Tree Configuration Thermosyphon Study - LEPTEN

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( r / r ) ln e i R7 = 2π k l (3) t c The R 6 resistance is related to the process of vapor condensation in the internal walls of the condenser. It is defined as the difference between the temperature of the saturated vapor and the temperature of the internal wall of the condenser divided by the rate of heat released during the condensation of the vapor. As presented by Incropera and de Witt 3 and Bejan 4 , among others, Nusselt was the first researcher to model the process of condensation on a vertical wall for the case of laminar liquid film flow and in the absence of shear stresses at the interface liquid-vapor. Faghri 1 presents a simple correlation for Nusselt’s laminar condensation model, which can be written in the following form: 1 3 * ⎛ Re f ⎞ Nu = 0.925 ⎜ ⎟ 4 (4) ⎝ ⎠ where the modified Nusselt number (Nu * ) and the liquid film Reynolds number (Re f ) are defined, respectively, as: 1 3 2 * h ⎡ ⎤ c ν l ⎛ ρ ⎞ Nu ⎢ ⎜ l ≡ ⎟⎥ (5) kl ⎢⎣ g ⎝ ρl − ρv ⎠⎥⎦ 4Q Re f = , (6) π d h µ i lv l Faghri 1 also presents extensions of the Nusselt laminar condensation theory that take into account for the shear stresses resulting from the vapor flowing in the opposite direction to the liquid film. According to Faghri 1 , the falling film is laminar for Re f < 30 and turbulent for Re f > 2000. For 30 < Re f < 1300 the flow is wavy-laminar, which means laminar flow with waves on the film surface. In most the applications, the falling film flow is turbulent or wavy-laminar and the heat transfer coefficients are larger than that in the laminar regime. However, the analytical modeling of these phenomena is difficult and normally correlations obtained from experimental data are used to predict the heat transfer coefficients during condensation. Faghri 1 and Mantelli et al. 2 , among others, present comparative studies between different correlations and models for the condensation heat transfer coefficients for applications in thermosyphons. Differences of up to one order of magnitude are observed between the different available correlations and models in the literature. Mantelli et al. (1999) also presented measurements of the condensation heat transfer coefficients obtained from a thermosyphon with similar characteristics to the one employed in this application. These authors show that the correlation proposed by Kaminaga et al. 5 was the one that better predicted their experimental data for the condenser. The correlation of Kaminaga et al. 5 can be written in the following form: hcdi 0.25 0.4 Nu ≡ = 25 Re f Prl , (7) kl The condensation resistance can be then computed as: 1 R6 = hc 2π ri l (8) c The R 2 resistance is related to the process of evaporation of the liquid. During the operation of the thermosyphon, the liquid pool generally does not fill the entire volume of the evaporator, or either, part of the internal wall of the evaporator is coated by the condensed film that returns from the condenser. Therefore, the thermal resistance of the evaporator is the association of two resistences in parallel: the resistance of evaporation of the condensed film and the resistance of evaporation of the liquid pool. The process of evaporation of a liquid film in a vertical wall is similar to the process of film condensation. The results of the Nusselt condensation laminar film theory has been used (Faghri 1 , Brost 6 ) to predict the exchange of heat associated to the evaporation of the liquid film on the walls of the evaporator. Other correlations have been proposed for the specific process of evaporation in thermosyphons based on the Nusselt theory and that take into account also for the effect of the diameter of the pipe and the presence of nucleate boiling at the interface between the tube wall and of the liquid film, as described by Faghri 1 . The heat transfer coefficients calculated according with the Nusselt theory are generally larger than observed experimentally in thermosyphons, because the condensed film is broken and transformed into rivulets of liquid, which dries part of the evaporator wall. On the other hand, when the evaporator heat flux is high, nucleate boiling occurs in the rivulets. When the vapor bubbles inside the liquid 4 American Institute of Aeronautics and Astronautics
