Ind. Eng. Chem. Res. 1987,26, 333-336 Silva, J. M.; Wallman, P. H.; Foss, A. S. Ind. Eng. Chem. Fundam. 1979, 18, 383. Stewart, W. E.; Sorensen, J. P. Paper presented a t the Proceedings of the 5th European Symposium on Chemical Reaction Engineering, Amsterdam, 1972. Swerling, P. J. J. Astronaut. Sei. 1959, 6 , 46.
333
Wong, C.; Bonvin, D.; Mellichamp, D. A.; Rinker, R. G. Chem. Eng. Sei. 1983, 38, 619.
Received for review March 25, 1985 Revised manuscript received February 13, 1986 Accepted March 28, 1986
Cooling of a Stretching Sheet in a Viscous Flow Binay K. Dutta Chemical Engineering Department, Calcutta University, Calcutta 700 009, India
Anadi S. Gupta* Mathematics Department, Indian Institute of Technology, Kharagpur 721 302, India
A hot, flat sheet issues from a thin slit into a n incompressible viscous fluid and is subsequently stretched in its own plane a t a velocity proportional to the distance from the slit. T h e cooling of the sheet in this extension zone is studied by solving the coupled heat-transfer problem in the fluid and the sheet. Variation of the sheet temperature with distance from the slit is found for several values of the Prandtl number and stretching speeds. I t is shown t h a t for a fixed Prandtl number, the surface temperature decreases with a n increase in the stretching speed.
1. Introduction Boundary-layer flow on moving solid surfaces is an important type of flow arising in a number of technical processes. This problem was first studied by Sakiadis (1961). The momentum and heat and mass transfers from such surfaces are determined by the structure of this layer. Due to the entrainment of the ambient fluid, this boundary layer is different from that in the Blasius flow past a flat plate. Erickson et al. (1966) extended this problem to the case when the transverse velocity a t the moving surface is non-zero and is such that similar solutions exist. These investigations have a bearing on the problem of a polymer sheet extruded continuously from a die and is based on the tacit assumption that the sheet is inextensible. However, situations may also arise in the polymer industry where one has to deal with flow over a stretching sheet. For example, in a melt-spinning process, the extrudate from the die is generally drawn and simultaneously stretched into a filament or sheet, which is then solidified through rapid quenching or gradual cooling by direct contact with water or chilled metal rolls. In fact, stretching imparts a unidirectional orientation to the extrudate, thereby improving its mechanical properties (Fisher, 1976). Several metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid, and in the process of drawing, these strips are sometimes stretched. Mention may be made of drawing, annealing, and tinning of copper wires. In all these cases, the properties of the final product depend to a great extent on the rate of cooling. McCormack and Crane (1973) gave a similar solution in a closed analytical form for steady two-dimensional, boundary-layer flow over a sheet which issues from a slit and is subsequently stretched at a velocity proportional to the distance from the slit. The corresponding nonsimilar solution for the same problem in the presence of a uniform free stream velocity was obtained by Danberg and Fansler (1976). Gupta and Gupta (1977) analyzed heat and mass transfer in the boundary layer over a stretching sheet held at constant temperature, the sheet being subject to suction or blowing. Similar studies for the temperature distribution in the flow a viscous, incompressible fluid caused by the stretching of an impermeable, nonporous sheet were made by Crane (1970) for prescribed surface temperature variation and 0888-5885/81/2626-0333$01.50/0
by Dutta et al. (1985) for uniform surface heat flux. On the other hand, Vleggaar (1977) investigated the cooling of a sheet that issues from a slit and is stretched at a velocity proportional to the distance from the slit by solving the conjugate problem of heat transfer inside the sheet and the fluid. However, this study suffers from the serious defect that the temperature distribution in the boundary layer was found for a constant surface temperature of the sheet, which was subsequently used to determine the longitudinal variation of the same surface temperature. The aim of this paper is to remove this anomaly through a rigorous analysis of the above conjugate heat-transfer problem. The effects of the Prandtl number and the stretching rate on the surface temperature are also investigated. 2. Mathematical Formulation Consider a broad, flat sheet issuing from a long, thin slit at x = 0 and y = 0 and, subsequently, being stretched a t a velocity proportional to the distance from the slit as in Figure 1. Assuming the boundary layer approximations, the equations of continuity, momentum, and energy in the steady flow near the sheet are in usual notations
