UNIVERSIDAD DE ORIENTENCLEO DE ANZOTEGUIESCUELA DE INGENIERA Y CIENCIAS APLICADASDEPARTAMENTO DE MECNICAFLUJO BIFSICO

HEAT TRANSFER MODEL OF FLOW IN PIPES HORIZONTAL

REALIZADO POR:

REVISADO POR:

NARVAEZ ANDREA C.I.: 21079954PROF.CAMARGO LINO

PERDOMO WILLIANNYS C.I.: 20398097

GIPE ANDERSON C.I.:20054894

PUERTO LA CRUZ, JULIO DE 2012

6. Heat transfer from two-phase flowHeat transfer during the flow of gas-liquid two-phase, is found widely in industrial operations. Occurs during the vaporization and condensation processes such as refrigerator systems, boilers, furnaces and capacitors, as well as in applications where the heating of the two phases occurs without significant phase shift. It has also been heat transfer two phase flow of special interest to the nuclear energy in connection with attempts to predict the course of a loss of coolant accident.

Most of the studies published in heat transfer two phase flow has been empirical in nature and independent flow pattern. As mentioned above, all the design variables of two phase flow strongly depend on the existing flow pattern, including the heat transfer process. Thus, the heat transfer for different flow patterns can differ markedly. Updated reviews of the literature on convective heat transfer two-phase flow are given by Kinm (2000) and Manabe (2001)

6.1 IntroductionPipe for the flow of turbulent phase, the correlations shown in Eq. 6.1 to 6.3 are commonly used to determine the coefficient of convective heat transfer.

For the correlation of Dittus and Boelter (1930).

..(Ec 6.1)

Correlatin Colburn (1933)

...(Ec.6.2)

Correlatin Sieder y Tate (1936)

......(Ec.6.3)

Where Nu is the Nusselt number, k is the thermal conductivity and B W are the bulk and wall viscosities at respective temperatures, Re is the Reynolds number and Pr is the Prandtl number, defined by

......................................... (Ec.6.4)

Note that the difference between the three correlations is practically negligible. A correlation was most recently issued by Petukhov (1970)

...................... (Ec.6.5)

Where f is the friction factor. In a modified version of the correlation, the RHS of Eq 6.5 is multiplied by the term ( / ) 0.25

Attempts to develop correlation for the two phase flow has been carried out using equations similar to phase current. The correlations developed are independent flow pattern. The simplest method is the use of inhomogeneous drive model, as suggested DeGance and Atherton (1970)

............................... (Ec.6.6)

Where all parameters are averaged on the basis of the non-sliding retaining fluid. This approach is best suited to dispersed bubble flow and is discussed in more detail in Section 6.3.

Davis and David (1964) suggest the following correlation is shown in Eq 6.7 for the determination of HTP the heat transfer coefficient two phases.

........................ (Ec.6.7)

Another approach to correlate data heat transfer two phase flow has been the use of the Lockhart and Martinelli parameter, X Several studies suggest a correlation follows:

................................ (Ec.6.8)

Where HL is the coefficient of heat transfer from the liquid phase (surface) and the parameter is XTT Lockhart and Martinelli turbulent to turbulent conditions, as shown in Eq. 6.9 (where x is the quality).

....................... (Ec.6.9)

Have proposed different values for the coefficients of Eq 6.8. Dengler and Addoms (1956) suggest C = 3.5 and n = 0.5, while Collier and Pulling (1962) used C = 2.5

............................ (Ec.6.10)

and n = 0.7. A similar correlation was proposed by Rounthwaite (1968), as shown in CD. 6.10.

Notice that Eq 6.10 is equal to 6.8 Eq if neglecting the effect of viscosity, ie, (ul / mg) 0.1 = 1, C = 1.8 and n = 0.77.Several investigators attempted to incorporate variable two-phase flow, such as superficial velocities or gas void fraction, the correlations of heat transfer coefficient of two phase flow. An example is the correlation of Dorresteijn (1970) for upstream and downstream given verticalby............. (Ec.6.11)And,

................... (Ec.6.11)

Another example of this approach is the Rezkallah and Sims (1987) upward vertical flow correlation, given as

............ (Ec.6.12)

.................. (Eq 6.12)

Barnea and Yacoub (1983) developed a mathematical model with an analytical solution based on the method of characteristics, to predict the heat transfer process to unstable slug flow in vertical upward pipe. The model provides a solution to the temperature of the gas phase and liquid phase in function of time and the axial position and temperature fluctuations in the pipe wall. The solutions are presented for both constant heat flux to constant wall temperature.Kaminsky (1999) derive methods of calculating an average coefficient of convective heat transfer in two-phase flow for all flow patterns except for annular flow. These methods are coupled explicitly with individual flow pattern hydrodynamic models for the prediction of condensate liquid and pressure drop. The comparison between the proposed methods: vertical and horizontal data reveals errors of 30%. The explicit link between the hydrodynamic and heat transfer ensures that the predictions are more robust, so that conditions can be applied beyond the range of experimental data.

More recently there have been two experimental and theoretical studies on convective heat transfer in two-phase flow. Kim (2000) developed a correlation coefficients for the prediction of heat transfer to gas-liquid flow in horizontal and vertical lines. The correlation has been approved as published and the data collected in this study. Manabe (2001) had acquired data convective transfer of heat to flow from one phase and two phases. Data were acquired in a tube of 5.25 cm diameter at 3101 kPa using crude oil and natural gas under cooling conditions. On the basis of experimental results, Manabe recommended Petukhov correlation (1970) for liquid phase flow, whereas for the gas flow were suggested phase correlations of Dittus and Boelter (1930) or Colburn (1933). Data from two phase flow measurements include average coefficient of heat transfer, bubble HTP vertical and intermittent streams, horizontal and vertical annular flow and stratified flow horizontally. The flow pattern dependent correlations of heat transfer coefficients were presented and evaluated in relation to the data, showing errors between 20 and 40%. As mentioned above, the following sections present studies only dependent mechanical model or data flow pattern of experimental local heat transfer in two-phase flow.

Stratified flow 6.2 Heat TransferThis analysis was presented in Taitel and Dukler (1985). A heat transfer scheme in stratified flow, including the relevant variables in a cross-sectional area is shown in Fig 6.1. The liquid phase zone is confined by the angle L while the gaseous phase is limited by the angle G. higher temperatures of liquid and gaseous phases are TL and TG, respectively. Tw () is the local pipe temperature of the wall surface, and qw () is the local heat flux. The coefficients of local heat transfer to the liquid phase and gas are hL () and hG () respectively. Note that the local variables defined, qw (), T w (), hL () and hG () depend on the angular position around the periphery of the tube .

6.2.1 physical phenomena in stratified flow, heat transfer gas and liquid phases are considered separately. This is due to the fact that the coefficient of heat transfer to the liquid phase, hL (), is often an order of magnitude greater than the coefficient of heat transfer to the gas phase, hG () of fact, a good approximation would be to transfer heat to the liquid phase only because the amount of pulse transferred to the gas phase is usually negligible. This estimate is true for conditions of large fluid retention and may not be the case for small water retention.

Although the heat transfer in the gas phase is negligible compared to the liquid phase is transferred, an interesting phenomenon in respect of the volume of gas temperatures and liquid, TG and TL. Although only a small amount of heat is transferred to the gas phase, there is a large increase in the temperature of the gas mass because the heat capacity of gas is much lower than the liquid. Furthermore, although a greater amount of heat is transferred to the liquid phase, the temperature increase is much smaller than the gas phase due to the heat capacity is greater in the liquid. As a result, when heat is transferred to the two phase flow, the temperature of the gas volume is always higher than the temperature of the liquid volume. This applies to other types of flows when it is not well mixed, and the region of liquid film flow of mud. Thus, sometimes the temperature increase of the volume of gas along the pipe can be of primary interest and importance, therefore not be neglected in the heat transfer gas.

