/;KUNDE
THE COLLEGE OF AERONAUTICS
CRANFIELD
THE RESPONSE TIME OF WIND TUNNEL
PRESSURE MEASURING SYSTEMS
ty
T H E C O L L E G E O F A E R O N A U T I C S C R A N F I E L D
The response time of wind tunnel p r e s s u r e measuring systems b y -G. M. Lilley, M.Sc. , D . I . C . , A . F . R . A e , S . , A. M.I. Mech. E. and D. Morton, D . C . A e . *
of the Department of Aerodynamics
SUMMARY
The time response of a wind tunnel p r e s s u r e measuring system, comprising a p r e s s u r e transducer of fixed volume and a length of
capillary tubing, is analysed and the results compared with experiments. It is shown that the approximate analysis of Kendall (1958), in which the friction losses at any given time are assumed equal to the steady state l o s s e s , has a wide range of validity, provided the L/R ratio for the capillary tube is large and the inlet and exit losses a r e included as equivalent lengths of the capillary tube.
*
Now a Research Engineer at Aircraft Research Association L t d . , Bedford.
Page Summary
Notation
Introduction 1 The sim.plified p r e s s u r e measurement 1
system
Assumptions 1 The approximate solution of Kendall (1958) 2
Unsteady flow in a capillary tube 4 Steady flow in a long capillary tube 8 The solution of the p r e s s u r e equation for 11
the unsteady flow in a r e s e r v o i r connected to a capillary tube
Apparatus used in the experiments 16
Description of tests 16 Discussion 16 Conclusions 17 Acknowledgement 18 References 18 Appendix 19 Table I 20 Figures
a, a speed of sound o c , c , c c o n s t a n t s 1 ' 2 ' 3 C, C p a r a m e t e r and i t s a v e r a g e value r e s p e c t i v e l y f( / R ) s e e equation 5.8 vrR^p^ k = -—r—— r e s p o n s e p a r a m e t e r 8 / J V L L length of c a p i l l a r y tube L effective length e ^ m r a t e of m a s s flow p p * Pa P . P 0 ^1 P = P = 0 p r e s s u r e p r e s s u r e at z = L a t m o s p h e r i c p r e s s u r e i n i t i a l and final p r e s s i p a - pe P^ „ 2 „ 2 Pi - Po 2 Pl Q r a t e of volume flow ( r , 0, z) c y l i n d r i c a l p o l a r c o - o r d i n a t e s R r a d i u s of c a p i l l a r y tube s L a p l a c e T r a n s f o r m o p e r a t o r t t i m e , a l s o non-dim.ensional t i m e t s t a b i l i s a t i o n t i m e s t t i m e constant ^^o
w axial velocity component
w axial velocity component at r = 0 w average velocity at any section
z distance along tube, also non-dimensional axial length (z/L) a see equation (7.12) 27rR^L V C X complex variable ju viscosity p density r wall shear stress
w
T see equation (7.4)
r value of r corresponding to t = 1
1. Introduction
In p r e s s u r e measuring systems for intermittent supersonic wind tunnels it is important that the response time of the system for
stabilization of the p r e s s u r e s is less than the overall available running time of the tunnel. Also in intermittent supersonic wind tunnel testing, it is often necessary to measure several p r e s s u r e s in a short t i m e . Due to the limited space available, in cases where the p r e s s u r e t r a n s -ducer has to be installed in the model itself, it is necessary to use a selector switch ( e . g . 'Scanivalve') to select the p r e s s u r e tappings in turn. To determine the speed at which such a selector switch may be operated and in order to gain useful information from such a system, a knowledge of the response tim.es of 'transducer - capillary tube'
combinations is required. The approximate calculation of the response time of a simplified system, comprising a capillary tube connected to a constant volume r e s e r v o i r , ha3 been given by Kendall (1958) and others. The purpose of the present note is to give a more exact t r e a t -m.ent of the response time for the same simplified system used by
Kendall, and to give the results of some experiments to check the theory. It should be noted that whereas Kendall considered the case of a sudden decrease in the applied p r e s s u r e to the transducer, for convenience, we have considered the case of a sudden increase in the applied p r e s s u r e to the transducer, although the theory is applicable to the case of a
sudden decrease in p r e s s u r e also.
2. The simplified p r e s s u r e rieasurement system
A capillary tube of diameter 2R and length L is connected to a r e s e r v o i r of volume V. The initial p r e s s u r e in the tube and r e s e r v o i r is p . At time t = 0 the p r e s s u r e at the tube inlet is raised to p^ . The problem is then to find the variation of p r e s s u r e with time in the r e s e r v o i r , and to find the stabilization time, which is the time taken for the p r e s s u r e in the r e s e r v o i r to r i s e within 1% of the p r e s s u r e difference (p, - p ).
