The dispersion of matter in turbulent flow through a pipe 447 by a point moving with the mean speed of flow to cover the distance must be determined. If this is Tm, the mean speed of flow is U = XjTm. 12 ( - ) To use the method it is therefore essential to know how to pick out the instant Tm by inspection of the conductivity-time curve measured at the point X. This problem was solved empirically by Allen & Taylor (1923), who first developed this method, by measuring the rate of discharge, from their water main by other methods and observing the point on the conductivity-time curve which corresponded to time X/U. They found in this way that Tm corresponded to the instant at which the conductivity at X was a maximum. This empirical result could not have been obtained if the salt had been carried down the pipe in laminar flow without lateral turbulent diffusion. In such a case it is a simple matter to calculate the distribution of concentration which results from any given initial distribution. If, for instance, a mass M of salt is initially concentrated uniformly on the plane = 0, the mean concentration (Taylor 1953a) at time t is uniform over a length u0t of pipe, where u0 is the maximum velocity in the centre of the pipe. Thus the mean concentration M per unit volume is C , where na2 is the area of cross-section of the pipe. The na2u0t concentration-time curve at the section = will therefore be represented by (7 = 0 (0 < t <X/uq), M (1*3) C = (t>X/u0).
448 Sir Geoffrey Taylor xx = x — Ut and by = u—U,U being the mean speed of flow. The mean-square deviation is then xl = 2 u 'H rm )d l (2-3) or, since t = XjU, 4 = r R(t)d£. (2-4) Bearing in mind (a) that U is uniform along a uniform pipe and (6) that the correla tion must disappear when X is sufficiently great, it seems that in a uniform pipe the spread, *Jx2, of an initially concentrated mass must increase as If, in any practical case, the spread is observed to increase in some other way than as Xi, this is a sure indication that the system considered is not simply a long straight pipe.
The dispersion of matter in turbulent flow through a pipe 449 11JO0 0-012 0-025 0-107 0-186 0-300 0-452 0-642 0-864 1-1051-342 1-544 1-658 1-6001-240 0-36801710-859 1-7532-3072-9713-758503 +++++++++++++++------- 10 X 0 0-05 0-39 1-25 1-90 2-65 3-34 4-16 4-71 4-92 4-58 3-52 1-013-331-053-810-1638-600-7960-062-718-7100 ++++++++++++-123 5 78 -------- 5 9 z)- 4-2 -4-25-4-19-4-01-3-72-3-50-3-24-2-96-2-63-2-25-1-83-1-35-0-85-0-20+ 0-55+ 1-54+ 2-85+ 3-41+ 4-12+ 5-115-86+ 6-87+ 8-60+ + GO ( 2 d 8 (z) J o 125491123553044579168807481178911706559443285613966353567800062446 f f 00-0-1-1-2-2-3-3-4-5-5-6-7-8-9-9-9-10-10-10-11-11- .2 - H 1 7 -^(z) 02-494-837-829-3710-2511-1512-4113-1813-7714-0915-2616-5017-6317-7415-9116-9318-3420-3522-4524-1922-23 Ooo e [ l b 5] a 2 T 4- 6f'(z)z2 —11858-543-639-433-830-228-526-424-723-223-724-926-929-030-936-947-267-788-4130255 00 (z)~ f [ X 173941426218599766017 5 0(z) 00-0210-0820-1790-2370-3030-3690-4350-4990-5570-6070-6440-6620-6550-6110-5140-4580-3880-3000-2540-1860-1100 - z d z) 01232025671360387609752949396905978461542757 4 zzf( J000-000-000-010-020-030-060-090-140-200-290-390-530-700-921-201-341-481-651-741-851-972-12 c 3 /'(*) 01-182-363-934-815-406-107-108-008-909-8011-614-017-221-025-031-241-662-583-32550 00 12 9600 2 f(z) 00050-230-530-751-011-291-622-002-422-893-404-054-805-797-107-668-379-360-111-122-85 00 111 1 z 00 100200300350400450500 550600650700750800-850900920 940 960 970-980 991 00.
450 Sir Geoffrey Taylor (A) The ‘ universal ’ distribution of velocity in a pipe will be assumed. It has been well verified by experiment (Goldstein 1938, p. 336) that the velocity u at radius r in a pipe of diameter 2 ais given by = /(*)> (3-1) where 2 = r/a. (3-2) The function f(z)is universal in the sense that it applies to all straight pipes with circular cross-section whether smooth or rough, provided the flow is fully turbulent. u0 is the velocity at the centre of the pipe and = (r0 (3-3) where r0 is the friction stress exerted by the turbulent fluid of density p on the wall.