(3) where the effects of natural convection and viscous dissipation are neglected in (3) and the fluid properties are assumed to remain constant over the temperature range of interest. The sheet material (of density p and specific heat C,) gradually cools down along the direction of motion, and a heat balance equation in the sheet gives the temperature, T2,of the sheet as the solution of
0 1987 American Chemical Society
334 Ind. Eng. Chem. Res., Vol. 26, No. 2, 1987
Since this relation holds for all x , equating the coefficient of x n gives
ne-"gn(v) - (1 - e-'9gn'(q) = plgnll(v)
Setting t = -Pe-q in eq 17 leads to the confluent hypergeometric equation
3
LStntching s u r t a a
td2gn + [l - P - t] dgn + ng, = 0 dt2 dt
Here Uo(x)and d,(x) denote the velocity and thickness of the sheet at a distance x from the slit. The factor 2 in eq 4 appears because both sides of the sheet are exposed to the fluid. Conservation of mass in the sheet demands
(5) In writing eq 4, any viscous dissipation inside the sheet material is neglected. Since the sheet thickness is assumed to be very small, the temperature distribution over any section of the sheet is taken to be uniform. The appropriate boundary conditions for the above conjugate heat transfer problem are u = Uo = y x u =0 T1 = T2(x) at y = 0 (6)
g , = B1F[-n, 1 - P, t]
T,-T,
asy-m
(w)'/2Xf(v)
v
P=
= BY
(Y/vP
gn(v) =
u = -(.T.v)1'2f(v)
To determine the constants A,,, we use eq 12 and 13 in the heat balance relation, eq 4. This leads to m
c nA,xn-'
(8)
(9)
where a prime denotes differentiation. Substitution of eq 9 in eq 2 gives f 12
- ff ' I =
f"'
(10)
whose solution satisfying f ( 0 ) = 0, f ' ( 0 ) = 1, and f ' ( m ) = 0 (which follow from eq 6 and 7) was given by McCormack and Crane (1973) as
f ( v ) = 1 - e-7
(20)
where y ( a , P) stands for the incomplete y function
n=O ?Xf'(V)
e-P"[P - n, P + 1, -Pe-q] F [ P - n; P + 1; -PI
In particular, for n = 0,
Hence, u =
(19)
(7)
where T , is the constant ambient temperature in the fluid. Equation 1 implies a stream function, $(x, y), such that u = a*/ay and u = +/ax with =
+ B,tPF[P - n, 1 + P , t ]
where B , and B2 are constants. Using the boundary conditions eq 14 and 15 in eq 19 yields
d,Uop = constant
u-0
(18)
Its general solution in terms of the confluent hypergeometric function, F , is
Figure 1. Sketch of the physical problem.
*
(17)
m
=C
n=O
Anxng,'(0)
(23)
where
c = 2klP/(PCpUO4
(24)
which is constant by virtue of eq 5. Equating the coefficients of x" in eq 23, we get
Repeated use of this recursion relation gives
(11)
The solution for the fluid temperature, T,, is assumed in the form
(26)
n
where the stands for the continued product. If the temperature of the sheet material at the slit ( x = 0, y = 0) is T,, then eq 13 leads to A, = ( T , - T,)/T,. Hence, from eq 12, 13, and 26, the temperature distributions in the surrounding fluid and the sheet are given by
n=O
while the sheet temperature is taken as T, - T,
--
- CA,x" T, n=O where A, are the constants. The continuity of temperature at the sheet, y = 0, given by eq 6 requires for n = 0, 1, 2, etc. Further, eq 7 demands
(14)
for n = 0 , 1, 2, etc. Substitution from eq 9, 11, and 12 in eq 3 gives
(15)
g,(O) = 1
gn(m) = 0
m
m
w e - 7 2 nAnx"-lgn(v) - ~ (- e-?) l C Anxngn'(v) = n=O
n=O
5
(v/P)P2 A,x"g,"(v) n=O
(16)
3. Convergence Characteristics of Temperature Distribution Since our main goal is to determine the temperature distribution in the stretching sheet, it is evident that the expression for sheet temperature given by eq 28 will be of little value unless the infinite series in this expression is convergent. We now establish the convergence of this series in the following manner.
Ind. Eng. Chem. Res., Vol. 26, No. 2 , 1987 335 When Kummer's transformation (Abramowitz and Stegun, 1965) for the confluent hypergeometric function given by
F [ a , b, 21 = ezF[b- a, b, -21
(29)
is used in eq 20, one finds
where
F [ n ,P
+ 1, PI
e = F [ n + 1, P + 1, PI Since F[a, b, X I = a(a
ax
1+-+-b b(b
+ 1) x2
+ 1) 2!
a(a +
b(b
x(m) -c
(31)
+ l ) ( a + 2) x3 + ... + l ) ( b + 2) 3!
