Anaesth. Intens. Care (1977), 5, 157
FLOW THROUGH VENOUS CANNULAE A. SELWYN* AND W. J. RUSSELL t Royal Postgraduate Medical School, London, U.K. SUMMARY
Fluid flow through modern cannulae is not simple and cannot be expressed in classical terms. Progressively increasing the length of a cannula diminishes flow predictably b1d altering the bore of the cannula does not give a fourth power improvement in flow but rather a linear one. A reasonable working value for the relative viscosity of ACD blood in infusions would seem to be 2·6 below about 125 mlfminute flow (saline). The decline in relative viscosity above this may be caused by turbulence in the saline flow. INTRODUCTION
Although the theory of flow in ideal situations was developed in the last century, few people have looked at the problem of flow during blood transfusion. In a study of the equipment available in 1951, Melrose and Shackman noted a difference between the expected flows and those obtained clinically. Some of the difference they attributed to turbulent flow but they emphasized that the equipment used in clinical practice is far from the ideal flow system. Thus the Hagen (1839)-Poiseuille (1840) law and Reynolds' critical flow equation (1883) are of limited use clinically. This is perhaps why Knight (1968) simply examined the performance of nine cannulae and catheters in delivering 500 ml of blood, Hartmann's and dextran 40 and suggested a method of resuscitating a hypovolaemic patient. He did not compare the flow performance observed with that which might have been expected in theory. Farman and Powell (1969) examined 22 cannulae and catheters and recorded the flow with blood and water. Although they mention the HagenPoiseuille equation they do not attempt to
* M.B.,
B.S., F.F.A.R.C.S., Associate Professor, Department of Anaesthesia, University of Massachusetts, Massachusetts. t M.B., B.S., F.F.A.R.C.S. (Eng.), F.F.A.R.A.C.S., D.I.C., Ph.D. (Lond.), WeUcome Foundation Research Fellow, Department of Anaesthesia and Intensive Care, Royal Adelaide Hospital. Address for reprints: Dr. W. J. Russell, Dept. Anaesthesia and Intensive Care, Royal Adelaide Hospital, North Terrace, Adelaide, S.A. 5000 Australia.
Anaesthesia and Intensive Care, Vol. V, No. 2, May, 1977
relate it to their observations. Similarly, Bell and Farman (1972) examined 54 cannulae for their flow performance with water, mentioned both the Hagen-Poiseuille equation and Reynolds' critical flow equation but did not attempt to relate them to their observations. It seems contradictory to mention the fundamental equations on flow and yet not to attempt to fit them to the observations. On TABLE 1 Cannulae Dimensions
Diameter mm Length mm External
Internal
52 52 52 52
2·2 1·7 1·4 1·1
1·6 1·3 1·0 0·7
·.
152 152 152 152
2·2 1·7 1·4 1·1
1·6 1·3 1·0 0·7
..
·.
70 162
1·45 1·45
1·0 1·0
..
·.
51 51 51 51
2·0 1·6 1·2 0·91
1·8 1·4 1·0 0·71
Angiocath
·.
57 57 32
2·0 1·6 1·1
1·47 1·12 0·69
Intracath
·.
305 610
1·5 1·5
1·0 1·0
Dannula ..
·.
Dannula ..
Stille Medicut
A.
158
SELWYN AND
this topic, probably the most detailed treatment which is aimed at anaesthetists is by :'IacIntosh, Mushin and Epstein (1963). They discuss the relevance of the flow equations and the additional important factor of the kinetic energy imparted to the fluid by the head of pressure.
BLOOD FLOW IN CATHETERSI1'00mm 1.0.1 t
60 t.
•
FLOW
•
W.
J.
RUSSELL
readings was taken. All flows were run at a constant 100 cm head of pressure by repeatedly adjusting the fluid level so that it remained a steady 100 cm above the tip of the cannula. RESULTS
The cannulae and catheters which were tested are shown in Table 1 together with their dimensions.