1 o
3. Assumptions
1. The flow is laminar and the distribution of velocity across any section of the tube is independent of the distance along the tube. 2. The velocity in the r e s e r v o i r is z e r o .
This assumption is reasonable since in most practical systems the Reynolds number of the pipe flow will be less than 1000 (based on radius).
3 . Inlet and outlet l o s s e s in the c a p i l l a r y tube a r e n e g l e c t e d . 4 . The flow i s i s o t h e r m a l . (See footnote after equation 5.4). 5. The flow d o e s not involve s l i p .
4 . The a p p r o x i m a t e solution of Kendall (1958)
In addition to the a s s m n p t i o n s l i s t e d under p a r a g r a p h 3 Kendall m a k e s the a s s u m p t i o n that the p r e s s u r e s q u a r e d d i s t r i b u t i o n along the tubing i s l i n e a r with d i s t a n c e for all t i m e . T h u s the flow along the tube at any t i m e i s d e t e r m i n e d from the end p r e s s u r e d i f f e r e n c e . If p i s the v a r i a b l e p r e s s u r e at z = L then the r a t e of nnass flow m(t) in the tube i s given by (see p a r a g r a p h 6 below):
(4.1) w h e r e m(t) = 16/ia^ o R = tube r a d i u s L = tube length ju = v i s c o s i t y P: -P' L
a = speed of sound (for i s o t h e r n a a l flow) o
The r a t e of change of fluid m a s s in the r e s e r v o i r of volume V i s
d V p ^ V d £ ^ J ^ dp
dt dt a dt o
s i n c e p = P a^ in i s o t h e r m a l flow.
Now the r a t e of m a s s flow in the tube m u s t equal the r a t e of change of fluid m a s s in the r e s e r v o i r , s o that
dp - £ R 1 PI' ' P " (4 3)
dt ~ 16MV L ^^'-^^ with p = p when t = 0.
The solution of 4.3 i s
p - p P - P^ / '^^P,
_ J = _ ! 2 exp - -——^ t 1 (4.4)
P, + P P, + PQ ^ ^ ^ ^ ^
P
If IL. ^v, 1 then for s m a l l t Po 2 - = I f 1 - exp(-kt) j (4.5) w h e r e k = fJL 8 liVL, P, S i m i l a r l y for l a r g e t i m e with —^ >> 1 2 ^ = 1 - 2 exp(-kt) (4.6) P,
If we define the s t a b i l i z a t i o n t i m e t a s the t i m e t a k e n for the p r e s s u r e in the r e s e r v o i r to r e a c h within 1% of the o v e r a l l p r e s s u r e difference (p - p ) 1 o T , 2 0 0 p In ( Ü- ) R - Po
*s ' Ï
< * • "
and when p • » 1 ^/Po t = ^ (4.8) sWe find for a i r at 15 C that
TTR^ L R^ Pi , ^„ , ^ 9 -1
8V L ' P
w h o r e p = a t m o s p h e r i c p r e s s u r e .
9,
In F i g u r e 1 the v a r i a t i o n of p r e s s u r e with t i m e i s shown t o g e t h e r with the s m a l l and l a r g e t i m e a p p r o x i m a t i o n s . In t h i s e x a m p l e
R L V P i / P a t s U n s t e a d y = = = = -0.1 cm 200 cm 3000 c . c . 1 Pi 14 s e e s . 'Po flow in a c a p i l l a r y = 1000 tube Let us u s e c y l i n d r i c a l p o l a r c o - o r d i n a t e s r , 6, z with z m e a s u r e d along the tube a x i s . If the r a d i a l and c i r c u m f e r e n t i a l components of the v e l o c i t y a r e z e r o the equations of continuity and motion b e c o m e i f w = w ( t , r , z ) a n d p : 3 p ( t , r , z)
Equation of Continuity 9p 9pw _ . i^w 9t 9 z Equations of Motion 0 = - r ^ (5.2)
^ +
'-fr
=
- ^ ^ /.V^w (5.3)
01 d z d z ^2 9^ 1 9 9^ w h e r e V^ s + _ _ _ + _2. o 2 r 9 r „ 2 9 r 9z and \i i s constant s i n c e t h e flow i s i s o t h e r m a l .If the z differential of 5.3 . i s s u b t r a c t e d from the t differential of 5.1 we obtain
9!£. . U P + 9JPW^ _ ^ ^, 9W_ ^5^4^ a t ' 9z^ 9z^ ^^
•if,
We can e l i m i n a t e the d e n s i t y p by substituting the i s o t h e r m a l r e l a t i o n . The r e a s o n for a s s u m i n g the flow i s i s o t h e r m a l i s that s i n c e a l l
v e l o c i t i e s a r e a s s u m e d s m a l l (in fact for l a r g e t i m e t h e y m u s t n e c e s s a r i l y be so - for v e r y s m a l l t i m e t h e y will in g e n e r a l not be so) the d i s s i p a t i o n e n e r g y and p r e s s u r e work t e r m s will be v e r y s m a l l s i n c e they a r e of O(w^). Hence the t e m p e r a t u r e changes m u s t n e c e s s a r i l y be snaall and we a r e justified in neglecting t h e m . It i s for t h i s r e a s o n that we do not need to i n t r o d u c e the e n e r g y equation into o u r formulation of t h i s p r o b l e m .
