the Tassios form of the Wilson equation, it can provide equilibrium data a t finite gaseous concentrations for design purposes.
Aij
=
= constants for log Pioequation constants for one-parameter Tassios-Wilson equa-
tion
= distance on recorder chart between Nz peak and solute peak, in. AH = heat of vaporization ALr = energy of vaporization F = carrier gas flow rate measures with soap-film flowmeter a t column outlet, cc/min gii = energy of interaction between molecules of type i equal to energy of vaporization, cal/g mol = energy of interaction between an i-j pair of molegij cules, cal/lb mol Hi0 = infinite dilution partition coefficient M = molecular weight of solvent M , = moles of s t a t i k a r y liquid phase per unit volume Pi0 = vapor pressure of pure i, mm, absolute Pi, = column inlet pressure, mm, absolute Pout = column outlet pressure, mm, absolute P , = vapor pressure of water a t Tf, mm, absolute R = gasconstant T = temperature, O K Tr = temperature of soap-film flowmeter V,” = corrected retention volume; volume of mobile phase passed into the column, ml/g of liquid v i , uj = molar volume, ml/g mol W = weight of stationary solvent liquid, g 2: = mole fraction of solute in solvent phase z = recorder chart speed, in./min 2 = gas compressibility factor
D
activity coefficient at infinite dilution
=
Ti0
literature Cited
Nomenclature
A, B, C
GREEKLETTERS
Balej, J., Regner, A., Collect. Czech. Chem. Commun., 21, 1545 (1956).
Batelle Memorial Institute, “Applicability of Organic Liquids to Gases,” Report No. PB-183513 (1969). de Carli, I?., Atti, Accad. Lincei, 6 , 523 (1926); Chem. Abstr.,
21,1449 (1926). E. I. Du Pont de Nemours and Co., personal communications, 1971. Everet, D. H., Stoddart, C. T. H., Trans. Faraday Soc., 57, 746 (1961). Hankinson, R., Lagfitt, B., and Tassios, D. P., Can. J. Chem., 50, 511 (1972). Hougen, 0. A., Watson, K. M. “Chemical Process Principles, Part 2, Thermodynamics,” Whey, New York, N. Y., 1947, p 489. James, A. T., Martin, A. J. P., Biochem. J., 50, 679 (1952). Kwantes, A., Rijnders, G.. W. A., “GM Chromatography,” D. H. Desty, Ed., Academic Press, London, 1958. Martire. D.. Pollara. L.. J . Chem. Ena. Data. 10.40 , - (1965). - -, Perry, J. H., “Chemical Engineering Handbook,” 4th ed, McGraw-Hill, New York, N.Y., 1963, pp 3-188. Porter, P. D., Deal, C. H., Stross, F. H., J. Amer. Chem. Soc., 78,2999 (1956). Sheets, M. R., Marchello, J. M., Hydrocarbon Process., 42, 99 \
,
,.
(1Qfi9\ L Y “Y
Stull, D. R., Ind. Eng. Chem., 39,517 (1947). Tassios, D. P., Hydrocarbon Process., 49, 7, 114 (1970). Tassios, D. P., AIChE J., 17, 1367 (1971). Tassios, , n-0 \ D. P., Ind. Eng. Chem., Process Des. Develop., 11, 43 3
(LYIL).
Wilson, G. M., J. Amer. Chem. SOC.,86,127 (1964). RECEIVED for review June 26, 1972 ACCEPTED February 22, 1973
A Computer Design Method for Vertical Thermosyphon Reboilers N. V. 1. S. Sarma,’ P. J. Reddy, and P. S. Murti” Regional Research Laboratory, Hyderabad-9, India
Recent improvements in fluid flow correlations have been incorporated in the design procedure for the vertical thermosyphon reboilers. A general purpose computer program has been developed for the design.
V e r t i c a l thermosyphon reboilers play a wide role in chemical industry and require rational design procedures to develop truly optimal equipment. Several flow patterns manifest themselves during heat transfer to a flowing two-phase boiling mixture in a thermosyphon reboiler depending upon the flow rates, physical properties of the components, pipe diameter, and orientation. This complicates the design to be based on sound, theoretical considerations of heat and momentum transfer, and prohibits any great accuracy. A fair amount of empiricism still becomes unavoidable in spite of the tremendous amount of work on two-phase flow that has been reported in the last two decades. 1 Present address, Reactor Research Centre, Kalpakkam, Tamil Nadu, India.
278
Ind. Eng. Chem. Process Des. Develop., Vol. 12,
No. 3, 1973
The various types of flow patterns that are encountered during the upward flow of cocurrent vapor-liquid mixture through a vertical thermosyphon reboiler tube are (i) bubble, (ii) slug, (iii) annular, and (iv) mist. These flow regimes occur in the order of increasing vaporization but are not sharply demarcated. Slug flow prevents the steady-state operation of the reboiler by sending alternate layers of liquid and vapor forcing the system to unsteady state operation and, for this reason, it is desirable to minimize the slug flow region. In mist flow, which is the extreme regime of two-phase flow, heat transfer rates are very poor due to the continuous gas phase and therefore is to be again avoided. A thorough knowledge of the various types of flow regimes, conditions of their onset, and the regions of occurrence is essential for a sound design.
