On the Calculation of Flow Parameters During Adiabatic Vaporization
Abstract Generated abstract
This paper examines the calculation of flow parameters during adiabatic vaporization of heated water, emphasizing the difference between isentropic and isoenthalpic pressure release in different evaporating elements. Using experimental measurements of pressure, temperature, flow velocity, and flow rate in expanding, cylindrical, and conoidal elements, it argues that isentropic thermodynamic relations are applicable only to suitably designed expanding nozzles, while cylindrical and similar elements exhibit little conversion to kinetic energy and behave closer to isoenthalpic vaporization. The study identifies distinct pressure and temperature zones, including an active vaporizing section near the outlet of cylindrical elements, and criticizes prior calculations based on incomplete measurements. A proposed thermal calculation method for expanding elements is compared with experiment and gives a flow-rate discrepancy of about 4.5 percent in the example presented.
Full Text
ON THE CALCULATION OF FLOW PARAMETERS DURING ADIABATIC VAPORIZATION
S. V. Polozhii
(Presented by Prof. I. N. Butakov, Doctor of Technical Sciences)
In [1] the shortcomings of modern studies of determining the flow rate during the outflow of heated water, associated with the phenomenon of adiabatic vaporization, have already been revealed. Adiabatic vaporization (without heat supply during the pressure drop of the heated liquid) in the limiting case can proceed in two ways: with an isentropic and with an isoenthalpic pressure drop of the heated water. In the first case, during the process of adiabatic vaporization, steam and the kinetic energy of the flow are generated. In the second case, the process proceeds with the production of steam without a substantial increase in kinetic energy. The possibility of realizing one or another process of adiabatic vaporization depends on the type of evaporating element and on the process conditions. For a heated liquid, an isoentropic process of adiabatic vaporization can be realized only in expanding evaporating elements (nozzles) of a certain design, which excludes a significant loss of kinetic energy in the process. In the theoretical case the velocity coefficient is equal to unity. In all other cases, adiabatic vaporization proceeds with a deviation from isoentropic vaporization, and the greater the deviation, the less perfect the evaporating elements. In cylindrical, conoidal, and diaphragm-type evaporating elements, isoenthalpic vaporization, or a process close to it, occurs, without substantial acquisition of kinetic energy.
In Fig. 1, for illustration, the experimental value of the kinetic energy (flow velocity) of adiabatic vaporization is shown for various evaporating elements as a function of the initial temperature of the heated water during pressure drop into a vacuum. Table 1 gives the geometric characteristics of several evaporating elements studied at the Department of Thermal Power Installations of the Tomsk Polytechnic Institute, and Fig. 2 shows a diagram of the experimental setup on which the study of adiabatic vaporization was carried out. Expanding evaporating element No. 1, at an initial temperature of the heated water equal to the saturation temperature, makes it possible to carry out adiabatic vaporization close to the isoentropic process. Cylindrical—
Fig. 1. Flow velocity during adiabatic vaporization as a function of the temperature of the heated water: No. 1—expanding, No. 3—conoidal, No. 8—cylindrical evaporating element; \(C_0\)—theoretical velocity under isentropic pressure release.
evaporating element No. 8 and the conoidal No. 3, on the contrary, do not permit a substantial increase in the energy of the flow and are representatives of evaporating elements of the isoenthalpic type (Fig. 1). To this group of evaporating elements there cannot be applied
Table 1
Main characteristics of the evaporating elements used in the article
| Number of evaporating element | Type of evaporating element | Minimum-section diameter, mm | Opening angle, degrees | Length of expanding part, mm | Length of cylindrical part, mm | Outlet-part diameter, mm |
|---|---|---|---|---|---|---|
| No. 1 | Expanding | 2.0 | 14 | 125 | 25 | 28 |
| No. 2 | Expanding | 2.18 | 36 | 40 | 20 | 28 |
| No. 3 | Conoidal | 1.15 | — | — | — | 1.15 |
| No. 8 | Cylindrical | 2.2 | — | — | 70 | 2.2 |
the thermodynamic relations of the isoentropic process of vaporization of the form:
\[ x=\frac{S'_0-S'_2}{S''_2-S'_2}, \]
\[ C_0=\sqrt{2\,(i'_0-i_2)}. \]
Therefore, an attempt to calculate the parameters of the flow during the discharge of heated water from cylindrical nozzles on the basis of the thermodynamic relations of the isoentropic process, as was done in [2], is incorrect. The publication of this article again compels us to return to the question of the methodology of theoretical and experimental investigations of phenomena associated with adiabatic vaporization.