low up, the rivulets are broken and liquid is spread all around the evaporator, wetting the entire internal wall and increasing the heat transfer coefficient (see Faghri 1 for more details). The R 3 and R 5 resistances shown in the equivalent circuit of Fig. 2 appear at the liquid-vapor interface in the evaporator and in the condenser, respectively. These resistances can generally be ignored (Brost 6 ). The resistance R 4 is associated with the drop of saturation temperature of the vapor due to the pressure drop of the vapor flow between the evaporator and the condenser. In general, this resistance can also be neglected. Finally, the R 8 resistance is due to the axial heat conduction between the evaporator and the condenser through the tube walls. It can be estimated as the average length that the heat crosses between the evaporator and the condenser, which is the summation of half the length of the evaporator, half the length of the condenser and the length of the adiabatic section, divided by the thermal conductivity and by the cross-section area of the tube walls: ( le + lc ) 2 + la R8 = (9) 2 2 ktπ ( re − ri ) In general, the axial conduction resistance R 8 is much larger than the other resistances of the equivalent circuit of Fig. 2, and, as it is in parallel with resistances R 1 , R 2 , R 3 , R 4 , R 5 , R 6 and R 7 , which are associated in series, it can be neglected. As a result, the thermosyphon total thermal resistance is given by the summation in series of the resistances R 1 , R 2 , R 6 and R 7 . The thermophysical properties appearing in the above equations must be estimated at the average temperature of the thermosyphon, given by the arithmetic mean of the average temperatures of the evaporator and of the condenser. The complete equivalent electric circuit for the tree configuration thermosyphon is presented in Fig. 3(a). In this figure, q [W] is the total heat input to the thermosyphon, while q 1 ,q 2 , q 3 and q 4 are the heat rejected by each one of the condensers. Similarly, T c,1 , T c,2 , T c,3 , T c,4 are the averages of the condensers temperatures. These temperatures are, a priori, different from each other, specially when the vapor is not supplied equally among the different condensers. However, as it will be shown later in the experimental study, the condensers temperatures are practically equal to each other. In this modeling it is assumed that all the condensers are at the same temperature. Neglecting resistances R 3 , R 4 and R 5 , according to the discussion in the preceding paragraphs, and neglecting the thermal path by axial conduction through the tube walls, i. e., assuming R 8 much larger than the other resistances (actually it is approximately 3 to 4 orders of magnitude higher), one obtain a much simpler equivalent circuit, which is shown in Fig. 3(b). Also, the resistances associated to the condensation process inside each condenser (R 6,1 , R 6,2 , R 6,3 , R 6,4 ) are assumed to be equal to each other (R 6 ). This assumption is made based on Eqs. (4) to (6), which show that the condensation resistance is dependent on the thermophysical properties of the working fluid, which are dependent on the temperature level and also dependent on the heat transfer rate released during the condensation process. Assuming equal temperature levels and equal heat transfer rate for all the condensers, one can assume that the condensation resistances are equal for all the condensers. 1 2 q = q + q + q + q 1 T [°C] T [°C] T [°C] T [°C] c,1 c,2 c,3 c,4 2 q [W] q [W] q [W] q [W] 3 3 4 4 R R6,1 R R 7 5,1 R7 R6,2 R 5,2 R7 R6,3 R 5,3 R7 R6,4 R 5,4 R7 6 R7 R6 R7 R6 R7 R6 q [W] T [°C] c R 4,1 R 4,2 R 4,3 R 4,4 R8,1 R 8,2 R R 8,3 8,4 T [°C] v R 3 R 2 R 2 R 1 R 1 T [°C] e T [°C] e q [W] (a) q [W] Figure 3. <strong>Tree</strong> configuration thermosyphon equivalent electric circuit: (a) complete circuit, (b) simplified circuit 5 American Institute of Aeronautics and Astronautics (b)
Page 1 and 2: Tree Configuration Thermosyphon Stu
Page 3: Te −Tc Rt = (1) Q where T e and T
Page 7 and 8: (a) (b) Figure 4. Experimental setu
Page 9 and 10: 90 T6, T7, T8, T9 80 70 T12, T17, T
Page 11 and 12: prediction was obtained using Eqs.
Page 13: References 1 Faghri, A., Heat Pipe

Tree Configuration Thermosyphon Study - LEPTEN

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