p " p a^ (5.5) o
T h u s
ËIP . ^2 9 ^ ^ l_PJL _ , a^ ^^ 1^
(5.6)-9 t ' ^ dz' dz' o a z
Some simplification r e s u l t s if we i n t e g r a t e the equations of motion and continuity with r e s p e c t to r^ and m a k e the assumiption that the r a d i a l v e l o c i t y d i s t r i b u t i o n i s independent of z . T h u s on w r i t i n g w ( z , t ) a s the a v e r a g e v e l o c i t y at any s e c t i o n we have
P w ( z , t) = j w dir^l^) (5.7) and if w ( r , z , t) = w ( z , t) f ( r , ) (5.8) I ti With C^ = f ( 7 ? ) d r ° 1 Cg = j tin) dv C °/9f
t h e i n t e g r a l f o r m of the e q u a t i o n s of con+inuity and m o t i o n b e c o m e
I f + - ^ = 0 (5.9)
9t 9 z 1 / 3 p w ^ 9 -^\ 9 p 2c3liw ^ ^ 9^ w and —J [ % + c „ — - p w l = - -^ - 1— + T (5.10) r e s p e c t i v e l y , on m a k i n g u s e of (5.5) and (5.2). F o r a p a r a b o l i c velocity d i s t r i b u t i o n c = 4 / 3 and c, = 4 . 2 3If we s u b t r a c t the z - d e r i v a t i v e of (5.10) from the t i m e d e r i v a t i v e of (5,9) t h e r e r e s u l t s
9^ p 2 9^p 2 C3 /i a p 9 ^ ^ 9 ^ w 9^ p w^
TT
'
^o
a,a
- ^a ¥ r - ^%
i ; 7 +^a g^,
(5.11) which i s the ' i n t e g r a l f o r m ' of (5.6), and i s the wave equation for
p r e s s u r e fluctuations in a tube allowing for v i s c o u s d i s s i p a t i o n . Of c o u r s e equation (5.11) can only b e solved when w i s r e p l a c e d by a function of p .
Now from (5.10) we find that
R ' 9 p _ i f / ' 9 p w _^ ^ 9p w^Y JUR" 9^W jU w = - — - - I — + Cj — + -. " ^3 9^ 2 c , a l \ 9 t «^ 9z y 2C3 - ^ 2 c , 9z 9 _ _2 \ 9t ^ dT. J 2C3 9 z ' 3=^0 (5.12) p R
and if R / L i s sufficiently s m a l l so that < 1 we m a y r e t a i n o
only t e r m s in R giving
R ' 9p R^ 9^p ^ R^ 9^p^ ^ n / o S / c - . o \ 2 C3 9Z 4 ^a g^3 2 2 6 z 9 t
3 3 o
If (5.13) i s s u b s t i t u t e d into e i t h e r (5.11) o r (5.9) and only t e r m s in R^ a r e r e t a i n e d we find that
(5.14) showing that p^ i s only p r o p o r t i o n a l to z in the s t e a d y c a s e .
If we m u l t i p l y (5.14) by 2p then
L£!
=
R l ^
^1£ ^
(5,5)
9 t 4/iC^ g^a
which i s a o n e - d i m e n s i o n a l diffusion equation, w h e r e the diffusivity
-—— , i s a function of p r e s s u r e . 9 p 9t (TV R^ 4 c^^ a 2 ^2 9 p 9 z 2
The solution of (5.15) is made more complicated by the form of 2
O
the boundary conditions. Thus on writing P = p^ - p^ with
P o (on 2 2 » P - Po , f 2 Pi dropping the b a r s a p '' 4 = on R^ r 1 ' ^o L t and Pi ;a L — z) ; z P _ Pi - z a^p 9z^ 2 Pl z / L (1.15) becomes (5.16)
with the boundary condition (associated with a step function change in p r e s s u r e at the tube inlet)
P = 0 t < 0 all z and p = p o p = p o 9 P 9 t 77 2 4 R c z = a l l z Pi u a Y 0 P Pi t t 9 P 9 z > 0 = CO at z = 1, for t > 1 (5J7) = 0 at z = 1 for r < 1
This last boundary condition a r i s e s from the necessary condition that the rate of m a s s flew out of the tube equals the rate of change of m a s s in the r e s e r v o i r of volume V.
