Effects of pipe angular velocity and oven configuration on tube temperature distribution in the radiative heating of PVC pipes

Effects of pipe angular velocity and oven configuration on tube temperature distribution in the... Several manufacturing processes in polymer industry aim at obtaining products by deforming preforms or sheets after a heat- ing process. A thorough knowledge of the operating parameters of such heating processes is fundamental to fulfill the often high production requirements with the least energy consumption and to avoid unacceptable defects in the final product. A common example of such an application is the end-forming process of polyvinyl chloride (PVC) tubes, which are enlarged at one end in order to allow pipes connections. The heating phase which comes before the deformation process is usually carried out in ovens equipped with short wave infrared lamps; to ensure uniform heating, pipes rotate with a given angular veloc- ity, which represents a fundamental parameter for the success of the whole manufacturing process. In this work, a transient analysis of the radiative heat exchange between rotating PVC pipes and infrared lamps in an oven for end-forming process has been conducted by means of a finite element model, in order to investigate the influence of cylinder angular velocity on the temperature distribution in the tube. Local view factors have been calculated for different oven configurations and have been expressed as a function of angular velocity, allowing pipe rotation to be simulated as a time-dependent boundary condi- tion, instead of using a moving mesh. Simulations were carried out for different tubes geometries and angular velocities and results were compared with the case of a uniformly irradiated tube in terms of temperature displacement. For a given oven configuration, the results obtained by the numerical model can be used to find a critical angular velocity over which further increase does not lead to appreciable improvements in temperature evenness. The effect of the lamps’ relative position was also investigated, showing a significant influence on critical angular velocities obtained. The model realized represents a potential tool to characterize the end-forming process in terms of critical angular velocity, leading to reductions in machine set-up time and product waste due to thermal failure. Keywords Pipes end-forming · Radative heating · Finite-element modeling List of symbols F View factor between finite extension elements i→j a Ratio between pipe external radius and lamp (–) radial position (–) g Control function (–) −1 −1 −2 −1 c Specific thermal capacity of PVC ( J kg K ) h Convection heat transfer coefficient ( Wm K ) −2 dF View factor between infinitesimal elements (–) I Heat flux ( Wm ) dj→di −1 −1 dF View factor between an infinitesimal element and k Thermal conductivity of PVC ( Wm K ) dj→i −1 a finite extension element (–) K Absorbtion coefficient of PVC ( m ) dF View factor between a finite extension element L Lamp extension in the plane perpendicular to i→dj and an infinitesimal element (–) cylinder axis (mm) L Lamp heating length (mm) L Lamp total length (mm) N Number of finite extension arcs along pipe outer * Marco Lorenzini circumference (–) marco.lorenzini@unibo.it N Number of external lamps in the oven (–) lamp Michael Lucchi N Number of quadrilateral elements on the outer El OC michael.lucchi2@unibo.it circumference of the pipe (–) N Total number of quadrilateral elements in the Department of Industrial Engineering, University El Tot of Bologna, via Fontanelle 40, Forlí, Italy grid (–) Vol.:(0123456789) 1 3 124 International Journal of Energy and Environmental Engineering (2018) 9:123–134 p Lamp facing period (s) r Pipe external radius P Lamp rated power (W) r Pipe mean radius r m −3 q Heat generation term ( Wm ) r Generic radial coordinate (mm) Introduction R Lamp radial position (mm) T Temperature (K) In the second half of the twentieth century, a rapid increase T Lamps switch off temperature when they are con- off in the production scales and in the variety and complex- trolled by a traditional pyrometer (K) ity of manufacturing processes characterized the evolution T Softening temperature of PVC (K) of the polymer industry [1]. In industrial applications like th Pipe thickness (mm) thermoforming, aimed at obtaining products by deform- x Generic coordinate along lamp extension (mm) ing polymeric preforms or sheets, the study of the ther- x Non-dimensional coordinate along lamp exten- mal stage allows to get a successful forming process and sion in a transversal plane perpendicular to directly affects the mechanical properties of the end prod- cylinder axis (–) uct [2]. Moreover, studying the heating stage in the design x Non-dimensional coordinate x calculated at the phase of such processes can lead to a significant increase first end of the lamp (–) in the process efficiency, with consequent energy savings x Non-dimensional coordinate x calculated at the and reduced environmental impact. Many authors developed second end of the lamp (–) models to analyze the thermal phase of Injection Stretch Greek letters Blowing Moulding (ISBM) technology. In [3] a finite ele- 2 −1 Thermal diffusivity of PVC ( m s ) ment model was realized for the thermal analysis of a pre- Lamp angular position (deg) form which rotates in front of infrared lamps located along Δ Angular extension of finite arcs on pipe outer its axis; the same industrial process has been investigated circumference (rad) also in [4] using the identical numerical approach of [3]; in ΔT Maximum temperature displacement with the latter case, however, the ray tracing method was adopted respect to the case of a perfectly uniform radia- to model the heat source term. The thermal phase represents tion (K) the most critical stage for the end-forming process of poly- ΔT R epresentative maximum temperature displace- vinyl chloride (PVC) tubes too, where one end undergoes a eval ment to be compared with the maximum allow- deformation process in order to allow piping connection. To able one (K) soften the material before deformation, the end of the tube ΔT Maximum allowable temperature displacement is slid into an oven equipped with infrared lamps and a rota- max in the heating stage of the end-forming process tion is imposed on the tube so as to ensure uniform heating. (K) Different kinds of defects (e.g. localized burns, wall thin- Emissivity of PVC (–) ning, excessive shrinkage and loss of elasticity) can occur in Angular position along pipe outer circumference the final product due to an uneven temperature distribution (rad) within the material, thus causing the pipe to be discarded Angle between the normal to cylinder outer [5]. Consequently, a fundamental parameter of a success- control surface at a point M and the line joining M to a ful end-forming process is represented by the tube angular generic point P on the ith lamp (rad) velocity. −3 Density of PVC ( kg m ) The heat transfer from rotating cylinders and between −2 −4 Stefan–Boltzmann constant ( Wm K ) concentric rotating cylinders has been the subject of several Time (s) research work, as reviewed in [6]: some authors studied the −1 Pipe angular velocity ( rad s ) problem of mixed convection on a horizontal rotating cylin- der both numerically [7], and experimentally [8]. In [9] an Subscripts experimental investigation on the convective heat transfer di Infinitesimal element on a lamp coefficient inside a rotating cylinder with an axial air flow dj Infinitesimal element on cylinder outer was conducted, while in [10] the transient heat conduction circumference in a rotating cylindrical shell exposed to a time-varying inci- i Finite extension lamp dent heat flux was numerically studied. j Finite extension arc on cylinder outer In this work, the transient analysis of the radiative heat circumference exchange between rotating PVC pipes and infrared lamps r Generic radial coordinate of a oven for end-forming process has been carried out r Pipe inner radius numerically, in order to investigate the influence of cylinder 1 3 International Journal of Energy and Environmental Engineering (2018) 9:123–134 125 angular velocity and oven configuration on the temperature softens the material thus allowing the deformation phase distribution within the tube. The approach adopted was to be carried out by means of pressurized air or mechani- the finite-element method, using COMSOL as modeling cal devices; the end of the process is a cooling step which environment; to obtain detailed temperature information in freezes the pipe in its final shape. The detailed analysis of both spatial and time domain, approaches like the lumped the end-forming heating phase is the focus of this research parameters modeling, often applied to the transient analysis work. at system level [11], could not be used. To avoid a moving mesh, pipe rotation was simulated by The oven assuming a stationary domain subjected to a time-dependent boundary condition. View factors were calculated locally as During manufacturing, heating usually takes place in ovens a function of the geometry and oven configuration and were equipped with infrared lamps arranged radially outside and expressed as a function of angular velocity. The influence inside the pipe, as shown in Fig. 1. The emitters are usually of angular velocity on temperature distribution both at the coupled quartz tubes of elliptical cross-section, which allow outer surface and within the pipe wall was investigated and to maximize the power output. These devices are chosen compared with the case of uniform heat flux to quantify the according to the type of polymer to be heated, selecting temperature unevenness. emission wavelength to match the absorption characteristics The authors began the analysis of the heating stage of of pipe material; therefore, they emit most of their radia- the end-forming process in [12], where the influence of tive power in narrow bands. Considering short wave quartz parameters such as angular velocity, convective heat transfer emitters, these have little more than 91% of their radiative coefficients and the presence of inner additional lamps on power limited to wavelengths below 4 μ m with about 62% the temperature unevenness within pipe wall was quantified below 2 μ m and about 29% within the 2–4 μ m band; there- for a fixed oven configuration, adopting the same approach fore, the emissivity of short-wave (SW) lamps can be con- proposed here. However, only one oven configuration was sidered uniform for all practical purposes [13]. In addition, considered and no information was given about how to the assumption of uniform planar source can be justified by choose the most suitable angular velocity values to carry the planar emitting surface of the luminary. Moreover, as out a proper end-forming process. In this work, two different shown in Fig. 1, the oven is usually provided with air vents oven configurations were investigated to analyze the influ- which allow the air propelled by a fan to flow through the ence of the lamps’ angular position on temperature uniform- heating chamber; thus, in combination with