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Pipeline drying using dehumidified air with low dew point temperature
PII:
Applied Thermal Engineering Vol. 18, No. 5, pp. 231±244, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-4311/98 $19.00 + 0.00 S1359-4311(97)00083-5
PIPELINE DRYING USING DEHUMIDIFIED AIR WITH LOW DEW POINT TEMPERATURE Syed Younus Ahmed, P. Gandhidasan* and A. A. Al-Farayedhi Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (Received 26 September 1997) AbstractÐThe presence of humidity may be detrimental to the operation of pipelines transporting natural gas or other petroleum products. In particular conditions water solidi®es or reacts chemically with hydrocarbons, forming hydrates. Such crystalline substances may cause obstruction of the lines and damage the equipment of the relevant facilities. A procedure for predicting the performance of drying a pipeline using dehumidi®ed air with low dew point is described in this paper. The mathematical model estimates the time required for the complete removal of moisture in the pipeline for the given operating conditions with simpli®ed assumptions. The governing equations are solved analytically as well as numerically and the results are brie¯y discussed in the paper. # 1998 Elsevier Science Ltd. All rights reserved KeywordsÐPipeline, drying, time, dehumidi®cation, modelling.
NOMENCLATURE A A1,a1,b1 C* C*0 CA/C C*sat D Dva F h h1 k L m MWv MWa NA,NB NA0 p P Q Re t T Ta Te U1 V V0 W W0 x Y Ysat
pipeline cross section area, m2 constants concentration of water vapor in dry air; kg of dry air/kg of water concentration of water vapor in dry air at inlet, kg of water/kg of dry air mole fraction concentration concentration of water vapor in dry air at saturation, kg of water/kg of dry air pipeline diameter, m diusion coecient, m2/s mass transfer coecient, kg/m2s convection coecient from air to water, W/m2K spatial step, m mass transfer coecient, kg/m2s length of pipeline, m time step, s molecular weight of vapor molecular weight of air evaporation rate per unit area, kg/m2s maximum evaporation rate per unit area, kg/m2s partial pressure, mm Hg total pressure, mm Hg mass ¯ow rate of dry air, kg/s Reynolds number time, s temperature, 8C or K ambient air temperature, 8C or K entrance temperature of pipeline, 8C or K some function velocity of dry air, m/s maximum velocity of dry air, m/s amount of water per unit length of pipe, kg/m initial amount of water per unit length of pipe, kg/m spatial variable, m mole fraction concentration mole fraction concentration at saturation
*Author to whom correspondence should be addressed. 231
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Subscripts sat 1 2 1
saturation component 1 component 2 ambient condition
Greek a1,d,o,s b1 r rF n Z
constants mass transfer coecient between solution ®lm and air stream, kg/h m2 (mmHg) density, kg/m3 density of pipe material, kg/m3 dynamic viscosity, kg/ms kinematic viscosity, m2/s in Equation (14)
INTRODUCTION
To verify the integrity of the installation of pipelines, especially in the oil and gas industry, hydrostatic testing is used. This involves the sealing of the sections of the pipe, ®lling with water and pressing to check the pipe strength (strength test) and leak tightness (leak test). Most of the water that remains in a pipeline after the hydrostatic test can be removed by successively passing a series of cylindrical or spherical inserts called pigs and this process is known as the pigging operation. This is performed in wet-gas pipelines to periodically remove liquid accumulation in the lower portion of hilly terrain pipelines or the water that sticks to the surface of the pipes. In the pigging operation of gas or petroleum pipelines, water in¯atable rubber or neoprene spheres are tightly ®tted and inserted into the pipe at regular time intervals and pushed. As the pig moves, a liquid zone forms and grows by scooping the liquid. After this operation a very thin layer of water (usually 100mm or less than this) still remains on the inside surface of the pipeline. This very thin layer of water present appears a small amount but when the total amount of water for the whole pipeline is calculated usually comes to thousands of kilograms. It is necessary to dry the inner surface of the pipeline, otherwise hydrates of hydrocarbons may form, resulting in obstruction of the pipes, measuring instruments, valves, etc. and hence in failures. In the transportation of jet fuel, the presence of moisture is entirely unacceptable. A special feature of pipelines for pumping carbon dioxide is the need to carefully remove the water after the hydrostatic test, since highly corrosive carbonic acid is formed due to the interaction of carbon dioxide with water. Therefore, before putting the pipeline into operation it has to be dried of the water present after the hydraulic testing. Drying of pipelines by dry air can be achieved by passing dehydrated air through the pipeline until the concentration of water vapor in the outgoing air reaches a minimum set value. At the downstream end of the pipeline, the air is discharged into the atmosphere. But the entrance must be provided with the equipment for dehydration and compression. Increasing the ¯ow rate of the dry air shortens the time required for drying, but requires larger dehydration and compression facilities. Dry air is rapidly enriched with water vapor as it ¯ows along the pipeline. But water absorption still continues, though at a slower rate, because of pressure drop due to head loss. In fact, the concentration of saturation increases as the pressure decreases. The prediction of the time history of the water content in the line is dicult if a proper method for its estimation is not available. Analytical as well as numerical methods are used in the present analysis for the same purpose. PIPELINE DRYING TECHNIQUES AND LITERATURE REVIEW
The presence of water before starting normal pipeline operation usually results from the hydrostatic testing of the pipelines. Most of the water used for the test can be removed by having sequences of pigs run through the lines. After this operation usually a small amount of water wets the pipe internal wall. This residual water may be eliminated by some drying processes.
