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Heat loss in insulated pipe the influence of thermal contact resistance: A case study
1359-8368(95)00028-3
ELSEVIER
Composites: Part B 27B (1996) 85-93 © 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
H e a t loss in i n s u l a t e d pipe t h e i n f l u e n c e of t h e r m a l c o n t a c t resistance: a case s t u d y
Michael A. Stubblefield, Su-Seng Pang* and Vic A. Cundyt Department of Mechanical Engineering. Louisiana State University. Baton Rouge. LA 70803. USA ~Oepartment of Mechanical Engineering. Montana State University. Bozeman. MT 59715. USA (Received 12 January 1994. accepted 28 July 1995) The thermal contact resistance is an important parameter in many heat loss problems. Determining the contact resistance for practical systems is quite complex due to the dependency of the relative geometry of the contacting surfaces. It is, therefore, difficult to make general contact resistance data available in the literature. In this paper, we first describe a simple model to predict the effect of contact resistance. This is followed by describing a simple device which can be used to measure thermal contact resistance for an insulated pipe system. The apparatus consists of a steel containment pipe exposed to saturated steam. The heat flux is determined by measuring steam condensate over a fixedperiod of time, while temperature measurements are obtained using standard type K thermocouples. The apparatus is calibrated using insulating materials with known thermal conductivities as they are necessary for the calibration and validation of the experimental setup. Once the device has been calibrated, the thermal contact resistance is determined for the insulating materials (standard fiberglass and calcium-silicate) using the electrical analog resistance method. It is shown also that the energy loss in a system may be affected by manipulating the contact resistance between the pipe and insulation. The effect of a small air gap to influence contact resistance is investigated. By placing spacers between the pipe and insulation, effectively producing a small air gap, we were able to significantly alter the contact resistance. A generalized optimization approach is also presented. The defined parameters are considered as a function of insulation cost and the cost due to the energy loss of the system. (Keywords: heat loss; contact resistance; insulated pipe; steam; energy loss)
INTRODUCTION Heat transfer between a pumped fluid and the environment, in general, occurs by convection and conduction. Consider the insulated pipe system shown in Figure 1. Energy transfer in this system occurs via: (1) Convection from the pumped fluid t o the inner wall of the pipe, (2) Conduction from the inner to the outer wall of the pipe, (3) Conduction from the outer wall o f the pipe to the insulating medium (involving contact resistance), (4) Conduction thrOugh the insulating medium, and (5) Convection to the environment. The problem may be modeled simply (as in the case of steady heat transfer using the electrical resistance analog) or more detailed (by considering the unsteady case where the solution to the unsteady conduction equation gives, * TO whom correspondenceshould be addressed
by closed form or numerical method, the temperature history of the system). In either case, the energy loss or gain from the system can be readily calculated, analytically from initial boundary conditions, once the temperature distribution is determined. In any realistic model the contact resistance will have to be simulated. This paper starts with an overview of existing techniques currently used to handle contact resistance. A simple model of the contact resistance is developed and verified using experimental data. Finally, the contact resistance for several pipe configurations currently used in the field is determined experimentally.
B A C K G R O U N D OF A N A L Y T I C A L M O D E L S Contact resistance arises because the surfaces are in actual contact over only a small percentage of the apparent area. M a n y of the previous studies of contact resistance were concerned with relatively small contacting surfaces. Those surfaces are often characterized as a
COMPOSITES,PART B Volume 27B Number 1 1996
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Heat loss in insulated pipe: M. A. Stubblefield et al. Environment Energy T r a n s fe r
4> Ins u lation ~ ~ ~ N ~ k N ~ ~ x ~ k ~ Containment
Pipe
~///////'/////////////~//////////~5
ii
Figure3 Temperaturediscontinuitydue to contact resistance14
S t e a d y Flow of-Hot,,. P u m p e d - Fluid Figure 1 Insulted pipe arrangement uniform distribution of micro contacts. Subsequent studies understood that this approach was only appropriate if the macroscopic nonuniformity o f the contacting surfaces was negligible. These macroscopic contacts may be identified as a direct effect of the surface waviness and/or deformation of the surface due to applied loadings. When nominally flat surfaces are held in contact under loading, the heat transfer between them is impeded by a surface contact resistance at the interface causing a temperature drop across the contacting surfaces as seen in Figure 21 . The degree o f the thermal resistance at the interface is a function of the actual contact area, pressure between the two surfaces, thermal and physical properties of the mating surfaces, and interstitial fluids present between the contacting surfaces. Past experimental investigations have demonstrated that for relatively smooth, nominally flat surfaces, the actual contact area is between 2 and 5% of the apparent contact area 2. In the study o f the actual contact, the associated asperities may be modeled as conical in shape. Analytical results 3 showed that the contact or constriction resistance was reduced significantly in the presence of a conducting fluid (see Figure 34). Heat can be transferred across the contacting interface via the following means: (a) solid-to-solid conduction at contact points, (b) conduction through the fluid in the gaps around the contacts, (c) free convection within the gap, (d) radiation across the gap, or (e) any combination of these. In actual application, these modes are interdependent and difficult to predict.
A
B
'
Figure2 Heat flow through a joint8
86
COMPOSITES PART B Volume 27B Number 1 1996
The contribution of solid-to-solid conduction is strongly dependent on the number and size of asperities in actual contact region. Several correlations between solid-to-solid conduction and the parameters affecting it have been developed over the past 20 years. Developmerits 5 have led to a correlation relating solid-to-solid conduction to the ultimate strength of the softer contacting material and a constant dependent upon the average surface roughness heights, and the radius of the contact spot. Other factors considered 6 are the mean junction temperature, a gap dimension parameter accounting for the roughness as well as the flatness deviation, and the variation of the above parameters with contact pressure. The effects of gas rarefaction and surface roughness as it relates to the thermal gap conductance were studied 7. The gap conductance was defined as inversely proportional to the effective gap thickness, which typically lies in the range of 0.1-100#m. Since the thickness o f the interstitial gap is usually much smaller than the other contact dimensions, only fluid conduction normal to the contact plane is usually considered. It is noted that the effective gap thickness is of the same order of magnitude as the mean free path of the gas molecules in contact for normal engineering applications. Free convection and radial conduction factors are usually considered negligible within the fluid. Several approaches have been developed, however, to determine free convective coefficients utilizing the interstitial gap distance. The so-called gap parameter 5 uses the ratio of the volume of the interface fluid to the contact area. Free convective effects were also related to the maximum heights of roughness of the contacting surfaces, approach of surfaces under loading, maximum thickness of the interstitial fluid layer, and the temperature jump distance 6. A review of contact resistance studies g suggests that surface parameters are a determining factor in thermal contact resistance. Also, recent studies evaluating conforming surfaces, along with the developed models, predict the real contact area. Recent studies consider the geometrical aspects of the contacting materials as well as the surface preparation. Several studies have been done to enhance the thermal contact conductance, the reciprocal of the contact resistance, across an interface. One method commonly used includes the use of a thermal grease between the contacting surfaces. A thermomechanical model for nominally flat rough contacting surfaces coated with a metallic layer allows for a coated joint to be reduced to
Heat loss in insulated pipe." M. A. Stubblefield et al. an equivalent bare joint using an effective microhardness and an effective thermal conductivity 9. An experimental investigation was conducted to study the effect metallic coatings have on thermal contact conductance 1. The hardness of the coating materials seemed to be the prominent parameter in the magnitude of this effect. This contact conductance was also noticed to be greatest at low contact pressures. The effects of metallic coatings on contacting surfaces at cryogenic temperatures was also studied 1°. The developed model requires only a knowledge of the surface roughness, thermal conductivities and microhardness values. As the temperature increased, the microhardness increased by 35% for the copper in the experimental study, which validates the incorporation of microhardness parameter when discussing cryogenic temperatures, One characteristic of thermal contact resistance at cryogenic temperatures is that it is nearly inversely proportional to the bulk thermal conductivity of the contacting material. Later, it was shown that beryllium reacted inversely to this postulation ll. A possible reason may be because at the interface more static imperfections became evident as compared to those in the bulk material, thus causing the difference between the bulk and interfacial thermal conductivity measurements. Another important factor in consideration of the magnitude of contact resistance is the phenomenon known as directional effect. This directional effect is a trait of thermal contacts to present a greater thermal resistance in one direction as compared to the reversed direction. The effects of thermal strain as a source of an enhanced directional effect were considered 12. From experimentation, it was noticed that the contact resistance at the interface of dissimilar metals is strongly dependent on the heat flux. Changes due to differing properties of contacting surfaces may also affect directional effects ~3. These results were later related with surface interface conditions such as the flatness and roughness of the contacting surfaces. Experimental studies ~4 suggest that the thermal contact resistance between similar metals was lesser when the heat flow was from the higher thermal conductivity metal. However, it was also believed 12 that for the same material pairs, the thermal strain should not cause directional effects to manifest. Thermal radiation involves energy emission from matter occurring at a given temperature. Unlike the transfer of energy by conduction or convection, radiation does not require the presence of a material medium. The radiative resistance is a complex parameter depending upon the geometry of the gap, the emissivities of the contacting solids and the side walls forming the enclosure, as well as the temperature distribution of the bounding surfaces. In general, for low temperature applications, neglecting radiation does not introduce significant error into the problem. There are a number of techniques available for the prediction of contact resistance. However, each approach is appropriate for only selected applications.
