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The inaccuracy of heat transfer characteristics for non-insulated and insulated spherical containers neglecting the influence of heat radiation
Energy Conversion and Management 52 (2011) 1612–1621
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
The inaccuracy of heat transfer characteristics for non-insulated and insulated spherical containers neglecting the influence of heat radiation King-Leung Wong a,⇑, José Luis León Salazar a, Leo Prasad b, Wen-Lih Chen a a b
Department of Mechanical Engineering, Kun-Shan University of Technology, 949, Da-Wan Road, Yung-Kang City, Tainan County 710, Taiwan, ROC Faculty of Built Environment, University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 27 July 2009 Received in revised form 31 May 2010 Accepted 6 October 2010 Available online 13 November 2010 Keywords: Insulation Non-insulation Sphere Heat radiation Emissivity Inaccuracy
a b s t r a c t In this investigation, the differences of heat transfer characteristics for insulated and non-insulated spherical containers between considering and neglecting the influence of heat radiation are studied by the simulations in some practical situations. It is found that the heat radiation effect cannot be ignored in conditions of low ambient convection heat coefficients (such ambient air) and high surface emissivities, especially for the non-insulated and thin insulated cases. In most practical situations when ambient temperature is different from surroundings temperature and the emissivity of insulation surface is different from that of metal wall surface, neglecting heat radiation will result in inaccurate insulation effect and heat transfer errors even with very thick insulation. However, the insulation effect considering heat radiation will only increase a very small amount after some dimensionless insulated thickness (such insulation thickness/radius =0.2 in this study), thus such dimensionless insulated thickness can be used as the optimum thickness in practical applications. Meanwhile, wrapping a material with low surface emissivity (such as aluminum foil) around the oxidized metal wall or insulation layer (always with high surface emissivity) can achieve very good insulated effect for the non-insulated or thin insulated containers. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Non-insulated and insulated cold/hot containers and ducts are widely used in the various industries and domestic devices. Because heat radiation equation contains the 4th order exponent of temperature, it is very difficult to obtain precise theoretical or numerical heat radiation solutions before the time when modern computing tools were not available. In the past, many heat transfer experts and scholars avoid this problem and argue, from their own experiences, that the heat radiation effect can be ignored when the temperature difference between non-insulated or insulated cold/hot containers and surrounding is small. As a result, heat radiation has been commonly neglected in many practical heat transfer applications even in cases involving low heat convection coefficients. For the non-insulated situations, log mean temperature difference (LMTD) method, which neglects the influence of heat radiation, is introduced in most heat transfer [such as 1,2], air conditioning, and refrigeration text books [such 3,4], and is
⇑ Corresponding author. Tel.: +886 62057121; fax: +886 62050509. E-mail address: [email protected] (K.-L. Wong). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.10.016
conventionally used to calculate the total heat transfer rate of heat exchangers. Recently, Hsien et al. [5] studied the complete heat transfer characteristics of non-insulated circular ducts considering heat radiation effect. They reported that in some practical situations, the heat radiation effect cannot be ignored when a noninsulated duct has high surface emissivities and the ambient air’s convection heat coefficient is low. Even in situations when the temperature difference between the fluid inside a non-insulated circular duct and the ambient air is very low, the errors generated by neglecting heat radiation are still very large and hence cannot be ignored. Unfortunately, it can be seen from Table 1 that most surface emissivities of oxidized metal are greater than 0.64; and Table 2 shows that the heat convection coefficients of ambient air with medium wind speed are with quite small. Consequently, the errors associated with the heat transfer characteristics obtained with conventional LMTD method applying to condensers and evaporators with heat exchanging in ambient air can be very large. Thus, LMTD method should not be applied to evaporators and condensers. To solve this problem, Wong et al. [6] developed a log mean heat transfer rate (LMHTR) method which considers the influence of heat radiation to calculate the exact heat transfer rate of heat exchangers. Their study showed that the results obtained by LMHTR method are the same as those obtained from
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Nomenclature
e e0 A1 A2 A3 EF EFa EFn hi ho hr HR KA Ks q q0 qa qa0 qc qc0
emissivity of insulation surface emissivity of container surface inner surface area of a spherical container external surface area of a spherical container external surface area of a insulated spherical container insulation effect insulation effect of an insulated spherical container considering heat radiation insulation effect of an insulated spherical container neglecting heat radiation inner heat convection coefficient external heat convection coefficient external radiation heat convection coefficient convective heat coefficient ratio conductivity of duct material conductivity of insulated material total heat transfer rate of insulated container neglecting heat radiation total heat transfer rate of non-insulated container neglecting heat radiation total heat transfer rate of insulated container considering heat radiation total heat transfer rate of non-insulated container considering heat radiation convection heat transfer rate of insulated container considering heat radiation convection heat transfer rate of non-insulated container considering heat radiation
LMTD method while surface emissivities equal to zero; and they indicated that it is risky to neglect the influence of heat radiation Table 1 The emissivities e of various substances from the manual of infrared temperature demonstrator [24]. Human skin Gold Silver Aluminum Copper Iron Stainless steel Steel Nickel Brick Carbon Concrete Glass Paint oil Paper, white Paper Plaster Rubber, black Wood, oak White ceramic Black painting Oil, lubricant
Soil Water
0.98 0.02 0.02 Weathered = 0.83; Foil (bright)=0.04 Disk, rough = 0.96 Polished = 0.05 Oxidized = 0.78 Cast(ox) = 0.64 Sheet, rusted = 0.69 Polished = 0.16 Oxidized = 0.85 Polished = 0.07 Oxidized = 0.79 Electro pole = 0.05 0.81 0.95 0.95 0.84–0.97 0.94 0.70 0.89 0.86 0.95 0.90 0.91 0.96 Film 0.03 mm = 0.27 Film 0.13 mm = 0.72 Thick = 0.82 Dry = 0.92 Saturated water = 0.95 Distilled = 0.96 Frost = 0.98 Snow = 0.85
qr qr0 QR r1 r2 r3 SR TD t1 t t/r2 T2 T2a T3 T3a Ti To Tsur
radiation heat transfer rate of insulated container considering heat radiation radiation heat transfer rate of non-insulated container considering heat radiation error of heat transfer rate generated by neglecting heat radiation inner radius of spherical container external radius of spherical container external radius of insulated spherical container error of surface temperature generated by neglecting heat radiation the surface temperature difference generated by neglecting heat radiation thickness of spherical container wall thickness of insulated layer dimensionless insulated thickness bare wall surface temperature generated by neglecting heat radiation bare wall surface temperature generated by considering heat radiation insulated surface temperature generated by neglecting heat radiation insulated surface temperature generated by considering heat radiation temperature of the fluid inside the duct temperature of the fluid outside the spherical container temperature of the outside surrounding
especially in the situations involving low ambient air convection coefficients and heat exchanger (such as condensers and evaporators) surfaces with high surface emissivity. Thus, in order to obtain more accurate results, in stead of the conventional LMTD method, the LMHTR method should be employed. However, investigations related to non-insulated heat transfer to ambient air neglected the heat radiation are still quite common. For example, Elsayed et al. [7] studied free convection of air around a constant
Table 2 Referred approximate values of convection heat transfer [13]. Approximate values of convection heat transfer, h (W m2 K1) Mode h (W m2 K1) Free convection Temp. diff. = 30 °C horizontal plate 0.3 in high in air Temp. diff. = 30 °C vertical plate 0.3 in high in air Horizontal cylinder, 2 cm diameter, in water Heat transfer across 1.5 cm vertical air gap with temp. diff. = 60 °C Forced convection Air flow at 2 m/s over 0.2-m square plate Air flow at 35 m/s over 0.75-m square plate Air at 2 atm flowing in 2.5 cm diameter tube at 10 m/s (=36 km/h) Water at 0.5 kg/s flow in 2.5 cm diameter tube Air flow across 5 cm diameter cylinder with velocity of 50 m/s(=180 km/h) Boiling water In a pool or container Flowing in a tube Condensation of water vapor, 1 atm Vertical surfaces Outside horizontal tubes
4.5 6.5 890 2.64
12 75 65 3500 180
2500–35,000 5000– 100,000 4000–11,300 9500–25,000
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heat flux elliptic tube without considering heat radiation; Lazaro et al. [8] investigated the PCM-air heat exchangers for free cooling applications in buildings neglecting the heat radiation effect; Karadag and Teke [9] developed a new approach relevant to floor Nusselt number in floor heating system without considering heat radiation; Linhui et al. [10] reported an experimental study on natural convective heat transfer from a vertical plate with discrete heat sources mounted on the back neglecting heat radiation; Chen et al. [11] solved the inverse problem in determining convection heat transfer coefficient of an annular fin without considering heat radiation; Chang et al. [12] investigated the heat pipe for cooling of electronic equipment neglecting heat radiation; and the list goes on. For the insulated situations, it is commonly shown in many heat transfer text books [such as 13–15] that heat radiation is neglected in insulated duct/container even in situations with ambient air at low heat convection coefficients. Meanwhile, the critical heat transfer does occur with ambient air at a very small heat convection coefficient. From Table 1, it can be seen that the surface emissivities of insulated material are greater than 0.8. Thus, the heat radiation should be considered in the analysis of critical heat transfer. But the critical heat transfer characteristics introduced in most heat transfer text books [such as 13–15] neglect the heat radiation. In addition, many investigations associated with insulation in open literature also neglected the influence of heat radiation, for example: Chen and Yang [16] used an iterative regularization method in estimating the transient heat transfer rate on the surface of the insulation layer of a double circular pipe; Ahmed et al. [17] investigated the heat transfer characteristics across the insulated walls of refrigerated truck trailers by the application of phase change materials; Karadag and Teke [18] investigated the floor Nusselt number in floor heating system for insulated ceiling conditions; Atayilmaz and Teke [19] obtained the experimental and numerical study of the natural convection from a heated horizontal cylinder wrapped with a layer of textile material; Chen et al. [20] found out the reliable one-dimensional approximate solution of insulated oval duct, Lee et al. [21] investigated the complete heat transfer solutions of an insulated regular polyhedron by using a RPSWT model, Hsien et al. [22] found out the reliable one-dimensional approximate solutions for insulated oblate spheroid containers, etc. Recently, Hsien et al. [5] conducted a comparative study on the heat transfer characteristics of an insulated circular duct considering and neglecting the influence of heat radiation. They found that the heat radiation effect cannot be ignored in conditions of thin insulation with low ambient convective heat coefficient and high surface emissivity. In order to highlight the importance of heat radiation for non-insulated and insulated ducts/containers in situation where heat convection coefficients of ambient air is small, this paper, a follow-up study to Hsien et al. [5], examines the differences in heat transfer characteristics of a non-insulated and insulated spherical container resulted by considering and neglecting the influence of heat radiation. Then the inaccuracy of heat transfer characteristics for a non-insulated and insulated spherical container neglecting the influence of heat radiation can be demonstrated.
