PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW

Transcription

1 PREDICTION OF MILKLINE FILL AND TRANSITION FROM STRATIFIED TO SLUG FLOW ABSTRACT: by Douglas J. Reineann, Ph.D. Assistant Professor of Agricultural Engineering and Graee A. Mein, Ph.D. Visiting Professor of Dairy Science and Veterinary Medicine University of Wisconsin-Madison Published in the 1995 Transactions of the ASAE. Vol. 38(3): Equations were developed to predict ilkline fill and the transition fro stratified to slug flow under noral ilking conditions. The equation to predict fill explained 93 percent of the variation in the experiental data for 48 to 98 ilklines sloped at 0.5 to percent. The prediction equation for the transition fro stratified to slug flow accounted for 98 percent of the variation in the experientally deterined transition points. INTRODUCTION Equations developed to describe open channel flow have been used to predict fill in ilklines (Gates, 1980, Gates et al, 1981, and 198). These studies used Manning's equation to relate ilkline diaeter, slope and fill with ilk flow. Manning's equation is derived fro a force balance on a fluid eleent between gravity and the friction between the fluid and pipe wall. The oentu transfer between the ilk and air at the free surface is neglected. Furtherore, it is usually assued that the flow is fully developed. In ost ilking systes, air flows in the ilkline above the ilk, generally in the sae direction as ilk flow. Air enters ilklines in two ways during ilking. A steady flow of air is introduced through designed air vents in clusters and soe types of ilk eters, and through leaks at fittings. Air adission through air vents is constant during the ilking process and accounts for 8 to 15 L/in (0.3 to 0.5 cf) per ilking unit. Air is also aditted interittently (transient air) when units are attached to the cow, or when cups slip or fall off. For a constant ilk flow rate, oentu transferred fro the steady air flow in the free space above the ilk acts to decrease ilkline fill under stratified flow conditions. Moentu transferred fro interittent air adission will either further reduce fill or cause slugging in the ilkline. Moentu transfer fro air to ilk increases as the depth of ilk in the pipe increases and the area available for air flow decreases. Air velocity can be an order of agnitude above ilk 1

2 velocity under noral operating conditions. Analyses by Gates (1980) showed that when the voluetric air to ilk flow ratio and pipe fill are high, the effects of oentu transfer are as large or larger than the opposing ilk/wall frictional force. These are the conditions encountered near the point of transition fro stratified to slug flow. Milk enters the ilkline fro individual ilking units at intervals of 65 to 10 c (7 to 48 inches) in ilking parlors and up to 40 c (96 inches) in round-the-barn pipeline systes. The ilk enters fro the ilk hose at high velocity copared with the flow velocity in the ilkline and in a direction perpendicular to the flow direction in the ilkline. This creates a region of intense ixing at each ilk inlet and induces resistance to the flow of ilk in the ilkline. It ay take up to 300 pipe diaeters for the flow to fully develop and for the effects of this ixing region to decay. The ilk flow rate in the ilkline increases nearer the receiver as ore units add ilk to the line. Maxiu fill is expected to occur just downstrea of the ilk inlet nearest the receiver, which is a region of undeveloped flow. This is the critical location for the developent of slug flow. Stratified flow is characterized by ilk flow in the lower portion of the pipe and air flow in the upper portion. Slug flow occurs when ilk fills the entire pipe cross section foring "slugs" of varying length. The objectives of this study were to develop equations to predict: 1) ilkline fill, considering the effects of ixing and airflow and, ) the onset of slugging in ilklines under noral operating conditions. METHODS An experiental study was perfored to deterine the point of transition fro stratified to slug flow conditions for 48, 73, and 98 ID (", 3" and 4" noinal) ilklines (Reineann and Mein, 1994). The experiental conditions and easureent points were chosen to replicate a worst-case scenario for the expected axiu fill and slugging conditions. Both steady and transient air flows were aditted to the ilkline in this experient. Tests were perfored over a range of steady air to ilk flow ratios fro 1.5 to 3.0 with ost tests perfored with a ratio of.:1. This represents the range of steady air to ilk flow ratios under peak flow conditions for coercial ilking units and fast ilking cows (Mein et al, 1993). Milkline fill was easured just upstrea fro the elbow nearest the receiver (1.5 downstrea of the nearest ilk inlet). The transition between stratified and slug flow was deterined for both steady air flow and a cobination of steady and transient air flows as encountered in ilking systes. MILKLINE FILL PREDICTION Manning's equation, developed to predict open channel flow, is usually presented in the = A /3 R n S 1/ following for:

3 where: = Voluetric ilk flow rate (L/in) A = Cross sectional area of ilk flow ( ) R = Hydraulic radius of ilk () S = Milkline slope (%) This equation is derived fro a force balance on a fluid eleent flowing in an open channel, neglecting interfacial forces at the ilk / air interface. Experiental data for stratified flow conditions for a 73 (3") ilkline were used to deterine the value of n in (1). The average value of n for 171 data points was 0.11 and the correlation coefficient (R ) was Calculated values of n are plotted against the air velocity in the free space above the ilk in Figure 1. At low air velocity, the value for n of 0.16 is considerably higher than values used for fully developed, open channel flow. Gates found a value of 0.09 in a study in which pipe fill was easured far fro the point of ilk entry, where flow was fully developed. The higher value of n found here at low air velocity is the effect of increased friction caused by ixing in the region of undeveloped flow near ilk inlets. Increased air velocity reduced the value of n. The for of the decrease (proportional to the square of the velocity) is as expected for frictional transfer of oentu fro air to ilk. The effects of gravity, ilk/wall friction and air/ilk oentu transfer were included in a force balance on a fluid eleent to give: f V S = g R f a ρ a (V a V ρ g Ra ) where: f = ilk / wall friction factor V = ilk flow velocity (/s) g = acceleration due to gravity (/s ) f a = air/ilk friction factor ρ a = air density (kg/ 3 ) V a = steady air velocity (/s) ρ = ilk density (kg/ 3 ) R a = hydraulic radius of air () Neglecting the ilk velocity copared with the air velocity in the second ter, this equation can be rearranged in the for of Manning's equation as: S = C R 1 4/3 A C R 4/3 a asteady A a 3