In the experiments of [2] the problem of investigating the physical and thermodynamic essence of the process is limited to the utmost. The experiments were set up in a very simplified manner and are not based on a correct theoretical hypothesis, as a result of which not only is the physical essence of the process not revealed, but even the changes in the temperature and pressure of the process and the factors affecting them remain unknown. All the experimental material in this work was obtained with an obviously insufficient variety and number of measurements. In the experiments only the pressure before the nozzle was measured; at some points along the length and the pressure of the medium into which the discharge occurs; the temperature of the water in the heater; and the flow rate of the heated and cold water. With such an arrangement of the experiment, the author of [2] was left unaware of the changes in pressure and temperature of the process along the entire length of the evaporating element, the distinctive character of the course of the process and its dependence on the type and shape of the evaporating elements, the initial and final parameters, etc. The same applies to the energy characteristic of the vaporization process and the factors affecting it, in particular, the temperature head, the type of evaporating elements, and so forth. The structure of the flow and the place of origin of the vapor phase were also not clarified, in view of the fact that the experiments were carried out in opaque evaporating elements. In the experiments even the temperature of the flow at the exit from the evaporating element was not measured. Nor was the pressure at the outlet section of the evaporating element measured. Instead of all this, the conclusions and statements in [2] are based on speculative considerations that have nothing in common with actual phenomena.
Fig. 2. Schematic of the experimental apparatus for studying adiabatic vapor formation under vacuum: 1 — housing, 2 — evaporator, 3 — reactive-force meter, 4 — water heater, 5 — measuring tank, 6 — condenser, 7 — water and condensate collector, 8 — vacuum pump, 9 — movable hollow measuring needle, 10 — thermocouple switch, 11 — pressure switch, 12, 13 — working and compensating thermoplates, \(p\) — pressure measurement point, \(t\) — temperature measurement point.
... phenomena in the process under investigation. This applies above all to the change in pressure in the cylindrical evaporating element and to the completion of the vapor-formation process.
As an illustration of the actual change in pressure and temperature in the evaporating elements, Figs. 3–5 present experimental curves of the variation of these parameters for several evaporating elements, at unchanged initial and final flow pressures and at different initial temperatures of the heated water. These experimental data clearly reveal the distinctive character of outflow
Fig. 3. Change in pressure during the process of adiabatic vapor formation in a cylindrical evaporating element as a function of the temperature of the heated water: \(A\) — inlet section, \(BC\) — steam-generating section, \(AB\) — liquid at saturation parameters. The subscript “0” denotes the pressure of the heated water, the subscript “1” the saturation pressure at the temperature of the heated water, the subscript “2” the final pressure downstream of the evaporating element, and the subscript “k” the critical pressure at the outlet section of the evaporating element.
in adiabatic vapor formation and its difference from the outflow of single-phase liquids.