Very few, in general, appear to be successful, even in consideration of the application for which the models were developed. Obviously, if one considers all modes of heat transfer, the analysis of thermal contact resistance is quite formidable. Indeed, this is the reason that most contact resistance problems are solved using empirical approaches. If one is to model the problem totally, an understanding of the multimode heat transfer phenomena is required and fundamental physical properties such as the thermal conductivities of the contacting solids and interstitial fluid, emissivity, hardness, gap thickness, etc., must be determined. Such a task is very difficult and usually many simplifying assumptions must eventually be made. A very sophisticated analytical model may be too problem specific to be of general use. On the other hand, a much simplified model requiring fewer input parameters might also fail to accurately predict some situations due to the inherent simplifications of the model. As a result, a compromise approach might be the desired alternative in many engineering applications. This is the approach we have chosen. In this approach, a relatively simple engineering model is proposed for specific pipe/insulation configurations. This model, macroscopic in nature, will include enough physics to accurately describe the contact resistance phenomena, while relying heavily on experimentation. It is hoped that such a model will provide a good tool for engineering optimization.
G O A L OF THIS W O R K The goal of this work is to develop a simple model which will accurately predict the heat transfer from an insulated pipe. A primary objective is to handle contact resistance in a relatively simple manner and verify the results experimentally. The model will be situation and geometry dependent. Another objective of the study is to develop a simple test rig to obtain the thermal physical properties of the pipe and insulating material. This apparatus will be used to measure the thermal contact resistance for insulated pipe systems. The final objective is to use the developed model to optimize the energy transfer for a given pipe/ insulation configuration. The optimization approach is based on an economic analysis of the piping system as a function of insulation cost and the subsequent cost due to the energy loss of the system.
ENGINEERING MODEL Contact resistance can be modeled using the electricalanalog approach with an apparent convective heat transfer coefficient, defined as R~o~ -
1
hconAa
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Heat loss in insulated pipe. M. A. Stubblefield et al. where: Rcon: the thermal contact resistance, hcon: the 'effective' convective coefficient for contact resistance, and Aa: the area based on the radius of the pipe/ insulation interface. In the following discussion, the assumptions listed below have been made: (1) the thermal conductivity and other physical properties are constant, (2) the contact is static, (3) the contribution from radiation in the contact resistance term is small and neglected, and (4) the heat flux is steady. Given these assumptions, the rate of heat transfer, Q, under steady conditions, is:
Q = UAAT
(2)
where U is an overall conductance based on the area A and A T is the temperature difference between the pumped-fluid and the ambient. The overall conductance, U, is obtained by determining the individual resistances associated with the chosen pipe/insulation configuration (see Figure 4). For the insulated pipe arrangement shown, these resistances are: R1: R2: R3: R4: Rs:
the convective resistance between the pumped-fluid and innermost pipe wall, the conductive resistance through the pipe, the contact resistance between the steel and insulating medium, the conductive resistance through the insulating medium, and the convective resistance between the insulation and ambient.
This resistive network, shown in Figure 4, is valid at any axial location along the pipe, where 1
RI, 5
-
-
hi,o(27rrl,31)
in r2 r] R2 = 27rkl~~
in r3 1
R3
r2
hcon(27rr21)
R4
-
-
and, hi: l: kl: hcon: k2: ho:
inner convective coefficient, length of pipe under consideration, thermal conductivity of containment pipe, the 'effective' convective coefficient for contact resistance thermal conductivity of insulation, and outer convective coefficient.
While this simple thermodynamic analysis appears to provide the energy loss from the pumped-fluid in a straightforward manner, the complexity of the contact resistance makes application of the analysis challenging. The challenge is to determine the 'effective' contact convective coeffÉcient, hcon. A specially designed test rig has been built to determine hcon. The rig is shown in Figure 5. In this work, we only consider steady-state values of contact resistance (nontime or temperature varying). The test rig is designed so that the temperature of the inner surface of the containment pipe is known very accurately. Other temperatures are also measured within the insulating material using conventional thermocouples. Heat loss through the pipe/insulation is also measured very accurately. Knowing the energy loss through the pipe and the temperature potential, the term hcon, is readily determined by considering the thermal network from the surface of the inner pipe to a location in the insulation where the temperature is measured (not necessarily the outer skin insulation temperature). The electrical analog for this network is: Q-
AT
(4)
R2 + R3 q- R 4
where A T is the temperature difference between resistances R 2 and R 4. The conductive resistances, R 2 and R4, are determined
Stainless Steel Containment__ Saturated Pipe \Steam Supply
Temperature Measurement -- Location with Known Separation
TAmbient
Tpumped- Pumped n.id Fluid
Containment Pipe
RI R~ R3 Tpumped-
TI
Fluid
Figure 4
88
R4
R5 Te
T.tmbient
Elcctrical analog network
COMPOSITES PART B Volume 27B Number 1 1996
(3)
27rk21
_ Figure 5
Insulation
Thermal conductivity/contact resistance test rig
Heat loss in insulated pipe: M. A. Stubblefield et al. by knowledge of the geometry of the pipe/insulation configuration, and the thermal conductivities of the pipe and the insulation. Only the contact resistance term remains unknown and in this term, hcon, is readily solved for.
EXPERIMENTAL APPARATUS The test rig developed for this work is shown in Figure 5. It is used not only to measure the contact resistance, but also to measure the thermal conductivity of the pipe and insulating materials. Materials from the field are used in the rig since the pipe/insulation pair determines the contact resistance. Installation methods are also the same as those used in the field since contact resistance is sensitive to surface contact pressures. In this test rig, saturated steam condenses on the inner surface of the containment pipe. The heat transfer rate is constant and is determined by measuring steam condensate as energy is transferred from the steam through the pipe/insulation configuration. A calibrated sight glass is used to measure the condensate level. Temperatures are obtained using thermocouples. Since saturated steam is condensing at the inner surface of the containment pipe, the convective coefficient between the steam and pipe is assumed large, and the inner pipe temperature is taken as the saturation temperature of the steam. In consideration of the thin wall approximation for pipes with high thermal conductivity, this temperature can also be approximated as the outer wall temperature of the containment pipe. Since the rig makes use of saturated steam and the energy measurement is based on condensed steam, it is necessary to determine the water content of the steam for each experiment (the steam quality). This is accomplished using standard separating/throttling calorimetry techniques.
Thermal conductivity experiment The thermal conductivity of the containment pipe and insulating material must be determined before the contact resistance can be obtained. While such data are available for many commonly used materials, we chose to measure the conductivity for the materials of our tests for two reasons. First, such measurements validate the experimental methodology in as much as the conductivities for the materials of our tests were given by the manufacturer. By comparing with the published values, the test rig and the methodology are examined. Second, in the future, we anticipate using materials for which there is little or no thermal property information. The thermal conductivity of the pipe and insulating materials is readily determined using the experimental test rig. Referring to Figure 5, thermocouples are accurately placed at a fixed distance within the insulating material. Knowing the energy transfer and geometry, results in a straightforward calculation for thermal conductivity using the following relationship:
Q = AT/Rx
(5)
where Rx is the conductive resistance of the insulation over the distance shown in Figure 5 and A T is the associated temperature difference. The heat transfer, Q, is measured, A T is measured; therefore, the only unknown in equation (5) is the thermal conductivity.
