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Heat and mass transfer in a circular tube subject to the boundary condition of the third kind is investigated. The closed form of temperature and concentration distributions, the local Nusselt number based on the total external heat transfer and convective heat transfer inside the tube, as well as the Sherwood number were obtained. The effects of Lewis number and Biot number on heat and mass transfer were investigated.
Coupled heat and mass transfer with sublimation or vapor deposition are often encountered in the lyophilization process, sublimation dehydration, cryopreservation industry, and food engineering processes [1–6]. For example, intracellular ice formation (IIF) is generally believed to be fatal to cells due to mechanical damage of the cellular ultrastructure either by the direct action or by the associated volumetric expansion during cell cryopreservation [7, 8]. Vitrification can be used to avoid IIF and potentiate the cell survival rate. Recently, the ultrafast cooling rate system is being developed in order to keep cell vitrification directly and avoid IIF damage when its cooling rate is higher enough . Coupled heat and mass transfer is encountered in the design of such ultrafast cooling system. Another application for sublimation of materials is the preparation of specimens using freeze-drying for a scanning electronic microscope (SEM) or a transmission electronic microscope (TEM) . Coupled forced convective heat and mass transfer have also been widely used in the field of heat and mass transfer enhancement. Therefore, better understanding the mechanisms of the coupled forced convective heat and mass transfer is important in the optimal design of high-efficient heat transfer system.
The theoretical solution of coupled forced heat and mass transfer between two thermally insulated parallel plates can be traced back to later 1960s by Sparrow and his co-workers [10, 11]. Kurosaki  obtained numerical solution of coupled forced convective heat and mass transfer between two uniformly heated parallel plates. Since heat and mass transfer in a circular tube is more useful than that between two parallel plates, Zhang and Chen  obtained an analytical solution of coupled laminar heat and mass transfer in a circular tube with uniform heat flux. Zhang  further analyzed the coupled forced convection heat and mass transfer in tube with the boundary condition of the third kind, which is a generic boundary condition because the boundary conditions of the first and second kind can be readily achieved by setting the Biot number to infinity (Bi → ∞) or zero (Bi → 0), respectively. Their results show that the Nusselt number based on the convective heat transfer inside the tube is identical to Sherwood number when the Lewis number is unity. In order to better understand the mechanisms of the coupled heat and mass transfer with external convections heating, it is necessary to further investigate the effects of Lewis number on the Nusselt and Sherwood Numbers. Therefore, coupled heat and mass transfer process in a circular tube at different Lewis, Biot number with the boundary condition of the third kind is theoretically investigated in this paper.
Figure 1 shows the physical model of the coupled heat and mass transfer problem under consideration. A circular tube with radius R is subject to external convective heating with a heat transfer coefficient, he, and temperature, Te. The internal flow of the tube is assumed to be a slug flow for which velocity is uniform and equal to the entrance velocity U. The concentration and temperature at the entrance are Co and To, respectively. The inner surface of the tube is coated by a thin layer of sublimable material, which can be sublimated under heating. In order to solve the problem analytically, the following assumptions are made:
The governing equations of heat and mass transfer at cylindered coordinates can be expressed as
subject to the following boundary conditions
where A and B are constant.
By defining the following dimensionless variables:
The detailed derivation of above solution is presented in the “Appendix”.
The heat and mass transfer can be evaluated by Nusselt and Sherwood number, respectively. There are two different Nusselt numbers: one of them calculated based on the total amount of heat transferred into the fluid and the other is based on the convective heat transfer coefficient inside the tube. Nusselt number based on the total heat supplied by the external heat transfer is:
Based on the heat transferred to the fluid inside the tube by convection, the Nusselt number can be obtained by
Similarly, the Sherwood number of mass transfer inside the tube can be calculated by
The following relationship between the Nusselt and Sherwood numbers can be obtained from Eq. 10
The mean dimensionless temperature and concentration can be obtained by
The dimensionless temperature and concentration gradient at the tube wall can be obtained by
Figure 2 shows the effects of the Biot number on the dimensionless wall temperature for Lew = 1.4. For very large Biot number, the dimensionless wall temperature drops drastically in a very short distance to zero, after which the inner wall temperature is identical to the temperature of the external flow. The dimensionless wall temperature gradually increases with decreasing Biot number. When Bi = 0.1, the dimensionless wall temperature become a linear function of X, which is similar to the characteristic of the boundary condition of the second kind. The mean dimensionless temperature profiles along the X-direction at different Biot numbers are shown in Fig. 3. For large Biot number, the dimensionless mean temperature decreases dramatically along the X-direction. When Bi = 0.1, the dimensionless mean temperature linearly change with X, which is consistent with the characteristics of the boundary condition of the second kind.