a proper surface ity and the results obtained were subsequently used in order temperature control, the operator can adjust the air flow to to determine the minimum values of angular velocity which prevent pipe burnout during the process. In this work, an allow to obtain the desired uniformity in temperature dis- oven with eight short-wave radially-arranged lamps has been tribution. Thus, the aim of this work is to create a design considered. Two additional lamps located inside the tube and instrument for the end-forming process of polymeric tubes, angularly displaced by 180 provide further thermal power allowing to determine the most suitable angular velocity when processing tubes with large diameters and thicknesses, values for a successful heating phase. Adding new develop- as shown in Fig. 1. A rotary motion is imposed on the tube ments to this work, like implementing a surface temperature with a certain angular velocity, which determines the angular control within the model, will lead to further analysis on the distribution of thermal radiation incident at the pipe outer end-forming heating stage, allowing to carry out a charac- surface. Infrared lamps with a rated power P = 1000 W, a terization of the process in terms of heating time. Consider- total length L = 430 mm and a heating length L = 300 mm t h ing the high pipe production rate of companies which work have been considered. in this industrial field and the high electric power installed (around 16 kW per oven), the creation of such a model can Material lead to more efficient process able to fulfil the production requirements with the least possible energy consumption. The end-forming process can be applied to pipes of different materials; in this study, the thermal analysis focused on pol- yvinyl chloride (PVC) tubes. Thermophysical characteristics −3 Outline of the end‑forming process considered for PVC are: density  = 1.44 g cm , t her mal −1 −1 conductivity k = 0.18 W m K , specific thermal capac- −1 −1 End-forming indicates a manufacturing operation through ity c = 1005 J kg K , softening temperature T = 80 C , which a tube is deformed at one end, so as to allow the surface emissivity  = 0.93 and absorbtion coefficient for −1 connection of several units to form long pipelines, used for infrared radiation K = 147 m . PVC has a semitranspar- the transport of fluids such as waste-water. First the pipe is ent behaviour in the emission band of SW lamps, mak- placed inside an oven to undergo the heating stage, which ing the absorption process of the radiation a volumetric 1 3 126 International Journal of Energy and Environmental Engineering (2018) 9:123–134 Fig. 1 Oven with 8 short-wave infrared lamps and 2 additional lamps positioned internally for larger-diameter tubes phenomenon which can be described using Lambert’s law, properties (density  , thermal conductivity k, thermal capac- as discussed in the next section. K represents a very impor- ity c , absorption coefficient in the infrared wavelengths K a p a tant material property because it directly affects the quan - and surface emissivity  ), pipe geometry (inner radius r and tity of energy absorbed by the material when considering outer radius r ) and oven configuration (lamp nominal power a certain kind of emitter. The value chosen for K has been and heating length, number of external and inner auxiliary established on the basis of information given by manufactur- lamps, lamps’ radial positions R , lamps’ angular positions ers working in this field. During the actual heating process,  , lamps’ length L, see Figs. 1, 2). While for the model with attention must be paid to the maximum temperature reached uniform radiation it is enough to calculate a global view in order to avoid thermal degradation and decomposition of factor between lamps and pipe on the basis of correlation the material; for PVC, decomposition generally manifests as available in literature [15], in order to simulate the radiative a dehydrochlorination process, which consists of hydrogen heat exchange between a rotating pipe and the lamps, the chloride production and, for commercial PVC, often starts angular distribution of the view factors as a function of tube at temperatures in the range 520–590 K, [14]. Moreover, outer radius and oven configuration must be determined, as when reaching a temperature level of about 450 K, burns will be explained in the next section. and surface whitening phenomena begin to appear, making the product unacceptable. View Factor Calculation The procedure for the calculation of local view factors for a Model given pipe geometry and oven configuration has been exten- sively reported in [12], where the local view factor along In order to analyze the effect of tube angular velocity on the the outer circumference of the pipe has been calculated on temperature distribution within the pipe wall, two differ - the basis of the theoretical fundamentals available in [15] ent models have been devised. One simulates the uniform for radiative heat exchange between two infinitely long sur - radiative heating of a static polymeric tube considering the faces. Figure 2 shows a sketch of the geometric configura- semitransparent behavior of PVC only, the other the radia- tion studied. The eight short-wave lamps in the oven for the tive heating of a rotating pipe considering the actual angu- end-forming process considered have L = 23  mm and are lar distribution of the radiation over the tube. The former positioned at a radial coordinate R = 143 mm, with a result- model can be seen as an asymptotic form of the latter (e.g. ing ratio L∕R equal to 0.16. As regards the lamps’ angular ⟶ +∞ ). For both models, the input data are the material positions  , a first analysis has been carried out considering 1 3 International Journal of Energy and Environmental Engineering (2018) 9:123–134 127 The quantity obtained in Eq. (3) represents the fraction of energy leaving an infinitesimal element on the external sur - face of the cylinder j at an angular position  which reaches the ith lamp of finite extension L. Reciprocity law has been subsequently used, as shown in Eq. (4), allowing to calculate the fraction of energy F i→dj leaving the ith lamp of finite extension L which reaches an infinitesimal element of extension r ⋅ d on the outer surface of the cylinder j at an angular position . r ⋅ d F = F (4) i→dj dj→i To have a distribution of the view factors on the outer side of the cylinder, the outer circumference must be discretized into N angular arcs, each of angular extension Δ = 2∕N . The fraction of energy leaving the ith lamp of finite extension L which reaches an angular arc of finite extension can r ⋅ Δ be calculated as: + x r 2 Fig. 2 Geometry for the calculation of local view factors along the F = dF d (5) i→j dj→di external circumference of cylinders − x To this aim, the outer circumference of the cylinder has been their current placement within a real oven (Fig. 1), in which discretized into N = 1000 arcs of equal angular extension ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ = 28 , 69 , 110 , 155 , 208 , 250 , 290 , 332 . Δ . Equation (5) has been numerically integrated by means In the following, the main steps described in [12] will be of the MATLAB function integral2, [16, 17]. Extending reported. the procedure described before to each arc, the view factor The portion of energy leaving an infinitesimal element on distribution on the outer circumference of a cylinder for a the external surface of the cylinder j at an angular position  given layout of the lamps was calculated. Figure 3 represents which reaches an infinitesimal element of extension dx on a the distribution of the local view factors for three different generic lamp i has been calculated as: external radii r in the case of a single lamp, Fig. 3a, and in the case of eight lamps angularly displaced as in the real dF = ⋅ d(sin( )) oven, Fig. 3b, when considering a ratio L∕R equal to 0.16. (1) dj→di control The validation of the numerical procedure to calculate the Referring to Fig. 2,  represents the angle between the local view factor profile has been carried out comparing the control global view factor, calculated as the sum of all local view normal to the outer surface of the pipe on a generic point M �����⃗ and the vector MP , where P is a generic point on the surface factors, with the value obtained by means of the correspond- ing expression available in the literature [15]. The numerical of the lamp. After introducing the non-dimensional quantities x = x∕R approach developed by the authors gave the same results as those obtained using the analytical expression, thus validat- and a = r ∕R , Eq. (1) takes the following form: e L ing our mathematical model. [cos( −  )− a − x(1 − acos( −  ))sin( −  )] L L L dF = (1 + g) dx dj→di 4 2 2 [1 + x + a + 2a(xsin( −  )− cos( −  )] L L Lambert’s law for the semitransparent behaviour (2) of PVC In Eq. (2) g represents a control function defined as ������⃗ �����⃗ g = sign(OM ∙ MP) , which accounts for the portion of cyl- Considering the semitransparent behaviour of PVC within inder surface in the shadow zone of the i-th lamp. the emission band of SW lamps, thermal irradiation I trans- Numerically integrating Eq. (2) between x =−L∕2R and 1 L ferred at a given radius r can be expressed as: x =+L∕2R leads to F : 2 L dj→i I = I ⋅ exp(−K ⋅ (r − r)) r r a e (6) Equation 6 describes Lambert’s law and relates the thermal F = dF (3) dj→i dj→di x irradiation I to the incident radiative flux I at the outer r re radius r , the coefficient of absorption K , the outer radius r e a e 1 3 128 International Journal of Energy and Environmental Engineering (2018) 9:123–134 0.02 source term due to internal lamps radiation in cylindrical r =75 mm coordinates can be calculated from: r =100 mm 0.018 r =125 mm ��� 0.016 q = K − ⋅ I ⋅ exp(−K ⋅ (r − r )) (8) a ri a i g Internal Lamps 0.014 I indicates the radiative heat flux incident at pipe inner sur - ri 0.012 face and r the inner radius. 