Pipeline drying with dehumidi®ed air
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Daib [1] used a slug method for drying the gas pipeline with a batch of methanol or glycol for freeze protection and hydrate inhibition. This procedure involves pushing a slug of methanol or glycol through the pipeline between two pigs or spheres. Factors aecting optimization and sizing of the batch are also discussed. Mathematical equations are developed for the optimum concentration and volume of the slug of methanol used to dry the pipeline. It is concluded in the analysis that the lowest slug concentration at the pipe must have a freezing point below the ambient temperature and the ®nal slug concentration must have a concentration high enough to act as a hydrate inhibitor. Comparing the three spheres method with the two spheres method, the use of three spheres is an improved method to keep the concentration of the methanol in the ®lms left on the pipe wall at a higher level than that of the two spheres method. Improvements such as optimizing the size and concentration of the slug to dry a pipeline by using the three spheres method show better results in terms of economy and operation [2]. Battara and Selandari [3] used the dry air method to dry a pipeline. Drying by air can be achieved by having a certain amount of previously dehydrated air run through the pipeline until the concentration of water vapor in the outgoing air reaches a minimum set value. Mathematical methods are used to simulate the process of drying. Partial dierential equations for the energy and mass balance are solved numerically with appropriate boundary conditions. The computer program Essic is also developed, which allows the fundamental parameters to be optimized. The results obtained by the Essic program proved to be competitive with the methanol swabbing and vacuum drying processes. In addition, this process does not require costly equipment and skilled labour, but attains very high drying levels. LaCasse and Ingvordsen [4] have used solid desiccant to produce dehumidi®ed air to dry gas pipelines. In this study simple equations of speci®c humidity dierence which accounts for only steady state conditions are used. Minimum time required to dry a pipeline is estimated at an inlet air dew point temperature of ÿ308C attained by refrigeration and desiccant dehumidi®cation methods. This system is simple in operation and less expensive but the analysis is not accurate since the mass transfer coecient between the airstream and water ®lm is not taken into account. Gorislavets and Sverdlov [5] have used a numerical method to analyze the drying of a pipeline by ventilative drying process. Thus many methods have been tried to ensure that no residual moisture remains in the pipelines. Some methods include the use of nitrogen, methanol, natural gas, and vacuum. However, because of economics, technical feasibility, and safety, many of these methods have proved to be less than satisfactory. The dry air method is considered to be cost eective as well as environmentally friendly. In this method atmospheric air is dried to a low dew point, thus providing a low vapor pressure to the dry air as it is introduced in the pipeline. This low vapor pressure air provides the driving force for drying the pipeline.
DEHUMIDIFIED AIR FOR PIPELINE DRYING
Dehumidi®ed air used for drying the pipeline can be achieved by any one or combination of the processes described below: Refrigeration process The ®rst and most common is the reduction of moisture in the air by means of reducing the temperature using refrigeration. By examination of the dew point line or saturation curve on the psychrometric chart, it can be seen that as the temperature of the air is lowered, the amount of moisture which it can hold is reduced considerably. Thus, by cooling the air below the dew point of the moisture contained in that air, one can condense out and remove some of the moisture vapor in liquid form. The reduction in air temperature is also limited by the freezing point of the water condensing on the cooling coil. Therefore achieving air with low dew point temperature becomes complicated with conventional refrigeration processes. The increase in power normally required in such equipment also make this process expensive.
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Air compression Air compression constitutes another method of reducing the moisture content in air. When air is compressed, the partial pressure of the water vapor in the water±air mixture is raised to the point where moisture can be condensed from the air at a higher temperature. This approach is sometimes quite practical for very small volumes of air to be dehumidi®ed, but the cost of compression equipment, the high power requirements and the amount of cooling water required for after-cooling make it very impractical for large volumes of air to be dehumidi®ed. Desiccant dehumidi®cation The most simple method of obtaining dry air is through the use of desiccant dehumidi®ers. These dehumidi®ers utilize solid or liquid desiccants which have an enormous natural anity for water. This paper deals with this process for dehumidi®cation of air for drying purposes because, extremely low dew points can be obtained in this way without the need of complex systems or controls.
PSYCHROMETRIC OF DEHUMIDIFICATION
The principle in the dry air method of drying pipelines consists of blowing a low dew point air into the pipeline. Moisture will be absorbed in the dry air stream since low dew point air has a low vapor pressure. This vapor pressure dierence between the moisture content in the pipeline and the dry air moisture content is the driving force for drying. The greater the dierence, the faster the pipeline will be dried. To obtain low dew point air, it is necessary to use both the mechanical refrigeration and the desiccant dehumidi®cation processes as shown in Fig. 1, and the psychrometric of drying process is shown in Fig. 2. The ®rst stage of the dehumidi®cation process uses mechanical refrigeration to chill the air below its dew point. Air is ®rst passed through the cooling coil, where it is sensibly cooled from state 1 to 2 as shown in Fig. 2 and the moisture is then removed by condensation from state 2 to 3. Usually air is cooled to approximately 48C. Below this temperature, the condensed moisture would freeze on the cooling coil stopping the dehumidi®cation process. Hence, the second stage of dehumidi®cation is accomplished by using a liquid desiccant dehumidi®er as shown in
Fig. 1. Schematic of pipeline drying process.
Pipeline drying with dehumidi®ed air
235
Fig. 2. Psychrometric of drying process.
Fig. 1. In the dehumidi®er, air is made to pass through a spray of liquid desiccant which is sprinkled from the top of the dehumidi®er. The desiccant is ecient at removing moisture and thus makes the air reach a ®nal dewpoint of ÿ208C to ÿ408C. The desiccant dehumidi®er can work at any temperature without freezing up. This process is shown as state 3 to 4 on the psychrometric plot.
MATHEMATICAL MODELLING
One of the most important aspects of drying technology, especially for industrial processes, is mathematical modelling of the drying processes. The purpose of modelling is to allow the engineer to choose the most appropriate method of drying for a given product as well as to choose suitable operating conditions. The principle of modelling is based on having a system of mathematical equations which completely characterize the system to be modelled. In particular the solution of these equations makes it possible to predict the parameters of the process as a function of time at any point based on given initial and boundary conditions. The drying process by using the dehumidi®ed air is schematized as shown in Fig. 3 by making some simpli®cations that are justi®ed by the fact that the vapor mass concentration in the ¯owing air±vapor mixture remains very low. In particular, the following assumptions are made:
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Fig. 3. Schematic of a pipeline.