it is clear that 0 < 0 < 1 for positive integral values of n because each term of the denominator of eq 31, starting from the second, is greater than the corresponding term of the numerator. Thus, (30) shows that for large values of n, k,'(O)/(n 1)1becomes less than unity for a given P. In situations of practical interest, the ambient fluid is air with a Prandtl number of P = 0.70, and for some other gases also P = 0(1), so that in such cases Ig,'(O)/(n 1)1 < 1, even for moderate values of n. Further, since 0 < 0 < 1, it can be seen from eq 30 that g,'(O)/(n + 1)< 0 for positive integral values of n. Thus, the series in eq 28 is an alternating series and by Leibnitz theorem (Piskunov, 1964) will converge for a given value of x (i.e., a t a fixed location on the sheet) provided Ig,'(O)/(n + 1)1< 1and Cx < 1since the last two inequalities ensure that the nth term of the series tends toward zero as n m. As shown above, the first inequality holds for fixed P and a large but finite value of n. Now consider the typical data for drawing a polyethylene sheet in air (Middleman, 1977). For air, kl = 6.5 X cal/(s.cm.OC), P = 0.7, and Y = 0.17 cm2/s. For polyethylene, k2 = 8.0 X cal/(s.cm.°C), p = 57 lb/ft3, and k z / p C p = 1.3 X cm2/s. Using Vo= 1m/s, d, = 0.05 cm, p = 0.5 s-', and the above data, we find from (24) that C = 7.246 X m-l. As pointed out by Vleggaar (1977), the major part of the stretching process and heat transfer takes place over a length, x , of about 0.5 m from the die. Thus, it follows from above that the condition, Cx < 1,is fulfilled, at least in the region near the die where the sheet velocity is low and is approximately proportional to x . Hence, the convergence of the series in (28) is established in a problem of practical interest.
Figure
2. Variation of the dimensionless sheet temperature, B,, along the sheet for several values of P with 9 = 0.5 s-l.
O**
t P-0.7
..
+
+
-
4. Discussion
Using the above data for drawing a polyethylene sheet in air, we have computed 8, (=(Tz(0)- TZ(X))/(TZ(O) - TJ) for several values of P and 7. Clearly dl defined above is 1- (T2- Tm)/(Tw - T,)where (Tz- T,)/(Tw - 2") is given by (28) with T, = Tz(0).Evidently the physical meaning of 8, is that it represents the fractional drop in the sheet temperature or fractional cooling. For the purpose of computation, we have considered six terms in series 28. It is well-known that for an alternating convergent series, the estimate of the error is of the order of the absolute value of the f i s t term neglected. Using this, we have found that the truncation error for the chosen values of the parameters ranges from 0.02% to 1.7% only. It is also worth mentioning that for Cx < 1, the convergence of series 28 is quite rapid. Figure 2 shows the variation of 8, with x
x(m)
-
Figure 3. Variation of the dimensionless sheet temperature, B,, along the sheet for several values of 9 with P = 0.7.
for different values of P with p = 0.5 s-'. It can be seen that a t a given position the rate of cooling of the sheet increases with P. Figure 3 displays the variation of 8, with x for several values of p with P = 0.7. It is found that a t a given location, an increase in the stretching rate, ?, results in enhanced cooling. Physically, this may perhaps be attributed to the fact that an increase in the stretching rate results in an increase in the entrainment of air, which carries away heat from the sheet. In actual practice, however, one would expect that the sheet temperature will be somewhat lower as heat transfer due to free convection and radiation is neglected in the analysis.
Nomenclature A,, = constants defined by eq 1.2 B,, B2 = constants defined by eq 19 C = constant defined by eq 24 C, = specific heat of the stretching sheet d, = thickness of the sheet f = function defined by eq 8 g, = function defined by eq 12 k, = thermal conductivity of the sheet P = Prandtl number of the fluid T I = fluid temperature T2 = sheet temperature T , = ambient fluid temperature t = variable given by -Pe-q U, = velocity of the sheet u,u = velocity components in the fluid x , y = Cartesian coordinates Greek Symbols
p = constant defined by eq 8 9 = stretching rate of the sheet Y = kinematic viscosity of the fluid p = density of the sheet material Ji = stream function for the fluid fl'ow
Ind. Eng. Chem. Res. 1987,26, 336-337
336
7 = dimensionless distance normal to the sheet 8 = constant defined by eq 31 8, = dimensionless sheet temperature
Literature Cited Abramowitz, M.; Stegun, I. A. Handbook of Mathematics Functions; Dover: New York, 1965. Crane, L. 2. Angew. Math. Phys. 1970, 21, 645. Danberg, J. E.; Fansler, K. S. Q. Appl. Math. 1976, 34, 305. Dutta, B. K.; Roy, P.; Gupta, A. S. Int. Commun. Heat Mass Transfer 1985, 12, 89. Erickson, L. E.; Fan, L. T.; Fox. V. G. Ind. Eng. Chem. Fundam. 1966, 5 , 19.