6.2.2 Definition of heat transfer variables. For stratified flow, as shown in Figure 6.1 and discussed above, varying the heat transfer coefficient around the periphery of the pipe. The coefficient of heat transfer local to the liquid phase is defined as:

.................................... (Ec.6.13)

Similarly, the coefficient of heat transfer local to the gas phase is defined as

..................................... (Ec.6.14)

The circumferential average coefficients of heat transfer fluid and gas can take place in two different ways. The first method is the average coefficient of heat transfer on the angle confined by the areaphase, , as

.... (Ec.6.15)

The second method is based on a local average heat flow and temperature of the wall of a phase, as shown in EQ 6.16 for the liquid phase,

.................................... (Ec.6.16)

Where:

....................................... (Ec.6.17)

and

..................................... (Ec.6.18)

The coefficient of average heat transfer from the gas phase can be defined similarly. In practice, there is negligible difference between the definition of heat transfer coefficient given by Eq circumferential average 6.15 or Eq.6.16.

Since the gas temperature is generally higher than the temperature of liquid can occur heat transfer between the two phases. The average heat transfer coefficient at the interface is given by

.................................... (Ec.6.19)

Where is the average heat flux at the interface.

6.2.3 General approaches. Two basic approaches may be used to predict the heat transfer process in stratified flow. The first approach, a simplified approach is to treat the gas and liquid phases as a single-phase flow, using the concept of hydraulic diameter. Therefore, the correlations available single-phase flow are such as Dittus and Boelter (1930), which can be used to determine the heat transfer coefficients for each of the phases. This is analogous to the processing of the pressure drop in stratified flow, where the calculated correlations of the shear stresses in the walls of gas and liquid flow pipe monophasic, applying hydraulic diameters for the phases. This approach is discussed in detail in the next section.

The second approach, a numerical approach, presented by Davis and Guzy (1979), is similar to your solution to the pressure drop and fluid retention. In this case, it is considered a heat flux of constant wall, which is solved according to the temperature profile of the liquid phase, by the expressions of Deissler and Von Karman for eddy viscosity (see Eqs. 5.71 and 5.72). Assumed that the temperature profile is a function only of the radial position, Roy, normal heat flow, and that the thermal diffusivity equal to the turbulent eddy viscosity. The gas phase, however, is treated as single phase flow in a duct using the single-phase flow correlations. This approach requires a numerical solution for determining the temperature profile of the liquid phase T = T (r) and the temperature TL of the liquid volume.

6.2.4 Simplified model. The following assumptions are made in the simplified model:

The heat transfer coefficients are constant liquid and gas, hl = const. and hG = const., averaged over the tube length. The coefficients of heat transfer fluid and gas can be calculated by the correlation of single-phase flow for the Nusselt number (as given in Eq. (6.1 to 6.5), using the hydraulic diameters of gas and liquid phases. Using the correlation Colburn (1933), the Nusselt numbers (and heat transfer coefficients) for gas and liquid phases can be determined, respectively,

...... (Ec.6.20)

and

The heat transfer coefficient of the interfaces, hi, is zero. (Another approach is hi = hG).

Two different boundary conditions used are constant wall temperature and constant heat flux, as detailed below.The two different boundary conditions that can be used are shown schematically in Fig 6.2. The case of constant wall temperature (case a) may occur, for example, when steam condenses on the outer tube wall. The second case (case b) is the case of constant heat flux, which occur during the uniform heating, as by electrical elements attached around the periphery of the pipe.

In case a constant wall temperature. Shows a diagram of the case wall temperature constant, only the liquid phase, in Fig 6.3. As shown in the LHS, the length of the pipe section is c and the liquid temperature increases from the inlet temperature (known) TLI. the outlet temperature (TBD), TLO. The temperature profile in function of the axial length L, is shown in the RHS. The heat transferred to the liquid phase is given by (using the log mean temperature)

................................. (Ec.6.21)

The heat transfer is related to the temperature rise by

................................ (Ec.6.22)

For a given set of flow conditions, including temperature wall, Tw and the intake temperature, TLI, the outlet temperature, TLO QL and the heat transfer can be determined by solving the simultaneous 6.21 and 6.22 Eq. The heat transfer coefficient is calculated from Eq hL 6.20 (or similar correlations). The heat transfer gas phase and temperature can be resolved in a similar manner.

Case B: constant heat flux. For constant heat flux is shown in Figure 6.4, only the liquid phase. In this case, as shown in the LHS, the wall temperature and the liquid temperature increases, TWL and to Two TLL to TLO, respectively. It is assumed that the wall temperature at any cross section along the pipe is constant around the periphery of the pipe. This means that the temperature of the pipe wall is a function only of the axial position

......................................... (Ec.6.23)

Note that this will occur for a high conductivity of the pipe wall due to a radial heat flux to the wall. Assuming that all heat is transferred to the liquid phase is obtained by an energy balance

........................ (Ec.6.24)

As QL, WL, SL CPL and are constant, it is assumed that the derivative of the temperature is constant (equal LHS) and that the wall temperature difference to liquid is constant (from equal RHS), given respectively by,

.............. (Ec.6.25)

Therefore, Tw and TL increase linearly along the pipe, as shown schematically in Fig RHS 6.4. integration of Eq. 6.24 is:

................. (Ec.6.26)

Eq. 6.26 to calculate the outlet temperature, TLO (from equality LHS) and the constant temperature difference between the wall and the liquid, Tw-TL (RHS equal). Therefore, the wall temperature of the inlet and outlet, and Two TWI can be determined. The wall temperature profiles and liquid are shown on the right side of Fig. 6.4.

6.3 Bubbles and dispersed flow heat transfer in bubbleThe non-slip homogeneous model can predict the behavior of hydrodynamic flow to dispersed bubble flow (see Section 2.1). In this model, the gas and liquid phases are assumed well mixed, so that no slippage occurs between the phases. Therefore, the two phase mixture is treated as a single phase fluid in a pseudo speed and the average physical properties. The physical properties are determined from a single gas phase and the liquid properties, on average based on liquid retention desizante not, L. the non-slip homogeneous model can be extended to predict the heat transfer process in the dispersed bubble flow.

This is done assuming, again, a single-phase flow and using a pseudo-correlations of the Nusselt number for determining the coefficient of heat transfer phase flow. Consequently, the thermal properties are calculated from the liquid phase mixture of gas and thermal properties, averaged on the basis of the non-sliding retaining fluid.

The Nusselt number of the mixture can be determined from equation modified Colbum (1933) for flow conditions of gas-liquid non-slip, which is given by

........................... (Ec.6.6)

The thermal properties (ie, thermal conductivity and thermal capacity of heat) are averaged based on the non-sliding retaining liquid, given respectively by

............................. (Ec.6.27)

And

.................... (Ec.6.27)

Note that the density and viscosity of the mixture does not slip and are defined by Eq.2.6 and 2.11, and non-slip Reynolds number is defined by EQ. 2.16 The non-slip Prandtl number is given by

................................. (Ec.6.28)

Using Eq 6.27 and 6.28 in Equation 6.6 is possible to determine the heat transfer coefficient mixing calculations HNS and heat transfer. The calculations for the cases of constant wall temperature and constant heat flux to follow a similar procedure presented above for the case of stratified flow (see Sec 6.2.4)

6.3.1 Constant wall temperature. The heat transferred to the mixture is given by (similar to 6.21 EC.)

................ (Ec.6.29)The heat transfer is related to the increase in temperature (similar to Eq. 6.22)

........................... (Ec.6.30)

The outlet temperature, and the heat transfer TMO, QM can be determined by solving Eq. 6.29 and 6.30 simultaneously with the heat transfer coefficient, the HNS calculated from Equation 6.6.

6.3.2 constant heat flux. For this case, the heat transfer process can be determined by (similar to Ec.6.26)

........... (Ec.6.31)

Because QM is given for this case, Equation 6.31 allows calculation of the outlet temperature of the mixture, TMO (equal LHS) and also the constant temperature difference in the wall of the mixture, TW-TM (from equal RHS). Thus, the inlet and outlet temperatures of the wall, and TwO TWI also be determined.

6.3.3 Bubble Flow. The same procedure as described above, can be used to determine the heat transfer coefficient for bubble flow. The only difference is the use of real fluid retention can be calculated from the bubble flow model (see Sec 4.3) instead of the no-slip liquid retention of the above equations.

