective thermal conductivity
6
6.3 Uncertainty Analysis The uncertainty in effective thermal conductivity is determined using Kline and McClintock method.
Therefore,
But,
And
Therefore,
Using the above equations the uncertainty in effective thermal conductivity is found to be 19.3%. 6.4 Comparison No.
Shape
Porosity
keff
keff
Percentage
keff
Numerical(W/mK)
Experimental(W/mK)
deviation (%)
Average (W/mK)
1
Rhomboidal
0.22
258.851
237.173
-9.14
248.012
2
Rhomboidal
0.30
236.48
197.482
-16.49
216.966
3
Rhomboidal
0.33
211.504
191.719
-10.32
201.612
4
Rhomboidal
0.37
185.798
173.794
-6.46
179.796
5
Circular
0.22
346.56
317.573
-8.36
332.066
6
Circular
0.30
328.07
301.472
-8.11
314.771
7
Circular
0.33
319.182
290.486
-8.99
301.834
8
Circular
0.37
252.304
227.601
-8.54
292.666
9
Rectangle
0.30
265.869
238.393
-10.33
252.131
10
Square
0.30
243.150
199.410
-17.98
221.280
Table 4: Comparison of Numerical and Experimental values
7
The average values are taken for further discussion. 6.5 Variation of kreff with porosity Effective thermal conductivity ratio is the ratio of thermal conductivity of perforated plate (keff) to the thermal conductivity of unperforated plate (k) given by:
where, k is taken as 400W/mK. The variation is shown graphically in Fig 5:
Figure 5: Variation of kreff with porosity The following correlations are obtained: 1.
For circular shape : keff = (0.9974 – 0.7182p)k
2.
For rhomboidal shape: keff = (0.9856 – 1.4855p)k
These equations are similar to the equation given by M J Nilles et al. in for perforations of equilateral triangular arrangement in the literature [5] given by: keff = (1- 1.16p)k 6.6 Variation of kreff with perforation shape The effective thermal conductivity ratio values of different plates shown in Table 6. No
Shape
Porosity
kreff
1
Circular
0.30
0.8301
2
Rectangular
0.30
0.7869
3
Square
0.30
0.7620
4
Rhomboidal
0.30
0.5424
Table 5: Variation of kreff with perforation shape 8
To understand the variation of kreff with the perforation shape, a generalised parameter, which is a characteristic of the perforation shape, is essential. It is found out on the principle that: 1.
Any perforation causes a breakage in the conduction path.
2.
The extent of breakage in the conduction path is given by the maximum vertical distance of the cut (l) and the maximum horizontal distance of the cut (b) for any shape of perforation.
The different perforation shapes given in Fig 1(a) are analysed base on the principle and is given in Table 6:
No
Shape
Porosity
kreff
l (mm)
b (mm)
1
Circular
0.30
0.8301
1
1
2
Rectangular
0.30
0.7869
1
2.25
3
Square
0.30
0.7620
1.5
1.5
4
Rhomboidal
0.30
0.5424
1.7
1
Table 6: Variation of kreff with l and b.
The following three observations are made from Table 6: 1.
As the vertical distance increases the kreff value decreases.
2.
For the same vertical distance (No. 1& 2), the kreff value decreases with increase in horizontal distance (b).
3.
The rate of decrease of kreff(0.8301 – 0.5424) with vertical distance ( 1 – 1.7) is more when compared to the rate of decrease of kreff (0.8301 – 0.7869) with horizontal distance (1 – 2.25).
i.e. as the length (l) increases from 1 to 1.7 the decrease in effective thermal conductivity is 0.2877 but as the breadth (b) increases from 1 to 2.25 the effective thermal conductivity decreases only by 0.0432. Based on the observation the following conclusion is made “The influence of (l) on kreff is more when compared to the influence of (b)”.
By trial and error method, a new parameter called perforation coefficient (dpc) is defined by: dpc = 0.75l + 0.25b The parameter is made non dimensional by taking the
ratio.
Where, D = Diameter of the perforated plate. 9
The variation is graphically shown in Fig 6.
Figure 6: Graph showing variation of kreff with D/dpc The correlation connecting kreff and dpc could be obtained in the form Y = aXb as follows.