'Effective' convective heat transfer coefficient for contact resistance Once the heat transfer and thermal conductivities of the system are determined, the 'effective' convection coefficient for contact resistance, hcon, is readily determined using equations (4) and (6). R2 --
in r2 rl 2~rkl l
R3
--
1
hcon (27rr2l)
R4 __
in r3 r2
27rk21
(6)
EXPERIMENTAL PROCEDURE The experimental procedure consists of the following steps: (1) obtain steam quality, (2) measure the thermal conductivity of the pipe and the insulation, and (3) measure the heat transfer rate and determine the contact resistance between the pipe and the insulating material, and consequently, the 'effective' convective coefficient for the contact resistance, hcon.
Steam quality In this work, both a separating and throttling calorimeter are used in series to determine the wetness of the steam. Typical steam qualities for the data shown are in the range from 79 to 90%.
EXPERIMENTAL RESULTS
Validation of the model and expertments The development of the model and experimental techniques is based on the fact that the total energy dissipated can be determined through the measurement of condensation of water. If the energy dissipation through the pipe section can be determined, the contact resistance can be calculated. The total energy dissipation consists of that through the test pipe section and the steam feedline to the pipe. The loss associated with the feedline must either be minimized or accounted for. In our approach, we chose to calibrate by accounting for the heat loss. In order to calibrate the energy loss through the line, an insulation foam with a given manufacture thermal conductivity was used. In this case, a calcium-silicate
COMPOSITES PART B Volume 27B Number 1 1996
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Heat loss in insulated pipe. M. A. Stubblefield et al. pipe insulation is used. It has a known thermal conductivity of 0.071W/m-K (0.041Btu/h-ft-F). The data obtained was for a piping system consisting of a containment pipe with 50.8 mm (2.0 in) nominal inside diameter, covered with 25.4mm (l in) thick calciumsilicate insulation. The pipe is 304 stainless steel schedule 40 with a wall thickness of 4 mm (0.16 in) and a length of 914ram (3.0ft). The thermal conductivity of the steel pipe is reported as 16.9W/m-K (9.8 Btu/h-ft-F). After 10 experimental trials, the total energy loss (including that through the test pipe and steam feedline), using the calcium-silicate insulation as a basis, was 115.0 -4- 16.0W (392.4 + 54.6Btu/h). Using the thermal conductivity provided by the manufacturer, the heat loss through the pipe is determined to be 45.1W (153.9 Btu/h). After the calibration procedure, the energy lost through the pipe system can be obtained by deducting the energy loss through the feedline from the total energy loss. In order to verify the model and experiments, another insulating foam, fiberglass, with a known thermal conductivity of 0.05-0.06W/m-K (0.0300.035 Btu/h-ft-F) was used in this part of the work. Two verifications were obtained. The first one was an insulated pipe system without an air gap, while the second case consisted of a small air gap between the containment pipe and the insulation. We did this because in later experiments, we will use an air gap to increase the effective insulation of the pipe system. The existence of the air gap should result in no change of the insulation thermal conductivity. Table 1 shows the results of the experimental tests for a steel pipe with fiberglass insulation, both with and without an air gap. Consider Table 1 with insulation only (no air gap). The thermal conductivity of the fiberglass insulation was calculated to be 0.052 i 0.006W/m-K (0,030 + 0.003 Btu/h-ft-F). When the air gap was introduced between the insulation and containment pipe, the thermal conductivity of the insulation was 0.052 :[: 0.001 W / m - K (0.030 5:0.0006 Btu/h-ft-F) (refer to Table 1). The errors between the measured value and the manufacturer reported value were 9.2 and 8.1%, respectively. Table I
Table 2
Experimental data for steel pipe with fiberglass insulation
Run #
X Quality
hcon W/m2-K
Rcon K/W
1 2 3 4 5 6 7 8
0.81 0.80 0.81 0.81 0.81 0.80 0.80 0.81
2.90 2.13 2.92 2.91 2.18 2.72 2.81 2.84
2.00 2.57 1.98 1.98 2.42 2.12 2.05 2.03
Mean ± standard dev.
0.81 ± 0.005
2.67 50.307
2.14 ± 0.211
The above results show that the average difference, with a magnitude of 0.057W/m-K, based on two different situations, was about 8.7%. As the manufacturer published value may also have some level of uncertainty and the exact value can also be considered very temperature dependent, the current model proposed, along with experimental setup, is deemed valid. While the focus of this study is on the contact resistance and the energy dissipation, it is obvious that once the energy loss through the line is calibrated, the model and device can be used to obtain the thermal conductivity of other insulating materials. Thermal contact resistance determination
Based on the experimental data, the value of the quality, x, calculated value of hcon and the thermal contact resistance, Rcon, for each experiment performed are shown in Tables 2 and 3. In the fiberglass configuration, the resistance due to contact was 2.14 K/W (1.13 F/h-Btu). The pipe resistance was 0.0007K/W (0.00037F/h-Btu) and the fiberglass insulation resistance was 1.85K/W (0.976F/.h-Btu). Therefore, the contact resistance is 54% of the total resistance. For the calcium-silicate configuration, the resistance due to the calcium insulation was 1.33 K/W (0.702 F/h-Btu) and the contact resistance was 1.20 K/W (0.633 F/h-Btu). In this case, the contact resistance is 47% of the total resistance.
Experimental data for steel pipe with fiberglass insulation Table 3
Experimental data for steel pipe with calcium-silicate insulation
% Error (no air gap)
kins W/m-K (with air gap)
% Error (with air gap)
Run #
x Quality
hcon W/mZ-K
Rcon K/W
Run #
kins W/m-K (no air gap)
1 2 3 4 5 6 7 8
0.057 0.042 0.057 0.057 0.043 0.053 0.055 0.056
0.0 26.3 10.0 0.0 24.6 7.0 3.5 1.8
0.051 0.053 0.053 0.053 0.053 0.051 0.054 0.051
10.5 7.0 7.0 7.0 7.0 10.5 5.3 10.5
1 2 3 4 5 6 7 8 9 10
0.86 0.86 0.87 0.85 0.89 0.86 0.86 0.87 0.86 0.86
3.0 9.2 6.6 4.4 7.0 2.9 6.4 6.6 4.4 4.4
1.94 0.63 0.88 1.32 0.82 1.97 0.90 0.87 1.31 1.31
Mean + standard dev.
0.052 ± 0.006
9.2 ± 10.6
0.052 ± 0.001
Mean • standard dev.
0.86 ± 0.010
5.50 ± 1.89
1.20 5_ 0.44
90
8.1 ± 1.94
COMPOSITES PART B Volume 27B Number 1 1996
Heat loss in insulated pipe, M. A. Stubblefield et al. Table 4 Experimental data for steel pipe/air gap/fiberglass insulation
70
0.12
Run #
x Quality
hcon W/m2-K
Rcon K/W
t-" o t~
1 2 3 4 5 6
0.83 0.84 0.84 0.85 0.84 0.84
3,07 3,20 3.20 3.84 3.20 3.06
1.88 1.80 1.80 1.50 1.80 1.89
.c
7 8
0.83 0.83
3.20 3.06
1.80 1.89
~"
M e a n :~ s t a n d a r d dev.
