EFFECTIVENESS OF THERMAL INSULATION OF COMPRESSED-AIR PIPELINES AS A MEASURE TO PREVENT THEIR FREEZING IN WINTER

Eng. B. P. AKILOV

In his article “On Methods of Drying Compressed Air,” published in the journal Promyshlennaya energetika, 1949, No. 7, the author attempted to prove that neither the terminal cooler of a compressor installation nor water separators on compressed-air pipelines can serve as a means of preventing the freezing of moisture in these pipelines and the entry of moisture into pneumatic machines.

The only measure that completely eliminates this phenomenon, which under the conditions of Siberia and the Far East causes great damage to the national economy of the USSR, consists in bringing the compressed air to the consumer in an “overheated” state, i.e., with humidity \(\varphi < 100\%\).

But if earlier a considerable part of the moisture was separated in one form or another in pipelines, terminal coolers, water separators, receivers, etc., then after insulation, it is clear, this moisture will enter the pneumatic machine completely and, if the temperature of the compressed air before it is not sufficiently high, will be released during expansion and interfere with its operation. Consequently, the relative humidity of the compressed air must be such as to preclude the release of moisture in the tool.

The purpose of the present work is to show that insulation of compressed-air pipelines, while providing a radical solution to the basic problem of their freezing, is at the same time a means of increasing the thermal potential of the working fluid—compressed air—and, on this basis, of obtaining a considerable saving of the electric power expended on compressing the air in compressors.

Let us trace the influence of the temperature of compressed air on its energetics. We shall consider two processes:

1. The application of a layer of insulation and, consequently, an increase in the temperature of the compressed air before the engine \(T'_1\) at constant pressure \(p_1\) and variable specific volume \(v'_1\).

2. The same at constant specific volume \(v_1\) and variable pressure \(p'_1\). As the basis for this consideration we shall take the theoretical cycle of an air piston machine, direct for the engine and reversed for the compressor. In this cycle the process of expansion (in the engine) or compression (in the compressor) follows a polytrope, while the processes of filling and exhaust (in the engine) and exhaust and suction (in the compressor) proceed along an isobar. We shall regard the clearance space as equal to zero.

The working cycle of some pneumatic engines is not distinguished by definiteness or theoretical simplicity, but there is no doubt that their cycles can, with a high degree of accuracy, be considered close to the adopted theoretical one. This may be said especially of pneumatic hammers, drilling machines, turbine-type machines, etc. (Kel’dyshev, Pneumatics).

The work of the theoretical polytropic cycle is expressed by the formulas

\[ L=\frac{m}{m-1}p_1v_1\left[1-\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}}\right] \tag{1} \]

or

\[ L=\frac{m}{m-1}p_1v_1\left[1-\left(\frac{v_1}{v_2}\right)^{m-1}\right], \tag{1a} \]

or

\[ L=\frac{m}{m-1}\left[p_1v_1-p_2v_2\right], \tag{1б} \]

where \(p_1, v_1\) and \(p_2, v_2\) are the parameters, respectively, of the beginning and end of expansion (engine), or of the end and beginning of compression (compressor).

Application of an insulating layer and increase in the temperature of compressed air before the engine at \(p_1=\mathrm{const}\).

As is known, at constant pressure \(p_1\) and mass delivery of compressed air \(G\), as the temperature of the compressed air increases, its volume increases. If, in this case, the filling volume of each engine remains unchanged, then the number of engines must increase.

What effect will the rise in the temperature of the compressed air have on the operation of an individual engine, the compressor, the pipelines, and the entire system as a whole?

Engine. By condition, the pressure \(p_1\) before the engine and the dimensions of the latter remain constant; consequently, points 1 and 2 of the engine cycle (Fig. 1) in the \(pV\) system do not change their coordinates. Since the polytropic exponent \(m\) and the pressure \(p_2\) also have not changed, point 3 will remain with its former coordinates. It follows from this that the power output of an individual existing engine has remained constant, i.e.

