Let us consider the classical transport problem of a fluid

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1 Entry Length Effects for Momentum, Heat and Mass Transfer in Circular Ducts with Laminar Flow Robert E. Hayes, 1 * Andrés Donoso-Bravo 2,3 and Joseph P. Mmbaga 1 1. Department of Chemical and Materials Engineering, University of Alberta, Edmonton, AB, T6G 2G6, Canada 2. INRIA-Chile. Communication and Information Research and Innovation Center (CIRIC), Avenida Apoquindo 2827, Piso 12, Las Condes, Santiago, Chile 3. Escuela de Ingeniería Bioquímica, Pontificia Universidad Católica de Valparaíso. General Cruz 34, Valparaíso, Chile This paper shows the results of a computational study relating to the calculation of heat and mass transfer coefficients in the entry region of a circular duct. Laminar flow with temperature dependent physical properties is used. The results are compared with the classical results obtained for a fluid with constant physical properties. It is seen that a plot of the Nusselt number versus the inverse Graetz number does not yield a unique curve, but rather depends on the channel diameter, wall temperature or flux value, and the inlet fluid velocity. The deviation from the curve obtained with constant physical properties increases as the velocity decreases. Keywords: heat transfer, entry length, developing flow, Navier-Stokes, boundary layer INTRODUCTION Let us consider the classical transport problem of a fluid entering a duct, with a focus on the developing flow region. It is generally known that when a fluid enters a duct, the no-slip velocity condition at the walls of the duct slows the fluid in the vicinity of the wall, the viscous effects spread the retardation inwards, until finally the developing boundary layers meet and the entire flow becomes a boundary layer. This intersection point defines the hydrodynamic entry length,. If a temperature difference is maintained between the surface of the duct and the fluid, heat transfer takes place, and a corresponding thermal boundary layer also develops. Depending on the nature of the wall boundary condition, a thermally fully developed condition may eventually be reached. The corresponding distance from the duct entrance is called the thermal entry length, denoted L T. The dimensionless temperature profile in the fully developed region does not vary along the axial distance. If a reaction occurs at the wall of the duct, a concentration boundary layer will develop in a similar manner to the temperature boundary layer. For a reaction that has a significant heat effect, the development of the concentration boundary layer will be tied to the thermal boundary layer. It is well known that the hydrodynamic entrance length depends on the Reynolds number (Re), the thermal entry length depends on the Re and Prandtl (Pr) numbers, and the concentration entrance length depends on the Re and Schmidt (Sc) numbers. The hydrodynamic and thermal entry lengths may have similar values (Pr near unity), or, for large Pr, the thermal entry length may be very much the larger of the two. For most fluids the ratio of the Pr and Sc (the Lewis number) is close to unity, and the thermal and concentration entry lengths should be similar in this case. For the case of laminar flow conditions for fluids with constant physical properties, this entry length problem is discussed in great detail in Shah and London [1] and Kays and Crawford. [2] The developing flow region is important for numerous engineering applications. Possibly because it is so well known, and is presented in all of the basic heat transfer textbooks, it appears that the commonly held view is that this problem is solved. However, this is not really the case, and, as will be discussed shortly, there are still papers being published today that reveal new insights into this situation. The motivation for performing the work described in this paper was an interest in the developing region of catalytic monolith reactors for combustion applications. This combined entry length problem involves a fluid with varying physical properties. However, the correlations developed for the study of the hydrodynamic and thermal entry lengths are based on simplified assumptions which often leads to incorrect conclusions, or are based on specific conditions and fluid properties, limiting the applicability of the solution. Despite the recent developments in computational power and resources, very few attempts had been made to study the entry length problems with a more realistic approach. The scope of this paper is therefore to revisit some aspects of this classical problem of the hydrodynamic and thermal entry lengths for laminar flow through