Transfer 6.4 Slug flow heatSeveral studies have been published on heat transfer in slug flow intermittently. These include Abou-know and Johnson (1952), Akimenko et al (1970), and Zarudnev Fedotkin (1970), fried (1954), Hughmark (1965), Johnson (1955), King (1952), Lunde (1961) and Oliver and Wright (1964). However, the results of these studies are presented as time and space averaged data over a slug unit. However, the intermittent flow is inherently unstable process with large variations with time of local flow rates, distribution of phase and phase velocities in any cross section. As a result, therefore expect large swings in the local heat flux at heat transfer coefficients, and in the wall temperature. In addition, the temperature of gas and liquid may differ substantially, which can be important for the designer.

The unsteady process of heat transfer in slug flow is highly dependent on the hydrodynamic flow. Dukler and Hubbard (1975) proposed a hydrodynamic model for slug flow (see Sect 3.4.1). Suggest that a typical slug unit consists of four zones (see Figure 6.5):1. a swirl that is mixed in front of the slug length, Lm, where the slow film before the slug is collected and mixed with the slug2. the main part of the slug of liquid, Ls, where the liquid moves as the flow occupy the entire pipe.3. An area of liquid film, Lf, where the liquid is poured from the back of the slug and decelerates. This fluid flows in a layered configuration, with the height and varying speed along a distance behind the slug.4. A region of gas flowing over the film pocket, of the same length LF.

The Dukler and Hubbard (1975) model (and other models of slug flow) allows the prediction of these characteristic lengths and the velocity distribution (ie, the slug, film speeds and gas-pocket). Because the flow characteristics of the three zones of liquid are different, one would expect the heat transfer process is also different. Based on the hydrodynamic model of slug, Niu and Dukler (1976) developed a model for

Figure 6.5. Scheme of the different areas of slug

Predict the timing and position depending on the temperature of the fluid and wall, and the heat flux and heat transfer coefficients. Shoham et al. (1982) measured the characteristics of local heat transfer for flow in horizontal pipes slug. The variation in time of the wall and fluid temperatures, heat flow and heat transfer coefficient for the different zones were reported slug flow. Also presented a qualitative theory to explain the substantial differences in the heat transfer coefficients measured between the top and bottom of the slug body. This study is presented below.

6.4.1 Experimental ProgramA schematic of the experimental design is shown in Figure 6.6. It represents a modification of the equipment, which was originally used to study the hydrodynamics of slug flow by Dukler and Hubbard (1975). Flow loop was 3.81 cm internal diameter and the working fluids were air and water.

Test section. Figure 6.7 shows the thermal test section, consisting of an inner diameter of 3.81 cm and 6.35 cm outer diameter, and a brass tubing of 1.76 m in length, heated by electrical elements. The tube was covered with a layer thickness of 0.5 mm asbestos and a copper sheet 0.25 mm thick on top of which were tied heaters, this combination of an insulator and a good conductor provided a uniform heat flux along the pipe and around its periphery. Electric heaters consist of tubular electric elements 27 of 2,025 W at 277 V. They received three-phase power to 480 V thyristor controller CSCR able to adjust the output power or 50 kW. A layer of insulation was tied around the entire thermal section.

The measuring station was located 15 cm from the rear end of the section of heat. In this location of cross section were measured outer wall temperature and eight inner thermocouples, four in the interior and the exterior wall four equally spaced along half the periphery, as shown in Figure 6.8. Three thermocouples were suspended in the pipe by a spider for measuring the fluid temperature cross section of the measuring station. The inlet temperatures of gas and liquid were measured by two thermocouples located at the entrance of the test section. The numbers assigned to each thermocouple is shown in Figure 6.8. All thermocouples were 0.08 cm XacTpack type T (copper-Constantan) being grounded junction except thermocouple 14, which had a knot exposed to achieve a fast transient response for measuring the outlet temperature of the gas bag.

Figure 6.6-flow diagram of experimental heat transfer (after Shoham et al., 1982).

Fig 6.7

Figure 6.7-Schematic of thermal test section

Figure 6.8. Thermocouple locations.

The process temperature measurement is complicated by the multichannel output variable thermocouples set in the system. A multiplexer was constructed especially for that purpose, capable of output channels 14 to a preselected rate up to 10,000 samples / s sampling. The level of direct current (DC) of each thermocouple was removed and recorded, while the level of alternating current (AC) of each channel is amplified and then fed to the multiplexer shows all 14 channels sequentially, a second output channel provides a sync pulse at the beginning of each cycle multiplexer. The system was verified by the data acquisition for single phase water flow. For these races, the outer wall temperatures IT T4 were uniform, as were the inner wall temperatures T5 to T8. As a result, the heat flux and heat transfer coefficient on the inner wall were constant around the periphery of the pipe. Calculations of heat transfer coefficients showed good agreement with the values predicted by the correlation of Colburn (1933).

Data reduction. The 14 thermocouple outputs were sampled by the multiplexer and recorded on analog tape. This was digitized analog recording, and data are stored on a digital tape. Each record consisted of a set of numbers 14, 14 corresponding to the thermocouple readings taken during one cycle of the multiplexer. For these tests, the sampling rate was 1,000 samples / s. Thus, the maximum time between the two readings of the same thermocouple was 0,014 s. This time period is small compared with the characteristic frequency of slimy process, the time-varying temperatures of the inner and outer walls of the tube, IT T8, were used as the boundary condition variable in time for the solution the equation of conduction in the pipe wall in transverse measurement, given by

. ......................... (Ec.6.32)

where = k / Cp is the thermal diffusivity. This equation is solved numerically using a finite difference technique assuming symmetry about the vertical axis of the tube. Once it determines the temperature field in the pipe, could calculate the local heat flux at the inner surface

................................ (Ec.6.33)

Where r1 is the inner radius of the thermal section brass tube. Because the fluid temperatures were measured at the same time, it was determined the heat transfer coefficient local

........................... (Ec.6.33)

Where T w is the wall temperature and TB is the temperature of the mass of gas and liquid. Were calculated heat transfer coefficients in this way in the upper and lower positions and in two intermediate positions around the periphery (at the location of the thermocouples of the inner wall, see Figure 6.8). Because the axial variation in temperature was relatively small, axial conduction was negligible.

6.4.2 Experimental Results. Thirty experimental runs were conducted varying the mass flow rates between 0.45 and 1.6 kg / s for water and between 0.0023 and 0.011 kg / s of air, these flow rates cover a wide range of flow conditions slug to "blow" at which the gas blown through the slimy liquid and the transition results in the annular flow.

Temperature distribution. The time-varying measures the distribution of wall temperature and a set of fluid flow conditions shown in Fig 6.9. The energy input for this run was 8.5 W/cm2. Temperatures are measured in time intervals of 0014 s for each thermocouple by the multiplexer. Plotting these data produces very smooth curves as shown in the figure, with little or no dispersion. Shown here, for clarity, only 9 of 14 temperatures measured in a plane normal to the axis of the pipe at the measuring station located 15 cm from the end of the test section of heat. The data show the following results:

1. The outer wall temperatures T1 and T4 are constant with time. Which is expected for the wall thickness used in the pipe. Under these conditions, changes in heat flow in the inner wall does not reach the outer wall before initiating the next cycle of slug.

2. The temperature in the upper interior wall, T5, experience wide fluctuations. When the gas phase passes over the measuring station, the incoming flow in the outer wall exceeds the gas flow transferred to the inner wall, whereby the difference in energy is stored in the wall when a slimy liquid arrives, the wall surface is tempered, and the stored energy is transferred to the body wall of slime. With thin wall (low thermal capacity) of other materials of brass (low thermal conductivity) and higher heat flux, temperature fluctuations can be expected inner wall of over 100 C.Figure 6.9-time variation in the measurement of fluid and wall temperatures (see thermocouple locations Fig. 6.8).

3. Temperature fluctuations similar but lesser amplitude will be held in the bottom of the inner wall, T8. This is because the bottom of the inner wall is always in contact with the liquid phase. Note that there are wide differences in temperature between the upper and lower walls. Because the top is regularly in contact with the gas phase, its temperature is always above the bottom wall. As a result, one can expect to take place in the peripheral wall pipes.

