Studying the Model that Predict and Monitor Constant Area Variable Pressure Drop in Orifice Plate

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1 MSE JOURNLS 04-Series: Modelling B; Vol. 83; N ; Submitted May 03; Revised Dec., 04; cceted Feb. 0, 04 Studying the Model that redict and Monitor Constant rea Variable ressure Dro in Orifice late *C.. Ukaka, and ** C. Orji * Deartment of Chemical/etrochemical Engineering **Deartment of Marine Engineering Rivers State University of Science and Technology, Nkolu,.M.B. 5080, ort Harcourt, Nigeria. (chukwuemeka4@yahoo.com) bstract Mathematical models for the design of neumatic roortional control to redict and monitor constant area variable ressure dro using different fluids were develoed and tested for an ice late. The different fluids used are water, Benzene, ethyl ethanol and butane. Results obtained from the investigation revealed that density, viscosity and flow rate are functional arameters that influence the ressure dro in the system. The roortional gain is also other functional arameters that govern and imrove effectiveness of control rocess by reducing the error value in the system. The develoed mathematical model of constant area, variable ressure dro was correlated with neumatic roortional control. Comuter simulation of the different fluids characteristics were carried out using C- rogramming language software develoed for this urose. The model develoed in this research work is found useful in monitoring and redicting the effect of functional arameters on the characteristics of different fluids flowing through an ice late. Keywords: neumatic, fluids, redict, monitor, ice late,. Introduction Orifice lates are devices widely used in industry for continuous measurement of the rate of fluid flow in ies. For the urose of design and develoment of mathematical model, they use the same rincile as a venture nozzle, namely Bemoulli s rincile which states that there is a relationshi between the ressure of the fluid and the velocity of the fluid, when the velocity increases, the ressure decreases and vice versa. In this study, the ice late was be used as the key device to roagate the mathematical model that will redict and monitor constant area, variable ressure dro using different fluids. It is a thin late with a hole at the middle usually laced in a ie in which fluid flows. When the fluid reaches the ice late with the hole in the middle, the fluid is forced to converge and ass through the small hole the oint of maximum convergence actually occurs shortly downstream of the hysical

2 ice, at the so called vena contracta oint, as shown in the Figure below. (Cunningham, 95; Ogoni and Ukaaka, 004; Creankolis, 993; erry and erry, 984; Yuxi, Zhen, Yamin, Riyao and Xi, 008 and Zhn, Ndegwa and Luo, 00). s it does so, the velocity and the ressure changes. Beyond the vena contracta, the fluid exands and the velocity ressure change once again. By measuring the difference in ressure between the normal ie section and at the vena the volumetric and mass flow rates can be obtained from Bernoulli s equation. (Jiang, Graham, ndre, Kelsalt and Brandon 00; Bradford, Tadassa, and Jin, 006; JIC, 000; Uyigue and Umoh 007; Fried, 000; Winkler, 997; and LENTECH, 005) The imortance of this study of using an ice late device rather than other instrument is that the ice affords a larger surface area er unit volume for mass transfer through the ies. They are readily used extensively for continuous measurement of fluid in ies and in some small river system to measure flow at locations where the river asses through a culvert or drain. Only a small number of rivers are aroriate for the use of the technology since the late must remain comletely immersed i.e the aroach ie must be full and the river must be substantially free of debris. In the natural environment large ice lates are used to control onward flow in flood relief dams, in these structures a low dam is laced across a river and in normal oeration the water flows through the ice late unimeded as the ice is substantially larger than the normal flow cross section. However, in floods the flow rate rises and floods out the ice late which can then only ass a flow determined by the hysical dimensions of the ice. Flow is then held back behind the low dam in a temorary reservoir which is slowly discharged through the ice when the flood subsides. (Chrysikooulos, Hsuan, Fyrillas and Lee, 003; Glaso, 980; l-marhoun, 003; Bergen, Kiko, and Weisenbor 999, Labede, 990; Halliburton services, 978 and Gerard, 998). The urose of this study is to redict and monitor the behaviour of hysical and chemical rocesses of different fluids as they flow in the ie of the ice with resect to a constant area and variable ressure dro. In this study, mathematical models were develoed and tested to monitor and redict the characteristics of the rocess as well as the functional arameters. To understand the concet of variable ressure dro and constant area over the ice under different fluid flow conditions and interret the values obtained by simulation and theoretically in the concets to industrial alications was conducted by various research grous. (Donald, 950; William, 989; Consider, 957; Byron, Warren and Edwin 976 and Eckman 958).