pipes, with and without reaction at the wall. The main focus is on the values of the heat transfer coefficients in the entry region, although we will discuss some new insight into the hydrodynamic entry length. Although some of the results presented here may be known, they are presented again both to provide a complete picture and as a demonstration of model validation. Our main motivation for this study is to advance the knowledge for the application of monolith reactors, which contain a catalytic washcoat on the channel walls. The role of the entry length region in the reaction is not completely understood. *Author to whom correspondence may be addressed. address: bob.hayes@ualberta.ca Can. J. Chem. Eng. 93: , Canadian Society for Chemical Engineering DOI /cjce Published online 10 March 2015 in Wiley Online Library (wileyonlinelibrary.com). VOLUME 93, MAY 2015 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 863

2 HYDRODYNAMIC ENTRY LENGTH FOR INCOMPRESSIBLE FLOW Consider the laminar flow of an incompressible Newtonian fluid through a circular duct. As noted earlier, a fluid entering with a flat velocity profile will adjust over the entry length to a parabolic velocity profile, as a result of the drag exerted by the walls. The entry length is usually expressed as a function of the Re and pipe diameter D in an equation of the form D ¼ C 1 Re: ð1þ solution domain was initially an axi-symmetric pipe containing an incompressible fluid. The domain was meshed using a structured mesh of quadrilateral elements, with a very fine mesh near the wall. The no-slip condition was applied at the wall, and the axis was a line of symmetry, as usual. For the first study, a flat velocity profile was imposed at the inlet. The criterion for fully developed flow was taken as the value at which the centreline velocity reached 98 % of the maximum value, which is the usual criterion used. The results are illustrated in Figure 1. Using non-linear regression analysis we obtained the following equation for the entry length: D ¼½ð0:5422Þ1:55 þð0:0464 ReÞ 1:55 Š 1=1:55 : ð5þ Numerous investigations have been conducted to quantify the value of the constant C 1. A list of the values computed in different studies, which vary from to 0.08, is given by Durst et al. [3] An early analytical work by Langhaar [4] suggested a value of for C 1, and many works use [3] Equation (1) does not, however, incorporate the role of diffusion in the axial direction of the flow. Taking this effect into account, Atkinson [5] proposed a relationship of the form D ¼ C 0 þ C 1 Re: ð2þ The values of the coefficients C 0 and C 1 were reported as 0.59 and respectively. Recent studies by Durst et al. [3] found similar values (0.619 and , respectively), but also indicated the fact that the non-linearity of the convection terms in the momentum balance equation yield condition that does not permit /D to be simply calculated by the addition of a constant to the linear relationship, particularly in the low Reynolds number regime. To eliminate this problem they used asymptote matching [6] to interpolate between the limiting solutions to yield D ¼½ð0:619Þ1:6 þð0:0567 ReÞ 1:6 Š 1=1:6 : ð3þ A brief summary of some correlations for laminar flow through pipes is given in Table 1. The steady laminar flow of a Newtonian fluid, ignoring the effect of bulk viscosity, is governed by the Navier-Stokes equation, which gives the velocity distribution. rv rv ¼r pi þ mðrv þðrvþ T Þ 2m 3 ðr vþ I Although historically the solution of the Navier-Stokes equation has presented challenges, and much early work uses many approximations, today it is almost trivial to develop a solution, especially in two dimensions. For this work the commercial finite element package COMSOL Multiphysics Version 4.4 was used. The Table 1. Available correlations for hydrodynamic entry length of laminar flow through pipes and channels Reference Approach Correlation ( /D) [4] [5] [3] [3] Analytical Re Numerical 0.59 þ Re Numerical þ Re Numerical [(0.619) 1.6 þ ( Re) 1.6 ] 1/1.6 ð4þ This result is similar to that obtained by Durst et al. [3] The role of diffusion is significant at low Reynolds number, at least up to a value of 10, with influences up to about 100. At very low Reynolds number, the entry length is independent of Re, and therefore diffusion is the only factor. At these low values of Re the entry length is less than one pipe diameter. However, at these low values of Re, it is doubtful that the inlet boundary condition of imposed flat velocity profile is realistic, and an inlet boundary condition that accounts for diffusion across the inlet should be used instead. The simplest method to allow for diffusion across the inlet