Heat Transfer from Helical Coils Immersed in Agitated Vessels K. K. SETH
Immersed helical coils can provide a most
E. P. STAHEL
practical method for heat transfer in circulating fluids. Advantages and limitations of the process equipment are discussed in this paper mmersed coils are used to provide heat transfer surface
I in process vessels and to augment available jacketed
surface, often in preference to jacketed surface because of lower cost and the ability to accommodate higher pressures in a coil or circulate fluids at higher velocities and thus attain higher heat transfer coefficients. Coils offer the most practical heat transfer surface for ceramiclined vessels in corrosive service. An immersed coil in an agitated vessel, with or without baffles, takes one of two forms (a) the helical type or (b) the vertical baffle type. I n process equipment shown in Figure 1, heat transfer is primarily conduction and convection. I n general, the two modes of heat transfer in such cases are superimposed. One mode may predominate depending on conditions. The rate of heat transfer to or from an agitated liquid mass in such process equipment is a function of the physical properties of the agitated liquid and the heating or cooling medium (inside the coil), the vessel geometry, coil geometry, and the degree of mixing. The resistance or film theory adequately describes this process of heat transfer.
Figure 7.
Agitated vessel with immersed helical coil for heat transfer
Equations to Estimate Heat Transfer (8) Case I. For simplicity, consider a situation where liquid in the vessel is being heated by a heating medium, such as steam on the inner side of the coil. An element of the coil wall is shown in Figure 2. If heat losses from the vessel wall are negligible, then at any instant under steady-state conditions inside the coil, we have the following : The condensing steam has a constant temperature, t 8 . The condensate forms a film of liquid in contact with the scale at a temperature, t f t . Beneath thewater film are deposits of scale and dirt at temperature, tWt,in contact with the coil inner wall. Similarly, adjacent to the outside wall of the coil is a film of scale at temperature, tfo.
Figure 2. Resistance to heat transfer through coil wall to a liquid being heated VOL. 6 1
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The agitated liquid in the vessel at this instant has an average temperature, to. The wall has a thickness, x , with inside and outside heat transfer areas, A,, and A,,, respectively. In the classical way we can write
Q
= htcAs(ts
-
=
(condensate film)
k
- Am(twt X
-
hliA,i(t,i - twt) (inside fouling film)
=
- tfo) = hocAo(t,o - to) (outside fouling (process liquid film) film)
= h,o&o(two
two)
(wall)
(1 1 which, in the case of the resistance of the outside of the coil controlling, reduces to -1= -
uo
Ao A +-+-+hlt A,* k Am
A, htc A,
A,
ox
h,o
&o
+ h,,1
(2)
where Uois the overall heat transfer coefficient based on the outside area, A,. For simplicity, assume that there is no fouling on either side of the coil wall. Hence,
1 Uo
+ -hoc1
Aox
- htcA As , +k Am
(3)
Case 11. When we have a situation where the resistance of the inside of the coil is controlling (liquid inside the coil, or chemical reaction inside the coil), then Equation 3 has the corresponding form
A ~ x +-+htc k Am
1 _ -- -1
ut
At hoc Ao
(4)
I t is customary to write the equation defining U(Q = UAAt) in differential form to account for temperature variation along the coil : dQ = UloodA(tin =
UiocdA(t,-
tout)loo
(5)
to)ioo
Similarly an analogous equation to 3 can be derived for Ul00 * Determination of h,, and hoc, From Equation 3 if conditions are such that hi, and k do not vary appreciably,
1 _ -P+-
UO
/3
where Also note P
= constant
1
N R ~ > 11.G. 'fe' can also be obtained for this region using Barua's ( 2 ) theoretical analysis and Kubair and Kuloor's (74) experimental data :
where VOL. 6 1
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fc = [Reynolds f8
number
(29)
is valid for the range Reynolds number ( d / D ) 2 > 6. They also recommend that for curved tubes
Equation 26 does not seem to indicate the effect of entry length and hence should be used with caution. The limitations in evaluating 'fiin Equation 28 have already been pointed out earlier. The above equation seems to indicate that for straight tube, the critical Reynolds number is zero. Therefore, all such equations need a little care in their use. Seban and McLaughlin (24)have compared the results obtained from their equations for two coils with those obtained from Jeschke's equation and report the ratio of heat transfer coefficients for coil and straight tube, employing Equations 26 and 29:
+
D/d
(1 3.5 OD)
17 104
1.20 1.03
N R ~=
104
1.27 1.06
r\k, =
For laminar flow, using empirical modification of Graetz theoretical equation, we have hi, - -d k
(LD)li3
- 0.01 16 fc(NRe)4/3(1V:pr)113
I4.'):(
(33)
where 'fc' is estimated using White's correlation and for turbulent flow, using Colburn analogy, we have
105
1.34 1.12
Thus there is little choice between the two recommendations. Mori and Nakayama (15-77) very recently have determined theoretically, as well as experimentally, the velocity and the temperature distribution over a cross section of fluid in a coil for both laminar and turbulent flow. The temperature profiles were found to be asymmetric as are the velocity profiles. Using the perturbation technique (15), they have obtained equations for N Nin~the laminar flow regime and for values of Npr less than and greater than one. After very careful and extensive analysis for heat transport in the turbulent region, they recommend for gases
and for liquids
48
They conclude, that, for coils, entrance effects are negligible and that the heat transfer data for a constant heat flux are the same as those for a constant wall temperature. A number of equations which predict friction factors and film heat transfer coefficients have also been reported (28). From studies in fluid flow7 in coils and analogy between heat transfer and momentum transfer, it seems that for a Prandtl number close to 1, the following may be more useful, and may offer more precise results (25).
