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Streaming currents in turbulent flows and metal capillaries
Boumans, A.A.
Physica X X l I I
1957
1027-1037
STREAMING CURRENTS IN T U R B U L E N T FLOWS AND METAL CAPILLARIES II. T H E O R Y
(2). C H A R G E T R A N S P O R T E D
BY THE FLOW OF LIQUID
by A. A. BOUMANS Stichting veer fundamenteel onderzoek der materie, Utrecht Synopsis The m e a n charge d e n s i t y / ] which appears if a liquid flows t h r o u g h a t u b e is calculated. The following dimensionless n u m b e r s are used. 1°. KR~, i.e. the ratio of the t u b e ' s hydraulic radius R~ and the diffuse layer thickness 1/K. 2 °. 12vK/v,, i.e. the ratio of the l a m i n a r sublayer thickness 12v/v, and the diffuse layer thickness l/K, where v is the k i n e m a t i c viscosity of the liquid and v , the friction v e l o c i t y as defined in 1.5. 3 °. ~ , K , i.e. the ratio of the distance t r a v e l e d with velocity v, in the relaxation time r and the diffuse layer thickness 1/K. 4 °. /~/p0, i.e. the ratio of the m e a n charge density/~ in the liquid and the m e a n charge d e n s i t y p0 = -- 2¢~/nR~ which appears in a l a m i n a r flow if KRh >~ 1, where ¢ is the dielectric c o n s t a n t of the liquid and ~ the wall potential. 5 °. R e y n o l d s n u m b e r Re = 2~Rh/v and friction n u m b e r Jl = - - ( 4 R h / s ~ 2 ) d p / d l where s is the specific density of the liquid, ~ the m e a n velocity across the tube cross section and dp/dl the pressure gradient along the tube axis. H y d r o d y n a m i c s teaches t h a t 2 is a function of Re, which is d r a w n in fig. 1-7. The following relations are deduced for KRh>~ 1. I n a l a m i n a r flow ~/po = 1. I n a t u r b u l e n t flow, as long as 12vK/v, >~ 1, ~/po = Re 2/64, b u t if 12vK/v, ~ 1, ~/po=KRh/4. The complete function is given in fig. II-1, which illustrates also the situation KRn ~ 1. I t is p r o v e d t h a t ~ is only slightly d e p e n d e n t on the shape of the charge distribution in the t u r b u l e n t core of t h e ' f l o w (i.e. on the value of ~v,K). In tubes of finite lenth L,/3 is influenced by the relaxation effect, b y a possible saturation of wall current and b y t h e fact t h a t the velocity profile close behind the t u b e e n t r a n c e differs f r o m t h a t farther in the tube. The consequences for KRI~>~ 1, illustrated in fig. 11-3, are the following. If LRh is small : ~ ~ L 2 for small L/Rh and/5 ~ L~/R~ for large L/Rh. If LRn is large: # ~ 1/Rh for small L/Rh and # ~ 1/R~ for large L/Rn. Similar relations are deduced for KRh ~ 1 and illustrated in fig. 1I-2. All results are valid for flow through round tubes as well as between parallel walls.
1. Preliminary remarks. D i f f e r e n t f r o m t h e m o r e u s u a l p r o c e d u r e , mean charge density/], potential
i n s t e a d of t h e s t r e a m i n g
V, is u s e d as c h a r a c t e r i s t i c
charge transported
quantity
the current I or the streaming describing
the amount
of
b y t h e f l o w . T h e r e a s o n is t h a t i t g i v e s a b e t t e r c o n n e c --
1027
--
1028
A . A . BOUMANS
tion with our experimental technique, and theoretically simpler expressions, /5 being independent of R.e for a laminar flow. The mean charge density in the flow is/5 = I/O. For two parallel walls:
I = 2Bfoa pvdx; Q = 2Bfod vdx
(II-la)
and for a round tube :
I = 2~tf~ pvrdr; Q --- 2z~foR vrdr
(II-lb)
For a purely laminar flow v is given b y (I-15a) of (I-15b) and by (I-4a) or (I-4b). For a turbulent flow between two parallel walls
/5, = (2B/Q) foa pvdx = (2By~Q) f~.a/. #l, d n = = (4~Re) {f{~ p~HdH + 2.5fi~.el" odin H + 2.2)dH}
(II-2a)
where pz describes the charge distribution in the laminar sublayer and pt in the turbulent core. If rv,K >~ 1 as is the case in a very badly conducting liquid, Oz =
-
-
(eK2~/4~) ch Kx/ch ~cd
according to (I-4a) and e~z~ sh ~d(1 12v/v,d) 4a Kd(1 12v/v,d)ch Kd -
Ot
-
-
-
according to (I-39a). If ,v,K ~ 1 like in most liquids, pl and pt are both given b y --(eK2~/4~) chKx/ch Kd. For a turbulent flow in a round tube:
2z~ Fr
2~tRv fv'R'v ( L pCP l
/5'=--Qjoovrdr= F 4 =
vH ) v,R all=
.p,H 1 C-R
If cv,K >~ 1, m = -- (~K2¢/4~t) Io(Kr)/Io(KR) according to (I-4b) and ex2~ 211(1 12v/v,R) 4~t ~R(1 -- 12v/v,R)Io(KR) -
-
according to (I-39b). If rv,K ~ 1, pt and Ot are both given b y --(eK2¢/4~) Io(Kr)/Io(~:R). Equations (II-2a) and (II-2b) may be worked out exactly, but the formulae obtained are not very convenient and must be approximated to get practicable expressions. The same results can be reached, however, in a simpler
CHARGE TRANSPORTED BY FLOW
1029
m a n n e r b y substituting formula (1-4), i.e. --(eKe~/4~)exp( - Ky) for pz if ,v.K >~ 1 and for pz and pt if ,v.K ~ 1 into the equations (II-2a) and (II-2b) instead of using the more complicated expressions. On the other h a n d simple expressions can be found if KRh ~ 1 as will be shown. Thus we are able to calculate ~ in a turbulent flow in the cases KRh >~ 1 and KR~ ~ ! respectively. In the case KRh m 1, however, calculation of ~ in a turbulent flow is much less simple and has not been performed here; conclusions on the values of/3 for KR~ ~ 1 can be made from the plots of ~(Re) which have been made for KRh ~ 1 and KRh>~ 1 (fig. I I - 1 ) . For the calculation of/3 in a purely laminar flow, the exact expressions (I-4a) and (I-4b) are used for p to calculate ~ for all KRh values. We will calculate ~ as a functions of liquid flow for different tubes and liquids. To get general expressions the results must be given in terms of dimensionless quantities. This has been reached b y calculating ~/p0 as a function of Re with KR~ as parameter and for the two extreme cases TV,K~I and ,v.K >~ 1. For p0 is more or less arbitrarily chosen the value which takes in a purely laminar flow if KRh ---- o% i.e. p0 = -- 2e~/~R~. The results which are plotted in fig. II-1 are valid for flow between parallel walls as well as through round tubes. In practice and especially in our experiments, KRh will be nearly always large with respect to 1, i.e. the tube diameter is m u c h larger t h a n the double layer thickness. So far only very long tubes have been considered. In comparatively short tubes three other effects influence #, namely the relaxation effect, a possible saturation of wall current and the fact t h a t the velocity profile close behind tube entrance differs from t h a t far in the tube. The influence of these effects on ~ leads to a dependence of # on tube length L and hydraulic radius Rh, for which the qualitative plots fig. 11-2 (KRh ~ 1) and fig. -11-3 (~R~ >> 1) are given.