0.01 Finite element model 0.008 0.006 After an expression for the calculation of local view factors along the outer circumference of the cylinder was obtained, 0.004 a finite element model accounting for cylinder rotation via 0.002 a time-dependent boundary condition applied to a stationary 0 domain was set-up in COMSOL . In the uniform radiation -150 -100 -50 050100 150 model, the heat source term depends on the radial coordinate θ (deg) only, while in the rotating cylinder model it depends both on (a) the radial and angular coordinates of a cylindrical reference system and is considered as time-varying with a time period 0.025 r =75 mm equal to 2∕ , where  is the angular velocity of the PVC r =100 mm r =125 mm e tube. The local heat source intensity at any given time instant was computed by the solver recalling the MATLAB function 0.02 written for the calculation of the local view factors. As reported in [12], the influence of axial thermal conduction on 0.015 the temperature distribution within the tube wall was inves- tigated developing a 3D model: in an opposite way to other thermal problems [18–22], simulations demonstrated how con- 0.01 jugate effects affect little the temperature field in the heated zone owing to the high conductive thermal resistance of the PVC tube. The analysis on the effects of tube angular velocity 0.005 on the temperature distribution has therefore been carried out over 2D domains. Equation (9) describes the thermal energy transport in a solid medium under transient conditions, con- -150 -100 -500 50 100 150 sidering a generic 2D domain in cylindrical coordinates, with θ (deg) r the radial coordinate, z the axial coordinate and  the angular (b) coordinate respectively. ��� Fig. 3 Local view factors. a Single lamp configuration. b Eight lamps q (r, , ) 2 2 1 T  T 1 T 1  T configuration (9) = + + + 2 2 2 k r r r r and the generic radius r. As previously stated, the radiative A comparison with the case of uniform radiative flux was heating of semitransparent media is a volumetric phenom- made in terms of the maximum temperature difference along enon and can therefore be simulated as a heat source term. the outer circumference, as well as the circumference defined Based on the discussion in [15], the heat source term due to by the mean radius and the inner circumference. The term external lamps radiation in cylindrical coordinates can be corresponding to internal generation q due to external expressed as: lamps radiation can be calculated by means of Eq.  7 and ��� depends on the radial coordinate r only in the model with q = + K ⋅ I ⋅ exp(−K ⋅ (r − r)) (7) a r a e r uniform radiation, while it is a function of r,  and time,  , when considering the view factor distribution due to pipe As discussed in “The oven” section, for tubes with large geometry and oven configuration for the calculation of I . re diameters and thicknesses additional lamps positioned on The heat generation due to internal lamps, when present, has the inner side of the pipes are used. When present, the heat 1 3 View Factor (-) View Factor (-) International Journal of Energy and Environmental Engineering (2018) 9:123–134 129 been considered uniform in  and expressed by means of Eq. Moreover, a grid-sensitivity analysis was conducted, con- (8). sidering quadrilateral Lagrange elements. To control grid When  =  = 0 s , the temperature at each point in the refinement, the number of elements on the outer circum- domain is uniform, T = 293.15  K. At the inner radius, ference of the cylinder, N , has been chosen as tunable 0 El OC assuming that no axial air flow was present, a boundary con- parameter. To establish the most suitable value of N , El OC dition of adiabaticity was considered, because of the very the temperature at the end of a 100 s simulation on a point low value of the convection heat transfer coefficient obtained placed in the middle of pipe wall ( r = r ) and at the angu- by means of the correlation experimentally given in [9] for a lar position  = 0 was monitored. The grid-sensitivity rotating horizontal cylinder. At the outer radius, a combined analysis was conducted on a tube with D = 250 mm and convective-radiative boundary condition was adopted, with thickness th = 13.4 mm since it represents one of the larg- an external convective coefficient h calculated with the cor- est processable pipe in terms of diameter and thickness and relation given by Etemad for natural convection on horizontal it is reasonably more affected by angular and radial ther - rotating cylinders [23], neglecting the presence of an axial air mal gradients than smaller pipes. In addition, three differ - flow over the pipe outer surface. In fact, the value of pipe sur - ent operating conditions were considered: stationary pipe, −1 face emissivity  and the convective heat transfer coefficients, rotating pipe with  = 1.87 rad s and pipe irradiated by a which contribute to the definition of the boundary conditions, perfectly uniform radiation. Figure 4 shows the results of do not have a significant influence on temperature unevenness the grid-sensitivity analysis; in particular, the temperature within the pipe, as shown in [12]. The temperatures of ambi- of the monitored point in the three operating conditions at ent air, T , and walls, T , were both considered at 293.15 K. the end of the simulation and the total number of elements air w in the grid, N , are reported. The case of rotating pipe El Tot represents the most critical one, showing grid independence Model reliability and sensitivity analysis for N ≥ 500 ; thus, 500 has been subsequently chosen as El OC the best value for N , since it was verified that it guaran- El OC Before starting with simulations aimed at quantifying the teed grid independence for smaller pipes too. For a pipe with temperature displacement with respect to the case of a per- outer diameter D = 250 mm and thickness th = 10.8 mm, fectly uniform radiation as a function of angular velocity which is the largest geometry investigated, a mesh formed and oven configuration, the reliability of the model has been by 3996 quadrilateral Lagrange elements and 4496 nodes investigated comparing the asymptotic solution of the time- was subsequently adopted. dependent 2D finite-element model in the case of uniform The temporal discretization method was the Alpha heat flux with the analytical solution obtained integrating the method and the maximum time-step adopted was 0.1  s, heat equation in its one-dimensional and axisymmetric form defined after a time-sensitivity analysis. Simulations aimed and in steady-state condition. Neglecting angular variations at quantifying the temperature unevenness within pipe wall in temperature field and imposing steady-state conditions, had a duration of 100 s. Eq. (9) becomes: ��� 2 q (r, , ) T 1 T (10) + + = 0 r r k Integrating Eq. (10) twice, considering external SW lamps only and assuming conditions of adiabaticity at the inner radius and a convective boundary condition at the outer radius leads to the steady-state temperature profile within pipe wall: -1 ω=0 rad s -1 I r 1 1 re i ω=1.87 rad s T(r)= T + I ⋅ ( + )− ⋅ exp(−K (r − r )) Uniform 2000 air re a e i kK h h r a e e e El Tot I r I re i re + ⋅ exp(−K (r − r )) ⋅ ln( )− ⋅ exp(−K (r − r)) a e i a e 340 k r kK e a (11) 100 150 200 250 300 350 400 450 500 550600 The asymptotic solution obtained by means of the time- N (-) El OC dependent 2D finite-element model showed perfect agree- ment with the analytical steady-state solution described by Fig. 4 Grid sensitivity analysis. Temperature on a point placed in the Eq. (11), giving maximum discrepancies lower than 0.03% middle of pipe wall (r = r and  = 0 ) and total number of elements over the whole domain. in the grid. D = 250 mm, th = 13.4 mm 1 3 T (K) rm El Tot 130 International Journal of Energy and Environmental Engineering (2018) 9:123–134 Results τ=0 s τ=20 s τ=40 s τ=60 s In Fig. 5 the temperature profile along the outer circumfer - τ=80 s τ=100 s ence of a stationary PVC pipe in a oven like that shown in Fig.  1 has been compared at different times with that due to uniform radiation. A tube with D = 250 mm and th = 10.8 mm has been investigated, since it represents one of the largest customary tube sizes in end-forming processes. −2 −1 An external convection coefficient h = 9 Wm K was calculated. Two additional internal SW lamps have been considered, in order to make the heating process quicker and more uniform, as usually happens in practice. The actual temperature on the outer circumference is strongly affected by the view factor distribution: at  = 40 s, the maximum temperature exceeds 500 K, closely approaching the limit 115 116 117 118 119 120 121 122123 124125 r (mm) of dehydrochlorination, with a maximum temperature dif- ference of 120 K with respect to the uniform radiation case. Fig. 6 Temperature distribution within pipe wall. D = 250  mm, This result fully justifies the use of handling systems for the e −1 th = 10.8 mm,  = 1.87 rad s rotation of tubes, in order to obtain uniform heating. Simulations at different angular velocities were carried Figure 7 depicts the maximum temperature displacement out for the same PVC tube. Figure 6 shows the temperature along the outer circumference, ΔT : as expected, this quan- profile within pipe wall when considering the maximum −1 tity decreases with increasing angular velocity. Moreover, angular velocity investigated (  = 1.87 rad s ). It can be curves show a nonlinear dependence of ΔT on the angular clearly seen that the maximum temperature shifts inwards velocity: the reduction of ΔT with increasing angular veloc- from the outer wall, because of the semitransparent behav- ity is more pronounced for low values of  . This implies ior of PVC with respect to short-wave radiation. As time the existence of a limiting angular velocity, over which a increases, the maximum temperature moves toward the inner further increase does not bring substantial reductions in the radius, because of the increase in convective heat exchange maximum temperature displacement. at the outer surface. The flat trend of curves at the inner Figure 8a, b show the same quantity along the circum- radius is a consequence of the assumption of adiabaticity, ference defined by the mean radius ( ΔT ) and along the because of the small influence of the drag effect caused by inner circumference ( ΔT ). A reduction in the maximum the tube rotation on the air motion, when axial air flow is absent. -1 ω=0.468 rad s -1 ω=0.935 rad s -1 -1 ω=1.400 rad s ω=0 rad s , τ=0 s 2.5 -1 -1 ω=0 rad s , τ=20 s ω=1.870 rad s -1 550 ω=0 rad s , τ=40 s Uniform, τ=0 s Uniform, τ=20 s Uniform, τ=40 s 1.5 350 0.5 0102030405060708090 100 012345 6 θ (rad) τ (s) Fig. 5 Temperature profile along the outer circumference of a station- Fig. 7 Maximum temperature displacement along the outer circum- ary cylinder. D = 250 mm, th = 10.8 mm ference. D = 250 mm, th = 10.8 mm e e 1 3 T (K) re ΔT (K) T (K) re r International Journal of Energy and Environmental Engineering (2018) 9:123–134 131 3 1 -1 -1 ω=0.468 rad s ω=0.468 rad s -1 -1 0.9 ω=0.935 rad s ω=0.702 rad s -1 -1 ω=1.400 rad s ω=0.935 rad s 2.5 -1 ω=1.870 rad s 0.8 0.7 0.6 1.5 0.5 0.4 0.3 0.2 0.5 0.1 0 0 0102030405060708090 100 0102030405060708090 100 τ (s) τ (s) (a) Fig. 9 Maximum temperature displacement along the outer circum- ference. D = 125 mm and th = 7.4 mm thick -1 ω=0.468 rad s -1 ω=0.935 rad s -1 ω=1.400 rad s 2.5 -1 ω=1.870 rad s angular velocity, a strong reduction in ΔT emerges with decreasing external diameter; this can be justified consider - ing the local view factor distribution, which becomes less affected by the angular position as the diameter of the pipe 1.5 decreases; see Fig. 3b. This means that for pipes of smaller dimensions, the limiting value of angular velocity takes lower values. A critical angular velocity can be defined as the mini- mum value of rotational speed which allows to maintain 0.5 the maximum temperature displacement below a certain threshold. For a given oven configuration, the model can be used to determine this critical value for different tube geom- 0102030405060708090 100 τ (s) etries, after a maximum allowable temperature displace- (b) ment ( ΔT ) along the outer circumference with respect max to a perfectly uniform heating process has been