On the Calculation of the Process of Adiabatic Expansion of Boiling Water
Abstract Generated abstract
This paper examines the calculation of heat drop during adiabatic expansion, or outflow, of boiling water, a quantity needed for determining velocity and flow rate. It shows that direct use of standard tabulated enthalpy and entropy values in the usual thermodynamic expression can give zero, negative, or widely scattered heat drops for small pressure differences, because the required differences are smaller than the accuracy and internal consistency of the tabular data. The author analyzes examples over several pressure ranges, tests smoothing and graphical processing, and proposes a graphical analytical approach based on the temperature entropy diagram. For small pressure differences, simplified formulas using temperature and entropy or enthalpy differences give consistent results, while for larger differences the paper recommends repeated calculations with graphical averaging.
Full Text
ON THE CALCULATION OF THE PROCESS OF ADIABATIC EXPANSION OF BOILING WATER
A. A. GURTSENOK
(Presented by Prof. Dr. Tech. Sci. G. I. Fuks)
The calculation of the velocity and flow rate in the adiabatic expansion (outflow) of boiling water is associated with determining the magnitude of the heat drop—the difference of enthalpies at the beginning and at the end of the process (Fig. 1) [1].
Fig. 1.
For an adiabatic process the following relations are valid:
\[ i_1 - i_2 = i_1' - (i_2' + xr), \tag{1} \]
\[ x = \frac{s_1' - s_2'}{s_2'' - s_2'}. \tag{2} \]
Since
then
\[ s_2''-s_2'=\frac{r}{T_2}, \tag{3} \]
\[ xr=T_2(s_1'-s_2'), \]
whence, after substitution into (1), we obtain
\[ i_1-i_2=i_1'-i_2'-T_2(s_1'-s_2')\ \frac{\mathrm{kcal}}{\mathrm{kg}} . \tag{4} \]
Here \(i_1'\), \(i_2'\) are the enthalpy of boiling water, respectively at the initial \(p_1\) and final \(p_2\) pressures, ata;1
\(x\) is the dryness fraction of the steam, kg/kg;
\(r\) is the heat of vaporization, kcal/kg;1
\(s_1'\), \(s_2'\), \(s_2''\) are, respectively, the entropy of boiling water and steam, kcal/kg·°K;1
\(T_2\) is the boiling temperature at \(p_2\), °K.
Equation (4) is convenient in that its right-hand side contains only tabulated quantities of the thermal parameters of boiling water.
In the table of thermodynamic properties of water and steam [2], the values of the thermal parameters are given with small intervals of variation of the argument (pressure, temperature). This makes it possible, when calculating the process of outflow of boiling water with a small pressure difference, to use the tabulated values of the parameters \(i'\), \(s'\) without interpolation.
In Fig. 2, a, the values of the heat-drop quantities \(i_1-i_2\) are presented, which are obtained in the calculation from (4) according to the data of tables [2].
Fig. 2.
Examples of the outflow of boiling water were calculated for a pressure difference of \(0.2\ \mathrm{kg/cm^2}\). For examples with an initial pressure of \(1\text{–}5\) ata this difference was twice the tabulated difference, while at \(5\text{–}10\) ata it was equal to the tabulated difference. From Fig. 2, a it is seen that in 19
in 45 cases the value of \(i_1-i_2\) turns out to be equal to zero or a negative quantity; in the remaining cases the quantity \(i_1-i_2\) is positive, but its numerical values give a wide scatter of points in the field \(i_1-i_2=f(p_1)\). A similar picture is obtained for examples with initial pressures of 20–200 ata (Table 1).