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
However, the equation for turbulent flow can be modified for Prandtl number larger than 1 using Prandtl, Von Karman, Martinelli, Seban, and Shimazaki, or Lyon analogy depending on the desired precision and range of Prandtl number. The accuracy of 'hi,' is dependent on 'f:. Thus more precise correlations for 'f: are needed, free from the limitations pointed out earlier. Conclusions
Theoretical heat transfer equations, with the experimental methods to obtain individual film coefficients, together with the available correlations for both outside and inside film coefficients have been discussed. Whenever possible the range and limitations of the variables involved have been indicated. Considerable work has been done on 'ho: for immersed coil in agitated tanks. Though there seems agreement on the use of dimensionless numbers for the general form of the correlation for Nusselt number, however, the proposed equations are only valid for geometrically similar systems. Even in this, certain variations in the exponents and coefficients are apparent. All the available correlations are ordinarily based on the methods of correlation for conduits and plane surfaces proposed by Colburn, and modified by Seider and Tate. Because of the wide variation in the coefficients established (Tables I1 and III), it is difficult to make a choice for the general situation for their use in practice. I t is indicated that the extensive available data be fitted empirically to the generalized correlation via Equation 17. It is also believed that the use of nunierical and computer methods in lieu of modified Wilson plot to arrive a t the values of 'hoc' or ‘hit' from the experimental observations should yield more accurate results.
Some of the gaps that need to be filled in this area have been indicated. Literature reveals that there is not much information (theoretical or experimental) available on the influence of geometry of the coil, on ‘&’. The available information seems inadequate and presents results which are widely divergent. I n laminar flow, heat transfer coefficients reveal an effect of thermal entry length; the entry effect ultimately terminates, and asymptotic coefficients can be estimated. Additional experimental data are clearly necessary properly to define the heat transfer inside the curved tubes. The available correlations for Nusselt number merely modify the Dittus-Boelter equation by a ratio (d/D) for turbulent regime. This is not the best approach. Apparently, the correlation may be given by a composite expression in which Nusselt number is some function of ‘d’ and ‘D’ and not necessarily of (d/D). Average ‘hi: for the periphery can be specified accurately from the usual analogy formulation, using friction factor appropriate for curved tubes. I t is believed that the use of Equations 32 through 34, based on the above, may yield fairly satisfactory results. This precision can further be improved by employing mathematically more rigorous and complex analogy relationships. The following work seems desirable in this area of heat transfer, viz: (i) to obtain generalized correlation for ‘hoc’ using extensive available data with numerical and computer techniques (ii) to study the effect of geometry of the coil on ‘hit' and to attempt to correlate and verify the suggested approach, additional experimental data are necessary
Nomenclature Unless otherwise stated, numbers in parentheses after description refer to equations in which symbols are used or thoroughly defined. Dimensions are given in terms of mass (M), length (L), time (e), and temperature (T).