0.83 ± 0.020
3.23 ± 0.240
1.80 ± 0.119
I
0.10 / o
60
,d
co
/ 50
0.08
"u o
E g,
0.06
Thermal - 40 Conductivity - - - H e a t Loss
stY" ~"
0.04
I 0.6
0.5
I 0.7
J 0.8
I 0.9
30 1.0
Steam Quality, X Figure 6 Thermal conductivity/heat loss versus steam quality
These results show that the test rig provides reproducible data over a range of steam qualities. F o r example, by varying the steam quality of a specific case and keeping all other factors constant, it is noticed that across a quality range from 0.5 to 1.0, the heat dissipation only changes by 35.0W ( l l 9 . 4 B t u / h ) and the corresponding thermal conductivity changes by 0.05 W / m - K (0.03 Btu/h-ft-F), as seen in Figure 6. The operating range of the quality during the experimental trials ranged from 0.8 to 0.9. Therefore, it is shown that the overall difference is only 7.0W (23.9Btu/h) and 0.01 W / m - K (0.006 Btu/h-ft-F) for the heat loss and the thermal conductivity, respectively. This shows that the
25
20 v
o~ t~
15
- -- C a l c i u m - S i l i c a t e
/
data presented are only slightly adherent to steam quality. As seen in Figure 7, as the insulation thickness increases, the total thermal resistance increases also, following a logarithmic behavior. Since the heat loss is inversely proportional to the thermal resistance, the heat loss will decrease in an exponential manner. It should be noted, however, that an optimum insulation thickness does not exist, rather a critical insulation thickness. Beyond this critical insulation thickness, the amount of heat loss actually increases. Conversely, the heat loss below this critical point decreases as shown in Figure 7. In any pipe/insulation configuration, contact resistance plays a role in the insulating effect. Obviously, the contact resistance is an important factor and therefore, it is reasonable to investigate methods to alter the contact resistance. Suppose it is desired to decrease energy transfer between a hot fluid and the environment. In this case, the contact resistance should be increased. We investigated this scenario by separating the insulation from the containment pipe. A small air gap, 8 m m (0.31 in), was i n t r o d u c e d to the system by placing a : spacer between the pipe and fiberglass insulation. We lumped the air gap effect into the contact resistance term. The experimental results are shown in Table 4. F o r this case, the contact resistance decreased from 2.33 to 1.87K/W (1.23-0.99F/h-Btu). The resistance resulting from this modified contact decreased by 20%. The desired result is an increase in the contact resistance. However, it is noticed that the inclusion of an air gap in the pipe system dropped the overall thermal resistance and increased the heat loss. This was not totally unexpected because the air gap was quite large. This points to the importance of the correct choice of insulation for the system. By modifying the conditions present at the pipe/insulation contact, the amount of heat dissipated can be affected.
J
/ OPTIMIZATION/ECONOMIC ANALYSIS 0
0
I
I
1
I
I
I
I
1
2
3
4
5
6
7
~8
Insulation Thickness (ram) Figure 7
Total thermal resistance versus insulation thickness
An optimal design of a required piping system can be obtained through a parameter study involving optimization theory. The use of optimization theory in the form of a cost/design function, which could be the cost of the piping system, or a combination of~he material cost and
COMPOSITES
PART B Volume27BNumber
t 1996
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Heat loss in insulated pipe: M. A. Stubblefield et al. the operating cost (fuel), can be minimized within certain constraints. These constraints could be the pipe diameter, initial input fluid temperature, foam thickness, etc. An example of the optimization process is described here. The general mathematical approach is to minimize the cost function, fo- This function can be fo = wlC1 +
w2C2
(7)
where w I and w2 are weighting functions, Cl and C 2 represent the material and energy costs, respectively. With an appropriate selection of the weighting functions, Wl and w2, the cost function can be minimized. An optimization approach to determine the optimal insulation thickness based on an economic analysis of the system is presented. Our approach considers the developed experimental setup of an insulated steel pipe, whose inner wall of the steel pipe is maintained at the temperature of the saturated steam. The following information is considered to be given: - - Total temperature difference across the system is 125K. - - ' E f f e c t i v e ' convective coefficient for the contact resistance between the steel pipe and the insulation, hcon, is 2.67 W/mZ-K. - - Thermal conductivity of the fiberglass insulation, kins, is 0.052 W/m-K. Cost/volume of the fiberglass insulation is governed by the function: - -
Cost/Volume = 0.1883/Volume + 1064.5 where the cost is in dollars and the volume of the insulation is in m 3. - - The energy cost of the system per kiloWatt-hour is $0.10. This value is obtained from the charged cost of an electric utility. It is noted that hcon and kins are derived from the experimental procedures, and also the function relating the cost/volume of the fiberglass insulation is derived from manufacturer cost. The objective function to be optimized finally takes the form of Cost = (0' 188--~3 + 1064.5) V
( $0.10 ~ (Rst O.OOIAT x . . . . + \~.,/ + Rcon + Rins) "
H r
(8)
where V: AT: Hr:
Rst, Rcon, Rins:
92
volume of insulation, m3; temperature difference across the pipe and the insulation thickness; continuous number of hours in operation; the thermal resistance of the steel pipe, at the contact of the insulation and the steel pipe, and of the insulation, respectively, K/W,
COMPOSITES PART B Volume 27B Number 1 1996
and Cj -
0.1883 V F 1064.5
w1 = V
w2 =
C2 --
$0.10 KW-Hrs
(0.001zXT)
Rst q--Rcon q- Rins
Hr
(9)
It can be seen that w~ and w2 are functions of the V, AT, Rst , Rcon and Rins, which are geometry dependent. The given example is essentially unconstrained when considering the limitations from hardware facilities and available space. These constraints can be factored in by using a constrained optimization approach, such as Lagrangian multipliers, to determine an optimum configuration, if needed.
SUMMARY AND CONCLUSIONS This preliminary study has shown the following: - - A simplified engineering model for thermal contact resistance has been proposed. A simple test rig has been designed and constructed to aid in model development and validation. Validity studies indicate that the developed model and test rig can provide good results. When comparing with an insulated foam with known thermal conductivity, the difference is 8.7%. - - R e l a t i v e contact resistances for two widely used insulating materials have been determined. The contact resistance is a significant portion of the total resistance to energy flow. - - One way to affect the energy loss is to increase foam thickness up to the governing criteria known as the critical insulation thickness. - - Second approach is to use an air gap. The contact resistance can be significantly affected by modifying contact resistance with an air gap. - -
- -
In conclusion, the results demonstrate the potential of the presented methodology to provide an optimal configuration for a given piping system. The contact resistance can be modified to either increase or decrease contact resistance. These results suggest that standard optimization techniques might be used in these types of design problems. It is also shown that the air-gap method has a high potential to affect energy loss. In addition, while the focus of this study is on the contact resistance and energy loss, the proposed model and device can also be used to obtain the thermal conductivity of any insulating material.
ACKNOWLEDGEMENTS This research has been partially funded by the Louisiana Board of Regents under contract numbers LEQSF(199295)-RD-B-11 and L E R S F (1994-97)-RD-B-02. The first author would like to acknowledge the support from
Heat loss in insulated pipe: M. A. Stubblefield et al. the Louisiana Board of Regents' Dean fellowship. The assistance of Mr Roger Conway and Mr Daniel Ferrell in c o n s t r u c t i n g the e x p e r i m e n t a l s e t u p is g r e a t l y appreciated.