established. To this aim, for each tube geometry and angular velocity Fig. 8 Maximum temperature displacement along the circumference defined by the mean radius (a) and along the inner circumference (b). investigated, ΔT must be compared to ΔT , whose value max r D = 250 mm, th = 10.8 mm is time-dependent, as clearly demonstrated in Figs. 7 and 9, because of the combined effects of conduction and convec- temperature displacement when moving to lower radii tion which become more pronounced at high temperatures, appears for each angular velocity investigated. This behavior favoring reduction in peak values of ΔT . In order to estab- is more pronounced at low angular velocities and it is due to lish a significant instantaneous value of ΔT for the com- the low thermal diffusivity of the material which dampens parison with ΔT , a time interval Δ =[ − p,  + p] has max 0 0 the amplitude of thermal perturbations and imposes a time been considered, where p is lamps facing period, defined as delay to it, which causes an increase in the temperature dis- p = 2∕( ⋅ N ) , where N is the number of external lamp lamp placement at the inner radius when time progresses. lamps in the oven, and  is the instant of time in which the A set of simulations with the same boundary condi- mean temperature along the outer circumference reaches the tions but without internal lamps was run for a tube with value T = 433.15 K, which represents a credible value of off D = 125 mm and th = 7.4 mm, which represents a medium- the switch-off temperature of lamps, when these are con- sized tube. Figure  9 shows the trend of ΔT for different trolled by a traditional pyrometer. The comparison value has values of angular velocities. Comparing the curves in Fig. 9 consequently been calculated as ΔT = max{ΔT()} , with eval with those in Fig.  7 and considering the same values of  ∈Δ . The value of ΔT can be established arbitrarily in max 1 3 ΔT (K) ΔT (K) ri rm ΔT (K) re 132 International Journal of Energy and Environmental Engineering (2018) 9:123–134 2.5 the process design stage: in this work, a maximum tempera- Not Equidistant ture displacement ΔT = 1 K has been considered. The max Equidistant analysis on critical angular velocity has been carried out on ΔT =1 K max three different pipe geometries and considering two different oven configurations in order to investigate the effects of the reciprocal angular positions of the lamps on critical veloc- 1.5 ity values. As to the latter aspect, the oven with external lamps positioned as described in “View factor calculation” section has been compared to one with eight equally-dis- placed SW lamps. Figure 10a–c show the trends of ΔT eval for three pipe geometries and for the two configurations of 0.5 lamps considered, comparing it to ΔT . It is clear how, max for a given geometry and oven configuration, the value of ΔT decreases when increasing angular velocity and tends eval 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -1 to flatten. The intersection between the curves relative to ω (rad s ) ΔT and ΔT identifies the value of critical angular eval max (a) velocity, over which a further increase in angular velocity 2.5 does not lead to significant improvements in the uniformity Not Equidistant of the temperature distribution. It is clear how higher radii Equidistant ΔT =1 K necessitate higher angular velocities to obtain a maximum max temperature displacement of 1 K with respect to the case of uniform radiation. Moreover, fixing pipe geometry, a sub- 1.5 stantial reduction in the values of ΔT and consequently eval in critical angular velocity is obtained when moving to a configuration with eight lamps displaced at uniform angular intervals. This tendency is stronger for pipes of smaller size. The larger reduction in critical angular velocity high- lighted for smaller pipes in the case of lamps equally-dis- 0.5 placed can be justified observing the heat flux profile on the pipes’ outer surface for the three geometries investigated and for the oven configurations studied, shown in Fig.  11a–c. 0 0.5 11.5 -1 In fact, with reference to Fig. 2, for a given lamps radial ω (rad s ) position R and extension L, it is clear how pipes with low (b) external diameter are more sensitive to the angular position 2.5 of the lamps, because of the overlap of the heat fluxes gen- Not Equidistant erated by every single lamp, which increases the deviation Equidistant ΔT =1 K max from the uniform radiation. 1.5 Conclusions A 2D finite-element model for the transient analysis of the radiative heat exchange between a rotating polymeric cylin- drical pipe and the SW infrared lamps of an oven for end- 0.5 forming has been devised in order to investigate the influence of cylinder angular velocity and oven configuration on the temperature distribution within the tube. Results obtained for different rotational velocities were compared with the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 -1 ω (rad s ) case of a uniformly irradiated tube. A MATLAB function (c) for the calculation of local view factor distribution given pipe’s outer radius and oven configuration has been devel- Fig. 10 Effect of the lamps’ angular positions on critical velocity. a Big-size oped. The code was validated comparing the global view pipe: D = 250  mm th = 10.8  mm. b Medium-size pipe: D = 180  mm e e factor calculated as the sum of all local view factors with the th = 10.7 mm. c Small-size pipe: D = 125 mm th = 7.4 mm 1 3 ΔT (K) ΔT (K) ΔT (K) eval eval eval International Journal of Energy and Environmental Engineering (2018) 9:123–134 133 Fig. 11 Effect of the lamps’ angular positions on heat flux distribu- ▸ tion along the outer circumference of pipes. a External diameter of Equidistant Not Equidistant 250  mm and thickness of 10.8  mm. b External diameter of 180  mm Uniform and thickness of 10.7 mm. c External diameter of 125 mm and thick- ness of 7.4 mm value obtained by means of the corresponding correlation given in [15]. Comparison between the temperature distri- butions on the outer circumference of a static tube obtained in the cases of uniform and actual radiation distribution justifies the use of rotation devices, in order to avoid high temperature peaks which could lead to the thermal failure of the material. Simulations carried out on different pipe geom- etries at varying values of angular velocity demonstrated -3 -2 -1 012 3 how increasing angular speed decreases the maximum tem- θ (rad) perature displacement with respect to the case of a uniformly (a) irradiated tube. Results showed a nonlinear dependence between maximum temperature displacement and angular Equidistant velocity, thus demonstrating the existence of a limiting value Not Equidistant over which a further increase in angular velocity brings little Uniform reduction in maximum temperature difference. The same analysis also highlighted how the low thermal diffusivity of the material dampens the maximum temperature difference when moving from the outer to the inner side of the pipe. For a fixed value of the angular velocity, lower values of maximum temperature difference were obtained for smaller tubes, because of the more uniform distribution of the local view factors with respect to the angular coordinate. Moreo- ver, as highlighted in [12], parameters which contribute to 10 the definition of boundary conditions, such as convective heat transfer coefficients and PVC surface emissivity, do not -3 -2 -1 012 3 influence temperature unevenness significantly over the pipe θ (rad) wall, since they just affect the mean temperature reached (b) at a certain radial position at a given time without varying the maximum temperature displacement with respect to the case of a perfectly uniform radiation. The model can be used Equidistant Not Equidistant as a tool to characterize the end-forming process of poly- Uniform meric tubes in terms of critical angular velocity, for different materials and oven configurations. Simulations showed that, for a given oven configuration, higher critical velocities are obtained for large-sized pipes, because of the much more uneven distribution of the heat flux on the outer circumfer - ence. Also, comparisons between two different oven con- figurations showed how relative angular positions of lamps 20 could lead to significant modification of the values of critical angular velocity, all the more for smaller pipes. To the best knowledge of the authors, this is the first numerical model able to characterize the heating stage of -3 -2 -1 012 3 the end-forming process in terms of critical angular velocity; θ (rad) an experimental campaign is planned to validate the numeri- (c) cal predictions and fine-tune the model. Finally, it should be mentioned that the addition of surface temperature control in the model can allow the design of optimal heating cycles 1 3 -2 -2 -2 Heat Flux (kW ·m ) Heat Flux (kW ·m ) Heat Flux (kW ·m ) 134 International Journal of Energy and Environmental Engineering (2018) 9:123–134 11. Lucchi, M., Lorenzini, M., Valdiserri, P.: Energy performance to achieve the desired pipe temperature in the most efficient of a ventilation system for a block of apartments with a ground way, meeting production requirements whilst saving energy source heat pump as generation system. J. Phys. Conf. Ser. 796(1), at the same time. 012034 (2017) 12. Lucchi, M., Lorenzini, M.: Transient analysis of the radiative heat- Open Access This article is distributed under the terms of the Creative ing of rotating PVC pipes in a oven for end-forming process. Appl. Commons Attribution 4.0 International License (http ://crea tive comm Therm. Eng. 129, 84 (2018) ons.org/licenses /b y/4.0/), which permits unrestricted use, distribution, 13. Heraeus group catalog: Infrared emitters for industrial processes. and reproduction in any medium, provided you give appropriate credit http s://apps .her a eus.com/IR_Pr od ucts _EN/mobi le/inde x.html to the original author(s) and the source, provide a link to the Creative #p=1. Accessed 16 Jan 2018 Commons license, and indicate if changes were made. 14. Yu, J., Sun, L., Ma, C., Qiao, Y., Yao, H.: Thermal degradation of PVC: a review. Waste Manag. 48, 300 (2016) 15. Howell, J., Siegel, R., Menguc, M.P.: Thermal Radiation Heat Transfer. CRC Press - Taylor & Francis Group, Boca Raton (2010) References 16. Shampine, L.: Vectorized adaptive quadrature in MATLAB. J. Comput. Appl. Math. 211, 131 (2008) 1. Bush, S.: Scale, order and complexity in polymer processing. 17. Shampine, L.: Matlab program for quadrature in 2D. Appl. Math. Proc. Inst. Mech. Eng. Part E J. Process. Mech. Eng. 214, 217 Comput. 202, 266 (2008) (2000) 18. Ahmad, W., Syed, K., Ishaq, M., Hassan, A., Iqbal, Z.: Numerical 2. Throne, J.L.: Technology of Thermoforming. Hanser, Munich study of conjugate heat transfer in a double-pipe with exponential (1996) fins using DGFEM. Appl. Therm. Eng. 111, 1184 (2017) 3. Luo, Y., Chevalier, L., Utheza, F., Nicolas, X.: Simplified model- 19. Aneesh, A., Sharma, A., Srivastava, A., Vyas, K., Chaudhuri, P.: ling of the infrared heating involving the air convection effect Thermal-hydraulic characteristics and performance of 3D straight before the injection stretch blowing moulding of PET preform. channel based printed circuit heat exchanger. Appl. Therm. Eng. Key Eng. Mater. 611–612, 844 (2014) 98, 474 (2016) 4. Cosson, B., Schmidt, F., Maoult, Y.L., Bordival, M.: Infrared heat- 20. Henning, T., Brandner, J., Schubert, K., Lorenzini, M., Morini, ing stage simulation of semi-transparent media (PET) using ray G.: Low-frequency instabilities in the operation of metallic multi- tracing method. Int. J. Mater. Form. 4(1), 1 (2011) microchannel evaporators. Heat Transf. Eng. 28, 834 (2007) 5. Sikora, R.: The effect of heating of PVC pipes on selected 21. Morini, G., Yang, Y., Lorenzini, M.: Experimental analysis of mechanical properties of pipe bells. Polimery 43, 384 (1998) gas micro-convection through commercial microtubes. Exp. Heat 6. Fénot, M., Bertin, Y., Dorignac, E., Lalizel, G.: A review of heat Transf. 25, 151 (2012) transfer between concentric rotating cylinders with or without 22. Yang, Y., Chalabi, H., Lorenzini, M., Morini, G.: The effect on the axial flow. Int. J. Therm. Sci. 50, 1138 (2011) Nusselt number of the nonlinear axial temperature distribution of 7. Costa, V., Raimundo, A.: Steady mixed convection in a differ - gas flows through microtubes. 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Effects of pipe angular velocity and oven configuration on tube temperature distribution in the radiative heating of PVC pipes