4. Between slugs, the temperature of the liquid film increases as the temperature rises bottom wall. Therefore, temperature T11liquid film varies in phase and direction of T8, the temperature at the bottom inner wall. Engine temperature is smaller, just after the passage of slug and increases with time, reflecting a decrease in heat transfer coefficient as the deceleration of the liquid film.

5. T9 shows the temperature measurement using the shielded thermocouple located within the thermal section in a position where it undergoes gas-liquid alternately. However, it is expected that the shielded thermocouple has a slow response in the air. A thermocouple junction exposed, T14 was installed at the outlet of the test section. As shown, the two temperatures coincide with each other after exposure times of 1 s, which is consistent with the anticipated delay. Therefore, T14 was used as an indicator of exhaust gas temperature real. The data show that even at these lower heat flux, temperature of liquid and gas differ by approximately 30 C.

6. The response time is short liquid thermocouple. Thus, the decrease indicated T11 reflects the passage of a liquid slug of the thermocouple. Note the small difference in temperature of the liquid, as indicated by T9 (upper slug) and T11 (bottom of slug). This suggests the existence of a temperature gradient in the body of the slug.

Table 6.1. A summary of the temperature measurements that will be useful in subsequent discussions. T5 and T8, temperatures in the center of the top and bottom wall are the arithmetic mean of the maximum and minimum temperatures during the passage of liquid slugs in place. TG, was the average was obtained from the thermocouple junction T14. TF represents the film of liquid taken from T11 as the film of liquid which passes through the thermocouple, where Ts is the arithmetic average of T9 and T11 taken during the entire time that the slug was present on the thermocouples.

Heat flow. A typical example illustrates an output of a computer showing the time variation of heat flow to the fluid in the upper and lower inner wall (Fig. 6.10).Unusual time intervals shown on the abscissa are related to the frequency multiplexing, in which a unit in the X axis corresponds to 0128 s. The solid curves were superimposed on a printing machine. For this run, the time interval between slugs was 1.92 s, corresponding to a frequency of slug Vs = 0.52 s-1. The passage of the slug is indicated more clearly from the heat flux at position 5, the interior of the upper wall. It takes 0.34 s for the liquid dribble pass through this thermocouple. At this time, the surface becomes exposed to the gaseous phase, and heat flow drops sharply. A pronounced peak in flow exists in the top front of the slug of liquid due to turbulent mixing. Also shown in Figure 6.10 heat flow at the bottom and inner wall. a comparison between the heat flow from the bottom and which is produced at the top shows that the flow is not uniform around the slug body. This comparison also provides a better view of the hydrodynamic processes:

1. The effect of turbulence in the mixture in front of the slug is more pronounced at the top of the pipe at the bottom. This is because the upper wall surface acts as a stagnation in the mixing process. Furthermore, the effects of turbulence mixing are not located in front of the slug, as previously visualized, but influencing the transfer process along the entire body of slug.2. At the bottom of the pipe, the fluid velocity decreases continuously and smoothly from the liquid slug to the film. The data show that the lower heat flux also decreases continuously. The process of change results in streams that flow around the wall and these are often the cause of the data spread in the area of film.

Experimental Fig. 6.1O-heat flux variable in time.

Heat transfer coefficients in the slug. With a computer data output were heat transfer coefficient at the top and bottom of the pipe, which corresponds to the heat flow data in Fig 6.10, is presented in Fig 6.11. This graph is typical of the results of all executions. As expected, there is shown a heat transfer coefficient increased in the mixing zone at the front of the slug. A nearly constant value shows the body of the slug of liquid, while the coefficient of the liquid film decreases as decelerates with distance behind the liquid slug, also as expected, in regions where contact with the gas surface, the coefficient of heat transfer decreases to a very low value. Two very interesting results emerge from the data:

1. The coefficients of heat transfer in the mixing zone at the front of the slug is surprisingly high. For this case, a maximum coefficient of more than 35,000 W/m2. C was observed in the bottom of the pipe, this value is greater than what would be expected to further condensation of film or nucleate boiling water.2. The coefficients in the bottom of the pipe is always higher than the top, and in many cases, this difference can be as large as a factor of 2.0.

Time average values of the heat transfer coefficient on the experimentally measured area of the mixture and in the body of the slug in the upper and lower positions were calculated and are listed in Table 6.2.

Figure 6.11 heat transfer coefficient experimental variation in time.The heat transfer coefficients predicted given in the table were calculated by the correlation of Colburn (1933) (ie Eq. 6.2) applied to the slug body, as given in the following manner

................ (Ec.6.35)

Note that Nuo is calculated for uniform wall temperature during the passage of the slug, the physical properties of the slug is calculated by averaging the properties of the liquid and gas phase by HLLS, retention of fluid in the slug body gives, as shown in Eq 6.36

......... (Ec.6.36)

And

.............. (Ec.6.37)

For these calculations, HLLS was obtained from the experimental measurements of Dukler and Hubbard (1975), the marked difference between the coefficients of heat transfer at the top and bottom can be easily observed in the nose (mixing zone) and the body slug. High rates of fluid and the low gas flow rates, the ratio of heat transfer coefficients of the top and bottom is low, as an example, for WL = 1.59 kg / s WG = 0.0023 kg / s, this ratio is 1.5, during these runs, the temperature variation around periphery of the inner tube was small, approximately 6 C, when there is a decrease in fluid flow relationship the coefficients of heat transfer increases. In the same gas flow rate, WG = 0.0023 kg / s, but with WL = 0.91 kg / s, the ratio is 2.3 and is approximately 3 to WL = 0.68 kg / s. Under these conditions, there is a substantial difference in temperature between the upper and lower walls of the inner tube, on the order of 30 C. The proportion of the heat transfer coefficient and bottom also increases with increasing gas flow rate at constant flow of liquid, WL = 0.91 kg / s with a low rate of gas flow WG = 0.0023 kg / s, has a ratio of 2.3, increasing to approximately 4 WG = 0.0091 kg / s.

The experimentally derived values of Nusselt number for slug body, Nus = hsd / ks, in places of the top and bottom depending on the Reynolds number of slug, Res = svsd / ms are shown in Fig points . 6.12. To calculate these parameters, the heat transfer coefficient was obtained from the experiments, as shown in Table 6.2, the speed was slug is Vs = VM, while the average physical properties were calculated using Eq 6.36 , assuming a uniform value HLLS on the top and bottom of the pipe. The number of Nussel for conditions of uniform wall temperature, Nuo is calculated by Equation 6.35, is shown as the dashed line. The data in the bottom of the pipe are greater NU0, while the top are generally low.

A representation similar to the mixing zone of slug shown in Fig 6.13. For this case, the average Nusselt number, NU0, is given by the EC. 6.35, with a coefficient of 0.023 substituting 0.03.

Fig. 6.12 Nusselt numbers for slug body.

Heat transfer coefficients in the liquid film. The heat transfer coefficients in the liquid film varies with distance, as expected, since the liquid is decelerated at a distance behind the slug. The experimental results for the cases of short, moderate and long lengths of film, Lf shown in Fig 6.14 as solid lines. In this figure, x is the distance along the film is measured from the back of the slug. The correlation of Colburn (1933) was used to calculate values for comparison, using the hydraulic diameter of the film and the properties of the liquid phase, as done in stratified flow (Eq. 6.20). The results are shown as dashed lines in Fig 6.14. The difference in results in the region immediately behind the slug body is given at the bottom of the pipe, the coefficient on the back of the slug is greater than predicted by equation 6.35, discussed above. When areas of film are long enough, these differences disappear.

Heat transfer coefficients in the gas bag. Due to the heat flows too low, the heat transfer coefficient in the gas bag can not be determined with the same procedure used for the liquid phase. Instead, we calculated the average coefficient of heat transfer of measurements of temperature increase of the gas bag in the thermal test section. A comparison of these measured values and calculated from the correlation of Colburn (1933), using the properties and the hydraulic diameter of the gas phase (Eq. 6.20), is given in Table 6.3. Caen experimental and predicted values from 5 to 50 W/m2. C.