(a) Length The flow through the seven devices all of internal diameter 1·0 mm was compared both for blood and saline. The flow with blood is shown in Figure 1 and the flow with saline in
40
SALINE FLOW IN CATHETERS 11-00mm LD.l 160
ml/min
20
•
120
•
FLOW 00
80 600 250 150 100 io ~ mm Length Ireciprocal plot! I.-The relationship between blood flow and catheter length for seven cannulae of 1·00 mm internal diameter. The abscissa is the reciprocal of the catheter length. The filled circles show the observed values. The filled diamonds show the values corrected for kinetic energy (see discussion). The correction does not improve the linear fit.
" 0
ml/min
0 0
FIGURE
When the kinetic energy is allowed for, it should be possible to test the equations and derive estimates for the parameters of apparatus used clinically. In this study, the data derived from the seven types of cannulae have been used to assess flow performance. METHOD
Two test fluids were used. One was 22 to 28 days old ACD blood with a haematocrit of 35%. The other was normal saline. Both fluids were tested at 22°C. Each fluid was passed through a standard blood giving set (Avon medical) to the cannula under test. The flow of each test was timed for one minute by stopwatch and the volume measured in a graduated glass cylinder. The mean of three
0
0
40
0
s
600 250 150
100
70
mm Length (reciprocal plotJ
50
FIGURE 2.-The relationship between saline flow and catheter length for seven cannulae of 1· 00 mm internal diameter. The abscissa is the reciprocal of the catheter len.gth. The circles show the observed values. The diamonds show the values corrected for kinetic energy (see discussion). The kinetic correction improves the linear fit.
Figure 2. The abscissae in both figures is drawn as the reciprocal of the catheter or cannula length as this would be expected to give a straight line if the flows were conforming to the Hagen-Poiseuille equation. The results for blood gives an excellent fit r=0·97, and the fit for saline is also quite good r =0· 9±. Anaesthesia and Intmsive Ca/B, Vol. V. No. 2, May, 1977
159
FLOW THROUGH CANNULAE
(b) Diameter
The flow through ten cannulae all between 51 and 57 mm length and of diameter ranging from o· 7 to 1· 8 mm was compared both with blood and saline. The relationship for both blood and saline is shown in Figure 3. There is an excellent linear correlation between diameter and flow for both fluids, r=0·99 for both.
=*
Saline Blood
20
'V
T
o
•
ANGIOCATH DANNUlA MEOICUT
o
•
100
ml/min
0·6
1-0 1-4 Diam.eter mm
1·8
FIGURE 3.-The relationship between observed flow and diameter for ten cannulae of between 51 and 57 mm length. Various diameters of three types of cannula are shown. The open symbols show the observed flow with saline and the filled symbols show the observed flows in blood. With both fluids there is an excellent positive linear relationship between the nominal internal diameter of the cannula and the observed flow.
DISCUSSION
(a) Length One form of the Hagen-Poiseuille equation is LlHd 4 LlH is the head of pressure q= x[ll d ~s r~dius. of tube [l
IS VISCOSIty
l x
is length of tube is constant is bulk flow
q
Hx=2·35x10-3
thus qoc1/l and a plot of l/l should be a straight line passing through the origin. Figure 1 shows the least squares regression line for seven catheters of 1·0 mm diameter. The comparatively fast flows mean that a large Anaesthesia and Intensive Care, Vol. V, No. 2, May, 1977
[~r
Hx=head of pressure in cm H 20 q=flow ml/min. d=diameter in mm. For laminar flow, more kinetic energy is imparted to the fluid and the effective head of pressure is doubled; Hx=4·7 x10- 3
o
FLOW
proportion of the pressure head is utilized in creating kinetic energy. The head of pressure which contributes to the kinetic energy with plug flow can be calculated as
u~r (MacIntosh
et al. 1963).