2.1. Cases with the influence of the heat radiation being neglected Here, we assume that heat transfer is steady state without heat generation inside solid materials. All thermal properties are uniform and independent of temperature, and hi and ho are constants. If the influence of external surface heat radiation is not considered, it can be seen from Fig. 1(a) that the total thermal resistance can be derived from heat transfer text book [such as 15] is:
X
Rth ¼
1 r1
r12
1 r2
r13
1 1 þ þ þ 4pK A 4p K S hi 4pr 21 h0 4pr 23
ð1Þ
and the relative total heat transfer rate neglecting heat radiation is:
Ti T0 T3 T0 q¼ P ¼ 1 Rth h 4pr 2 0
ð2Þ
3
Eq. (2) implies that heat transfer rate q and insulated surface temperature T3 of situations without considering the influence of external surface heat radiation can be obtained. From Fig. 2(a), the heat transfer rate of non-insulated spherical container without considering the influence of outside heat radiation can be written as:
2. Problem formulation Insulated and non-insulated spherical containers are shown in Figs. 1 and 2, respectively. Both spherical containers are exposed to a surrounding temperature of Tsur, and have a wall thickness of t1, conductivity KA, bare wall surface emissivity e0. The insulation layer has thickness t, conductivity KS and surface emissivity e. The internal and external fluids have convection heat transfer coefficients hi and ho, temperatures Ti and To, respectively.
Fig. 1. An insulated spherical container and relative parameters: (a) neglecting heat radiation and (b) considering heat radiation.
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where T3a is the actual surface temperature considering heat radiation. Then from Fig. 1(b), the surface convective heat transfer becomes:
qc ¼ h0 4pr 23 ðT 3a T 0 Þ
ð6Þ
and the surface radiation heat transfer rate is:
qr ¼ re4pr 23 T 43a T 4sur
ð7Þ
Comparing Eqs. (2) and (6), under the conditions of e – 0, T3a – T3, qc – q, the energy conservation among qa, qc and qr can be written as:
qa ¼ qc þ qr
ð8Þ
Therefore, the qa, qc and qr and T3a can be readily deduced from Eqs. (5)–(8). With both q and qa being solved, the error of heat transfer rate generated by neglecting heat radiation effect can be defined as:
q 100% QR ¼ 1 qa
ð9Þ
The equivalent heat convection coefficient of heat radiation [15] is defined from Eq. (7) as:
hr ¼ re T 23a þ T 2sur ðT 3a þ T sur Þ
ð10Þ
Here, hr is normally used to compare with the heat convection coefficient ho which in turn shows how significant the effect of radiation is. The ratio between equivalent heat convection coefficient of heat radiation and original heat convection coefficient is defined as:
HR ¼
hr hr 4pr23 ðT 3a T sur Þ q 100% ¼ r 100% 100% h0 qc h0 4pr23 ðT 3a T 0 Þ
ð11Þ
where Tsur is always close to To. Similar to Eq. (9), the error of surface temperature generated by neglecting heat radiation effect, based on T3 and T3a using Celsius temperature scale, is defined as:
SR ¼
Fig. 2. A non-insulated spherical container and relative parameters: (a) neglecting heat radiation and (b) considering heat radiation.
Ti T0
q0 ¼ 1 hi 4pr 21
þ
1 1 r 1 þr 2
4pK A
ð3Þ
Eqs. (2) and (3) lead to an insulated effect EFn of an insulated spherical container comparing with non-insulated situation neglecting heat radiation being:
q 100% EF n ¼ 1 q0
ð4Þ
From Fig. 1, when the influence of external surface heat radiation is considered, the complete heat transfer rate from the surface of an insulated spherical container can be expressed as:
1 hi 4pr21
þ
1 1 r 1 r 2
4pK A
þ
1 1 r2 r 3
4pK S
T3 100% T 3a
ð5Þ
ð12Þ
In the case of the insulated cold spherical container, condensed water may be formed if the insulated surface temperature is less than the dew point of ambient air. Therefore, predicting the insulated surface temperature is very important. Dimensionless SR tends to reduce the magnitude of error; hence SR is not suitable to demonstrate the heat characteristic of cold spherical container. Instead, the surface temperature difference is used and is written as:
ð13Þ
While the influence of external surface heat radiation is considered, the complete heat transfer rate from the surface of a noninsulated spherical container shown in Fig. 2(b) can be written as:
qa0 ¼
2.2. Case with the influence of heat radiation being considered
qa ¼
1
TD ¼ T 3a T 3
þ h 41pr2 0 2
T T i 3a
T i T 2a 1 hi 4pr 21
þ
ð14Þ
1 1 r 1 r 2
4pK A
where T2a is the actual surface temperature of a non-insulated spherical container shown in Fig. 2(b) with the influence of heat radiation taken into account. Additionally, its surface convective heat transfer rate is:
qc0 ¼ ho 4pr 23 ðT 2a T o Þ
ð15Þ
and the surface radiation heat transfer rate is:
qr0 ¼ re4pr23 T 42a T 4sur
ð16Þ
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Table 3 Referred approximate values of thermal conductivities [13].
The heat balance can be expressed as:
qa0 ¼ qc0 þ qr0
Thermal conductivity of various materials at 20 °C Metals Material Copper (pure) Aluminum (pure) Carbon steel, l% C Carbon steel (18%Cr, 8%Ni) Cast iron
K (W m1 K1) 386 204 73–77 43 16
Nonmetallic solids Glass, window Plaster, gypsum Metal lath Woof lath Teflon Asphalt Wood fiber sheet Wool Glass fiber Building brick common Building brick face Concrete, cinder Stone, 1–2–4 mix
0.78 0.48 0.4 0.28 0.35 0.7 0.047 0.038 0.035 0.69 1.32 0.76 1.37
Graphite, pyrolytic Perpendicular to layers Polyethylene Polypropylene Polyvinylchloride Rubber, hard
5.6 0.33 0.16 0.09 0.1
ð17Þ
Similarly, qco, qro, qao and T2a of non-insulated spherical container considering heat radiation can be readily obtained from Eqs. (14)– (17). And finally, the insulated effect EFa of an insulated spherical container comparing with non-insulated situation when heat radiation is considered can be obtained by:
q EF a ¼ 1 a 100% qa0
ð18Þ
3. Calculation of heat transfer results The one-dimensional heat transfer solutions of the insulated and non-insulated spherical containers are exact. These solutions are calculated by one dimensional LabVIEW [23] programming, which provide an easy and friendly interface to input all needed parameters and also offer live graph, table data, and automatic connection with excel. With this handy tool, it is possible to obtain accurate values with one-dimensional spherical heat equation. The major parameter in this program is the insulation thickness, and according to the input insulation thickness, Eqs. (5) and (8) and equations from (14) to (17) are solver iteratively until the difference of heat transfer rates obtained between two consecutive iterations is less than 104. The iteration process converges quickly
ºC
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ºC
Fig. 3. The relations between QR and t/r2 in the situation of Ti = 100 °C, KA = 77W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
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ºC
Fig. 4. The relations between HR and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
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and only takes a few seconds to complete, therefore, with each change of parameters, accurate solutions can be returned in a very short period of time. According to the emissivities shown in Table 1, e = 0.8 and 0.9 and e = 0.1 and 0.2 are adopted to represent the cases of high and low surface emissivity, respectively. Also Table 2 shows that the natural convection coefficients of air are below 10 W m2 K1; and even in the cases with a very high air speed, the forced convection coefficients of air are less than 100 W m2 K1. Therefore, hi = 30 W m2 K1 can be used to represent the situation of low/medium convection coefficients inside the spherical container, and ho = 8.3 and 10 W m2 K1 are chosen to represent the natural or low forced convection coefficients of ambient air. Meanwhile, the natural convection coefficients of water are more than 890 W m2 K1, and the values of convection coefficients are between 2500 and 35,000 W m2 K1 for boiling water in a pool or container. Nevertheless, in insulated cases, convection coefficients may become smaller, thus hi = 800 W m2 K1 is chosen to represent the cases of high convection coefficients of hot or cold liquid (may be other than water) inside the insulated spherical container. In practical applications, carbon steel is the common material applied to construct containers. According to the Table 3, KA = 77 W m1 K1 is used as the carbon steel spherical container’s conductivity; and the insulated material conductivity with the value KS = 0.035 W m1 K1shown in Table 3 is adopted in this study. In order to check if the computer results are reliable, the following measures are adopted:
(a) Let surface emissivity e = 0, i.e., heat radiation does not exist and HR = hr/ho = 0; check if the resulted QR being close to zero. (b) Let surface emissivity e = 1 and external convection coefficient ho = 50,000 W/m2 K, i.e., heat radiation effect becomes very small by comparing the very big heat convection. And HR = hr/ho becomes very small; check if the resulted and QR being close to zero. 4. Results and discussions For the results to be generalized, they are shown in dimensionless parameters except for TD, the surface temperature difference between considering and neglecting heat radiation effect for insulated cold spherical container. This is because that the value of difference in dimensionless temperatures is very small, and it might result in a misunderstanding of physical meaning among readers. The first investigation is on the case of insulated hot spherical containers with the following thermal and geometrical parameters: Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 at different ambient-air conditions: (a) To = Tsur = 30 °C and e = e0 and (b) To = 30 °C, Tsur = 32 °C, e = e0 or e – e0, respectively. Figs. 3–6 shows the heat transfer rate error (QR), convective coefficients ratio (HR), error of surface temperature (SR), and insulated effect (EF) are affected by dimensionless insulated thickness (t/r2), internal convection coefficient (hi), insulated surface emissivity (e), and metal wall surface emissivity (e0).