REFERENCES 1 2 3 4
Kang, T.K., Peterson, G.P. and Fletcher, L.S.J. Heat Transfer 1990, 112, 864 Yovanovich, M.M. in "Heat transfer', (Eds C.L. Tien, V.P. Carey and J.K. Ferrel), 1986, Vol. 1, pp. 35 45 Madhusudana, C.V. AIAA J. 1980, 18, 1261 Incropera, F.P. and D.P. DeWitt 'Introduction to Heat Transfer,' 2nd edn. John Wiley, New York, 1990
5 6 7 8 9 10 11 12 13 14
Fried, E. in 'Thermal Conductivity' (Ed. R.P. Tye), 1969, Vol. 2, pp. 253 274 Madhusudana, C.V. and Fletcher, L.S. AIAA J. 1986, 24, 510 Song, S., Yovanovich, M.M. and Goodman, F . O . J . Heat Transfer 1993, 115, 533 Fletcher, L.S.J. Heat Transfer 1988, 110, 1059 Antonetti, V.W. and Yovanovich, M.M.J. Heat Transfer 1985, 107, 513 Ochterbeck, J.M., Peterson, G.P. and Fletcher, L.S.J. Heat Transfer 1992, 114, 21 Maddren, J. and Marschall, E. J. Spacecraft Rockets 1995, 32(3), 469 Clausing, A.M. Int. J. Heat Mass Transfer 1966, 9, 791 Lewis, D,V. and Perkins, H.C. Int. J. Heat Mass Transfer 1968, 11, 1371 Thomas, T.R. and Probert, S.D. Int. J. Heat Mass Transfer 1970, 13, 789
COMPOSITES PART B Volume 27B Number 1 1996
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ELSEVIER
Composites: Part B 27B (1996) 85-93 © 1995 Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-8368/96/$15.00
H e a t loss in i n s u l a t e d pipe t h e i n f l u e n c e of t h e r m a l c o n t a c t resistance: a case s t u d y
Michael A. Stubblefield, Su-Seng Pang* and Vic A. Cundyt Department of Mechanical Engineering. Louisiana State University. Baton Rouge. LA 70803. USA ~Oepartment of Mechanical Engineering. Montana State University. Bozeman. MT 59715. USA (Received 12 January 1994. accepted 28 July 1995) The thermal contact resistance is an important parameter in many heat loss problems. Determining the contact resistance for practical systems is quite complex due to the dependency of the relative geometry of the contacting surfaces. It is, therefore, difficult to make general contact resistance data available in the literature. In this paper, we first describe a simple model to predict the effect of contact resistance. This is followed by describing a simple device which can be used to measure thermal contact resistance for an insulated pipe system. The apparatus consists of a steel containment pipe exposed to saturated steam. The heat flux is determined by measuring steam condensate over a fixedperiod of time, while temperature measurements are obtained using standard type K thermocouples. The apparatus is calibrated using insulating materials with known thermal conductivities as they are necessary for the calibration and validation of the experimental setup. Once the device has been calibrated, the thermal contact resistance is determined for the insulating materials (standard fiberglass and calcium-silicate) using the electrical analog resistance method. It is shown also that the energy loss in a system may be affected by manipulating the contact resistance between the pipe and insulation. The effect of a small air gap to influence contact resistance is investigated. By placing spacers between the pipe and insulation, effectively producing a small air gap, we were able to significantly alter the contact resistance. A generalized optimization approach is also presented. The defined parameters are considered as a function of insulation cost and the cost due to the energy loss of the system. (Keywords: heat loss; contact resistance; insulated pipe; steam; energy loss)
INTRODUCTION Heat transfer between a pumped fluid and the environment, in general, occurs by convection and conduction. Consider the insulated pipe system shown in Figure 1. Energy transfer in this system occurs via: (1) Convection from the pumped fluid t o the inner wall of the pipe, (2) Conduction from the inner to the outer wall of the pipe, (3) Conduction from the outer wall o f the pipe to the insulating medium (involving contact resistance), (4) Conduction thrOugh the insulating medium, and (5) Convection to the environment. The problem may be modeled simply (as in the case of steady heat transfer using the electrical resistance analog) or more detailed (by considering the unsteady case where the solution to the unsteady conduction equation gives, * TO whom correspondenceshould be addressed
by closed form or numerical method, the temperature history of the system). In either case, the energy loss or gain from the system can be readily calculated, analytically from initial boundary conditions, once the temperature distribution is determined. In any realistic model the contact resistance will have to be simulated. This paper starts with an overview of existing techniques currently used to handle contact resistance. A simple model of the contact resistance is developed and verified using experimental data. Finally, the contact resistance for several pipe configurations currently used in the field is determined experimentally.
B A C K G R O U N D OF A N A L Y T I C A L M O D E L S Contact resistance arises because the surfaces are in actual contact over only a small percentage of the apparent area. M a n y of the previous studies of contact resistance were concerned with relatively small contacting surfaces. Those surfaces are often characterized as a
COMPOSITES,PART B Volume 27B Number 1 1996
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Heat loss in insulated pipe: M. A. Stubblefield et al. Environment Energy T r a n s fe r
4> Ins u lation ~ ~ ~ N ~ k N ~ ~ x ~ k ~ Containment
Pipe
~///////'/////////////~//////////~5
ii
Figure3 Temperaturediscontinuitydue to contact resistance14
S t e a d y Flow of-Hot,,. P u m p e d - Fluid Figure 1 Insulted pipe arrangement uniform distribution of micro contacts. Subsequent studies understood that this approach was only appropriate if the macroscopic nonuniformity o f the contacting surfaces was negligible. These macroscopic contacts may be identified as a direct effect of the surface waviness and/or deformation of the surface due to applied loadings. When nominally flat surfaces are held in contact under loading, the heat transfer between them is impeded by a surface contact resistance at the interface causing a temperature drop across the contacting surfaces as seen in Figure 21 . The degree o f the thermal resistance at the interface is a function of the actual contact area, pressure between the two surfaces, thermal and physical properties of the mating surfaces, and interstitial fluids present between the contacting surfaces. Past experimental investigations have demonstrated that for relatively smooth, nominally flat surfaces, the actual contact area is between 2 and 5% of the apparent contact area 2. In the study o f the actual contact, the associated asperities may be modeled as conical in shape. Analytical results 3 showed that the contact or constriction resistance was reduced significantly in the presence of a conducting fluid (see Figure 34). Heat can be transferred across the contacting interface via the following means: (a) solid-to-solid conduction at contact points, (b) conduction through the fluid in the gaps around the contacts, (c) free convection within the gap, (d) radiation across the gap, or (e) any combination of these. In actual application, these modes are interdependent and difficult to predict.
A
B
'
Figure2 Heat flow through a joint8
86
COMPOSITES PART B Volume 27B Number 1 1996
The contribution of solid-to-solid conduction is strongly dependent on the number and size of asperities in actual contact region. Several correlations between solid-to-solid conduction and the parameters affecting it have been developed over the past 20 years. Developmerits 5 have led to a correlation relating solid-to-solid conduction to the ultimate strength of the softer contacting material and a constant dependent upon the average surface roughness heights, and the radius of the contact spot. Other factors considered 6 are the mean junction temperature, a gap dimension parameter accounting for the roughness as well as the flatness deviation, and the variation of the above parameters with contact pressure. The effects of gas rarefaction and surface roughness as it relates to the thermal gap conductance were studied 7. The gap conductance was defined as inversely proportional to the effective gap thickness, which typically lies in the range of 0.1-100#m. Since the thickness o f the interstitial gap is usually much smaller than the other contact dimensions, only fluid conduction normal to the contact plane is usually considered. It is noted that the effective gap thickness is of the same order of magnitude as the mean free path of the gas molecules in contact for normal engineering applications. Free convection and radial conduction factors are usually considered negligible within the fluid. Several approaches have been developed, however, to determine free convective coefficients utilizing the interstitial gap distance. The so-called gap parameter 5 uses the ratio of the volume of the interface fluid to the contact area. Free convective effects were also related to the maximum heights of roughness of the contacting surfaces, approach of surfaces under loading, maximum thickness of the interstitial fluid layer, and the temperature jump distance 6. A review of contact resistance studies g suggests that surface parameters are a determining factor in thermal contact resistance. Also, recent studies evaluating conforming surfaces, along with the developed models, predict the real contact area. Recent studies consider the geometrical aspects of the contacting materials as well as the surface preparation. Several studies have been done to enhance the thermal contact conductance, the reciprocal of the contact resistance, across an interface. One method commonly used includes the use of a thermal grease between the contacting surfaces. A thermomechanical model for nominally flat rough contacting surfaces coated with a metallic layer allows for a coated joint to be reduced to
Heat loss in insulated pipe." M. A. Stubblefield et al. an equivalent bare joint using an effective microhardness and an effective thermal conductivity 9. An experimental investigation was conducted to study the effect metallic coatings have on thermal contact conductance 1. The hardness of the coating materials seemed to be the prominent parameter in the magnitude of this effect. This contact conductance was also noticed to be greatest at low contact pressures. The effects of metallic coatings on contacting surfaces at cryogenic temperatures was also studied 1°. The developed model requires only a knowledge of the surface roughness, thermal conductivities and microhardness values. As the temperature increased, the microhardness increased by 35% for the copper in the experimental study, which validates the incorporation of microhardness parameter when discussing cryogenic temperatures, One characteristic of thermal contact resistance at cryogenic temperatures is that it is nearly inversely proportional to the bulk thermal conductivity of the contacting material. Later, it was shown that beryllium reacted inversely to this postulation ll. A possible reason may be because at the interface more static imperfections became evident as compared to those in the bulk material, thus causing the difference between the bulk and interfacial thermal conductivity measurements. Another important factor in consideration of the magnitude of contact resistance is the phenomenon known as directional effect. This directional effect is a trait of thermal contacts to present a greater thermal resistance in one direction as compared to the reversed direction. The effects of thermal strain as a source of an enhanced directional effect were considered 12. From experimentation, it was noticed that the contact resistance at the interface of dissimilar metals is strongly dependent on the heat flux. Changes due to differing properties of contacting surfaces may also affect directional effects ~3. These results were later related with surface interface conditions such as the flatness and roughness of the contacting surfaces. Experimental studies ~4 suggest that the thermal contact resistance between similar metals was lesser when the heat flow was from the higher thermal conductivity metal. However, it was also believed 12 that for the same material pairs, the thermal strain should not cause directional effects to manifest. Thermal radiation involves energy emission from matter occurring at a given temperature. Unlike the transfer of energy by conduction or convection, radiation does not require the presence of a material medium. The radiative resistance is a complex parameter depending upon the geometry of the gap, the emissivities of the contacting solids and the side walls forming the enclosure, as well as the temperature distribution of the bounding surfaces. In general, for low temperature applications, neglecting radiation does not introduce significant error into the problem. There are a number of techniques available for the prediction of contact resistance. However, each approach is appropriate for only selected applications.
