On Thermal Insulation of the Air Duct of Compressor Installations

B. M. Titov, A. N. Kabakov

To prevent carbon formation and explosion in a compressor installation, it is recommended that the temperature of the compressed air be limited to \(140^\circ\mathrm{C}\) [1]. As the air moves through the pipeline, it is cooled, with the temperature changing according to a logarithmic law [2]

\[ \ln \frac{T_1 - T_0}{T_2 - T_0} = \frac{k_{\mathrm{tr}}\cdot l}{c_p \cdot G}, \tag{1} \]

where \(T_1\) is the temperature at the beginning of the pipeline, \({}^\circ\mathrm{K}\);

\(T_2\) is the temperature at the end of the pipeline, \({}^\circ\mathrm{K}\);

\(T_0\) is the ambient temperature, \({}^\circ\mathrm{K}\);

\(k_{\mathrm{tr}}\) is the heat-transfer coefficient from the compressed air to the surrounding medium, kcal/hour m \({}^\circ\mathrm{K}\);

\(l\) is the length of the pipeline, m;

\(c_p\) is the heat capacity of the compressed air, kcal/kg \({}^\circ\mathrm{K}\);

\(G\) is the air flow rate, kg/hour.

According to the temperature regime, the air-duct network can be divided into two sections. In the first section, the compressed air is cooled to the ambient temperature; in the second, isothermal flow of the air takes place.

The first section of the pipeline is of interest. For specific conditions, its length can be determined from formula (1).

\[ l = \frac{c_p}{k_{\mathrm{tr}}}\,G \ln \frac{T_1 - T_0}{T_2 - T_0}. \tag{2} \]

The working capacity and the weight flow rate of compressed air at the consumers vary in proportion to temperature [3]

\[ \frac{G_1}{G_2} = \frac{L_2}{L_1} = \frac{T_2}{T_1}, \tag{3} \]

where \(G_1, G_2\) are the flow rates of compressed air at \(T_1\) and \(T_2\), kg/hour;

\(L_1, L_2\) are the working capacities of the air, respectively at \(T_1\) and \(T_2\), kgm/kg.

Thus, to perform one and the same work, less hot air is required by the amount

\[ \Delta G = G\left(\frac{T_1}{T_2} - 1\right). \tag{4} \]

To maintain the temperature of the compressed air, thermal insulation of the air duct may be used. The heat losses in the pipeline, expressed through the air parameters, are

\[ \Delta Q = c_p G (T_1 - T_2). \tag{5} \]

The same heat losses, expressed through the parameters of the pipeline, are

\[ \Delta Q = k'_{\mathrm{tr}} l' \Delta T_{\mathrm{av}}, \tag{6} \]

where \(k'_{\mathrm{tr}}\) is the heat-transfer coefficient from the compressor air to the environment through the pipeline with an insulating covering;
\(l'\) is the length of the insulated pipeline, m;
\(\Delta T_{\mathrm{av}}\) is the mean logarithmic temperature difference between the compressor air and the environment.

\[ \Delta T_{\mathrm{av}} = \frac{T_1 - T_2}{\ln \dfrac{T_1 - T_2}{T_2 - T_0}} . \tag{7} \]

From equations (5), (6), (7), we have

\[ c_p G (T_1 - T_2) = k'_{\mathrm{tr}} l' \frac{T_1 - T_2}{\ln \dfrac{T_1 - T_2}{T_2 - T_0}} . \tag{8} \]

Hence

\[ l' = \frac{ c_p G \ln \dfrac{T_1 - T_0}{T_2 - T_0} }{ k'_{\mathrm{tr}} }. \tag{9} \]

From formulas (2) and (9) it is evident that, all other quantities being equal, the length of the first section of the pipeline depends on the heat-transfer coefficients

\[ l' = l \frac{k_{\mathrm{tr}}}{k'_{\mathrm{tr}}}. \tag{10} \]

The heat-transfer coefficients \(k_{\mathrm{tr}}\) and \(k'_{\mathrm{tr}}\) are determined from known formulas [4, p. 106].

The costs of insulating the air duct are

\[ S_{\mathrm{ins}} = \frac{100 + \psi + \varphi}{100}\, \pi (d_2 + \delta_{\mathrm{ins}})\,\delta_{\mathrm{ins}} a_m l' + \pi (d_2 + 2\delta_{\mathrm{ins}}) a_n l' = \]

\[ = \pi l' \left[ a_m \delta_{\mathrm{ins}}^{2}(a_m d_2 + 2a_n) + a_n d_2 \right], \tag{11} \]

where \(\psi\) is the cost of applying the insulation, %;
\(\varphi\) is the cost of repairing the insulation over its service life, %;
\(\delta_{\mathrm{ins}}\) is the insulation thickness, m;
\(a_m\) is the cost of the insulating material, rubles/m\(^3\);
\(a_n\) is the cost of the covering layer, rubles/m\(^3\).