6.7 Bi – Variable correlation A general correlation relating effective thermal conductivity ratio (kreff) and other two parameters namely, porosity (p) and perforation coefficient is determined using regression analysis in MS Excel as follows:
The co-efficient of determination (R2) value is found to be 0.9872 and the significance F value of the regression analysis is 2.3*10-7. Thus the fit is good. The parity plot is given in Fig 7.
Figure 7: Parity plot The plot shows that the predicted values lie within an error bar of 5%. 10
7. Conclusions 1.
The effective thermal conductivity of the perforated plates are determined numerically and experimentally.
2.
The effect of porosity on effective thermal conductivity is studied. It is found that keff decreases with increase in porosity.
3.
The correlations connecting keff and porosity are determined and found to be close with the works in literature.
4.
The reason for the decrease in thermal conductivity of perforated plates is inferred to be a cut in the conduction path due to the presence of perforation.
5.
A new parameter called perforation coefficient (dpc) is coined, which quantifies the cut in the conduction path irrespective of the shape of perforation.
6.
The effective thermal conductivity ratio (kreff) is found to decrease with decrease in D/dpc ratio.
7.
A new correlation is obtained relating kreff and D/dpc as kreff =
8.
A
general
bi
–
variable
correlation
is
found
. using
regression
analysis,
given
by
.
8. References [1] Shah,R.K, and Sekulic, D.P., "Fundamentals of Heat Exchanger Design", John Wiley & Sons, 1-73, 2003. [2] Krishnakumar, K and Venkatarathanam, G., “Transient testing of perforated plate matrix heat exchanger”, Cryogenics, 43(3), 101 – 109, 2003. [3] Venkatarathanam, G and Sarangi, S., “Matrix heat exchangers and their applications in Cryogenic systems”, 30(11), 907 – 918, 1990. [4] John, A.K., Krishnakumar, K and Rathish, T.R., “Numerical method to determine effective thermal conductivity of perforated plate matrix heat exchanger surfaces and its experimental validation”, IJSER, 5(5), 1079 – 1084, 2014. [5] M J Nilles, M E Calkins, M.L Dingus and J.B Hendriks, “Heat transfer and flow friction in perforated plate heat exchangers”, Experimental thermal and fluid science, 10, 238 – 247, 1995.
11
TABLE AND FIGURE CAPTIONS
Table 1: List of Experiments Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30. Figure 2: Geometry of perforated square plate taken for numerical analysis Figure 3: Mesh profile Figure 4: Experimental setup Table 2: Numerical values of effective thermal conductivity Table 3: Experimental values of effective thermal conductivity Table 4: Comparison of Numerical and Experimental values Figure 5: Variation of kreff with porosity Table 5: Variation of kreff with perforation shape Table 6: Variation of kreff with l and b. Figure 6: Graph showing variation of kreff with D/dpc Figure 7: Parity plot
12
FIGURES
Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30.
Figure 2: Geometry of perforated square plate taken for numerical analysis
Figure 3: Mesh profile
13
Figure 4: Experimental setup
Figure 5: Variation of kreff with porosity
Figure 6: Graph showing variation of kreff with D/dpc
14
Figure 7: Parity plot
15
RESEARCH HIGHLIGHTS 1.
The effective thermal conductivity (keff) of the perforated plates is determined.
2.
Porosity and perforation geometry are the two influencing parameters.
3.
The correlations connecting keff and porosity are determined.
4.
New parameter called perforation coefficient (dpc) is coined to account for geometry.
5. A bi – variable correlation is obtained.
19
S1359-4311(16)30927-9 http://dx.doi.org/10.1016/j.applthermaleng.2016.06.024 ATE 8435
To appear in:
Applied Thermal Engineering
Received Date: Revised Date: Accepted Date:
28 February 2016 3 June 2016 4 June 2016
Please cite this article as: P. Navaneethakrishnan, K. Krishnakumar, A.K. John, Effect of Perforation Shape and Porosity on Effective Thermal Conductivity of Matrix Heat Exchanger Plates, Applied Thermal Engineering (2016), doi: http://dx.doi.org/10.1016/j.applthermaleng.2016.06.024
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effect of Perforation Shape and Porosity on Effective Thermal Conductivity of Matrix Heat Exchanger Plates P Navaneethakrishnana1, K Krishnakumarb Anish K Johnc a
b
Student, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
Professor, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
c
Asst.Professor, Department of Mechanical Engineering College of Engineering, Trivandrum, Kerala 695 016, India.