According to our assumptions we have not considered in detail the wave motion set up in the tube as a result of the step-function change in the inlet p r e s s u r e . We see from equation (5.6) that the shock wave propagated down the tube would be attenuated and decelerated, and that the p r e s s u r e would not begin to r i s e in the r e s e r v o i r until a time
t = L / a after the initiation of the shock wave. Consequent reflections of the shock wave would result in further changes in p r e s s u r e but these changes in p r e s s u r e will be small, and in any case the times over which they occur will be small compared with the stabilisation time provided L/R » 1.
Hence provided we accept the fact that for o u r p u r p o s e the d e t a i l s of the wave motion in the tube a r e not r e q u i r e d in d e t a i l , we s e e that our p r o b l e m r e d u c e s from a p r e s s u r e wave problena to a p r e s s u r e diffusion p r o b l e m .
Before the solution of (5.16) t o g e t h e r with (5.17) i s a t t e m p t e d we will r e t u r n to the c o r r e s p o n d i n g s t e a d y flow p r o b l e m in o r d e r to justify the a p p r o x i m a t e f o r m s used for w a b o v e .
6. Steady flow in a long c a p i l l a r y tube
F o r s t e a d y a x i - s y n a m e t r i c flow in a tube of constant r a d i u s R t h e equations of continuity and motion a r e
Equation of Continuity 9p w _ _ /R i ^ d z Equations of Motion 9p w^ 9p , „2 ,„ n\ -^ —r— = ^ :rr + ^^ W (6.2) dz 9z 9p 9r (6.3)
Equation (6.1) shows that Pw i s a function of r only while equation (6.3) shows that p i s a function of z only. If the flow i s i s o t h e r m a l p = pa^ , s a y , and then
o 9pw'^ - w dp 9z ^ dz (6.4) with (6.2) b e c o m i n g / / „ ^ ' i P u. /^ 9 /^„ 9 w \ , „ 9^w * a / dz r 9r V 917 9 z (6.5) If r and w a r e t a k e n a s the r e f e r e n c e d e n s i t y and v e l o c i t y
O = - 1 1 - w ^ a " o d /p w o o d ^ / L + R R
1 _1
r 9 r 9 w / w s o \ 9 r z 9 w/w R o L^ (6.6) w h e r e r = r / R , and Ri -L
r 9 r 9w/w 9 F a n d p„w„ R o o M o 9 ( ^ / L ) 'a("/L) J
It i s a s s u m e d that a r e the s a m e o r d e r of Rnnagnitude. It follows that when w / a « 1 and — « 1 equation (6.5)
o j i ^ a p p r o x i m a t e s t o 0 = d p i" 9w dz r 9 r V" 9r T h e f i r s t i n t e g r a l of (6.7) with r e s p e c t to r i s r dp , 9w
° = - 2 d^ -^^ 97
and the second i n t e g r a l i s
w^ - w = d p 4:11 dz
w h e r e w i s the axial velocity at r = 0. R^ dp Hence w. w w 4^ dz 'Ow^ 9 r r=R R dp 2 dz — = 1 - rVz w^ R w = w.,/2
a s in s t a n d a r d P o i s e u i l l e flow except t h a t d p / d z i s not c o n s t a n t . (See equation 6.15 below).
T h e r a t e of volunae flow Q = TTR W TTR dp 8/J dz (6.7) (6.8) (6.9) (6.10) (6.11) (6.12) (6.13) ( 6 J 4 )
If the flow i s i s o t h e r m a l p = ^ a^ , s a y , and then the r a t e of m a s s flow i s
4 2
ra = PrrRw = - ^ i - ^ , p - (6.15)
16/J a^ dz
U n d e r s t e a d y flow conditions the r a t e of m a s s flow will be constant so that if the p r e s s u r e s at z = 0 and z = L a r e p, and p ^ r e s p e c t i v e l y
2 * ^ vR' Pi - P ,„ , „ , m = (6.16) and TTR" 16 ^ a^ o R^ 16/iz K^ , 2 * 2 Pf - P L 2 2 Pl - P P V / R + P n - n) w(z) = 16/iz P Tj^ / n + D \ (6.17) w h e r e a s u n d e r conditions of constant d e n s i t y - R^ w = -X (p - p) = c o n s t . (6.18) 8 iu z 1 Hence a s p •• p in equation (6.17)
w -• -T: (p - p) and w i s then independent of z .
8 /i z 1
In t h e l a t t e r c a s e equation (6.7) i s exact and not a p p r o x i m a t e s i n c e then w = w(r) only with
9w „ , 9^ w
P w —— = 0 and —r- = 0. 9z 9z^
We s e e from (6.14) that u n d e r s t e a d y flow conditions
w = - - ^ ^ (6.19)
on dz
and h e n c e the a p p r o x i m a t i o n to w naade in p a r a g r a p h 5 for the u n s t e a d y c a s e i s e s s e n t i a l l y that of r e p l a c i n g w by i t s value in the s t e a d y c a s e , allowing for the fact that the v e l o c i t y d i s t r i b u t i o n in the u n s t e a d y c a s e will not be e x a c t l y p a r a b o l i c .