is to add a reservoir in front of the pipe, which can be modelled as an extension with slip conditions at the walls. These results are also shown in Figure 1. In this case, the entry length is given by D ¼½ð0:3194Þ2:57 þð0:0425 ReÞ 2:57 Š 1=2:57 : ð6þ It is clear that the diffusion at the entrance affects the solution, although the effect only has major significance at relatively small Reynolds number. THERMAL ENTRY LENGTH FOR CONSTANT PROPERTY FLUID The thermal development length is more complicated to deal with than the hydrodynamic length. As noted in the introduction, the thermal entry length depends on both the Prandtl and the Reynolds numbers. Furthermore, there are two cases to deal with. In the first, there is a combined or simultaneous development, where both entry lengths are of similar size, and both develop at the same time. This situation arises when there is heating from the inlet of the pipe, and the Prandtl number is not large. Alternatively, if there is an unheated starting length or if the Prandlt number is very large, then the flow can be considered to be hydrodynamically well developed over the entire length, and there is only a thermal entry length to consider. There are two issues of interest in the developing region. The first is the size of the entry length and the second is the value of the Nusselt number in the developing region. The entry length is usually defined in a similar manner to the hydrodynamic length, without considering the diffusion term, that is L T D ¼ C 1 Re Pr: ð7þ In this case the constant is often assigned a value of around [7] The steady state temperature distribution can be obtained by solving the convection equation rðkrtþ rc P v rt ¼ 0: ð8þ 864 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 93, MAY 2015

3 Figure 1. Hydrodynamic entry length plotted as a function of the Reynolds number for an incompressible fluid. Figure 2. Thermal entry length plotted as a function of the Reynolds number for an incompressible fluid with constant physical properties. Regardless of the wall boundary condition, the mean bulk temperature will continue to increase as long as a temperature difference exists, thus fully developed flow is defined as the point at which the temperature profile becomes constant. We can define a dimensionless temperature as Q ¼ T S T : ð9þ T S T m Fully developed flow then occurs when this quantity is constant. For our purposes, we defined fully developed flow as occurring at 98 % of the final value. In a similar manner to the hydrodynamic entry length, we obtained the following equation for the entry length in the absence of the reservoir: L T D ¼½ð0:5761Þ2:34 þð0:0358 Re PrÞ 2:34 Š 1=2:34 : ð10þ With a reservoir added, the equation is L T D ¼½ð0:5149Þ2:26 þð0:0383 Re PrÞ 2:26 Š 1=2:26 : ð11þ The entry length as a function of Reynolds number is shown in Figure 2. Overall, the effect of adding this reservoir is seen to be minor. HEAT TRANSFER COEFFICIENT IN THE ENTRY REGION CONSTANT PROPERTY FLUID The main interest in the thermally developing region is usually the value of the heat transfer coefficient, which in turn is expressed in terms of the Nusselt number. The Nusselt number is Nu ¼ hd k : ð12þ In the classical graphs, the Nusselt number plotted against the inverse Graetz number, defined as ðgzþ 1 z ¼ D Re Pr : ð13þ Most classical correlations presented in heat transfer textbooks give values for the Nusselt number averaged over a given length, either for a combined entry length or thermal entry length only. [7] It is often more desirable to have an expression for the local Nu number, especially when modelling reactors. In the first instance, we calculate the entry length Nusselt numbers for the constant wall temperature and constant wall flux cases for a fluid of constant physical properties and a Prandtl number of 0.7. It is well known that for the case of the combined entry length problem that the entry length, and the heat transfer coefficient depends on the value of Pr. Here we choose 0.7 to be close to the value for air, which is the fluid used in the next section and is typical for gas phase reactions. The energy conservation equation was solved as described previously, and the heat transfer coefficient was calculated from the computed temperature field. The mixing cup temperature is defined as T m ¼ R R 0 R v z r C P Trdr : ð14þ v z r C P rdr 0 The heat transfer coefficient is then defined in the usual way: ht ð S T m Þ ¼ k dt dr : r¼0 ð15þ The right hand side of Equation (15) is the heat flux at the wall, which was calculated internally by COMSOL. The mixing cup temperature was calculated from the temperature field using MATLAB. The Nu and Gz were then computed and the results plotted. The results for the local Nu are plotted in Figure 3 for the two cases (constant wall temperature and flux). We note that several workers [8,9,10] have proposed correlations for Nu for nonreacting fluids for the constant wall temperature case (Nu T ) and the constant wall flux conditions (Nu H ) with the generic form as follows: Nu ¼ A 1 þ BðGzÞ n exp C Gz : ð16þ VOLUME 93, MAY 2015 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 865