There is no completely satisfactory method for prediction of the flow regimes in vertical cocurrent two-phase flow. Griffith and Wallis (1961) and Govier, et al. (1957), presented charts for the identification of flow patterns. The charts of Govier, et al., are helpful for the prediction, a t least approximately, of the individual types of flow patterns existing in the pipe for the given flow conditions. Orkiszewski (1967), based on a large amount of experimental data on cocurrent two-phase vertical flow and on the results of Griffith and Wallis (1961), Duns and Ros (1963), and Kicklin, et al. (1962), formulated some useful correlations for the identification of flow patterns using dimensionless numbers. Mechanism of Flow in Vertical Thermosyphon Reboilers
Figure 1 shows a sketch of the vertical thermosyphon reboiler including the flow arrangement between the reboiler and the distillation column. Liquid from the holdup section A of the distillation column passes through the inlet leg of the reboiler, enters the bottom channel, and then it is distributed uniformly to the tubes. The tubes are heated by the flux supplied by the heating medium, usually steam, which is passed through the shell of the reboiler. The process fluid entering the tubes is below its boiling temperature due to static head effects in the tube and heat losses. Once the liquid enters the reboiler tube, it receives the sensible heat from the heat flux supplied. This occurs up to point C where the temperature of the liquid equals the saturation temperature corresponding to the pressure a t that point. The length BC in Figure 1 represents sensible heating zone. At the point C nucleate boiling sets in resulting in the vaporization of the liquid which continues throughout the rest of the length (Ltp) of the tube. Thus, the length CD (Lt,) in Figure 1 represents the two-phase region. Circulation Rate
The total pressure drop resulting from the flow of a boiling two-phase vapor-liquid mixture through the reboiler tube is the sum of the contributions due to the following three effects: (1) static pressure drop due to elevation; (2) pressure drop due to friction; and (3) pressure drop due to increase in momentum of the two-phase vapor-liquid mixture. The total pressure gradient can be represented as
For a system with specified physical properties, value of heat flux, and tube geometry, the flow through the reboiler tube is controlled by the pressure balance between inlet and outlet legs (Le., between AB and BE in Figure 1).In order to establish the circulation rate, under the above conditions, it is necessary to calculate (1) the total pressure drop in the reboiler tube, and (2) the pressure drop in inlet and exit pipings. Heat Flux
As the vapor-liquid mixture flows through the reboiler tube, the flow is not continuous and smooth but is rather pulsating. The amplitude of the pulsations is controlled by the rate of heat transfer and hence on impressed heat flux for a given system. At sufficiently high values of heat flux, the amplitude of the pulsation is large enough so that the downflow pulse sucks the vapor into the compartment below the tubes resulting in “vapor locking.” This condition has been called a “surge point” by Lee, et al. (1956), and corresponds
to the maximum permissible heat flux and critical temperature difference to be used in any reboiler design. Correlations for predicting maximum permissible heat flux and maximum heat transfer coefficient were given by Lee, et al. It is normal practice to choose a certain fraction of this maximum flux as the operable value. Nucleate Boiling and Forced Convective Heat Transfer
Once the liquid attains the saturation temperature corresponding to the pressure prevailing a t C, nucleate boiling sets in with vapor existing as a second phase and the quantity of vapor increases along the tube length. The single-phase region represents heat transfer by convection and, if the temperature difference between the tube wall and saturation temperature of liquid is greater than the minimum required for nucleate boiling, then the phenomenon of nucleate boiling takes place a t the tube wall with the collapse of bubbles in the subcooled liquid. I n both single-phase and two-phase regions, heat transfer can be represented by the superposition of heat fluxes due to forced convection and nucleate boiling. For a given boil-up rate and reboiler outlet condition, the circulation rate and the quality of vapor depend upon the heat flux and the geometry of the reboiler loop. Similarly, heat transfer coefficient on the process side is a function of the characteristics of the reboiler and the physical properties of the process fluid. I n general, it can be concluded that heat and momentum transfers are interconnected and, for a given reboiler loop, there will be only one set of conditions which give a balance between the two transfer processes. Design Methods Available in literature
Several design methods are available in literature but those due to Fair (1960, 1963) and Hughmark (1961, 1962, 1964, 1969) have received wide acceptance as they deal with various aspects of vertical thermosyphon reboilers in great detail. The method of Fair (1960) follows stepwise procedure for hydrodynamic and heat transfer calculations along the tube length using incremental vaporization and makes use of twophase flow correlations for gas-liquid flow developed by Lockhart and Martinelli (1949). All liquid single-phase values are made use of for the calculations of two-phase pressure drops and heat transfer coefficients. The method involves a nestled series of trial and error calculations using design charts. Fair, while using the Lockhart-Martinelli parameter , did not originally introduce any factor to account for vertical flow, but subsequently modified (1963) the parameter by introducing the Froude number as suggested by Davis (1963). Correlation for the calculation of liquid holdups was presented in terms of the Lockhart-Martinelli parameter but is limited to liquid mass rates equal to or higher than 50 lb/sec ft2. This is due to the fact that for mass flow rates lower than 50 lb/sec ft2, the magnitude of deviation in the Lockhart-Martinelli parameter is a function of total mass velocity (1961). This method also assumes isothermal conditions inside the reboiler tube and constant temperature difference between the wall and the fluid which may not lead to accurate results. No generalized correlation for the calculation of boiling heat transfer coefficient was used but its selection from the available data on boiling was suggested. Hughmark’s approach (1961, 1964) is more or less similar to that of Fair except for the correlations used for the hydrodynamic and heat transfer calculations. Polynomial relationships in terms of no-slip volume fraction, in-place holdup, and Froude number have been used for the calculation of the Ind. Eng. Cham. Process Des. Develop, Vol. 12, No. 3, 1 9 7 3