Equation (6.16) gives the expression used by Kendall (see 4.1 . above) for the rate of m a s s flow through the tube.
7. The solution of the p r e s s u r e equation for the unsteady flow in a r e s e r v o i r connected to a capillary tube.
It was shown above that the approximate p r e s s u r e equation is
4 c u a L 3 o R' P^ 9 P 9 t w i t h t h e b o u n d a r y c o n d i t i o n s P = 0 P = P o P -- P _ P Pi t < 0 z = 0 a l l z 9^
a
p 2 z a l l z t > 0 t = CO (7.1) (7.2) 9 P 9t = 0 -n o t a t i o -n T T R ' ' P^ 2 P = P 0 <t p P1 2 - P o <1 '^ t 9z , w i t h P 3 1 2 = Pi z = 2 - P o o p; p^' and t and z are non-dimensional quantities, z = 0 corresponds to theopen mouth of the capillary tube while at z = 1 the tube, of length L, is connected to the r e s e r v o i r of volume V, having the initial and final p r e s s u r e s of p and p respectively. The significance of t = 1 is that according to our assvimptions the shock wave of initial strength p - p at time t = 0 takes a time t = 1 (non-dimensional time) to travel the length L of the capillary tube.
An approximate solution of the non-linear equation (7.1) with the non-linear boundary conditions (7.2) can be obtained if we assume that the t e r m p/P| on the right hand side of (7.1) can be replaced by a function of t only. Thus if p*(t) is the value of p at z = 1 then let
p ^ C ( t ) p'^(t) ^^ 3j Pi Pi w h e r e C i s a n a d j u s t a b l e p a r a m e t e r , w h o s e v a l u e i s of t h e o r d e r of u n i t y . If t ^2 ^ * r R P , C P , , , ' ^ ' d t ' ( 7 . 4 ) ( 7 . 5 ) 4 c Aia L p 3 o 1 o t h e n ( 7 . 1 ) a n d ( 7 . 2 ) b e c o m e r e s p e c t i v e l y 9 P _ 9 ^ P d z w i t h t h e b o u n d a r y c o n d i t i o n s P = 0 T < 0 a l l z P = 0 z = 0 T > 0 P =* P a l l Z T = oo , ^ _, o ( 7 . 6 ) 9 P * - V - = 0 Z = l 0 < T < T 9r 1 9 P * = - y - g ^ z = 1 r >r^ , _,* p * ^ - Po 27r R L . , X u w h e r e P = " , y = —— i s a s s i u n e d t o h a v e PS^ V C . a c o n s t a n t v a l u e , b y r e p l a c i n g C b y i t s a v e r a g e v a l u e C , a n d R Pi c p * , , — a t . ^1 J 4 c^a. L, p, o 3 o "^1 If s i s t h e L a p l a c e T r a n s f o r m o p e r a t o r t h e ' s u b s i d i a r y e q u a t i o n ' i s P . i ^ ( 7 . 7 ) s d z CO -— / —ST w h e r e P = ; P e d r , and t h e b o u n d a r y c o n d i t i o n s a r e r e p l a c e d b y
P = P / s z = O o
P * = - X ^ ^ z = 1 s dz
( 7 . 8 )
The solution of ( 7 . 7) satisfying ( 7 . 8 ) i s
_ s i n h ( l - z ) V s + 4 ^ c o 3 h ( l - z ) V s P / P ^ = ^ 1 | _ ( 7 . 9 ) s I sinh -/s + y e ^ cosh V"s ~j and when z = 1 s r , F / P ^ = ^ (7.10) s I / s sinh -fs + y e ^ c o s h V s j F r o m the i n v e r s i o n t h e o r e m we obtain P * / p -- y f e ^ ' ^ " ^ - ' d ^ o 27ri
e - ico 'X VX sinh V\ + y e"'' cosh-/X = 0 0 < T < T, (7.11) exp ( - gy r )
y _ '
- 1 - 2y /__ 2 y= 1 1/ 1 " ^ 1(1 + y e )sin «^^+«^(1 + 2yr^e )cosaj, for r > T^
w h e r e + a , ± a . . . a r e the r o o t s of the equation
a^T.
