On the Calculation of Mixing Processes of Ideal Gases
Abstract Generated abstract
This paper presents calculation procedures for the mixing of ideal gases using molar quantities and tabulated thermodynamic properties. It derives energy balance relations for mixing in a flow, at constant volume, and by feeding gas into a volume already occupied by another gas, including cases with kinetic energy changes and heat exchange. The author gives trial calculation methods for determining mixture temperature and then pressure or volumetric flow rate from the ideal gas equation, with simplified approximate formulas for adiabatic mixing when molar heat capacities are similar. The paper also notes practical interpolation of tabulated enthalpies for engineering calculations.
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ON THE CALCULATION OF MIXING PROCESSES OF IDEAL GASES
G. I. Fuks
§ 1
The calculation of mixing processes of ideal gases is simplified if it is carried out in moles and tables of the thermodynamic properties of gases are used [1]. Calculations of mixing in a flow are performed using the mathematical expression of the first law in the form:
\[ Q = ni-\left(n_1 i_1+n_2 i_2+\ldots\right) +\left[ nM\frac{c^2}{2} -\left(n_1M_1\frac{c_1^2}{2}+n_2M_2\frac{c_2^2}{2}+\ldots\right) \right], \tag{1}^{1)} \]
where \(n, n_1, n_2,\ldots\) are the numbers of moles of the mixture and of the first, second, ... gases being mixed;
\(M, M_1, M_2,\ldots\) are the molar weights of the mixture and of the first, second, ... gases being mixed;
\(i, i_1, i_2,\ldots\) are the enthalpies of one mole of the mixture and of the first, second, ... gases, respectively (before mixing);
\(c, c_1, c_2,\ldots\) are the velocities of the formed mixture and also of the first, second, ... gases (before mixing);
\(Q\) is the heat supplied during mixing.
From (1) we have:
\[ i+M\frac{c^2}{2} = \frac{n_1}{n}\left(i_1+M_1\frac{c_1^2}{2}\right) + \frac{n_2}{n}\left(i_2+M_2\frac{c_2^2}{2}\right) +\ldots +\frac{Q}{n}. \tag{2} \]
The quantity \(i+M\dfrac{c^2}{2}\) represents the total enthalpy of the moving gas per mole (or the stagnation enthalpy). Let us introduce the notation:
\[ (i)=i+M\frac{c^2}{2}, \]
\[ (i_1)=i_1+M_1\frac{c_1^2}{2}. \tag{3}^{2)} \]
\[ \ldots\ldots\ldots \]
\(^{1)}\) The transcription of the formulas is given in the SI system. The calculation relations are given in the MKGSS system in the notes.
\(^{2)}\) In the MKGSS system \((i)=i+AM\dfrac{c^2}{2g}\), etc.
Then
\[ (i)=\frac{n_1}{n}(i_1)+\frac{n_2}{n}(i_2)+\ldots+\frac{Q}{n}. \tag{4} \]
In calculating mixing, the quantities on the right-hand side are specified, or are determined directly from tables of thermodynamic properties of gases. It is also assumed that the pressure of the resulting mixture, \(P\), is specified; it must be lower than the minimum of the pressures of the gases being mixed, \(P_1, P_2,\ldots\).
A second expression for the total enthalpy of the mixture can be written if it is taken into account that all its components have the same temperature and the same velocity. Therefore
\[ (i)=\frac{n_1}{n} i_1^{0}+\frac{n_2}{n} i_2^{0}+\ldots+M\frac{c^2}{2}, \tag{5} \]
where \(i_1^{0}, i_2^{0}\) are the enthalpies of a mole of the first, second, ... gases, respectively, at the temperature of the mixture.
The left-hand side of (5) is calculated from (4). The temperature of the mixture is calculated by trial. We specify an assumed temperature of the mixture \(t\). This makes it possible:
- To find \(i_1^{0}, i_2^{0}\ldots\) from a table of thermodynamic properties of gases.
- To find the volumetric flow rate of the gas mixture \(V\) from the equation of state:
\[ pV=8314\,nT \tag{6} \]
and, consequently, knowing the cross section of the flow, to determine its velocity \(c\).
The calculation is completed when the temperature has been found at which the right- and left-hand sides of (5) coincide.