Fig. 1. Cycle of a pneumatic engine in \(p,V\)-coordinates in the process \(p_1=\mathrm{const}\)

\[ N_0=\mathrm{const}. \]

In formula (1), all quantities except \(v_1'\) remained constant, i.e.

\[ L = kv_1'. \]

From the expression for the power for an individual engine

\[ N_0 = kv_1'G_0 \]

it follows that

\[ G_0=\frac{N_0}{kv_1'} . \tag{2} \]

In other words, the weight flow rate of air to the existing engines decreased in inverse proportion to the increase in specific volume and, consequently, in temperature \(T_1'\). But since the total weight delivery of the compressors \(G\) remained constant, new engines are needed in order to use the surplus compressed air that arose as a result of the reduced air consumption by the existing engines.

The power output by these additional engines will be equal to

\[ \Delta N=\frac{m-1}{m}\,G\,p_1\,v_1 \left(\frac{v_1'}{v_1}-1\right) \left[1-\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}}\right]. \tag{3} \]

Pipelines. The magnitude of the pressure drop in pipelines is expressed by the formula

\[ \Delta p=\lambda\,\frac{l}{d}\,\frac{w^2}{2g}\,\gamma_1', \tag{4} \]

replacing \(\gamma_1'\) by \(\frac{1}{v_1'}\), \(w\)—by \(\frac{Gv_1'}{F}\), \(\lambda\,\frac{l}{d}\cdot\frac{1}{2g}\cdot\frac{1}{F^2}\)—by the coefficient \(a\), we obtain

\[ \Delta p=aG^2v_1'. \tag{5} \]

The methods of increasing the number of new engines in order to increase the consumption of compressed air, as a process adopted to fulfill the specified condition of maintaining constant pressure, may be very diverse. Of all these methods, two should be noted that have opposite effects on the magnitude of the pressure drop.

1. The additional engines receive their own separate pipeline from the compressor installation. In this case expression (5) takes the form:

\[ \Delta p=aG_0,\quad G_0v_1'=a_1G_0, \tag{5a} \]

since

\[ v_1'G_0=V=\text{const}. \]

Since the air consumption by the existing engines has decreased, the magnitude of the pressure drop in the compressed-air network will consequently also decrease.

1. The additional engines are supplied with compressed air from the existing pipelines. Here the weight flow rate \(G\) remained constant, but the specific volume \(v_1'\) increased in proportion to the temperature of the compressed air; consequently, the pressure drop will also increase in the same proportion.

In real conditions both of these cases will occur simultaneously. Moreover, the magnitude of the pressure drop itself is usually small and amounts to \(0.2\text{–}0.6\) atm. Taking these circumstances into account, one may ignore the influence of temperature on the magnitude of the resistance of pipelines.

Compressor. Under constant suction conditions, pressure \(p_1\), mass delivery \(G\), and polytropic exponent \(m\), the compressor cycle will not change; consequently, the power consumed by the compressor will remain independent of the air temperature upstream of the engine.

Thus, the only result of applying insulation to the pipelines and increasing the temperature of the compressed air upstream of the engines, under the condition of constant pressure \(p_1\) and mass flow rate \(G\), is an increase in the power delivered, realized in the additional engines, directly proportional to the increase in temperature:

\[ \Delta N_1= \frac{m}{m-1}\,G p_1 v_1 \left(\frac{v_1'}{v_1}-1\right) \left[ 1-\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}} \right]. \]

A special case of the preceding case—the application of insulation and an increase in the temperature of the compressed air at constant pressure and constant number and power of the engines

Engine. From the equality

\[ G_0=\frac{N_0}{k v_1'} \]

it follows that the mass flow rate of compressed air at constant engine power decreases inversely proportional to the specific volume \(v_1'\) or to the temperature \(T_1'\).

Pipelines. The pressure drop in the pipelines will decrease in proportion to the decrease in mass flow rate \(G_0\) (5a). But, taking into account the small magnitude of this decrease, we may neglect it.

Compressor. As in the preceding case, the conditions of the compressor cycle remain unchanged; but since the mass delivery of the compressors has changed from \(G\) to \(G_0\), the power consumed by the compressors will also change by the amount:

\[ \Delta N= \frac{m}{m-1}\,G p_1 v_1^\circ \left(1-\frac{G_0}{G}\right) \left[ 1-\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}} \right], \]

where \(v_1^\circ\) is the specific volume of the air at the end of the compression process.