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Engineering; Renewable and Green Energy
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Abstract

Several manufacturing processes in polymer industry aim at obtaining products by deforming preforms or sheets after a heat- ing process. A thorough knowledge of the operating parameters of such heating processes is fundamental to fulfill the often high production requirements with the least energy consumption and to avoid unacceptable defects in the final product. A common example of such an application is the end-forming process of polyvinyl chloride (PVC) tubes, which are enlarged at one end in order to allow pipes connections. The heating phase which comes before the deformation process is usually carried out in ovens equipped with short wave infrared lamps; to ensure uniform heating, pipes rotate with a given angular veloc- ity, which represents a fundamental parameter for the success of the whole manufacturing process. In this work, a transient analysis of the radiative heat exchange between rotating PVC pipes and infrared lamps in an oven for end-forming process has been conducted by means of a finite element model, in order to investigate the influence of cylinder angular velocity on the temperature distribution in the tube. Local view factors have been calculated for different oven configurations and have been expressed as a function of angular velocity, allowing pipe rotation to be simulated as a time-dependent boundary condi- tion, instead of using a moving mesh. Simulations were carried out for different tubes geometries and angular velocities and results were compared with the case of a uniformly irradiated tube in terms of temperature displacement. For a given oven configuration, the results obtained by the numerical model can be used to find a critical angular velocity over which further increase does not lead to appreciable improvements in temperature evenness. The effect of the lamps’ relative position was also investigated, showing a significant influence on critical angular velocities obtained. The model realized represents a potential tool to characterize the end-forming process in terms of critical angular velocity, leading to reductions in machine set-up time and product waste due to thermal failure. Keywords Pipes end-forming · Radative heating · Finite-element modeling List of symbols F View factor between finite extension elements i→j a Ratio between pipe external radius and lamp (–) radial position (–) g Control function (–) −1 −1 −2 −1 c Specific thermal capacity of PVC ( J kg K ) h Convection heat transfer coefficient ( Wm K ) −2 dF View factor between infinitesimal elements (–) I Heat flux ( Wm ) dj→di −1 −1 dF View factor between an infinitesimal element and k Thermal conductivity of PVC ( Wm K ) dj→i −1 a finite extension element (–) K Absorbtion coefficient of PVC ( m ) dF View factor between a finite extension element L Lamp extension in the plane perpendicular to i→dj and an infinitesimal element (–) cylinder axis (mm) L Lamp heating length (mm) L Lamp total length (mm) N Number of finite extension arcs along pipe outer * Marco Lorenzini circumference (–) marco.lorenzini@unibo.it N Number of external lamps in the oven (–) lamp Michael Lucchi N Number of quadrilateral elements on the outer El OC michael.lucchi2@unibo.it circumference of the pipe (–) N Total number of quadrilateral elements in the Department of Industrial Engineering, University El Tot of Bologna, via Fontanelle 40, Forlí, Italy grid (–) Vol.:(0123456789) 1 3 124 International Journal of Energy and Environmental Engineering (2018) 9:123–134 p Lamp facing period (s) r Pipe external radius P Lamp rated power (W) r Pipe mean radius r m −3 q Heat generation term ( Wm ) r Generic radial coordinate (mm) Introduction R Lamp radial position (mm) T Temperature (K) In the second half of the twentieth century, a rapid increase T Lamps switch off temperature when they are con- off in the production scales and in the variety and complex- trolled by a traditional pyrometer (K) ity of manufacturing processes characterized the evolution T Softening temperature of PVC (K) of the polymer industry [1]. In industrial applications like th Pipe thickness (mm) thermoforming, aimed at obtaining products by deform- x Generic coordinate along lamp extension (mm) ing polymeric preforms or sheets, the study of the ther- x Non-dimensional coordinate along lamp exten- mal stage allows to get a successful forming process and sion in a transversal plane perpendicular to directly affects the mechanical properties of the end prod- cylinder axis (–) uct [2]. Moreover, studying the heating stage in the design x Non-dimensional coordinate x calculated at the phase of such processes can lead to a significant increase first end of the lamp (–) in the process efficiency, with consequent energy savings x Non-dimensional coordinate x calculated at the and reduced environmental impact. Many authors developed second end of the lamp (–) models to analyze the thermal phase of Injection Stretch Greek letters Blowing Moulding (ISBM) technology. In [3] a finite ele- 2 −1 Thermal diffusivity of PVC ( m s ) ment model was realized for the thermal analysis of a pre- Lamp angular position (deg) form which rotates in front of infrared lamps located along Δ Angular extension of finite arcs on pipe outer its axis; the same industrial process has been investigated circumference (rad) also in [4] using the identical numerical approach of [3]; in ΔT Maximum temperature displacement with the latter case, however, the ray tracing method was adopted respect to the case of a perfectly uniform radia- to model the heat source term. The thermal phase represents tion (K) the most critical stage for the end-forming process of poly- ΔT R epresentative maximum temperature displace- vinyl chloride (PVC) tubes too, where one end undergoes a eval ment to be compared with the maximum allow- deformation process in order to allow piping connection. To able one (K) soften the material before deformation, the end of the tube ΔT Maximum allowable temperature displacement is slid into an oven equipped with infrared lamps and a rota- max in the heating stage of the end-forming process tion is imposed on the tube so as to ensure uniform heating. (K) Different kinds of defects (e.g. localized burns, wall thin- Emissivity of PVC (–) ning, excessive shrinkage and loss of elasticity) can occur in Angular position along pipe outer circumference the final product due to an uneven temperature distribution (rad) within the material, thus causing the pipe to be discarded Angle between the normal to cylinder outer [5]. Consequently, a fundamental parameter of a success- control surface at a point M and the line joining M to a ful end-forming process is represented by the tube angular generic point P on the ith lamp (rad) velocity. −3 Density of PVC ( kg m ) The heat transfer from rotating cylinders and between −2 −4 Stefan–Boltzmann constant ( Wm K ) concentric rotating cylinders has been the subject of several Time (s) research work, as reviewed in [6]: some authors studied the −1 Pipe angular velocity ( rad s ) problem of mixed convection on a horizontal rotating cylin- der both numerically [7], and experimentally [8]. In [9] an Subscripts experimental investigation on the convective heat transfer di Infinitesimal element on a lamp coefficient inside a rotating cylinder with an axial air flow dj Infinitesimal element on cylinder outer was conducted, while in [10] the transient heat conduction circumference in a rotating cylindrical shell exposed to a time-varying inci- i Finite extension lamp dent heat flux was numerically studied. j Finite extension arc on cylinder outer In this work, the transient analysis of the radiative heat circumference exchange between rotating PVC pipes and infrared lamps r Generic radial coordinate of a oven for end-forming process has been carried out r Pipe inner radius numerically, in order to investigate the influence of cylinder 1 3 International Journal of Energy and Environmental Engineering (2018) 9:123–134 125 angular velocity and oven configuration on the temperature softens the material thus allowing the deformation phase distribution within the tube. The approach adopted was to be carried out by means of pressurized air or mechani- the finite-element method, using COMSOL as modeling cal devices; the end of the process is a cooling step which environment; to obtain detailed temperature information in freezes the pipe in its final shape. The detailed analysis of both spatial and time domain, approaches like the lumped the end-forming heating phase is the focus of this research parameters modeling, often applied to the transient analysis work. at system level [11], could not be used. To avoid a moving mesh, pipe rotation was simulated by The oven assuming a stationary domain subjected to a time-dependent boundary condition. View factors were calculated locally as During manufacturing, heating usually takes place in ovens a function of the geometry and oven configuration and were equipped with infrared lamps arranged radially outside and expressed as a function of angular velocity. The influence inside the pipe, as shown in Fig. 1. The emitters are usually of angular velocity on temperature distribution both at the coupled quartz tubes of elliptical cross-section, which allow outer surface and within the pipe wall was investigated and to maximize the power output. These devices are chosen compared with the case of uniform heat flux to quantify the according to the type of polymer to be heated, selecting temperature unevenness. emission wavelength to match the absorption characteristics The authors began the analysis of the