(II-Ib)
2,~/~R~, it is found for a respectively t h a t for a flow
pdPo ---- 1 -- th •d/Kd
(II-3a)
2. Mean charge density in the/low. If laminar flow b y workirig out between two parallel walls:
(II-la)
and
p0 =
--
and for a flow in a round tube:
~dpo = I2(KR)/Io(KR)
(II-3b)
Calculation shows t h a t for the same KRh, i.e. for R = ~ / 8 ~ d , the values of (II-3a) and (II-3b) never differ more t h a n a few percents. For KRh >~ 1 the expressions ( I I - 3 a ) and ( I I - 3 b ) reduce to:
pl/pO =
1
(II-4)
and for KRh ~ 1 to:
~z/p0 = ~(~Rh) 2
(II-5)
1030
A.A. BOUMANS
For a turbulent flow and with KRh >~ 1, the integrals in (II-2a) and (II-2b) m u s t be evaluated; this m a y be done b y using the formulas: f e -on In H d H = -- c -1 in H . e -on + c -1 E i ( - - cH) ; [ e - o n H in H d H = - - c-Z(H In H + c -1 in H + c-z) e -oH + c-2 Ei (-- cH) ; f e -oH d H = - - c -1 e-oH; f H e - a I d H = - - c -1 e -oH (c-1 + H); f H~ e - c n d H = - - c -1 e-cH(2/c 2 + 2H/c + Ha); f ln H d H = H (in H - - 1);
In H d H = {Hg'(ln H -- ½);
fH
where c is a constant. If, in accordance with I-5, d in the turbulent core is replaced b y Rh/2, the following results are found. W i t h p~ = pw e x p ( - - Ky) and pt = pw e x p ( - - Ky)"
f2'mHdM=--pw(v*~2[(l+12~V~exp(--12K~')--1~(II-6a) \ KV I
V,
U,
/
r +Ei(--12KV)--Ei(--~)+(ln v*Rh +2.2)exp(--~)] v, 2v Pdin H + 2.2)dH = -- p w - -
,d 12
-- (in 12 +
2.2) exp
-- 12
KV
+ ?.),
(II-7a,
and with pt = pro: pt(In'H+2.2)dH=pm #12
In
~
.
+0.5
--12(111 12+1.2)
(II-8a)
'V
For the integrals of (II-2b) is found: with
pz = pw e x p ( - - Ky) and pt = pw exp(-- Ky)"
f:~p,H1
:-~)dH:--pw(v~v)2[{(,--~)+12?**(1
KR
v--~/] X
X exp ( - - 1 2 _: : ) - - ( 1 - - ~ ) ]
r
pt(inH+2.2)
(II-6b)
dH=
1
dis = - - p w
X exp
--12
v,E
Kv +
--
{
(In 1 2 + 2 . 2 )
1 --
Ei
(
1
12v
v,R
-- 12
1 (lnV*R+3.2)exp(--KR)~ KR v
1 ) KR
--
--Ei(--~R)
1 1 --K-R-J
X
-(II-7b)
1031
CHARGE T R A N S P O R T E D B Y F L O W
and with pt = pro:
f~¢m~ pt(ln H + 2.2) (1 -- ,,H/v,R) dH = =pm{(v,R/2v) (ln(v,R/v) +0.7) -- 12(11112+ 1.2) + (72v/v,R)(ln 12+ 1.7)) (II-Sb) Since v,Rh/v = Re%/~/4%/2 is always > 120 and KRh is supposed >> 1, the equations (II-6a) and (II-6b) reduce to the same expression, as is also true of (II-7a) and (II-7b) and for (II-Sa) and (II-Sb). Thus the following expressions are obtained for (II-2a) and (II-2b). For ,v,K ~ 1" fit = 4(Iz + 2.512)/Re
]
I
and for ~-V,K >~ 1 : fit =
4(11 + 2.513)/Re
(11-2)
where I1 I2
=
--
I
p_w\Kv/ L\
1 +
1 2 Kv
exp
v,/
pw v, { _ ( l n l 2 + 2 . 2 ) e x p ( _ KV 13 =
v,Rh
p ~ - 2v
--
12--
v,
--
1
;
(11-6)
12ff__v_V)+Ei(_ 12 K v ) } ; V,
v,
/
v,R~
In - -
(11-7) (II-8)
v
These equations are valid for a flow between two parallel walls as well as for a flow in a round tube. With pw = -- e'~2~/4~ = ~(KRn)2Po and, according to (I-39a) and (I-39b) :
pro= Pm
eK2¢ sh b 4z~b ch a
pw
exp b exp(b ~ pw b exp a a
-
-
exp(-- 12~:v/v,) .
a) =
pw KR
~,¢2¢11(#). 1/~- exp t~ exp(fl -- =) z~I0(~/ ~ 2pwV-~ ~ p ~ ~ 2pw
--
2 exp(-- 12Kv/v,) =
pw
KR
'
from which it follows that for parallel walls and for round tubes: pm = pw 2 exp(-- 12~/v,)/KRh, the equations (II-2) give with (II-6), (II-7) and (II-8) for the mean charge density in a turbulent flow the following expressions: In a well conducting turbulent liquid (rv,x ~ 1):
~t
p0
Re2 64
1 --
1+1.64 KRh
exp - - 6 7 . 9 R e % / 2 / +
+ 1 4 . 1 7 _ ,~R~ Ei ( - - 6 7 . 9 ReA/2KRh)} /xeVz
(II-9)
I032
A.A. BOUMANS
and in a badly conducting turbulent liquid (zv,K ~ 1): p0
64
El_{,
Re/a)14.1Re~/~l ~R~ ~ "I ( ~R ~ ) ] (II-10) J exp --67.9 Rex/jl
The :function E i ( " x ) i s tabulated in ref. 7, where also approximations for very small or very large x are given. Plotting (I1-9) and (II-I0), as is done in fig. 11-1, informs us t h a t there is no large difference between (11-9) and (II410). This is plausible, because the velocity profile of the turbulent core is rather flat, so it does not make much difference whether the charge in that_ part of the flow is distributed homogeneously or not. ,o3.