From the condition for the process of heat loss through the pipeline walls at constant pressure, it follows that

\[ G_0=G\frac{v_1}{v_1'}; \qquad v_1^\circ=v_1'\frac{T_1''}{T_1'}, \]

where \(T_1''\) is the temperature of the air at the compressor outlet. From the condition of the compression process in the compressor:

\[ T_1''=T_1\left(\frac{p_1}{p_2}\right)^{\frac{m-1}{m}}. \]

From the condition of increasing the thickness of the insulation layer at constant pressure:

\[ T_1'=T_1\frac{v_1'}{v_1}, \]

where \(T_1\) is the temperature of the compressed air upstream of the engine without pipeline insulation, equal to the outside-air temperature. Taking all these relations into account, we obtain:

\[ \Delta N_2= \frac{m}{m-1}\,G p_1 v_1 \left(1-\frac{v_1}{v_1'}\right) \left[ \left(\frac{p_1}{p_2}\right)^{\frac{m-1}{m}}-1 \right]. \tag{6} \]

This saving in the power consumed by the compressor is also the sole result of insulating the pipelines at constant pressure and constant number and power of the engines.

Application of insulation and increase in temperature at constant specific volume of the compressed air before the engine

Using a similar method, it can be established that in this case, with constant compressor delivery, the following changes will occur:

1. The saving in engine power is determined by the expression

\[ \Delta N_3'=\frac{m}{m-1}G p_1 v_1 \left(\frac{p_1'}{p_1}-1\right) \left[1-\frac{1}{m}\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}}\right]. \tag{7} \]

1. The increase in the power expended by the compressors for compressing the air as a result of the increase in pressure will be:

\[ \Delta N_3''=\frac{m}{m-1}G p_1 v_1 \left(\frac{p_1}{p_2}\right)^{\frac{m-1}{m}} \left[\left(\frac{p_1'}{p_1}\right)^{\frac{m-1}{m}}-1\right]. \tag{8} \]

Consequently, the energy saving from the use of insulation at variable pressure and constant specific volume will be:

\[ \Delta N_3=\frac{m}{m-1}G p_1 v_1 \left\{ \left(\frac{p_1'}{p_1}-1\right) \left[1-\frac{1}{m}\left(\frac{p_2}{p_1}\right)^{\frac{m-1}{m}}\right] - \left(\frac{p_1}{p_2}\right)^{\frac{m-1}{m}} \left[ \left(\frac{p_1'}{p_1}\right)^{\frac{m-1}{m}}-1 \right] \right\}. \tag{9} \]

The relations (3), (6), (7), (8), and (9) are shown in the form of curves in Fig. 2. It is evident from the diagram that insulation is most advantageous in the case where constant pressure is maintained in the compressed-air network, with no increase in the number and power of the pneumatic engines. For example, under the suction and discharge conditions given above, and at a compressed-air temperature before the pneumatic engine of \(120^\circ\text{C}\), it is possible to reduce compressor delivery and, consequently, the power consumed by them by 46%.

The least advantageous application of insulation is at constant specific volume and increased pressure, since the very considerable increase in the power output of the pneumatic engines is to a large extent absorbed by the increase in the power expended by the compressors for compressing the air at elevated pressure. This is explained, first, by the fact that compressed air at the increased pressure does not expand completely in the engine, but only to the pressure \(p_2' = p_2 \frac{T_1'}{T_1}\), exceeding \(p_2\) by an amount proportional to the ratio of the absolute temperatures before the engine.

However, what most strongly affects the reduction in the benefit of insulation at variable pressure is the circumstance that, when the end-of-compression pressure is increased and the specific volume of the compressed air before the engine is constant, pipeline insulation is not the same process for the compressor and the compressed-air pipelines.

This excess power expended by the compressor, in relation to the energy saving obtained in the process of insulating the pipelines at constant pressure, at \(T_1'=120^\circ\text{C}\) before the engine amounts to 40%, whereas the work not utilized owing to the high back pressure \(p_2\) in the engine amounts to only 5.5%.

Consequently, the application of insulation will be most advantageous only in the case where the compressed-air pressure remains the same as before

application of insulation. In the absence of new consumers of compressed air, this condition will require shutting down some of the compressors and reducing the supply of air to the network. At the same time it must be borne in mind that the work delivered by the motors, together with the pressure, has remained constant. Consequently, the consumers suffer no harm from this.