The annual reduction in operating costs when using hot air is

\[ \Delta \mathcal{E} = V n_{\mathrm{yr}} a_v f \left[ \left(\frac{T_1}{T_2} - 1\right) + \left( 1 - \sqrt{ \frac{T_0}{T_2}\frac{R_0}{R_2} } \right) \right], \tag{12} \]

where \(V\) is the compressor capacity, m\(^3\)/hour;

\(n_{\text{yr}}\) — number of hours of operation of the compressor installation per year;
\(a_{\text{v}}\) — cost of \(1\ m^3\) of compressed air, rubles/\(m^3\);
\(R_0,\ R_2\) — gas constants at temperatures \(T_0\) and \(T_2\);
\(f\) — coefficient taking account of leaks in the air-duct network.

Payback period of the pipeline insulation

\[ C_{\text{ins}}=\frac{S_{\text{ins}}}{\Delta E} = \frac{100+\psi+\varphi}{100}\cdot \frac{ \pi C_p \gamma V\left[a_m\delta_{\text{ins}}^2+\delta_{\text{ins}}(a_m d_2+2a_{\text{p}})a_{\text{p}}d_2\right] }{ \kappa_{\text{pr}}\cdot V\cdot n_{\text{yr}}\cdot a_{\text{v}} } \times \frac{ \ln \dfrac{T_1-T_0}{T_2-T_0} }{ f\left[\left(\dfrac{T_1}{T_2}-1\right)+\left(1-\sqrt{\dfrac{T_0}{T_2}\cdot\dfrac{R_0}{R_2}}\right)\right] }. \tag{13} \]

Denoting by

\[ A= \frac{100+\psi+\varphi}{100}\cdot \frac{ \pi C_p\gamma\left[a_m\delta_{\text{ins}}^2+\delta_{\text{ins}}(a_m d_2+2a_{\text{p}})a_{\text{p}}d_2\right] }{ \kappa_{\text{pr}}\cdot n_{\text{yr}}\cdot a_{\text{v}} }, \tag{14} \]

we obtain

\[ C_{\text{ins}}= \frac{ A\cdot \ln \dfrac{T_1-T_0}{T_2-T_0} }{ f\left[\left(\dfrac{T_2}{T_0}-1\right)+\left(1-\sqrt{\dfrac{T_2}{T_0}\cdot\dfrac{R_2}{R_0}}\right)\right] }. \tag{15} \]

In formula (14), the most advantageous thickness \(\delta_{\text{ins}}\) is determined by the method of payback of the last layers of insulation [5, p. 123].

Fig. 1. Design diagram of the air-duct network

Example. Calculate the expediency of insulating a mine air-duct network (Fig. 1). In the calculation take the following data:

number of hours of operation of the compressor installation per year \(n_{\text{yr}}=7000\) h;

temperature at the outlet from the compressor \(T_1=413^\circ\mathrm{K}\);

temperature of the mine environment \(T_0=298^\circ\mathrm{K}\);

cost of \(1\ m^3\) of an insulating layer of plastered mineral-wool shells — 53.34 rubles;

thermal-conductivity coefficient of the insulating layer,

\[ \lambda=0.052+0.00016t_{\text{av}},\quad \frac{\text{kcal}}{m\cdot h\cdot deg} \]

cost of painting the insulation — 0.28 rubles/\(m^2\);

leakage of compressed air from the pneumatic network — 20%;

standard service life of the insulation — 8 years.

We take the cost of installing and repairing the thermal insulation as, respectively, 15 and 10% of the cost of insulating the air duct.

The calculations show (Table 1) that, for mine conditions with a large extent and branching of the pneumatic network, in the presence of

Calculation of the effectiveness of thermal insulation of the air duct (for Example 1)

Table 1

a large number of low-power portable pneumatic mechanisms (drills, jackhammers, etc.), insulation of compressed-air ducts proves in most cases to be impractical. Thermal insulation of mine air ducts may be expedient when compressed air supplies high-power pneumatic mechanisms connected to the pneumatic network close to the compressor installation.

Under factory conditions, with a comparatively short pneumatic network and stationary equipment, insulation of air ducts is economically advantageous. It is proposed that the expediency of insulating compressed-air ducts for specific conditions be determined by the method set forth in this article.

LITERATURE

1. V. E. Gordeev, V. F. Kotov, A. I. Serbikov, Ya. K. Proshin. On explosions in air piston compressors and mains. Promyshlennaya energetika, No. 12, 1964.

2. M. A. Matveev. Temperature drop of compressed air in mine air ducts. Gornyi zhurnal, No. 4, 1951.

3. V. I. Shishkin. Reduction of compressed-air consumption by consumers. Promyshlennaya energetika, No. 4, 1964.

4. A. S. Ilyichev. Mine Pneumatic Installations, vol. 1. Ugletekhizdat, 1953.

5. S. V. Khizhnyakov. Practical Calculations of Thermal Insulation of Industrial Equipment and Pipelines. Energiya Publishing House, Moscow–Leningrad, 1964.