6.4.3. An approximate theory for heat transfer slimy flow. Experimental data in table 6.2 and fig. 6.12 and 6.13 show that the heat transfer coefficients significantly differ between the upper and lower body slug. These differences may be attributed to two effects: (1) the existence of a higher concentration of bubbles in the upper part of the slug and (2) the existence of a significant temperature difference between the surfaces of upper and lower wall which are put in wetted.

Figure 6.14 Experimental and heat transfer coefficients predicted in the region of the liquid film.

The presence of a higher concentration of bubbles in the upper surface can be expected to decrease the heat transfer coefficient due to its effect on the local average flow properties. However, the heat transfer experiments was carried out in the rate of gas and liquid, so that the slug was free from bubbles (HLLS = 1), this showed that there are still marked differences. Moreover, calculating the heat transfer coefficient by Eq 6.35 HLLS with different values in the bottom and top showed that the effect of a non-uniform concentration of gas bubbles in the coefficients of heat transfer is small compared to the observed differences.

Fully developed flow in pipes phase, the existence of the variation in circumferential heat flux or temperature around the periphery of pipe has been shown to have a substantial effect on the coefficients of local heat transfer. Solutions have been proposed for laminar and turbulent flow in pipes (Reynolds, 1960, 1963, Sparrow and Lin, 1963; Rapier, 1972; Gartner et al., 1974; and Schmidt and Sparrow, 1978). These solutions are not applicable here because the slug flow is mainly a problem in the region of heat input to the temperature of the periphery of the wall. This situation exists because it increases the temperature of upper wall during the passage of the gas bag, just before the arrival of each slug flow.

To explain this phenomenon in a slug, is considered a short cylindrical element of liquid placed first in the nose of the slug, the intense mixture produces a uniform temperature, as the slime IT moves down the tube, the cylindrical member moves back on the nose, until it moves to the back of the slug. During this period of residence in the slime, the element is exposed to different temperatures around the perimeter. As a first approximation, the wall temperature is assumed to be constant in the axial direction (average value in the length of the slug), for any given circumferential position. The problem is then to predict h (x, ), where x is the distance traveled by the element in the nose of the slug, until moving to the back, and is the peripheral coordinate.

Not yet developed a solution to this problem of pipe turbulent flow. As a means to estimate this effect, Shoham et al., (1982) presented a solution for the case of the laminar slug flow between parallel plates, which are held constant but different temperatures. The objective is to find the relationship Nu1/Nuo Nu2/Nuo and the top and bottom plates, respectively, to explore the tendency, in comparison with the data of the same proportions of turbulent flow pipe. Note that NU0 is for conditions of uniform wall / isothermal top and bottom. In this approach, the energy equation is solved for the laminar flow of slug to a speed v between two parallel plates spaced at a distance, e. The temperatures of the bottom and top plate TB and TT (TT> TB), respectively, are constant in the flow direction x, as shown schematically in Figure 6.15. The y coordinate is measured from the bottom plate and the liquid enters at x = 0 with a uniform temperature, TI.

Transfer Fig. 6.15 heat slug in laminar flow between parallel plates.

Heat equation for the case of two dimensions is

................................ (Ec.6.37)

where a = k / ( Cp) = / . Using an overlay technique, for the convenience of solution (assigning 1 to the bottom plate and top plate 2) establishing the form of dimensionless temperature profile.

.................... (Ec.6.38)

Where

.................... (Ec.6.39)

The boundary conditions

and

........................ (Ec.6.40)

Is the dimensionless solution

....... (Ec.6.41)

where x '= (x / l) / Pe, y' = y / l and Pe = l Cp / k (Pe = l / ). As an example, the solution of a case where the initial temperature is 1 = 0.5 is given in Fig 6.16. Note that at x = 0.181, disappears the derivative of the temperature in the bottom plate.

The Nusselt number in the lower and upper plates respectively, are

.................. (Ec.6.42)

Figure 6.16. Temperature profile for laminar flow between parallel plates with different initial temperatures of 1 = 0.5.

Where = (T - TI) / (TT - IT). value of the integration can be found from Eq 6.41 to between y 'from 0 to 1.

...... (Ec.6.43)

Because the numerators in Eq 6.42 are obtained by differentiating Eq 6.41, the resulting Nusselt numbers depend only 1 and x ', a typical solution for the Nusselt numbers of top and bottom, calculated for the case where 1 = 0.5 (for the temperature profile of Fig 6.16), this is shown in Figure 6.17. The dashed line shows the result of the temperatures of the plate equal to Nuo, fully developed flow between convergent plates great lengths in the value of 2 / 2, as expected. When there is a temperature difference between the plates, the Nusselt number of the top plate, Nu2, monotonously decreases, as shown and finally approaches a value of 2.0 (for the case of 1 = 0.5). However, the Nusselt number for the bottom plate, nu1 behaves quite differently. Between the inlet region and a value of x '= 0.09, nu1 passes through a minimum, approaching infinity at x' = 0.09. At this point, the average temperature approaches the temperature of the bottom wall, although the results of energy transfer coefficient of the temperature gradient near the wall can be expected to be larger. Thus, in most of the region 0 0.09, the Nusselt number is negative, with additional heating, 1 or exceeds the average temperature becomes higher than the lowest temperature of the wall, although following the heat transfer fluid. For large values of x ', the heat transfer direction changes by which the energy moves from the bottom in the wall of the liquid, and as a result, the heat transfer coefficient and the Nusselt number become positive. Finally, the values of length of the plate are sufficiently large, the two Nusselt numbers again become equal.

Figure 6.17 - Nusselt numbers at the top and bottom of slug flow between parallel plates with different initial temperatures ( 1 = 0.5).

It is convenient to consider the results of the theory of the ratio of Nusselt numbers, and observe Nu2/Nuo Nu1/Nuo and its variation with respect to local temperature difference.

The results are shown in Figure 6.18. Nuo values as a function of x 'can be calculated from Eq 6.42 setting 1 = 1.0. Interestingly, in this coordinate system, for values of x ' 0.1, the ratio approaches a single value for a given and is essentially independent of x'.

With the premise that the area of the region behind the slug mix is characterized by the length parameter, x ' 0.1, and O for all locations along the slug body is now possible to compare the coefficients of heat transfer experimental and theoretical. The temperature data in Table 6.1 were used to calculate for each execution, which in turn was used to determine Nu2/Nuo and nu1 / Nuo Figure 6.18. The theoretical values resulting from Nu2 (solid line) and nu1 (dashed line) are shown in Figure 6.12. The trend is provided in accordance with the experiment, and especially for nu1 at the bottom, the quantitative agreement is reasonably satisfactory. At the top, the role of gas and is especially important for data Nu2 dispersion is more important. A similar comparison is made in the slug flow in Fig 6.13.

Therefore, it appears that the difference in the coefficients of heat transfer located between the bottom and the top tube observed in the slug flow can be explained on the basis of theoretical analysis, results from the fact that each slug is in fact a region developing thermal input. The temperature differences which fall within the calculation of heat transfer coefficient are substantially different between the upper and lower walls due to the presence of a higher wall temperature above the top of each slug. Model using a single input stream in the region can be approximated slug correctly to the difference in the coefficients of heat transfer in the two locations.

Figure 6.18-dependence of the ratio of Nusselt numbers of upper and lower proportion of the local temperature difference.

6.4.4 A model of heat transfer for slug flow in horizontal pipes. Niu and Dulder (1976) developed a model for predicting heat transfer processes in unstable slug flow. The model allows the prediction of: (1) the time variation in position along the pipe or (2) the axial variation in an instant in time the temperature of liquids and gases, temperature inside the wall at the top and bottom of the pipe and the heat flow for each phase.

Physical phenomena. The heat transfer in slug flow is essentially a transient process. Consider point A in Fig 6.19, which is fixed in position on the inner wall of the pipe at any arbitrary axial location. If a heat source located in the outer wall, then the heat transfer is carried out in the liquid. however, the instantaneous rate of transfer will vary depending on the fluid (gas or liquid) in contact with the point A and in the local fluid velocities.