1. The dry air behaves like an ideal gas, that is we can assume that the gas containing the moisture obeys all the laws of ideal gases. 2. The pressure pro®le along the pipeline can be calculated as for the same air ¯ow rate in a single-phase steady ¯ow without evaporation. 3. In¯uence of the temperature drop of the air due to the moisture evaporation from the pipe wall on the drying time is neglected. 4. Wall temperature is assumed to be equal to the ambient temperature. 5. Initially the whole system is assumed to be at a temperature equal to the atmospheric temperature. 6. Film thickness on the pipeline wall is assumed to be uniform. 7. All the terms for kinetic energy and the action of gravity are neglected. The mass conservation on the element Dx shown in the ®gure can be written as: Accumulation = Input ÿ output + generation ÿ destruction As there is no generation or the destruction term in the drying process, the above equation reduces with the ®rst two terms, @W ÿpDNA @t
1
@ pDDxNA ÿ rAVC : rADxC rAVC x xDx @t
2
and
After simpli®cation, rA
@C @C ÿQ pDNA ; @t @x
3
where Q = rAV. The above Equations (1) and (3) express the mass conservation of water, in the liquid and vapor phases, respectively, and they can be solved analytically as well as numerically with the following initial and boundary conditions: W x; 0 W0 ;
4
; C x; 0 Csat
5
C 0; t C0 ;
6
@C L; t 0: @x
7
The expression describing the rate of mass transfer in the saturated section for a binary gaseous system with one diusing component is given [6] as, ! NA NA NA NB ÿ CA2 =C ; 8 F ln NA NA NA NB NA NB ÿ CA1 =C
Pipeline drying with dehumidi®ed air
237
where CA2/C and CA1/C represent the mole fraction concentration in the stream core and the saturation at pipe wall, respectively. CA/C = XA for liquids, and CA/C = YA for gases. In the present analysis we consider only the transfer of liquid component A from the water ®lm on the pipe wall to the dry gas given by component B. Therefore, NA 1: NA NB
9
1 ÿ CA2 =C NA F ln : 1 ÿ CA1 =C
10
Since NB=0, Equation (8) reduces to,
By analogy with Equation (10), the mass ¯ux of water vapor in the saturated section can be found from the equation given below: 1ÿY : 11 NA F ln 1 ÿ Ysat The molar content of moisture at the pipe wall is determined in turn, as the limiting content expressed in molar fraction [6]. It can be given as, Ysat
Psat : P ÿ Psat
The mass transfer coecient F is determined from the expression [7], 0:023MWv Dva r z 0:8 Re : F MWa D Dva
12
13
The exponent z in Equation (13) depends on Reynolds number. As the ¯ow is assumed to be turbulent, it can be taken as unity. Therefore, Equation (13) reduces to F
0:023MWv Z 0:8 Re : MWa D
14
In order to use Equation (11) one has to determine the molar fraction Y of water vapor in the stream core. This can be determined in terms of the mass fraction C* of water vapor by solving Equations (1) and (3) along with initial and boundary conditions. PREDICTION OF THE VARIATION OF CONCENTRATION
Analytical method In order to determine the variation of concentration of water vapor in the dry air, a linear functional form for NA is suggested as given below, because the transform cannot be used due to the presence of non linearity in NA. Equations representing the mass conservation of water in the liquid and vapor phases are solved analytically using the Laplace transforms for the variation of concentration of the water vapor in the drying air, and for predicting the drying time required to dry a pipeline of given length for dierent operating parameters. The rate of mass transfer term NA can be assumed as a linear function as, N A a1 b 1 C ;
15
where a1 and b1 are constants and can be determined using the equation given below: NA b1 Psat ÿ P1 ; where
16
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Y. A. Syed et al.
b1
0:622k kg=hm2 mm Hg P
17
and h 950 J=kg K k
18
@C Q @C pDNA ÿ : @t rA rA @x
19
Equation (3) can be rewritten as,
By substituting the value of NA in Equation (19), it reduces to, @C @C ÿd o a1 b1 C ; @t @x
20
where d
Q rA
21
o
pD : rA
22
and
Applying Laplace Transform to Equation (19), @C @C oL a1 b1 C : ÿd L @t @x Taking Laplace Transform on the transient term of Equation (23), @C L SC x; s ÿ C x; 0: @t
23
24
Substituting Equation (24) in Equation (23), we get, ÿd SC x; s ÿ Csat
ha i @C 1 o b1 C : @x s
The above equation can be simpli®ed as, d oa1 : d s ÿ b1 o C Csat s dx
25
26
The solution of this equation [8] is,
where
c cp cc
27
s ÿ b1 o cc A1 exp ÿ x ; d
28
and cp can be calculated as, Zx oa1 Csat ob1 ÿ s cp exp ÿ U1 ÿ x dU1: ds d d u0 Solving the above equations and substituting the value of cc and cp in Equation (27), we get,
29
Pipeline drying with dehumidi®ed air
C
ob1 x=d Csat e
a1 ob1 t a1 ob1 tÿx=d ob1 t ob1 tÿx=d ob1 x=d e ÿ 1 Csat e ÿ e ÿ 1 Csat e : e b1 b1
239
30
The variation of concentration of water vapor in the dry air can be determined for dierent x and t by using the above equation.
Numerical method The numerical solution for the system of Equations (1) and (3) is also carried out using semiimplicit ®nite dierence scheme. In this scheme, the equation is juxtaposed in two dierences @C Q @C ; @t Ar @x
31
@C Q @C pDNA : Ar @x @t Ar
32
and
In Equation (31) the value of C* is determined at an intermediate time level without involving the nonlinear term in NA. In Equation (32) value of C* is determined at the next time level, in this way the solution marches in the forward direction. Let us take the following non-dimensional parameters, C ; Csat
33
V j ; V0
34
f
x ; L
35
NA ; NA0
36
t ; t0
37
W : W0
38
Z c
t C
After non-dimensionalization Equations (1) and (3) reduce to, @f 3 3 1 @f ÿa1 f Z b1 1 237:6 ln 1 ÿ 0:0323f @t 4 4 4 @Z
39
and @C ÿsc: @t The Equations (31) and (32) after non-dimensionalization reduce to, @f 3 3 1 @f ÿa1 f Z ; @t 4 4 4 @Z and
40
41
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Y. A. Syed et al.
Fig. 4. Concentration variation of a pipeline at 20th minute.
@f 3 3 1 @f ÿa1 f Z b1 1 237:6 ln 1 ÿ 0:0323f: @t 4 4 4 @Z
42
The discretization of Equations (41) and (42) is done by considering a three time level solution method, with fni, fni + 1 and fni + 1 representing the values at ®rst, intermediate and ®nal time level respectively. n1 ÿ fn 3 3 1 fni1 ÿ fniÿ1 f i i ÿa1 fni i h1 0: 43 Dt 2h1 4 4 4 Discretization of Equation (41) results as, 3 n 3 1 fni1 ÿ fniÿ1 n1 fni : ih fi ÿDt a1 fi 2h1 4 4 4
44
Equation (42) can be discretized as, " n1 n1 !# n fi1 ÿ fiÿ1 fn1 ÿ f 3 3 1 i i n1 ÿa1 f b1 1 237:6 ln 1 ÿ 0:0323fni : 45 i h1 Dt 2h1 4 i 4 4 On simplifying the above equation reduces to, " n1 n1 !# 3 n1 3 1 f i1 ÿ fiÿ1 n1 fi ÿDt a1 fi Dt b1 1 237:6 ln 1 ÿ 0:0323fni fni ; i h1 2h1 4 4 4 46 Cn1 ÿ Cni i ÿsc: Dt
47
The set of Equations (44), (46) and (47) are solved numerically with the boundary and initial conditions.