Table 1
Pressure difference during outflow \(p_1-p_2=2\ \mathrm{kgf/cm^2}\)
| \(p_1\), ata | 20 | 22 | 60 | 62 | 80 | 82 | 120 | 122 | 160 | 162 | 180 | 182 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| from (4) | 1.0 | 0.6 | 0.8 | 1.3 | 1.0 | −0.1 | 0.5 | 0.2 | 0.7 | 0.0 | 1.1 | 1.21 |
| \((i_1-i_2)\cdot 10\) (6) | 0.31 | 0.26 | 0.06 | 0.05 | 0.034 | 0.033 | 0.02 | 0.02 | 0.015 | 0.015 | 0.014 | 0.014 |
| \(\dfrac{\mathrm{kcal}}{\mathrm{kgf}}\) (8) | 0.31 | 0.26 | 0.05 | 0.05 | 0.035 | 0.033 | 0.02 | 0.02 | 0.015 | 0.015 | 0.015 | 0.015 |
The value of the pressure difference of \(2\ \mathrm{kgf/cm^2}\) (Table 1) exceeds, for examples with \(p_1=20\text{–}22\) ata, the tabular pressure-difference value by a factor of 4, and for examples with \(p_1=60\text{–}80\) ata by a factor of 2. From Table 1 it is seen that in two examples the quantity \(i_1-i_2\) proved to be equal to zero or to a negative quantity; in the remaining cases the value of \(i_1-i_2\) is positive, but its numerical values fluctuate over a wide range.
From the examples given in Fig. 2, a, and Table 1, it is seen that calculating by (4) the magnitude of the heat drop \(i_1-i_2\) during the outflow of boiling water, with a pressure difference close to the tabular difference of the argument, gives either clearly erroneous results or results of low reliability with respect to accuracy (the field of “curve” 2, a).
The reason for obtaining the indicated results from calculation by (4) is that the magnitude of the heat drop \(i_1-i_2\) in the examples considered will be, in absolute value, smaller than the accuracy of the values of the enthalpies of boiling water with which this quantity is given in the tables. The presence of clearly erroneous results (\(i_1-i_2\) equal to zero or to a negative quantity) shows that the tabular values of the enthalpy and entropy of boiling water are given in the tables without the required consistency between them.
To verify the statement made, the author compiled a table of first and second differences of enthalpy and entropy with a step of the argument difference (pressure) of \(0.2\ \mathrm{kgf/cm^2}\) for the entire interval of initial pressures 1–10 ata [3]. As a result it turned out that incorrect results are observed in the values of the second differences. For enthalpies, the values of the second differences in 19 cases out of 45 are equal to zero or negative. The same is observed in 11 cases for entropies.
It is characteristic to note that incorrect values of the second differences were obtained in 15 examples out of 19 for enthalpies and in 7 examples out of 11 for entropies for the same values of the argument for which erroneous values of the heat drop were obtained (Fig. 2, a).
Table 2 gives the values of the first differences obtained by the author of the article by smoothing the tabular data by means of graphical interpolation. There are also presented the values of the heat-drop quantities \(i_1-i_2\), which are obtained when calculating by (4) from the smoothed data. From Table 2 it is seen that clearly erroneous results are obtained in only three examples. However, the numerical results still contain large errors.
Table 2
| $P_1$, ata | $(i'_1-i'_2)\cdot 10^2$, kcal/kg | $(S'_1-S'_2)\cdot 10^4$, kcal/kg °K | $(i_1-i_2)10^2$, kcal/kg |
|---|---|---|---|
| 1,2 | 519 | 139 | 2 |
| 1,4 | 451 | 119 | 2 |
| 1,6 | 403 | 105 | 3 |
| 1,8 | 366 | 94 | 3 |
| 2,0 | 335 | 85 | 4 |
| 2,2 | 309 | 78 | 3 |
| 2,4 | 287 | 71,8 | 2 |
| 2,6 | 269 | 66,3 | 4 |
| 2,8 | 253 | 62 | 4 |
| 3,0 | 238 | 58,6 | 2 |
| 3,2 | 227 | 55,5 | 2 |
| 3,4 | 218 | 52,5 | 2 |
| 3,6 | 208 | 49,8 | 2 |
| 3,8 | 196 | 47,4 | 1 |
