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Convection-radiation from a continuously moving, variable thermal conductivity sheet or rod undergoing thermal processing
International Journal of Thermal Sciences 50 (2011) 1523e1531
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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Convection-radiation from a continuously moving, variable thermal conductivity sheet or rod undergoing thermal processing A. Aziz a, *, Robert J. Lopez b, c a
Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA c Maplesoft, Waterloo, Ontario, Canada N2V 1K8 b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 September 2010 Received in revised form 23 March 2011 Accepted 24 March 2011 Available online 22 April 2011
A numerical study of the heat transfer process in a continuously moving sheet or rod of variable thermal conductivity that loses heat by simultaneous convection and radiation is reported. The process is governed by five dimensionless parameters, namely Peclet number Pe, thermal conductivity parameter a, convection-conduction parameter Nc, radiation-conduction parameter Nr, and sink-to-base temperatureratio, qa. The effects of these parameters on the temperature distribution, base heat conduction rate, advection and surface heat loss are illustrated and explained. The dimensionless length parameter, L*, that the material must traverse to cool to within one percent of the sink temperature is determined for various combinations of the five parameters. This information gives the designer the fabrication time if the material is moved at a certain speed or it gives the speed if the processing of the material is to be completed in a fixed period of time. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Thermal processing Moving sheet or rod Convection-radiation Variable thermal conductivity Processing time
1. Introduction In processes such as extrusion, glass fiber drawing, hot rolling and casting, the material being manufactured is processed thermally by allowing it to exchange heat with the ambient while it is in continuous motion [1e8]. The purpose of the thermal treatment is to cool the material to a desirable temperature before it is spooled or removed. The velocity of the material can be extremely low (few centimeters per hour) such as in crystal growth or very fast (few meters per second) as in optical fiber drawing. As the material at high temperature emerges from a furnace or a die, it is exposed to the colder ambient and a transient conduction process accompanied by surface heat loss is initiated. For the slow-moving material, the initial transient dies out with the passage of time and the process quickly attains a steady state. However, for a fast-moving material, the temperature distribution may continue to evolve with time during the entire duration of thermal processing. Studies of fluid flow and heat transfer from a heated moving surface may be classified into three groups. The first group comprises of papers in which the fluid flow over the surface is assumed to be either induced by the motion of the surface in an
* Corresponding author. E-mail address: [email protected] (A. Aziz). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.03.014
otherwise quiescent fluid or the flow is driven by an independent source. Assuming a specific thermal boundary condition at the surface, the applicable continuity, momentum and energy equations are solved to obtain the velocity and temperature fields in the fluid. The final outcome is the surface heat transfer data such as the convection heat transfer coefficient. This group of papers does not address the problem of thermal transport in the moving material itself. The number of papers belonging to this group is vast, covering Newtonian and non-Newtonian fluids, surface mass transfer, magnetic and electric effects, different thermal boundary conditions, combined free and forced convection, etc. References [9e12] provide a representative sample of such studies chosen from the recent literature. The second group of papers considers the thermal transport in the fluid and in the moving material concurrently and treats the process as a conjugate conduction-convection process. Karwe and Jaluria [13] considered the flow field generated due to the movement of a continuous heated surface and used the CrankeNicolson finite difference method to compute the temperature fields in the fluid and in the moving material. They also derived analytical solutions for the situation when the surface convection heat transfer coefficient is known a priori. This work was later extended to include buoyancy effects in the flow field induced by the motion of heated surface, i.e., a mixed convection situation [14,15]. As expected, the effect of buoyancy was found to be more significant
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Nomenclature Ac a c cp G h k ka L L* Nc Nr P Pe q
cross-sectional area of the rod or sheet, m2 dimensionless thermal conductivity parameter, b Tb a constant specific heat of the material, J/kg K irradiation on the plate or rod from the surroundings, W/m2 convection heat transfer coefficient, W/m2 K thermal conductivity at temperature T, W/m K thermal conductivity at temperature Ta,W/m K distance from point of emergence to the point where the adiabatic condition is applied, m adjustable length parameter, dimensionless convection-conduction parameter, dimensionless radiation-conduction parameter, dimensionless perimeter of the cross-section of rod or sheet, m Peclet number, dimensionless surface heat loss by combined convection and radiation, W
when the surface moved in a vertical direction than when it moved in a horizontal direction. Choudhry and Jaluria [16] studied the forced convection induced when a cylindrical rod moved centrally along the axis of a channel. Two scenarios were considered: one in which a uniform flow in the channel assisted the flow induced by the motion of the rod; and the other with a uniform flow opposing the flow induced by the motion of the rod. For the opposing flows, the surface heat transfer rate was reduced due to the creation of a recirculation zone in the fluid adjacent to the surface of the rod. Other conjugate conduction-convection studies have been reported by Lee and Tsai [17], Char, Chen, and Cleaver [18], and Mendez and Trevino [19], among others. The third group of papers assumes that the convection heat transfer coefficient is known and focuses exclusively on the thermal transport in the moving material. The mathematical model governing the temperature distribution in the moving material is then developed by establishing an energy balance between heat conduction, heat advection due to motion, and the surface heat loss. The resulting ordinary differential equation for the steady state or the partial differential equation for the transient state, is then solved analytically or numerically. For example, Choudhry and Jaluria [20] obtained a double series solution for the twodimensional, transient temperature distribution in a moving rod or a flat plate moving with a constant velocity and losing heat by convection to the ambient fluid through a constant heat transfer coefficient. The same problem was solved numerically in a then contemporary paper by Jaluria and Singh [21]. This paper quantified the effect of Biot number (characterizing the surface convective heat loss) and Peclet number (characterizing the speed of the moving material) on the temperature distribution in the material and the surface heat loss. The focus of the present paper is to include the effect of temperature dependent thermal conductivity of the moving material in the analysis and add a radiative component to the surface heat loss mechanism. To the best of our knowledge, these improvements have not been investigated in depth in the literature. With the inclusion of radiation and variable thermal conductivity, three new parameters, in addition to the Biot number and Peclet number, emerge, namely a thermal conductivity parameter, a radiation-conduction parameter, and an environment temperature parameter. For pure convection, the effect of environment temperature is usually absorbed in defining the dimensionless
Q Qa Qc T Ta Ts Tb U X x
a b 3 q qa qb r s
surface heat loss, dimensionless advection component of heat loss, dimensionless base heat conduction, dimensionless local temperature, K sink temperature for convection, K effective sink temperature for radiation, K temperature at the base (point of emergence), K speed of moving rod or sheet, m/s distance from the point of emergence, dimensionless distance from the point of emergence, m surface absorptivity, dimensionless or thermal diffusivity of the material, m2/s a measure of thermal conductivity variation with temperature, K1 surface emissivity, dimensionless dimensionless local temperature dimensionless sink temperature dimensionless base temperature density of material, kg/m3 StefaneBoltzmann constant ¼ 5.67 108 W/m2 K4
temperature but such simplification is not possible when radiation is included in the analysis. The results to be presented will highlight the effect of these five parameters on the temperature distribution in the moving material, the surface heat loss, and, most importantly, the time needed for the material to reach thermal equilibrium with the ambient. The last piece of information allows the designers to fix, for example, the distance between the point of emergence of the material from the die or furnace to the delivery station if the material needs to be cooled to within 1% of the environment temperature. 2. Mathematical analysis Consider the thermal processing of a plate or a rod of crosssectional area A and perimeter P while it moves horizontally with a constant speed U as shown in Fig. 1. The hot plate or rod emerges from a die or furnace or an intermediate station at a constant temperature Tb. The motion of the plate or rod may induce a flow field in an otherwise quiescent surrounding medium or alternatively, the plate or rod may experience an externally driven flow over its surface. Either way, the plate or rod is exposed to a colder surrounding medium and loses heat by convection and radiation. The radiative component would play a more prominent role if the forced convection is weak or absent or when only natural convection occurs. If the plate or rod experiences irradiation G on its surface from the surroundings and if the absorptivity of the plate or rod is a, the rate of radiation absorption is aG. If the emissivity of the plate or rod is 3 and then the equivalent sink temperature is given by Ts4 ¼ aG=3s [22]. To avoid the
Fig. 1. A moving rod or sheet with surface convection and radiation.