Pipeline drying with dehumidi®ed air
241
Fig. 5. The eect of ambient temperature and the water ®lm thickness.
In the numerical solution of problems of the ¯uid ¯ow, the steps of the dierence grid are chosen in accordance with the necessary conditions of stability, which are found by the conditional setting of certain unknown functions of the system. In systems in which the thermophysical properties are assumed to be constant, dry air behaves like an ideal gas and the spatial steps are equal in all directions.
Fig. 6. Eect of volumetric ¯ow rate and the dew point temperature.
242
Y. A. Syed et al.
Fig. 7. The eect of pipeline length and the dew point temperature.
By satisfying the conditions of stability the time dierence step is de®ned as, h1 m : V
48
Depending on the above stability criteria, the spatial step, h1 is selected. RESULTS AND DISCUSSION
In order to predict the drying time required to dry the pipeline of given length for dierent operating parameters, both analytical as well as the numerical methods are studied. Analytical results are illustrated in Figs 4±7. Figure 8 compares the results of analytical as well as numerical studies for a short length (20 m) of pipeline. It is found appropriate to compare these two results in Fig. 8 since both solutions are obtained with the same operating parameters. Table 1 summarizes the pipeline dimensions and dierent operating parameters. Figure 4 illustrates the dynamics of the drying process and it shows the variation of water vapor concentration in dry air as it travels along the pipeline. It can be seen that the change in the concentration is rapid at the entrance of the pipeline. This can be explained by the fact that near the entrance the evaporation of moisture from the pipe wall occurs very rapidly due to the large dierence between the concentration of water vapor near the pipe wall and the air stream until it gets saturated as shown by a horizontal line. In order to determine the eect of various operating parameters on the process of pipeline drying, a parametric study is conducted using the analytical method. The study is initiated by predicting the behavior of the drying process under the in¯uence of ambient temperature and the water ®lm thickness. It is evident from Fig. 5 that the drying time increases with the increase in water ®lm thickness but decreases with the increase in ambient temperature. This may be attributed to the fact that as the ambient temperature increases, potential for mass transfer from the water ®lm to the air increases and hence the drying time decreases. Figure 6 shows the variation in drying behavior as a function of volumetric ¯ow rate with the dew point temperature as a parameter. It can be seen from the ®gure that as the dew point tem-
Pipeline drying with dehumidi®ed air
243
Fig. 8. Concentration variation of a pipeline at 55th second.
perature of dry air is decreased from ÿ208C to ÿ308C and ÿ408C the drying time is reduced and the behavior at these temperatures becomes closer to each other because of the higher level of dryness. This can be explained by the fact that the potential for mass transfer increases with decrease in air dew point temperatures as evident from the ®gure. In Fig. 7, the eect of length of pipeline and the air dew point temperature on the history of the drying process is shown. As illustrated in Fig. 7, the drying process is continuous in the sense that the pipeline is dried from its initial moisture content to the ®nal value with a constant rate period. As expected, with an increase in the length of pipe the duration of the total drying period increases. Figure 8 compares the present numerical solution with the analytical solution for a pipeline of length 20 m at the speci®ed operating parameters. It is evident from the ®gure that the results obtained from both the analyses for the concentration variation of dry air are comparable at short lengths of pipeline, since the numerical analysis is carried out for short length because in the case of ventilative drying the special stability condition Equation (48) is imposed on the solution of Equations (40)±(42), which limits the choice of h1, the spatial step. This is explained by the fact that evaporation of the moisture from the pipe wall occurs very rapidly due to the large dierence in the concentration. And, if a fairly large spatial step is chosen, the dry air may already reach the saturated state in the ®rst spatial step, and this may occur at any point of this step (depending on the evaporation rate and the chosen length of the step). The spatial step must therefore be chosen so as to eliminate this situation.
Table 1. Pipeline dimension and operating parameters Pipeline diameter (D) Ambient temperature Dry air entrance temperature Inlet pressure (Pi) Outlet pressure (P0) Volumetric ¯ow rate of dry air (Vfr) Length of pipeline (L) Water ®lm thickness Dry air dew point temperature (Tdp)
460 mm 208C 158C 5 bar 1 bar 5000±15000 m3/h 50±150 km 50±100 mm ÿ208CÐ ÿ 408C
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Y. A. Syed et al.
CONCLUSIONS
Drying using dehumidi®ed air with low dew point temperature is one of the most widely used methods for removing residual water after the hydrostatic testing. The problem of pipeline drying was solved using the analytical and numerical methods. The analytical method with simpli®ed assumptions using Laplace transforms is used to ®nd the drying time required to dry a pipeline. The numerical method is also used for predicting the drying time for a pipe length of 20 m and the results obtained by these two methods show good agreement. It is found that the thickness of the water ®lm, the volumetric ¯ow rate of dry air, the dew point temperature of the dry air and the length of the pipeline in¯uence the time required for drying the pipeline. AcknowledgementsÐThe authors are grateful for the ®nancial support and facilities provided by the King Fahd University of Petroleum and Minerals.
REFERENCES 1. S. Y. Daib, Determining volume and concentration to dry gas pipelines. Oil and Gas Journal 81, 80±83 (1983). 2. S. Y. Daib, Techniques for drying pipelines by the three-spheres method. Oil and Gas Journal 81, 112±116 (1983). 3. V. Battara and B. Selandari, Mathematical model predicts performance of pipeline drying with air. Oil and Gas Journal 82, 114±116 (1984). 4. G. A. LaCasse and T. Ingvordsen, Desiccant drying of gas pipelines. Material Performance 27, 848±851 (1988). 5. V. M. Gorislavets and A. A. Sverdlov, Numerical investigation of the process of ventilative drying of a pipeline. Journal of Engineering Physics 60, 615±623 (1991). 6. Treybal, R. E., Mass Transfer Operations. McGraw-Hill, New York (1980). 7. T. H. Chilton and A. P. Colburn, Mass transfer coecients. Industrial and Engineering Chemistry 26, 1183± 1187 (1934). 8. Trim, D. W., Applied Partial Dierential Equations. PWS-Kent Publishing Company, Boston (1989).