The shaded area bounded by a continuous curve showing the liquid limits at time T0. At that moment, the gas passes through the surface at point A, and the rate of heat transfer fluid is low. If the rate of energy input to the outer wall exceeds the rate transferred to the fluid in the inner wall, then it can be expected by increasing the wall temperature at point A. As the slug moves downstream, the film contains at front and emerges in the rear. Eventually, the front of the slug will move to a position showing the limits of liquid by the dashed line, ie at time T1. at this instant, the speed of heat transfer (from the wall at point A liquid) is considerably greater than the time T0 because of these reasons: (1) much higher thermal conductivity of the liquid phase, in comparison with the gas , (2) the surface temperature is greater under constant conditions of fluid flow due to a previous contact with the gas phase, and (3) the liquid temperature is lower than the gas phase. Thus, the wall temperature falls rapidly. Thus, near the top of the pipe, the wall temperature can be expected to oscillate with the frequency of the slug.

Fig 6.19-Outline of a slug unit advancing on time.

At point B situated on the inner surface of the bottom wall, one can expect a variation in the rate of heat transfer and temperature similar to take place, although not as dramatically. At time T0, the speed of the liquid film is low (much smaller than the slug speed), resulting in a low rate of heat transfer, on the other hand, at time T1 the slug of liquid is moved past the point B and heat transfer rate is higher (approximately proportional to the square of the speed). therefore in this location, one can expect a variation in the rate of heat transfer and fluid temperature and the wall.There are two additional complications associated with the unstable hydrodynamic flow during the slimy:

1. When the gas phase passes through the point A while the liquid film passes point B, the upper wall temperature exceeds the temperature of the bottom wall and peripheral heat transfer takes place through the wall of the top to the bottom of the pipe.2. After passage of the slug, during the flow of the liquid film, the covered portion of the perimeter of the wall by the liquid film changes with time g. Thus, the calculation of the total energy transferred to the gas and liquid phases requires not only the calculated heat transfer coefficient for each phase, but, a determination of the fraction of the perimeter covered by the liquid at each point in space time. This information may be provided by a hydrodynamic model drooling.

Clearly, this unstable nature of heat transfer process is characterized not only by the behavior of the hydrodynamic slug flow and fluid properties, but also by the thickness and thermal properties of the pipe wall. The objective of this study is to provide the variation in position along the pipe or the axial variation in a time instant of the following variables:

Temperature of the liquid and gas volume. temperature of the inner wall at the top and bottom of the pipe. Heat flow for each phase.This time-dependent information will allow the determination of space and time variables or average total heat transferred to the fluid during a specified time interval for a given set of flow conditions, including the pipe diameter, flow, and physical properties phase.

Focus. The methodology for the study of this simulation is shown in Fig6.20. The input information necessary for the simulator is:

1. the data necessary to calculate the hydrodynamics of slug flow, according to a model like the Dulder and Hubbard (1975) (see Sect 3.4.1).

Simulator Fig.6.20 flow heat transfer slimy.

2. The data needed to calculate the thermal behavior of the pipe wall. The driving force for heat transfer is closely coupled to the volume of the pipe wall. Therefore, it is necessary to provide the data in the pipe wall including wall thickness, thermal conductivity and specific heat.3. Correlation coefficient of heat transfer between the pipe wall and adjacent gas or liquid phases.4. The data of the thermal boundary conditions on the outer surface of the wall.The simulator is built in two parts. Part A uses the model Dukler and Hubbard (1975) to calculate the hydrodynamic characteristics of a slug unit. Part B calculates the thermal efficiency using as input the output of part A and the thermal input data (sections 2, 3 and 4 given above).

Can occur four different mechanisms for heat transfer in a slug unit:

1. In the mixing zone at the front of a slug, the heat transfer coefficient to be controlled by eddies associated with the rapid collection and mixing of the film. In this region, the swirl velocity distributions expected for complete flow line should be significantly distorted. The heat transfer coefficient of this zone must be much greater, as demonstrated by the higher speed of transfer of momentum.

2. In the body of the slug of liquid, studies show that the hydrodynamics of the distortion in the wall is similar to that expected in a full pipe flow. Furthermore, the gas transmission has been demonstrated that move with the liquid slug velocity horizontal flow, therefore, the same correlations used for heat transfer fluid in turbulent flow pipe apply here, too.

3. In the liquid film behind the slug flow is similar to stratified configuration. For stratified flow, the friction factor can be calculated as an approximation, using single-phase flow correlations in pipe using the concept of hydraulic diameter (see Sect 3.2.1). By analogy, it is assumed that the same approach is applied to heat transfer, this was confirmed by experimental research Shoharn et al (1982). In their study demonstrated that, except in the region immediately behind the slug body, the heat transfer coefficient of the liquid film can be calculated from flow pipe correlations, such as correlations Colbum (1933) or Sieder and Tate (1936), always used the concept of hydraulic diameter.4. In the gas bag on the liquid film uses the same approach as in Article 3 (for the liquid film). Thus, one can determine the heat transfer coefficient of the gas bag via the flow pipe correlations using the hydraulic diameter of the gas bag.

As a first approximation, the special relationship that should apply to the mixing zone is ignored and the form of the correlation (1933) of Colbum for heat transfer is supposed to apply to different regions of indications.

................... (Ec.6.45)

Where dH is the hydraulic diameter and the subscript i refers to the location phase. Next is the application of the EC. 6.45 to the different areas of slug:

i = l: means the slug, where DH1 is the ID of the pipe, and EC. EQ reduced to 6.35 6.45, whereby the physical properties are determined from the EQ 6.36. i = 2: refers to the liquid film region behind the slug. Here, DH2 is the hydraulic diameter of the film (which varies along the region of film), given by the HLTBAp DH2 = 4 / SF, where SF is the wetted perimeter of the film. For this case, EQ EQ reduced to 6.45 6.20 (for the film) and physical properties for this case are those of the liquid phase. i = 3: is the gas behind the slug flow over the liquid film so that the hydraulic diameter is DH3 = 4 (1 - HLTB) Rev / SG, where SG is the perimeter humidified gas. EQ. 6.45 in this case, EQ is reduced to 6.20 (for gas), and uses the properties of the gas phase.

It should be noted that CD. 6.45 is based on data taken during steady state operations, with a pipe wall temperature uniform. Thus, application of this correlation to the slug flow conditions is carried out as a practical approach. In reality, each slug body is a thermal entrance region of temperature variation on the peripheral wall of pipe. As a result, the different heat transfer coefficients produced in the upper and lower body slug experimentally confirmed in a later study (Shoham et al., 1982). In the region of film of liquid / gas-hole, the conditions are also unstable. However, the heat transfer in the region of film can be treated by an approximation of pseudo steady state where the heat transfer coefficient is determined by the instantaneous speed and instantaneous average hydraulic diameter (this was confirmed by Shoham, 1982 .)

Thermal conditions of the limit. Is necessary to specify the boundary temperature conditions in the outer wall of the tube. In this simulation, it is possible to configure the heat flow or the outside temperature as constant or variable along the pipe or with time.

Modeling. The heat transfer process is modeled by an Eulerian approach / Lagrange mixed. The pipe, whose limits are fixed over time, is divided by a series of planes normal to the axis. This defines M segments any of which may contain (a) a homogeneous liquid mixture of gas if the slug is present (b) separate liquid film at the bottom and the gas phase at the top if that particular segment is occupied by region of the film of liquid / gas bag behind the slug. The length of each segment, Ax, is selected so that an integer number of segments fit into a slug unit, as shown in Fig 6.21. Within an entire axial segment, the wall is divided into 12 sections, as shown in Fig 6.22. The thickness of the wall is divided into three rings of equal width and the circumference is divided into four sectors, each including 1/4 of the total perimeter of pipe. The advantage of the symmetry of such a division is evident.