~/%
_~~,,=
I0
lOlL
o taae~ll not ~ t l d ~l > RICHL
for
¢~ometer
~R
tu~
tO 2
1~3 Reo.lt..' t(~4
1~,
,(~6
,;7
t~8
1()9
Re
Fig. II-1. #/OOa s a function of Re for different values of KRh. The curves are valid for two parallel walls as well as for a round tube. As long as the diffuse electrical layer lies fully within the laminar sublayer (KRh >~ R e ~ / ~ / 4 8 ~ 2 , i.e. up to a certain Re) (11-9) and (II-10) reduce to: jSdpo = Re,t/64
(II-11)
while for large Re (II-9) and (II-10) give: lim (~dP0) = ¼KR~
(II- 12)
Re-+oo
The last mentioned result corresponds with the fact that for Re--->oo the laminar sublayer thickness reduces to zero and the charge in the turbulent core equals the total charge in the flow which is equal to the inverse of the wall charge; per length unit this charge is 2z~Rq. For Re ---, oo charge distribution always becomes homogeneous, so # = p = 2z~Rq/zcR 2 = s~:~/2z~R = = poKR/4. These results for mean charge density in a turbulent flow are
CHARGE TRANSPORTED BY FLOW
103~
valid only if KR~ >~ 1 and if one of the two extreme situations exists, i.e. charge distribution in the turbulent core equals laminar distribution, in which case (11-9) is valid, or charge distribution in the turbulent core is homogeneous, in which case (11-10) holds. Because (11-9) and (II-10) do not differ much, charge distributions between these extremes would not give much different results. If, however, the condition KRh >~ ! is not fulfilled, then the derivation given above is no longer valid. Inspection of fig. II-1 would make us expect that for •Rh ~ I ~t/po will have values of the order of ! to 2 times ~dP0. The exact values could be calculated by using the equations p = --(s~2~/4~) ch ~x/ch Kd
(I-4a)
p = --(,~2~/4~) Io(Kr)/Io(,,R)
(I-4b)
or
instead of the approximation p = -- (eK2¢/4~)/exp(--Ky)
(1-4)
but the result is not going to be striking. For KRa ~ ! diffuse layer thickness is so large with respect to tube diameter that charge distribution is always homogeneous, also in a laminar flow; in that case ~ t = ~ = p w = - - e K 2 ( / 4 g , or, in accordance with (11-5): ~t/po = ~(KRh) ~"
(11-12)
C o n c l u s i o n . For KRa ~ 1 always ~/po = (KRa)2/8. For KRa ~ 1 and purely laminar flow, /3/p0 is given b y (II- 3a) or (II-3b) which are actually valid for all values of KRa. For KRa >~ 1 the following cases m a y be discerned. Purely laminar flow: ~/p0 = 1. Turbulent flow: ~/po is given b y (II-9) or (II-10) (or some value between); as long as the diffuse layer is fully within the laminar sublayer (I 2 w / v . >~ 1) we have f/po = Re 2/64; if the larger part of the diffuse layer extends into the turbulent core (12w/v, ~ 1) we have ~/p0 = KRa/4. All these results have been plotted in fig. II-I. 3. Ratio o/pressure P and streaming potential V. Although the measurement of streaming current is emphasized here, it is interesting to see in what circumstances the streaming potential behaves such that P / V is expected to be given b y the well known formula of Helmholtz-Smoluchowski (ref. 5, p. 83 and ref. 6, p. 426). The streaming potential is the potential difference built up between the ends of the tube, if a stationary state occurs between streaming current and back current through the liquid resistance. Without regard to the higher conductivity near the wall, and if the
1034
A.A. BOUMANS
resistance of the tube material is infinite, according to Ohm's law #~=crV/L, or, with Re~ = - - 8 R ~ P / ~ L : V / P = -- 8 R ~ / ~ R e ~ . If p = poReA/64 (laminar flow with KRh >~ 1 or turbulent flow where the diffuse layer is fully within the laminar sublayer) the formula of HelmholtzSmoluchowski is found: vIP =
--R~polS~
= ~14~
=
.~1~.
That V / P is independent of tube shape is in accordance with Smoluchowski's theory (ref. 4, p. 380). If ~Rh ~ 1 or if Re is so large that the diffuse layer extends into the turbulent core of the flow, then V / P will be smaller. If the ends of the tube are connected through a resistance V/P will also be smaller.