Applied Thermal Engineering Vol. 18, No. 5, pp. 231±244, 1998 # 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-4311/98 $19.00 + 0.00 S1359-4311(97)00083-5
PIPELINE DRYING USING DEHUMIDIFIED AIR WITH LOW DEW POINT TEMPERATURE Syed Younus Ahmed, P. Gandhidasan* and A. A. Al-Farayedhi Mechanical Engineering Department, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia (Received 26 September 1997) AbstractÐThe presence of humidity may be detrimental to the operation of pipelines transporting natural gas or other petroleum products. In particular conditions water solidi®es or reacts chemically with hydrocarbons, forming hydrates. Such crystalline substances may cause obstruction of the lines and damage the equipment of the relevant facilities. A procedure for predicting the performance of drying a pipeline using dehumidi®ed air with low dew point is described in this paper. The mathematical model estimates the time required for the complete removal of moisture in the pipeline for the given operating conditions with simpli®ed assumptions. The governing equations are solved analytically as well as numerically and the results are brie¯y discussed in the paper. # 1998 Elsevier Science Ltd. All rights reserved KeywordsÐPipeline, drying, time, dehumidi®cation, modelling.
NOMENCLATURE A A1,a1,b1 C* C*0 CA/C C*sat D Dva F h h1 k L m MWv MWa NA,NB NA0 p P Q Re t T Ta Te U1 V V0 W W0 x Y Ysat
pipeline cross section area, m2 constants concentration of water vapor in dry air; kg of dry air/kg of water concentration of water vapor in dry air at inlet, kg of water/kg of dry air mole fraction concentration concentration of water vapor in dry air at saturation, kg of water/kg of dry air pipeline diameter, m diusion coecient, m2/s mass transfer coecient, kg/m2s convection coecient from air to water, W/m2K spatial step, m mass transfer coecient, kg/m2s length of pipeline, m time step, s molecular weight of vapor molecular weight of air evaporation rate per unit area, kg/m2s maximum evaporation rate per unit area, kg/m2s partial pressure, mm Hg total pressure, mm Hg mass ¯ow rate of dry air, kg/s Reynolds number time, s temperature, 8C or K ambient air temperature, 8C or K entrance temperature of pipeline, 8C or K some function velocity of dry air, m/s maximum velocity of dry air, m/s amount of water per unit length of pipe, kg/m initial amount of water per unit length of pipe, kg/m spatial variable, m mole fraction concentration mole fraction concentration at saturation
*Author to whom correspondence should be addressed. 231
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Y. A. Syed et al.
Subscripts sat 1 2 1
saturation component 1 component 2 ambient condition
Greek a1,d,o,s b1 r rF n Z
constants mass transfer coecient between solution ®lm and air stream, kg/h m2 (mmHg) density, kg/m3 density of pipe material, kg/m3 dynamic viscosity, kg/ms kinematic viscosity, m2/s in Equation (14)
INTRODUCTION
To verify the integrity of the installation of pipelines, especially in the oil and gas industry, hydrostatic testing is used. This involves the sealing of the sections of the pipe, ®lling with water and pressing to check the pipe strength (strength test) and leak tightness (leak test). Most of the water that remains in a pipeline after the hydrostatic test can be removed by successively passing a series of cylindrical or spherical inserts called pigs and this process is known as the pigging operation. This is performed in wet-gas pipelines to periodically remove liquid accumulation in the lower portion of hilly terrain pipelines or the water that sticks to the surface of the pipes. In the pigging operation of gas or petroleum pipelines, water in¯atable rubber or neoprene spheres are tightly ®tted and inserted into the pipe at regular time intervals and pushed. As the pig moves, a liquid zone forms and grows by scooping the liquid. After this operation a very thin layer of water (usually 100mm or less than this) still remains on the inside surface of the pipeline. This very thin layer of water present appears a small amount but when the total amount of water for the whole pipeline is calculated usually comes to thousands of kilograms. It is necessary to dry the inner surface of the pipeline, otherwise hydrates of hydrocarbons may form, resulting in obstruction of the pipes, measuring instruments, valves, etc. and hence in failures. In the transportation of jet fuel, the presence of moisture is entirely unacceptable. A special feature of pipelines for pumping carbon dioxide is the need to carefully remove the water after the hydrostatic test, since highly corrosive carbonic acid is formed due to the interaction of carbon dioxide with water. Therefore, before putting the pipeline into operation it has to be dried of the water present after the hydraulic testing. Drying of pipelines by dry air can be achieved by passing dehydrated air through the pipeline until the concentration of water vapor in the outgoing air reaches a minimum set value. At the downstream end of the pipeline, the air is discharged into the atmosphere. But the entrance must be provided with the equipment for dehydration and compression. Increasing the ¯ow rate of the dry air shortens the time required for drying, but requires larger dehydration and compression facilities. Dry air is rapidly enriched with water vapor as it ¯ows along the pipeline. But water absorption still continues, though at a slower rate, because of pressure drop due to head loss. In fact, the concentration of saturation increases as the pressure decreases. The prediction of the time history of the water content in the line is dicult if a proper method for its estimation is not available. Analytical as well as numerical methods are used in the present analysis for the same purpose. PIPELINE DRYING TECHNIQUES AND LITERATURE REVIEW
The presence of water before starting normal pipeline operation usually results from the hydrostatic testing of the pipelines. Most of the water used for the test can be removed by having sequences of pigs run through the lines. After this operation usually a small amount of water wets the pipe internal wall. This residual water may be eliminated by some drying processes.