Figures Figures44 and and55 depict the effects of the Biot number on the variations of dimensionless inner wall concentration and mean concentrations along the X-direction. When Bi = 0.1, the concentration decreases very quickly in the entrance region (X ≤ 0.1). The dimensionless wall and mean concentration become linear functions of X for X > 0.1, which is consistent with the results obtained by boundary condition of the second kind. For large Biot number, the dimensionless wall concentration is almost uniformly equal to zero, and the dimensionless mean concentration also decreases rapidly. This means that the dimensionless inner wall concentration is equal to the saturation concentration of the local temperature. Comparing Figs. Figs.22 and and4,4, one can conclude that the effects of the Biot number on the distributions of dimensionless inner wall concentration are more sensitive than that on the dimensionless inner wall temperature. It can also be concluded from Figs. Figs.33 and and55 that the effect of Biot number on the dimensionless concentration is similar to that on the dimensionless mean temperature.
Figure 6 presents the variation of the local Nusselt number based on the total heat supplied by the external flow at different Biot number. For low Biot number (Bi = 0.1 and 1), the Nusselt number decrease quickly when X ≤ 0.02, but when X > 0.02, the Nusselt number increase with X till they becomes constant, which are 15.9 for Bi = 0.1 and 15.4 for Bi = 1 when X ≥ 0.25. For high Biot number such as Bi = 10 and 100, the Nusselt number decrease dramatically at the entrance region and reaches constant when X ≥ 0.1. Figure 7 shows the variation of the local Nusselt number based on the convective heat transfer inside the tube. Comparing Fig. 7 with Fig. 6, although the Nusselt number based on the external heat transfer is much high than that based on the convective heat transfer inside the tube, the effects of Biot number on Nusselt number based on the convective heat transfer inside the tube is similar to that on the Nusselt number based on the external heat transfer. The entrance effect on the Nusselt number of internal flow is very sensitive and the fully developed Nusselt number decrease with increasing Biot number—their values are 8.0, 7.7, 6.8, and 6.5 for Bi = 0.1, 1, 10, and 100, respectively. The effect of Biot number on the Sherwood number is illustrated in Fig. 8. Due to the entrance effect, the local Sherwood Number drop dramatically when X < 0.1 and they reach constant after X > 0.1. For fully developed conditions, The Sherwood number decrease with Biot number increase and the values of Sh are 8.0, 7.6, 6.3, and 5.8 when Bi = 0.1, 1, 10, and 100, respectively.
Figure 9 shows the effect of Lewis number on the Nusselt number based on the heat transfer of the external flow at Bi = 1 and L = 1. As shown in Fig. 9, the Nue decrease very quickly and reach constant when X > 0.1 for low Lewis Number (Lew = 0.4 and 1), however, with Lewis number increase, the Nue drop when X < 0.05 and then increase till reach constant (the constant value is around 15) for any Lewis number in the range of 0.4 ~ 5. The effect of the Lewis number on the Nue mainly concentrates on the entrance region. For Bi = 1 and L = 1, the Lewis number effect on the Nusselt number based on the convective heat transfer inside the tube has been illustrated in Fig. 10. The local Nusselt number decrease dramatically at the entrance region (X < 0.1) and become constant in the fully developed region when Lew = 0.4 and 1. The different Lewis number effect on the Nui mainly focus on the entrance region, and the effect of Lewis number on Nui in developed region can be neglected.
The effect of Lewis number on the Sherwood number is shown in Fig. 11. For the Lew = 0.4, the Sherwood number decreases when X < 0.02, and increases till reach constant at X > 0.15. For the Lew = 1, 2 and 5, the Sherwood number drop quickly at the entrance region and arrives constant at the fully developed region. The Sherwood number decrease with increasing Lewis number in the fully developed region. Comparing Figs. Figs.1010 and and11,11, it can be found that the values of Nui are identical to the values of Sh when Lew = 1 since Eq. 12 is identical to Eq. 13 when Lew = 1.
Coupled heat and mass transfer in a circular tube subject to external convection was analytically studied and the closed form solutions for temperature and concentration distribution, the local Nusselt number and local Sherwood numbers were obtained. When Lew = 1.4 and X < 0.2, the Nue decreases at first (X < 0.02) and then increases till reaches constant along the X-direction at lower Bi while it decreases when Bi is high (Bi ≥ 10). When X ≥ 0.2, Nue keeps constant for all case and the Nue decreases with the increasing Bi. The effect of Bi on the Nui is similar to that on the Nue. For any Biot number, Sh decreases dramatically along the X-direction in the entrance region (X < 0.1), while they keep constant and decrease with Bi increase in the fully developed region (X ≥ 0.1). The results also show that the effects of the Lewis number on the Nue, Nui, and Sh mainly concentrate on the entrance region.
where, F1 is an unspecified constant and β is the eigenvalue.
By using separation of variables method, the solution of can be written as
where λ is the eigenvalue. Equation 26 can be rewritten as
The general solution of Eq. 27 is
Therefore, the dimensionless concentration solution becomes
where, F2 is another unspecified constant.
Obviously, F1 = F2 when Lew = 1, thus,
When Lew ≠ 1, Eq. 34 should be replaced by
Thus, θ and can be written as
The equation should satisfy the boundary conditions of Eq. 8, i.e.,
Anjun Jiao, Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.
Yuwen Zhang, Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.
Hongbin Ma, Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211, USA.
John Critser, Department of Veterinary Biology, University of Missouri, Columbia, MO 65211, USA.