4 Figure 3. Entry length Nusselt numbers for a fluid with constant physical properties at a Pr of 0.7, simultaneously developing boundary layers, with reservoir. Similar equations have been proposed for ducts of non-circular cross section. [11,12] The constants in the equation proposed by the various workers are summarized in Table 2. For this work, the value of constant A is equal to the Nu value for fully developed flow. The remaining three constants were determined by nonlinear regression analysis. Our results for fluids of constant properties are consistent with other results presented in the literature, and thus we can have confidence that our methods are correctly implemented. We also state here, without further discussion, that the results we observed were essentially the same with and without the presence of the reservoir at the entrance (data not shown). Furthermore, we ran simulations with a variety of fluid velocities and tube diameters, and in all cases the curve was the same, as it should be. HEAT TRANSFER COEFFICIENT IN THE ENTRY REGION VARIABLE PROPERTY FLUID As mentioned, the case for incompressible flow with constant physical properties has been well studied. However, and perhaps somewhat surprisingly, the case for a variable density gas with non-constant physical properties has not, to the best of our knowledge, been addressed in detail. We now present some results obtained for a gas flow in a circular channel with simultaneously developing thermal and hydrodynamic boundary layers. For illustration purposes, the properties of air were used, expressed in terms of temperature dependent polynomial expressions found in the COMSOL database. There are two cases of interest, namely those of constant wall temperature and constant wall flux, and both were tested. Table 2. Constants for the equation for entry length Nusselt numbers from the literature and this work Nu T Nu H A B C n A B C n Reference [9] [10] This work [8] Figure 4. Entry length Nusselt numbers for air entering a 1 cm diameter circular tube from a reservoir. The wall temperature is 800 K and the entering temperature is 620 K. The values for constant physical properties are shown for reference purposes. We consider first the case of air entering a 1 cm diameter circular tube with a constant wall temperature. As discussed earlier, a reservoir was placed prior to the tube to ensure the correct imposition of the inlet boundary condition. The fluid entering the reservoir was at a constant 623 K. Figure 4 shows the variation of Nu with reciprocal Gz for a series of inlet velocities, with the values for the constant properties at a Pr of 0.7 included for reference purposes. The wall temperature was 800 K. It is seen that the value of the heat transfer coefficient in the inlet region is a function of velocity, and that it increases in value as the velocity decreases. For low velocities the heat transfer coefficient in the entry region is significantly higher than that observed for the case of constant physical properties. As the inlet velocity increases, the Nu versus Gz 1 curve moves closer to the constant property case, until at a velocity of around 1.2 m/s the line is virtually the same as for the case of constant physical properties. The change in the heat transfer coefficient is related to the change in the physical properties with temperature, especially the density. As the velocity increases, the overall temperature increases in the fluid drops, giving less change, and thus pushing the curve closer to the constant properties line. Interestingly, we do not see a large change in the heat transfer coefficient when the wall temperature is changed. Figure 5 shows the effect on the heat transfer coefficient when the wall temperature was reduced to 700 K. For a velocity of 0.6 m/s the curve is essentially unchanged, which is not surprising because we are almost at the constant property line. For the smaller velocity of 0.05 m/s the heat transfer coefficient is reduced slightly, reflecting a smaller variation in the physical properties. We have seen that the Nu versus Gz 1 curve depends in essence on the amount of change in the physical properties. We can preserve the inlet Reynolds number the same by