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Daib [1] used a slug method for drying the gas pipeline with a batch of methanol or glycol for freeze protection and hydrate inhibition. This procedure involves pushing a slug of methanol or glycol through the pipeline between two pigs or spheres. Factors aecting optimization and sizing of the batch are also discussed. Mathematical equations are developed for the optimum concentration and volume of the slug of methanol used to dry the pipeline. It is concluded in the analysis that the lowest slug concentration at the pipe must have a freezing point below the ambient temperature and the ®nal slug concentration must have a concentration high enough to act as a hydrate inhibitor. Comparing the three spheres method with the two spheres method, the use of three spheres is an improved method to keep the concentration of the methanol in the ®lms left on the pipe wall at a higher level than that of the two spheres method. Improvements such as optimizing the size and concentration of the slug to dry a pipeline by using the three spheres method show better results in terms of economy and operation [2]. Battara and Selandari [3] used the dry air method to dry a pipeline. Drying by air can be achieved by having a certain amount of previously dehydrated air run through the pipeline until the concentration of water vapor in the outgoing air reaches a minimum set value. Mathematical methods are used to simulate the process of drying. Partial dierential equations for the energy and mass balance are solved numerically with appropriate boundary conditions. The computer program Essic is also developed, which allows the fundamental parameters to be optimized. The results obtained by the Essic program proved to be competitive with the methanol swabbing and vacuum drying processes. In addition, this process does not require costly equipment and skilled labour, but attains very high drying levels. LaCasse and Ingvordsen [4] have used solid desiccant to produce dehumidi®ed air to dry gas pipelines. In this study simple equations of speci®c humidity dierence which accounts for only steady state conditions are used. Minimum time required to dry a pipeline is estimated at an inlet air dew point temperature of ÿ308C attained by refrigeration and desiccant dehumidi®cation methods. This system is simple in operation and less expensive but the analysis is not accurate since the mass transfer coecient between the airstream and water ®lm is not taken into account. Gorislavets and Sverdlov [5] have used a numerical method to analyze the drying of a pipeline by ventilative drying process. Thus many methods have been tried to ensure that no residual moisture remains in the pipelines. Some methods include the use of nitrogen, methanol, natural gas, and vacuum. However, because of economics, technical feasibility, and safety, many of these methods have proved to be less than satisfactory. The dry air method is considered to be cost eective as well as environmentally friendly. In this method atmospheric air is dried to a low dew point, thus providing a low vapor pressure to the dry air as it is introduced in the pipeline. This low vapor pressure air provides the driving force for drying the pipeline.
DEHUMIDIFIED AIR FOR PIPELINE DRYING
Dehumidi®ed air used for drying the pipeline can be achieved by any one or combination of the processes described below: Refrigeration process The ®rst and most common is the reduction of moisture in the air by means of reducing the temperature using refrigeration. By examination of the dew point line or saturation curve on the psychrometric chart, it can be seen that as the temperature of the air is lowered, the amount of moisture which it can hold is reduced considerably. Thus, by cooling the air below the dew point of the moisture contained in that air, one can condense out and remove some of the moisture vapor in liquid form. The reduction in air temperature is also limited by the freezing point of the water condensing on the cooling coil. Therefore achieving air with low dew point temperature becomes complicated with conventional refrigeration processes. The increase in power normally required in such equipment also make this process expensive.
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Air compression Air compression constitutes another method of reducing the moisture content in air. When air is compressed, the partial pressure of the water vapor in the water±air mixture is raised to the point where moisture can be condensed from the air at a higher temperature. This approach is sometimes quite practical for very small volumes of air to be dehumidi®ed, but the cost of compression equipment, the high power requirements and the amount of cooling water required for after-cooling make it very impractical for large volumes of air to be dehumidi®ed. Desiccant dehumidi®cation The most simple method of obtaining dry air is through the use of desiccant dehumidi®ers. These dehumidi®ers utilize solid or liquid desiccants which have an enormous natural anity for water. This paper deals with this process for dehumidi®cation of air for drying purposes because, extremely low dew points can be obtained in this way without the need of complex systems or controls.
PSYCHROMETRIC OF DEHUMIDIFICATION
The principle in the dry air method of drying pipelines consists of blowing a low dew point air into the pipeline. Moisture will be absorbed in the dry air stream since low dew point air has a low vapor pressure. This vapor pressure dierence between the moisture content in the pipeline and the dry air moisture content is the driving force for drying. The greater the dierence, the faster the pipeline will be dried. To obtain low dew point air, it is necessary to use both the mechanical refrigeration and the desiccant dehumidi®cation processes as shown in Fig. 1, and the psychrometric of drying process is shown in Fig. 2. The ®rst stage of the dehumidi®cation process uses mechanical refrigeration to chill the air below its dew point. Air is ®rst passed through the cooling coil, where it is sensibly cooled from state 1 to 2 as shown in Fig. 2 and the moisture is then removed by condensation from state 2 to 3. Usually air is cooled to approximately 48C. Below this temperature, the condensed moisture would freeze on the cooling coil stopping the dehumidi®cation process. Hence, the second stage of dehumidi®cation is accomplished by using a liquid desiccant dehumidi®er as shown in
Fig. 1. Schematic of pipeline drying process.
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Fig. 2. Psychrometric of drying process.
Fig. 1. In the dehumidi®er, air is made to pass through a spray of liquid desiccant which is sprinkled from the top of the dehumidi®er. The desiccant is ecient at removing moisture and thus makes the air reach a ®nal dewpoint of ÿ208C to ÿ408C. The desiccant dehumidi®er can work at any temperature without freezing up. This process is shown as state 3 to 4 on the psychrometric plot.
MATHEMATICAL MODELLING
One of the most important aspects of drying technology, especially for industrial processes, is mathematical modelling of the drying processes. The purpose of modelling is to allow the engineer to choose the most appropriate method of drying for a given product as well as to choose suitable operating conditions. The principle of modelling is based on having a system of mathematical equations which completely characterize the system to be modelled. In particular the solution of these equations makes it possible to predict the parameters of the process as a function of time at any point based on given initial and boundary conditions. The drying process by using the dehumidi®ed air is schematized as shown in Fig. 3 by making some simpli®cations that are justi®ed by the fact that the vapor mass concentration in the ¯owing air±vapor mixture remains very low. In particular, the following assumptions are made:
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Fig. 3. Schematic of a pipeline.