The dispersion of matter in turbulent flow through a pipe 451 (B) Reynolds’s analogy will be assumed to be true. According to this assumption the transfer of matter, heat and momentum by turbulence are exactly analogous. If e is the coefficient of transfer, Reynolds’s analogy may be expressed in the present case by the equations r m G== d^^ (3*4) ^ drd r Here r is the shear stress at radius r, m is the rate of radial transfer of matter of concentration C.
452 Sir Geoffrey Taylor Substituting for e from (4*2), (4*3) becomes d i z 2 dC aW \ (4-4) dz \/'(z) dx v* Here x is measured along the pipe from a fixed point. It is convenient to use axes which move with the mean speed of flow. Thus the new co-ordinate is defined by x = xx+Ut, (4-5) and (4-4) becomes 0 / z2 0C\ J = ,j(/(Z)-4-25) | | +^ g ) J > (4-6) 0z\/'(z) (dC\ dC dC where (4-7) \ 0^ ) i dt+U dx is the rate of change of Cta a point which moves with velocity U.
The dispersion of matter in turbulent flow through a pipe 453 A virtual coefficient of diffusion K would transfer matter across a section at rate d£7 — Kna2 ——, so that matter is transferred across planes which move with the mean da^ speed of flow as though it were being dispersed by a virtual coefficient of diffusion K = 1006a?;*. (4-16) Estimate of the effect of longitudinal component in turbulent diffusion In general it is found in turbulent systems that there is a strong tendency to isotropy. To estimate the order of the effect of longitudinal turbulent diffusion on the longitudinal dispersion it is sufficient to assume that the coefficient of longi tudinal diffusion is equal to e, the coefficient of lateral diffusion. The rate of transfer of matter across a plane owing to longitudinal diffusion is therefore * * ¥ r ezdz, da; Jo and from (4’2) e = av*z/f dC C1 z2 so that Q' = ZnaPv* r_£ (4-17) dxjJoof/''<iz) n z* Using the figures in column 6 of table 1 ioTf the value of J dz was found to be 0-026, so that the mean coefficient of diffusion due to the longitudinal com ponents of turbulent velocity is Q’ K' ~— _ 0-052a?;*, (4-18) AC * nac This is small compared with K, but it is additive, so that the corrected value of K allowing for longitudinal diffusion is K = (10-06 + 0-05) a?;* = 10- la?;*. (4-19) An expression for K equivalent to (4-18) was found by Worster (1952). He noted that it is of order j-^th of the value necessary to account for the dispersion observed in pipe lines. He suggested that an expression analogous to (4-18), but with the numerical factor increased about 100-fold might be used as a ‘working formula’ for predicting dispersion. It will be seen that there is a good theoretical basis for this suggestion. The formula (4-19) was given earlier (Taylor 19536) without proof and with the factor 8-98 instead of 10-1.
454 Sir Geoffrey Taylor For smooth pipes v^U depends only on the Reynolds number. By definition 'V 1 /j ij= X jJ y The coefficient y, which is usually used to express the results of experi ments or friction in pipes, is defined by ro = \ypu%, (5-2) so that (5-3) Table 2 U K log10i? R UK l°g10i? R 12 3-150 1-41 x 103 22 5-181 1-52 x 105 13 3-361 2-30 x 103 23 5-378 2-39 x 106 14 3-570 3-72 x 103 24 5-573 3-74 x 105 15 3-777 5-99 x 103 25 5-767 5-85 x 105 16 3-982 9-60 x 103 26 5-961 9-15 x 105 17 4-185 1-53 x 104 27 6-157 1-44 x 106 18 4-387 2-44 x 104 28 6-347 2-22 x 106 19 4-587 3-86 x 104 29 6-550 3-55 x 106 20 4-786 6-11 x 104 30 6-731 5-38 x 106 21 4-984 9-64 x 104 Figubk 2.
The dispersion of matter in turbulent flow through a pipe 455 The relationship between y and the Reynolds number = may be expressed by the equation (Goldstein 1938, p. 338, eq. (19)) = — 0*40 + 4*00 log10i? + 2-00 log10y. (5*4) From this equation the value of R corresponding to any given value of y can be found.