4 asteady = steady voluetric air flow rate easured at standard conditions (L/in) A a = cross sectional area for air flow ( ) C 1 = ilk friction coefficient C = air friction coefficient This equation can be siplified by substituting the following expression for the hydraulic radii R = 0.35 D F 1/ = 0.79 D F Ra for circular pipe cross sections: where: D = Inner diaeter of pipe () F = Fraction of pipe cross-sectional area filled with ilk The expression for the hydraulic radius of the air is exact. The expression for the hydraulic radius of the liquid phase is within 1% of actual value for fills fro 0 to 80 percent. Substituting (4) and (5) in (3) and rearranging gives: 16/3 8/3 asteady F = C 3 S D F +C 4 (1 F ) Milk fill fro 30 to 70 percent is the range in which the transition to slug flow will occur for the type of air flows encountered in ilking systes. Experiental data for 48, 73, and 98 inner diaeter pipes sloped at 0.5, 1.0 and.0 percent with fills fro 0 to 70 percent were regressed to deterine the coefficients C 3 and C 4 in equation (6) with the following result: C 3 =.4 x 10-6 C 4 = 1.4 x 10 - Nuber of data points (n) = 171 Correlation Coefficient (R ) = 0.93 TRANSITION FROM STRATIFIED TO SLUG FLOW Flow pattern aps are coonly use to define the boundary between flow regies in two phase flow. The boundary lines are coonly expressed in ters of superficial liquid and air velocities. Superficial velocity is obtained by dividing the voluetric flow rate of the phase (liquid or gas) by the total pipe cross sectional area. Regression analysis was perfored to deterine the coefficient and exponents of a transition boundary between stratified and slug flow for ilk lines. Data fro 48, 73, and 98 inner diaeter pipes sloped at 0.5, 1.0 and.0 percent were used (n = 33, R = 0.88). 4/3 4

5 0.016 = s V 1.03 V Sa 0.9 S where: d = Internal pipe diaeter () V Sa = Superficial air velocity (/s) = Total voluetric air flow rate (steady + transient air) easured at ilking syste vacuu / total pipe cross-sectional area. V S = Superficial ilk velocity (/s) = Voluetric ilk flow rate / total pipe cross sectional area. This equation was siplified to the following for without significant reduction in prediction accuracy (n = 33, R = 0.86): V Sa d s d = V S 0.83 An easier to use for of (8) using voluetric ilk and voluetric air flow rate at standard conditions was also developed (n = 33, R =.98): 8.9(10 ) = 6 atotal S d 5 where: atotal = Total (steady + transient) voluetric air flow rate per ilkline slope, easured at standard conditions (L/in) Steady air to ilk flow ratios between 1.5:1 and 3:1 were tested experientally as this covers the range of claw air adission and peak ilk flow rates coonly encountered in practice. Over this range of steady air to ilk flow, experientally deterined ilk flow rate at the flow transition points varied by less than 5%. To siplify the calculations, therefore, a constant steady air to ilk ratio of.:1 (e.g. 10 L/in steady air flow per 4.5 L/in ilk flow) was used atotal = atransient +. to derive as follows: where: atransient = Transient voluetric air flow rate per ilkline slope, easured at standard conditions (L/in) Table I. Exaple predicted flow conditions at the transition fro stratified to slug flow for a 73 ilkline with steady air to ilk flow ratio of., using (9) and (10). 5

6 Milkline Slope (%) Steady Air Flow Rate (L/in) Transient Air Flow Rate (L/in) Total Air Flow Rate (L/in) Predicted Milk Flow Rate (L/in) Equation (10) was then substituted in (9) and solving for the positive root of the quadratic equation for (Table I). Experientally deterined transition points are plotted with predicted values using (9) and (10) in Figures - 4. SUMMARY AND CONCLUSIONS The effect of ixing at the point of ilk entry produces a significant increase in pipe wall friction and fill copared with fully developed flow. The effects of oentu transfer between air and ilk on ilkline fill are significant in the range of interest for ilking systes. Manning's equation and other versions of fully developed, open channel flow equations are not accurate predictors of ilkline flow dynaics because of these two effects. A new equation (6) was developed to account for these two effects. This equation explained 93 percent of the variation in the experiental data for 48 to 98 ilklines sloped at 0.5 to percent. An equation was also developed to predict the transition fro stratified to slug flow (9). This equation accounted for 98 percent of the variation in the experientally deterined transition points. REFERENCES Gates, R.S., Two Phase Flow in Milking Pipelines. Unpublished Masters Thesis, Cornell University, Ithaca NY. 175 pp. Gates, R.S., R. Sagi, and N.R. Scott, Theoretical considerations in sizing ilk pipelines. Trans. ASAE, Vol. 4, No. 6, pp Gates, R.S., R. Sagi, and R.W. Guest, 198. Criteria for optiizing size and configuration of ilk pipelines. Journal Dairy Science 65, Mein, G.A., D.J. Reineann and S.B. Spencer, Milkline sizing: recent research and recoendations. Proc. 3nd Annual Conference of the National Mastitis Council, Kansas City, MO, USA. Reineann, D.J., and G.A Mein, Transition fro stratified to slug flow in ilklines. Transactions of the ASAE, Vol. 37 6

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