It is possible therefore to calculate the head of pressure which is dissipated by the esistance to flow as H-Hx where for these experiments H=100 cm H 20 . If it is assumed that the flow is linearly related to the pressure then the head of pressure can be standardized back to 100 cm so that the flow which requires this pressure to simply overcome the resistance for the given conditions can be 100 calculated as qc=q X 100-Hx qc=the corrected
flow in mljmin. Applying this correction to the blood flow/ length relationship does not alter the correlation (both r= +0· 97) however the correction does move the linear relationship nearer to the origin. The correction to the saline flow/length relationship does give an improved fit (Figure 2). (b) Diameter
Here the practical problem seems surprisingly wide of the theory. Even allowing for kinetic energy, in the clinical situation, the increased flow with increasing internal diameter is far less than that expected if flow is proportional to the fourth power of the diameter. The flow through ten cannulae of about the same length (5751 mm) shows an excellent linear relationship both for blood and saline (Figure 3 +0, 990, 0·993 respectively). This is obviously a pragmatic relationship as extrapolation would suggest no flow through a cannula of 0·4 mm internal diameter. However it appears that over the common clinical range of diameters, intravenous devices deliver a flow which is linearly related to their final tip diameter such that for 50 mm length devices an increase in blood flow of about 12 ml/min can be expected for every 0·1 mm increase in internal diameter. Thus if a 1·0 mm diameter cannula about 50 mm long is used instead of a 0·7 mm one,
A.
160
SELWYN AND
the blood flow is better by about 35 ml/minute ; if a 1·3 mm cannula is used a further 35 ml/ minute improvement in flow is seen. If the fourth power relationship was effective and all resistance were in the cannulae a change from 0'7 to 1·3 mm diameter should flow improvement from 23 ml/minute to about 165 ml/minute, i.e. an improvement of about 140 ml/minute. Much of the extra energy which becomes available when the resistance is reduced appears to be taken up by an increase in kinetic energy. The difference between the flows observed and the expected flows with changes in the diameter are greater than can be simply attributed to the resistance of the giving set. (c) Relative flow/viscosity From the observed flows for blood and saline and assuming laminar flow it is possible to derive the energy for the resistance alone expressed as an estimated head of pressure (H-HK) for each cannula with blood and with saline by means of the Hagen-Poiseuille equation. K!1H1
for a given cannula [1.1 =viscosity of saline (approx. =water~l'O poise 20°C) K!1H2 q2=~ [1.2=viscosity of blood q1=~
thus
~=q1~HH2 =:=relative viscosity blood.
[1.1 q1iJ. 2 This closely resembles the technique of Pirofsky (1953) for estimating blood viscosity in man. He suggested a figure of 2 ·16 centipoise for venous blood (Haematocrit 35%). This was venous blood from an arm vein and so probably about 30°C. A slightly lower value of about 2·0 centipoise for fresh ACD blood would be expected due to dilution (diluted 13%). This corresponds quite well with the relative viscosity of 2·6 found here for lower flows when about a 25% increase in viscosity is expected because of the lower temperature used in this study (Langstroth 1919). The decline at higher flows probably relates to the development of turbulence in the saline. As the critical velocity is affected by the viscosity such that increased viscosity will give a higher critical velocity (Reynolds 1883), blood should show less turbulence than saline. Thus the development of turbulent flow increases the apparent viscosity of the saline and this decreases the relative viscosity. Probably this is the reason that the relative viscosity can be seen generally to decline as the flow increases above 125 ml/min
W.
J.
RUSSELL
saline flow (Figure 4). It seems that at higher flows commonly found in clinical practice with the intravenous devices available some degree of turbulent flow is likely.
..