4. Tubes o/[inite length. H y d r o n a m i c a l r e ' h a r k s . The transition from purely laminar into turbulent flow happens, according to fig. 1-7 at Re = Recr,t = 2300; this, however, is a minimum value; in tubes with very little entrance perturbation Recrt~ m a y be a 10 times higher (ref. 2, p. 572-575 and ref. 3, p. 11 !-118). The transition begins far into the tube, at large l/R, With increasing velocity the transition point moves back to the entrance; this means that close to the entrance Re/r** is larger than far inside the tube; Re,m is a decreasing function of l/R. Especially for relatively short, wide tubes Recr,t m a y be much larger than 2300 (ref. 2, p. 67-68). At the entrance (l = 0) of the tube the velocity profile is flat; with increasing l the profile (I-15a), (I-15b) or (1-18) and (1-19) starts from the wall, the flat part in t h e middle getting smaller, until at l/R > 0.06 Re (for a purely laminar flow) and l/R > 50-200 (for a turbulent flow) (ref. 1, p. 362) the profile is formed completely• Moreover, at a small distance beyond the transition from a wider into a narrower tube with sharp edge a contraction of the flow occurs, the value of which is dependent on the diameter ratio of the tubes and on Re (re L 3, p. 83-87). In tubes of finite length the mean charge density p in the liquid differs from the /5 in infinitely long tubes because: 1. the velocity profile close to tube entrance differs from that in infinitely long tubes; 2. wall current density m a y exceed the value of condition (I-14); 3. in the time of relaxation which is also a characteristic time for double !ayer formation the liquid m a y pass a considerable part of tube length. These factors will be looked into one b y one. I n f l u e n c e of t h e v e l o c i t y p r o f i l e . In the extreme case of a very short, wide tube, the velocity profile is v = v0 and, between parallel walls: 2B q eK~ = - - q 2Bd d 4~d th Kd
1035
CHARGE TRANSPORTED BY FLOW
for round tubes: =--q
2=R =R 2 --
2q R --
eke I0(KR) 2=R II(KR)
Now for KRh < 1 and for parallel walls as well as for round tubes:
(11-14) as in the case of infinitely long tubes, while for KRh >> 1 between parallel walls: = - - sK¢/4azd
(II-15a)
= - - s,~/2=R
(II- 15b)
and in round tubes:
The flat profile v = vo at tube entrance is found only if the opening is abrupt, so this also describes the influence of tube opening if it has a sharp edge. In an extremely short tube (a diaphragm) not only the velocity profile, but also the charge distribution will differ from their shape in a long tube as a result of an edge effect. Field strength and therefore q and ~ have higher values than within a long tube. So in a more exact calculation it must be taken into account whether the flat velocity profile is perceptible farther into the tube than the influence of tube edge on the electric field. The above calculation is but a rough estimation. I n f l u e n c e of w a l l c u r r e n t . According to equation (1-14) charge distribution is not influenced b y wall current if the wall current density satisfies the equation J ~ aK$. We shall now examine under what circumstances the streaming current can b~ fully supplied b y the wall without influencing charge distribution. Therefore J will be compared with the mean density of the pure streaming current ]str. This is for KRh >~ l, if L is tube length: Re~
2e~ _ 8R~P
ra~ P
Combination with J ~ aK~ (I-14) yields: J/ds,r < K~? L
r
P
(11-16)
If the streaming current is fully supplied b y the wall current, then: 2 B L J = 2BdJstr or 2 ~ R L J = ~ R % J s t r . So the streaming current m a y be fully supplied b y the wall current while the distribution of charge in the liquid remains just the same as in the case of the pure streaming current, if: R~ 2L
- -
m/ L r P
4---
(11-17)
1036
A.A.
BOUMANS
This means that with a given tube and liquid the pressure and so the liquid flow must remain under a certain value. If a number of tubes of different length are taken but all with the same dimater then, at constant Re, the condition (11-17) will be fulfilled only if the tube length exceeds a certain value. Shorter tubes will have a larger wall current density, thus a double layer which is influenced b y it; the streaming current will be smaller than in the longer tubes and will therefore show a saturation effect. If there is saturation, its influence will be roughly given b y I ~ R h L , so for constant Re: I ~'~ Q
RhL Q
RhL R n R e ~'~ L
(11-18)
I n f l u e n c e of r e l a x a t i o n . The separation of charge at tube entrance takes place in the characteristic time 7. If within this time the liquid has passed a considerable fraction of tube length, t~ae double layer in the tube will not be fully formed and the streaming current will be smaller than in a very long tube. The length l passed b y the liquid in t i m e , is: l r e t = z ' g = (e/4~a)(vRe/2Rh), so fi will then be of the order /Smax{1 -- exp(-- L/lrez)) m #maxL/lre* ' ~ L R h
(11-19)
with constant Re. If the diffuse layer lies far within the laminar sublayer, the length lrel will be governed b y the much smaller velocity v close to the wall, therefore lrez will in that case be smaller. 5. ~ as a [unction o/ tube d i m e n s i o n s L and R~. B y summarizing the
foregoing results it will be possible to show qualitatively how, for a given liquid and wall material, at constant Re, ~ depends on tube length L and hydraulic radius Rh. In very long tubes, i.e. if L / R ~ is very large then for constant Re, one has fi "~ po ~'~ 1/R~ if KRn >~ 1 since ~l = po = - - 2eCIz~R~, (11-4)
and
fit = poRe2~64 = - - (2e~/nR~)Re~/64 ( I I - 1 1 )
while ~ is independent of L and Rh if KRh ~ 1 because ~z - - ~, = po(,~Rh)~/8
= ~K2C/4~ (11-5) and (II-13).
In extremely short tubes, i.e. if L / R h is very small, then the fact that the velocity profile is rather flat, results in p ,~ 1/Rn if KRh >~ 1 since fit = pt = - - eKC/4xd ( I I - 1 5 a )
o r ~z = ~t - - - sK~/2z~R (II-15b)
and ~ constant if KRj, < 1 because ,ol = /St = -- e~¢2~'/4:~ (11-14)
1037
CHARGE TRANSPORTED B Y FLOW
Moreover, in tubes with comparatively small internal surface LRh saturation of wall current will cause an additional dependence on tube dimensions, roughly given by /5 , ~ L (II-18), while in short tubes relaxation causes an additional dependence /5 ~-~ LRh. (II-19). These results may be taken together into the following tables. From these tables the figures II-2 and II-3 can be constructed; these figures show qualitatively the expected shape of curves for constant /5 in the (L, Rh)-
plane. ~9
kk~gRh~
Rh
~l~log L/R h
F i g . I I - 2 . Curves of c o n s t a n t /5 in t h e (Rn, L ) - p l a n e f o r KRn ~ 1, at c o n s t a n t Re.
Fig. II-3. Curves of constant /5 in the
(Rn, L)-plane for KRh>~1 at constant Re.
Another consequence of the saturation effect and of relaxation is, that it tends to keep I as a.function of Re constant, therefore/5 ---=I/Q,~I/Q,~I/Re; thus it might happen that /5 rises less than is calculated for infinitely long tubes, or falls even, with increasing Re. KRn~ 1 t
LRh small
[
LRn large
KRh>~I [
LRn small 1
L/Rh small
/5 ~ L. LRh L2Rh
/5 independent of L and Rh
L/Rh small
/5 ~
L/l?t, large
/5 ~ L . LRn L2Rh
/5 independent of L and Ri,
largeL/Rn
P~~ - ~ n1
Rh L~
]
LRn large
.L.LRn
/5 ~ l/Rh
.L.LRn
/5 ~ l / R n 2
~LZ/Rh Received 29-8-57.
REFERENCES 1) S c h l i c h t i n g , H., Grenzschichttheorie, Braun, Karlsruhe (1950). 2) S c h i l l e r , L., Phys. Z. 26 (1925), 64-69 and 566-595. 3) Wi e n, W. and H ar InS, F., Handbuch der Experimentalphysi k I V 4, Akad. Verlagsgesellschaft, Leipzig (1932). 4) G r a e t z , L., Handbuch der Elektrizit~it und des Magnetismus II, Barth, Leipzig (1921). 5) Symposiuna Grenslaagverschijnselen, Kon. V'laamse Acad. Wetensch. (1947). 6) R u t g e r s , A. J., Physische scheikunde II, Noordhoff, Groningen (1947). 7) FI i.igge, W., Four-place tables of transcendental functions, Pergamon press, London (1954).