A
= characteristic area or area of heat transfer of the coil tube,
B
= number of blades on impeller, dimensionless
C
=
C
=
d
D
= =
Di
=
f
=
L2
\
F
= =
= =
h
=
AH
= =
J k
L M N
NO P
Q
= = =
= = = =
heat capacity at constant pressure, per unit mass L2/€PT (or Btu/lb, deg F ) clearance, distance from lower impeller to bottom of the vessel, L coil tube diameter, L coil helix diameter, L impeller diameter, L friction factor (Fanning), dimensionless frictional drag (24), ML/@ gravitational acceleration, L/e2 gravitational conversion factor (lb,/lbf)(ft/sec2) working height of vessel or liquid height, L individual heat transfer coefficient, M/WT (or Btu/hr ft2 deg F ) pressure loss due to friction in feet of liquid, L baffle width, L thermal conductivity, M L / e 3 T (or Btu/hr f t deg F) impeller blade length, L mass of material, M impeller speed, rate of rotation, 8-1 a constant, dimensionless pitch of coil, L rate of thermal energy flow across a surface, ML2/fY (or Btu/hr)
R
= number of baffles, or number of turns in a coil, dimension-
less pitch of impeller, L temperature, T T tank diameter, L At driving force for thermal energy flow, temperature difference, T U = overall heat transfer coefficient, M/WT (or Btu/hr ft2 deg F ) V = mass average velocity, L/B W = impeller blade width, L x = coil wall thickness, L K , K‘, K”, A,, Ao’, a,, a*’, bo, bo’, co, co’, do, e,, fo, go, h,, p , q, s ’ are exponents or otherwise constants, dimensionless a!, a! ’, p, 0 ’ and ?r, exponent or otherwise constants, dimensionless p = fluid density, M/L3 8 = time, 8 p = viscosity, M/L8
s t
= = = =
Dimensionless Groups
N R ~= Reynolds number, ( D v p / ~ ) NN” = Nusselt number, ( h D / k ) Npr = Prandtl number, ( c p / k ) Subscripts av = average b = bulk fluid fi = inside film = outside film fo i = inside iC = inside of the coil loc = local 0 = outside oc = outside of the coil S = condensing steam, or straight tube W = a t the wall = inside wall of the helical coil Wi = outside wall of the helical coil WO REFERENCES ( 1 ) Adler, M., Z . Angew. Math. Mech., 14, 257-275 (1934). (2) Barua, S. N., Quart. J . Mech. Appl. Math. 16, 61 (1963). (3) Brooks, G., and Su, G. J., Chem. Eng. Progr., 55, (IO), 54 (1959). (4) Chapman, F. S., Dallenbach, H. R., and Holland, F.A., Trans. Inst. Chrm. (London), 42, 398 (1964). (5) Chilton, T., H., Drew, T. B., and Jebens, R. H., IND. ENO. CHEM.,36, 510 (1944). (6) Cummings, G. H., and West, A. S., ibid., 42,2303 (1950). (7) Dunlap I. R., and Rushton, J. H., Chem. Eng. Progr., 49, Symposium Series NO.5, 137 (195i). (8) Holland, F. A,, and Chapman, F. S., “Liquid Mixing and Processing i n Stirred Tanks,” Reinhold, New York,N. Y.,1966. (9) Inglesent, H., and Storrow, J. A,, Ind. Chem., 2 6 , 313 (1950). (IO) Ito, H., J. Basic Eng., Trunr. A.S.M.E.D., 81, 123-4 (1959). (11) Jeschke, D., Ver deut. Ins., 69 (1926); Z . Ver deut. h g . Ergnnrungsheft, 24, 1 (1925). (12) Jha, R. K., and Rao, M. R., Int. J . Heat Mass Transfer, 9, 63 (19G6). (13) Kraussold, H., Chem.-Ing.-Tech., 29, 177 (1951). (14) Kubair, V., and Kuloor, N. R., Int. J.Heat Mass Transfw, 9, 63 (1966). (15) Mori, Y., and Nakayama, W., ibid., 8, 67 (1965). (16) Mori, Y.,and Nakayama, W., ibid., 10, 37 (1967). (17) Mori, Y.,and Nakayama, W., &id., p 661. (18) Oldshue, J. Y., and Gretton, A. T., Chem. Eng. Progr., 50, 615 (1954). (19) Pierce, D. E., andTerry, P. B., Chem. @? Met. Eng., 30, 872 (1924). (20) Pratt, N. H., Truns. Inst. Chem. Eng. (London), 25, 163 (1947). (21) Rhodes, F. H., IND.ENC.CHEM.,26, 944 (1934). (22) Rushton, J. H., Costich, A. W., and Evertt, H.J., Chem. Eng. Prog., 46, 395 (1950). (23) Rushton, J. H., Costich, E. W., and Evertt, H . J., ibid., 46, 467 (1950). (24) Seban, R. A., and McLaughlin, E. F., Int. J . Heat Mars Transfer, 6, 387-95 (1963). (25) Seth, K. K., M.Sc. Thesis (Tech.), Bombay University, India, 1962. (26) Skelland, A. H., and Dabrowski, J. E., J. Birmingham Univ. Chem. E ~ sot., ~ . 14 (3), 82 (1963). (27) S iers H in “Technical Data on Fuel ” T h e British National Committee Worpd Pdwe;‘Conference, London, 1937, p. 5 6 . (28) Srinivasan, P. S., Nandapurkar, S. S., and Holland, F. A., Chem. Eng., 5 , 113 (1968). (29) Uhl V. W and Gray, J. B. “Mixing Theory and Practice,” Vol. I, Chap v, Acaderkc Pre‘is, New York and’london, 1966. (30) White, A. M., and Brenner, E., A.I.Ch.E., 30, 565 (1934). (31) White, A. M., Brenner, E., Philips, G. A., and Morrison, M. S., ibid., 30, 570 (1934). (32) White, C. M., Proc. Roy. SOC.(London) Ser. A., 123, 645-63 (1929). (33) Wilson, B., Trans. A S M E , 37,47 (1915).
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