| 4,0 | 189 | 45 | 1 |
| 4,2 | 181 | 43,2 | 1 |
| 4,4 | 173 | 41,4 | 5 |
| 4,6 | 167 | 34,3 | 2 |
| 4,8 | 164 | 38,5 | 2 |
| 5,0 | 158 | 37,2 | 3 |
| 5,2 | 153 | 35,7 | 2 |
| 5,4 | 149 | 34,5 | 2 |
| 5,6 | 145 | 33,4 | 2 |
| 5,8 | 140 | 32,2 | 2 |
| 6,0 | 136 | 31,3 | 2 |
| 6,2 | 133 | 30,5 | 2 |
| 6,4 | 130 | 29,7 | 2 |
| 6,6 | 126 | 29,0 | 0 |
| 6,8 | 123 | 28,2 | 1 |
| 7,0 | 120 | 27,5 | 0 |
| 7,2 | 118 | 26,8 | 1 |
| 7,4 | 116 | 26,2 | 1 |
| 7,6 | 113 | 25,6 | 0 |
| 7,8 | 111 | 25,0 | 1 |
| 8,0 | 109 | 24,4 | 1 |
| 8,2 | 107 | 23,9 | 1 |
| 8,4 | 105 | 23,4 | 2 |
| 8,6 | 103 | 22,9 | 3 |
| 8,8 | 101 | 22,5 | 2 |
| 9,0 | 99 | 22,1 | 2 |
| 9,2 | 98 | 21,7 | 1 |
| 9,4 | 97 | 21,3 | 1 |
| 9,6 | 96 | 21,0 | 1 |
| 9,8 | 95 | 20,7 | 2 |
| 10,0 | 94 | 20,4 | 2 |
To eliminate errors, a procedure was tried in which a series of calculations of the quantity \(i_1-i_2\) from equation (4) was carried out for different values of the pressure difference \(p_1-p_2\), for a specified value of the back pressure \(p_2\), with graphical processing of the results (Fig. 3).
The graphs clearly show the degree of deviation of the values of \(i_1-i_2\) obtained from calculation by (4) from the mean values. However, for examples of outflow with a small pressure difference, the mean values obtained are unreliable. For these cases, the following graphical-analytical method using a \(T\)—\(S\) diagram proved to be better.
It is known that if the state of boiling water at the beginning of the process is characterized by point 1 (Fig. 1), then the area \(12'041\) corresponds, in scale, to the numerical value of the enthalpy \(i_1\), while the area \(22'0432\) corresponds to the enthalpy at the end of the process, \(i_2\) [4]. The magnitude of the enthalpy difference \(i_1-i_2\), accordingly, is equal to the area of figure 123.
Fig. 3.
For examples of outflow with a small pressure difference, it may be assumed that the segment of the boundary curve 1–3 is a straight line. Then one may write
\[ F_{\text{pl }123}=\tfrac{1}{2}(T_1-T_2)(s'_1-s'_2), \tag{5} \]
whence
\[ i_1-i_2=\tfrac{1}{2}(T_1-T_2)(s'_1-s'_2). \tag{6} \]
Similarly, from area \(3'312'\) we obtain
\[ i'_1-i'_2=\tfrac{1}{2}(T_1+T_2)(s'_1-s'_2). \tag{7} \]
Taking (7) into account, expression (6) is rewritten in the form
\[ i_1-i_2=\frac{T_1-T_2}{T_1+T_2}(i'_1-i'_2). \tag{8} \]
In Fig. 2, b, the values of $i_1 - i_2$, obtained by calculation from (6) for the same examples as for the curve in Fig. 2, a, are presented. The very same results (within the accuracy of the data in Fig. 2) are obtained from (8).
From Fig. 2, b it is seen that, in all examples, the values of $i_1 - i_2$ fall on a single curve in the field of the dependence $i_1 - i_2 = f(p_1)$. Similar results of calculations from (6) and (8) are given in Table 1.
Conclusions
-
In tables of the thermodynamic properties of water and steam, the values of the parameters of boiling water are given with less accuracy than the magnitude of their difference within the interval of the argument-value difference.
-
For determining the heat drop in the adiabatic expansion (outflow) of boiling water from existing tables, the most convenient methods are: for small pressure differences—calculation by equation (6) or (8); for large differences—by a series of calculations using equation (4), with graphical processing of the results.
REFERENCES
-
Gurchenok A. A. On the calculation of parameters in a flow during the outflow of boiling water. Izv. TPI, vol. 110, 1962.
-
Vukalovich M. P. Thermodynamic Properties of Water and Steam. Mashgiz, 1958.
-
Bezikovich Ya. S. Approximate Calculations. GITL, 1941.
-
Litvin A. M. Technical Thermodynamics. GEI, 1956.