Physica X X l I I
1957
1027-1037
STREAMING CURRENTS IN T U R B U L E N T FLOWS AND METAL CAPILLARIES II. T H E O R Y
(2). C H A R G E T R A N S P O R T E D
BY THE FLOW OF LIQUID
by A. A. BOUMANS Stichting veer fundamenteel onderzoek der materie, Utrecht Synopsis The m e a n charge d e n s i t y / ] which appears if a liquid flows t h r o u g h a t u b e is calculated. The following dimensionless n u m b e r s are used. 1°. KR~, i.e. the ratio of the t u b e ' s hydraulic radius R~ and the diffuse layer thickness 1/K. 2 °. 12vK/v,, i.e. the ratio of the l a m i n a r sublayer thickness 12v/v, and the diffuse layer thickness l/K, where v is the k i n e m a t i c viscosity of the liquid and v , the friction v e l o c i t y as defined in 1.5. 3 °. ~ , K , i.e. the ratio of the distance t r a v e l e d with velocity v, in the relaxation time r and the diffuse layer thickness 1/K. 4 °. /~/p0, i.e. the ratio of the m e a n charge density/~ in the liquid and the m e a n charge d e n s i t y p0 = -- 2¢~/nR~ which appears in a l a m i n a r flow if KRh >~ 1, where ¢ is the dielectric c o n s t a n t of the liquid and ~ the wall potential. 5 °. R e y n o l d s n u m b e r Re = 2~Rh/v and friction n u m b e r Jl = - - ( 4 R h / s ~ 2 ) d p / d l where s is the specific density of the liquid, ~ the m e a n velocity across the tube cross section and dp/dl the pressure gradient along the tube axis. H y d r o d y n a m i c s teaches t h a t 2 is a function of Re, which is d r a w n in fig. 1-7. The following relations are deduced for KRh>~ 1. I n a l a m i n a r flow ~/po = 1. I n a t u r b u l e n t flow, as long as 12vK/v, >~ 1, ~/po = Re 2/64, b u t if 12vK/v, ~ 1, ~/po=KRh/4. The complete function is given in fig. II-1, which illustrates also the situation KRn ~ 1. I t is p r o v e d t h a t ~ is only slightly d e p e n d e n t on the shape of the charge distribution in the t u r b u l e n t core of t h e ' f l o w (i.e. on the value of ~v,K). In tubes of finite lenth L,/3 is influenced by the relaxation effect, b y a possible saturation of wall current and b y t h e fact t h a t the velocity profile close behind the t u b e e n t r a n c e differs f r o m t h a t farther in the tube. The consequences for KRI~>~ 1, illustrated in fig. 11-3, are the following. If LRh is small : ~ ~ L 2 for small L/Rh and/5 ~ L~/R~ for large L/Rh. If LRn is large: # ~ 1/Rh for small L/Rh and # ~ 1/R~ for large L/Rn. Similar relations are deduced for KRh ~ 1 and illustrated in fig. 1I-2. All results are valid for flow through round tubes as well as between parallel walls.
1. Preliminary remarks. D i f f e r e n t f r o m t h e m o r e u s u a l p r o c e d u r e , mean charge density/], potential
i n s t e a d of t h e s t r e a m i n g
V, is u s e d as c h a r a c t e r i s t i c
charge transported
quantity
the current I or the streaming describing
the amount
of
b y t h e f l o w . T h e r e a s o n is t h a t i t g i v e s a b e t t e r c o n n e c --
1027
--
1028
A . A . BOUMANS
tion with our experimental technique, and theoretically simpler expressions, /5 being independent of R.e for a laminar flow. The mean charge density in the flow is/5 = I/O. For two parallel walls:
I = 2Bfoa pvdx; Q = 2Bfod vdx
(II-la)
and for a round tube :
I = 2~tf~ pvrdr; Q --- 2z~foR vrdr
(II-lb)
For a purely laminar flow v is given b y (I-15a) of (I-15b) and by (I-4a) or (I-4b). For a turbulent flow between two parallel walls
/5, = (2B/Q) foa pvdx = (2By~Q) f~.a/. #l, d n = = (4~Re) {f{~ p~HdH + 2.5fi~.el" odin H + 2.2)dH}
(II-2a)
where pz describes the charge distribution in the laminar sublayer and pt in the turbulent core. If rv,K >~ 1 as is the case in a very badly conducting liquid, Oz =
-
-
(eK2~/4~) ch Kx/ch ~cd
according to (I-4a) and e~z~ sh ~d(1 12v/v,d) 4a Kd(1 12v/v,d)ch Kd -
Ot
-
-
-
according to (I-39a). If ,v,K ~ 1 like in most liquids, pl and pt are both given b y --(eK2~/4~) chKx/ch Kd. For a turbulent flow in a round tube:
2z~ Fr
2~tRv fv'R'v ( L pCP l
/5'=--Qjoovrdr= F 4 =
vH ) v,R all=
.p,H 1 C-R
If cv,K >~ 1, m = -- (~K2¢/4~t) Io(Kr)/Io(KR) according to (I-4b) and ex2~ 211(1 12v/v,R) 4~t ~R(1 -- 12v/v,R)Io(KR) -
-
according to (I-39b). If rv,K ~ 1, pt and Ot are both given b y --(eK2¢/4~) Io(Kr)/Io(~:R). Equations (II-2a) and (II-2b) may be worked out exactly, but the formulae obtained are not very convenient and must be approximated to get practicable expressions. The same results can be reached, however, in a simpler
CHARGE TRANSPORTED BY FLOW
1029
m a n n e r b y substituting formula (1-4), i.e. --(eKe~/4~)exp( - Ky) for pz if ,v.K >~ 1 and for pz and pt if ,v.K ~ 1 into the equations (II-2a) and (II-2b) instead of using the more complicated expressions. On the other h a n d simple expressions can be found if KRh ~ 1 as will be shown. Thus we are able to calculate ~ in a turbulent flow in the cases KRh >~ 1 and KR~ ~ ! respectively. In the case KRh m 1, however, calculation of ~ in a turbulent flow is much less simple and has not been performed here; conclusions on the values of/3 for KR~ ~ 1 can be made from the plots of ~(Re) which have been made for KRh ~ 1 and KRh>~ 1 (fig. I I - 1 ) . For the calculation of/3 in a purely laminar flow, the exact expressions (I-4a) and (I-4b) are used for p to calculate ~ for all KRh values. We will calculate ~ as a functions of liquid flow for different tubes and liquids. To get general expressions the results must be given in terms of dimensionless quantities. This has been reached b y calculating ~/p0 as a function of Re with KR~ as parameter and for the two extreme cases TV,K~I and ,v.K >~ 1. For p0 is more or less arbitrarily chosen the value which takes in a purely laminar flow if KRh ---- o% i.e. p0 = -- 2e~/~R~. The results which are plotted in fig. II-1 are valid for flow between parallel walls as well as through round tubes. In practice and especially in our experiments, KRh will be nearly always large with respect to 1, i.e. the tube diameter is m u c h larger t h a n the double layer thickness. So far only very long tubes have been considered. In comparatively short tubes three other effects influence #, namely the relaxation effect, a possible saturation of wall current and the fact t h a t the velocity profile close behind tube entrance differs from t h a t far in the tube. The influence of these effects on ~ leads to a dependence of # on tube length L and hydraulic radius Rh, for which the qualitative plots fig. 11-2 (KRh ~ 1) and fig. -11-3 (~R~ >> 1) are given.