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introduction of an additional variable, a common sink temperature Ta (¼Ts) < Tb is assumed for both convection and radiation. The surface of the moving material is assumed to be gray and diffuse with a constant emissivity 3. The convective flow in the surrounding medium provides a constant heat transfer coefficient h over the entire surface of the moving material. Since the material undergoing the treatment experiences a large change in its temperature during the process, the change in its thermal conductivity may be as much as 100%. For most materials, the thermal conductivity of the material k increases linearly with temperature. Thus we assume
k ¼ ka ½1 þ bðT Ta Þ
(1)
where ka is the thermal conductivity of the material at temperature Ta and b is a measure of thermal conductivity variation with temperature. It is assumed that the initial transient, which occurs when the material is exposed to the coolant, dies out quickly and the processing takes place under steady state conditions. This assumption has been made in several studies, including the one by Karwe and Jaluria [13]. The transverse Biot number encountered in material processing applications are usually less than 1.0 [11]. Even if the convective cooling is very strong, the value is not likely to exceed 2.0. The onedimensional heat conduction model when used with the Biot number 2.0 gives predictions of the cooling process that are within 3% of the two-dimensional conduction model [23]. Even when Bi is large as 6, the error between the one-dimensional and twodimensional fin solutions is about 4%. The effect of radiation on the transverse heat conduction may be evaluated by linearizing the radiation and calculating the Biot number based on the combined effective heat transfer coefficient. Since the largest values Nc and Nr used in the computations is 2.5 each, the effective Biot number is about 6.0 which as stated a little earlier entails an error of 4% at most in using the one-dimensional assumption. In view of this, the results presented in this paper are not going to be affected in any significant way by taking transverse conduction into account. The steady state energy balance for the material moving with a constant speed and losing heat by simultaneous convection and radiation may be written as
3sP 4 4 1 dT d dT hP ðT Ta Þ T Ta U ½1 þ bðT Ta Þ ¼0 a dx dx dx ka A ka A (2) where a ¼ ka/r cp is the thermal diffusivity of the material, r is the density and cp is the specific heat. The last term on the left in Eq. (2) is the advection term. The axial coordinate x is measured from the slot from which the material emerges and comes in contact with the surrounding fluid. The solution of Eq. (2) for a constant thermal conductivity (b ¼ 0) and no radiation (3 ¼ 0) for a semi-infinite fin with a constant base temperature and a finite temperature at x ¼ N has been presented by Jiji [24]. With the introduction of the following dimensionless variables,
q ¼ T=Tb ; qa ¼ Ta =Tb ; L* ¼ PL=A; X ¼ xL* =L; a ¼ bTb ; Nc ¼ hA=Pka ; Nr ¼ 3sATb3 =Pka ; Pe ¼ UA=aP
(3)
where L is the length between the point of emergence (x ¼ 0) and the point where the temperature gradient in the material is zero or where the temperature is within a specified fraction of the ambient temperature depending on which boundary condition is chosen; Nc is the convection-conduction number (essentially the Biot number); Nr is the radiation-conduction number; and Pe is the Peclet number, which represents the dimensionless speed of the moving material, Eq. (2) takes the following form
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d dq dq 4 4 ½1 þ aðq qa Þ Nc ðq qa Þ Nr q qa Pe ¼0 dX dX dX (4) In physical terms, the Peclet number is the ratio of energy advected by the material through its motion and the energy conducted through the material [2]. The parameter L* allows the length of the material exposed to the surrounding fluid to be adjusted. When the material is stationary, Pe ¼ 0 and Eq. (4) reduces to that of a stationary fin. The solution of Eq. (2) is sought under the following boundary conditions.
X ¼ 0; q ¼ 1 X ¼ L* ;
dq ¼ 0 or q ¼ cqa dX
(5a) (5b,c)
where the parameter L* must be specified to obtain the solution. The choice of insulated boundary condition at x ¼ L or X ¼ L* may not reflect the physics in its entirety but such a choice has been made in the pioneering papers by Karwe and Jaluria [13], Choudhry and Jaluria [20] and Jaluria and Singh [21] and found to provide satisfactory approximations to the more realistic condition except when L* and Pe are small. However, we did carry out the computations where this condition was replaced by a condition qðL* Þ ¼ cqa where c is constant. Sample calculations for c ¼ 0.99, that is, at X ¼ L*, the material has cooled down to 1% of the sink temperature will be presented and discussed later. The problem depicted in Fig. 1 (region 2) may be extended to include a more realistic boundary condition at x ¼ 0 by coupling it to the upstream (region1) and downstream (region 3) thermal fields. For the case of constant thermal conductivity and no surface heat loses in regions 1 and 3 and only convective heat loss in region 2, the three energy equations can be solved analytically in terms of the exponential functions as shown in Smith et al. [25]. When the temperatures in the three regions are plotted (see Fig. 17.4 on p.1239 in [25]), it is seen that for Pe ¼ 1 and 10, the dimensionless temperature q at x ¼ 0 is virtually unity. Even at Pe ¼ 0.1, its value is 0.91. In view of this, the assumption of q ¼ 1 at x ¼ 0 is justified within a 9% error at Pe ¼ 0.1 and with negligible error for Pe ¼ 1 and higher. It is unlikely that the inclusion of radiative heat loss in region 2 and/or the effect of temperature dependent thermal conductivity would invalidate the assumption of a constant temperature at x ¼ 0. There are other assumptions made in writing the heat transfer model (Eqs. (4) and (5)) that require some explanation. First, the surface of the moving rod or sheet is assumed to be diffuse, gray. We could have accommodated a temperature dependent emissivity in our model but we concluded from a separate work by the first author on a stationary convective-radiative fin in which the surface emissivity was a linear function of temperature that its effect on the thermal performance was small compared with the effect of temperature dependent thermal conductivity [26]. The inclusion of a spectral, directional emissivity would have deviated us from the main focus of this work and consequently it was judged to be beyond the scope of the present contribution. Often in the calculations of simultaneous convection and radiation from a surface, the radiation term is linearized and the two processes are described by a combined effective heat transfer coefficient. Such a simplification can introduce error of over 30% in some cases [27] and was therefore not pursued here. When only convective surface loss occurs in region 2, the dimensionless temperature q could be defined as q ¼ ðT Ta Þ=ðTb Ta Þ to eliminate the parameter qa, but this is not feasible when radiation is present [28].
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3. Numerical solutions and results 3.1. Temperature profiles The solutions of the boundary value problem represented by Eqs. (4) and (5) were obtained for L* ¼ 1, 2, and 3 using the dsolve (differential equation solver) command in the symbolic algebra package Maple 13 [29] and specifying the numeric option. Maple 13 uses a fourth-fifth order RungeeKuttaeFehlberg (rkf45) procedure as the default numeric solver for initial value problems, but does not use it in a shooting technique for the numeric solution of boundary value problems. For these problems Maple 13 uses a second-order difference scheme combined with an order bootstrap technique with mesh-refinement strategies: the difference scheme is based on either the trapezoid or midpoint rules; the order improvement/accuracy enhancement is either Richardson extrapolation or a method of deferred corrections. Each figure to be presented will show the effect of one parameter on the temperature distributions in the moving material. Fig. 2 shows the effect of the thermal conductivity parameter a on the temperature distributions for Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5, and qa ¼ 0.5. For each L*, the bottom curve corresponds to a ¼ 0 (constant thermal conductivity) and the top curve corresponds to a ¼ 1 (thermal conductivity at the base temperature is 50% higher than the thermal conductivity at the environment temperature). As the parameter a increases, the average thermal conductivity of the material increases and as expected, the result is the gradual increase in the local temperature. This behavior is exhibited for all three values of L*. The time t taken by the material to move a distance L* is t ¼ L*/U. For a fixed U, the time period for which the moving material is exposed to the surrounding fluid is proportional to L*. For smaller values of L*, the duration of exposure to the surrounding fluid is shorter and hence the material stays at relatively higher temperatures. Alternatively, one may argue that for smaller values of L*, the temperatures are higher because the energy (conduction and advection) entering at X ¼ 0 is lost over
Fig. 2. Effect of variable thermal conductivity on temperature distribution in the moving material.