Abstract
The effective thermal conductivity of a matrix heat exchanger plate is used to determine the heat transfer characteristics of the matrix exchanger in transient testing methods. In this paper, the influence of two parameters namely porosity, which is an indication of the void space and perforation shape, which is the geometry of the void, on effective thermal conductivity is analysed numerically and experimentally. The perforated copper plates of square, rhomboidal, rectangular and circular shapes of different porosities are used. A new parameter called perforation coefficient which generalises the different perforation shapes is coined. Correlations are developed using regression analysis to understand the effect of these parameters on effective thermal conductivity.
Keywords: Effective thermal conductivity; Matrix heat exchanger plates; Correlations; Perforation coefficient; Porosity; Numerical analysis; Experimental Analysis; Design and Manufacturing.
1. Introduction
The matrix heat exchangers [1] are high effectiveness heat exchangers used in cryogenic applications. Its increased effectiveness is due to the increased surface area of perforations and decrease in axial conduction due to alternate stacking of low thermal conductivity stainless steel spacers. Heat transfer characteristics of matrix heat exchangers are obtained by conducting single blow transient test [2]. The result of the test along with the effective thermal conductivity is used to calculate the effectiveness of the heat exchanger. Thus determination of effective thermal conductivity of the perforated plates is important. The plates can be made with different shapes of perforation and with different porosities. Thus the perforation shape and porosity become important influencing parameters. Porosity is the ratio of volume of the void spaces to the total volume of the plate.
1
Corresponding author. Tel.: +918547374786; E-mail address: [email protected]
1
The paper describes in detail the design and manufacturing of the plates, numerical and experimental methodologies and setups used to determine effective thermal conductivity (keff) of the plates, the study also develops a generalised parameter for the perforation shapes based on observations and arrives at correlations which quantifies the effect of the parameters.
2. Design and Manufacturing
The plates are made of copper with 5cm diameter and 0.3mm thickness, i.e. the material and thickness of the plates are considered to be constant parameters. The following set of analysis is done numerically and experimentally as provided in Table 1. Experiment
Parameters fixed
Parameter varied
Remarks
1
Shape: Rhomboidal
Porosity
(0.22, 0.30, 0.33, 0.37)
2
Shape: Circular
Porosity
(0.22, 0.30, 0.33, 0.37)
3
Porosity: 0.30
Shape
Circle, Rhomboid, Square, Rectangle
Table 1: List of Experiments To study the effect of porosity, plates of same shape but with different porosities are chosen. The porosity values are shown in brackets. To study the effect of perforation geometry, plates of same porosity (0.30) but of different shape of perforation are taken. The dimensions of individual perforations are shown in Fig 1(a). Four geometries namely circular, square, rectangular and rhomboidal are used. A drawing of the plate with porosity 0.30 and of rhomboidal shape is shown in Fig 1(b). Nomenclature keff
Effective thermal conductivity (W/mK)
kreff
Effective thermal conductivity ratio
p
Porosity
dpc
Perforation coefficient
D
Diameter of the perforated plate (mm)
T1
Temperature at the left end of the circular plate (⁰ C)
T2
Temperature at the right end of the circular plate (⁰ C)
w
Uncertainty in measurement
q
Heat flux (W/cm2)
dx
Gradient length (mm)
2
Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30. The manufacturing method used for the production of plates is photochemical milling [3]. In this process the pattern showing the desired profile is drawn to a convenient scale and photographed, to make a mask of exact size, on transparent plastic. Copper sheets coated with photo resist are exposed from an appropriate light source through the transparent mask. Exposed sheets are then developed and etched in chemical baths to get the desired plates. This method is used as the plate thickness is less than 0.5mm.
3. Methodology of investigation.
The methodology is to apply a known constant heat flux at the left end (Fig 2) and a constant temperature at the right end of the plate, so that the heat flux passes radially. All the other surfaces except the interior of the pores are insulated. Interior of the pores are assumed to have natural convection due to still air. It could then be solved for the end temperatures. Knowing the temperature difference across the plate ends (dT) and the length dx, the effective thermal conductivity (keff) is found using the Fourier conduction equation.