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Fig. 5. The relations between SR and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
ºC
Fig. 6. The relations between EF and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1: (a) To = Tsur = 30 °C with e = e0 and (b) To = 30 °C, Tsur = 32 °C with e – e0.
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Fig. 7. The relations between QR and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
Figs. 3, 4 and 6 show that QR and HR decrease, and EF increases as t/r2 increases. The higher the e and hi are, the larger the QR, HR and EF will be. The plots in Fig. 3(a) indicate that in the condition of To = Tsur = 30 °C and e = e0, QR gets close to 0 as t/r2 approaches to 1. This phenomenon, however, does not happen at the condition of To = 30 °C, Tsur = 32 °C, and e = e0 as shown by the plots given in Fig. 3(b). Here QR gets close to 3% as t/r2 approaches to 1; thus in the condition of To – Tsur, neglecting the heat radiation produces error in QR even with very thick insulation. Fig. 4 suggests that HR value becomes larger at t/r2 = 0. It is noticeable from Fig. 4(a) that even at t/r2 = 1, HR is about 50% for e = 0.8 and HR ; 12% for e = 0.2 under the condition of To = Tsur = 30 °C. Fig. 4(b) also shows that even at t/r2 = 1, HR reaches 58% for e = 0.8 and HR is 7.5% for e = 0.2 in the condition of To = 30 °C and Tsur = 32 °C. HR represents the ratio between heat radiation and heat convection as shown in Eq. (11), thus heat radiation effect cannot be neglected if HR > 5%. Fig. 5 shows that SR exists, and its absolute value reaches maximum around t/r2 = 0.025; the absolute value of SR is generally proportional to e. If the value of e is fixed, smaller hi tends to result in larger absolute value of SR before t/r2 ; 0.025 then the trend reverses after that value. Fig. 6(a) shows that for same values of hi and e = e0, EFa values at the condition of e = e0 – 0 (considering heat radiation) are greater than that at e = e0 = 0 (neglecting heat radiation), meaning neglecting heat radiation will underestimate insulation effect. The slope of EFa (considering heat radiation) becomes much smaller after t/r2 = 0.2, and the value of EFa is within 3% of between t/r2 = 0.2 and t/r2 = 1; thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in practical applications. The most meaningful practical characteristics of non-insulated and insulated spherical container are shown in Fig. 6(b). Fig. 6(b)
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Fig. 8. The relations between HR and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
shows that at t/r2 = 0, the value of EFa (;36%) at e = 0.1, e0 = 0.79, and hi = 800 W m2 K1 is much greater than that (EFa ; 3%) at e = 0.9, and with the same values of e0 and hi; meanwhile EFa (;24%) at e = 0.1, e0 = 0.79, and hi = 30 W m2 K1 is much greater than that (EFa ; 3%) at e = 0.9, and with the same values of e0 and hi. Hence, it can be concluded that for non-insulated and thin insulated cases, wrapping an aluminum foil (e = 0.04 as shown in Table 1) around the oxidized metal wall or insulation layer (e at high values as shown in Table 1) can result in excellent insulated effect. This practice works if t/r2 5 0.15. The slope of EFa becomes much smaller after t/r2 = 0.2, and the value of EFa is within 3% between t/r2 = 0.2 and t/r2 = 1. Therefore, t/r2 = 0.2 can be treated as the optimum dimensionless insulation thickness in practical applications. When t/r2 = 0.0025 and for same values of hi, EFa value at e – e0 > 0 (considering heat radiation) is greater than that at e = e0 = 0 (neglecting heat radiation). Thus neglecting heat radiation will result in inaccurate insulation effect in most situations. The second simulation is to examine insulated cold spherical containers with the following thermal and geometrical parameters: Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 at different ambient-air conditions: (a) To = Tsur = 30 °C with e = e0 and (b) To = 30 °C, Tsur = 32 °C with e = e0 and e – e0, respectively. Figs. 7–10 give the heat transfer rate error (QR), convective coefficients ratio (HR), the difference of surface temperature (TD), and insulated effect (EF) are affected by the dimensionless insulated thickness (t/r2), internal convection coefficient (hi), insulated surface emissivity (e), and metal wall surface emissivity (e0). The characteristics observed in Figs. 7–10 are very similar to those in Figs. 3–6. It can be seen in Figs. 7, 8 and 10 that QR decreases, but EF and HR increase as t/r2 increases. The
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Fig. 9. The relations between TD and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
higher the e and hi are, the larger the QR, HR and EFa will be. Fig. 7(a) shows under the condition of To = Tsur = 30 °C and e = e0, QR approaches zero value as t/r2 approaches 1. But this is not the case for the condition of To = 30 °C, Tsur = 32 °C, and e = e0 as shown in Fig. 7(b). Fig. 7(b) shows that in the above condition, QR gets close to 5% as t/r2 approaches 1. This suggests that in practical situation of To – Tsur, neglecting the heat radiation produces errors in QR even with very thick insulation. Fig. 8 shows that HR value remains stable at t/r2 = 0 but begins to increase at t/r2 = 1. It can be seen in Fig. 8(a), at t/r2 = 1, HR is about 60% for e = 0.8 and about 15% for e = 0.2 in the condition of To = Tsur = 30 °C. In Fig. 8(b), even at t/ r2 = 1, HR is about 70% for e = 0.8 and about 8% for e = 0.2 in situation of To = 30 °C and Tsur = 32 °C. HR represents the ratio between heat radiation and heat convection as shown in Eq. (11), and heat radiation effect cannot be neglected if HR > 5%. The results indicate that heat radiation cannot be neglected within the whole range of insulation thickness. In Fig. 9, TD reaches its maximum value near t/r2 ; 0.025, and the absolute value of TD becomes higher with larger e; but if the value of e is fixed, smaller hi tends to result in larger absolute value of TD at some point before the maximum TD then the trend reverses after that point. Fig. 10(a) shows that for the same values of hi and e = e0, EFa values of e = e0 – 0 (considering heat radiation) are larger than those of e = e0 = 0 (neglecting heat radiation). This means that neglecting heat radiation will result in inaccurate the insulation effect. The slope of EFa becomes much smaller after t/r2 = 0.2, and the value of EFa between t/r2 = 0.2 and t/r2 = 1 is within 4%, thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in practical application. In the second set of cases, the most significant characteristics of non-insulated and insulated spherical container are shown in
ºC
Fig. 10. The relations between EF and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1: (a) To = Tsur = 30 °Cwith e = e0 and (b) To = 30 °C, Tsur = 32°Cwith e – e0.