heating stage of of pipe material; therefore, they emit most of their radia- the end-forming process in [12], where the influence of tive power in narrow bands. Considering short wave quartz parameters such as angular velocity, convective heat transfer emitters, these have little more than 91% of their radiative coefficients and the presence of inner additional lamps on power limited to wavelengths below 4 μ m with about 62% the temperature unevenness within pipe wall was quantified below 2 μ m and about 29% within the 2–4 μ m band; there- for a fixed oven configuration, adopting the same approach fore, the emissivity of short-wave (SW) lamps can be con- proposed here. However, only one oven configuration was sidered uniform for all practical purposes [13]. In addition, considered and no information was given about how to the assumption of uniform planar source can be justified by choose the most suitable angular velocity values to carry the planar emitting surface of the luminary. Moreover, as out a proper end-forming process. In this work, two different shown in Fig. 1, the oven is usually provided with air vents oven configurations were investigated to analyze the influ- which allow the air propelled by a fan to flow through the ence of the lamps’ angular position on temperature uniform- heating chamber; thus, in combination with a proper surface ity and the results obtained were subsequently used in order temperature control, the operator can adjust the air flow to to determine the minimum values of angular velocity which prevent pipe burnout during the process. In this work, an allow to obtain the desired uniformity in temperature dis- oven with eight short-wave radially-arranged lamps has been tribution. Thus, the aim of this work is to create a design considered. Two additional lamps located inside the tube and instrument for the end-forming process of polymeric tubes, angularly displaced by 180 provide further thermal power allowing to determine the most suitable angular velocity when processing tubes with large diameters and thicknesses, values for a successful heating phase. Adding new develop- as shown in Fig. 1. A rotary motion is imposed on the tube ments to this work, like implementing a surface temperature with a certain angular velocity, which determines the angular control within the model, will lead to further analysis on the distribution of thermal radiation incident at the pipe outer end-forming heating stage, allowing to carry out a charac- surface. Infrared lamps with a rated power P = 1000 W, a terization of the process in terms of heating time. Consider- total length L = 430 mm and a heating length L = 300 mm t h ing the high pipe production rate of companies which work have been considered. in this industrial field and the high electric power installed (around 16 kW per oven), the creation of such a model can Material lead to more efficient process able to fulfil the production requirements with the least possible energy consumption. The end-forming process can be applied to pipes of different materials; in this study, the thermal analysis focused on pol- yvinyl chloride (PVC) tubes. Thermophysical characteristics −3 Outline of the end‑forming process considered for PVC are: density  = 1.44 g cm , t her mal −1 −1 conductivity k = 0.18 W m K , specific thermal capac- −1 −1 End-forming indicates a manufacturing operation through ity c = 1005 J kg K , softening temperature T = 80 C , which a tube is deformed at one end, so as to allow the surface emissivity  = 0.93 and absorbtion coefficient for −1 connection of several units to form long pipelines, used for infrared radiation K = 147 m . PVC has a semitranspar- the transport of fluids such as waste-water. First the pipe is ent behaviour in the emission band of SW lamps, mak- placed inside an oven to undergo the heating stage, which ing the absorption process of the radiation a volumetric 1 3 126 International Journal of Energy and Environmental Engineering (2018) 9:123–134 Fig. 1 Oven with 8 short-wave infrared lamps and 2 additional lamps positioned internally for larger-diameter tubes phenomenon which can be described using Lambert’s law, properties (density  , thermal conductivity k, thermal capac- as discussed in the next section. K represents a very impor- ity c , absorption coefficient in the infrared wavelengths K a p a tant material property because it directly affects the quan - and surface emissivity  ), pipe geometry (inner radius r and tity of energy absorbed by the material when considering outer radius r ) and oven configuration (lamp nominal power a certain kind of emitter. The value chosen for K has been and heating length, number of external and inner auxiliary established on the basis of information given by manufactur- lamps, lamps’ radial positions R , lamps’ angular positions ers working in this field. During the actual heating process,  , lamps’ length L, see Figs. 1, 2). While for the model with attention must be paid to the maximum temperature reached uniform radiation it is enough to calculate a global view in order to avoid thermal degradation and decomposition of factor between lamps and pipe on the basis of correlation the material; for PVC, decomposition generally manifests as available in literature [15], in order to simulate the radiative a dehydrochlorination process, which consists of hydrogen heat exchange between a rotating pipe and the lamps, the chloride production and, for commercial PVC, often starts angular distribution of the view factors as a function of tube at temperatures in the range 520–590 K, [14]. Moreover, outer radius and oven configuration must be determined, as when reaching a temperature level of about 450 K, burns will be explained in the next section. and surface whitening phenomena begin to appear, making the product unacceptable. View Factor Calculation The procedure for the calculation of local view factors for a Model given pipe geometry and oven configuration has been exten- sively reported in [12], where the local view factor along In order to analyze the effect of tube angular velocity on the the outer circumference of the pipe has been calculated on temperature distribution within the pipe wall, two differ - the basis of the theoretical fundamentals available in [15] ent models have been devised. One simulates the uniform for radiative heat exchange between two infinitely long sur - radiative heating of a static polymeric tube considering the faces. Figure 2 shows a sketch of the geometric configura- semitransparent behavior of PVC only, the other the radia- tion studied. The eight short-wave lamps in the oven for the tive heating of a rotating pipe considering the actual angu- end-forming process considered have L = 23  mm and are lar distribution of the radiation over the tube. The former positioned at a radial coordinate R = 143 mm, with a result- model can be seen as an asymptotic form of the latter (e.g. ing ratio L∕R equal to 0.16. As regards the lamps’ angular ⟶ +∞ ). For both models, the input data are the material positions  , a first analysis has been carried out considering 1 3 International Journal of Energy and Environmental Engineering (2018) 9:123–134 127 The quantity obtained in Eq. (3) represents the fraction of energy leaving an infinitesimal element on the external sur - face of the cylinder j at an angular position  which reaches the ith lamp of finite extension L. Reciprocity law has been subsequently used, as shown in Eq. (4), allowing to calculate the fraction of energy F i→dj leaving the ith lamp of finite extension L which reaches an infinitesimal element of extension r ⋅ d on the outer surface of the cylinder j at an angular position . r ⋅ d F = F (4) i→dj dj→i To have a distribution of the view factors on the outer side of the cylinder, the outer circumference must be discretized into N angular arcs, each of angular extension Δ = 2∕N . The fraction of energy leaving the ith lamp of finite extension L which reaches an angular arc of finite extension can r ⋅ Δ be calculated as: + x r 2 Fig. 2 Geometry for the calculation of local view factors along the F = dF d (5) i→j dj→di external circumference of cylinders − x To this aim, the outer circumference of the cylinder has been their current placement within a real oven (Fig. 1), in which discretized into N = 1000 arcs of equal angular extension ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ = 28 , 69 , 110 , 155 , 208 , 250 , 290 , 332 . Δ . Equation (5) has been numerically integrated by means In the following, the main steps described in [12] will be of the MATLAB function integral2, [16, 17]. Extending reported. the procedure described before to each arc, the view factor The portion of energy leaving an infinitesimal element on distribution on the outer circumference of a cylinder for a the external surface of the cylinder j at an angular position  given layout of the lamps was calculated. Figure 3 represents which reaches an infinitesimal element of extension dx on a the distribution of the local view factors for three different generic lamp i has been calculated as: external radii r in the case of a single lamp, Fig. 3a, and in the case of eight lamps angularly displaced as in the real dF = ⋅ d(sin( )) oven, Fig. 3b, when considering a ratio L∕R equal to 0.16. (1) dj→di control The validation of the numerical procedure to calculate the Referring to Fig. 2,  represents the angle between the local view factor profile has been carried out comparing the control global view factor, calculated as the sum of all local view normal to the outer surface of the pipe on a generic point M �����⃗ and the vector MP , where P is a generic point on the surface factors, with the value obtained by means of the correspond- ing expression available in the literature [15]. The numerical of the lamp. After introducing the non-dimensional quantities x = x∕R approach developed by the authors gave the same results as those obtained using the analytical expression, thus validat- and a = r ∕R , Eq. (1) takes the following form: e L ing our mathematical model. [cos( −  )− a − x(1 − acos( −  ))sin( −  )] L L L dF = (1 + g) dx dj→di 4 2 2 [1 + x + a + 2a(xsin( −  )− cos( −  )] L L Lambert’s law for the semitransparent behaviour (2) of PVC In Eq. (2) g represents a control function defined as ������⃗ �����⃗ g = sign(OM ∙ MP) , which accounts for the portion of cyl- Considering the semitransparent behaviour of PVC within inder surface in the shadow zone of the i-th lamp. the emission band of SW lamps, thermal irradiation I trans- Numerically integrating Eq. (2) between x =−L∕2R and 1 L ferred at a given radius r can be expressed as: x =+L∕2R leads to F : 2 L dj→i I = I ⋅ exp(−K ⋅ (r − r)) r r a e (6) Equation 6 describes Lambert’s law and relates the thermal F = dF (3) dj→i dj→di x irradiation I to the incident radiative flux I at the outer r re radius r , the coefficient of absorption K , the outer radius r e a e 1 3 128 International Journal of Energy and Environmental Engineering (2018) 9:123–134 0.02 source term due to internal lamps radiation in cylindrical r =75 mm coordinates can be calculated from: r =100 mm 0.018 r =125 mm ��� 0.016 q = K − ⋅ I ⋅ exp(−K ⋅ (r − r )) (8) a ri a i g Internal Lamps 0.014 I indicates the radiative heat flux incident at pipe inner sur - ri 0.012 face and r the inner radius. 