simultaneously reducing the diameter and increasing the inlet velocity. Figure 6 shows a comparison of the results obtained for 1 mm and 1 cm diameter ducts with a constant inlet Reynolds number (velocity in the 1 mm duct was ten times that in the 1 cm diameter duct). Even though the inlet Reynolds is the same, the deviation from the constant properties line is greater when the duct diameter is smaller. We have thus far considered only the case of constant wall temperature with variable properties. To make it short, we observed similar trends when the case of constant wall flux was used. That is, we saw some dependence on the value of the flux, as 866 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 93, MAY 2015

5 Figure 5. Entry length Nusselt numbers for air entering a 1 cm diameter circular tube from a reservoir. The wall temperature is either 700 or 800 K and the entering temperature is 620 K. The values for constant physical properties are shown for reference purposes. Figure 7. Entry length Nusselt numbers for air entering a 1 mm diameter circular tube with a constant wall flux of 2500 W/m2 from a reservoir. The heat transfer coefficient increases as the velocity drops. The values for constant physical properties are shown for reference purposes. well as the velocity and diameter. A sample of the results is shown in Figure 7, where we present the results from a circular duct of 1 mm in diameter, a constant wall flux of 2500 W/m 2 and various inlet velocities. The line for the constant properties case for constant wall flux is also shown for reference purposes. As with the constant wall temperature case, we see that the inlet heat transfer coefficients increase as the velocity decreases. It is clear from the results shown that the determination of the heat transfer coefficient in the inlet region of a pipe for a fluid with variable properties is much more complicated than that for a fluid with constant properties. The implications for the case of chemical reaction are discussed in the following sections. REACTION IN THE ENTRY REGION Consider the so-called tube wall reactor, where a catalyst is located at the walls of the channel. Sometimes the catalyst is contained within a thin washcoat, such as the case of the automotive catalytic converter. Owing to our interest in monolith reactors, we will consider a duct of 1 mm in diameter (typical monolith cell size). To keep it simple, we consider a simplified case of a circular duct with the catalyst on the wall, thus there is no internal transport resistance to contend with. There have been numerous studies of both the entry length and asymptotic heat and mass transfer coefficients for this reactor type, that have addressed a number of issues including channel shape, presence of washcoat, etc. [12 17] In a wall catalyzed reactor, the boundary condition is usually neither constant wall temperature nor constant wall flux, and the heat and mass transfer coefficients typically vary between the values obtained for these two wall boundary conditions, and depend on the rate of reaction. Typically, the local Nu value in the system is approximated by interpolating between Nu T and Sh H using the interpolation formula of Brauer and Fetting: [18] Nu Nu H Nu T Nu H ¼ Da Nu : ð17þ ðda þ NuÞNu T The Damköhler number, Da, is defined for an arbitrary reaction of component A by Da ¼ ð R AÞ S D : ð18þ 4 C A;S D AB Figure 6. Entry length Nusselt numbers for air entering a circular tube with a constant wall temperature from a reservoir. Two diameters were used with a constant inlet Reynolds number. The values for constant physical properties are shown for reference purposes. The operating fluid in many tube wall reactors is a gas, and has reactions with strong heat effects. Therefore, the fluid properties, especially the density, will exhibit large changes during operation. In view of the results discussed earlier for gas phase systems, a few results were obtained for a reacting system. We consider a first order reaction, and for the purpose of illustration assume that the reactant is methane with an excess of oxygen, following Hayes and Kolaczkowski. [15] The reaction rate expressed in terms of the wall surface area is ð R A Þ S ¼ exp mol C A T m 2 s : ð19þ An inlet gas temperature of 700 K was used to ensure that the reaction was initiated at the entrance. A typical result is shown in Figure 8 for three methane inlet concentrations, assuming that the VOLUME 93, MAY 2015 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 867