Fig. 10(b). It can be seen that at t/r2 = 0, the value of EFa (;28%) at e = 0.1, e0 = 0.79, and hi = 800 W m2 K1 is much greater than that (EFa ; 2%) at e = 0.9 and with same values of e0 and hi. Another example is that the value of EFa (;21%) at e = 0.1, e0 = 0.79, and hi = 30 W m2 K1 is much greater than that (EFa ; 2%) at e = 0.9 and with same values of e0 and hi. The results indicate that for non-insulated and thin insulated cases, wrapping an aluminum foil (e = 0.04 as shown in Table 1) around the oxidized metal wall or insulation layer (e at high values as shown in Table 1) can achieve very good insulated effect; this advantage exists until t/r2 5 0.15. The slope of EFa becomes much smaller after t/r2 = 0.2, the value of EFa is within 4% between t/r2 = 0.2 and t/r2 = 1, thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in the practical applications. While t/r2 = 0.0025 and for same values of hi, the values of EFa at e – e0 > 0 (considering heat radiation) are greater than those at e = e0 = 0 (neglecting heat radiation), suggesting that neglecting heat radiation will underestimate the insulation effect in most situations. The data shown in Tables 4 and 5 serve to demonstrate the inaccuracy of heat transfer characteristics for insulated and non-insulated spherical containers neglecting the influence of heat radiation. It can be seen that the values of considering heat radiation are very different from those of neglecting heat radiation. For the non-insulated case, data at t/r2 = 0 with e0 = 0.9 listed in Table 4 show that qa is 367.0 W m2, which is larger than q(=256.7 W m2), and qc is 215.8 W m2, which is quite different from q. Thus, q is obtained by neglecting heat radiation but its value does not stand for heat convection. Due to the effect of heat radiation, T2a is 72.93 °C, a smaller value than T2 (=81.07 °C). This is because that qc ¼ ho 4pr 22 (T2a To) is not equal to q ¼ ho 4pr 22 (T2-To). Since q
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Table 4 Ti = 100 °C; To = Tsur = 30 °C, KA = 77 W m1 K1; Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm, t1 = 10 mm, ho = 10 W m2 K1, e = e0 = 0.9. t (mm)
t/R2
qa (W m2)
qr (W m2)
qc (W m2)
q (W m2)
QR (%)
HR (%)
T3a (°C)
T3 (°C)
SR (%)
0 0.5 1.0 1.5 2.5 3.5 5.0 7.5 10 30 40 50 60 80 100 200
0 0.003 0.005 0.008 0.013 0.018 0.025 0.038 0.05 0.15 0.2 0.25 0.3 0.4 0.5 1
367.0 318.6 281.9 253.1 210.7 180.8 149.6 116.7 96.17 42.38 34.05 28.83 25.25 20.67 17.86 12.09
151.2 129.2 113.0 100.5 82.39 69.97 57.22 44.10 36.05 15.55 12.45 10.52 9.21 7.52 6.49 4.39
215.8 189.3 168.9 152.6 128.3 110.9 92.37 72.63 60.12 26.83 21.59 18.30 16.05 13.15 11.37 7.70
256.7 233.3 213.9 197.5 171.5 151.7 129.5 104.6 87.99 41.01 33.23 28.29 24.87 20.45 17.71 12.05
30.06 26.77 24.12 21.95 18.59 16.11 13.40 10.43 8.50 3.23 2.40 1.88 1.52 1.08 0.81 0.31
70.09 68.27 66.89 65.81 64.23 63.11 61.94 60.72 59.96 57.98 57.68 57.49 57.37 57.21 57.12 56.94
72.93 67.48 63.27 59.92 54.90 51.30 47.49 43.42 40.85 34.04 32.98 32.33 31.89 31.33 31.01 30.38
81.07 76.18 72.13 68.72 63.28 59.15 54.53 49.33 45.88 36.17 34.59 33.60 32.93 32.08 31.57 30.60
11.16 12.90 14.00 14.69 15.28 15.29 14.82 13.59 12.31 6.27 4.87 3.93 3.26 2.36 1.81 0.71
Table 5 Ti = 20 °C; To = Tsur = 30 °C, KA = 77 W m1 K1; Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm, t1 = 10 mm, ho = 8.3 W m2 K1, e = e0 = 0.9. t (mm)
t/R2
qa (W m2)
qr (W m2)
qc (W m2)
q (W m2)
QR (%)
HR (%)
T3a (°C)
T3 (°C)
TD (°C)
0 0.5 1.0 1.5 2.5 3.5 5.0 7.5 10 30 40 50 60 80 100 200
0 0.003 0.005 0.008 0.013 0.018 0.025 0.038 0.05 0.15 0.2 0.25 0.3 0.4 0.5 1
221.7 198.4 179.4 163.8 139.5 121.5 102.0 80.68 66.99 30.03 24.18 20.50 17.98 14.73 12.73 8.63
81.28 73.64 67.27 61.89 53.35 46.90 39.72 31.75 26.52 12.09 9.76 8.29 7.27 5.97 5.16 3.50
140.4 124.7 112.2 101.9 86.13 74.63 62.25 48.94 40.46 17.94 14.42 12.21 10.70 8.76 7.57 5.12
159.5 146.8 136.0 126.7 111.6 99.8 86.33 70.69 60.06 28.77 23.42 19.99 17.61 14.52 12.60 8.59
28.04 25.99 24.20 22.62 19.97 17.84 15.34 12.39 10.34 4.20 3.15 2.48 2.02 1.44 1.09 0.42
57.89 59.04 59.97 60.74 61.95 62.84 63.81 64.87 65.55 67.38 67.66 67.84 67.96 68.11 68.20 68.37
3.65 0.25 3.38 5.94 9.86 12.72 15.80 19.10 21.20 26.75 27.60 28.13 28.48 28.93 29.19 29.69
8.24 5.01 2.28 0.07 3.90 6.88 10.30 14.26 16.94 24.79 26.10 26.93 27.50 28.22 28.66 29.49
4.58 5.26 5.66 5.87 5.96 5.84 5.49 4.84 4.26 1.96 1.50 1.19 0.98 0.70 0.54 0.21
is obtained by neglecting heat radiation, its value does not stand for heat convection, and qr(=151.2 W m2) cannot be neglected while comparing with the value of qc (=215.8 W m2); the resulted HR ; qr/qc is 70.09%. If it is neglected, a very big QR (=30.06%) is produced for non-insulated case. For the insulated case in Table 4, data at t/r2 = 1 with e0 = 0.9 show that qa is 12.09 W m2, which is very close to q(=12.05 W m2). However, qc is 7.07 W m2, whose value is very different from q. Due to the effect of heat radiation, T3a is 30.38 °C, a smaller value than T3(=30.60 °C). This is because that qc ¼ ho 4pr 23 (T3a-To) is not equal to q ¼ ho 4pr23 (T3-To). Thus, q is obtained by neglecting heat radiation but its value does not stand for heat convection. Also qr(=4.39 W m2) cannot be neglected while comparing with the value of qc (=7.07 W m2) because the resulted HR ; qr/qc would be 56.94%. It is interesting that a very small QR (=0.31%) is produced for thick insulated case (t/r2 = 1) even when heat radiation plays a very important role. It can be explained that both T3a(=30.38 °C) and T3(=30.60 °C) are very close to To and Tsur (both temperatures are assumed to be 30 °C) in the very thick insulated case (t/r2 = 1). Thus the difference between qa(=12.09 W m2) and q(=12.05 W m2) is quite small. The above explanations can be applied to all data shown in Tables 4 and 5.
5. Conclusion This paper shows that the influence of the heat radiation effect is very important in situation where the convection coefficient of
ambient air is of low value. Neglecting heat radiation will result in large errors for non-insulated and thin insulated spherical containers with a high surface emissivity. Meanwhile, using computer programming (like LabVIEW [23] in this study) which takes the heat radiation effect into account in a common PC can easily obtain highly accurate results. Since taking heat radiation effect into account by modern computers only requires some additional trivial effort, the heat radiation should not be neglected in any theoretical analysis of the heat transfer characteristics of non-insulated and insulated ducts and containers. Acknowledgment The authors would like to thank the National Science Council of Taiwan, ROC for the financial support of this study, which was completed under the project numbered NSC-97-2221-E168-044MY2. References [1] Bejan A. Heat transfer. John Wiley & Sons Inc.; 1993. p. 42–5. [2] Kreith F, Bohn MS. Principles of heat transfer. 5th ed. Harper International Edition; 1993. p. 21–41. [3] Dossat RJ. Principles of refrigeration. 3rd ed. Prentice Hall; 1991. p. 167–78. [4] Chatenever R. Air conditioning and refrigeration for the professional. John Wiley & Sons Inc.; 1988. p. 445–8. [5] Hsien TL, Wong KL, Yu SJ. The inaccuracy of heat transfer characteristics of insulated and non-insulated circular duct while neglecting the influence of heat radiation. Energy Convers Manage 2009;50(11):2856–63.
K.-L. Wong et al. / Energy Conversion and Management 52 (2011) 1612–1621 [6] Wong KL, Ke MT, Ku SS. The log mean heat transfer rate method of heat exchanger considering the influence of heat radiation. Energy Convers Manage 2009;50(11):2693–8. [7] Elsayed AO, Ibrahim EZ, Elsayed SA. Free convection from a constant heat flux elliptic tube. Energy Convers Manage 2003;44(15):2445–53. [8] Lazaro A, Dolado P, Marin JM, Zalba B. PCM-air heat exchangers for free-cooling applications in buildings: experimental results of two real-scale prototypes. Energy Conver Manage 2009;50(3):439–43. [9] Karadag R, Teke I. New approach relevant to floor Nusselt number in floor heating system. Energy Convers Manage 2008;49(5):1134–40. [10] Linhui C, Huaizhang T, Yanzhong L, Dongbin Z. Experimental study on natural convective heat transfer from a vertical plate with discrete heat sources mounted on the back. Energy Convers Manage 2006;47(18):3447–55. [11] Chen WL, Yang YC, Lee HL. Inverse problem in determining convection heat transfer coefficient of an annular fin. Energy Convers Manage 2007;48(4): 1081–8. [12] Chang YW, Cheng CH, Wang JC, Chen SL. Heat pipe for cooling of electronic equipment. Energy Convers Manage 2008;49(11):3398–404. [13] Holman JP. Heat transfer. ninth edition of international edition. McGraw-Hill Inc.; 2002. p. 26–37. [14] Incropera FP, DeWitt DP, Theodore LB, Adrienne SL. Fundamentals of heat and mass transfer. 6th ed. John Wiley & Sons Inc.; 2007. p. 116–25. [15] Yunus AC. Heat transfer-a practical approach. second edition in SI units. McGraw-Hill Inc.; 2004. p. 151–5.