The change in pressure of the process of adiabatic vaporization in a cylindrical evaporating element, as a function of the temperature of the heated water at unchanged pressure, occurs in two stages: in the inlet and outlet parts of the evaporating element. Moreover, the more the water is subcooled relative to the saturation temperature at the initial pressure, the greater the share of the pressure drop that falls on the outlet section. When the temperature of the heated water is equal to or lower than the saturation temperature at the ambient pressure, the entire pressure drop falls only on the inlet section, as in the outflow of cold water. When the temperature of the heated water exactly corresponds to the saturation temperature at the pressure at the entrance to the evaporating element, the entire pressure reduction falls on the outlet part of the cylindrical evaporating element. The pressure drop in the inlet section occurs down to the saturation pressure corresponding to the initial temperature of the heated water. This pressure is maintained along the length of the cylindrical part—if the frictional resistance of the evaporating element is neglected—up to the section where vaporization begins in the outlet part of the evaporating element. The pressure reduction in the inlet section characterizes the presence of subcooling of the liquid relative to the saturation temperature at its pressure at the entrance to the evaporating element. The change in pressure in the inlet and outlet sections proceeds differently. In the inlet section there is a sharp change in pressure (vertically, Fig. 3), always down to the saturation pressure at the given temperature of the heated water. In the outlet section the pressure falls according to a curvilinear dependence over some section of the evaporating element, and not to the final pressure of the medium, but to a certain critical pressure, whose value also depends on the initial temperature of the heated water at unchanged ambient pressure and unchanged water pressure at the entrance to the evaporating element. The critical pressure in the outlet cross section of the evaporating element is the higher, the closer the initial temperature of the heated water approaches the saturation temperature at the water pressure at the entrance to the evaporating element. The pressure drop from the critical pressure to the ambient pressure occurs in the free space outside the evaporating element. Another important consequence of a detailed study of the temperature and pressure of the process of adiabatic vaporization in cylindrical evaporating elements is the discovery of an active vaporizing section in the outlet part, at a distance of 4–6 mm from the outlet cross section of the evaporating element for the given magnitude of the pressure drop (Figs. 3, 4) and experimental conditions.
In [2, 3] neither the pressure nor the temperature was measured in the active section of the evaporating element, and this feature of the process remained unknown to the author of the works under consideration. The pressure-drop scheme in [2] (Fig. 2) and in [3] (Fig. 1) does not correspond to the actual process. The change in temperature of the vaporization process in a cylindrical evaporating element (Fig. 4) occurs only in the outlet part and also depends on the initial temperature of the heated water at unchanged inlet pressure and ambient pressure. The pressure drop in the inlet section does not cause vaporization, since the pressure decreases in this section only to the saturation temperature reached by the heated water in the inlet section. In the cylindrical part, an equilibrium state is maintained between pressure and temperature, and vaporization likewise does not occur, owing to the small hydraulic resistance of this section. The vapor-generating (active) part, as already noted, is only the outlet section of the cylindrical evaporating element, in which, in accordance with the drop in
pressure occurs, as does a drop of the temperature to the critical value within the evaporative element, and from the critical value to the ambient temperature outside the evaporative element. Simultaneous measurement of the pressure and temperature in the active section showed a relatively high agreement between the measured temperature and the saturation temperature found from steam tables from the measured pressure (Fig. 5). Such a peculiarity of the variation of pressure and temperature in cylindrical evaporative elements with a small pressure drop is very noteworthy and shows how incorrect the methodology is for the experimental investigation of the flow rate of heated water in [3] and in other works cited in [1], when in the experiments only the pressure was measured: before the nozzle, the ambient pressure, or additionally at certain random points along the length of the nozzle. In works [2, 3] not only is an inappropriate methodology of experimental investigations allowed, but an impermissible treatment of the data of their own experiments is also used. In one case the experimental data are supplemented by fabricated notions of those portions of the process on which no
Fig. 4. Change in the temperature of the process of adiabatic vaporization in a cylindrical evaporative element as a function of the temperature of the heated water: \(A\) — inlet section, \(AB\) — section of the equilibrium state of the pressure and saturation temperature, \(BC\) — vapor-generating section. The indices “0,” “1,” “2,” “\(k\)” denote, respectively, the initial temperature of the heated water, the saturation temperature, the temperature downstream of the evaporative element, and the critical temperature in the outlet section of the evaporative element.
Figure labels: \(P_{\text{bar}}\); distance from the evaporator throat, mm; evaporator No. 2; evaporator No. 1; No. 2; No. 1.