a shorter distance. If the value of L* were to be increased sufficiently, a condition would be reached in which the end (X ¼ L*) temperature would very nearly equal the temperature of the surrounding fluid. The words “very nearly” are used because it is only when L* approaches infinity that the material would be in thermal equilibrium with the surrounding fluid. A detailed discussion of this important point will be offered in a later section. Fig. 3 illustrates how the temperature profiles in the material are affected by the changes in the sink temperature. The other parameters were fixed at a ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5. These values of the parameters are representative of some wire drawing applications [6,11]. As expected, the temperatures in the material progressively decrease as the sink temperature is lowered. This trend can be seen for all three values of L*. In each set of curves, the bottom curve corresponds to qa ¼ 0.2 and the subsequent curves correspond to qa ¼ 0.3, 0.4, 0.5, and 0.6 (top curve) in that order. Again, higher temperature are observed when L* is smaller and the time period for which the material is exposed to the surrounding fluid is shorter. The effect of qa is most profound on the temperature at the end of the cooling process as indicated by the relatively larger spread between the curves at the terminal points. The effect of Peclet number Pe (dimensionless speed) on the temperature distribution is illustrated in Fig. 4 with the other parameters fixed at a ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5 and qa ¼ 0.5. As Pe increases, that is, as the material moves faster, the time for which the material is exposed to the environment gets shorter, the temperatures are consequently higher, and their variation more gradual than when the material moves at a slower speed. This conclusion applies to temperature profiles for all three values of L*. Attention is now turned to the effects of surface heat transfer mechanisms on the temperature distribution in the moving material. Fig. 5 provides the temperature profiles when the convection-conduction parameter Nc is assigned values of 0, 0.5, 1.0, 1.5, and 2.0 with the remaining parameters fixed at a ¼ 0.5, Nr ¼ 0.25, qa ¼ 0.5, and Pe ¼ 0.5. The top curve for each value of L* applies to Nc ¼ 0 (no convection) but since Nr ¼ 0.25, it represents the case of weak pure radiating cooling of the material. Other
Fig. 3. Effect of sink temperature on the temperature distribution in the moving material.
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Fig. 4. Effect of Peclet number on the temperature distribution in the moving material.
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Fig. 6. Effect of radiation-conduction parameter on the temperature distribution in the moving material.
curves represent the cases of simultaneous convection and radiation loss from the surface. As Nc increases, that is, as convection gets stronger, the cooling becomes more effective, promoting lower temperatures in the material. As L* increases, the cooling period increases, which is reflected in the further lowering of the temperatures in the material. Fig. 6 shows the temperature profiles when the radiationconduction parameter Nr is allowed to vary with the rest of the parameters fixed at a ¼ 0.5, Nc ¼ 0.25, qa ¼ 0.5, and Pe ¼ 0.5. For each value of L*, the top curve corresponds to zero radiation but since Nc ¼ 0.25, it represents a circumstance in which the material
is cooled by weak pure convection. Other curves represent different cases of combined convective-radiative cooling. As the radiative transport becomes stronger, the radiative cooling becomes more effective, which in turn causes the lowering of temperatures in the material. The effect is similar to what was observed in Fig. 5 with respect to the strengthening of the convective transport mechanism. It should be noted that the curves corresponding to L* ¼ 1 in Figs. 3e6 may be of theoretical interest only and may not arise in real thermal processing situations.
Fig. 5. Effect of convection-conduction parameter on the temperature distribution in the moving material.
Fig. 7. Effect of variable thermal conductivity on advection, base conduction and surface heat loss.
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Qa ¼ Peð1 qð1ÞÞ
Fig. 8. Effect of sink temperature on advection, base conduction and surface heat loss.
3.2. Surface heat loss A quantity of fundamental interest in thermal processing applications is the amount of surface heat loss from the moving material to its surrounding. The surface heat loss can be computed as follows. The energy entering at x ¼ 0 consists of conduction heat transfer plus the energy advected. A portion of this energy is lost to the surrounding fluid and the remainder is advected at x ¼ L. The energy conducted at x ¼ L is zero in view of the insulated boundary condition at x ¼ L. This energy balance can be expressed as
kA
dT ð0Þ þ rUAcp Tb q ¼ rUAcp Tð1Þ dx
(6)
where T(1) is the temperature at the exit (i.e., x ¼ L) and q is the surface heat loss. Solving for q and expressing it in dimensionless form, we have
Q ¼
qL ¼ Q c þ Qa ka Tb A
(7)
where
Qc ¼ ½1 þ aðqð0Þ qa Þ and
dq ð0Þ dX
(8)
(9)
where q(0) and q(1) are the dimensionless temperatures at X ¼ 0 and X ¼ 1, respectively. Eq. (8) gives the conduction energy entering at X ¼ 0 that is lost to the surrounding fluid, while Eq. (9) gives the advection energy that is lost to the surrounding. Eqs. (7)e(9) were incorporated in the Maple code written to generate the temperature profiles that were discussed in the preceding section. For consistency, the values of Qc, Qa and Q were computed using the same values of parameters that were used in generating Figs. 2e6. In all these calculations, the value of L* was fixed at unity. Fig. 7 illustrates the effect of Peclet number on advection Qa, the base heat conduction, Qc and the surface heat loss, Q. The effect of variable thermal conductivity is indicated by three parametric values of a, namely 0, 0.5 and 1.0. As the Peclet number increases, advected heat loss (bottom set of curves) increases but the base heat conduction rate (middle set of curves) decreases. The increase in advected heat loss offsets the decrease in base heat conduction and the surface heat loss (top set of curves) increases slightly with the increase in Peclet number. The increase in surface heat loss (the total energy input into the system) with the Peclet number was also observed by Jaluria and Singh [21]. The effect of increase in thermal conductivity with temperature, i.e., increase in parameter a is to reduce advected heat loss, increase the base heat conduction and increase the surface heat loss or the total energy input into the moving system. Fig. 8 is similar to Fig. 7 except that the sink temperature Ta is now chosen as a parameter on the curves. As the sink temperature Ta increases, both advected heat loss and base heat conduction decrease, which in turn is reflected in a decrease in the surface heat loss or the total energy input into the system. The decrease in surface heat loss with an increase in the sink temperature is the result of the decrease in the temperature difference driving the flow of energy from the moving surface to the surroundings. Fig. 9 shows how advected heat loss, base heat conduction, and surface heat loss are affected by the changes in Peclet number Pe and the radiation-conduction parameter Nr when the other parameters are kept fixed. The three solid lines represent the case of zero radiation, i.e., Nr ¼ 0. As Pe increases, advection (Qa) increases but the base conduction (Qc) decreases. At Pe z 1.8, Qa and Qc are equal. The surface heat loss is very slightly affected by Pe. When the radiation-conduction parameter is set at Nr ¼ 0.5 (see dotted curves), advection, base heat conduction, and the surface heat loss are all higher compared with the case of Nr ¼ 0. The curves for Nr ¼ 0.5 exhibit the same pattern as seen in the case of Nr ¼ 0,
Fig. 9. Effect of radiation-conduction parameter on advection, base conduction and surface heat loss.