(II-Ib)
2,~/~R~, it is found for a respectively t h a t for a flow
pdPo ---- 1 -- th •d/Kd
(II-3a)
2. Mean charge density in the/low. If laminar flow b y workirig out between two parallel walls:
(II-la)
and
p0 =
--
and for a flow in a round tube:
~dpo = I2(KR)/Io(KR)
(II-3b)
Calculation shows t h a t for the same KRh, i.e. for R = ~ / 8 ~ d , the values of (II-3a) and (II-3b) never differ more t h a n a few percents. For KRh >~ 1 the expressions ( I I - 3 a ) and ( I I - 3 b ) reduce to:
pl/pO =
1
(II-4)
and for KRh ~ 1 to:
~z/p0 = ~(~Rh) 2
(II-5)
1030
A.A. BOUMANS
For a turbulent flow and with KRh >~ 1, the integrals in (II-2a) and (II-2b) m u s t be evaluated; this m a y be done b y using the formulas: f e -on In H d H = -- c -1 in H . e -on + c -1 E i ( - - cH) ; [ e - o n H in H d H = - - c-Z(H In H + c -1 in H + c-z) e -oH + c-2 Ei (-- cH) ; f e -oH d H = - - c -1 e-oH; f H e - a I d H = - - c -1 e -oH (c-1 + H); f H~ e - c n d H = - - c -1 e-cH(2/c 2 + 2H/c + Ha); f ln H d H = H (in H - - 1);
In H d H = {Hg'(ln H -- ½);
fH
where c is a constant. If, in accordance with I-5, d in the turbulent core is replaced b y Rh/2, the following results are found. W i t h p~ = pw e x p ( - - Ky) and pt = pw e x p ( - - Ky)"
f2'mHdM=--pw(v*~2[(l+12~V~exp(--12K~')--1~(II-6a) \ KV I
V,
U,
/
r +Ei(--12KV)--Ei(--~)+(ln v*Rh +2.2)exp(--~)] v, 2v Pdin H + 2.2)dH = -- p w - -
,d 12
-- (in 12 +
2.2) exp
-- 12
KV
+ ?.),
(II-7a,
and with pt = pro: pt(In'H+2.2)dH=pm #12
In
~
.
+0.5
--12(111 12+1.2)
(II-8a)
'V
For the integrals of (II-2b) is found: with
pz = pw e x p ( - - Ky) and pt = pw exp(-- Ky)"
f:~p,H1
:-~)dH:--pw(v~v)2[{(,--~)+12?**(1
KR
v--~/] X
X exp ( - - 1 2 _: : ) - - ( 1 - - ~ ) ]
r
pt(inH+2.2)
(II-6b)
dH=
1
dis = - - p w
X exp
--12
v,E
Kv +
--
{
(In 1 2 + 2 . 2 )
1 --
Ei
(
1
12v
v,R
-- 12
1 (lnV*R+3.2)exp(--KR)~ KR v
1 ) KR
--
--Ei(--~R)
1 1 --K-R-J
X
-(II-7b)
1031
CHARGE T R A N S P O R T E D B Y F L O W
and with pt = pro:
f~¢m~ pt(ln H + 2.2) (1 -- ,,H/v,R) dH = =pm{(v,R/2v) (ln(v,R/v) +0.7) -- 12(11112+ 1.2) + (72v/v,R)(ln 12+ 1.7)) (II-Sb) Since v,Rh/v = Re%/~/4%/2 is always > 120 and KRh is supposed >> 1, the equations (II-6a) and (II-6b) reduce to the same expression, as is also true of (II-7a) and (II-7b) and for (II-Sa) and (II-Sb). Thus the following expressions are obtained for (II-2a) and (II-2b). For ,v,K ~ 1" fit = 4(Iz + 2.512)/Re
]
I
and for ~-V,K >~ 1 : fit =
4(11 + 2.513)/Re
(11-2)
where I1 I2
=
--
I
p_w\Kv/ L\
1 +
1 2 Kv
exp
v,/
pw v, { _ ( l n l 2 + 2 . 2 ) e x p ( _ KV 13 =
v,Rh
p ~ - 2v
--
12--
v,
--
1
;
(11-6)
12ff__v_V)+Ei(_ 12 K v ) } ; V,
v,
/
v,R~
In - -
(11-7) (II-8)
v
These equations are valid for a flow between two parallel walls as well as for a flow in a round tube. With pw = -- e'~2~/4~ = ~(KRn)2Po and, according to (I-39a) and (I-39b) :
pro= Pm
eK2¢ sh b 4z~b ch a
pw
exp b exp(b ~ pw b exp a a
-
-
exp(-- 12~:v/v,) .
a) =
pw KR
~,¢2¢11(#). 1/~- exp t~ exp(fl -- =) z~I0(~/ ~ 2pwV-~ ~ p ~ ~ 2pw
--
2 exp(-- 12Kv/v,) =
pw
KR
'
from which it follows that for parallel walls and for round tubes: pm = pw 2 exp(-- 12~/v,)/KRh, the equations (II-2) give with (II-6), (II-7) and (II-8) for the mean charge density in a turbulent flow the following expressions: In a well conducting turbulent liquid (rv,x ~ 1):
~t
p0
Re2 64
1 --
1+1.64 KRh
exp - - 6 7 . 9 R e % / 2 / +
+ 1 4 . 1 7 _ ,~R~ Ei ( - - 6 7 . 9 ReA/2KRh)} /xeVz
(II-9)
I032
A.A. BOUMANS
and in a badly conducting turbulent liquid (zv,K ~ 1): p0
64
El_{,
Re/a)14.1Re~/~l ~R~ ~ "I ( ~R ~ ) ] (II-10) J exp --67.9 Rex/jl
The :function E i ( " x ) i s tabulated in ref. 7, where also approximations for very small or very large x are given. Plotting (I1-9) and (II-I0), as is done in fig. 11-1, informs us t h a t there is no large difference between (11-9) and (II410). This is plausible, because the velocity profile of the turbulent core is rather flat, so it does not make much difference whether the charge in that_ part of the flow is distributed homogeneously or not. ,o3.