The simulation proceeds by solving simultaneous equations of energy to the wall and the fluid starting at t = O when a distribution of temperature or specified heat flux is applied to the outer wall. The initial temperature of the wall and the liquid in all segments is set equal to the fluid inlet temperature. A computational time step, Dt, select the At = Ax / VTB. Therefore, if the film of liquid in the segment N (see Fig.6.21) is considered to time t, the energy balance is calculated for a time interval Dx, where the film moves the segment in the N + 1 time t + Dt. Thus, it is known the position of the liquid at the beginning and end of each time interval Dt, and the temperature change can be related to the driving forces of the fixed locations of particular surface to which said element is exposed . in a similar manner, the energy balance in gas phase and slug body is executed for each control volume, and then the fluids are advanced to the next segment.

Figure 6.21 Axial segments per unit slug breakthrough method to control the volume.

Figure 6.22. Wall thickness segments

Energy balance for the liquid film. The increase in temperature of the liquid film during the time Dt, N moves from segment to segment N + 1, the result of: (1) the energy is transferred from the inner wall, EW and (2) energy net is transferred by convection, EC. Because the gas and liquid temperatures in the film in the region are different, the energy transfer can also take place between the gas phase and the liquid film through the interface. However, the relative velocity between gas and liquid is not great, and this transfer process is neglected. Consider the film, as shown in the cross section of Fig 6.22, the fraction of the area occupied by liquid, HLTB as well as all other parameters hydrodynamic hydrodynamic model can be obtained.

1. Fluid transfer wall.if

............. (Ec.6.46)

if

............... (Ec.6.47)

and if

(Ec.6.48)

2. Convective transfer. As a fluid element moves from segment N to N + 1, receives the liquid moves from the element in front of him and lose water to the element that follows, all in accordance with the hydrodynamic model. each of these fluid streams carries energy and the net transfer of energy contributing to the accumulation of energy in the fluid eIemento N. The volumetric rate of liquid thrown from anything that moves at a speed, HTLV, is (VTB - VLTB) ApHLTB. Therefore, the net energy is caused by convection in time, Dt

(Ec.6.49)

3. Balance the equation. Transfer mechanisms contribute to changes in temperature of the liquid film, given by EQ.6.50

............. (Ec.6.50)

Energy balance for the gas bag. Similar mechanisms for transferring power to a gas element as it moves from the segment N to N + 1, as presented below.

1. FLUID TRANSFER TO WALLif

(Ec.6.51)

if

(Ec.6.52)

And if

... (Ec.6.52)

2 - convective transfer, the speed of the gas bag has been demonstrated by: (Dukler and Hubbard, 1975)

............................ (Ec.6.53)

Where C = Co - l (Co equal to 1.2 or 2 and laminar to turbulent flow, respectively), therefore, the flow of gas from the preceding element, as is moved from N to N + 1, to become:

.................. (Ec.6.54)

Note that if HLLS = 1.0, this term is zero and there is no input of convective heat transfer to the gas. In the general case,

..... (Ec.6.55)

3 - Balance Equation. The temperature change of the gas-pocket is related to the heat transfer processes as

......... (Ec.6.56)

Energy balance for the drool.1. Wall to fluid transfer.

(Ec.6.57)2. Convective transfer...... (Ec.6.58)

3. The equilibrium equation

........................ (Ec.6.59)

Energy balance of the pipe wall. Consider a single ring in a sector of the wall of the included angle = / 2 in segment N (eg, rings 3, 6 or 9 in Fig 6.22). Four possible transfer processes can occur: (1) radial transfer between hoops (2) radial transfer from the inner wall of fluids (3) axial transfer between longitudinally spaced rings, and (4) transfer through the peripheral wall. Of course, there are transfer from the source located in the outer wall. Consider the outer ring segment 9, as shown in Fig 6.22. This ring segment expands in Fig 6.23.

The energy transfer process in each time period Dt shown in EQ. 6.60 through 6.64, assuming a known heat flow in the outer wall.

.................. (Ec.6.60)

(Ec.6.61)

(Ec.6.62)and

(Ec.6.63)

Where w is the thickness of the segment, therefore, the energy equation for the ring element is:

For the other rings, similar equations are developed.

Calculation procedure: The calculations are initiated by setting the fluid in all elements and the pipe wall temperature of the inlet fluid. At time t = O, the power is turned on and started the calculation. X = position is located so that the plane coincides with the front of a slug at time t = o. After each time step Dt, the fluids are advanced elements and repeat the calculations.

Fig 6.23 Energy balance of wall segment No. 9.eventually, the fluid and wall elements reach a cyclic steady state condition, ie, the temperature will range over time, but in any location in space, these cycles repeten exactly. This point in the calculation is determined by comparing the temperatures in the last segment of a slug after two successive cycles. When this temperature difference does not exceed a certain tolerance small , the computer stops and prints the results.

Results of a typical simulation. A computer program was developed on the basis of the proposed model. As an example of the results, consider the following conditions, which reproduce one of the executions (1975) by Dukler and Hubbard.

WL: 2.22 lbm / sWG: 0.00555 lbm / s

HLLs: 1r1 (ID): 0.06 ftvs: 1.15 s-1Fluid: Water-Air

For purposes of these energy calculations, we selected the following conditions.

Wall thickness: 0.01 ftFlow in the outer wall: 10 Btu/ft2s (independent of x)Pipe Material: Stainless SteelInlet temperature: 200 F (independent of t).

The hydrodynamic flow behavior was calculated by the simulator A (see Figure 6.20) using the model (1975) by Dukler and Hubbard, with the following results.

LV: 8.84 ft / sLS: 2.42ftVTB: 10.17 ft / sVs: 8.44 ft / sHLTB (x): see Figure 6.24.VLTB (x): See Fig 6.24.

To obtain these results, we selected the following increments:

X = 0,805 ft (well, 11 segments)At = Ax / VTB = 0.079 s.

One result of this simulation are shown in Fig 6.25, where temperatures are shown as functions of axial position. The heavy solid curve designates the temperature of the liquid phase and the thick broken curve represents the gas. The thin curves represent different sectors of the wall and rings identified in Figure 6.22. Consider first the very large differences in the temperatures of the gas and liquid in the region of lquido-pelucla/gas-bolsillo, in this case as much as 50 F. The fact that the temperature difference between gas and liquid is large questions the validity of the hypothesis that there is no heat transfer takes place between phases.The coupling between the wall and the fluid capacity is very clearly seen in curve 3, which represents the temperature of the inner ring of the upper wall. For these conditions, wall temperature oscillations are also about 50 F and are decaying into rings that are closer to the outer diameter of the pipe. A typical time history of temperatures in a fixed position along the pipe shown in Fig 6.26. The observed frequency coincide exactly with the frequency of the slug.Figure 6.27 shows the typical comparison between the proposed model and experimental data reported by Shoham et al. (1982). As shown, the pipe wall temperature are higher than expected experimental temperatures by 25%. On the other hand, the temperatures of gas and liquid are provided in accordance with experimental data within 1%

Figure 6.24

Figure 6.24-values predicted by VLTS hlts and Dukler and Hubbard Hydrodynamic mode (1975).

Figure 25.6-temperature prediction as a function of the distance (see Fig 6.22 for wall segment locations).

6.4 Unified model for predicting flowing temperature in wells and pipelines

A unified general equation for predicting the flow temperature in wells and pipes, is applicable to the entire range of angles, was presented by Alves et al. (1992). The equations are transformed into the equation for ideal gas or incompressible liquid for wells Ramey (1962) and the equation for pipe and Brandon Coulter (1979) with appropriate assumptions. This study also proposes an approximate method to calculate the flow coefficient Joule-Thomson oil.

Fig. 6.26-Prediction of temperature as a function of time (see Fig 6.22 for wall segment locations).

Figure 6.27-Comparison of experiments and predictions of temperature profiles (see Fig 6.8 for thermocouple locations).

6.5.1 Introduction. The temperature distribution and flow pipe wells often predicted by different methods. The method normally used Ramey (1962) to predict the temperature distribution (vertical) of the well. This method incorporates rigorously the complex process of transient heat transfer between the well and reservoir. Method Ramey, however, is limited to ideal gas or incompressible fluid. The equation of Coulter and Bardon (1979) is commonly used for the prediction of the temperature (horizontal) of tubing. It takes into account a more rigorous thermodynamic behavior of the fluid, entering the Joule-Thomson coefficient. Although equation of Coulter and Bardon (1979) was derived originally for the gas flow can also be used for the flow of liquid phase or two phase flow.