279
h
00
280
h
a
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
h
0
8
a
+
5
two-phase pressure drop in reboiler tubing, top channel, and exit piping. Modified Bankoff's correlation (1962) was employed for the calculation of in-place liquid and gas holdups. Further, Chen's (1966) correlation, which takes into account the effect of flow on the boiling rate, was employed for the calculation of the boiling heat transfer coefficient. This type of correlation takes into consideration the variation of physical properties of the process fluid and is helpful in calculating point values of the boiling heat transfer coefficient along the length of the tube. It may be concluded that both the above methods did not explicitly take into account the various flow regimes for the hydrodynamic and heat transfer calculations and thus perhaps suffer from lack of sound theoretical basis. But efforts are still required to set up a calculating procedure from a more theoretical approach so that it can overcome the limits of extrapolation. Proposed Method
An attempt has been made in this paper to present a different design procedure based on the consideration of individual flow patterns for the hydrodynamic and heat transfer calculations. The heat transfer part of the design deals with the prediction of nucleate boiling and forced convective heat transfer coefficients in the sensible heating zone and the twophase region. The equations framed by Orkiszewski (1967) for vertical upward cocurrent two-phase flow are used in this proposed method because of the large amount of experimental data used in arriving a t correlations and of a more accurate way of defining flow regime boundaries. Orkiszewski made a critical analysis of the results of Griffith and Wallis (1961) and Duns and Ros (1963) on pressure drop with his bank of petroleum production data and then chose the best method to include in his correlations; if none was accurate enough, he developed relations of his own. Considering the basic mechanical energy equation and making use of flow regime knowledge and a large amount of empirical data, Orkiszewski established relations for friction pressure gradient, two-phase density, and acceleration term in each flow regime. Some of the equations of Orkiszewski are used as rearranged by DeGance and Atherton (1970) so as to prepare a general design procedure and computer program for vertical thermosyphon reboilers. Until more refined expressions are available, these are believed to represent the available data on two-phase flow better. The various equations, on flow pattern basis, that are to be used for the hydrodynamic design are shown in Tables 1-111. Design Procedure
1. Process Requirements. The following parameters are normally fixed for a reboiler: (a) boil-up rate; (b) reboiler outlet pressure, temperature, and composition; (c) material of construction for the tube and choice of particular tube geometry; and (d) heating medium. Also, tube length may be chosen based on locational considerations. 2. Physical Property Data. I n order to obtain the point values of physical properties of the process fluid along the length of the tube, i t is necessary to have equations representing physical properties as functions of either temperature or pressure (for a single component). For this, liquid and gas densities (p1, p p ) , viscosities (PI, pa), liquid and vapor specific heat (Cpl, Cpv), liquid and vapor thermal conductivity (kl, k y ) , latent heat of vaporization (A), and surface tension (u) data are related to either temperature Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1 9 7 3
281
Table II.
Equations for the Calculation of Inplace Holdups
Description
Equation
Dimensionless number Dimensionless number Gas velocity number Liquid velocity number Parameter Parameter Parameter Parameter Parameter Parameter
No.
Nd = 120.9Dt(pi/~)"* N i = 0.1573(~(1/2.42)(l/plua)1/4 Ngy = 1.938Vag(pi/u)1/4 Niv = 1.938vsi(p~/r)'/' Neeo = (NgvNl'~")/Nd*~" Nhoid = (N1v/N~v'~~'~)(P/14~65)~~~Cn(lO~/N~) $ = 1 exp[6.6598 8.8173 In N,,, 3.7693(1n 0.5359(1n Nseo)'] $ = 1.00 for Ne,, < 0.01 $ = 1.82 for Neeo > 0.09 Cn = exp[-4.895 - 1.0775 In N i 0.80822(1n N I ) ~ 0.1597(1n Nl)' 0.01019(ln N J 4 ] Cn = 0.0115 for N1 > 0.4 Cn 7 0.00195 for N1 > 0.002 RL = $[exp(-3.6372 0.8813(1n Nhoid) 0.1335(1n Nhoid)' 0.018534(1n NhO1d)' 0.001066(ln Nhold)'] RL = $ for Nhold > 4000 RL = 0.02633 for Nhoid < 0.1 RL = A, for RL < A,
+
+
+
+
-
Parameter Parameter Inplace liquid holdup Inplace liquid holdup Inplace liquid holdup Inplace liquid holdup
+
+
5 6 7 8 9 10 11 12 13 14 15 16 17
18 19 20
Table 111. Equations Used in the Hydrodynamic Part of the Design Description
Saturation temperature when bulk boiling starts Calculated value of saturation pressure where bulk boiling starts on the assumption of the sensible heating zone length Calculated value of saturation temperature corresponding to the point where bulk boiling starts on the basis of the assumed value for the sensible heating zone length Mean temperature Latent heat of vaporization corresponding to mean temperature Exit vapor quality Number of tubes Fluid velocity in inlet piping Reynolds number for the flow through inlet piping Equation for Moody friction factor for the flow through inlet piping Equivalent length of inlet piping Pressure a t the end of element considered Temperature corresponding to the pressure at the end of the element considered Mean of temperatures a t the begining and end of the element Heat supplied to the element of length aL Latent heat corresponding to T, Incremental quality corresponding to the element of length AL Vapor quality a t the end of the element Superficial liquid velocity a t the beginning of the element Superficial liquid velocity at the end of element Superficial gas velocity a t the beginning of the element 282 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
Equation
Tmo = Pas =
No.