~/%
_~~,,=
I0
lOlL
o taae~ll not ~ t l d ~l > RICHL
for
¢~ometer
~R
tu~
tO 2
1~3 Reo.lt..' t(~4
1~,
,(~6
,;7
t~8
1()9
Re
Fig. II-1. #/OOa s a function of Re for different values of KRh. The curves are valid for two parallel walls as well as for a round tube. As long as the diffuse electrical layer lies fully within the laminar sublayer (KRh >~ R e ~ / ~ / 4 8 ~ 2 , i.e. up to a certain Re) (11-9) and (II-10) reduce to: jSdpo = Re,t/64
(II-11)
while for large Re (II-9) and (II-10) give: lim (~dP0) = ¼KR~
(II- 12)
Re-+oo
The last mentioned result corresponds with the fact that for Re--->oo the laminar sublayer thickness reduces to zero and the charge in the turbulent core equals the total charge in the flow which is equal to the inverse of the wall charge; per length unit this charge is 2z~Rq. For Re ---, oo charge distribution always becomes homogeneous, so # = p = 2z~Rq/zcR 2 = s~:~/2z~R = = poKR/4. These results for mean charge density in a turbulent flow are
CHARGE TRANSPORTED BY FLOW
103~
valid only if KR~ >~ 1 and if one of the two extreme situations exists, i.e. charge distribution in the turbulent core equals laminar distribution, in which case (11-9) is valid, or charge distribution in the turbulent core is homogeneous, in which case (11-10) holds. Because (11-9) and (II-10) do not differ much, charge distributions between these extremes would not give much different results. If, however, the condition KRh >~ ! is not fulfilled, then the derivation given above is no longer valid. Inspection of fig. II-1 would make us expect that for •Rh ~ I ~t/po will have values of the order of ! to 2 times ~dP0. The exact values could be calculated by using the equations p = --(s~2~/4~) ch ~x/ch Kd
(I-4a)
p = --(,~2~/4~) Io(Kr)/Io(,,R)
(I-4b)
or
instead of the approximation p = -- (eK2¢/4~)/exp(--Ky)
(1-4)
but the result is not going to be striking. For KRa ~ ! diffuse layer thickness is so large with respect to tube diameter that charge distribution is always homogeneous, also in a laminar flow; in that case ~ t = ~ = p w = - - e K 2 ( / 4 g , or, in accordance with (11-5): ~t/po = ~(KRh) ~"
(11-12)
C o n c l u s i o n . For KRa ~ 1 always ~/po = (KRa)2/8. For KRa ~ 1 and purely laminar flow, /3/p0 is given b y (II- 3a) or (II-3b) which are actually valid for all values of KRa. For KRa >~ 1 the following cases m a y be discerned. Purely laminar flow: ~/p0 = 1. Turbulent flow: ~/po is given b y (II-9) or (II-10) (or some value between); as long as the diffuse layer is fully within the laminar sublayer (I 2 w / v . >~ 1) we have f/po = Re 2/64; if the larger part of the diffuse layer extends into the turbulent core (12w/v, ~ 1) we have ~/p0 = KRa/4. All these results have been plotted in fig. II-I. 3. Ratio o/pressure P and streaming potential V. Although the measurement of streaming current is emphasized here, it is interesting to see in what circumstances the streaming potential behaves such that P / V is expected to be given b y the well known formula of Helmholtz-Smoluchowski (ref. 5, p. 83 and ref. 6, p. 426). The streaming potential is the potential difference built up between the ends of the tube, if a stationary state occurs between streaming current and back current through the liquid resistance. Without regard to the higher conductivity near the wall, and if the
1034
A.A. BOUMANS
resistance of the tube material is infinite, according to Ohm's law #~=crV/L, or, with Re~ = - - 8 R ~ P / ~ L : V / P = -- 8 R ~ / ~ R e ~ . If p = poReA/64 (laminar flow with KRh >~ 1 or turbulent flow where the diffuse layer is fully within the laminar sublayer) the formula of HelmholtzSmoluchowski is found: vIP =
--R~polS~
= ~14~
=
.~1~.
That V / P is independent of tube shape is in accordance with Smoluchowski's theory (ref. 4, p. 380). If ~Rh ~ 1 or if Re is so large that the diffuse layer extends into the turbulent core of the flow, then V / P will be smaller. If the ends of the tube are connected through a resistance V/P will also be smaller.
4. Tubes o/[inite length. H y d r o n a m i c a l r e ' h a r k s . The transition from purely laminar into turbulent flow happens, according to fig. 1-7 at Re = Recr,t = 2300; this, however, is a minimum value; in tubes with very little entrance perturbation Recrt~ m a y be a 10 times higher (ref. 2, p. 572-575 and ref. 3, p. 11 !-118). The transition begins far into the tube, at large l/R, With increasing velocity the transition point moves back to the entrance; this means that close to the entrance Re/r** is larger than far inside the tube; Re,m is a decreasing function of l/R. Especially for relatively short, wide tubes Recr,t m a y be much larger than 2300 (ref. 2, p. 67-68). At the entrance (l = 0) of the tube the velocity profile is flat; with increasing l the profile (I-15a), (I-15b) or (1-18) and (1-19) starts from the wall, the flat part in t h e middle getting smaller, until at l/R > 0.06 Re (for a purely laminar flow) and l/R > 50-200 (for a turbulent flow) (ref. 1, p. 362) the profile is formed completely• Moreover, at a small distance beyond the transition from a wider into a narrower tube with sharp edge a contraction of the flow occurs, the value of which is dependent on the diameter ratio of the tubes and on Re (re L 3, p. 83-87). In tubes of finite length the mean charge density p in the liquid differs from the /5 in infinitely long tubes because: 1. the velocity profile close to tube entrance differs from that in infinitely long tubes; 2. wall current density m a y exceed the value of condition (I-14); 3. in the time of relaxation which is also a characteristic time for double !ayer formation the liquid m a y pass a considerable part of tube length. These factors will be looked into one b y one. I n f l u e n c e of t h e v e l o c i t y p r o f i l e . In the extreme case of a very short, wide tube, the velocity profile is v = v0 and, between parallel walls: 2B q eK~ = - - q 2Bd d 4~d th Kd
1035
CHARGE TRANSPORTED BY FLOW
for round tubes: =--q
2=R =R 2 --
2q R --
eke I0(KR) 2=R II(KR)
Now for KRh < 1 and for parallel walls as well as for round tubes:
(11-14) as in the case of infinitely long tubes, while for KRh >> 1 between parallel walls: = - - sK¢/4azd
(II-15a)
= - - s,~/2=R
(II- 15b)
and in round tubes:
The flat profile v = vo at tube entrance is found only if the opening is abrupt, so this also describes the influence of tube opening if it has a sharp edge. In an extremely short tube (a diaphragm) not only the velocity profile, but also the charge distribution will differ from their shape in a long tube as a result of an edge effect. Field strength and therefore q and ~ have higher values than within a long tube. So in a more exact calculation it must be taken into account whether the flat velocity profile is perceptible farther into the tube than the influence of tube edge on the electric field. The above calculation is but a rough estimation. I n f l u e n c e of w a l l c u r r e n t . According to equation (1-14) charge distribution is not influenced b y wall current if the wall current density satisfies the equation J ~ aK$. We shall now examine under what circumstances the streaming current can b~ fully supplied b y the wall without influencing charge distribution. Therefore J will be compared with the mean density of the pure streaming current ]str. This is for KRh >~ l, if L is tube length: Re~
2e~ _ 8R~P
ra~ P
Combination with J ~ aK~ (I-14) yields: J/ds,r < K~? L
r
P
(11-16)
If the streaming current is fully supplied b y the wall current, then: 2 B L J = 2BdJstr or 2 ~ R L J = ~ R % J s t r . So the streaming current m a y be fully supplied b y the wall current while the distribution of charge in the liquid remains just the same as in the case of the pure streaming current, if: R~ 2L
- -
m/ L r P
4---
(11-17)
1036
A.A.