4. Numerical Investigation
The numerical investigation [4] is done on a finite element scheme on Ansys workbench 15.0. The perforated plates are modelled in Autodesk Inventor 2015 and a steady state thermal analysis is done on the imported model.
3
4.1 Geometry Geometry of one of the perforated plate used for numerical analysis is shown in Fig 2. The plate is inscribed in a rectangular block of dimensions 100mm x 50mm x 0.3mm. A similar model is created for all the other plates. The model is then imported to Ansys for meshing.
Figure 2: Geometry of perforated square plate taken for numerical analysis 4.2 Meshing The geometry is meshed with solid 90 element. It is a 20 node element having temperature degree of freedom at its nodes. The element has compatible temperature shapes and is well suited to model curved boundaries. A grid independence test is conducted for all the meshes. The mesh profile is shown in Fig 3.
Figure 3: Mesh profile 4.3 Boundary conditions The following boundary conditions are given to the meshed geometry. 1.
A constant heat flux of 2.64 W/cm2 (
2.
A constant temperature of 30oC given to the right face (face CD).
3.
Perfectly insulated condition given to the top, bottom, front and back faces.
4.
Convective condition of 5 W/m2K at mean temperature 30oC given to the inner surfaces of the pores.
) given to the left face (face AB).
4
5. Experimental Investigation 5.1 Experimental setup A schematic of the experimental setup is given Fig 4. The constant heat flux to the system is provided using a Nichrome heating coil and is regulated by an auto transformer. K type thermocouples are used to take the temperature at the ends of the plate. Constant temperature condition on the right face is achieved by circulating cooling water in a small water reservoir which takes away the heat. Agilent 34970 bench link data logger is used as the data acquisition system (DAQ) for recording the steady state temperature in every 10 seconds.
Figure 4: Experimental setup
5.2 Procedure The experimental procedure is as follows: 1.
Prepare the plates, wire and thermocouple connections as shown in Fig.4.
2.
Pour water to submerge the plate and insert thermometer as shown.
3.
Switch on the Autotransformer and turn the knob to about 100V.
4.
Switch on the DAQ and set to take the readings in every 10 seconds.
5.
After reaching a steady state note down the temperatures displayed on its monitor.
6.
Repeat the above steps for other plates.
The following initial readings are taken and the heat flux is calculated. Voltage Applied (V) = 104.6V. Current (I) = 38.9 mA. Area of cross section of the support plate (A) = 50mm * 1mm = 50 * 10-6m2. Heat flux =
83400W/m2.
5
6. Result and Discussion 6.1 Numerical Results No.
Shape
Porosity
T1
T2
ΔT
Δx
keff
(⁰ C)
(⁰ C)
(⁰ C)
(mm)
(W/mK)
1
Rhomboidal
0.22
113.4596
62.465
50.994
50
258.851
2
Rhomboidal
0.30
117.712
61.3311
56.3808
50
236.48
3
Rhomboidal
0.33
123.662
61.252
62.4101
50
211.504
4
Rhomboidal
0.37
132.267
60.5048
71.7622
50
185.798
5
Circular
0.22
100.573
62.1003
38.4724
50
346.56
6
Circular
0.30
102.286
61.6446
40.6414
50
328.07
7
Circular
0.33
103.138
61.3643
41.7734
50
319.182
8
Circular
0.37
113.5048
61.187
52.3178
50
252.304
9
Rectangle
0.30
111.640
61.4901
50.1499
50
265.869
10
Square
0.30
116.201
61.3652
54.8358
50
243.150
Table 2: Numerical values of effective thermal conductivity 6.2 Experimental Results No.