3 This study had contributed extensively to knowledge as it rovides a better way of controlling onward flow in flood relief dams in some of our natural environment. It also gives the design engineer the oortunity of using mathematical model to measure flow at location where rivers asses through culverts or drains. The scoe of the work involves the flow of different fluids through an ice and designing and develoing a mathematical model that redict and monitor the constant area and variable ressure dro, and this develoed models were related to the neumatic roortional controller. The rincile involves lacing a suitable fixed area flow restriction in the ie. This restriction causes a ressure dro which varies with the flow-rate. Thus, measurement of the ressure dro by means of a suitable differential ressure icku allows flow rate measurement. In the case of ice is mainly used for the determination of the above arameters with different fluids flowing through the ie.. Materials and Methods. Density Measurement/Samling The density of the various substance used for the investigation was determined using hydrometer. The various fluid used for the laboratory exeriment was obtained in the Deartment of chemical etrochemical Engineering, Rivers State University of Science and Technology Nkolu, ort Harcourt in Nigeria.. Comutational rocedure The develoed model was simulated using the exerimentally determine values inconjection with theoretically assumed values of flow rate 8m 3 /s, 0m 3 /s, 7m 3 /s and 4m 3 /s, cross sectional area of ice m and cross sectional area of ie 5m. The exerimental values of densities of various fluid is given as water 873.8kg/m 3, ethyl ethanol 785.kg/m 3 and Butane ch 5 CH 5O H o 000kg/m 3 benzene c 6 H 6 C 4 H 0 599kg/m 3 all samles analyzed at 5 o C. The exerimental and the theoretical data were fed into the develoed equations (5) and (7) to determine the functional arameters..3 Exerimental rocedures The method involves lacing a suitable fixed area flow restriction in the ie of an ice the restriction causes a ressure dro which varies with flow-rate as the different fluids flow through the ie. Thus measurement of the ressure dro by means of a suitable differential

4 ressure dro ick u allows flow rate measurement. Several emirical correlations exist in this methods to redict how the ice will oerate relating ressure dro and assuming steady-state, incomressible (constant fluid density), in viscid, laminar flow in a horizontal ie with negligible friction losses. One of those most useful equations is the Bernoulli s equation which reduces to an equation relating the conservation of energy between two oints on the same streamline as will be shown in the mathematical model. ie wall Flow in d3 d d Flow out ie wall ie wall Figure : Systematic diagram of fluid flow in ie system.4 The Model Mathematical concet was used in develoing the model for this aer based on flow characteristics such as V gz V gz V 0 gz constant () If the elevation effects are negligible, if the otential energy term gz becomes zero (gz 0), then equation () reduces to, V constant () Multily through equation () by, we have V constant (3)

5 V V (4) From continuity equation, we have that V V (5) V V (6) From equation (6) we have V (7) V (8) Substituting equations (7) and (8) into equation (4), we have (9) Simlifying equation (9), we have (0) () () ( ) ( ) ( ) (3) Multiling through equation (3) by /, we have ( ) (4) ( ) (5)

6 ( ) (6) ( ) (7) Δ (8) Where, Δ ressure dro. ll other arameters remain the same equation (7) is the therefore the mathematical model that is related to the neumatic roortional controller, which can be reresented in form of flow diagram as shown in Figure. Figure : roortional neumatic Controller The outut ressure can be exressed as; o K c E (9) Recalling equation (7) and making outut ressure the subject of formula, we have (0) ( )