0.01 Finite element model 0.008 0.006 After an expression for the calculation of local view factors along the outer circumference of the cylinder was obtained, 0.004 a finite element model accounting for cylinder rotation via 0.002 a time-dependent boundary condition applied to a stationary 0 domain was set-up in COMSOL . In the uniform radiation -150 -100 -50 050100 150 model, the heat source term depends on the radial coordinate θ (deg) only, while in the rotating cylinder model it depends both on (a) the radial and angular coordinates of a cylindrical reference system and is considered as time-varying with a time period 0.025 r =75 mm equal to 2∕ , where  is the angular velocity of the PVC r =100 mm r =125 mm e tube. The local heat source intensity at any given time instant was computed by the solver recalling the MATLAB function 0.02 written for the calculation of the local view factors. As reported in [12], the influence of axial thermal conduction on 0.015 the temperature distribution within the tube wall was inves- tigated developing a 3D model: in an opposite way to other thermal problems [18–22], simulations demonstrated how con- 0.01 jugate effects affect little the temperature field in the heated zone owing to the high conductive thermal resistance of the PVC tube. The analysis on the effects of tube angular velocity 0.005 on the temperature distribution has therefore been carried out over 2D domains. Equation (9) describes the thermal energy transport in a solid medium under transient conditions, con- -150 -100 -500 50 100 150 sidering a generic 2D domain in cylindrical coordinates, with θ (deg) r the radial coordinate, z the axial coordinate and  the angular (b) coordinate respectively. ��� Fig. 3 Local view factors. a Single lamp configuration. b Eight lamps q (r, , ) 2 2 1 T  T 1 T 1  T configuration (9) = + + + 2 2 2 k r r r r and the generic radius r. As previously stated, the radiative A comparison with the case of uniform radiative flux was heating of semitransparent media is a volumetric phenom- made in terms of the maximum temperature difference along enon and can therefore be simulated as a heat source term. the outer circumference, as well as the circumference defined Based on the discussion in [15], the heat source term due to by the mean radius and the inner circumference. The term external lamps radiation in cylindrical coordinates can be corresponding to internal generation q due to external expressed as: lamps radiation can be calculated by means of Eq.  7 and ��� depends on the radial coordinate r only in the model with q = + K ⋅ I ⋅ exp(−K ⋅ (r − r)) (7) a r a e r uniform radiation, while it is a function of r,  and time,  , when considering the view factor distribution due to pipe As discussed in “The oven” section, for tubes with large geometry and oven configuration for the calculation of I . re diameters and thicknesses additional lamps positioned on The heat generation due to internal lamps, when present, has the inner side of the pipes are used. When present, the heat 1 3 View Factor (-) View Factor (-) International Journal of Energy and Environmental Engineering (2018) 9:123–134 129 been considered uniform in  and expressed by means of Eq. Moreover, a grid-sensitivity analysis was conducted, con- (8). sidering quadrilateral Lagrange elements. To control grid When  =  = 0 s , the temperature at each point in the refinement, the number of elements on the outer circum- domain is uniform, T = 293.15  K. At the inner radius, ference of the cylinder, N , has been chosen as tunable 0 El OC assuming that no axial air flow was present, a boundary con- parameter. To establish the most suitable value of N , El OC dition of adiabaticity was considered, because of the very the temperature at the end of a 100 s simulation on a point low value of the convection heat transfer coefficient obtained placed in the middle of pipe wall ( r = r ) and at the angu- by means of the correlation experimentally given in [9] for a lar position  = 0 was monitored. The grid-sensitivity rotating horizontal cylinder. At the outer radius, a combined analysis was conducted on a tube with D = 250 mm and convective-radiative boundary condition was adopted, with thickness th = 13.4 mm since it represents one of the larg- an external convective coefficient h calculated with the cor- est processable pipe in terms of diameter and thickness and relation given by Etemad for natural convection on horizontal it is reasonably more affected by angular and radial ther - rotating cylinders [23], neglecting the presence of an axial air mal gradients than smaller pipes. In addition, three differ - flow over the pipe outer surface. In fact, the value of pipe sur - ent operating conditions were considered: stationary pipe, −1 face emissivity  and the convective heat transfer coefficients, rotating pipe with  = 1.87 rad s and pipe irradiated by a which contribute to the definition of the boundary conditions, perfectly uniform radiation. Figure 4 shows the results of do not have a significant influence on temperature unevenness the grid-sensitivity analysis; in particular, the temperature within the pipe, as shown in [12]. The temperatures of ambi- of the monitored point in the three operating conditions at ent air, T , and walls, T , were both considered at 293.15 K. the end of the simulation and the total number of elements air w in the grid, N , are reported. The case of rotating pipe El Tot represents the most critical one, showing grid independence Model reliability and sensitivity analysis for N ≥ 500 ; thus, 500 has been subsequently chosen as El OC the best value for N , since it was verified that it guaran- El OC Before starting with simulations aimed at quantifying the teed grid independence for smaller pipes too. For a pipe with temperature displacement with respect to the case of a per- outer diameter D = 250 mm and thickness th = 10.8 mm, fectly uniform radiation as a function of angular velocity which is the largest geometry investigated, a mesh formed and oven configuration, the reliability of the model has been by 3996 quadrilateral Lagrange elements and 4496 nodes investigated comparing the asymptotic solution of the time- was subsequently adopted. dependent 2D finite-element model in the case of uniform The temporal discretization method was the Alpha heat flux with the analytical solution obtained integrating the method and the maximum time-step adopted was 0.1  s, heat equation in its one-dimensional and axisymmetric form defined after a time-sensitivity analysis. Simulations aimed and in steady-state condition. Neglecting angular variations at quantifying the temperature unevenness within pipe wall in temperature field and imposing steady-state conditions, had a duration of 100 s. Eq. (9) becomes: ��� 2 q (r, , ) T 1 T (10) + + = 0 r r k Integrating Eq. (10) twice, considering external SW lamps only and assuming conditions of adiabaticity at the inner radius and a convective boundary condition at the outer radius leads to the steady-state temperature profile within pipe wall: -1 ω=0 rad s -1 I r 1 1 re i ω=1.87 rad s T(r)= T + I ⋅ ( + )− ⋅ exp(−K (r − r )) Uniform 2000 air re a e i kK h h r a e e e El Tot I r I re i re + ⋅ exp(−K (r − r )) ⋅ ln( )− ⋅ exp(−K (r − r)) a e i a e 340 k r kK e a (11) 100 150 200 250 300 350 400 450 500 550600 The asymptotic solution obtained by means of the time- N (-) El OC dependent 2D finite-element model showed perfect agree- ment with the analytical steady-state solution described by Fig. 4 Grid sensitivity analysis. Temperature on a point placed in the Eq. (11), giving maximum discrepancies lower than 0.03% middle of pipe wall (r = r and  = 0 ) and total number of elements over the whole domain. in the grid. D = 250 mm, th = 13.4 mm 1 3 T (K) rm El Tot 130 International Journal of Energy and Environmental Engineering (2018) 9:123–134 Results τ=0 s τ=20 s τ=40 s τ=60 s In Fig. 5 the temperature profile along the outer circumfer - τ=80 s τ=100 s ence of a stationary PVC pipe in a oven like that shown in Fig.  1 has been compared at different times with that due to uniform radiation. A tube with D = 250 mm and th = 10.8 mm has been investigated, since it represents one of the largest customary tube sizes in end-forming processes. −2 −1 An external convection coefficient h = 9 Wm K was calculated. Two additional internal SW lamps have been considered, in order to make the heating process quicker and more uniform, as usually happens in practice. The actual temperature on the outer circumference is strongly affected by the view factor distribution: at  = 40 s, the maximum temperature exceeds 500 K, closely approaching the limit 115 116 117 118 119 120 121 122123 124125 r (mm) of dehydrochlorination, with a maximum temperature dif- ference of 120 K with respect to the uniform radiation case. Fig. 6 Temperature distribution within pipe wall. D = 250  mm, This result fully justifies the use of handling systems for the e −1 th = 10.8 mm,  = 1.87 rad s rotation of tubes, in order to obtain uniform heating. Simulations at different angular velocities were carried Figure 7 depicts the maximum temperature displacement out for the same PVC tube. Figure 6 shows the temperature along the outer circumference, ΔT : as expected, this quan- profile within pipe wall when considering the maximum −1 tity decreases with increasing angular velocity. Moreover, angular velocity investigated (  = 1.87 rad s ). It can be curves show a nonlinear dependence of ΔT on the angular clearly seen that the maximum temperature shifts inwards velocity: the reduction of ΔT with increasing angular veloc- from the outer wall, because of the semitransparent behav- ity is more pronounced for low values of  . This implies ior of PVC with respect to short-wave radiation. As time the existence of a limiting angular velocity, over which a increases, the maximum temperature moves toward the inner further increase does not bring substantial reductions in the radius, because of the increase in convective heat exchange maximum temperature displacement. at the outer surface. The flat trend of curves at the inner Figure 8a, b show the same quantity along the circum- radius is a consequence of the assumption of adiabaticity, ference defined by the mean radius ( ΔT ) and along the because of the small influence of the drag effect caused by inner circumference ( ΔT ). A reduction in the maximum the tube rotation on the air motion, when axial air flow is absent. -1 ω=0.468 rad s -1 ω=0.935 rad s -1 -1 ω=1.400 rad s ω=0 rad s , τ=0 s 2.5 -1 -1 ω=0 rad s , τ=20 s ω=1.870 rad s -1 550 ω=0 rad s , τ=40 s Uniform, τ=0 s Uniform, τ=20 s Uniform, τ=40 s 1.5 350 0.5 0102030405060708090 100 012345 6 θ (rad) τ (s) Fig. 5 Temperature profile along the outer circumference of a station- Fig. 7 Maximum temperature displacement along the outer circum- ary cylinder. D = 250 mm, th = 10.8 mm ference. D = 250 mm, th = 10.8 mm e e 1 3 T (K) re ΔT (K) T (K) re r International