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[16] Chen WL, Yang YC. An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe. Energy Convers Manage 2009;50(12):3096–103. [17] Ahmed M, Meade O, Medina MA. Reducing heat transfer across the insulated walls of refrigerated truck trailers by the application of phase change materials. Energy Convers Manage 2010;51(3):383–92. [18] Karadag R, Teke I. Investigation of floor Nusselt number in floor heating system for insulated ceiling conditions. Energy Convers Manage 2007;48(3):967–76. [19] Atayilmaz SO, Teke I. Experimental and numerical study of the natural convection from a heated horizontal cylinder wrapped with a layer of textile material. Int Commun Heat Mass Transfer 2010;37(1):58–67. [20] Chen W, Wong KL, Hsien TL, Huang CT. Reliable one-dimensional approximate solution of insulated oval duct. Energy Convers Manage 2008;49(8):2214–24. [21] Lee JF, Wong KL, Chen WL, Ku SS. Complete heat transfer solutions of an insulated regular polyhedron by using a RPSWT model. Energy Convers Manage 2005;46:2232–57. [22] Hsien TL, Wong KL, Chen WL, Siao CC. Reliable one-dimensional approximate solutions for insulated oblate spheroid containers. Energy Convers Manage 2009;50(8):1871–8. [23] LabVIEW 7.1 Edition. National Instruments Corporation Product. [24] Manual of IR Thermography Device: ThermaCAM™ E45, FLIR Systems, Sweden Product.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
The inaccuracy of heat transfer characteristics for non-insulated and insulated spherical containers neglecting the influence of heat radiation King-Leung Wong a,⇑, José Luis León Salazar a, Leo Prasad b, Wen-Lih Chen a a b
Department of Mechanical Engineering, Kun-Shan University of Technology, 949, Da-Wan Road, Yung-Kang City, Tainan County 710, Taiwan, ROC Faculty of Built Environment, University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 27 July 2009 Received in revised form 31 May 2010 Accepted 6 October 2010 Available online 13 November 2010 Keywords: Insulation Non-insulation Sphere Heat radiation Emissivity Inaccuracy
a b s t r a c t In this investigation, the differences of heat transfer characteristics for insulated and non-insulated spherical containers between considering and neglecting the influence of heat radiation are studied by the simulations in some practical situations. It is found that the heat radiation effect cannot be ignored in conditions of low ambient convection heat coefficients (such ambient air) and high surface emissivities, especially for the non-insulated and thin insulated cases. In most practical situations when ambient temperature is different from surroundings temperature and the emissivity of insulation surface is different from that of metal wall surface, neglecting heat radiation will result in inaccurate insulation effect and heat transfer errors even with very thick insulation. However, the insulation effect considering heat radiation will only increase a very small amount after some dimensionless insulated thickness (such insulation thickness/radius =0.2 in this study), thus such dimensionless insulated thickness can be used as the optimum thickness in practical applications. Meanwhile, wrapping a material with low surface emissivity (such as aluminum foil) around the oxidized metal wall or insulation layer (always with high surface emissivity) can achieve very good insulated effect for the non-insulated or thin insulated containers. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Non-insulated and insulated cold/hot containers and ducts are widely used in the various industries and domestic devices. Because heat radiation equation contains the 4th order exponent of temperature, it is very difficult to obtain precise theoretical or numerical heat radiation solutions before the time when modern computing tools were not available. In the past, many heat transfer experts and scholars avoid this problem and argue, from their own experiences, that the heat radiation effect can be ignored when the temperature difference between non-insulated or insulated cold/hot containers and surrounding is small. As a result, heat radiation has been commonly neglected in many practical heat transfer applications even in cases involving low heat convection coefficients. For the non-insulated situations, log mean temperature difference (LMTD) method, which neglects the influence of heat radiation, is introduced in most heat transfer [such as 1,2], air conditioning, and refrigeration text books [such 3,4], and is
⇑ Corresponding author. Tel.: +886 62057121; fax: +886 62050509. E-mail address: [email protected] (K.-L. Wong). 0196-8904/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2010.10.016
conventionally used to calculate the total heat transfer rate of heat exchangers. Recently, Hsien et al. [5] studied the complete heat transfer characteristics of non-insulated circular ducts considering heat radiation effect. They reported that in some practical situations, the heat radiation effect cannot be ignored when a noninsulated duct has high surface emissivities and the ambient air’s convection heat coefficient is low. Even in situations when the temperature difference between the fluid inside a non-insulated circular duct and the ambient air is very low, the errors generated by neglecting heat radiation are still very large and hence cannot be ignored. Unfortunately, it can be seen from Table 1 that most surface emissivities of oxidized metal are greater than 0.64; and Table 2 shows that the heat convection coefficients of ambient air with medium wind speed are with quite small. Consequently, the errors associated with the heat transfer characteristics obtained with conventional LMTD method applying to condensers and evaporators with heat exchanging in ambient air can be very large. Thus, LMTD method should not be applied to evaporators and condensers. To solve this problem, Wong et al. [6] developed a log mean heat transfer rate (LMHTR) method which considers the influence of heat radiation to calculate the exact heat transfer rate of heat exchangers. Their study showed that the results obtained by LMHTR method are the same as those obtained from
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Nomenclature
e e0 A1 A2 A3 EF EFa EFn hi ho hr HR KA Ks q q0 qa qa0 qc qc0
emissivity of insulation surface emissivity of container surface inner surface area of a spherical container external surface area of a spherical container external surface area of a insulated spherical container insulation effect insulation effect of an insulated spherical container considering heat radiation insulation effect of an insulated spherical container neglecting heat radiation inner heat convection coefficient external heat convection coefficient external radiation heat convection coefficient convective heat coefficient ratio conductivity of duct material conductivity of insulated material total heat transfer rate of insulated container neglecting heat radiation total heat transfer rate of non-insulated container neglecting heat radiation total heat transfer rate of insulated container considering heat radiation total heat transfer rate of non-insulated container considering heat radiation convection heat transfer rate of insulated container considering heat radiation convection heat transfer rate of non-insulated container considering heat radiation
LMTD method while surface emissivities equal to zero; and they indicated that it is risky to neglect the influence of heat radiation Table 1 The emissivities e of various substances from the manual of infrared temperature demonstrator [24]. Human skin Gold Silver Aluminum Copper Iron Stainless steel Steel Nickel Brick Carbon Concrete Glass Paint oil Paper, white Paper Plaster Rubber, black Wood, oak White ceramic Black painting Oil, lubricant
Soil Water
0.98 0.02 0.02 Weathered = 0.83; Foil (bright)=0.04 Disk, rough = 0.96 Polished = 0.05 Oxidized = 0.78 Cast(ox) = 0.64 Sheet, rusted = 0.69 Polished = 0.16 Oxidized = 0.85 Polished = 0.07 Oxidized = 0.79 Electro pole = 0.05 0.81 0.95 0.95 0.84–0.97 0.94 0.70 0.89 0.86 0.95 0.90 0.91 0.96 Film 0.03 mm = 0.27 Film 0.13 mm = 0.72 Thick = 0.82 Dry = 0.92 Saturated water = 0.95 Distilled = 0.96 Frost = 0.98 Snow = 0.85
qr qr0 QR r1 r2 r3 SR TD t1 t t/r2 T2 T2a T3 T3a Ti To Tsur
radiation heat transfer rate of insulated container considering heat radiation radiation heat transfer rate of non-insulated container considering heat radiation error of heat transfer rate generated by neglecting heat radiation inner radius of spherical container external radius of spherical container external radius of insulated spherical container error of surface temperature generated by neglecting heat radiation the surface temperature difference generated by neglecting heat radiation thickness of spherical container wall thickness of insulated layer dimensionless insulated thickness bare wall surface temperature generated by neglecting heat radiation bare wall surface temperature generated by considering heat radiation insulated surface temperature generated by neglecting heat radiation insulated surface temperature generated by considering heat radiation temperature of the fluid inside the duct temperature of the fluid outside the spherical container temperature of the outside surrounding
especially in the situations involving low ambient air convection coefficients and heat exchanger (such as condensers and evaporators) surfaces with high surface emissivity. Thus, in order to obtain more accurate results, in stead of the conventional LMTD method, the LMHTR method should be employed. However, investigations related to non-insulated heat transfer to ambient air neglected the heat radiation are still quite common. For example, Elsayed et al. [7] studied free convection of air around a constant