Very few, in general, appear to be successful, even in consideration of the application for which the models were developed. Obviously, if one considers all modes of heat transfer, the analysis of thermal contact resistance is quite formidable. Indeed, this is the reason that most contact resistance problems are solved using empirical approaches. If one is to model the problem totally, an understanding of the multimode heat transfer phenomena is required and fundamental physical properties such as the thermal conductivities of the contacting solids and interstitial fluid, emissivity, hardness, gap thickness, etc., must be determined. Such a task is very difficult and usually many simplifying assumptions must eventually be made. A very sophisticated analytical model may be too problem specific to be of general use. On the other hand, a much simplified model requiring fewer input parameters might also fail to accurately predict some situations due to the inherent simplifications of the model. As a result, a compromise approach might be the desired alternative in many engineering applications. This is the approach we have chosen. In this approach, a relatively simple engineering model is proposed for specific pipe/insulation configurations. This model, macroscopic in nature, will include enough physics to accurately describe the contact resistance phenomena, while relying heavily on experimentation. It is hoped that such a model will provide a good tool for engineering optimization.
G O A L OF THIS W O R K The goal of this work is to develop a simple model which will accurately predict the heat transfer from an insulated pipe. A primary objective is to handle contact resistance in a relatively simple manner and verify the results experimentally. The model will be situation and geometry dependent. Another objective of the study is to develop a simple test rig to obtain the thermal physical properties of the pipe and insulating material. This apparatus will be used to measure the thermal contact resistance for insulated pipe systems. The final objective is to use the developed model to optimize the energy transfer for a given pipe/ insulation configuration. The optimization approach is based on an economic analysis of the piping system as a function of insulation cost and the subsequent cost due to the energy loss of the system.
ENGINEERING MODEL Contact resistance can be modeled using the electricalanalog approach with an apparent convective heat transfer coefficient, defined as R~o~ -
1
hconAa
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87
Heat loss in insulated pipe. M. A. Stubblefield et al. where: Rcon: the thermal contact resistance, hcon: the 'effective' convective coefficient for contact resistance, and Aa: the area based on the radius of the pipe/ insulation interface. In the following discussion, the assumptions listed below have been made: (1) the thermal conductivity and other physical properties are constant, (2) the contact is static, (3) the contribution from radiation in the contact resistance term is small and neglected, and (4) the heat flux is steady. Given these assumptions, the rate of heat transfer, Q, under steady conditions, is:
Q = UAAT
(2)
where U is an overall conductance based on the area A and A T is the temperature difference between the pumped-fluid and the ambient. The overall conductance, U, is obtained by determining the individual resistances associated with the chosen pipe/insulation configuration (see Figure 4). For the insulated pipe arrangement shown, these resistances are: R1: R2: R3: R4: Rs:
the convective resistance between the pumped-fluid and innermost pipe wall, the conductive resistance through the pipe, the contact resistance between the steel and insulating medium, the conductive resistance through the insulating medium, and the convective resistance between the insulation and ambient.
This resistive network, shown in Figure 4, is valid at any axial location along the pipe, where 1
RI, 5
-
-
hi,o(27rrl,31)
in r2 r] R2 = 27rkl~~
in r3 1
R3
r2
hcon(27rr21)
R4
-
-
and, hi: l: kl: hcon: k2: ho:
inner convective coefficient, length of pipe under consideration, thermal conductivity of containment pipe, the 'effective' convective coefficient for contact resistance thermal conductivity of insulation, and outer convective coefficient.
While this simple thermodynamic analysis appears to provide the energy loss from the pumped-fluid in a straightforward manner, the complexity of the contact resistance makes application of the analysis challenging. The challenge is to determine the 'effective' contact convective coeffÉcient, hcon. A specially designed test rig has been built to determine hcon. The rig is shown in Figure 5. In this work, we only consider steady-state values of contact resistance (nontime or temperature varying). The test rig is designed so that the temperature of the inner surface of the containment pipe is known very accurately. Other temperatures are also measured within the insulating material using conventional thermocouples. Heat loss through the pipe/insulation is also measured very accurately. Knowing the energy loss through the pipe and the temperature potential, the term hcon, is readily determined by considering the thermal network from the surface of the inner pipe to a location in the insulation where the temperature is measured (not necessarily the outer skin insulation temperature). The electrical analog for this network is: Q-
AT
(4)
R2 + R3 q- R 4
where A T is the temperature difference between resistances R 2 and R 4. The conductive resistances, R 2 and R4, are determined
Stainless Steel Containment__ Saturated Pipe \Steam Supply
Temperature Measurement -- Location with Known Separation
TAmbient
Tpumped- Pumped n.id Fluid
Containment Pipe
RI R~ R3 Tpumped-
TI
Fluid
Figure 4
88
R4
R5 Te
T.tmbient
Elcctrical analog network
COMPOSITES PART B Volume 27B Number 1 1996
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27rk21
_ Figure 5
Insulation
Thermal conductivity/contact resistance test rig
Heat loss in insulated pipe: M. A. Stubblefield et al. by knowledge of the geometry of the pipe/insulation configuration, and the thermal conductivities of the pipe and the insulation. Only the contact resistance term remains unknown and in this term, hcon, is readily solved for.
EXPERIMENTAL APPARATUS The test rig developed for this work is shown in Figure 5. It is used not only to measure the contact resistance, but also to measure the thermal conductivity of the pipe and insulating materials. Materials from the field are used in the rig since the pipe/insulation pair determines the contact resistance. Installation methods are also the same as those used in the field since contact resistance is sensitive to surface contact pressures. In this test rig, saturated steam condenses on the inner surface of the containment pipe. The heat transfer rate is constant and is determined by measuring steam condensate as energy is transferred from the steam through the pipe/insulation configuration. A calibrated sight glass is used to measure the condensate level. Temperatures are obtained using thermocouples. Since saturated steam is condensing at the inner surface of the containment pipe, the convective coefficient between the steam and pipe is assumed large, and the inner pipe temperature is taken as the saturation temperature of the steam. In consideration of the thin wall approximation for pipes with high thermal conductivity, this temperature can also be approximated as the outer wall temperature of the containment pipe. Since the rig makes use of saturated steam and the energy measurement is based on condensed steam, it is necessary to determine the water content of the steam for each experiment (the steam quality). This is accomplished using standard separating/throttling calorimetry techniques.
Thermal conductivity experiment The thermal conductivity of the containment pipe and insulating material must be determined before the contact resistance can be obtained. While such data are available for many commonly used materials, we chose to measure the conductivity for the materials of our tests for two reasons. First, such measurements validate the experimental methodology in as much as the conductivities for the materials of our tests were given by the manufacturer. By comparing with the published values, the test rig and the methodology are examined. Second, in the future, we anticipate using materials for which there is little or no thermal property information. The thermal conductivity of the pipe and insulating materials is readily determined using the experimental test rig. Referring to Figure 5, thermocouples are accurately placed at a fixed distance within the insulating material. Knowing the energy transfer and geometry, results in a straightforward calculation for thermal conductivity using the following relationship:
Q = AT/Rx
(5)
where Rx is the conductive resistance of the insulation over the distance shown in Figure 5 and A T is the associated temperature difference. The heat transfer, Q, is measured, A T is measured; therefore, the only unknown in equation (5) is the thermal conductivity.