(~)G>(ot)(LSH)/(C*I)(wt)
(Pin
f Ti
- APs~)/144
22
23
T,, = Taat(P.,)
1 (Ti T m = ~ [ X = X(Tm)
+ Tea)
21
+
(Tsa
+ Te) 2
1
24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
Table 111 (Continued). Equations Used in Hydrodynamic Parl of the Design Superficial gas velocity at the end of Vsgz = (Gt) ( z d / P g a the element Mean superficial liquid velocity Val = (Val, Vs1,)/2 Mean superficial gas velocity Vag = (Vsg, Vsg,)/2 No slip velocity corresponding to the Vns = V s g Vs1 flow in the element Slip ratio corresponding to the flow in A s = VsdVns the element Slip corresponding to the conditions at the end of the element Acceleration term
+ ++
42 43 44 45 46 47 48
Total pressure gradient for the element Total pressure drop in the element Total pressure drop in the two-phase region
49 50 51
Total pressure drop in the reboiler tubing
52
Equations for the Calculations of Pressure Drop in Exit Piping Mean value of pressures corresponding P m = (Pn+z Pe/144)/2 to entry and outlet to exit piping Temperature corresponding to P, Tm = Ta,t(Pm) No slip density Pna = h e P l (1 - Xse)Pg No slip viscosity fins = haefi1/3600 (1 - xse)fig/36OO
+ + + (E) (E)+ (2)[(I - ~ e ) z / ~ g e ~
53 54
55 56
Parameter
@ =
Reynolds number
58
Parameter f e defined
Retpe = [(B)(De)(Gte)/finsl 1 - 2 log [e/(3.7)(De)
Parameter Parameter
.(Asel
q
60 61
+ 2.51/Retped&I = = 1.0 - [ln (Ase/q)l = 1.28 + 0.478 In (hae) + 0.444(1n Xse)' + 0.094(ln + 0.00843(1n hse)4
x=
ftpe/fe
57
59
XaJ3
Two-phase friction factor for constant slip flow in exit piping
ftpe
=
fea(xse)
62
Frictional pressure gradient
83
Total frictional pressure drop in exit piping
64
or pressure and best-fitted least-squares polynomials are obtained for use in the computer program. Assumptions
The following assumptions are made in the design procedure outlined later. (1) Constant heat flux conditions exist throughout the length of the reboiler tube; variation of heat flux can also be introduced easily in the procedure. (2) The top tube sheet of the reboiler is in line with the liquid level in the fractionator, a geometrical parameter which can easily be met in the fabrication of the reboiler. (3) A constant slip model in proposed for exit piping including top channel. (4) At any point in the two-phase region along the tube, thermodynamic equilibrium exists between saturated liquid and vapor corresponding to the pressure a t that point, a valid assumption in practice. (5) Linear temperature variation of the process fluid is assumed along the length of the tube as shown in Figure 2. (6) The fluid entering the reboiler has the same temperature as the liquid vapor mixture leaving the reboiler and entering the fractionator, i.e., the system is perfectly insulated.
(7) A certain amount of flashing is to be expected when the two-phase mixture flows up the reboiler tube; but this is neglected in comparison with the amount being vaporized from zone to zone. Based on the above assumptions, operational data and a knowledge of the physical properties, the following are calculated: (1) circulation rate based on pressure drops in inlet and outlet legs (including reboiler tubing) such that a balance is achieved between them; and (2) the heat transfer area and hence the number of tubes from the known tube geometry. Method of Calculation
The procedure adopts stepwise calculations along the length of the tube using incremental lengths. The following steps outline the method. (1) Assume a value for the design heat flux (q, BTU/sec ft2) as a certain percentage of the maximum allowable flux. Calculate the maximum flux using the correlation proposed by Lee, et al. (2) Assume a value for the tube inlet velocity ( V I ,ft/sec). (2) Fix the length (L) and diameter (Dt) of the tube. (4) Calculate the length of the sensible heating zone (LsH) starting with an initially assumed value for it and using Ind. Eng. Cham. Process Des. Develop., Vol. 12, No. 3, 1 9 7 3
283
Figure 1 . Schematic diagram of vertical therrnosyphon reboiler
drop. (d) Calculate the point values of in-place holdups a t RG)using eq 5-20. the beginning and end of element (EL, (8) Repeat the procedure given in step 7 for all the elements and obtain the two-phase pressure drop in the reboiler tube applying eq 50 and 51. (9) Obtain the total pressure drop, including the pressure drop in the sensible heating zone, using eq 52. (10) Calculate the frictional pressure gradient and hence the total pressure drop in the exit piping using eq 53-64. (11) Calculate the pressure drop in the outlet leg of the reboiler, which is equal to the sum of the pressure drops in the reboiler tube and exit piping. (12) Find out the difference between the pressures a t entry to the reboiler and the pressure drop in the outlet leg. The residual pressure drop should converge to the pressure below the bottom tray in the distillation column. (13) If convergence is achieved, the assumed tube inlet velocity gives the circulation rate inside the tube. Otherwise] the calculations are to be repeated using a new assumed value for tube inlet velocity every time until convergence within a specified value is achieved. Heat Transfer
Figure 2. Temperature profile inside reboiler tube: y axis, temperature of process Fluid; x axis, distance from tube inlet