Journal of Energy and Environmental Engineering (2018) 9:123–134 131 3 1 -1 -1 ω=0.468 rad s ω=0.468 rad s -1 -1 0.9 ω=0.935 rad s ω=0.702 rad s -1 -1 ω=1.400 rad s ω=0.935 rad s 2.5 -1 ω=1.870 rad s 0.8 0.7 0.6 1.5 0.5 0.4 0.3 0.2 0.5 0.1 0 0 0102030405060708090 100 0102030405060708090 100 τ (s) τ (s) (a) Fig. 9 Maximum temperature displacement along the outer circum- ference. D = 125 mm and th = 7.4 mm thick -1 ω=0.468 rad s -1 ω=0.935 rad s -1 ω=1.400 rad s 2.5 -1 ω=1.870 rad s angular velocity, a strong reduction in ΔT emerges with decreasing external diameter; this can be justified consider - ing the local view factor distribution, which becomes less affected by the angular position as the diameter of the pipe 1.5 decreases; see Fig. 3b. This means that for pipes of smaller dimensions, the limiting value of angular velocity takes lower values. A critical angular velocity can be defined as the mini- mum value of rotational speed which allows to maintain 0.5 the maximum temperature displacement below a certain threshold. For a given oven configuration, the model can be used to determine this critical value for different tube geom- 0102030405060708090 100 τ (s) etries, after a maximum allowable temperature displace- (b) ment ( ΔT ) along the outer circumference with respect max to a perfectly uniform heating process has been established. To this aim, for each tube geometry and angular velocity Fig. 8 Maximum temperature displacement along the circumference defined by the mean radius (a) and along the inner circumference (b). investigated, ΔT must be compared to ΔT , whose value max r D = 250 mm, th = 10.8 mm is time-dependent, as clearly demonstrated in Figs. 7 and 9, because of the combined effects of conduction and convec- temperature displacement when moving to lower radii tion which become more pronounced at high temperatures, appears for each angular velocity investigated. This behavior favoring reduction in peak values of ΔT . In order to estab- is more pronounced at low angular velocities and it is due to lish a significant instantaneous value of ΔT for the com- the low thermal diffusivity of the material which dampens parison with ΔT , a time interval Δ =[ − p,  + p] has max 0 0 the amplitude of thermal perturbations and imposes a time been considered, where p is lamps facing period, defined as delay to it, which causes an increase in the temperature dis- p = 2∕( ⋅ N ) , where N is the number of external lamp lamp placement at the inner radius when time progresses. lamps in the oven, and  is the instant of time in which the A set of simulations with the same boundary condi- mean temperature along the outer circumference reaches the tions but without internal lamps was run for a tube with value T = 433.15 K, which represents a credible value of off D = 125 mm and th = 7.4 mm, which represents a medium- the switch-off temperature of lamps, when these are con- sized tube. Figure  9 shows the trend of ΔT for different trolled by a traditional pyrometer. The comparison value has values of angular velocities. Comparing the curves in Fig. 9 consequently been calculated as ΔT = max{ΔT()} , with eval with those in Fig.  7 and considering the same values of  ∈Δ . The value of ΔT can be established arbitrarily in max 1 3 ΔT (K) ΔT (K) ri rm ΔT (K) re 132 International Journal of Energy and Environmental Engineering (2018) 9:123–134 2.5 the process design stage: in this work, a maximum tempera- Not Equidistant ture displacement ΔT = 1 K has been considered. The max Equidistant analysis on critical angular velocity has been carried out on ΔT =1 K max three different pipe geometries and considering two different oven configurations in order to investigate the effects of the reciprocal angular positions of the lamps on critical veloc- 1.5 ity values. As to the latter aspect, the oven with external lamps positioned as described in “View factor calculation” section has been compared to one with eight equally-dis- placed SW lamps. Figure 10a–c show the trends of ΔT eval for three pipe geometries and for the two configurations of 0.5 lamps considered, comparing it to ΔT . It is clear how, max for a given geometry and oven configuration, the value of ΔT decreases when increasing angular velocity and tends eval 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 -1 to flatten. The intersection between the curves relative to ω (rad s ) ΔT and ΔT identifies the value of critical angular eval max (a) velocity, over which a further increase in angular velocity 2.5 does not lead to significant improvements in the uniformity Not Equidistant of the temperature distribution. It is clear how higher radii Equidistant ΔT =1 K necessitate higher angular velocities to obtain a maximum max temperature displacement of 1 K with respect to the case of uniform radiation. Moreover, fixing pipe geometry, a sub- 1.5 stantial reduction in the values of ΔT and consequently eval in critical angular velocity is obtained when moving to a configuration with eight lamps displaced at uniform angular intervals. This tendency is stronger for pipes of smaller size. The larger reduction in critical angular velocity high- lighted for smaller pipes in the case of lamps equally-dis- 0.5 placed can be justified observing the heat flux profile on the pipes’ outer surface for the three geometries investigated and for the oven configurations studied, shown in Fig.  11a–c. 0 0.5 11.5 -1 In fact, with reference to Fig. 2, for a given lamps radial ω (rad s ) position R and extension L, it is clear how pipes with low (b) external diameter are more sensitive to the angular position 2.5 of the lamps, because of the overlap of the heat fluxes gen- Not Equidistant erated by every single lamp, which increases the deviation Equidistant ΔT =1 K max from the uniform radiation. 1.5 Conclusions A 2D finite-element model for the transient analysis of the radiative heat exchange between a rotating polymeric cylin- drical pipe and the SW infrared lamps of an oven for end- 0.5 forming has been devised in order to investigate the influence of cylinder angular velocity and oven configuration on the temperature distribution within the tube. Results obtained for different rotational velocities were compared with the 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91 -1 ω (rad s ) case of a uniformly irradiated tube. A MATLAB function (c) for the calculation of local view factor distribution given pipe’s outer radius and oven configuration has been devel- Fig. 10 Effect of the lamps’ angular positions on critical velocity. a Big-size oped. The code was validated comparing the global view pipe: D = 250  mm th = 10.8  mm. b Medium-size pipe: D = 180  mm e e factor calculated as the sum of all local view factors with the th = 10.7 mm. c Small-size pipe: D = 125 mm th = 7.4 mm 1 3 ΔT (K) ΔT (K) ΔT (K) eval eval eval International Journal of Energy and Environmental Engineering (2018) 9:123–134 133 Fig. 11 Effect of the lamps’ angular positions on heat flux distribu- ▸ tion along the outer circumference of pipes. a External diameter of Equidistant Not Equidistant 250  mm and thickness of 10.8  mm. b External diameter of 180  mm Uniform and thickness of 10.7 mm. c External diameter of 125 mm and thick- ness of 7.4 mm value obtained by means of the corresponding correlation given in [15]. Comparison between the temperature distri- butions on the outer circumference of a static tube obtained in the cases of uniform and actual radiation distribution justifies the use of rotation devices, in order to avoid high temperature peaks which could lead to the thermal failure of the material. Simulations carried out on different pipe geom- etries at varying values of angular velocity demonstrated -3 -2 -1 012 3 how increasing angular speed decreases the maximum tem- θ (rad) perature displacement with respect to the case of a uniformly (a) irradiated tube. Results showed a nonlinear dependence between maximum temperature displacement and angular Equidistant velocity, thus demonstrating the existence of a limiting value Not Equidistant over which a further increase in angular velocity brings little Uniform reduction in maximum temperature difference. The same analysis also highlighted how the low thermal diffusivity of the material dampens the maximum temperature difference when moving from the outer to the inner side of the pipe. For a fixed value of the angular velocity, lower values of maximum temperature difference were obtained for smaller tubes, because of the more uniform distribution of the local view factors with respect to the angular coordinate. Moreo- ver, as highlighted in [12], parameters which contribute to 10 the definition of boundary conditions, such as convective heat transfer coefficients and PVC surface emissivity, do not -3 -2 -1 012 3 influence temperature unevenness significantly over the pipe θ (rad) wall, since they just affect the mean temperature reached (b) at a certain radial position at a given time without varying the maximum temperature displacement with respect to the case of a perfectly uniform radiation. The model can be used Equidistant Not Equidistant as a tool to characterize the end-forming process of poly- Uniform meric tubes in terms of critical angular velocity, for different materials and oven configurations. Simulations showed that, for a given oven configuration, higher critical velocities are obtained for large-sized pipes, because of the much more uneven distribution of the heat flux on the outer circumfer - ence. Also, comparisons between two different oven con- figurations showed how relative angular positions of lamps 20 could lead to significant modification of the values of critical angular velocity, all the more for smaller pipes. To the best knowledge of the authors, this is the first numerical model able to characterize the heating stage of -3 -2 -1 012 3 the end-forming process in terms of critical angular velocity; θ (rad) an experimental campaign is planned to validate the numeri- (c) cal predictions and fine-tune the model. Finally, it should be mentioned that the addition of surface temperature control in the model can allow the design of optimal heating cycles 1 3 -2 -2 -2 Heat Flux (kW ·m ) Heat Flux (kW ·m ) Heat Flux (kW ·m ) 134 International Journal of Energy and Environmental Engineering (2018) 9:123–134 11. Lucchi, M., Lorenzini, M., Valdiserri, P.: Energy performance to achieve the desired pipe temperature in the most efficient of a ventilation system for a block of apartments with a ground way, meeting production requirements whilst saving energy source heat pump as generation system. J. Phys. Conf. Ser. 796(1), at the same time. 012034 (2017) 12. Lucchi, M., Lorenzini, M.: Transient analysis of the radiative heat- Open Access This article is distributed under the terms of the Creative ing of rotating PVC pipes in a oven for end-forming process. Appl. Commons Attribution 4.0 International License (http ://crea tive comm Therm. 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International Journal of Energy and Environmental EngineeringSpringer Journals

Published: Jan 23, 2018

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