Table 2 Referred approximate values of convection heat transfer [13]. Approximate values of convection heat transfer, h (W m2 K1) Mode h (W m2 K1) Free convection Temp. diff. = 30 °C horizontal plate 0.3 in high in air Temp. diff. = 30 °C vertical plate 0.3 in high in air Horizontal cylinder, 2 cm diameter, in water Heat transfer across 1.5 cm vertical air gap with temp. diff. = 60 °C Forced convection Air flow at 2 m/s over 0.2-m square plate Air flow at 35 m/s over 0.75-m square plate Air at 2 atm flowing in 2.5 cm diameter tube at 10 m/s (=36 km/h) Water at 0.5 kg/s flow in 2.5 cm diameter tube Air flow across 5 cm diameter cylinder with velocity of 50 m/s(=180 km/h) Boiling water In a pool or container Flowing in a tube Condensation of water vapor, 1 atm Vertical surfaces Outside horizontal tubes
4.5 6.5 890 2.64
12 75 65 3500 180
2500–35,000 5000– 100,000 4000–11,300 9500–25,000
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heat flux elliptic tube without considering heat radiation; Lazaro et al. [8] investigated the PCM-air heat exchangers for free cooling applications in buildings neglecting the heat radiation effect; Karadag and Teke [9] developed a new approach relevant to floor Nusselt number in floor heating system without considering heat radiation; Linhui et al. [10] reported an experimental study on natural convective heat transfer from a vertical plate with discrete heat sources mounted on the back neglecting heat radiation; Chen et al. [11] solved the inverse problem in determining convection heat transfer coefficient of an annular fin without considering heat radiation; Chang et al. [12] investigated the heat pipe for cooling of electronic equipment neglecting heat radiation; and the list goes on. For the insulated situations, it is commonly shown in many heat transfer text books [such as 13–15] that heat radiation is neglected in insulated duct/container even in situations with ambient air at low heat convection coefficients. Meanwhile, the critical heat transfer does occur with ambient air at a very small heat convection coefficient. From Table 1, it can be seen that the surface emissivities of insulated material are greater than 0.8. Thus, the heat radiation should be considered in the analysis of critical heat transfer. But the critical heat transfer characteristics introduced in most heat transfer text books [such as 13–15] neglect the heat radiation. In addition, many investigations associated with insulation in open literature also neglected the influence of heat radiation, for example: Chen and Yang [16] used an iterative regularization method in estimating the transient heat transfer rate on the surface of the insulation layer of a double circular pipe; Ahmed et al. [17] investigated the heat transfer characteristics across the insulated walls of refrigerated truck trailers by the application of phase change materials; Karadag and Teke [18] investigated the floor Nusselt number in floor heating system for insulated ceiling conditions; Atayilmaz and Teke [19] obtained the experimental and numerical study of the natural convection from a heated horizontal cylinder wrapped with a layer of textile material; Chen et al. [20] found out the reliable one-dimensional approximate solution of insulated oval duct, Lee et al. [21] investigated the complete heat transfer solutions of an insulated regular polyhedron by using a RPSWT model, Hsien et al. [22] found out the reliable one-dimensional approximate solutions for insulated oblate spheroid containers, etc. Recently, Hsien et al. [5] conducted a comparative study on the heat transfer characteristics of an insulated circular duct considering and neglecting the influence of heat radiation. They found that the heat radiation effect cannot be ignored in conditions of thin insulation with low ambient convective heat coefficient and high surface emissivity. In order to highlight the importance of heat radiation for non-insulated and insulated ducts/containers in situation where heat convection coefficients of ambient air is small, this paper, a follow-up study to Hsien et al. [5], examines the differences in heat transfer characteristics of a non-insulated and insulated spherical container resulted by considering and neglecting the influence of heat radiation. Then the inaccuracy of heat transfer characteristics for a non-insulated and insulated spherical container neglecting the influence of heat radiation can be demonstrated.
2.1. Cases with the influence of the heat radiation being neglected Here, we assume that heat transfer is steady state without heat generation inside solid materials. All thermal properties are uniform and independent of temperature, and hi and ho are constants. If the influence of external surface heat radiation is not considered, it can be seen from Fig. 1(a) that the total thermal resistance can be derived from heat transfer text book [such as 15] is:
X
Rth ¼
1 r1
r12
1 r2
r13
1 1 þ þ þ 4pK A 4p K S hi 4pr 21 h0 4pr 23
ð1Þ
and the relative total heat transfer rate neglecting heat radiation is:
Ti T0 T3 T0 q¼ P ¼ 1 Rth h 4pr 2 0
ð2Þ
3
Eq. (2) implies that heat transfer rate q and insulated surface temperature T3 of situations without considering the influence of external surface heat radiation can be obtained. From Fig. 2(a), the heat transfer rate of non-insulated spherical container without considering the influence of outside heat radiation can be written as:
2. Problem formulation Insulated and non-insulated spherical containers are shown in Figs. 1 and 2, respectively. Both spherical containers are exposed to a surrounding temperature of Tsur, and have a wall thickness of t1, conductivity KA, bare wall surface emissivity e0. The insulation layer has thickness t, conductivity KS and surface emissivity e. The internal and external fluids have convection heat transfer coefficients hi and ho, temperatures Ti and To, respectively.
Fig. 1. An insulated spherical container and relative parameters: (a) neglecting heat radiation and (b) considering heat radiation.
K.-L. Wong et al. / Energy Conversion and Management 52 (2011) 1612–1621
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where T3a is the actual surface temperature considering heat radiation. Then from Fig. 1(b), the surface convective heat transfer becomes:
qc ¼ h0 4pr 23 ðT 3a T 0 Þ
ð6Þ
and the surface radiation heat transfer rate is:
qr ¼ re4pr 23 T 43a T 4sur
ð7Þ
Comparing Eqs. (2) and (6), under the conditions of e – 0, T3a – T3, qc – q, the energy conservation among qa, qc and qr can be written as:
qa ¼ qc þ qr
ð8Þ
Therefore, the qa, qc and qr and T3a can be readily deduced from Eqs. (5)–(8). With both q and qa being solved, the error of heat transfer rate generated by neglecting heat radiation effect can be defined as:
q 100% QR ¼ 1 qa
ð9Þ
The equivalent heat convection coefficient of heat radiation [15] is defined from Eq. (7) as:
hr ¼ re T 23a þ T 2sur ðT 3a þ T sur Þ
ð10Þ
Here, hr is normally used to compare with the heat convection coefficient ho which in turn shows how significant the effect of radiation is. The ratio between equivalent heat convection coefficient of heat radiation and original heat convection coefficient is defined as:
HR ¼
hr hr 4pr23 ðT 3a T sur Þ q 100% ¼ r 100% 100% h0 qc h0 4pr23 ðT 3a T 0 Þ
ð11Þ
where Tsur is always close to To. Similar to Eq. (9), the error of surface temperature generated by neglecting heat radiation effect, based on T3 and T3a using Celsius temperature scale, is defined as:
SR ¼
Fig. 2. A non-insulated spherical container and relative parameters: (a) neglecting heat radiation and (b) considering heat radiation.
Ti T0
q0 ¼ 1 hi 4pr 21
þ
1 1 r 1 þr 2
4pK A
ð3Þ
Eqs. (2) and (3) lead to an insulated effect EFn of an insulated spherical container comparing with non-insulated situation neglecting heat radiation being:
q 100% EF n ¼ 1 q0
ð4Þ
From Fig. 1, when the influence of external surface heat radiation is considered, the complete heat transfer rate from the surface of an insulated spherical container can be expressed as:
1 hi 4pr21
þ
1 1 r 1 r 2
4pK A
þ
1 1 r2 r 3
4pK S
T3 100% T 3a
ð5Þ
ð12Þ
In the case of the insulated cold spherical container, condensed water may be formed if the insulated surface temperature is less than the dew point of ambient air. Therefore, predicting the insulated surface temperature is very important. Dimensionless SR tends to reduce the magnitude of error; hence SR is not suitable to demonstrate the heat characteristic of cold spherical container. Instead, the surface temperature difference is used and is written as:
ð13Þ
While the influence of external surface heat radiation is considered, the complete heat transfer rate from the surface of a noninsulated spherical container shown in Fig. 2(b) can be written as:
qa0 ¼
2.2. Case with the influence of heat radiation being considered
qa ¼
1
TD ¼ T 3a T 3
þ h 41pr2 0 2
T T i 3a
T i T 2a 1 hi 4pr 21
þ
ð14Þ
1 1 r 1 r 2
4pK A
where T2a is the actual surface temperature of a non-insulated spherical container shown in Fig. 2(b) with the influence of heat radiation taken into account. Additionally, its surface convective heat transfer rate is:
qc0 ¼ ho 4pr 23 ðT 2a T o Þ
ð15Þ
and the surface radiation heat transfer rate is:
qr0 ¼ re4pr23 T 42a T 4sur
ð16Þ
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Table 3 Referred approximate values of thermal conductivities [13].