50
100
150 FLOW ml/min
200
FIGURE 4.-The relationship between the relative . . (ViSCOSity of blOOd) vIscosity . . f r as calculated from the vIscosity 0 sa me corrected flows of blood and saline and the observed flow of saline. There is a progressive decline in the relative viscosity above a flow of about 125 ml/min. ACKNOWLEDGEMENT
The secretarial assistance of Miss E. K. Hilton is gratefully acknowledged. REFERENCES
Bell, G. T., and Farman, J. V. (1972): "Disposable Venous Cannnlae ", Brit. J. Hasp. Med., 8, 49-58 (Suppl.). Farman, J. V., and Powell, D. (1969) : " The Performance of Disposable Venous Catheters, Needles and Cannulae ", Brit. J. Hasp. Med., 2,37-45 (Suppl. 2). Hagen, G. H. (1839): "Uber Die Bewegung des Wassers in Engen Zylinderischen Kohren ", Ann. Phys. Lpz., 46, 423-44-2. Knight, R. J. (1968) : " Flow-rates Through Disposable Intravenous Cannulae ", Lancet, 2, 665-667. Langstroth, L. (1919): "Blood Viscosity. I. Conditions Affecting the Viscosity of Blood After Withdrawal from the Body", J. Exp. Med., 30, 597-606. MacIntosh, R., Mushin, W. W., and Epstein, H. G. (1963): Physics for the A naesthetist. Oxford. Blackwell. Melrose D. G., and Shackman, R. (1951): "Fluid Mechanics and Dynamics of Transfusion: Rapid Replacement of Severe Blood Loss ", Lancet, 1, 1144-1147. Pirofsky, B. (1953): "The Determination of Blood Viscosity in Man by a Method Based on Poiseuille's Law", J. Clin. Invest., 32, 292-298. Poiseuille, J. L. M. (1840): "Recherches Experimentales sur le Movements des Liquids dans les Tubes de Tres Petits Diametres ", Comptes rend. Acad. d. Se., 11, 961-1041. Reynolds, O. (1883) : " An Experimental Investigation of the Circumstances which Determine Whether a Motion of Water shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels ", Phil. Trans. (Land.)., 174,935-982.
Anaesthesia and Intensive Care, Vol. V, No. 2, lVlay, 1977
FLOW THROUGH VENOUS CANNULAE A. SELWYN* AND W. J. RUSSELL t Royal Postgraduate Medical School, London, U.K. SUMMARY
Fluid flow through modern cannulae is not simple and cannot be expressed in classical terms. Progressively increasing the length of a cannula diminishes flow predictably b1d altering the bore of the cannula does not give a fourth power improvement in flow but rather a linear one. A reasonable working value for the relative viscosity of ACD blood in infusions would seem to be 2·6 below about 125 mlfminute flow (saline). The decline in relative viscosity above this may be caused by turbulence in the saline flow. INTRODUCTION
Although the theory of flow in ideal situations was developed in the last century, few people have looked at the problem of flow during blood transfusion. In a study of the equipment available in 1951, Melrose and Shackman noted a difference between the expected flows and those obtained clinically. Some of the difference they attributed to turbulent flow but they emphasized that the equipment used in clinical practice is far from the ideal flow system. Thus the Hagen (1839)-Poiseuille (1840) law and Reynolds' critical flow equation (1883) are of limited use clinically. This is perhaps why Knight (1968) simply examined the performance of nine cannulae and catheters in delivering 500 ml of blood, Hartmann's and dextran 40 and suggested a method of resuscitating a hypovolaemic patient. He did not compare the flow performance observed with that which might have been expected in theory. Farman and Powell (1969) examined 22 cannulae and catheters and recorded the flow with blood and water. Although they mention the HagenPoiseuille equation they do not attempt to
* M.B.,
B.S., F.F.A.R.C.S., Associate Professor, Department of Anaesthesia, University of Massachusetts, Massachusetts. t M.B., B.S., F.F.A.R.C.S. (Eng.), F.F.A.R.A.C.S., D.I.C., Ph.D. (Lond.), WeUcome Foundation Research Fellow, Department of Anaesthesia and Intensive Care, Royal Adelaide Hospital. Address for reprints: Dr. W. J. Russell, Dept. Anaesthesia and Intensive Care, Royal Adelaide Hospital, North Terrace, Adelaide, S.A. 5000 Australia.