Fig. 5. Change in the pressure and temperature of the adiabatic vaporization process for expanding evaporating elements.
corresponding measurements of the process parameters discussed above. In Fig. 6, a, a diagram is shown of the process sections identified from experiment [3] and conjectured (solid and dashed curves, respectively). In the other case, a complete departure is permitted from the experimental data [3] identified by the same author, which is found in the subsequent work [2] (Fig. 6, b). As is seen from this figure, contrary to experiment the pressure drop is now represented by the author of [2] in a completely different way. The entire pressure drop of the process from $P_1$ to $P_2$ has been transferred to the inlet section, while the temperature change is represented as a linear function of the length of the nozzle (the dashed curve in Fig. 6, a). Moreover, the supporting experimental data for such a “law” of change in the process temperature are the measurement of the temperature at only one point—in the boiler, and then at a considerable distance from the inlet section of the nozzle. Having constructed such a process scheme, which does not exist in nature and does not correspond even to any degree to his own experimental data, the author of [2] attempts to convince the reader of the reliability of his “theoretical” conclusions for calculating the parameters in the flow during the discharge of heated water, of the correctness of his assumptions about the degree of perfection of the process, the residence time of the water in the nozzle, etc. Such theoretical studies cannot claim to solve the posed problem; they merely introduce confusion into this question. In this connection it is also necessary to note the erroneous interpretation in [2] of the length of the evaporating element as the principal factor in the degree of completion of the vaporization process. Such a hypothesis connects adiabatic vaporization not with the actual driving forces of the process—the magnitude and character of the pressure release and the temperature head—but only with the residence time of the water in the nozzle. All the conclusions and inferences in [2] are based on this assumption. The incorrectness of such a position becomes evident not only from consideration of the actual character of the change in pressure and temperature in the cylindrical nozzles of our experiments, which were discussed above (Figs. 3 and 4), but also from the experimental data of the author of [3].
The decrease in the flow rate of heated water with increasing nozzle length in the experiments of [2] (Fig. 5) is in fact explained not by a difference in the residence time of the water inside the nozzle, as is asserted in [2], but by the increase in hydraulic resistance in the longer nozzles at unchanged initial pressure and ambient pressure (Table 2). From these experiments it follows that the relative decrease in the flow rate of cold and heated water as a function of nozzle length, at unchanged $P_1$ and $P_2$,
Table 2
Dependence of the flow rate of cold and boiling water on the length of a cylindrical nozzle [2] (Fig. 5).
| Figure number in [2] | Length of cylindrical nozzle, mm | Cold water: flow rate of cold water, g/cm²·s | Cold water: flow rate relative to nozzle 2, % | Boiling water: flow rate of boiling water, g/cm²·s | Boiling water: flow rate relative to nozzle 2, % | Decrease in heated-water flow rate $\dfrac{G_{\mathrm{hot}}}{G_{\mathrm{cold}}}$ |
|---|---|---|---|---|---|---|
| 5-г | 250 | 0.97 | 100 | 0.326 | 100 | 0.33 |
| 5-д | 500 | 0.70 | 71.5 | 0.235 | 71.8 | 0.33 |
| 5-е | 1500 | 0.50 | 51.5 | 0.167 | 51.2 | 0.33 |
has one and the same value and does not depend on the length of the cylindrical nozzle. Consequently, vaporization at unchanged initial and final pressure of the process proceeds identically in all three nozzles.
Fig. 6. a) Diagram of the variation of pressure and temperature in a cylindrical nozzle during the outflow of heated water according to the data of [3].
b) The same according to the data of [2]. \(A\)—location of pressure and temperature measurement in the experiments. \(B, C, D, E\)—locations of pressure measurement in the experiments. Solid curves—change in the flow pressure according to the measurements. Dash-dotted curves—change in the process pressure according to the assumption of the author of [2, 3]. Dashed-dotted curves—change in the process temperature according to the assumption of the author of [2, 3]. \(P_1, P_2\)—measured pressure in the boiler and beyond the nozzle, \(t_{\mathrm{in}}\)—measured temperature in the boiler, \(l\)—nozzle length.
nozzles of different lengths and does not depend on the residence time of the water in the nozzle. The actual picture of the process was not revealed by the author of work [2] in his own experiments only because he did not pay attention to the change in the flow rate of cold water with increasing nozzle length and had a preconceived erroneous opinion about the factors affecting the process of vapor formation and its flow rate during the outflow of heated water.