eq 21 and 23 given in Table 111. The value of L S Hfor which convergence is achieved between the values of T,,,and T,, within specified limits gives the length of sensible heating zone. (5) Calculate the number of tubes (St)using eq 24-27. (6) Assume a certain value for pressure drop (APJ in the inlet leg and using eq 28-31 calculate the equivalent length (L,) of the inlet piping. This assumption enables the process engineer to fix the equivalent length of the inlet piping depending upon the reboiler location and process requirements. ( 7 ) Divide the total length of the tube in the two-phase region (Lt,) into a number of elements, each of length AL. Starting with the first element and knowing the conditions a t the beginning of the element adopt the following procedure to identify the type of flow regime prevailing in the element and calculate in-place holdups and the total pressure drop in the reboiler tubing. (a) Initially assume a value for the pressure drop in the element. Using eq 32-47 and the flow regime equations given in Table I, identify the type of flow regime prevailing in the element. (b) After identifying the type of flow regime, apply the appropriate equations given in Table I and calculate frictional pressure gradient ( v ) and two-phase density (ptp). Calculate the acceleration term (AC) and hence the total pressure drop in the element using eq 48-50. (c) Compare the calculated value of pressure drop with the assumed value to test for convergence. If convergence is obtained within a specified limit, calculations are t o be switched over to the next element; otherwise, calculatio~isare to be repeated lqith a different assumed value for pressure 284 Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1 9 7 3
There are two modes of heat transfer taking place in a reboiler tube: (1) heat transfer due to nucleate boiling, and (2) heat transfer due to convection. I n the single-phase region, the liquid will be subjected to surface boiling if the temperature difference between the tube wall ( T w ) and the saturation temperature (T,,)reaches the limit for nucleate boiling. The total heat flux (a) is equal to the sum of the contributions due to nucleate boiling and convective flow. If the temperature difference is below that for nucleate boiling the heat transfer is entirely due to convection. Similarly, in the two-phase region, the heat transfer is due to the combination of the above two effects. However, the effect of nucleation diminishes whereas forced convective heat transfer predominates as the boiling vapor-liquid mixture flows up the tube. Finally, a point is reached where complete suppression of nucleation takes place. Above this point, the heat transfer is entirely due to forced convection. The present work employs a method to test for nucleation and uses correlations given in Table IV to find the nucleate boiling, forced convective and hence the overall heat transfer coefficients. Design Procedure
The heat transfer calculations associated with the design are (1) prediction of heat transfer coefficients in the sensible heating zone and the two-phase region for an assumed value of heat flux; (2) calculation of combined film heat transfer coefficient on the process side; (3) prediction of overall heat transfer coefficient based on calculated value of combined film coefficient and the heating side film coefficient taking into consideration the fouling factors on shell and tube sides and the resistance of the metal of the tube; and (4)based on the overall temperature difference and overall heat transfer coefficient] calculation of the heat flux and comparing it with the assumed value. The design procedure for the heat transfer part runs along the same lines as that of Hughmark (1969). (a) Assumptions. I n addition to those mentioned in the hydrodynamic part of the design, i t is assumed that the bulk and saturation temperatures are equal. (b) Heat Transfer Coefficient. Test for Nucleation. Hsu (1961) suggested that the relative thickness of the hydrodynamic layer (&) and thermal boundary layer ( ~ T H ) Can
Table IV. Equations Used in the Heat Transfer Part of the Design No.
Description
Hydrodynamic boundary layer thickness
65
Thermal boundary layer thickness
66
Nucleate boiling heat transfer coefficient
67
Convective heat transfer coefficient in single-phase region
68 69 70
Total heat transfer coefficient in sensible heating zone Single phase heat transfer coefficient for total flow Dengler's equation for two phase convective heat transfer coefficient Reciprocal of Lockhart-Martinelli par ameter Hughmarks equation for convective heat transfer coefficient in twophase region Two-phase heat transfer coefficient in the region RL > 0.985 Total heat transfer coefficient in twophase region considering single element Combined film heat transfer coefficient on process side Overall heat transfer coefficient based on inside surface Overall temperature difference (see Figure 2)
71
!% hi
(A)
0.5
=
3.5
73 74 75
79
") + + Ltp)
Calculated value of heat flux
be taken as the basis to test for the nucleation in convective flow. It was stated that nucleation occurs when the thickness of the hydrodynamic boundary layer is greater than that of the thermal boundary layer. Later, as a criterion for suppression of nucleation in forced convective vaporization, Collier and Pulling (1962) suggested that the hydraulic laminar sublayer of a single-phase turbulent flow is smaller than HSU'S thermal boundary layer for incipient boiling. Equations 65 and 66 as suggested by Hughmark (1969) are used to test for nucleation. Nucleate Boiling Heat Transfer Coefficient ( h ~ )I n. order to calculate nucleate boiling heat transfer coefficient ( h N ) both in single-phase and two-phase regions eq 67 given in Table IV is used. This was developed by Chen (1966) modifying Forster and Zuber's (1955) equation with the inclusion of a factor S to account for the suppression effect of moving fluid on the boiling rate. The value of S has been taken as 0.25 as suggested by Hughmark (1969).