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that is, as Pe increases, Qa increases, Qc decreases, and Q increases. The results for Nr ¼ 1 are similar to those for Nr ¼ 0.5 except that advection, base heat conduction, and the surface heat loss are all much higher compared with the case of Nr ¼ 0.5. This is understandable in view of the fact that as radiation gets stronger, it must manifest itself in enhanced surface heat loss. Figs. 10e13 give the surface heat loss data, i.e., Q, for many combination values of the parameters a, Nc, Nr, Pe and qa. Fig. 10 shows that as convection and radiation increase either individually or together, the surface heat loss increases as expected. It can be seen from Fig. 11 that as the sink is raised, the temperature difference driving the heat flow is reduced causing a reduction in surface heat loss. In Fig. 12, we illustrate the effect of the convection parameter on the surface heat loss. For a fixed Nr, as the convection gets stronger, the surface heat loss increases. The effect is similar when the convection parameter is kept fixed and radiation parameter is allowed to increase (Fig. 13). 3.3. Thermal equilibrium The temperature to which the material needs to be cooled, i.e., T(L*) depends on the desired characteristics of the material at the conclusion of thermal processing. As noted earlier, the parameter L* gives the fabrication time if the material is moved at a certain speed or alternatively, it gives the speed if the processing of the material is to be completed in a prescribed period of time. Because the degree to which the material is to be cooled varies from process to process, it gives rise to a large of number of possibilities. For the sake of illustration here, it was assumed that the material is to be cooled to within 1% of the environment temperature. A total of 19 graphs depicting the values of L* as a function of various parameters were generated. A selection of these graphs is presented here. Fig. 14 shows the variation of L* with the thermal conductivity parameter a for parametric values of qa ¼ 0.2, 0.3, 0.4, 0.5 and 0.6 with Nc ¼ Nr ¼ 0.25 and Pe ¼ 0.5. These values of Nc and Nr represent weak surface convection and radiation. The parameter L* increases if the variation of thermal conductivity with
temperature is larger. The smaller the environment temperature, the larger the time required for the material to attain thermal equilibrium with the environment. This is clearly indicated by the larger values of L* at smaller values of qa. From the graphs not shown here, it is found that the value of L* increases progressively as the Peclet number increases. At Pe ¼ 2.5 with the other parameters fixed as in Fig. 14, the value of L* is 58 at a ¼ 1.0, so that the thermal conductivity at the base temperature is twice the thermal conductivity at the surroundings temperature.
Fig. 10. Effect of convection-conduction parameter, Nc and radiation-conduction parameter, Nr on surface heat loss.
Fig. 12. Effect of convection-conduction parameter, Nc and sink temperature, qa on surface heat loss.
Fig. 11. Effect of thermal conductivity parameter, a and sink temperature, qa on surface heat loss.
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Fig. 13. Effect of radiation-conduction parameter, Nr and sink temperature, qa on surface heat loss.
The case of strong radiation and weak convection is illustrated in Fig. 15. The values of the different parameters used are the same as in Fig. 14 except that Nr has been increased from 0.25 to 2. A comparison of Fig. 15 with Fig. 14 shows that the effect of strong radiation is to reduce the values of L*. This reduction may be attributable to the fact that the bigger surface heat loss would result in a quicker attainment to thermal equilibrium. Fig. 16 gives the results for the case of strong convection and weak radiation. Except for Nc, which is now increased from 0.25 to 2.0, the rest of the
Fig. 14. Effect of thermal conductivity parameter a and sink temperature qa on parameter L*.
Fig. 15. Effect of thermal conductivity parameter, a and sink temperature, qs on parameter L*: case of weak convection and strong radiation.
parameters are fixed at the values used in Fig. 14. A comparison of Figs. 14e16 reveals that the effect of convection on the attainment of thermal equilibrium is much stronger than that of radiation. The values of L* in Fig. 16 (strong convection, weak radiation) are significantly lower than the values of L* in Fig. 15 (strong radiation, weak convection). Thus the strength of convective heat loss plays comparatively a more influential role than the strength of radiative heat loss. Fig. 17 reveals that there is a further reduction in the value of L* when both convection and radiation are simultaneously strong.
Fig. 16. Effect of thermal conductivity parameter, a and sink temperature, qa on parameter L*: case of weak radiation and strong convection.
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Acknowledgement The authors appreciate the valuable and constructive comments of the reviewers. Their input has improved the clarity and the technical quality of the paper. References
Fig. 17. Effect of thermal conductivity parameter, a and sink temperature, qa on parameter L*: case of strong radiation and strong convection.
4. Conclusions A numerical study of the heat transfer process in a continuously moving sheet or rod of variable thermal conductivity losing heat by simultaneous convection and radiation has been conducted. The process is governed by five dimensionless parameters: Peclet number Pe, thermal conductivity parameter a, convectionconduction parameter Nc, radiation-conduction parameter Nr, and sink-to-base temperature-ratio, qa. The study led to the following conclusions when all parameters except one were allowed to vary. (1) The effect of an increase in the thermal conductivity parameter a is an increase in the local temperature for the material, and in base heat conduction and surface heat loss, but a decrease in advected energy loss. (2) As the Peclet number increases, it increases the local temperature, and the advected heat loss and surface heat loss, but decreases the base heat conduction. (3) As the surface convection and/or radiation gets stronger, the effect is to lower the local temperature, increase the base heat conduction, advected energy loss and surface heat loss. (4) The dimensionless parameter c characterizing the attainment (to within a 1% difference) of thermal equilibrium increases as the Peclet number increases but decreases as the surface convection and/or radiation increase. The effect of convection on L* is more pronounced than the effect of radiation. The parameter c also increases as the thermal conductivity parameter increases and as the sink temperature decreases. (5) A comparison of present numerical results with the experimental data has not been possible because the experimental data for the precise thermal processing situation analyzed in this work could not be found in the literature.
[1] Z. Tadmor, I. Klein, Engineering Principles of Plasticating Extrusion, Polymer Science and Engineering Series. Van Nostrand Reinhold Company, New York, 1970. [2] E.G. Fisher, Extrusion of Plastics. John Wiley, New York, 1976. [3] T. Altan, S. Oh, H. Gegel, Metal Forming Fundamentals and Applications. American Society of Metals, Metals Park, OH, 1979. [4] S. Kalpakjian, Manufacturing Engineering and Technology. Addison Wesley, Reading, MA, 1989. [5] J. Szekely, Fluid Flow Phenomena in Metals Processing. Academic Press, New York, 1979. [6] Y. Jaluria, Transport from continuously moving materials undergoing thermal processing Chapter Four, , In: Annual Reviews of Heat Transfer, vol. 4. Academic Press, 1992, pp. 187e245. [7] D. Poulikakos, Transport Processes in Material Processing, Advances in Heat Transfer, vol. 18, Academic Press, New York, 1996. [8] Y. Jaluria, Thermal processing of materials: from basic research to engineering, ASME Journal of Heat Transfer 125 (2003) 957e979. [9] R. Cortell, Flow and heat transfer in a moving fluid over a moving flat surface, Theoretical and Computational Fluid Dynamics 21 (2007) 435e446. [10] X. Zhang, L. Zheng, J. He. Skin friction and heat transfer on a continuous moving flat surface in power fluids, Proc. ASME Summer Heat Transfer Conference, 2003, pp. 419e425. [11] H. C-Chen, Convection cooling of a continuously moving surface in manufacturing processes, Journal of Materials Processing Technology 138 (2003) 332e338. [12] E.M. Sparrow, J.P. Abraham, Universal solutions for the streamwise variation of the temperature of a moving sheet in the presence of a moving fluid, International Journal of Heat and Mass Transfer 48 (2005) 3047e3056. [13] M.V. Karwe, Y. Jaluria, Thermal transport from a heated moving surface, ASME Journal of Heat Transfer 108 (1986) 728e733. [14] M.V. Karwe, Y. Jaluria, Fluid flow and mixed convection transport from a plate in rolling and extrusion processes, ASME Journal of Heat Transfer 110 (1988) 635e661. [15] M.V. Karwe, Y. Jaluria, Numerical simulation of thermal transport associated with a continuously moving flat sheet in materials processing, ASME Journal of Heat Transfer 113 (1991) 612e619. [16] S.R. Choudhury, Y. Jaluria, Forced convective heat transfer from a continuously moving heated cylindrical rod in material processing, ASME Journal of Heat Transfer 116 (1994) 724e734. [17] S.L. Lee, J.S. Tsai, Cooling of a continuously moving sheet of finite thickness in the presence of natural convection, International Journal of Heat and Mass Transfer 33 (1990) 457e464. [18] I. M-Char, C.K. Chen, J.W. Cleaver, Conjugate forced convection heat transfer from a continuously moving flat sheet, International Journal of Heat and Fluid Flow 11 (1990) 257e261. [19] F. Mendez, C. Trevino, Heat transfer analysis on a moving flat sheet emerging into a quiescent fluid, Journal of Thermophysics and Heat Transfer 16 (2002) 373e378. [20] S.R. Choudhury, Y. Jaluria, Analytical solution for the transient temperature distribution in a moving rod or plate of finite length with surface heat transfer, International Journal of Heat and Mass Transfer 37 (1994) 1193e1205. [21] Y. Jaluria, A.P. Singh, Temperature distribution in a moving material subjected to surface energy transfer, Computer Methods in Applied Mechanics and Engineering. 