A comprehensive study of adiabatic vapor formation under vacuum, carried out by the author at the Department of Thermal-Power Installations of TPI in order to determine experimentally the maximum possible dependences and factors affecting the process of adiabatic vapor formation, made it possible to understand this question more deeply and, in particular, to reveal the energy characteristic of the process, the change in the parameters and the degree of perfection of adiabatic vapor formation, to determine the velocity and certain other features of vapor formation during the outflow of heated water. Absence—
Table 3
Method of thermal calculation of adiabatic vapor formation and comparison
of it with experimental data
| Name of quantities | Expanding EE No. 1 | |
|---|---|---|
| Experimental data | Initial pressure and temperature of the heated water \(P_0\), bar; \(t_0\), °C | 1.025; 84.6 |
| Experimental data | Pressure and temperature in the outlet section of the evaporating element (EE), \(P_2\), bar; \(t_2\) | 0.0408; 29.0 |
| Experimental data | Pressure and temperature beyond the evaporating element \(P_{2\mathrm{cp}}\), bar; \(t_{2\mathrm{cp}}\), °C | 0.0408; 29.0 |
| Experimental data | Outflow velocity at the exit from the evaporating element \(C_2\), m/sec | 107.2 |
| Experimental data | Flow rate of the stream \(g_2\), kg/sec | 0.0243 |
| Theoretical calculations | Heat drop in the evaporating element \(\displaystyle h_s=i'_0-i'_2+(s'_0-s'_2)T_{\mathrm{H}2}\), kJ | 25.76 |
| Theoretical calculations | Outflow velocity \(\displaystyle C_s=\sqrt{2h_s}\), m/sec | 218 |
| Theoretical calculations | Relative quantity of steam \(\displaystyle x_s=\frac{(s'_0-s'_2)\,T_{\mathrm{H}2}}{r_p}\), kg/kg | 0.0872 |
| Calculated data | Velocity coefficient of the EE \(\displaystyle \varphi=\frac{C_2}{C_s}\) | 0.491 |
| Calculated data | Quantity of steam \(\displaystyle x_p=x_s+(1-\varphi)^2\frac{h_s}{r_2}\), kg/kg | 0.0946 |
| Calculated data | Specific volume of the stream at the exit from the EE \(\displaystyle v_p=v'_2+x_p(v''_2-v'_2)\), m³/kg | 3.26 |
| Calculated data | Flow rate of the stream \(\displaystyle g_p=\rho\cdot C_2\cdot f_2\), kg/sec | 0.0232 |
| Comparison of experiment and calculation | Relative calculated flow rate \(\displaystyle \beta_p=\frac{g_p}{g_2}\) | 0.955 |
| Comparison of experiment and calculation | Discrepancy between experimental and calculated flow rate \(\Delta g=(1-\beta)\cdot100\), % | 4.5 |
The absence of any substantial discrepancy between the changes in pressure and temperature of the process, together with other characteristics, confirms the thermodynamic nature of adiabatic vaporization and the possibility, with accuracy sufficient for practical purposes, of calculating the parameters of this process in expanding evaporating elements from thermodynamic relationships, in the same way as is done in calculating the parameters and processes of wet steam (adiabatic condensation), while taking into account the energetic directionality and certain specific features of these processes obtained experimentally: the pressure drop in the liquid phase without vaporization in subcooled water, the magnitude of the pressure release within the evaporating element, and a number of other factors. For example, for expanding evaporating element No. 1, Table 3 gives the procedure for the theoretical calculation of the process of adiabatic vaporization and its comparison with experimental data. The notation of the quantities in the formulas is standard. The index “0” denotes the initial parameters, “2” the parameters at the outlet section of the evaporating element, “2sr” the parameters of the medium outside the evaporating element, “s” the theoretical value for an isentropic pressure release, and, finally, the index “p” the calculated value.
REFERENCES
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Polozhii S. V. On the process of vaporization during the outflow of heated water. Izv. vuzov SSSR—Energetika, No. 2, 1960.
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Gurchenok A. A. On the calculation of parameters in a flow during the outflow of boiling water. Izv. TPI, vol. 110, 1962.
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Gurchenok A. A. Investigation of the process of outflow of boiling water through diaphragms and cylindrical nozzles. Izv. TPI, vol. 101, 1958.