80
81
Convective Heat Transfer Coefficients. I n order to calculate convective heat transfer coefficients in the sensible heating zone and the two-phase region, eq 68-74 are used. Equation 73 has been developed by Dengler (1962), and eq 74 has been suggested by Hughmark (1969) for convective heat transfer in a two-phase flow. Method of Calculation. The stepwise method followed in the hydrodynamic part of the design is continued for heat transfer calculations also. The point values of pressures, vapor qualities, and in-place liquid holdups determined in the hydrodynamic part of the design, for a n assumed value of heat flux and tube inlet velocity for which pressure balance is achieved, are made use of in this part. The following steps are indicated for the heat transfer part of the design. (1) Calculate the average values of pressure, vapor, quality, and liquid holdup in each selected increment of the tube. Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
285
Calc R L , R C b , f t p , T f b and
+=, talc
Pb from
cqns (1,5),(1.3),(1.6),(1.7) and 1.4 respectively
rmand
N~ from cqns ( 4 . 3 ) and ( 4 . 5 )
Nw-.005
Figure 3. Flow diagram for bubble flow
COlC t , , rt.and T f t from
to Return
eqns (3.41, (3.3) and (3.5) respectively
Figure
4. Flow diagram for transistion flow
1
t
-
Figure 6. Flow diagram for mist flow
+
Calc Nhold from eq(1O)
Figure 7. Flow diagram for friction factor Calc
V from e q ( l 0
Figure 5. Flow diagram for holdup
(2) Using the average properties and eq 65 and 66, carry out the tests for nucleation both in the sensible heating zone and the two-phase region. (3) Calculate h~ and hc both in the sensible heating zone and the two-phase regions using eq 67-70 and 72-76, respectively . (4) Calculate the total film heat transfer coefficient ( h s ~in ) the sensible heating zone and the total two-phase heat transfer coefficient (ht) for each selected increment of the tube using eq 71 and 77. (5) Calculate the combined film heat transfer coefficient (hp) on the process side as a weighted average of the heat transfer coefficients in the sensible heating zone and the two-phase regions making use of the eq 78. (6) Calculate the overall heat transfer coefficient ( U ) , overall temperature difference (AT' overall), and hence heat flux (qD) using the eq 79-81. (7) If the assumed and calculated values of heat flux significantly differ from each other, the entire loop including hydrodynamic and heat transfer calculations is to be repeated with different assumed value of heat flux until convergence is achieved. 286 Ind. Eng. Chem. Process Des. Develop., Vol. 1 2, No. 3, 1973
Figure 8. Flow diagram for slug flow
Computer Program
h general purpose computer program was developed in Fortran IV incorporating both the hydrodynamic and heat transfer aspects of thermosyphon reboiler design. The program in Fortran 11, separately for hydrodynamic and heat transfer parts, was initially run on IBM 1620 extensively and, later, the integrated Fortran IV program was run on IBM 7044 with 1401 satellite and on IBM 360 Model 44
program
-
SLUG
Calc Po, Tfo from subpraqram TRANS ">O CaIc
Pa,rf0 from
proqram
I
'
07
I
I
Figure 1 1. Flow diagram for hydrodynamic part: identification of flow regimes and calculation of T and po
*Ot
Figure 9. Flow diagram for hydrodynamic part: calculation of Nt, L i , etc.
MIST
sub-
APaE AP,
-h
+ and APT from rqns 1511 t o (641
NOITEN
I
I -
I
I
Cole Nb
f r o m 8q l t . 2 1
Calc hold up RLp2, t r r a t l n q c ,ft
XT+1
t
RG22
, p t and
A s as polnt values a t
Figure 12. Flow diagram for hydrodynamic part: calculation of pressure drops Figure 10. Flow diagram for hydrodynamic part: calculation of dimensionless numbers for identification of flow regimes
without a satellite. The following six subprograms have been used in the master program. (1)
SUBROUTINE HOLD-UP
(2)
FUNCTIOX FF
(3) SUBROUTIXE BUBBLE (4) SUBROUTINE SLUG (5) SUBROUTINE TRANS (6) SUBROUTINE MIST
to calculate in-place liquid and gas holdups to calculate Moody friction factor to calculate the two-phase densities and frictional pressure gradients in various flow regimes
The main and subprograms are presented as flow charts in Figures 3-13.
The program for the hydrodynamic part calculates the pressure drops in the inlet and outlet legs and checks for pressure balance within a specified limit. Each iteration involves the calculation of the length of the sensible heating , number of tubes (Nt),the inlet piping length zone ( L s H )the ( Li ) , pressure drops, vapor qualities, in-place holdups, etc. As an option, the type of flow regime prevailing in each element and the calculated values of the above can be printed for each iteration. This enables the process engineer to make necessary alterations in the design so as to minimize slug flow and avoid the mist flow region. Important design variables such as length of the tube ( L ) , diameter of the tube (Dt), heat flux, etc., appear in the input list and hence the designer is placed in an advantageous position to vary any of these parameters to arrive a t a design satisfying the reInd. Eng. Chem. Process Der. Develop,, Vol. 12, No, 3, 1973
287
Figure 15. Heat flux vs. tube inlet velocity: y axis, tube inlet velocity, ft/sec; x axis, heat flux BTU/hr ft* Symbol Cole hv iron q 7 2 for vapor & hip from rq 76
0
X 0
L
Tube geometry
Pi, Ib/fP
8 ft X 0.75 in. 8 ft X 0.75in. 8 ft X 1 in.
10 35
35
Cole U and qc trom r nr IP to 81
Figure 13. Flow diagram for heat transfer pari
b
0.2 1000 I
3000
4000
5000
6000
7000
8000
1
9000
Figure 14. Heat flux vs. two-phase heat transfer coefficient: y axis, two-phase heat transfer coefficient, BTU/hr ftz OF; x axis, heat flux BTU/hr ftz Symbol
Tube geometry
Pi, Ib/fP
0 X
8 ft X 0.75 in. 8 ft X 0.75 in. 8 ft X 1 in.
10 35 35
0
quirements. The program involves a number of interconnected loops, each one adopting iterative calculations based on simple trial and error procedures. Depending upon the required accuracy, the user can vary the number of elements into which the length of the two-phase region of the tube is divided. The program for the heat transfer calculations carried out the tests for nucleation and computes local heat transfer coefficient for each element of the two-phase region. As the change in heat flux has its effect on the hydrodynamic part of the design, both hydrodynamic and heat transfer parts can be easily looped together. Detailed flow diagrams are given in Figures 3-13, in which, for an assumed heat flux, converged steam temperature are found. 288
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 3, 1973
1
2
3
I
4
6
7
0
9
10
11
12
I
Figure 16. Variation of liquid holdup along the reboiler tube at a flux = 6000 and constant pressure drop = 35 Ibr/ftz: y axis, liquid holdup; x axis, distance from tube inlet (ft) Symbol
VI
Tube geometry
x%
Nt
0
1.55 2.05 4.15
8 ft X 1 in. 8 f t X 0.75in. 12 ft X 1 in.