The heat balance can be expressed as:
qa0 ¼ qc0 þ qr0
Thermal conductivity of various materials at 20 °C Metals Material Copper (pure) Aluminum (pure) Carbon steel, l% C Carbon steel (18%Cr, 8%Ni) Cast iron
K (W m1 K1) 386 204 73–77 43 16
Nonmetallic solids Glass, window Plaster, gypsum Metal lath Woof lath Teflon Asphalt Wood fiber sheet Wool Glass fiber Building brick common Building brick face Concrete, cinder Stone, 1–2–4 mix
0.78 0.48 0.4 0.28 0.35 0.7 0.047 0.038 0.035 0.69 1.32 0.76 1.37
Graphite, pyrolytic Perpendicular to layers Polyethylene Polypropylene Polyvinylchloride Rubber, hard
5.6 0.33 0.16 0.09 0.1
ð17Þ
Similarly, qco, qro, qao and T2a of non-insulated spherical container considering heat radiation can be readily obtained from Eqs. (14)– (17). And finally, the insulated effect EFa of an insulated spherical container comparing with non-insulated situation when heat radiation is considered can be obtained by:
q EF a ¼ 1 a 100% qa0
ð18Þ
3. Calculation of heat transfer results The one-dimensional heat transfer solutions of the insulated and non-insulated spherical containers are exact. These solutions are calculated by one dimensional LabVIEW [23] programming, which provide an easy and friendly interface to input all needed parameters and also offer live graph, table data, and automatic connection with excel. With this handy tool, it is possible to obtain accurate values with one-dimensional spherical heat equation. The major parameter in this program is the insulation thickness, and according to the input insulation thickness, Eqs. (5) and (8) and equations from (14) to (17) are solver iteratively until the difference of heat transfer rates obtained between two consecutive iterations is less than 104. The iteration process converges quickly
ºC
ºC
ºC
ºC
Fig. 3. The relations between QR and t/r2 in the situation of Ti = 100 °C, KA = 77W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
ºC
ºC
Fig. 4. The relations between HR and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
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and only takes a few seconds to complete, therefore, with each change of parameters, accurate solutions can be returned in a very short period of time. According to the emissivities shown in Table 1, e = 0.8 and 0.9 and e = 0.1 and 0.2 are adopted to represent the cases of high and low surface emissivity, respectively. Also Table 2 shows that the natural convection coefficients of air are below 10 W m2 K1; and even in the cases with a very high air speed, the forced convection coefficients of air are less than 100 W m2 K1. Therefore, hi = 30 W m2 K1 can be used to represent the situation of low/medium convection coefficients inside the spherical container, and ho = 8.3 and 10 W m2 K1 are chosen to represent the natural or low forced convection coefficients of ambient air. Meanwhile, the natural convection coefficients of water are more than 890 W m2 K1, and the values of convection coefficients are between 2500 and 35,000 W m2 K1 for boiling water in a pool or container. Nevertheless, in insulated cases, convection coefficients may become smaller, thus hi = 800 W m2 K1 is chosen to represent the cases of high convection coefficients of hot or cold liquid (may be other than water) inside the insulated spherical container. In practical applications, carbon steel is the common material applied to construct containers. According to the Table 3, KA = 77 W m1 K1 is used as the carbon steel spherical container’s conductivity; and the insulated material conductivity with the value KS = 0.035 W m1 K1shown in Table 3 is adopted in this study. In order to check if the computer results are reliable, the following measures are adopted:
(a) Let surface emissivity e = 0, i.e., heat radiation does not exist and HR = hr/ho = 0; check if the resulted QR being close to zero. (b) Let surface emissivity e = 1 and external convection coefficient ho = 50,000 W/m2 K, i.e., heat radiation effect becomes very small by comparing the very big heat convection. And HR = hr/ho becomes very small; check if the resulted and QR being close to zero. 4. Results and discussions For the results to be generalized, they are shown in dimensionless parameters except for TD, the surface temperature difference between considering and neglecting heat radiation effect for insulated cold spherical container. This is because that the value of difference in dimensionless temperatures is very small, and it might result in a misunderstanding of physical meaning among readers. The first investigation is on the case of insulated hot spherical containers with the following thermal and geometrical parameters: Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 at different ambient-air conditions: (a) To = Tsur = 30 °C and e = e0 and (b) To = 30 °C, Tsur = 32 °C, e = e0 or e – e0, respectively. Figs. 3–6 shows the heat transfer rate error (QR), convective coefficients ratio (HR), error of surface temperature (SR), and insulated effect (EF) are affected by dimensionless insulated thickness (t/r2), internal convection coefficient (hi), insulated surface emissivity (e), and metal wall surface emissivity (e0).
ºC
ºC
ºC
ºC ºC
Fig. 5. The relations between SR and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
ºC
Fig. 6. The relations between EF and t/r2 in the situation of Ti = 100 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 10 W m2 K1: (a) To = Tsur = 30 °C with e = e0 and (b) To = 30 °C, Tsur = 32 °C with e – e0.
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ºC
ºC
ºC
ºC
Fig. 7. The relations between QR and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
Figs. 3, 4 and 6 show that QR and HR decrease, and EF increases as t/r2 increases. The higher the e and hi are, the larger the QR, HR and EF will be. The plots in Fig. 3(a) indicate that in the condition of To = Tsur = 30 °C and e = e0, QR gets close to 0 as t/r2 approaches to 1. This phenomenon, however, does not happen at the condition of To = 30 °C, Tsur = 32 °C, and e = e0 as shown by the plots given in Fig. 3(b). Here QR gets close to 3% as t/r2 approaches to 1; thus in the condition of To – Tsur, neglecting the heat radiation produces error in QR even with very thick insulation. Fig. 4 suggests that HR value becomes larger at t/r2 = 0. It is noticeable from Fig. 4(a) that even at t/r2 = 1, HR is about 50% for e = 0.8 and HR ; 12% for e = 0.2 under the condition of To = Tsur = 30 °C. Fig. 4(b) also shows that even at t/r2 = 1, HR reaches 58% for e = 0.8 and HR is 7.5% for e = 0.2 in the condition of To = 30 °C and Tsur = 32 °C. HR represents the ratio between heat radiation and heat convection as shown in Eq. (11), thus heat radiation effect cannot be neglected if HR > 5%. Fig. 5 shows that SR exists, and its absolute value reaches maximum around t/r2 = 0.025; the absolute value of SR is generally proportional to e. If the value of e is fixed, smaller hi tends to result in larger absolute value of SR before t/r2 ; 0.025 then the trend reverses after that value. Fig. 6(a) shows that for same values of hi and e = e0, EFa values at the condition of e = e0 – 0 (considering heat radiation) are greater than that at e = e0 = 0 (neglecting heat radiation), meaning neglecting heat radiation will underestimate insulation effect. The slope of EFa (considering heat radiation) becomes much smaller after t/r2 = 0.2, and the value of EFa is within 3% of between t/r2 = 0.2 and t/r2 = 1; thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in practical applications. The most meaningful practical characteristics of non-insulated and insulated spherical container are shown in Fig. 6(b). Fig. 6(b)
ºC
ºC
Fig. 8. The relations between HR and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
shows that at t/r2 = 0, the value of EFa (;36%) at e = 0.1, e0 = 0.79, and hi = 800 W m2 K1 is much greater than that (EFa ; 3%) at e = 0.9, and with the same values of e0 and hi; meanwhile EFa (;24%) at e = 0.1, e0 = 0.79, and hi = 30 W m2 K1 is much greater than that (EFa ; 3%) at e = 0.9, and with the same values of e0 and hi. Hence, it can be concluded that for non-insulated and thin insulated cases, wrapping an aluminum foil (e = 0.04 as shown in Table 1) around the oxidized metal wall or insulation layer (e at high values as shown in Table 1) can result in excellent insulated effect. This practice works if t/r2 5 0.15. The slope of EFa becomes much smaller after t/r2 = 0.2, and the value of EFa is within 3% between t/r2 = 0.2 and t/r2 = 1. Therefore, t/r2 = 0.2 can be treated as the optimum dimensionless insulation thickness in practical applications. When t/r2 = 0.0025 and for same values of hi, EFa value at e – e0 > 0 (considering heat radiation) is greater than that at e = e0 = 0 (neglecting heat radiation). Thus neglecting heat radiation will result in inaccurate insulation effect in most situations. The second simulation is to examine insulated cold spherical containers with the following thermal and geometrical parameters: Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 at different ambient-air conditions: (a) To = Tsur = 30 °C with e = e0 and (b) To = 30 °C, Tsur = 32 °C with e = e0 and e – e0, respectively. Figs. 7–10 give the heat transfer rate error (QR), convective coefficients ratio (HR), the difference of surface temperature (TD), and insulated effect (EF) are affected by the dimensionless insulated thickness (t/r2), internal convection coefficient (hi), insulated surface emissivity (e), and metal wall surface emissivity (e0). The characteristics observed in Figs. 7–10 are very similar to those in Figs. 3–6. It can be seen in Figs. 7, 8 and 10 that QR decreases, but EF and HR increase as t/r2 increases. The
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ºC
ºC
ºC ºC
ºC
Fig. 9. The relations between TD and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1 with e = e0: (a) To = Tsur = 30 °C and (b) To = 30 °C, Tsur = 32 °C.