1. The dry air behaves like an ideal gas, that is we can assume that the gas containing the moisture obeys all the laws of ideal gases. 2. The pressure pro®le along the pipeline can be calculated as for the same air ¯ow rate in a single-phase steady ¯ow without evaporation. 3. In¯uence of the temperature drop of the air due to the moisture evaporation from the pipe wall on the drying time is neglected. 4. Wall temperature is assumed to be equal to the ambient temperature. 5. Initially the whole system is assumed to be at a temperature equal to the atmospheric temperature. 6. Film thickness on the pipeline wall is assumed to be uniform. 7. All the terms for kinetic energy and the action of gravity are neglected. The mass conservation on the element Dx shown in the ®gure can be written as: Accumulation = Input ÿ output + generation ÿ destruction As there is no generation or the destruction term in the drying process, the above equation reduces with the ®rst two terms, @W ÿpDNA @t
1
@ pDDxNA ÿ rAVC : rADxC rAVC x xDx @t
2
and
After simpli®cation, rA
@C @C ÿQ pDNA ; @t @x
3
where Q = rAV. The above Equations (1) and (3) express the mass conservation of water, in the liquid and vapor phases, respectively, and they can be solved analytically as well as numerically with the following initial and boundary conditions: W x; 0 W0 ;
4
; C x; 0 Csat
5
C 0; t C0 ;
6
@C L; t 0: @x
7
The expression describing the rate of mass transfer in the saturated section for a binary gaseous system with one diusing component is given [6] as, ! NA NA NA NB ÿ CA2 =C ; 8 F ln NA NA NA NB NA NB ÿ CA1 =C
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where CA2/C and CA1/C represent the mole fraction concentration in the stream core and the saturation at pipe wall, respectively. CA/C = XA for liquids, and CA/C = YA for gases. In the present analysis we consider only the transfer of liquid component A from the water ®lm on the pipe wall to the dry gas given by component B. Therefore, NA 1: NA NB
9
1 ÿ CA2 =C NA F ln : 1 ÿ CA1 =C
10
Since NB=0, Equation (8) reduces to,
By analogy with Equation (10), the mass ¯ux of water vapor in the saturated section can be found from the equation given below: 1ÿY : 11 NA F ln 1 ÿ Ysat The molar content of moisture at the pipe wall is determined in turn, as the limiting content expressed in molar fraction [6]. It can be given as, Ysat
Psat : P ÿ Psat
The mass transfer coecient F is determined from the expression [7], 0:023MWv Dva r z 0:8 Re : F MWa D Dva
12
13
The exponent z in Equation (13) depends on Reynolds number. As the ¯ow is assumed to be turbulent, it can be taken as unity. Therefore, Equation (13) reduces to F
0:023MWv Z 0:8 Re : MWa D
14
In order to use Equation (11) one has to determine the molar fraction Y of water vapor in the stream core. This can be determined in terms of the mass fraction C* of water vapor by solving Equations (1) and (3) along with initial and boundary conditions. PREDICTION OF THE VARIATION OF CONCENTRATION
Analytical method In order to determine the variation of concentration of water vapor in the dry air, a linear functional form for NA is suggested as given below, because the transform cannot be used due to the presence of non linearity in NA. Equations representing the mass conservation of water in the liquid and vapor phases are solved analytically using the Laplace transforms for the variation of concentration of the water vapor in the drying air, and for predicting the drying time required to dry a pipeline of given length for dierent operating parameters. The rate of mass transfer term NA can be assumed as a linear function as, N A a1 b 1 C ;
15
where a1 and b1 are constants and can be determined using the equation given below: NA b1 Psat ÿ P1 ; where
16
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b1
0:622k kg=hm2 mm Hg P
17
and h 950 J=kg K k
18
@C Q @C pDNA ÿ : @t rA rA @x
19
Equation (3) can be rewritten as,
By substituting the value of NA in Equation (19), it reduces to, @C @C ÿd o a1 b1 C ; @t @x
20
where d
Q rA
21
o
pD : rA
22
and
Applying Laplace Transform to Equation (19), @C @C oL a1 b1 C : ÿd L @t @x Taking Laplace Transform on the transient term of Equation (23), @C L SC x; s ÿ C x; 0: @t
23
24
Substituting Equation (24) in Equation (23), we get, ÿd SC x; s ÿ Csat
ha i @C 1 o b1 C : @x s
The above equation can be simpli®ed as, d oa1 : d s ÿ b1 o C Csat s dx
25
26
The solution of this equation [8] is,
where
c cp cc
27
s ÿ b1 o cc A1 exp ÿ x ; d
28
and cp can be calculated as, Zx oa1 Csat ob1 ÿ s cp exp ÿ U1 ÿ x dU1: ds d d u0 Solving the above equations and substituting the value of cc and cp in Equation (27), we get,
29
Pipeline drying with dehumidi®ed air
C
ob1 x=d Csat e
a1 ob1 t a1 ob1 tÿx=d ob1 t ob1 tÿx=d ob1 x=d e ÿ 1 Csat e ÿ e ÿ 1 Csat e : e b1 b1
239
30
The variation of concentration of water vapor in the dry air can be determined for dierent x and t by using the above equation.
Numerical method The numerical solution for the system of Equations (1) and (3) is also carried out using semiimplicit ®nite dierence scheme. In this scheme, the equation is juxtaposed in two dierences @C Q @C ; @t Ar @x
31
@C Q @C pDNA : Ar @x @t Ar
32
and
In Equation (31) the value of C* is determined at an intermediate time level without involving the nonlinear term in NA. In Equation (32) value of C* is determined at the next time level, in this way the solution marches in the forward direction. Let us take the following non-dimensional parameters, C ; Csat
33
V j ; V0
34
f
x ; L
35
NA ; NA0
36
t ; t0
37
W : W0
38
Z c
t C
After non-dimensionalization Equations (1) and (3) reduce to, @f 3 3 1 @f ÿa1 f Z b1 1 237:6 ln 1 ÿ 0:0323f @t 4 4 4 @Z
39
and @C ÿsc: @t The Equations (31) and (32) after non-dimensionalization reduce to, @f 3 3 1 @f ÿa1 f Z ; @t 4 4 4 @Z and
40
41
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Fig. 4. Concentration variation of a pipeline at 20th minute.
@f 3 3 1 @f ÿa1 f Z b1 1 237:6 ln 1 ÿ 0:0323f: @t 4 4 4 @Z
42
The discretization of Equations (41) and (42) is done by considering a three time level solution method, with fni, fni + 1 and fni + 1 representing the values at ®rst, intermediate and ®nal time level respectively. n1 ÿ fn 3 3 1 fni1 ÿ fniÿ1 f i i ÿa1 fni i h1 0: 43 Dt 2h1 4 4 4 Discretization of Equation (41) results as, 3 n 3 1 fni1 ÿ fniÿ1 n1 fni : ih fi ÿDt a1 fi 2h1 4 4 4
44
Equation (42) can be discretized as, " n1 n1 !# n fi1 ÿ fiÿ1 fn1 ÿ f 3 3 1 i i n1 ÿa1 f b1 1 237:6 ln 1 ÿ 0:0323fni : 45 i h1 Dt 2h1 4 i 4 4 On simplifying the above equation reduces to, " n1 n1 !# 3 n1 3 1 f i1 ÿ fiÿ1 n1 fi ÿDt a1 fi Dt b1 1 237:6 ln 1 ÿ 0:0323fni fni ; i h1 2h1 4 4 4 46 Cn1 ÿ Cni i ÿsc: Dt
47
The set of Equations (44), (46) and (47) are solved numerically with the boundary and initial conditions.