Fisher, E. G. Extrusion of Plastics, 3rd ed.; Newnes-Butterworths: London, 1976. Gupta, P. S.; Gupta, A. S. Can. J. Chem. Eng. 1977, 55, 744. McCormack, P. D.; Crane, L. Physical Fluid Dynamics; Academic: New York, 1973. Middleman, S. Fundamentals of Polymer Processing; McGraw-Hill: New York, 1977. Piskunov, N. Differential and Integral Calculus; Mir: Moscow, 1964. Sakiadis, B. C. AIChE J . 1961, 7, 26. Vleggaar, J. Chem. Eng. Sci. 1977, 32, 1517. Received f o r review May 3, 1985 Accepted June 2, 1986
Solubility of Mercury in Normal Alkanes H. Lawrence Clever* and Marian Iwamoto Solubility Research a n d Information Project, Department of Chemistry, Emory University, Atlanta, Georgia ,70322
The equation In S (molality) = 5.1059 - 4970.90/T reproduces within experimental error the solubility of mercury in normal alkanes of carbon number 5-10 over the 273.15-336.15 K temperature interval. T h e molal solubility of mercury appears t o be independent of the normal alkane carbon number a t all temperatures. T h e equation can be used with caution to estimate the solubility of mercury in normal alkanes of other carbon numbers at other temperatures. During a review of the literature of the solubility of mercury in organic liquids, it was observed that the molal solubility (mol kg-l) of mercury is nearly independent of the solvent a t a given temperature for the C5-Cl0 normal alkanes between 273.15 and 336.15 K. The molar solubility (mol L-l) is also nearly independent of the normal alkane solvent but at a slightly larger uncertainty. The mole fraction solubility shows a systematic change with solvent. The relationship does not apply to branched alkanes. The solubility of mercury in the branched chains depends on the chain branching and is smaller than in the normal alkane of the same carbon number. Figure 1shows a plot of the natural logarithm of molal solubility against the inverse of the absolute temperature. There are 58 points from 10 papers. The papers are not completely independent. Several (Moser and Voigt, 1957a,b; Klehr and Voigt, 1960; Spencer, 1967; Spencer and Voigt, 1968) are from the same laboratory. Another (Kuntz and Mains, 1964) reference their values to the mercury in hexane solubility value at 298.15 K of Moser and Voigt (1957). The experimental values from Okouchi and Sasaki (1981,1983), Reichardt and Bonhoeffer (1931),and Vogel and Gjaldbaek (1974) appear to be independent experiments. All of the experimental values will be published in a volume of the Solubility Series under preparation. Table I gives the average molal solubility at each temperature. At four of the temperatures, dropping one value from the average greatly improves the standard deviation of the average. These values are compared with the smoothed values from an equation obtained from a linear regression of 53 of the 58 points shown in Figure 1. The equation is
Table I. Solubility of Mercury in Normal Alkanes. Comparison of the Average Experimental Molal Solubility with the Calculated Molal Solubility and the Calculated Molar Solubility lo6 M solubility, mol L-' lo6 m solubility, mol kg-' T. K exDtl av. f SD" no. calcd ea 1 calcd ea 2 1.45 2.3 f 0.4 4 2.06 273.15 2.1 f 0.1 3 4 2.86 1.99 2.8 f 0.2 278.15 2.70 3.9 f 0.2 3.92 5 283.15 3.63 5.31 4 5.6 f 0.3 288.15 7 7.13 4.83 6.9 f 0.3 293.15 14 9.4 f 0.6 9.48 6.37 298.15 9.6 f 0.1 13 8.32 12.4 f 0.4 8 12.48 303.15 10.78 15.3 f 1.3 4 16.28 308.15 15.9 f 0.4 3 21.06 13.84 6 20.9 f 1.5 313.15 5 21.4 f 0.9 17.6 27.0 1 18.7 318.15 62.4 39.8 82.8 1 336.15
In S (molal) = (5.1059 f 0.1576) - (4970.90 f 46.40)/T (1)
with a standard error about the regression line of 5.7 x lo-'. Smoothed molar solubilities from eq 2 are in Table I. The equations above are considered reliable for the experimental range of 5-10 carbon normal alkanes at 273.15-336.15 K. The equations can be used with caution to estimate the solubility of mercury in normal alkanes and
with a standard error about the regression line of 4.0 x lo-'. The constants of the equation relate to thermodynamic changes for the transfer of 1mol of mercury from the liquid
(I
SD = standard deviation.
metal to the hypothetical 1 m solution of AHl = 41.3 f 0.4) kJ mol-' and ASl = (42.5 f 1.3) J K-' mol-'. The treatment assumes the same values for all normal alkanes. The molar solubility fits a similar pattern but with a little larger uncertainty. A linear regression of the concentrations gives In S (molar) = (4.2390 f 0.2472) - (4830.90 f 72.78)/T (2)
0888-588518712626-0336$01.50/0 0 1987 American Chemical Society