Anaesthesia and Intensive Care, Vol. V, No. 2, May, 1977
relate it to their observations. Similarly, Bell and Farman (1972) examined 54 cannulae for their flow performance with water, mentioned both the Hagen-Poiseuille equation and Reynolds' critical flow equation but did not attempt to relate them to their observations. It seems contradictory to mention the fundamental equations on flow and yet not to attempt to fit them to the observations. On TABLE 1 Cannulae Dimensions
Diameter mm Length mm External
Internal
52 52 52 52
2·2 1·7 1·4 1·1
1·6 1·3 1·0 0·7
·.
152 152 152 152
2·2 1·7 1·4 1·1
1·6 1·3 1·0 0·7
..
·.
70 162
1·45 1·45
1·0 1·0
..
·.
51 51 51 51
2·0 1·6 1·2 0·91
1·8 1·4 1·0 0·71
Angiocath
·.
57 57 32
2·0 1·6 1·1
1·47 1·12 0·69
Intracath
·.
305 610
1·5 1·5
1·0 1·0
Dannula ..
·.
Dannula ..
Stille Medicut
A.
158
SELWYN AND
this topic, probably the most detailed treatment which is aimed at anaesthetists is by :'IacIntosh, Mushin and Epstein (1963). They discuss the relevance of the flow equations and the additional important factor of the kinetic energy imparted to the fluid by the head of pressure.
BLOOD FLOW IN CATHETERSI1'00mm 1.0.1 t
60 t.
•
FLOW
•
W.
J.
RUSSELL
readings was taken. All flows were run at a constant 100 cm head of pressure by repeatedly adjusting the fluid level so that it remained a steady 100 cm above the tip of the cannula. RESULTS
The cannulae and catheters which were tested are shown in Table 1 together with their dimensions.
(a) Length The flow through the seven devices all of internal diameter 1·0 mm was compared both for blood and saline. The flow with blood is shown in Figure 1 and the flow with saline in
40
SALINE FLOW IN CATHETERS 11-00mm LD.l 160
ml/min
20
•
120
•
FLOW 00
80 600 250 150 100 io ~ mm Length Ireciprocal plot! I.-The relationship between blood flow and catheter length for seven cannulae of 1·00 mm internal diameter. The abscissa is the reciprocal of the catheter length. The filled circles show the observed values. The filled diamonds show the values corrected for kinetic energy (see discussion). The correction does not improve the linear fit.
" 0
ml/min
0 0
FIGURE
When the kinetic energy is allowed for, it should be possible to test the equations and derive estimates for the parameters of apparatus used clinically. In this study, the data derived from the seven types of cannulae have been used to assess flow performance. METHOD
Two test fluids were used. One was 22 to 28 days old ACD blood with a haematocrit of 35%. The other was normal saline. Both fluids were tested at 22°C. Each fluid was passed through a standard blood giving set (Avon medical) to the cannula under test. The flow of each test was timed for one minute by stopwatch and the volume measured in a graduated glass cylinder. The mean of three
0
0
40
0
s
600 250 150
100
70
mm Length (reciprocal plotJ
50
FIGURE 2.-The relationship between saline flow and catheter length for seven cannulae of 1· 00 mm internal diameter. The abscissa is the reciprocal of the catheter len.gth. The circles show the observed values. The diamonds show the values corrected for kinetic energy (see discussion). The kinetic correction improves the linear fit.
Figure 2. The abscissae in both figures is drawn as the reciprocal of the catheter or cannula length as this would be expected to give a straight line if the flows were conforming to the Hagen-Poiseuille equation. The results for blood gives an excellent fit r=0·97, and the fit for saline is also quite good r =0· 9±. Anaesthesia and Intmsive Ca/B, Vol. V. No. 2, May, 1977
159
FLOW THROUGH CANNULAE
(b) Diameter
The flow through ten cannulae all between 51 and 57 mm length and of diameter ranging from o· 7 to 1· 8 mm was compared both with blood and saline. The relationship for both blood and saline is shown in Figure 3. There is an excellent linear correlation between diameter and flow for both fluids, r=0·99 for both.