a t a n a = y e ^ (7.12)
o
Now in o u r p r o b l e m y = — , which i s roughly the r a t i o of
V C 2^
- a T
the tube v o l u m e to the r e s e r v o i r v o l u m e , and in g e n e r a l y e << 1, In t h i s c a s e «^ = y e ^ ^ with sin a^ ~ a^ and cos a^ ^ \, and on r e t a i n i n g only t h e f i r s t t e r m in the s e r i e s in (7.11) and r e p l a c i n g t by r e a l t i m e ,
P P ^ = O O < t < L / a w h e r e k = 2 c^nV L exp - =7-k_ C C p dt L / a for t 5 / a (7.13)
The solution to (7.13) e a s i l y follows When C i s a c o n s t a n t . On differentiating both s i d e s of (7.13) with r e s p e c t to t and r e a r r a n g i n g we find that 6< t < L, dt 0 k 2 , * 2
7a,
for t 2= L ( 7 . 1 4 ) / a ,which we found above w a s the a p p r o x i m a t e equation d e r i v e d by K e n d a l l . The solution of ( 7 , 1 4 ) i s t h e r e f o r e Pi P^ + P = Pi " P o P + P '^l ^ o exp k (t - L / a ) o and the s t a b i l i s a t i o n t i m e , for p / p >> 1, i s
'^i ^o
5.3 ^ L_ s k a
(7.15)
(7.16)
It world a p p e a r at f i r s t sight a l i t t l e s u r p r i s i n g that o u r v a r i o u s a s s u m p t i o n s , all of which s e e m j u s t i f i a b l e , m e r e l y add up to K e n d a l l ' s l o c a l s t e a d y flow a p p r o x i m a t i o n . The only modification to K e n d a l l ' s r e s u l t i s the addition of the t e r m L / a .
o
However the solution to (7.13) i s b a s e d on the fact that C i s a c o n s t a n t . But when C i s taken a s a function of t i m e and noting that C / C i s l e s s than unity, and t a k e s on i t s s m a l l e s t v a l u e s for s m a l l v a l u e s of t - L / a ,
we s e e that the effective value of k in (7.15) will be l e s s * than K e n d a l l ' s k
for s m a l l v a l u e s of t - L / a , and v/ill a p p r o a c h its value for l a r g e t i m e . T h i s p a r t of the difference between the t h e o r e t i c a l and e x p e r i m e n t a l c u r v e s shown in F i g . 8 m a y be due to t h i s o v e r e s t i m a t e of k at s m a l l t i m e s .
T h e above solution can be extended to the c a s e of a s e r i e s of t u b e s of different l e n g t h s and d i a m e t e r s by r e d u c i n g t h e m to an 'equivalent length' of constant d i a m e t e r tubing. In addition allowance can be m a d e for e n t r y and exit l o s s e s , a s well a s the l o s s at a rapid expansion, by adding 'equivalent l e n g t h s ' of constant d i a m e t e r tubing. F o r e x a m p l e , if in addition to a tube of r a d i u s R and length L t h e r e a r e a n u m b e r of t u b e s of r a d i u s R. e a c h of length L . , then the 'equivalent length' of tube of r a d i u s R i s = L +
Fh
R_ R. 1 (7.17)I
4./2R
t
Lz ^
,//^<'i t f T l
-Hp-^« LuThe equivalent length of tubing of r a d i u s R for the above a r r a n g e m e n t m a k i n g a l l o w a n c e s for e n t r y , exit and r a p i d expansion l o s s e s i s
R R"
L . L . ( ^ R . ) \ ^ . ff . L ( 1 . 4 8 . ^ , w h e r e R P^w^R
( 7 . 1 8 )
and suffix ' i ' d e n o t e s conditions at t h e e n t r y to t h e tube of r a d i u s R*. . . .
It might b e noted that if the e n t r y hole ( s a y a d i a m e t e r of 2R ) i s of different d i a m e t e r than the tube of d i a m e t e r 2R, the t e r m in b r a c k e t s in (7.18) i s r e p l a c e d by R, 1.48 R_ R: o o Ri
In most p r e s s u r e systems for supersonic wind tunnels, where R <1000, the last term can be neglected. Further details of the application of the 'equivalent tube' method to the response of p r e s s u r e measuring systems . should be made to Kendall (1958) and Heyser (1958).
8. Apparatus used in the experiments
The basic apparatus consisted of a Langham Thompson type U P . 4 / 150/325 No. 806 p r e s s u r e transducer having a range of ±15 p . s . i . and a chamber volume of 0,0941 cu.in. (1540 mm^). This was connected in turn to various lengths of steel capillary tubing having a bore of 1 ntim. diameter. The other end of the capillary tube was connected to the vacuum chamber of a 'Speedivac' vacuum pump and the chamber was sealed off
with a cellophane diaphragm (Fig. 2). The output of the Langham Thompson p r e s s u r e transducer was connected to a 'Tektronix' type 545 oscilloscope fitted with a 'Polaroid' camera for permanent recording.
The p r e s s u r e in the capillary tube and transducer chamber was first reduced to approximately 10 mm of miercury. The subsequent bursting of the diaphragm produced a good approximation to a step input of p r e s s u r e to the system. Also fitted to the vacuum chamber was an S . L . M . P Z . 1 4 piezoelectric p r e s s u r e transducer. This was used initially to measure the p r e s s u r e in the vacuum chamber but l a t e r , due to its very rapid response, it was used to trigger the oscilloscope. In the latter case it was used with a direct coupled amplifier.