higher the e and hi are, the larger the QR, HR and EFa will be. Fig. 7(a) shows under the condition of To = Tsur = 30 °C and e = e0, QR approaches zero value as t/r2 approaches 1. But this is not the case for the condition of To = 30 °C, Tsur = 32 °C, and e = e0 as shown in Fig. 7(b). Fig. 7(b) shows that in the above condition, QR gets close to 5% as t/r2 approaches 1. This suggests that in practical situation of To – Tsur, neglecting the heat radiation produces errors in QR even with very thick insulation. Fig. 8 shows that HR value remains stable at t/r2 = 0 but begins to increase at t/r2 = 1. It can be seen in Fig. 8(a), at t/r2 = 1, HR is about 60% for e = 0.8 and about 15% for e = 0.2 in the condition of To = Tsur = 30 °C. In Fig. 8(b), even at t/ r2 = 1, HR is about 70% for e = 0.8 and about 8% for e = 0.2 in situation of To = 30 °C and Tsur = 32 °C. HR represents the ratio between heat radiation and heat convection as shown in Eq. (11), and heat radiation effect cannot be neglected if HR > 5%. The results indicate that heat radiation cannot be neglected within the whole range of insulation thickness. In Fig. 9, TD reaches its maximum value near t/r2 ; 0.025, and the absolute value of TD becomes higher with larger e; but if the value of e is fixed, smaller hi tends to result in larger absolute value of TD at some point before the maximum TD then the trend reverses after that point. Fig. 10(a) shows that for the same values of hi and e = e0, EFa values of e = e0 – 0 (considering heat radiation) are larger than those of e = e0 = 0 (neglecting heat radiation). This means that neglecting heat radiation will result in inaccurate the insulation effect. The slope of EFa becomes much smaller after t/r2 = 0.2, and the value of EFa between t/r2 = 0.2 and t/r2 = 1 is within 4%, thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in practical application. In the second set of cases, the most significant characteristics of non-insulated and insulated spherical container are shown in
ºC
Fig. 10. The relations between EF and t/r2 in the situation of Ti = 20 °C, KA = 77 W m1 K1, Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm and ho = 8.3 W m2 K1: (a) To = Tsur = 30 °Cwith e = e0 and (b) To = 30 °C, Tsur = 32°Cwith e – e0.
Fig. 10(b). It can be seen that at t/r2 = 0, the value of EFa (;28%) at e = 0.1, e0 = 0.79, and hi = 800 W m2 K1 is much greater than that (EFa ; 2%) at e = 0.9 and with same values of e0 and hi. Another example is that the value of EFa (;21%) at e = 0.1, e0 = 0.79, and hi = 30 W m2 K1 is much greater than that (EFa ; 2%) at e = 0.9 and with same values of e0 and hi. The results indicate that for non-insulated and thin insulated cases, wrapping an aluminum foil (e = 0.04 as shown in Table 1) around the oxidized metal wall or insulation layer (e at high values as shown in Table 1) can achieve very good insulated effect; this advantage exists until t/r2 5 0.15. The slope of EFa becomes much smaller after t/r2 = 0.2, the value of EFa is within 4% between t/r2 = 0.2 and t/r2 = 1, thus t/r2 = 0.2 can be used as the optimum dimensionless insulation thickness in the practical applications. While t/r2 = 0.0025 and for same values of hi, the values of EFa at e – e0 > 0 (considering heat radiation) are greater than those at e = e0 = 0 (neglecting heat radiation), suggesting that neglecting heat radiation will underestimate the insulation effect in most situations. The data shown in Tables 4 and 5 serve to demonstrate the inaccuracy of heat transfer characteristics for insulated and non-insulated spherical containers neglecting the influence of heat radiation. It can be seen that the values of considering heat radiation are very different from those of neglecting heat radiation. For the non-insulated case, data at t/r2 = 0 with e0 = 0.9 listed in Table 4 show that qa is 367.0 W m2, which is larger than q(=256.7 W m2), and qc is 215.8 W m2, which is quite different from q. Thus, q is obtained by neglecting heat radiation but its value does not stand for heat convection. Due to the effect of heat radiation, T2a is 72.93 °C, a smaller value than T2 (=81.07 °C). This is because that qc ¼ ho 4pr 22 (T2a To) is not equal to q ¼ ho 4pr 22 (T2-To). Since q
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Table 4 Ti = 100 °C; To = Tsur = 30 °C, KA = 77 W m1 K1; Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm, t1 = 10 mm, ho = 10 W m2 K1, e = e0 = 0.9. t (mm)
t/R2
qa (W m2)
qr (W m2)
qc (W m2)
q (W m2)
QR (%)
HR (%)
T3a (°C)
T3 (°C)
SR (%)
0 0.5 1.0 1.5 2.5 3.5 5.0 7.5 10 30 40 50 60 80 100 200
0 0.003 0.005 0.008 0.013 0.018 0.025 0.038 0.05 0.15 0.2 0.25 0.3 0.4 0.5 1
367.0 318.6 281.9 253.1 210.7 180.8 149.6 116.7 96.17 42.38 34.05 28.83 25.25 20.67 17.86 12.09
151.2 129.2 113.0 100.5 82.39 69.97 57.22 44.10 36.05 15.55 12.45 10.52 9.21 7.52 6.49 4.39
215.8 189.3 168.9 152.6 128.3 110.9 92.37 72.63 60.12 26.83 21.59 18.30 16.05 13.15 11.37 7.70
256.7 233.3 213.9 197.5 171.5 151.7 129.5 104.6 87.99 41.01 33.23 28.29 24.87 20.45 17.71 12.05
30.06 26.77 24.12 21.95 18.59 16.11 13.40 10.43 8.50 3.23 2.40 1.88 1.52 1.08 0.81 0.31
70.09 68.27 66.89 65.81 64.23 63.11 61.94 60.72 59.96 57.98 57.68 57.49 57.37 57.21 57.12 56.94
72.93 67.48 63.27 59.92 54.90 51.30 47.49 43.42 40.85 34.04 32.98 32.33 31.89 31.33 31.01 30.38
81.07 76.18 72.13 68.72 63.28 59.15 54.53 49.33 45.88 36.17 34.59 33.60 32.93 32.08 31.57 30.60
11.16 12.90 14.00 14.69 15.28 15.29 14.82 13.59 12.31 6.27 4.87 3.93 3.26 2.36 1.81 0.71
Table 5 Ti = 20 °C; To = Tsur = 30 °C, KA = 77 W m1 K1; Ks = 0.035 W m1 K1, r1 = 190 mm, r2 = 200 mm, t1 = 10 mm, ho = 8.3 W m2 K1, e = e0 = 0.9. t (mm)
t/R2
qa (W m2)
qr (W m2)
qc (W m2)
q (W m2)
QR (%)
HR (%)
T3a (°C)
T3 (°C)
TD (°C)
0 0.5 1.0 1.5 2.5 3.5 5.0 7.5 10 30 40 50 60 80 100 200
0 0.003 0.005 0.008 0.013 0.018 0.025 0.038 0.05 0.15 0.2 0.25 0.3 0.4 0.5 1
221.7 198.4 179.4 163.8 139.5 121.5 102.0 80.68 66.99 30.03 24.18 20.50 17.98 14.73 12.73 8.63
81.28 73.64 67.27 61.89 53.35 46.90 39.72 31.75 26.52 12.09 9.76 8.29 7.27 5.97 5.16 3.50
140.4 124.7 112.2 101.9 86.13 74.63 62.25 48.94 40.46 17.94 14.42 12.21 10.70 8.76 7.57 5.12
159.5 146.8 136.0 126.7 111.6 99.8 86.33 70.69 60.06 28.77 23.42 19.99 17.61 14.52 12.60 8.59
28.04 25.99 24.20 22.62 19.97 17.84 15.34 12.39 10.34 4.20 3.15 2.48 2.02 1.44 1.09 0.42
57.89 59.04 59.97 60.74 61.95 62.84 63.81 64.87 65.55 67.38 67.66 67.84 67.96 68.11 68.20 68.37
3.65 0.25 3.38 5.94 9.86 12.72 15.80 19.10 21.20 26.75 27.60 28.13 28.48 28.93 29.19 29.69
8.24 5.01 2.28 0.07 3.90 6.88 10.30 14.26 16.94 24.79 26.10 26.93 27.50 28.22 28.66 29.49
4.58 5.26 5.66 5.87 5.96 5.84 5.49 4.84 4.26 1.96 1.50 1.19 0.98 0.70 0.54 0.21
is obtained by neglecting heat radiation, its value does not stand for heat convection, and qr(=151.2 W m2) cannot be neglected while comparing with the value of qc (=215.8 W m2); the resulted HR ; qr/qc is 70.09%. If it is neglected, a very big QR (=30.06%) is produced for non-insulated case. For the insulated case in Table 4, data at t/r2 = 1 with e0 = 0.9 show that qa is 12.09 W m2, which is very close to q(=12.05 W m2). However, qc is 7.07 W m2, whose value is very different from q. Due to the effect of heat radiation, T3a is 30.38 °C, a smaller value than T3(=30.60 °C). This is because that qc ¼ ho 4pr 23 (T3a-To) is not equal to q ¼ ho 4pr23 (T3-To). Thus, q is obtained by neglecting heat radiation but its value does not stand for heat convection. Also qr(=4.39 W m2) cannot be neglected while comparing with the value of qc (=7.07 W m2) because the resulted HR ; qr/qc would be 56.94%. It is interesting that a very small QR (=0.31%) is produced for thick insulated case (t/r2 = 1) even when heat radiation plays a very important role. It can be explained that both T3a(=30.38 °C) and T3(=30.60 °C) are very close to To and Tsur (both temperatures are assumed to be 30 °C) in the very thick insulated case (t/r2 = 1). Thus the difference between qa(=12.09 W m2) and q(=12.05 W m2) is quite small. The above explanations can be applied to all data shown in Tables 4 and 5.
5. Conclusion This paper shows that the influence of the heat radiation effect is very important in situation where the convection coefficient of
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