If the change in kinetic energy during mixing can be neglected, then instead of (4) we have:
\[ i=\frac{n_1}{n} i_1+\frac{n_2}{n} i_2+\ldots+\frac{Q}{n} \tag{7} \]
and instead of (5)
\[ i=\frac{n_1}{n} i_1^{0}+\frac{n_2}{n} i_2^{0}+\ldots \tag{8} \]
From (7) the enthalpy of a mole of the mixture is determined. From (8), by trial, the temperature of the mixture is determined.
If, in addition, heat exchange is absent in the process of mixing and the molar heat capacities of the gases being mixed differ little from one another, then from (7) and (8) we obtain:
\[ t=\frac{n_1 t_1+n_2 t_2+\ldots}{n_1+n_2+\ldots}, \tag{9} \]
*) In the MKGSS system
\[ (i)=\frac{n_1}{n}i_1^{0}+\frac{n_2}{n}i_2^{0}+\ldots+AM\frac{c^2}{2g}. \]
*) In the MKGSS system
\[ pV=848\,nT. \]
\[ T=\frac{n_1T_1+n_2T_2+\cdots}{n_1+n_2+\cdots}. \tag{9'} \]
From the last relation, using the equation of state (6), one can obtain:
\[ pV=p_1V_1+p_2V_2+\cdots . \tag{10} \]
From (10) the volumetric flow rate of the mixture \(V\) is determined directly.
When heat exchange is present, it may be assumed that at first the mixing process proceeds without heat exchange, and the preliminary temperature of the mixture is calculated from (9). Therefore a value close to the true value of the final temperature is obtained from the relation
\[ t^1 \simeq t+\frac{Q}{nC_p}, \tag{11} \]
where \(C_p\) is the mean heat capacity of a mole of the mixture at constant pressure in the temperature interval from \(t\) to \(t'\). Since relations (9) and (11) are approximate and the role of heat exchange in the mixing process is usually small, a sufficiently accurate value of \(C_p\) can be obtained from the equation:
\[ C_p \sim \frac{n_1}{n}\left(\frac{\Delta i_1^{100}}{100}\right) +\frac{n_2}{n}\left(\frac{\Delta i_2^{100}}{100}\right)+\cdots , \tag{12} \]
where \(\Delta i_1^{100}, \Delta i_2^{100}\ldots\) is the change in enthalpy of the first, second, etc. gases per \(100^\circ\text{C}\) near the temperature \(t\).
The volumetric flow rate of the gas \(V\) is calculated from (6). In all cases it is advisable first to carry out the calculation by the approximate relations (9) or (9′), (10) and (11), since then solution by trial using (4), (5), (7) and (8) is greatly simplified.
§2.
For calculating mixing at constant volume, the mathematical expression of the first law is used in the form:
\[ Q=nu-(n_1u_1+n_2u_2+\cdots), \tag{13} \]
where \(u, u_1, u_2,\ldots\) are the internal energy of a mole of the mixture, and also of a kilomole of the first, second, … gases being mixed.
From (13) we have:
\[ u=\frac{n_1}{n}u_1+\frac{n_2}{n}u_2+\cdots+\frac{Q}{n}. \tag{14} \]
In compiling tables of thermodynamic properties [1] it is assumed that the enthalpies of all gases are equal to zero at \(t=0^\circ\text{C}\). It may be considered that the internal energies of all gases have these same values. Then, proceeding from the general relation \(i=u+pv\), with account of the equation of state of a gas, we obtain for a mole of gas
\[ u=i-8314\,t\ \frac{\text{J}}{\text{kmole}}. \tag{15} \]
*) In the MKGSS system
\[ u=i-1.986\,t\ \frac{\text{kcal}}{\text{kmole}\cdot\text{deg}}. \]
Taking this relation into account, from (14) we have:
\[ i-8314t=\frac{n_1}{n}(i_1-8314t_1)+\frac{n_2}{n}(i_2-8314t_2)+\ldots+\frac{Q}{n}. \tag{16}\,^{6)} \]
In calculating mixing, all quantities on the right-hand side must be considered known, since \(i_1, i_2,\ldots\) are determined from the table of thermodynamic properties by the temperatures of the gases being mixed \(t_1, t_2,\ldots\). To calculate the temperature of the mixture it must be taken into account that the enthalpy of the mixture is
\[ i=\frac{n_1}{n}i_1^0+\frac{n_2}{n}i_2^0+\ldots \tag{7} \]
The calculation is carried out by trial. We specify an assumed mixture temperature \(t\) and, using (7), compute the corresponding value of \(i\). The correctness of the choice of \(t\) is checked by (16).