Shape
Porosity
T1
T2
ΔT
Δx
keff
(⁰ C)
(⁰ C)
(⁰ C)
(mm)
(W/mK)
1
Rhomboidal
0.22
56.2586
37.4875
18.771
53
237.173
2
Rhomboidal
0.30
60.1850
37.8023
22.3827
53
197.482
3
Rhomboidal
0.33
61.3127
38.2571
23.0556
53
191.719
4
Rhomboidal
0.37
64.6877
39.2542
25.4335
53
173.794
5
Circular
0.22
51.5315
37.8755
13.6560
52
317.573
6
Circular
0.30
51.6405
37.2721
13.9684
52
301.472
7
Circular
0.33
54.1988
38.9823
15.2165
52
290.486
8
Circular
0.37
57.0753
39.0210
19.0543
52
227.601
9
Rectangle
0.30
56.5180
38.3262
18.1918
52
238.393
10
Square
0.30
60.2911
38.1247
22.1663
53
199.410
Table 3: Experimental values of effective thermal conductivity
6
6.3 Uncertainty Analysis The uncertainty in effective thermal conductivity is determined using Kline and McClintock method.
Therefore,
But,
And
Therefore,
Using the above equations the uncertainty in effective thermal conductivity is found to be 19.3%. 6.4 Comparison No.
Shape
Porosity
keff
keff
Percentage
keff
Numerical(W/mK)
Experimental(W/mK)
deviation (%)
Average (W/mK)
1
Rhomboidal
0.22
258.851
237.173
-9.14
248.012
2
Rhomboidal
0.30
236.48
197.482
-16.49
216.966
3
Rhomboidal
0.33
211.504
191.719
-10.32
201.612
4
Rhomboidal
0.37
185.798
173.794
-6.46
179.796
5
Circular
0.22
346.56
317.573
-8.36
332.066
6
Circular
0.30
328.07
301.472
-8.11
314.771
7
Circular
0.33
319.182
290.486
-8.99
301.834
8
Circular
0.37
252.304
227.601
-8.54
292.666
9
Rectangle
0.30
265.869
238.393
-10.33
252.131
10
Square
0.30
243.150
199.410
-17.98
221.280
Table 4: Comparison of Numerical and Experimental values
7
The average values are taken for further discussion. 6.5 Variation of kreff with porosity Effective thermal conductivity ratio is the ratio of thermal conductivity of perforated plate (keff) to the thermal conductivity of unperforated plate (k) given by:
where, k is taken as 400W/mK. The variation is shown graphically in Fig 5:
Figure 5: Variation of kreff with porosity The following correlations are obtained: 1.
For circular shape : keff = (0.9974 – 0.7182p)k
2.
For rhomboidal shape: keff = (0.9856 – 1.4855p)k
These equations are similar to the equation given by M J Nilles et al. in for perforations of equilateral triangular arrangement in the literature [5] given by: keff = (1- 1.16p)k 6.6 Variation of kreff with perforation shape The effective thermal conductivity ratio values of different plates shown in Table 6. No
Shape
Porosity
kreff
1
Circular
0.30
0.8301
2
Rectangular
0.30
0.7869
3
Square
0.30
0.7620
4
Rhomboidal
0.30
0.5424
Table 5: Variation of kreff with perforation shape 8
To understand the variation of kreff with the perforation shape, a generalised parameter, which is a characteristic of the perforation shape, is essential. It is found out on the principle that: 1.
Any perforation causes a breakage in the conduction path.
2.
The extent of breakage in the conduction path is given by the maximum vertical distance of the cut (l) and the maximum horizontal distance of the cut (b) for any shape of perforation.
The different perforation shapes given in Fig 1(a) are analysed base on the principle and is given in Table 6:
No
Shape
Porosity
kreff
l (mm)
b (mm)
1
Circular
0.30
0.8301
1
1
2
Rectangular
0.30
0.7869
1
2.25
3
Square
0.30
0.7620
1.5
1.5
4
Rhomboidal
0.30
0.5424
1.7
1
Table 6: Variation of kreff with l and b.
The following three observations are made from Table 6: 1.
As the vertical distance increases the kreff value decreases.
2.
For the same vertical distance (No. 1& 2), the kreff value decreases with increase in horizontal distance (b).
3.
The rate of decrease of kreff(0.8301 – 0.5424) with vertical distance ( 1 – 1.7) is more when compared to the rate of decrease of kreff (0.8301 – 0.7869) with horizontal distance (1 – 2.25).
i.e. as the length (l) increases from 1 to 1.7 the decrease in effective thermal conductivity is 0.2877 but as the breadth (b) increases from 1 to 2.25 the effective thermal conductivity decreases only by 0.0432. Based on the observation the following conclusion is made “The influence of (l) on kreff is more when compared to the influence of (b)”.