41 (1983) 145e156. [22] A.J. Chapman, Fundamentals of Heat Transfer. Macmillan, 1987, p. 671. [23] A.D. Kraus, A. Aziz, J.R. Welty, Extended Surface Heat Transfer. John Wiley, 2001, pp. 700e705. [24] L.M. Jiji, Heat Conduction. Begell House, New York, 2000. [25] R.N. Smith, C.H. Doumanidis, R. Pitchumani, Heat transfer in manufacturing and materials processing. in: A. Bejan, A.D. Kraus (Eds.), Heat Transfer Handbook. John Wiley, 2003 Chapter 17. [26] A. Aziz, Performance of a convective-radiative rectangular fin with temperature dependent surface emissivity, under preparation. [27] A. Aziz, Heat Conduction with Maple. R.T. Edwards, Philadelphia, PA, 2006. [28] E.M. Sparrow, E.R. Niewerth, Radiating, conducting and convecting fins: numerical and linearized solutions, International Journal of Heat and Mass Transfer 11 (1968) 377e379. [29] Maple 13. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Convection-radiation from a continuously moving, variable thermal conductivity sheet or rod undergoing thermal processing A. Aziz a, *, Robert J. Lopez b, c a
Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA Rose-Hulman Institute of Technology, Terre Haute, IN 47803, USA c Maplesoft, Waterloo, Ontario, Canada N2V 1K8 b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 September 2010 Received in revised form 23 March 2011 Accepted 24 March 2011 Available online 22 April 2011
A numerical study of the heat transfer process in a continuously moving sheet or rod of variable thermal conductivity that loses heat by simultaneous convection and radiation is reported. The process is governed by five dimensionless parameters, namely Peclet number Pe, thermal conductivity parameter a, convection-conduction parameter Nc, radiation-conduction parameter Nr, and sink-to-base temperatureratio, qa. The effects of these parameters on the temperature distribution, base heat conduction rate, advection and surface heat loss are illustrated and explained. The dimensionless length parameter, L*, that the material must traverse to cool to within one percent of the sink temperature is determined for various combinations of the five parameters. This information gives the designer the fabrication time if the material is moved at a certain speed or it gives the speed if the processing of the material is to be completed in a fixed period of time. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Thermal processing Moving sheet or rod Convection-radiation Variable thermal conductivity Processing time
1. Introduction In processes such as extrusion, glass fiber drawing, hot rolling and casting, the material being manufactured is processed thermally by allowing it to exchange heat with the ambient while it is in continuous motion [1e8]. The purpose of the thermal treatment is to cool the material to a desirable temperature before it is spooled or removed. The velocity of the material can be extremely low (few centimeters per hour) such as in crystal growth or very fast (few meters per second) as in optical fiber drawing. As the material at high temperature emerges from a furnace or a die, it is exposed to the colder ambient and a transient conduction process accompanied by surface heat loss is initiated. For the slow-moving material, the initial transient dies out with the passage of time and the process quickly attains a steady state. However, for a fast-moving material, the temperature distribution may continue to evolve with time during the entire duration of thermal processing. Studies of fluid flow and heat transfer from a heated moving surface may be classified into three groups. The first group comprises of papers in which the fluid flow over the surface is assumed to be either induced by the motion of the surface in an
* Corresponding author. E-mail address: [email protected] (A. Aziz). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.03.014
otherwise quiescent fluid or the flow is driven by an independent source. Assuming a specific thermal boundary condition at the surface, the applicable continuity, momentum and energy equations are solved to obtain the velocity and temperature fields in the fluid. The final outcome is the surface heat transfer data such as the convection heat transfer coefficient. This group of papers does not address the problem of thermal transport in the moving material itself. The number of papers belonging to this group is vast, covering Newtonian and non-Newtonian fluids, surface mass transfer, magnetic and electric effects, different thermal boundary conditions, combined free and forced convection, etc. References [9e12] provide a representative sample of such studies chosen from the recent literature. The second group of papers considers the thermal transport in the fluid and in the moving material concurrently and treats the process as a conjugate conduction-convection process. Karwe and Jaluria [13] considered the flow field generated due to the movement of a continuous heated surface and used the CrankeNicolson finite difference method to compute the temperature fields in the fluid and in the moving material. They also derived analytical solutions for the situation when the surface convection heat transfer coefficient is known a priori. This work was later extended to include buoyancy effects in the flow field induced by the motion of heated surface, i.e., a mixed convection situation [14,15]. As expected, the effect of buoyancy was found to be more significant
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Nomenclature Ac a c cp G h k ka L L* Nc Nr P Pe q
cross-sectional area of the rod or sheet, m2 dimensionless thermal conductivity parameter, b Tb a constant specific heat of the material, J/kg K irradiation on the plate or rod from the surroundings, W/m2 convection heat transfer coefficient, W/m2 K thermal conductivity at temperature T, W/m K thermal conductivity at temperature Ta,W/m K distance from point of emergence to the point where the adiabatic condition is applied, m adjustable length parameter, dimensionless convection-conduction parameter, dimensionless radiation-conduction parameter, dimensionless perimeter of the cross-section of rod or sheet, m Peclet number, dimensionless surface heat loss by combined convection and radiation, W
when the surface moved in a vertical direction than when it moved in a horizontal direction. Choudhry and Jaluria [16] studied the forced convection induced when a cylindrical rod moved centrally along the axis of a channel. Two scenarios were considered: one in which a uniform flow in the channel assisted the flow induced by the motion of the rod; and the other with a uniform flow opposing the flow induced by the motion of the rod. For the opposing flows, the surface heat transfer rate was reduced due to the creation of a recirculation zone in the fluid adjacent to the surface of the rod. Other conjugate conduction-convection studies have been reported by Lee and Tsai [17], Char, Chen, and Cleaver [18], and Mendez and Trevino [19], among others. The third group of papers assumes that the convection heat transfer coefficient is known and focuses exclusively on the thermal transport in the moving material. The mathematical model governing the temperature distribution in the moving material is then developed by establishing an energy balance between heat conduction, heat advection due to motion, and the surface heat loss. The resulting ordinary differential equation for the steady state or the partial differential equation for the transient state, is then solved analytically or numerically. For example, Choudhry and Jaluria [20] obtained a double series solution for the twodimensional, transient temperature distribution in a moving rod or a flat plate moving with a constant velocity and losing heat by convection to the ambient fluid through a constant heat transfer coefficient. The same problem was solved numerically in a then contemporary paper by Jaluria and Singh [21]. This paper quantified the effect of Biot number (characterizing the surface convective heat loss) and Peclet number (characterizing the speed of the moving material) on the temperature distribution in the material and the surface heat loss. The focus of the present paper is to include the effect of temperature dependent thermal conductivity of the moving material in the analysis and add a radiative component to the surface heat loss mechanism. To the best of our knowledge, these improvements have not been investigated in depth in the literature. With the inclusion of radiation and variable thermal conductivity, three new parameters, in addition to the Biot number and Peclet number, emerge, namely a thermal conductivity parameter, a radiation-conduction parameter, and an environment temperature parameter. For pure convection, the effect of environment temperature is usually absorbed in defining the dimensionless
Q Qa Qc T Ta Ts Tb U X x
a b 3 q qa qb r s