If it is assumed that heat exchange during mixing is absent \((Q=0)\) and that the heat capacities for a kilomole of the gases are close to one another, then for calculating the mixing temperature we obtain:
\[ t=\frac{n_1t_1+n_2t_2+\ldots}{n_1+n_2+\ldots}, \tag{9} \]
\[ T=\frac{n_1T_1+n_2T_2+\ldots}{n_1+n_2+\ldots}. \tag{9'} \]
of the same form as for mixing in a stream. Relation (10) is also retained, according to which in this case the pressure of the mixture is calculated.
When heat exchange is present in the mixing process, first, by (9) or (9′), the temperature of the mixture in the absence of heat exchange is calculated. Then the heating (cooling) of the resulting mixture is taken into account in accordance with the magnitude \(Q\). The computational relation for determining the mixing temperature will be:
\[ t' \simeq t+\frac{Q}{nC_v}, \tag{17} \]
where \(C_v\) is the mean heat capacity of a kilomole of the mixture in the temperature interval from \(t\) to \(t'\).
If it is taken into account that the calculations by (9) and (17) are approximate and that the influence of heat exchange in the mixing process is secondary, then the quantity \(C_v\) may be calculated from the relation:
\[ C_v \simeq \frac{n_1}{n}\left(\frac{\Delta i_1^{100}}{100}\right) +\frac{n_2}{n}\left(\frac{\Delta i_2^{100}}{100}\right) +\ldots -8314\ \frac{\mathrm{J}}{\mathrm{kmol}\,K} \tag{18}\,^{7)} \]
\(^{6)}\) In the MKGSS system
\[ i-1{,}968t= \frac{n_1}{n}(i_1-1{,}968t_1) +\frac{n_2}{n}(i_2-1{,}968t_2) +\ldots+\frac{Q}{n}. \]
\(^{7)}\) In the MKGSS system
\[ C_v \simeq \frac{n_1}{n}\left(\frac{\Delta i_1^{100}}{100}\right) +\frac{n_2}{n}\left(\frac{\Delta i_2^{100}}{100}\right) +\ldots -1{,}986\ \frac{\mathrm{kcal}}{\mathrm{kmol}\cdot\mathrm{deg}}. \]
After calculating the temperature of the mixture, its pressure is determined from the equation of state (6).
§ 3.
The relation for calculating the temperature when mixing by feeding \(n_2\) moles of gas at \(p_2\) and \(t_2\) into a volume occupied by \(n_1\) moles of gas at \(p_1\) and \(t_1\) has the form:
\[ i - 8314\,t = \frac{n_1}{n}\left(i_1 - 8314\,t_1\right) + \frac{n_2}{n} i_2 + \ldots + \frac{Q}{n}, \tag{19}^* \]
where \(i, t\) are the enthalpy and temperature of the resulting mixture;
\(i_1\) is the enthalpy of the gas initially present in the mixing volume; \(i_2\) is the enthalpy of the gas introduced into the mixing volume.
In the calculation, all quantities on the right-hand side are specified or are determined quite simply from tables of the thermodynamic properties of gases. On the other hand, the enthalpy of the resulting mixture can be calculated from (8).
An assumed temperature of the mixture \(t\) is specified, and the value of its enthalpy is calculated from (8). The correctness of the specified temperature is checked by (19). The pressure of the resulting mixture \(P\) is calculated from the equation of state (6), since the volume of the resulting mixture is known. In this case, for the indicated mixing scheme, \(P_2 < P < P_1\).
When calculating the enthalpy of individual gases and their mixtures for intermediate (between tabulated) temperature values, it should be taken into account that, within an interval of up to \(100^\circ\mathrm{C}\), linear interpolation of enthalpy is entirely permissible for engineering calculations.
LITERATURE
- M. P. Vukalovich, V. A. Kirillin, S. A. Remizov, V. S. Siletskii, V. N. Timofeev. Thermodynamic Properties of Gases. 1953.
\[ {}^*\text{ In the MKGSS system} \]
\[ i - 1.986\,t = \frac{n_1}{n}\left(i_1 - 1.986\,t_1\right) + \frac{n_2}{n} i_2 + \ldots + \frac{Q}{n} \]