6 occurs at the wall, it is therefore not completely straightforward to compute the appropriate value for the Nu number. The values obtained for constant physical properties should be used as a guideline only. It should perhaps be emphasized that the asymptotic values achieved for fully developed flow with a chemical reaction and variable properties are the same as for the classical case without reaction. Figure 8. Entry length Nusselt numbers for methane and air entering a 1 mm diameter circular tube with a catalyst on the wall. The effect of methane concentration is shown at inlet velocities of (a) 1 m/s and (b) 6 m/s. The values for constant wall temperature and flux at the appropriate velocity are shown for reference. reactor is adiabatic. Figure 8a shows the result at 1 m/s inlet velocity and Figure 8b shows the result at 6 m/s. On each graph are also plotted the curves for constant wall temperature and constant wall flux for the appropriate velocity. It is obvious that the concentration has a negligible effect on the heat transfer coefficient. In both cases the curves for the reacting case are close to those for the constant wall flux situation, which is considered typical when the reaction is kinetically controlled. The key point to make, however, is that the curves with reaction are also far from the result obtained when the physical properties are held constant. This observation has important implications for the calculation of heat and mass transfer coefficients in the entrance region of ducts when a fluid with non-constant physical properties is involved. CONCLUDING REMARKS In this paper, we have studied the influence of variable (temperature dependent) physical properties on the value of the Nusselt number in the entry region of a circular tube. The Nu number does not have the same value in this region as the corresponding value for constant physical properties. The difference in the two values depends on such factors as the fluid velocity, the overall temperature difference between the wall and the fluid, and the tube diameter. For the case when a chemical reaction NOMENCLATURE A Constant B Constant C Constant C 0 Constant C 1 Constant C A,S Surface concentration of reactant, mol/m 3 C P Constant pressure heat capacity, J/(kg K) D Diameter, m D AB Molecular diffusion coefficient, m 2 /s D a Damköhler number G z Graetz number h Heat transfer coefficient, W/(m 2 K) k Fluid thermal conductivity, W/(m K) Hydrodynamic entry length, m L T Thermal entry length, m n Exponent Nu Nusselt number Nu H Nusselt number with constant surface flux Nu T Nusselt number with constant surface temperature p Pressure, Pa Pr Prandtl number r Radial coordinate, m R Tube radius, m ( R A ) S Surface reaction rate, mol/(m 2 s) Re Reynolds number T Temperature, K T m Mixing cup temperature, K T s Surface temperature, K v Velocity, m/s v z Axial velocity, m/s z Axial coordinate, m Greek symbols Q Dimensionless temperature r Fluid density, kg/m 3 m Viscosity, Pa s ACKNOWLEDGEMENTS Partial financial support for this work was provided by NSERC. The work of Andrés Donoso-Bravo was funded by Fondecyt de Iniciacion project n and supported by CIRIC -INRIA- Chile (EP BIONATURE) through Innova Chile Project Code: 10CE REFERENCES [1] R. K. Shah, A. L. London, Laminar Flow Forced Convection in Ducts, John Wiley & Sons, New York [2] W. M. Kays, M. E. Crawford, Convective Heat and Mass Transfer, McGraw-Hill, New York THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING VOLUME 93, MAY 2015

7 [3] F. Durst, S. Ray, B. Ünsal, O. A. Bayoumi, J. Fluids Eng. 2005, 127, [4] H. L. Langhaar, J. Appl. Mech. 1942, 9, 55. [5] B. Atkinson, M. P. Brocklebank, C. C. H. Card, J. M. Smith, AIChE J. 1969, 15, 548. [6] S. W. Churchill, R. Usagi, AIChE J. 1972, 18, [7] F. P. Incropera, D. P. Dewitt, Introduction to Heat Transfer,5 th edition, John Wiley & Sons, Hoboken, USA [8] V. Grigull, H. Tratz, Int. J. Heat Mass Transfer 1965, 8, 669. [9] E. Tronconi, P. Forzatti, AIChE J. 1992, 38, 201. [10] R. E. Hayes, S. T. Kolaczkowski, Introduction to Catalytic Combustion, Gordon and Breach, Reading, UK [11] G. Groppi, A. Belloli, E. Tronconi, P. Forzatti, AIChE J. 1995, 41, [12] G. Groppi, E. Tronconi, Chem. Eng. Sci. 1997, 52, [13] N. Gupta, V. Balakotaiah, Chem. Eng. Sci. 2001, 56, [14] V. Balakotaiah, D. H. West, Chem. Eng. Sci. 2002, 57, [15] R. E. Hayes, S. T. Kolaczkowski, Catal. Today 1999, 47, 295. [16] R. E. Hayes, S. T. Kolaczkowski, W. J. Thomas, J. Titiloye, Proc. R. Soc. London 1995, A448, 321. [17] R. E. Hayes, S. T. Kolaczkowski, W. J. Thomas, J. Titiloye, I&EC Research 1996, 35, 406. [18] H. W. Brauer, F. Fetting, Chem. Eng. Technol. 1966, 38, 30. Manuscript received May 22, 2014; revised manuscript received July 8, 2014; accepted for publication July 17, VOLUME 93, MAY 2015 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING 869