The use of different methods for pipes and shafts, each of which has its own limitations, it is undesirable for the practical design. This chapter presents a general and unified equation for predicting the temperature of the flow. Can be applied to pipes or injection wells and production under single or two phase flow, especially angle between models from horizontal to vertical, with models of fluid or oil composition.

Previous studies 6.5.2. Predicting a rigorous temperature distribution flowing wells and pipelines is complex. Requires the simultaneous solution of mass, momentum and energy conservation equations. The solution is complicated by the thermal interaction between the flow and the environment, especially in the reservoir. Thus, a rigorous analysis for the solution of this problem is not currently possible. In the past, attempts have been numerical algorithms or approximate analytical solutions.

The numerical algorithms apply a double iterative procedure, temperature and pressure to solve the three conservation equations simultaneously, which requires knowledge of the thermodynamic behavior of the fluid. Gould (1979), Gregory and Aziz (1978), Furukawa et al (1986) and Goyon et al (1988) proposed these computational algorithms.

Several researchers have proposed solutions for predicting approximate analytical temperature. Explicit expressions for the temperature distribution of the fluid is obtained by making strong assumptions about the geometry of the pipe, heat transfer with the environment and the thermodynamic behavior of the fluid.

Schorre (1954) conducted a pioneering study for the prediction of temperature on horizontal gas pipes. Its equation for explicit temperature results in a continuously decreasing temperature profile along the pipe, it never reaches an equilibrium value. Coulter and Bardon (1979) modified the equation Schorre. The modified equation predicts a temperature profile of fluid which asymptotically approaches a temperature slightly below the surrounding temperature. In the equation of Coulter and Bardon (1979), the thermodynamic behavior of the fluid is taken into account more rigorously. However, the assumptions of the heat transfer in steady state with a constant temperature environment and limit horizontal flow with this method are made to a single pipe.

Ramey (1962) proposed a new method for predicting the temperature in wells. This method coupled with the mechanisms of heat transfer in the well with the transient thermal behavior of the reservoir. The equations of temperature for the injection of hot incompressible liquid phase or single phase ideal gas flow have been derived. Satter (1965) later included the effect of changes phase during operation of steam injection. In Ramey method, the transient thermal behavior of the reservoir is determined by solving the problem of radial heat conduction in an infinite cylinder. Resistance to heat flow in the well, caused by the presence of the tube wall, pipe insulation, fluid in the crown shell and tube wall and the casing and cement, are incorporated in a heat transfer coefficient in general. Willhite (1967) proposed a method for determining this coefficient. Shiu and Beggs (1980) developed an empirical correlation for the production of 10 oil wells to determine the distance of relaxation defined by Ramey.

Their work is really an attempt to avoid the complex calculation of the coefficient of total heat transfer in the well and the behavior of the transient heat transfer reservoir. Although this correlation simplifies the method of Ramey, caution should be used as an approximation. Sharma et al (1989) modified the Ramey equation for the case of producing wells drilling with a heater. Finally, Sagar et al (1991) developed a simplified method for predicting the temperature suitable for hand calculations on the basis of field data. All these methods include strong assumptions relating to the thermodynamic behavior of fluids and, therefore, are applicable only to limited operational conditions.6.5.3. Model development. The derivation of the general and unified equation for predicting the temperature is performed by applying the conservation laws of mass, momentum and energy balances on a differential control volume of a pipe. The resulting differential equation is integrated under simplifications, however sounds like a guess. Applying the steady-state mass, momentum and energy balances in the differential control volume leads to

(Ec.6.65)

(Ec.6.66)

and

(Ec.6.67)

Where Q is the heat flow and e is the internal energy per unit mass. Using the mass balance (Eq. 6.65) can be reduced further in Eq 6.66 and 6.67, respectively,

(Ec.6.68)

(Ec.6.69)0

(Ec.6.70)The transfer of heat to the surroundings can be expressed by the concept of heat transfer coefficient overall U as

Equation 6.71

Where TE is the temperature of the external environment. Must be carefully considered the geometrical configuration of the widenings or pipe and all heat transfer mechanisms involved in the determination of U. It is a general expression for U

Equation 6.72

For the prediction of the temperature widenings, the first term within the brackets, 1/Uo represents the heat transfer mechanisms in the well, while the second term, f (t) / kE represents the transient heat transfer in the reservoir, for predicting pipe temperature is generally considered the first term only. Equation 6.70 and 6.71 can be combined to produce

Equation 6.73

At this point, the analysis is rigorous and similar to that presented by other authors. Temperature prediction can be carried out strictly, if so provided a method for determining the enthalpy of liquid in each condition of pressure and temperature. This is the case, for example, when known fluid composition and may be generated by a simulator thermodynamic Vapor-liquid-equilibrium (VLE) an enthalpy chart. In many cases, this information is not available, and recommended another approach. The enthalpy gradient can be written in terms of temperature and pressure gradients as

Equation 6.74

6.73 and 6.74 Eq combining results in

Equation 6.75

6.75 Eq can be rewritten as follows most convenient

Equation 6.76

Define a relaxation distance, A, proposed by Ramey (1962).

Equation 6.77

and a dimensionless parameter, as

Equation 6.78

Equation 6.76 can be rewritten as

Equation 6.79

So far, only math maneuvers have been made to the enthalpy equation, and the analysis was done rigorously without simplification. Now, assume that the surrounding temperature is a linear function of depth, which can be expressed as

Equation 6.80

Substituting Equation 6.80 into Equation 6.79 gives

Equation 6.81

If, for a given tube segment, U, Cp, , gE, 0, vdv / dL, and dp / dL can be considered approximately constant, Eq 6.81 can be integrated, yielding an explicit equation for temperature, originally developed by Alves (1987).

Equation 6.82

Equation 6.82 is general and can be applied to any angle for single-phase flow or two phases. The average values of Cp and N, and the pressure derivative dp / dl; depends on the average pressure and temperature of the tube. Thus, an iterative procedure is necessary for the calculations, the pressure gradient can be determined by any method of two-phase flow. For fluid compounds may be generated tables to give enthalpy values of Cp and . To the crude, however, a method is necessary to approximate the values of these parameters.

6.5.4 Approximation for oil. For the more general case of a two-phase flow, the total enthalpy of a mixture is the sum of the enthalpies of the individual phases. Therefore, the derivative can be written as enthalpy

(Ec.6.83)

..................... (Ec.6.84)

The enthalpy derived for each stage is given by

Equation 6.85

and

Equation 6.86

Where

Equation 6.87

And(Ec.6.88)Are obtained by applying the thermodynamic behavior of real gas and liquid as incompressible fluid assumption

Equation 6.89

and

Equation 6.90

Therefore, Equation 6.85 and 6.86 become

Equation 6.91

And

Equation 6.92

Substituting Eq 6.91 and 6.92 in 6.84 yields equalization

Equation 6.93

Rearranging Equation 6.93 gives

Equation 6.94

So for a mixture of two phases, as expected, the mean heat capacity is

Equation 6.95

and becomes the Joule-Thomson coefficient average

Equation 6.96

O

Equation 6.97

To a liquid phase NS L = 1 and = L> 6.97 Eq is the same as Equation 6.90 for an incompressible fluid. Also, for a gas phase, L = O and Eq NS g = 6.97 is equivalent to Equation 6.89 for a real gas. The heat capacity of water and hydrocarbons do not vary greatly over a wide range of temperatures. Therefore, average values of Cp can be easily obtained and used for any widening or pipe. Correlations for the gas compressibility factor, z, are available in the literature. Therefore, the values of n can be easily evaluated with 6.97 EQ. As can be seen, very good approximation of Cp and 1) can be determined without any assumptions severe.

6.5.5 Results and discussion. An important feature of the overall EQ 6.82 is that degenerates into the equation (1979) Coulter and Bardon (pipe) or equation Ramey (1962) (for ideal gas or