BOUMANS
This means that with a given tube and liquid the pressure and so the liquid flow must remain under a certain value. If a number of tubes of different length are taken but all with the same dimater then, at constant Re, the condition (11-17) will be fulfilled only if the tube length exceeds a certain value. Shorter tubes will have a larger wall current density, thus a double layer which is influenced b y it; the streaming current will be smaller than in the longer tubes and will therefore show a saturation effect. If there is saturation, its influence will be roughly given b y I ~ R h L , so for constant Re: I ~'~ Q
RhL Q
RhL R n R e ~'~ L
(11-18)
I n f l u e n c e of r e l a x a t i o n . The separation of charge at tube entrance takes place in the characteristic time 7. If within this time the liquid has passed a considerable fraction of tube length, t~ae double layer in the tube will not be fully formed and the streaming current will be smaller than in a very long tube. The length l passed b y the liquid in t i m e , is: l r e t = z ' g = (e/4~a)(vRe/2Rh), so fi will then be of the order /Smax{1 -- exp(-- L/lrez)) m #maxL/lre* ' ~ L R h
(11-19)
with constant Re. If the diffuse layer lies far within the laminar sublayer, the length lrel will be governed b y the much smaller velocity v close to the wall, therefore lrez will in that case be smaller. 5. ~ as a [unction o/ tube d i m e n s i o n s L and R~. B y summarizing the
foregoing results it will be possible to show qualitatively how, for a given liquid and wall material, at constant Re, ~ depends on tube length L and hydraulic radius Rh. In very long tubes, i.e. if L / R ~ is very large then for constant Re, one has fi "~ po ~'~ 1/R~ if KRn >~ 1 since ~l = po = - - 2eCIz~R~, (11-4)
and
fit = poRe2~64 = - - (2e~/nR~)Re~/64 ( I I - 1 1 )
while ~ is independent of L and Rh if KRh ~ 1 because ~z - - ~, = po(,~Rh)~/8
= ~K2C/4~ (11-5) and (II-13).
In extremely short tubes, i.e. if L / R h is very small, then the fact that the velocity profile is rather flat, results in p ,~ 1/Rn if KRh >~ 1 since fit = pt = - - eKC/4xd ( I I - 1 5 a )
o r ~z = ~t - - - sK~/2z~R (II-15b)
and ~ constant if KRj, < 1 because ,ol = /St = -- e~¢2~'/4:~ (11-14)
1037
CHARGE TRANSPORTED B Y FLOW
Moreover, in tubes with comparatively small internal surface LRh saturation of wall current will cause an additional dependence on tube dimensions, roughly given by /5 , ~ L (II-18), while in short tubes relaxation causes an additional dependence /5 ~-~ LRh. (II-19). These results may be taken together into the following tables. From these tables the figures II-2 and II-3 can be constructed; these figures show qualitatively the expected shape of curves for constant /5 in the (L, Rh)-
plane. ~9
kk~gRh~
Rh
~l~log L/R h
F i g . I I - 2 . Curves of c o n s t a n t /5 in t h e (Rn, L ) - p l a n e f o r KRn ~ 1, at c o n s t a n t Re.
Fig. II-3. Curves of constant /5 in the
(Rn, L)-plane for KRh>~1 at constant Re.
Another consequence of the saturation effect and of relaxation is, that it tends to keep I as a.function of Re constant, therefore/5 ---=I/Q,~I/Q,~I/Re; thus it might happen that /5 rises less than is calculated for infinitely long tubes, or falls even, with increasing Re. KRn~ 1 t
LRh small
[
LRn large
KRh>~I [
LRn small 1
L/Rh small
/5 ~ L. LRh L2Rh
/5 independent of L and Rh
L/Rh small
/5 ~
L/l?t, large
/5 ~ L . LRn L2Rh
/5 independent of L and Ri,
largeL/Rn
P~~ - ~ n1
Rh L~
]
LRn large
.L.LRn
/5 ~ l/Rh
.L.LRn
/5 ~ l / R n 2
~LZ/Rh Received 29-8-57.
REFERENCES 1) S c h l i c h t i n g , H., Grenzschichttheorie, Braun, Karlsruhe (1950). 2) S c h i l l e r , L., Phys. Z. 26 (1925), 64-69 and 566-595. 3) Wi e n, W. and H ar InS, F., Handbuch der Experimentalphysi k I V 4, Akad. Verlagsgesellschaft, Leipzig (1932). 4) G r a e t z , L., Handbuch der Elektrizit~it und des Magnetismus II, Barth, Leipzig (1921). 5) Symposiuna Grenslaagverschijnselen, Kon. V'laamse Acad. Wetensch. (1947). 6) R u t g e r s , A. J., Physische scheikunde II, Noordhoff, Groningen (1947). 7) FI i.igge, W., Four-place tables of transcendental functions, Pergamon press, London (1954).