In practice, of course, this process cannot be carried out in pure form, since the growth of the consumers will cause an increase in the power of their motors; moreover, some increase in the compressed-air pressure is also inevitable. All these corrections to reality force us

Fig. 2. Effect of the temperature of compressed air on the magnitude of the additional power under various conditions

subsequently to treat the insulation process at constant pressure and constant compressor delivery (case 1) as a certain average process.

Thus, it is quite obvious that insulation of compressed-air pipelines is not only the most effective means of preventing their freezing, but at the same time yields a considerable saving of electric power expended on compressing the air.

Theoretically, it would be most desirable to bring the compressed air from the compressor to the motor at a temperature as close as possible to the temperature at the compressor outlet. However, this would require exceptionally large expenditures on insulation work, which might absorb all the benefit from the insulation. Therefore the temperature of the compressed air will inevitably be lower than that at which it leaves the compressor.

There is, evidently, such a temperature of the compressed air at which the use of insulation is most advantageous.

The energy produced by compressors is accounted for in terms of their volumetric capacity, reduced to normal suction conditions. Therefore, the additional energy obtained as a result of using the heat of compressed air must likewise be represented in terms of the volumetric delivery of the compressors:

\[ \Delta V_0=\frac{\Delta N}{L}v_0, \]

where \(L\) is the work expended in compressing one kilogram of air,
\(v_0\) is the specific volume of air under normal atmospheric conditions.
The benefit from using the additional power will be equal to:

\[ E=6\Delta V_0 A c \ \text{rub/year}, \tag{10} \]

where \(A\) is the coefficient of utilization of the compressors in hours per year,
\(c\) is the cost of \(1\ \text{nm}^3\) of compressed air.
The annual costs for maintaining the insulation are determined by the expression:

\[ I=\frac{\pi d_1^2}{4}\left[\left(\frac{d_2}{d_1}\right)^2-1\right]li\frac{pc_1+pc_2}{100}, \tag{11} \]

where \(d_1\) and \(d_2\) are the inner and outer diameters of the insulation layer;
\(l\) is the length of the pipeline;
\(i\) is the cost of a unit volume of insulation;
\(pc_1\) is depreciation charges, in percent;
\(pc_2\) is charges for maintenance and repair of the insulation, in percent.

The optimum temperature \(T'_1\) and the corresponding diameter \(d_2\) can be established by comparing the costs of maintaining the insulation with the saving in electric power obtained from the insulation.

To find the dependence of \(\dfrac{d_2}{d_1}\) on the temperature \(T'_1\), we shall use the equations of heat transfer and of the heat balance of losses.

The heat losses during the flow of compressed air through the pipeline will be equal to

\[ Q=3600\frac{V_0}{v_0}c_p(T''_1-T'_1). \tag{12} \]

The equation of heat transfer through the pipeline wall:

\[ Q=kF\Delta T. \tag{13} \]

Heat-transfer coefficient:

\[ k=\frac{1}{\dfrac{1}{\alpha_1}+\dfrac{d_1}{2\lambda}\ln\dfrac{d_2}{d_1}+\dfrac{1}{\alpha_2}\cdot\dfrac{d_1}{d_2}}, \tag{14} \]

where \(\alpha_1\) is the coefficient of heat transfer from the compressed air to the steel wall of the pipeline;
\(\alpha_2\) is the coefficient of heat transfer from the outer wall of the insulation to the atmospheric air;
\(\lambda\) is the coefficient of thermal conductivity of the insulation. The resistance to heat transfer through the steel wall of the pipeline is neglected because of its comparatively small magnitude.

The logarithmic temperature difference is equal to:

\[ \Delta T = -\frac{T_1''-T_1'}{\ln \dfrac{T_1''-T_1}{T_1'-T_1}}, \tag{15} \]

where \(T_1''\) and \(T_1'\) are the temperatures of the compressed air at the beginning and at the end of the pipeline,
\(T_1\) is the temperature of the outside air.