By trial and error method, a new parameter called perforation coefficient (dpc) is defined by: dpc = 0.75l + 0.25b The parameter is made non dimensional by taking the
ratio.
Where, D = Diameter of the perforated plate. 9
The variation is graphically shown in Fig 6.
Figure 6: Graph showing variation of kreff with D/dpc The correlation connecting kreff and dpc could be obtained in the form Y = aXb as follows.
6.7 Bi – Variable correlation A general correlation relating effective thermal conductivity ratio (kreff) and other two parameters namely, porosity (p) and perforation coefficient is determined using regression analysis in MS Excel as follows:
The co-efficient of determination (R2) value is found to be 0.9872 and the significance F value of the regression analysis is 2.3*10-7. Thus the fit is good. The parity plot is given in Fig 7.
Figure 7: Parity plot The plot shows that the predicted values lie within an error bar of 5%. 10
7. Conclusions 1.
The effective thermal conductivity of the perforated plates are determined numerically and experimentally.
2.
The effect of porosity on effective thermal conductivity is studied. It is found that keff decreases with increase in porosity.
3.
The correlations connecting keff and porosity are determined and found to be close with the works in literature.
4.
The reason for the decrease in thermal conductivity of perforated plates is inferred to be a cut in the conduction path due to the presence of perforation.
5.
A new parameter called perforation coefficient (dpc) is coined, which quantifies the cut in the conduction path irrespective of the shape of perforation.
6.
The effective thermal conductivity ratio (kreff) is found to decrease with decrease in D/dpc ratio.
7.
A new correlation is obtained relating kreff and D/dpc as kreff =
8.
A
general
bi
–
variable
correlation
is
found
. using
regression
analysis,
given
by
.
8. References [1] Shah,R.K, and Sekulic, D.P., "Fundamentals of Heat Exchanger Design", John Wiley & Sons, 1-73, 2003. [2] Krishnakumar, K and Venkatarathanam, G., “Transient testing of perforated plate matrix heat exchanger”, Cryogenics, 43(3), 101 – 109, 2003. [3] Venkatarathanam, G and Sarangi, S., “Matrix heat exchangers and their applications in Cryogenic systems”, 30(11), 907 – 918, 1990. [4] John, A.K., Krishnakumar, K and Rathish, T.R., “Numerical method to determine effective thermal conductivity of perforated plate matrix heat exchanger surfaces and its experimental validation”, IJSER, 5(5), 1079 – 1084, 2014. [5] M J Nilles, M E Calkins, M.L Dingus and J.B Hendriks, “Heat transfer and flow friction in perforated plate heat exchangers”, Experimental thermal and fluid science, 10, 238 – 247, 1995.
11
TABLE AND FIGURE CAPTIONS
Table 1: List of Experiments Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30. Figure 2: Geometry of perforated square plate taken for numerical analysis Figure 3: Mesh profile Figure 4: Experimental setup Table 2: Numerical values of effective thermal conductivity Table 3: Experimental values of effective thermal conductivity Table 4: Comparison of Numerical and Experimental values Figure 5: Variation of kreff with porosity Table 5: Variation of kreff with perforation shape Table 6: Variation of kreff with l and b. Figure 6: Graph showing variation of kreff with D/dpc Figure 7: Parity plot
12
FIGURES
Figure 1: (a) Dimensions of individual geometrical shapes, (b) Drawing of plate with rhomboidal shape and porosity 0.30.
Figure 2: Geometry of perforated square plate taken for numerical analysis
Figure 3: Mesh profile
13
Figure 4: Experimental setup
Figure 5: Variation of kreff with porosity
Figure 6: Graph showing variation of kreff with D/dpc
14
Figure 7: Parity plot
15
RESEARCH HIGHLIGHTS 1.
The effective thermal conductivity (keff) of the perforated plates is determined.
2.
Porosity and perforation geometry are the two influencing parameters.
3.
The correlations connecting keff and porosity are determined.
4.
New parameter called perforation coefficient (dpc) is coined to account for geometry.
5. A bi – variable correlation is obtained.
19