22.26 21.23 12.48
172 216 114
0
X
For each iteration of the velocity loop, it requires 10 sec of the IBM 7044 computer time and 13 sec on system 360. The number of iterations depends greatly on the chosen initial tube inlet velocity which should not be far from the converged velocity. The program was applied to the example of “propane column reboiler” of Fair (1960). Results and Discussion
The results of design calculations for the propane column reboiler example, with different combinations of tube geometry, heat flux, and inlet pressure drops are given in Table V, and shown graphically in Figures 14-17. It is observed from Figures 14 and 15, that for a given heat flux and tube geometry, both process heat transfer co-
~~
Table V. Reboiler Conditions for Propane at 401 psia (164OF) and for a Given Capacity of 17,500 Ib/hr Tube geometry (BW G, Steel)
VI, ft/rec
4000
8 ft X 0.75 in. 8 f t X 1 in. 16 ft X 1 in.
1.1 0.65 4.2
26.37 35.38 10.97
335.7 330.3 356.1
6000
8 ft X 0 . 7 5 in. 8 ft X 1 in. 12 ft X 1 in.
2.05 1.55 4.05
21.23 22.66 12.48
364.0 360.0 371.5
8000
8 ft X 0 . 7 5 in. 8 ft X 1 in. 12 ft x 1 in.
3.2 2.1 4.7
18.13 21.91 14.69
381.6 376.8 385.6
Flux BTU/hr ft2
efficient and liquid circulation rate show a decreasing tendency with increasing inlet pressure drop. The decrease in velocity (VI) is attributable to decreased available pressure a t the inlet of the reboiler for the hydrodynamic balance. From Table V, it is noted that for a given boil up rate, an increase in V1 is accompanied by a decrease in the quality of the mixture. I n view of the opposing character of these two parameters, the heat transfer coefficient may show a large, or negligible variation. I n the example given, the qualities obtained are, in general, low and hence the value of h, is largely dependent on VI. Thus, though there is an increase in the value of quality with a decrease in the value of VI, the cumulative effect of both these tends to decrease the value of h,. Further, it may be noted that the inlet velocity decreases with heat flux and tube length, and decreases with tube diameter, other conditions remaining the same. The effect of tube geometry on liquid holdup along the tube length is shown in Figure 16. Figure 17 shows the variation of the process side heat transfer coefficient along the tube length. I n all the cases, the slug flow region is encountered almost throughout the two-phase region and the curves are approximately S shape. As is evident from the Figure 17, the variation in the heat transfer coefficient is not appreciable a t the start of the twophase region due to fewer bubbles; hence, the liquid velocity as well as the thickness of the sublayer practically remains as in a single-phase flow. Along the tube length, the bubbles increase in number resulting in slugs which are larger a t higher velocity, causing an acceleration of the liquid film thereby reducing its thickness. The downward flow of some of the liquid due to slug flow and also the agitational effect of the vapor generated provided the necessary turbulence. As a result of the contributions by the above factors, the middle part of the curve shows a steep rise in the coefficient. In the upper part of the curve, the increase in h, is not as steep. This may possibly be due to the fact that the development of slugs is nearly complete and further reduction in the thickness of the laminar sublayer is marginal. The little increase may be attributed to the stirring action on the liquid film due to evaporation that occurs in the slugs and also the turbulence created by the downflow of the liquid carried by the slugs. The attention of the authors was drawn by Fair to the experimental results of Shellene, et al. (1968), on a nearcommercial reboiler with a heat transfer surface of 110 ft2. I t is our intention to test and compare the existing methods of design on these data and offer, in a separate communication, any improvements in the empirical correlations so as to
Wt
% vaporized
U
IS00 1800
1100 1600
1100 1400
Figure 17. Variation of heat transfer coefficient along the reboiler tube: y axis, heat transfer coefficient; x axis, distance from tube inlet (ft) Symbol
0
A 0 X
Tube geometry
8ft X 8 ft X 16 ft X 8ft X 8ft X
Flux
0.75in. 4000 1 in. 4000 1 in. 4000 0 . 7 5 i n . 6000 0.75in. 6000
Nt
VI
Pi
X%
324 257 129 216 216
1.1 0.65 4.2 2 05 1.65
35 35 35 35 45
26.37 35.38 10.97 21.23 26.37
enable the designer to use the computer program presented here with minor modifications, if any. Conclusions
A method is proposed for the design of a thermosyphon reboiler on a digital computer which enables the print out of statements a t every stage describing the flow regime a t that stage. This method was tested on the worked out example of Fair and a satisfactory agreement was obtained. The proposed method can be employed for a new design or to evaluate the performance of an existing reboiler. Ac knowldgmentr
The authors wish to express their thanks to Dr. James R. Fair, Monsanto Chemical Co., St. Louis, Mo., for kindly reviewing the manuscript. The authors are also grateful to Shri M. Suresh Kumar and Miss P. Suhasini for their assistance in the computer calculations and to Dr. G. S. Sidhu, Director, Regional Research Laboratory, Hyderabad, India, for the encouragement given during this work. Nomenclature