The cooling surface is determined by the expression:

\[ F=\pi d_1 l. \tag{16} \]

Equating the right-hand sides of equations (12) and (13) to one another and substituting for \(k\) and \(\Delta T\) their values from equations (14) and (15), we solve the resulting new equation with respect to \(\dfrac{1}{k}=R\).

\[ \frac{1}{\alpha_1} + \frac{d_1}{2\lambda}\ln\frac{d_2}{d_1} + \frac{1}{\alpha_2}\cdot\frac{d_1}{d_2} = R = \frac{\pi d_1 l v_0} {3600 V_0 \ln \dfrac{T_1''-T_1}{T_1'-T_1}}. \tag{17} \]

This equation, which represents the dependence of the outside diameter of the insulation \(d_2\) on the temperature of the compressed air before the engine \(T_1'\), is best solved graphically.

Fig. 3. Dependence of the thickness of the insulation layer on a compressed-air pipeline on its temperature before the pneumatic tool:
\(\alpha_2=20\ \text{kcal}/\text{m}^2\cdot\text{h}^\circ\text{C}\); \(\alpha_2=12\ \text{kcal}/\text{m}^2\cdot\text{h}^\circ\text{C}\);
\(\lambda=0.065\ \text{kcal}/\text{m}\cdot\text{h}^\circ\text{C}\); \(l=10.00\ \text{m}\); \(T_1=15^\circ\text{C}\); \(T_1''=163^\circ\text{C}\); \(C_p=0.311\ \text{kcal}/\text{kg}^\circ\text{C}\); \(v_0=500\ \text{nm}^3/\text{min}\); \(d_1=150\ \text{mm}\).

To do this, we proceed as follows: in the coordinates \(R—T_1'\) we plot curve 2 (Fig. 3), representing the right-hand side of equation (17).

Then, in the coordinates \(R—\dfrac{d_2}{d_1}\), which have the axis \(R\) in common with the axis \(R\) of the system

$R - T_1'$, we construct curve 1, representing the left-hand side of equality (17). For convenience, both systems are placed in one quadrant, with the axis

\[ \frac{d_2}{d_1} \]

arranged upward in the opposite direction.

The method for constructing the dependence of $\dfrac{d_2}{d_1}$ on $T_1'$ is clear from the diagram (curve 3).

Having established the dependence

\[ \frac{d_2}{d_1}=f_3(T_1'), \]

we construct curves representing equations (10)—$э=F_1(T_1')$ and (11)—$I=F_2(T_1')$.

The first of these represents a straight line (Fig. 4), and the second a curve that rises steeply at high temperatures $T_1'$, which is explained by the sharp increase in the cross-section of the insulation as the diameter $d_2$ increases.

Fig. 4. Determination of the optimum temperature of compressed air before the tool

Despite the opposite signs of the functions $э$ and $I$, for convenience they are also placed in one quadrant.

By algebraically adding both functions $э$ and $I$, we obtain, as a result, a third function $э-I=F_3(T_1')$, giving the final solution of the problem of the economic advisability of applying insulation to compressed-air pipelines.

It should be noted that, in constructing these curves, the following quantities were taken as the initial data:

$\alpha_1=20\ \text{kcal}/\text{m}^2\text{h}^{\circ}\text{C};\ \lambda=0.065\ \text{kcal}/\text{mh}^{\circ}\text{C};\ \alpha_2=12\ \text{kcal}/\text{m}^2\text{h}^{\circ}\text{C};$ pipeline length $l=10000\ \text{m};\ c_p=0.311\ \text{kcal}/\text{kg}^{\circ}\text{C};\ V_0=500\ \text{nm}^3/\text{min};\ d_1=150\ \text{mm};\ A=7000\ \text{h/year};\ c=0.0055\ \text{rub}/\text{m}^3;\ i=600\ \text{rub}/\text{m}^3;\ pc_1=100\%;\ pc_2=20\%.$

Under these conditions, the cost of the insulation is recovered in one year at the very high temperature $T_1'=120^{\circ}\text{C}$, and at $T_1'=96^{\circ}\text{C}$, moreover, it gives the maximum saving in electric power.

Thus, the use of insulation, while completely eliminating freezing of pipelines and contributing to improved operation of the pneumatic

tool, fully pays for itself and can, moreover, yield a net saving of electrical energy in the very first year of operation.

REFERENCES

Zhukovsky V. S. — Technical Thermodynamics.

Keldysh V. A. — Pneumatics.

Polikovsky V. I. — Fans, Compressors, and Air Blowers.

Akidov B. P. — On methods of drying compressed air. Journal Industrial Power Engineering, No. 7, 1949.

Noted Misprints