9. Description of tests
The t e s t s were made with capillary tube lengths from zero ( i . e . the transducer mounted flush with the wall of the vacuum chamber) to a maximum of 68 in. The longer tubes comprised 2 or 3 shorter lengths butted together and sealed with short pieces of rubber tubing. The results are shown in Table 1 and Figures 3 to 6 inclusive.
10. Discussion
The response of the S . L . M . transducer to the bursting of the diaphragm is shown in Fig. 3. It can be seen that the p r e s s u r e in the vacuum chamber reaches its final value (atmospheric) after 300 micro seconds and then, after about 5 overshoots, finally settles down to a steady value after 2.5 milliseconds. This is the 'step input' to the capillary tubing.
Since the stabilisation tim.e, that is the time required for the p r e s s u r e to reach 99% of its final value, is difficult to measure in practice, a more
accurate measure of the response is found from the determ.ination of the 'time constant', t , of the system. We will define the 'time constant' as the time required for the p r e s s u r e to reach its final value at the maxinrnm response r a t e . It v/iU be seen from F i g s . 4, 5 and 6 that the measurement of the maxim.um slope can be made with reasonable accuracy.
Fig. 7 shows that the experimental results agree with our predictions and those of Kendall, provided the tube length is not too short and the inlet and exit losses a r e included. When the tube length is zero the above theory is not applicable but a simple calculation for this case (see Appendix)
shows that the 'time constant' will be about 12 milliseconds, in agreement with the experimental r e s u l t s . On tl^e othsr hand, Fig. 8 shov/s that the experimental time variation of the p r e s s u r e in the r e s e r v o i r at small times differs greatly from the theoretical. A partial explanation for this difference lies in the inexactness of our approximations for small times especially the neglect of the initial wave motion. It has already been explained above that a more exact solution of our equations should give a smaller slope at small times than that shown in Fig. 8, and this would have the effect of bringing the theoretical curve nearer the experimental curve. The difference between the theory and experiment is further demons4t.ated in Fig. 9 and in this case it would appear that we have overestimated the losses due to inlet and exit.
The high frequency signals superimposed on both the Langham Thompson and S.L.M^ transducer t r a c e s are due to the gauges 'ringing'. The ringing frequencies are seen to be 1500 c/s for the Langham Thompson and 5000 c / s for the S . L . M . t r a n s d u c e r s .
When the diaphragms were burst they always shattered leaving small pieces of cellophane fi-ee to block up the end of the capillary tube. Although the possibility of this occurring was reduced by placing the end of the
capillary tube under the centre of the diaphragm, it did occur in cases
where the transducer v/as mounted flush with the wall of the vacuum chamber. Fortunately when this happened it could easily be detected from the
oscilloscope record (see Fig. 5d (49); Fig. 6d (44); F i g s . 4b and 4c (81)).
11. Conclusions
The response of a p r e s s u r e measuring system to a step-function input is analysed and it is shown that the approximate solution of Kendall (1958) has a wide range of validity. This solution gives results in reasonable agreement with experiment provided the length/diameter ratio of the capillary tubing is sufficiently large and allowance is made for the inlet and exit l o s s e s .
12. Acknowledgement
The authors wish to express their thanks to Mr. J . R. Busing for assistance in formulating the experiment and in the analysis of the experimental data.
13. References
1. Kendall, J . M . Optimized design of systems for
measuring low p r e s s u r e s in supersonic wind tunnels.
(Presented at AGARD meeting, London, March 1958).
2. Heyser, A, Development of p r e s s u r e measuring devices for a blow-down wind tunnel at the DVL.
(Presented at AGARD meeting, London, March 1958).
The response time of a zero length capillary tube p r e s s u r e system
When a p r e s s u r e transducer is subjected to a step function input the response time will be a function of the orifice size and chamber volume. If we assume the orifice discharge coefficient is 0.64, then on equating the rate of m a s s flow +hrough the orifice to the rate of increase of m a s s in the chamber we find that
where R and V a r e the orifice radius and chamber volume respectively andp V is the rate of m a s s flow per unit area of orifice. In the case where the applied p r e s s u r e ratio exceeds the critical p r e s s u r e for sonic
conditions at the orifice we have, for small times that
dp ^ 0.64. y R TTR^ 2 . .