surface heat loss, dimensionless advection component of heat loss, dimensionless base heat conduction, dimensionless local temperature, K sink temperature for convection, K effective sink temperature for radiation, K temperature at the base (point of emergence), K speed of moving rod or sheet, m/s distance from the point of emergence, dimensionless distance from the point of emergence, m surface absorptivity, dimensionless or thermal diffusivity of the material, m2/s a measure of thermal conductivity variation with temperature, K1 surface emissivity, dimensionless dimensionless local temperature dimensionless sink temperature dimensionless base temperature density of material, kg/m3 StefaneBoltzmann constant ¼ 5.67 108 W/m2 K4
temperature but such simplification is not possible when radiation is included in the analysis. The results to be presented will highlight the effect of these five parameters on the temperature distribution in the moving material, the surface heat loss, and, most importantly, the time needed for the material to reach thermal equilibrium with the ambient. The last piece of information allows the designers to fix, for example, the distance between the point of emergence of the material from the die or furnace to the delivery station if the material needs to be cooled to within 1% of the environment temperature. 2. Mathematical analysis Consider the thermal processing of a plate or a rod of crosssectional area A and perimeter P while it moves horizontally with a constant speed U as shown in Fig. 1. The hot plate or rod emerges from a die or furnace or an intermediate station at a constant temperature Tb. The motion of the plate or rod may induce a flow field in an otherwise quiescent surrounding medium or alternatively, the plate or rod may experience an externally driven flow over its surface. Either way, the plate or rod is exposed to a colder surrounding medium and loses heat by convection and radiation. The radiative component would play a more prominent role if the forced convection is weak or absent or when only natural convection occurs. If the plate or rod experiences irradiation G on its surface from the surroundings and if the absorptivity of the plate or rod is a, the rate of radiation absorption is aG. If the emissivity of the plate or rod is 3 and then the equivalent sink temperature is given by Ts4 ¼ aG=3s [22]. To avoid the
Fig. 1. A moving rod or sheet with surface convection and radiation.
A. Aziz, R.J. Lopez / International Journal of Thermal Sciences 50 (2011) 1523e1531
introduction of an additional variable, a common sink temperature Ta (¼Ts) < Tb is assumed for both convection and radiation. The surface of the moving material is assumed to be gray and diffuse with a constant emissivity 3. The convective flow in the surrounding medium provides a constant heat transfer coefficient h over the entire surface of the moving material. Since the material undergoing the treatment experiences a large change in its temperature during the process, the change in its thermal conductivity may be as much as 100%. For most materials, the thermal conductivity of the material k increases linearly with temperature. Thus we assume
k ¼ ka ½1 þ bðT Ta Þ
(1)
where ka is the thermal conductivity of the material at temperature Ta and b is a measure of thermal conductivity variation with temperature. It is assumed that the initial transient, which occurs when the material is exposed to the coolant, dies out quickly and the processing takes place under steady state conditions. This assumption has been made in several studies, including the one by Karwe and Jaluria [13]. The transverse Biot number encountered in material processing applications are usually less than 1.0 [11]. Even if the convective cooling is very strong, the value is not likely to exceed 2.0. The onedimensional heat conduction model when used with the Biot number 2.0 gives predictions of the cooling process that are within 3% of the two-dimensional conduction model [23]. Even when Bi is large as 6, the error between the one-dimensional and twodimensional fin solutions is about 4%. The effect of radiation on the transverse heat conduction may be evaluated by linearizing the radiation and calculating the Biot number based on the combined effective heat transfer coefficient. Since the largest values Nc and Nr used in the computations is 2.5 each, the effective Biot number is about 6.0 which as stated a little earlier entails an error of 4% at most in using the one-dimensional assumption. In view of this, the results presented in this paper are not going to be affected in any significant way by taking transverse conduction into account. The steady state energy balance for the material moving with a constant speed and losing heat by simultaneous convection and radiation may be written as
3sP 4 4 1 dT d dT hP ðT Ta Þ T Ta U ½1 þ bðT Ta Þ ¼0 a dx dx dx ka A ka A (2) where a ¼ ka/r cp is the thermal diffusivity of the material, r is the density and cp is the specific heat. The last term on the left in Eq. (2) is the advection term. The axial coordinate x is measured from the slot from which the material emerges and comes in contact with the surrounding fluid. The solution of Eq. (2) for a constant thermal conductivity (b ¼ 0) and no radiation (3 ¼ 0) for a semi-infinite fin with a constant base temperature and a finite temperature at x ¼ N has been presented by Jiji [24]. With the introduction of the following dimensionless variables,
q ¼ T=Tb ; qa ¼ Ta =Tb ; L* ¼ PL=A; X ¼ xL* =L; a ¼ bTb ; Nc ¼ hA=Pka ; Nr ¼ 3sATb3 =Pka ; Pe ¼ UA=aP
(3)
where L is the length between the point of emergence (x ¼ 0) and the point where the temperature gradient in the material is zero or where the temperature is within a specified fraction of the ambient temperature depending on which boundary condition is chosen; Nc is the convection-conduction number (essentially the Biot number); Nr is the radiation-conduction number; and Pe is the Peclet number, which represents the dimensionless speed of the moving material, Eq. (2) takes the following form
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d dq dq 4 4 ½1 þ aðq qa Þ Nc ðq qa Þ Nr q qa Pe ¼0 dX dX dX (4) In physical terms, the Peclet number is the ratio of energy advected by the material through its motion and the energy conducted through the material [2]. The parameter L* allows the length of the material exposed to the surrounding fluid to be adjusted. When the material is stationary, Pe ¼ 0 and Eq. (4) reduces to that of a stationary fin. The solution of Eq. (2) is sought under the following boundary conditions.
X ¼ 0; q ¼ 1 X ¼ L* ;
dq ¼ 0 or q ¼ cqa dX
(5a) (5b,c)
where the parameter L* must be specified to obtain the solution. The choice of insulated boundary condition at x ¼ L or X ¼ L* may not reflect the physics in its entirety but such a choice has been made in the pioneering papers by Karwe and Jaluria [13], Choudhry and Jaluria [20] and Jaluria and Singh [21] and found to provide satisfactory approximations to the more realistic condition except when L* and Pe are small. However, we did carry out the computations where this condition was replaced by a condition qðL* Þ ¼ cqa where c is constant. Sample calculations for c ¼ 0.99, that is, at X ¼ L*, the material has cooled down to 1% of the sink temperature will be presented and discussed later. The problem depicted in Fig. 1 (region 2) may be extended to include a more realistic boundary condition at x ¼ 0 by coupling it to the upstream (region1) and downstream (region 3) thermal fields. For the case of constant thermal conductivity and no surface heat loses in regions 1 and 3 and only convective heat loss in region 2, the three energy equations can be solved analytically in terms of the exponential functions as shown in Smith et al. [25]. When the temperatures in the three regions are plotted (see Fig. 17.4 on p.1239 in [25]), it is seen that for Pe ¼ 1 and 10, the dimensionless temperature q at x ¼ 0 is virtually unity. Even at Pe ¼ 0.1, its value is 0.91. In view of this, the assumption of q ¼ 1 at x ¼ 0 is justified within a 9% error at Pe ¼ 0.1 and with negligible error for Pe ¼ 1 and higher. It is unlikely that the inclusion of radiative heat loss in region 2 and/or the effect of temperature dependent thermal conductivity would invalidate the assumption of a constant temperature at x ¼ 0. There are other assumptions made in writing the heat transfer model (Eqs. (4) and (5)) that require some explanation. First, the surface of the moving rod or sheet is assumed to be diffuse, gray. We could have accommodated a temperature dependent emissivity in our model but we concluded from a separate work by the first author on a stationary convective-radiative fin in which the surface emissivity was a linear function of temperature that its effect on the thermal performance was small compared with the effect of temperature dependent thermal conductivity [26]. The inclusion of a spectral, directional emissivity would have deviated us from the main focus of this work and consequently it was judged to be beyond the scope of the present contribution. Often in the calculations of simultaneous convection and radiation from a surface, the radiation term is linearized and the two processes are described by a combined effective heat transfer coefficient. Such a simplification can introduce error of over 30% in some cases [27] and was therefore not pursued here. When only convective surface loss occurs in region 2, the dimensionless temperature q could be defined as q ¼ ðT Ta Þ=ðTb Ta Þ to eliminate the parameter qa, but this is not feasible when radiation is present [28].