'Effective' convective heat transfer coefficient for contact resistance Once the heat transfer and thermal conductivities of the system are determined, the 'effective' convection coefficient for contact resistance, hcon, is readily determined using equations (4) and (6). R2 --
in r2 rl 2~rkl l
R3
--
1
hcon (27rr2l)
R4 __
in r3 r2
27rk21
(6)
EXPERIMENTAL PROCEDURE The experimental procedure consists of the following steps: (1) obtain steam quality, (2) measure the thermal conductivity of the pipe and the insulation, and (3) measure the heat transfer rate and determine the contact resistance between the pipe and the insulating material, and consequently, the 'effective' convective coefficient for the contact resistance, hcon.
Steam quality In this work, both a separating and throttling calorimeter are used in series to determine the wetness of the steam. Typical steam qualities for the data shown are in the range from 79 to 90%.
EXPERIMENTAL RESULTS
Validation of the model and expertments The development of the model and experimental techniques is based on the fact that the total energy dissipated can be determined through the measurement of condensation of water. If the energy dissipation through the pipe section can be determined, the contact resistance can be calculated. The total energy dissipation consists of that through the test pipe section and the steam feedline to the pipe. The loss associated with the feedline must either be minimized or accounted for. In our approach, we chose to calibrate by accounting for the heat loss. In order to calibrate the energy loss through the line, an insulation foam with a given manufacture thermal conductivity was used. In this case, a calcium-silicate
COMPOSITES PART B Volume 27B Number 1 1996
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Heat loss in insulated pipe. M. A. Stubblefield et al. pipe insulation is used. It has a known thermal conductivity of 0.071W/m-K (0.041Btu/h-ft-F). The data obtained was for a piping system consisting of a containment pipe with 50.8 mm (2.0 in) nominal inside diameter, covered with 25.4mm (l in) thick calciumsilicate insulation. The pipe is 304 stainless steel schedule 40 with a wall thickness of 4 mm (0.16 in) and a length of 914ram (3.0ft). The thermal conductivity of the steel pipe is reported as 16.9W/m-K (9.8 Btu/h-ft-F). After 10 experimental trials, the total energy loss (including that through the test pipe and steam feedline), using the calcium-silicate insulation as a basis, was 115.0 -4- 16.0W (392.4 + 54.6Btu/h). Using the thermal conductivity provided by the manufacturer, the heat loss through the pipe is determined to be 45.1W (153.9 Btu/h). After the calibration procedure, the energy lost through the pipe system can be obtained by deducting the energy loss through the feedline from the total energy loss. In order to verify the model and experiments, another insulating foam, fiberglass, with a known thermal conductivity of 0.05-0.06W/m-K (0.0300.035 Btu/h-ft-F) was used in this part of the work. Two verifications were obtained. The first one was an insulated pipe system without an air gap, while the second case consisted of a small air gap between the containment pipe and the insulation. We did this because in later experiments, we will use an air gap to increase the effective insulation of the pipe system. The existence of the air gap should result in no change of the insulation thermal conductivity. Table 1 shows the results of the experimental tests for a steel pipe with fiberglass insulation, both with and without an air gap. Consider Table 1 with insulation only (no air gap). The thermal conductivity of the fiberglass insulation was calculated to be 0.052 i 0.006W/m-K (0,030 + 0.003 Btu/h-ft-F). When the air gap was introduced between the insulation and containment pipe, the thermal conductivity of the insulation was 0.052 :[: 0.001 W / m - K (0.030 5:0.0006 Btu/h-ft-F) (refer to Table 1). The errors between the measured value and the manufacturer reported value were 9.2 and 8.1%, respectively. Table I
Table 2
Experimental data for steel pipe with fiberglass insulation
Run #
X Quality
hcon W/m2-K
Rcon K/W
1 2 3 4 5 6 7 8
0.81 0.80 0.81 0.81 0.81 0.80 0.80 0.81
2.90 2.13 2.92 2.91 2.18 2.72 2.81 2.84
2.00 2.57 1.98 1.98 2.42 2.12 2.05 2.03
Mean ± standard dev.
0.81 ± 0.005
2.67 50.307
2.14 ± 0.211
The above results show that the average difference, with a magnitude of 0.057W/m-K, based on two different situations, was about 8.7%. As the manufacturer published value may also have some level of uncertainty and the exact value can also be considered very temperature dependent, the current model proposed, along with experimental setup, is deemed valid. While the focus of this study is on the contact resistance and the energy dissipation, it is obvious that once the energy loss through the line is calibrated, the model and device can be used to obtain the thermal conductivity of other insulating materials. Thermal contact resistance determination
Based on the experimental data, the value of the quality, x, calculated value of hcon and the thermal contact resistance, Rcon, for each experiment performed are shown in Tables 2 and 3. In the fiberglass configuration, the resistance due to contact was 2.14 K/W (1.13 F/h-Btu). The pipe resistance was 0.0007K/W (0.00037F/h-Btu) and the fiberglass insulation resistance was 1.85K/W (0.976F/.h-Btu). Therefore, the contact resistance is 54% of the total resistance. For the calcium-silicate configuration, the resistance due to the calcium insulation was 1.33 K/W (0.702 F/h-Btu) and the contact resistance was 1.20 K/W (0.633 F/h-Btu). In this case, the contact resistance is 47% of the total resistance.
Experimental data for steel pipe with fiberglass insulation Table 3
Experimental data for steel pipe with calcium-silicate insulation
% Error (no air gap)
kins W/m-K (with air gap)
% Error (with air gap)
Run #
x Quality
hcon W/mZ-K
Rcon K/W
Run #
kins W/m-K (no air gap)
1 2 3 4 5 6 7 8
0.057 0.042 0.057 0.057 0.043 0.053 0.055 0.056
0.0 26.3 10.0 0.0 24.6 7.0 3.5 1.8
0.051 0.053 0.053 0.053 0.053 0.051 0.054 0.051
10.5 7.0 7.0 7.0 7.0 10.5 5.3 10.5
1 2 3 4 5 6 7 8 9 10
0.86 0.86 0.87 0.85 0.89 0.86 0.86 0.87 0.86 0.86
3.0 9.2 6.6 4.4 7.0 2.9 6.4 6.6 4.4 4.4
1.94 0.63 0.88 1.32 0.82 1.97 0.90 0.87 1.31 1.31
Mean + standard dev.
0.052 ± 0.006
9.2 ± 10.6
0.052 ± 0.001
Mean • standard dev.
0.86 ± 0.010
5.50 ± 1.89
1.20 5_ 0.44
90
8.1 ± 1.94
COMPOSITES PART B Volume 27B Number 1 1996
Heat loss in insulated pipe, M. A. Stubblefield et al. Table 4 Experimental data for steel pipe/air gap/fiberglass insulation
70
0.12
Run #
x Quality
hcon W/m2-K
Rcon K/W
t-" o t~
1 2 3 4 5 6
0.83 0.84 0.84 0.85 0.84 0.84
3,07 3,20 3.20 3.84 3.20 3.06
1.88 1.80 1.80 1.50 1.80 1.89
.c
7 8
0.83 0.83
3.20 3.06
1.80 1.89
~"
M e a n :~ s t a n d a r d dev.