dt 1.728 a^ V ^ ^^"^^ On the assumption that a^ is constant, the time constant t , when
p^ » p ^ , is given by
t =^^^1^122- ^ ~ (A.3)
c 0.64. y a , ^R^
In our experiment V = 1540 mm and R = 0.5 mm so that t ~ 12 millisecs
c
F i g . No. Record No. Length of 1 m m V e r t i c a l s c a l e H o r i z o n t a l s c a l e b o r e c a p i l l a r y m V / c n a m s / c m tube ( i n . ) 10 10 10 2 10 0.2 5 2 5 2 5 2 5 2 3a 3b 3c 4a 4b 4c 5a 5b 5c 5d 6a 6b 6c 6d 64 63 62 61 -60 ) 59 ) 58 ) 43 ) 42 ) 41 ) 48 47 46 51 50 49 40 ) 39 ) 38 ) 54 53 52 57 56 55 45 ) 44 ) 0 0 0 0.01 0.01 0.01 9.9 5.8 9.8 13.8 18.8 42.9 26.8 22.8 31.0 55.4 50.9 46.9 67.9 ',?" 63.9 59.9 67.88 5 5 5 5 5 5 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10 5 10
H
o-a
0 6
0 - 4
HG.I. RESERVOIR PRESSURE AS A FUNCTION OF TIME (teNDALL)
S.L.M. PREAMPUFIER TEKTRONIX TYPE 112 DIRECT (X3UPLE0 AMPLIFIER TRKXXRING , SIGNAL
ri)
TEKTRONIX TYPE 545 CR.OL < CELLOPHANE HAPHRACM INLET CHAMBER S I M PZI4 CirrSTM. PRESSURE TRANSDUCERT^W
" = ^ — • I m m RESERVOIR.I m m BORC CAPILLARY TUBE
1
TO SPEEOrakC VACUUM PUMP LANGHAM THOMPSON PRESSURE TRANSDUCER TYPE U R 4 / I S O / 3 2 5 N f 8 0 6F i g . 3b T i m e b a s e 2 m s / c m . /A1WWMMMMWN>*4' ' ' ^y, 63 F i g . 3c T i m e b a s e 200 ; j 3 / c m . 62 T i m e F I G . 3 I N L E T CHAMBER P R E S S U R E RESPONSE TO C E L L O P H A N E DIAPHRAGM BURST ( S . L . M . P Z 14) ( V e r t i c a l S c a l e 10 m V / c m . ) F i g . 4a F i g . 4b F i g . 4c T i m e " ^ > As F i g . 4a but with o r i f i c e p a r t i a l l y blocked by d i a p h r a g m f r a g m e n t .
y
FIG. 4 TIME VARIATION OF RESERVOIR PRESSURE L = 0.01 in.
T i m e b a s e 2 m s / c m . ( V e r t i c a l S c a l e 5 m V / c m . )
59
FIG. 5b L = 5.8 in. T i m e b a s e 5 m s / c m .
FIG. 5 TIME VARIATION OF RESERVOIR PRESSURE (Vertical Scale 5 m V / c m . )
13.8 i n . 18.8 i n . 46 FIG. 5d Time base 10 m s / c m . 42.9 i n . 51 26.8 i n . 50 22.8 i n . 49
FIG. 5 TIME VARIATION OF RESERVOIR PRESSURE (Vertical Scale 5 m V / c m . )
T i m e b a s e 10 m s / c m . F I G . 6b T i m e b a s e 10 m s / c m . 54 50.9 in. 53 46.9 in. 52
F I G . 6 TIME VARIATION O F RESERVOIR P R E S S U R E ( V e r t i c a l S c a l e 5 m V / c m . )
67.9 in. 63.9 in. 59.9 in. FIG. 6d L = 67.88 in. T i m e b a s e 10 m s / c m . 45
FIG. 6 TIME VARIATION OF RESERVOIR PRESSURE (Vertical Scale 5 m V / c m . )
3 < a. < o z 6 0 SO 4 0 3 0 2 0 l O / -yh / / o ^
X-/ / / o / /
T
T
oeA
/Y^
© EXPERIMENT ~~— THEORY fAPPROXWATE ALLOWANCE INCLUDED FOR ENTRY AND EXIT LOSSES} -FINAL PRESSURE ATMOSPHERK
lO 2 0 3 0 4 0 SO 6 0 T I M E CONSTANT t e MILU-SECONDS
TO
FIG. 7. VARIATION OF TIME CONSTANT WITH CAPILLARY TUBE LENGTH.
M E T A L T U B E Imm INSIDE D I A M E T E R
LIXT
FIG.8> COMPARISON BETWEEN THEORY AND EXPERIMENT
TEST N9 S3 (FIG. 6 0
SO
J *o
»-> 3 0 2 0 1 0 EXIT LOSSES) F INAL PRESS /y
• -URE ATMO 0 SPHERIC ^^P\
^ ^ >1 z'
r . .. X ^ _
," 10 2 0 3 0 4 0 SO 6 0 7 0 8 0 STABILISATION TIME t g MILLI-SECONDSFIG.9. ^WIATION OF STABILISATION TIME WITH CAPILLARY TUBE LENGTH
METAL TUBE Imm INSIDE DIAMETER
9 0 1 0 0