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3. Numerical solutions and results 3.1. Temperature profiles The solutions of the boundary value problem represented by Eqs. (4) and (5) were obtained for L* ¼ 1, 2, and 3 using the dsolve (differential equation solver) command in the symbolic algebra package Maple 13 [29] and specifying the numeric option. Maple 13 uses a fourth-fifth order RungeeKuttaeFehlberg (rkf45) procedure as the default numeric solver for initial value problems, but does not use it in a shooting technique for the numeric solution of boundary value problems. For these problems Maple 13 uses a second-order difference scheme combined with an order bootstrap technique with mesh-refinement strategies: the difference scheme is based on either the trapezoid or midpoint rules; the order improvement/accuracy enhancement is either Richardson extrapolation or a method of deferred corrections. Each figure to be presented will show the effect of one parameter on the temperature distributions in the moving material. Fig. 2 shows the effect of the thermal conductivity parameter a on the temperature distributions for Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5, and qa ¼ 0.5. For each L*, the bottom curve corresponds to a ¼ 0 (constant thermal conductivity) and the top curve corresponds to a ¼ 1 (thermal conductivity at the base temperature is 50% higher than the thermal conductivity at the environment temperature). As the parameter a increases, the average thermal conductivity of the material increases and as expected, the result is the gradual increase in the local temperature. This behavior is exhibited for all three values of L*. The time t taken by the material to move a distance L* is t ¼ L*/U. For a fixed U, the time period for which the moving material is exposed to the surrounding fluid is proportional to L*. For smaller values of L*, the duration of exposure to the surrounding fluid is shorter and hence the material stays at relatively higher temperatures. Alternatively, one may argue that for smaller values of L*, the temperatures are higher because the energy (conduction and advection) entering at X ¼ 0 is lost over
Fig. 2. Effect of variable thermal conductivity on temperature distribution in the moving material.
a shorter distance. If the value of L* were to be increased sufficiently, a condition would be reached in which the end (X ¼ L*) temperature would very nearly equal the temperature of the surrounding fluid. The words “very nearly” are used because it is only when L* approaches infinity that the material would be in thermal equilibrium with the surrounding fluid. A detailed discussion of this important point will be offered in a later section. Fig. 3 illustrates how the temperature profiles in the material are affected by the changes in the sink temperature. The other parameters were fixed at a ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5, Pe ¼ 0.5. These values of the parameters are representative of some wire drawing applications [6,11]. As expected, the temperatures in the material progressively decrease as the sink temperature is lowered. This trend can be seen for all three values of L*. In each set of curves, the bottom curve corresponds to qa ¼ 0.2 and the subsequent curves correspond to qa ¼ 0.3, 0.4, 0.5, and 0.6 (top curve) in that order. Again, higher temperature are observed when L* is smaller and the time period for which the material is exposed to the surrounding fluid is shorter. The effect of qa is most profound on the temperature at the end of the cooling process as indicated by the relatively larger spread between the curves at the terminal points. The effect of Peclet number Pe (dimensionless speed) on the temperature distribution is illustrated in Fig. 4 with the other parameters fixed at a ¼ 0.5, Nc ¼ 0.5, Nr ¼ 0.5 and qa ¼ 0.5. As Pe increases, that is, as the material moves faster, the time for which the material is exposed to the environment gets shorter, the temperatures are consequently higher, and their variation more gradual than when the material moves at a slower speed. This conclusion applies to temperature profiles for all three values of L*. Attention is now turned to the effects of surface heat transfer mechanisms on the temperature distribution in the moving material. Fig. 5 provides the temperature profiles when the convection-conduction parameter Nc is assigned values of 0, 0.5, 1.0, 1.5, and 2.0 with the remaining parameters fixed at a ¼ 0.5, Nr ¼ 0.25, qa ¼ 0.5, and Pe ¼ 0.5. The top curve for each value of L* applies to Nc ¼ 0 (no convection) but since Nr ¼ 0.25, it represents the case of weak pure radiating cooling of the material. Other
Fig. 3. Effect of sink temperature on the temperature distribution in the moving material.
A. Aziz, R.J. Lopez / International Journal of Thermal Sciences 50 (2011) 1523e1531
Fig. 4. Effect of Peclet number on the temperature distribution in the moving material.
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Fig. 6. Effect of radiation-conduction parameter on the temperature distribution in the moving material.
curves represent the cases of simultaneous convection and radiation loss from the surface. As Nc increases, that is, as convection gets stronger, the cooling becomes more effective, promoting lower temperatures in the material. As L* increases, the cooling period increases, which is reflected in the further lowering of the temperatures in the material. Fig. 6 shows the temperature profiles when the radiationconduction parameter Nr is allowed to vary with the rest of the parameters fixed at a ¼ 0.5, Nc ¼ 0.25, qa ¼ 0.5, and Pe ¼ 0.5. For each value of L*, the top curve corresponds to zero radiation but since Nc ¼ 0.25, it represents a circumstance in which the material
is cooled by weak pure convection. Other curves represent different cases of combined convective-radiative cooling. As the radiative transport becomes stronger, the radiative cooling becomes more effective, which in turn causes the lowering of temperatures in the material. The effect is similar to what was observed in Fig. 5 with respect to the strengthening of the convective transport mechanism. It should be noted that the curves corresponding to L* ¼ 1 in Figs. 3e6 may be of theoretical interest only and may not arise in real thermal processing situations.
Fig. 5. Effect of convection-conduction parameter on the temperature distribution in the moving material.
Fig. 7. Effect of variable thermal conductivity on advection, base conduction and surface heat loss.
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Qa ¼ Peð1 qð1ÞÞ
Fig. 8. Effect of sink temperature on advection, base conduction and surface heat loss.