AC AD
= =
acceleration term lower limit for exit vapor quality
Ind. Eng. Cham. Process Des. Develop., Vol. 12, No. 3, 1 9 7 3
289
AE
= accuracy limit for pressure balance between inlet and outlet legs AG = step length for velocity, ft/sec AH = accuracy limit for the calculation of temperature at which nucleate boiling occurs ALAS = initially assumed value for the length of the sensible heating zone, ft A t = cross-sectional area of the reboiler tube, i7Dt2/4, f t 2 C, = specific heat, BTU/lb O F D = inside diameter, ft C,, = liquid specific heat in the sensible heating zone, BTU/lb O F FAC = conversion factor for gc f = Moody friction factor g = gravitational constant, 32.2 ft/sec2 gc = conversion factor, 32.2 1bbI ft/lbf sec2 Gt = mass velocity, 1b/ftB2sec h = heat transfer coefficient, BTU/hr ft2OF h.985 = heat transfer coefficient at RG = 0.985, BTU/hr
ft2O F HH1 = step length for the calculation of the heating medium temperature k = thermal conductivity, BTU/hr ft O F kli = liquid thermal conductivity in the sensible heating zone, BTU/hr ft O F KP,K 3 = step lengths for the calculation of the sensible heating zone length and the pressure drop in each element K., = accuracy limit for pressure balance in each element K 5 = accuracy limit for flux balance L = length of reboiler tube, ft Le, L , = equivalent lengths of exit and inlet pipings, ft m = maximum number of iterations for the velocity loop n = number of elemental lengths in the two-phase region N t = number of tubes N , , N 2 = parameters defined by eq 2.6 and 2.7 Nb = bubble flow number (dimensionless) N lrn = mist flow number (dimensionless) N 1 , = slug flow number (dimensionless) Ngv = gas velocity number (dimensionless) N , = parameter defined by eq 4.5 P , = pressure at entry to the distillation column, lbr/ft2 Pi, = pressure at entry to reboiler, lbr/ft2 Pi,, = ( L p l i - A P , ) , lbr/ft2 P, = Prandtl number (dimensionless) P , = saturation pressure corresponding to wall temperature lbr/in.2 PI,,PZ2 = pressure at the beginning and end of each element, 1br/h2 q, qo = assumed and calculated values of heat flux, BTU/hr f t 2 Q ,= heat supplied in the two-phase region (Lt,p/rDt), BTU/hr rh, = fouling resistances on the heating and process sides, hr ft2 OF/BTU S = suppression factor, 0.25 T = temperature, O F U = overall heat transfer coefficient, BTU/hr ft2O F VASS = assumed value for tube inlet velocity, ft/sec VI = tube inlet velocity, ft/sec V , = parameter defined by eq 2.8 and 2.9 wt = total mass flow rate through reboiler tube, lb/sec wtve = boil-up rate, lb/hr x = vapor quality
rp
GREEKLETTERS y = parameter defined by eq 2.11 I‘ = parameter defined by eq 2.12-2.15 AL = elemental length considered, ft AP = pressure drop, lbr/in.* or lbr/ft2
290 Ind. Eng. Chern. Process Des. Develop., Vol. 12, No. 3, 1973
AQ
ATw
= heat supplied per element, BTU/hr = difference between the wall and
saturation
temperature, O F AX = increment in quality per element dP/bL = pressure gradient, lbf/ft3 e = absolute roughness X = latent heat of vaporization, BTU/lb p = viscosity, lb,/ft h r v = kinematic viscosity, ft2/hr A = constant, 3.1416 p = density, lbt/ft3 U, G = surface tension, dyn/cm, lbr/ft 71 = frictional pressure gradient, lbr/ft3 = parameter defined by eq 11-13
+
SUBSCRIPTS c = convective e = exit, exit piping 1, L = liquid i = inlet, inlet piping m = mean, mist ns = no slip N = nucleate boiling P = processside ss, sat = saturation SH = sensible heating S t = steam side t = tube, total t p = two-phase v, V, g, G = vapor Literature Cited
Bankoff, S. G., AIChE J., 8(1), 63 (1962). Chen, J. C., Ind. Eng. Chem., Process Des. Develop., 5, 322 (1966). Collier, J. G., Pulling, D. J., “Heat Transfer to Two-phase GasLiquid Systems,” Part 11: Further data on Steam-Water Mixtures, U.K. Report No. AERE-R-3809 (1962). Davis, W. J., Brit. Chem. Eng., 8 , 462 (1963). DeGance, A. E., Atherton, R. W., Chem. Eng., Part I V , (New York), 77,95 (1970).
DeGance, A. E., Atherton, R. W., Chem. Eng., Part V I , (New York),77,87 (1970). Dengler, C. E., Sc.D. Thesis in Chemical Engineering, MIT, 1962.
Duns,-H., Jr., Ros, N. C. J., “Vertical Flow of Gas and Liquid Mixtures from Bore Holes,” Paper No. 22-FD6, Proceedings of World Petroleum Congress, Section 11, Frankfurt, Germany, June 19-26,1963. Fair, J. R., Petrol. Refiner, 39, 105 (1960). Fair. J. R.. Chem. Ena.. PartI. (New York). 70. 119 (1963). Fair; J. R.; Chem. En;.: Part 11,(New Yorid), 70, 101%(1963). Forster, H. K., Zuber, N., AIChE J.,1, 531 (1955). Govier, G. W., Radford, B. A., Dunn, J. S. C., Can. J . Chem. Eng., 35, 58 (1957).
Griffith, P., Wallis, G. B., Trans. ASME, C83, 307 (1961). Hsu, Y. Y., “On the Range of Active Nucleation Cavities on a Heatine Surface.” No. 61-WA-177. Dresented at the Winter Ann Hughmark, G. A., Hughmark, G. A., Huihmark; G. A,; Chem. En;. Prugr., 65,‘67 (1969). Lee, D. C., Dorsey, J. W., Moore, G. Z., Mayfield, F. D., Chem. Eng. Progr., 52, 160 (1956).
Lockhart, R. W., Martinelli, R. C., Chem. Engr. Progr., 45, 39 (1949).
Nicklin, D. J., et al., Trans. Inst. Chem. Eng. (London), 40, 61 (1962).
Orkiszewski, J., J . Petrol. Technol., 829 (1967). Shellene, K. R., Sternling, C. V., Church, D. M., Synder, N. H.,
C h a . Eng. Progr., Symp. Ser., 64,82, 102 (1968). for review July 10, 1972 RECEIVED ACCEPTEDFebruary 13, 1973