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Fig. 5. The eect of ambient temperature and the water ®lm thickness.
In the numerical solution of problems of the ¯uid ¯ow, the steps of the dierence grid are chosen in accordance with the necessary conditions of stability, which are found by the conditional setting of certain unknown functions of the system. In systems in which the thermophysical properties are assumed to be constant, dry air behaves like an ideal gas and the spatial steps are equal in all directions.
Fig. 6. Eect of volumetric ¯ow rate and the dew point temperature.
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Fig. 7. The eect of pipeline length and the dew point temperature.
By satisfying the conditions of stability the time dierence step is de®ned as, h1 m : V
48
Depending on the above stability criteria, the spatial step, h1 is selected. RESULTS AND DISCUSSION
In order to predict the drying time required to dry the pipeline of given length for dierent operating parameters, both analytical as well as the numerical methods are studied. Analytical results are illustrated in Figs 4±7. Figure 8 compares the results of analytical as well as numerical studies for a short length (20 m) of pipeline. It is found appropriate to compare these two results in Fig. 8 since both solutions are obtained with the same operating parameters. Table 1 summarizes the pipeline dimensions and dierent operating parameters. Figure 4 illustrates the dynamics of the drying process and it shows the variation of water vapor concentration in dry air as it travels along the pipeline. It can be seen that the change in the concentration is rapid at the entrance of the pipeline. This can be explained by the fact that near the entrance the evaporation of moisture from the pipe wall occurs very rapidly due to the large dierence between the concentration of water vapor near the pipe wall and the air stream until it gets saturated as shown by a horizontal line. In order to determine the eect of various operating parameters on the process of pipeline drying, a parametric study is conducted using the analytical method. The study is initiated by predicting the behavior of the drying process under the in¯uence of ambient temperature and the water ®lm thickness. It is evident from Fig. 5 that the drying time increases with the increase in water ®lm thickness but decreases with the increase in ambient temperature. This may be attributed to the fact that as the ambient temperature increases, potential for mass transfer from the water ®lm to the air increases and hence the drying time decreases. Figure 6 shows the variation in drying behavior as a function of volumetric ¯ow rate with the dew point temperature as a parameter. It can be seen from the ®gure that as the dew point tem-
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Fig. 8. Concentration variation of a pipeline at 55th second.
perature of dry air is decreased from ÿ208C to ÿ308C and ÿ408C the drying time is reduced and the behavior at these temperatures becomes closer to each other because of the higher level of dryness. This can be explained by the fact that the potential for mass transfer increases with decrease in air dew point temperatures as evident from the ®gure. In Fig. 7, the eect of length of pipeline and the air dew point temperature on the history of the drying process is shown. As illustrated in Fig. 7, the drying process is continuous in the sense that the pipeline is dried from its initial moisture content to the ®nal value with a constant rate period. As expected, with an increase in the length of pipe the duration of the total drying period increases. Figure 8 compares the present numerical solution with the analytical solution for a pipeline of length 20 m at the speci®ed operating parameters. It is evident from the ®gure that the results obtained from both the analyses for the concentration variation of dry air are comparable at short lengths of pipeline, since the numerical analysis is carried out for short length because in the case of ventilative drying the special stability condition Equation (48) is imposed on the solution of Equations (40)±(42), which limits the choice of h1, the spatial step. This is explained by the fact that evaporation of the moisture from the pipe wall occurs very rapidly due to the large dierence in the concentration. And, if a fairly large spatial step is chosen, the dry air may already reach the saturated state in the ®rst spatial step, and this may occur at any point of this step (depending on the evaporation rate and the chosen length of the step). The spatial step must therefore be chosen so as to eliminate this situation.
Table 1. Pipeline dimension and operating parameters Pipeline diameter (D) Ambient temperature Dry air entrance temperature Inlet pressure (Pi) Outlet pressure (P0) Volumetric ¯ow rate of dry air (Vfr) Length of pipeline (L) Water ®lm thickness Dry air dew point temperature (Tdp)
460 mm 208C 158C 5 bar 1 bar 5000±15000 m3/h 50±150 km 50±100 mm ÿ208CÐ ÿ 408C
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CONCLUSIONS
Drying using dehumidi®ed air with low dew point temperature is one of the most widely used methods for removing residual water after the hydrostatic testing. The problem of pipeline drying was solved using the analytical and numerical methods. The analytical method with simpli®ed assumptions using Laplace transforms is used to ®nd the drying time required to dry a pipeline. The numerical method is also used for predicting the drying time for a pipe length of 20 m and the results obtained by these two methods show good agreement. It is found that the thickness of the water ®lm, the volumetric ¯ow rate of dry air, the dew point temperature of the dry air and the length of the pipeline in¯uence the time required for drying the pipeline. AcknowledgementsÐThe authors are grateful for the ®nancial support and facilities provided by the King Fahd University of Petroleum and Minerals.
REFERENCES 1. S. Y. Daib, Determining volume and concentration to dry gas pipelines. Oil and Gas Journal 81, 80±83 (1983). 2. S. Y. Daib, Techniques for drying pipelines by the three-spheres method. Oil and Gas Journal 81, 112±116 (1983). 3. V. Battara and B. Selandari, Mathematical model predicts performance of pipeline drying with air. Oil and Gas Journal 82, 114±116 (1984). 4. G. A. LaCasse and T. Ingvordsen, Desiccant drying of gas pipelines. Material Performance 27, 848±851 (1988). 5. V. M. Gorislavets and A. A. Sverdlov, Numerical investigation of the process of ventilative drying of a pipeline. Journal of Engineering Physics 60, 615±623 (1991). 6. Treybal, R. E., Mass Transfer Operations. McGraw-Hill, New York (1980). 7. T. H. Chilton and A. P. Colburn, Mass transfer coecients. Industrial and Engineering Chemistry 26, 1183± 1187 (1934). 8. Trim, D. W., Applied Partial Dierential Equations. PWS-Kent Publishing Company, Boston (1989).