=*
Saline Blood
20
'V
T
o
•
ANGIOCATH DANNUlA MEOICUT
o
•
100
ml/min
0·6
1-0 1-4 Diam.eter mm
1·8
FIGURE 3.-The relationship between observed flow and diameter for ten cannulae of between 51 and 57 mm length. Various diameters of three types of cannula are shown. The open symbols show the observed flow with saline and the filled symbols show the observed flows in blood. With both fluids there is an excellent positive linear relationship between the nominal internal diameter of the cannula and the observed flow.
DISCUSSION
(a) Length One form of the Hagen-Poiseuille equation is LlHd 4 LlH is the head of pressure q= x[ll d ~s r~dius. of tube [l
IS VISCOSIty
l x
is length of tube is constant is bulk flow
q
Hx=2·35x10-3
thus qoc1/l and a plot of l/l should be a straight line passing through the origin. Figure 1 shows the least squares regression line for seven catheters of 1·0 mm diameter. The comparatively fast flows mean that a large Anaesthesia and Intensive Care, Vol. V, No. 2, May, 1977
[~r
Hx=head of pressure in cm H 20 q=flow ml/min. d=diameter in mm. For laminar flow, more kinetic energy is imparted to the fluid and the effective head of pressure is doubled; Hx=4·7 x10- 3
o
FLOW
proportion of the pressure head is utilized in creating kinetic energy. The head of pressure which contributes to the kinetic energy with plug flow can be calculated as
u~r (MacIntosh
et al. 1963).
It is possible therefore to calculate the head of pressure which is dissipated by the esistance to flow as H-Hx where for these experiments H=100 cm H 20 . If it is assumed that the flow is linearly related to the pressure then the head of pressure can be standardized back to 100 cm so that the flow which requires this pressure to simply overcome the resistance for the given conditions can be 100 calculated as qc=q X 100-Hx qc=the corrected
flow in mljmin. Applying this correction to the blood flow/ length relationship does not alter the correlation (both r= +0· 97) however the correction does move the linear relationship nearer to the origin. The correction to the saline flow/length relationship does give an improved fit (Figure 2). (b) Diameter
Here the practical problem seems surprisingly wide of the theory. Even allowing for kinetic energy, in the clinical situation, the increased flow with increasing internal diameter is far less than that expected if flow is proportional to the fourth power of the diameter. The flow through ten cannulae of about the same length (5751 mm) shows an excellent linear relationship both for blood and saline (Figure 3 +0, 990, 0·993 respectively). This is obviously a pragmatic relationship as extrapolation would suggest no flow through a cannula of 0·4 mm internal diameter. However it appears that over the common clinical range of diameters, intravenous devices deliver a flow which is linearly related to their final tip diameter such that for 50 mm length devices an increase in blood flow of about 12 ml/min can be expected for every 0·1 mm increase in internal diameter. Thus if a 1·0 mm diameter cannula about 50 mm long is used instead of a 0·7 mm one,
A.
160
SELWYN AND
the blood flow is better by about 35 ml/minute ; if a 1·3 mm cannula is used a further 35 ml/ minute improvement in flow is seen. If the fourth power relationship was effective and all resistance were in the cannulae a change from 0'7 to 1·3 mm diameter should flow improvement from 23 ml/minute to about 165 ml/minute, i.e. an improvement of about 140 ml/minute. Much of the extra energy which becomes available when the resistance is reduced appears to be taken up by an increase in kinetic energy. The difference between the flows observed and the expected flows with changes in the diameter are greater than can be simply attributed to the resistance of the giving set. (c) Relative flow/viscosity From the observed flows for blood and saline and assuming laminar flow it is possible to derive the energy for the resistance alone expressed as an estimated head of pressure (H-HK) for each cannula with blood and with saline by means of the Hagen-Poiseuille equation. K!1H1
for a given cannula [1.1 =viscosity of saline (approx. =water~l'O poise 20°C) K!1H2 q2=~ [1.2=viscosity of blood q1=~
thus
~=q1~HH2 =:=relative viscosity blood.