3.2. Surface heat loss A quantity of fundamental interest in thermal processing applications is the amount of surface heat loss from the moving material to its surrounding. The surface heat loss can be computed as follows. The energy entering at x ¼ 0 consists of conduction heat transfer plus the energy advected. A portion of this energy is lost to the surrounding fluid and the remainder is advected at x ¼ L. The energy conducted at x ¼ L is zero in view of the insulated boundary condition at x ¼ L. This energy balance can be expressed as
kA
dT ð0Þ þ rUAcp Tb q ¼ rUAcp Tð1Þ dx
(6)
where T(1) is the temperature at the exit (i.e., x ¼ L) and q is the surface heat loss. Solving for q and expressing it in dimensionless form, we have
Q ¼
qL ¼ Q c þ Qa ka Tb A
(7)
where
Qc ¼ ½1 þ aðqð0Þ qa Þ and
dq ð0Þ dX
(8)
(9)
where q(0) and q(1) are the dimensionless temperatures at X ¼ 0 and X ¼ 1, respectively. Eq. (8) gives the conduction energy entering at X ¼ 0 that is lost to the surrounding fluid, while Eq. (9) gives the advection energy that is lost to the surrounding. Eqs. (7)e(9) were incorporated in the Maple code written to generate the temperature profiles that were discussed in the preceding section. For consistency, the values of Qc, Qa and Q were computed using the same values of parameters that were used in generating Figs. 2e6. In all these calculations, the value of L* was fixed at unity. Fig. 7 illustrates the effect of Peclet number on advection Qa, the base heat conduction, Qc and the surface heat loss, Q. The effect of variable thermal conductivity is indicated by three parametric values of a, namely 0, 0.5 and 1.0. As the Peclet number increases, advected heat loss (bottom set of curves) increases but the base heat conduction rate (middle set of curves) decreases. The increase in advected heat loss offsets the decrease in base heat conduction and the surface heat loss (top set of curves) increases slightly with the increase in Peclet number. The increase in surface heat loss (the total energy input into the system) with the Peclet number was also observed by Jaluria and Singh [21]. The effect of increase in thermal conductivity with temperature, i.e., increase in parameter a is to reduce advected heat loss, increase the base heat conduction and increase the surface heat loss or the total energy input into the moving system. Fig. 8 is similar to Fig. 7 except that the sink temperature Ta is now chosen as a parameter on the curves. As the sink temperature Ta increases, both advected heat loss and base heat conduction decrease, which in turn is reflected in a decrease in the surface heat loss or the total energy input into the system. The decrease in surface heat loss with an increase in the sink temperature is the result of the decrease in the temperature difference driving the flow of energy from the moving surface to the surroundings. Fig. 9 shows how advected heat loss, base heat conduction, and surface heat loss are affected by the changes in Peclet number Pe and the radiation-conduction parameter Nr when the other parameters are kept fixed. The three solid lines represent the case of zero radiation, i.e., Nr ¼ 0. As Pe increases, advection (Qa) increases but the base conduction (Qc) decreases. At Pe z 1.8, Qa and Qc are equal. The surface heat loss is very slightly affected by Pe. When the radiation-conduction parameter is set at Nr ¼ 0.5 (see dotted curves), advection, base heat conduction, and the surface heat loss are all higher compared with the case of Nr ¼ 0. The curves for Nr ¼ 0.5 exhibit the same pattern as seen in the case of Nr ¼ 0,
Fig. 9. Effect of radiation-conduction parameter on advection, base conduction and surface heat loss.
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that is, as Pe increases, Qa increases, Qc decreases, and Q increases. The results for Nr ¼ 1 are similar to those for Nr ¼ 0.5 except that advection, base heat conduction, and the surface heat loss are all much higher compared with the case of Nr ¼ 0.5. This is understandable in view of the fact that as radiation gets stronger, it must manifest itself in enhanced surface heat loss. Figs. 10e13 give the surface heat loss data, i.e., Q, for many combination values of the parameters a, Nc, Nr, Pe and qa. Fig. 10 shows that as convection and radiation increase either individually or together, the surface heat loss increases as expected. It can be seen from Fig. 11 that as the sink is raised, the temperature difference driving the heat flow is reduced causing a reduction in surface heat loss. In Fig. 12, we illustrate the effect of the convection parameter on the surface heat loss. For a fixed Nr, as the convection gets stronger, the surface heat loss increases. The effect is similar when the convection parameter is kept fixed and radiation parameter is allowed to increase (Fig. 13). 3.3. Thermal equilibrium The temperature to which the material needs to be cooled, i.e., T(L*) depends on the desired characteristics of the material at the conclusion of thermal processing. As noted earlier, the parameter L* gives the fabrication time if the material is moved at a certain speed or alternatively, it gives the speed if the processing of the material is to be completed in a prescribed period of time. Because the degree to which the material is to be cooled varies from process to process, it gives rise to a large of number of possibilities. For the sake of illustration here, it was assumed that the material is to be cooled to within 1% of the environment temperature. A total of 19 graphs depicting the values of L* as a function of various parameters were generated. A selection of these graphs is presented here. Fig. 14 shows the variation of L* with the thermal conductivity parameter a for parametric values of qa ¼ 0.2, 0.3, 0.4, 0.5 and 0.6 with Nc ¼ Nr ¼ 0.25 and Pe ¼ 0.5. These values of Nc and Nr represent weak surface convection and radiation. The parameter L* increases if the variation of thermal conductivity with
temperature is larger. The smaller the environment temperature, the larger the time required for the material to attain thermal equilibrium with the environment. This is clearly indicated by the larger values of L* at smaller values of qa. From the graphs not shown here, it is found that the value of L* increases progressively as the Peclet number increases. At Pe ¼ 2.5 with the other parameters fixed as in Fig. 14, the value of L* is 58 at a ¼ 1.0, so that the thermal conductivity at the base temperature is twice the thermal conductivity at the surroundings temperature.
Fig. 10. Effect of convection-conduction parameter, Nc and radiation-conduction parameter, Nr on surface heat loss.
Fig. 12. Effect of convection-conduction parameter, Nc and sink temperature, qa on surface heat loss.
Fig. 11. Effect of thermal conductivity parameter, a and sink temperature, qa on surface heat loss.
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Fig. 13. Effect of radiation-conduction parameter, Nr and sink temperature, qa on surface heat loss.
The case of strong radiation and weak convection is illustrated in Fig. 15. The values of the different parameters used are the same as in Fig. 14 except that Nr has been increased from 0.25 to 2. A comparison of Fig. 15 with Fig. 14 shows that the effect of strong radiation is to reduce the values of L*. This reduction may be attributable to the fact that the bigger surface heat loss would result in a quicker attainment to thermal equilibrium. Fig. 16 gives the results for the case of strong convection and weak radiation. Except for Nc, which is now increased from 0.25 to 2.0, the rest of the
Fig. 14. Effect of thermal conductivity parameter a and sink temperature qa on parameter L*.
Fig. 15. Effect of thermal conductivity parameter, a and sink temperature, qs on parameter L*: case of weak convection and strong radiation.
parameters are fixed at the values used in Fig. 14. A comparison of Figs. 14e16 reveals that the effect of convection on the attainment of thermal equilibrium is much stronger than that of radiation. The values of L* in Fig. 16 (strong convection, weak radiation) are significantly lower than the values of L* in Fig. 15 (strong radiation, weak convection). Thus the strength of convective heat loss plays comparatively a more influential role than the strength of radiative heat loss. Fig. 17 reveals that there is a further reduction in the value of L* when both convection and radiation are simultaneously strong.
Fig. 16. Effect of thermal conductivity parameter, a and sink temperature, qa on parameter L*: case of weak radiation and strong convection.
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Acknowledgement The authors appreciate the valuable and constructive comments of the reviewers. Their input has improved the clarity and the technical quality of the paper. References
Fig. 17. Effect of thermal conductivity parameter, a and sink temperature, qa on parameter L*: case of strong radiation and strong convection.
4. Conclusions A numerical study of the heat transfer process in a continuously moving sheet or rod of variable thermal conductivity losing heat by simultaneous convection and radiation has been conducted. The process is governed by five dimensionless parameters: Peclet number Pe, thermal conductivity parameter a, convectionconduction parameter Nc, radiation-conduction parameter Nr, and sink-to-base temperature-ratio, qa. The study led to the following conclusions when all parameters except one were allowed to vary. (1) The effect of an increase in the thermal conductivity parameter a is an increase in the local temperature for the material, and in base heat conduction and surface heat loss, but a decrease in advected energy loss. (2) As the Peclet number increases, it increases the local temperature, and the advected heat loss and surface heat loss, but decreases the base heat conduction. (3) As the surface convection and/or radiation gets stronger, the effect is to lower the local temperature, increase the base heat conduction, advected energy loss and surface heat loss. (4) The dimensionless parameter c characterizing the attainment (to within a 1% difference) of thermal equilibrium increases as the Peclet number increases but decreases as the surface convection and/or radiation increase. The effect of convection on L* is more pronounced than the effect of radiation. The parameter c also increases as the thermal conductivity parameter increases and as the sink temperature decreases. (5) A comparison of present numerical results with the experimental data has not been possible because the experimental data for the precise thermal processing situation analyzed in this work could not be found in the literature.
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