0.83 ± 0.020
3.23 ± 0.240
1.80 ± 0.119
I
0.10 / o
60
,d
co
/ 50
0.08
"u o
E g,
0.06
Thermal - 40 Conductivity - - - H e a t Loss
stY" ~"
0.04
I 0.6
0.5
I 0.7
J 0.8
I 0.9
30 1.0
Steam Quality, X Figure 6 Thermal conductivity/heat loss versus steam quality
These results show that the test rig provides reproducible data over a range of steam qualities. F o r example, by varying the steam quality of a specific case and keeping all other factors constant, it is noticed that across a quality range from 0.5 to 1.0, the heat dissipation only changes by 35.0W ( l l 9 . 4 B t u / h ) and the corresponding thermal conductivity changes by 0.05 W / m - K (0.03 Btu/h-ft-F), as seen in Figure 6. The operating range of the quality during the experimental trials ranged from 0.8 to 0.9. Therefore, it is shown that the overall difference is only 7.0W (23.9Btu/h) and 0.01 W / m - K (0.006 Btu/h-ft-F) for the heat loss and the thermal conductivity, respectively. This shows that the
25
20 v
o~ t~
15
- -- C a l c i u m - S i l i c a t e
/
data presented are only slightly adherent to steam quality. As seen in Figure 7, as the insulation thickness increases, the total thermal resistance increases also, following a logarithmic behavior. Since the heat loss is inversely proportional to the thermal resistance, the heat loss will decrease in an exponential manner. It should be noted, however, that an optimum insulation thickness does not exist, rather a critical insulation thickness. Beyond this critical insulation thickness, the amount of heat loss actually increases. Conversely, the heat loss below this critical point decreases as shown in Figure 7. In any pipe/insulation configuration, contact resistance plays a role in the insulating effect. Obviously, the contact resistance is an important factor and therefore, it is reasonable to investigate methods to alter the contact resistance. Suppose it is desired to decrease energy transfer between a hot fluid and the environment. In this case, the contact resistance should be increased. We investigated this scenario by separating the insulation from the containment pipe. A small air gap, 8 m m (0.31 in), was i n t r o d u c e d to the system by placing a : spacer between the pipe and fiberglass insulation. We lumped the air gap effect into the contact resistance term. The experimental results are shown in Table 4. F o r this case, the contact resistance decreased from 2.33 to 1.87K/W (1.23-0.99F/h-Btu). The resistance resulting from this modified contact decreased by 20%. The desired result is an increase in the contact resistance. However, it is noticed that the inclusion of an air gap in the pipe system dropped the overall thermal resistance and increased the heat loss. This was not totally unexpected because the air gap was quite large. This points to the importance of the correct choice of insulation for the system. By modifying the conditions present at the pipe/insulation contact, the amount of heat dissipated can be affected.
J
/ OPTIMIZATION/ECONOMIC ANALYSIS 0
0
I
I
1
I
I
I
I
1
2
3
4
5
6
7
~8
Insulation Thickness (ram) Figure 7
Total thermal resistance versus insulation thickness
An optimal design of a required piping system can be obtained through a parameter study involving optimization theory. The use of optimization theory in the form of a cost/design function, which could be the cost of the piping system, or a combination of~he material cost and
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Heat loss in insulated pipe: M. A. Stubblefield et al. the operating cost (fuel), can be minimized within certain constraints. These constraints could be the pipe diameter, initial input fluid temperature, foam thickness, etc. An example of the optimization process is described here. The general mathematical approach is to minimize the cost function, fo- This function can be fo = wlC1 +
w2C2
(7)
where w I and w2 are weighting functions, Cl and C 2 represent the material and energy costs, respectively. With an appropriate selection of the weighting functions, Wl and w2, the cost function can be minimized. An optimization approach to determine the optimal insulation thickness based on an economic analysis of the system is presented. Our approach considers the developed experimental setup of an insulated steel pipe, whose inner wall of the steel pipe is maintained at the temperature of the saturated steam. The following information is considered to be given: - - Total temperature difference across the system is 125K. - - ' E f f e c t i v e ' convective coefficient for the contact resistance between the steel pipe and the insulation, hcon, is 2.67 W/mZ-K. - - Thermal conductivity of the fiberglass insulation, kins, is 0.052 W/m-K. Cost/volume of the fiberglass insulation is governed by the function: - -
Cost/Volume = 0.1883/Volume + 1064.5 where the cost is in dollars and the volume of the insulation is in m 3. - - The energy cost of the system per kiloWatt-hour is $0.10. This value is obtained from the charged cost of an electric utility. It is noted that hcon and kins are derived from the experimental procedures, and also the function relating the cost/volume of the fiberglass insulation is derived from manufacturer cost. The objective function to be optimized finally takes the form of Cost = (0' 188--~3 + 1064.5) V
( $0.10 ~ (Rst O.OOIAT x . . . . + \~.,/ + Rcon + Rins) "
H r
(8)
where V: AT: Hr:
Rst, Rcon, Rins:
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volume of insulation, m3; temperature difference across the pipe and the insulation thickness; continuous number of hours in operation; the thermal resistance of the steel pipe, at the contact of the insulation and the steel pipe, and of the insulation, respectively, K/W,
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and Cj -
0.1883 V F 1064.5
w1 = V
w2 =
C2 --
$0.10 KW-Hrs
(0.001zXT)
Rst q--Rcon q- Rins
Hr
(9)
It can be seen that w~ and w2 are functions of the V, AT, Rst , Rcon and Rins, which are geometry dependent. The given example is essentially unconstrained when considering the limitations from hardware facilities and available space. These constraints can be factored in by using a constrained optimization approach, such as Lagrangian multipliers, to determine an optimum configuration, if needed.
SUMMARY AND CONCLUSIONS This preliminary study has shown the following: - - A simplified engineering model for thermal contact resistance has been proposed. A simple test rig has been designed and constructed to aid in model development and validation. Validity studies indicate that the developed model and test rig can provide good results. When comparing with an insulated foam with known thermal conductivity, the difference is 8.7%. - - R e l a t i v e contact resistances for two widely used insulating materials have been determined. The contact resistance is a significant portion of the total resistance to energy flow. - - One way to affect the energy loss is to increase foam thickness up to the governing criteria known as the critical insulation thickness. - - Second approach is to use an air gap. The contact resistance can be significantly affected by modifying contact resistance with an air gap. - -
- -
In conclusion, the results demonstrate the potential of the presented methodology to provide an optimal configuration for a given piping system. The contact resistance can be modified to either increase or decrease contact resistance. These results suggest that standard optimization techniques might be used in these types of design problems. It is also shown that the air-gap method has a high potential to affect energy loss. In addition, while the focus of this study is on the contact resistance and energy loss, the proposed model and device can also be used to obtain the thermal conductivity of any insulating material.
ACKNOWLEDGEMENTS This research has been partially funded by the Louisiana Board of Regents under contract numbers LEQSF(199295)-RD-B-11 and L E R S F (1994-97)-RD-B-02. The first author would like to acknowledge the support from
Heat loss in insulated pipe: M. A. Stubblefield et al. the Louisiana Board of Regents' Dean fellowship. The assistance of Mr Roger Conway and Mr Daniel Ferrell in c o n s t r u c t i n g the e x p e r i m e n t a l s e t u p is g r e a t l y appreciated.
REFERENCES 1 2 3 4
Kang, T.K., Peterson, G.P. and Fletcher, L.S.J. Heat Transfer 1990, 112, 864 Yovanovich, M.M. in "Heat transfer', (Eds C.L. Tien, V.P. Carey and J.K. Ferrel), 1986, Vol. 1, pp. 35 45 Madhusudana, C.V. AIAA J. 1980, 18, 1261 Incropera, F.P. and D.P. DeWitt 'Introduction to Heat Transfer,' 2nd edn. John Wiley, New York, 1990
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Fried, E. in 'Thermal Conductivity' (Ed. R.P. Tye), 1969, Vol. 2, pp. 253 274 Madhusudana, C.V. and Fletcher, L.S. AIAA J. 1986, 24, 510 Song, S., Yovanovich, M.M. and Goodman, F . O . J . Heat Transfer 1993, 115, 533 Fletcher, L.S.J. Heat Transfer 1988, 110, 1059 Antonetti, V.W. and Yovanovich, M.M.J. Heat Transfer 1985, 107, 513 Ochterbeck, J.M., Peterson, G.P. and Fletcher, L.S.J. Heat Transfer 1992, 114, 21 Maddren, J. and Marschall, E. J. Spacecraft Rockets 1995, 32(3), 469 Clausing, A.M. Int. J. Heat Mass Transfer 1966, 9, 791 Lewis, D,V. and Perkins, H.C. Int. J. Heat Mass Transfer 1968, 11, 1371 Thomas, T.R. and Probert, S.D. Int. J. Heat Mass Transfer 1970, 13, 789
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