[1.1 q1iJ. 2 This closely resembles the technique of Pirofsky (1953) for estimating blood viscosity in man. He suggested a figure of 2 ·16 centipoise for venous blood (Haematocrit 35%). This was venous blood from an arm vein and so probably about 30°C. A slightly lower value of about 2·0 centipoise for fresh ACD blood would be expected due to dilution (diluted 13%). This corresponds quite well with the relative viscosity of 2·6 found here for lower flows when about a 25% increase in viscosity is expected because of the lower temperature used in this study (Langstroth 1919). The decline at higher flows probably relates to the development of turbulence in the saline. As the critical velocity is affected by the viscosity such that increased viscosity will give a higher critical velocity (Reynolds 1883), blood should show less turbulence than saline. Thus the development of turbulent flow increases the apparent viscosity of the saline and this decreases the relative viscosity. Probably this is the reason that the relative viscosity can be seen generally to decline as the flow increases above 125 ml/min
W.
J.
RUSSELL
saline flow (Figure 4). It seems that at higher flows commonly found in clinical practice with the intravenous devices available some degree of turbulent flow is likely.
..
50
100
150 FLOW ml/min
200
FIGURE 4.-The relationship between the relative . . (ViSCOSity of blOOd) vIscosity . . f r as calculated from the vIscosity 0 sa me corrected flows of blood and saline and the observed flow of saline. There is a progressive decline in the relative viscosity above a flow of about 125 ml/min. ACKNOWLEDGEMENT
The secretarial assistance of Miss E. K. Hilton is gratefully acknowledged. REFERENCES
Bell, G. T., and Farman, J. V. (1972): "Disposable Venous Cannnlae ", Brit. J. Hasp. Med., 8, 49-58 (Suppl.). Farman, J. V., and Powell, D. (1969) : " The Performance of Disposable Venous Catheters, Needles and Cannulae ", Brit. J. Hasp. Med., 2,37-45 (Suppl. 2). Hagen, G. H. (1839): "Uber Die Bewegung des Wassers in Engen Zylinderischen Kohren ", Ann. Phys. Lpz., 46, 423-44-2. Knight, R. J. (1968) : " Flow-rates Through Disposable Intravenous Cannulae ", Lancet, 2, 665-667. Langstroth, L. (1919): "Blood Viscosity. I. Conditions Affecting the Viscosity of Blood After Withdrawal from the Body", J. Exp. Med., 30, 597-606. MacIntosh, R., Mushin, W. W., and Epstein, H. G. (1963): Physics for the A naesthetist. Oxford. Blackwell. Melrose D. G., and Shackman, R. (1951): "Fluid Mechanics and Dynamics of Transfusion: Rapid Replacement of Severe Blood Loss ", Lancet, 1, 1144-1147. Pirofsky, B. (1953): "The Determination of Blood Viscosity in Man by a Method Based on Poiseuille's Law", J. Clin. Invest., 32, 292-298. Poiseuille, J. L. M. (1840): "Recherches Experimentales sur le Movements des Liquids dans les Tubes de Tres Petits Diametres ", Comptes rend. Acad. d. Se., 11, 961-1041. Reynolds, O. (1883) : " An Experimental Investigation of the Circumstances which Determine Whether a Motion of Water shall be Direct or Sinuous, and of the Law of Resistance in Parallel Channels ", Phil. Trans. (Land.)., 174,935-982.
Anaesthesia and Intensive Care, Vol. V, No. 2, lVlay, 1977