7 Equation (0) can be written as ( ) () () (3) ( )( ) Δ (4) (5) Equating equation (9) and (5), that is therefore we have o K C E (6) - o K C E (7)

8 Thus o Δ therefore equation (7) can be written as Δ I K E c (8) 3. Results and Discussion The results obtained from the investigation are resented in Tables and Figures as shown below. The various functional arameters were evaluated and results obtained resented in this aer. The result resented in Figure 3 illustrates the variation in ressure dro of the system in resent of neumatic roortional control as well as absent of neumatic roortional control with variation in flow rate. Increase in ressure dro was observed for both systems with increase in flow rate. The ressure dro was higher un system with the absent of neumatic roortional control than the one with the resent of neumatic roortional control for water. From the results, shown it can be deduced that there is variation in the ressure dro of the different fluids as the flow rate varies. n increase in the flow rate shows an increase in the ressure dro if the different fluid flowing through the ie. lso, the deviation in densities of the different fluids shows a variation in the ressure dro. Fluids with reduced densities had a lower ressure dro. Hence the lower the density the lower the ressure dro and the higher density the higher he ressure dro. There the density of the fluids is directly roortional to the ressure dro at constant area. Table. Exerimentally and theoretically values of functional arameters of different fluids. S/N Fluids Density at Flow rate ressure ressure

9 5 o C (m 3 /S) (kg/m 3 ) Water Benzene Ethyl ethanol Butane dro Δ (ka) dro Δ (ka) (m 6 /s ) ressure dro Δ and Δ (ka) ressure dro Delta (ka) ressure dro Delta ' (ka) Flow rate (m 6 /s ) Figure 3 Grah of ressure dro Δ and Δ (ka) versus flow rate (m 6 /s ) for water

10 5 ressure dro Delta (ka) ressure dro Delta ' (ka) ressure dro Δ and Δ (ka) Flow rate (m 3 /s) Figure 4: Grah of ressure dro Δ and Δ (ka) versus flow rate (m 3 /s) for Benzene In Figure 4, it is seen that both ressure dro increase with increase in flow rate. The variation in concentration of the ressure dro can be attributed to the variation in the flow rate. The ressure dro is higher for the normal oeration of fluid flow characteristics where the actual flow rate was evaluated in the absent of roortional neumatic control involved. The aim of the neumatic roortional control is to hel reduce the ressure dro in the system as well as the error range value for benzene. 5 ressure dro Delta (ka) ressure dro Delta ' (ka) ressure dro Δ and Δ (ka) Flow rate (m 3 /s) Figure 5: Grah of ressure dro Δ and Δ (ka) versus flow rate (m 6 /s ) for Ethyl ethanol Figure 5 illustrates the relationshi between the ressure dro in a fluid flow system of normal characteristics that is in the absent of roortional neumatic control as well as the

11 rocess that involved roortional neumatic control system. The results obtained sowed that minimum error is observed in the ressure dro with the system in resent of roortional neumatic control than the system in the absent of roortional neumatic control in rocess fluid flow of ethyl ethanol. The variation in the concentration of ressure dro for both systems can be attributed to variation in the flow rate of fluid in the ie. 0 8 ressure dro Delta (ka) ressure dro Delta ' (ka) ressure dro Δ and Δ (ka) Flow rate (m 6 /s ) Figure 6: Grah of ressure dro Δ and Δ (ka) versus flow rate (m 6 /s ) for Butane Figure 6 illustrate the relationshi between the ressure dros of the both rocess with flow rate characteristic of butane. Increase in ressure dro for both systems was observed with increase in flow rate of butane. The variation in the ressure dro for both systems can be attributed to the variation in the flow rate characteristics of fluid (butane). Conclusion In conclusion, the flow of fluids through thin-late ices romotes increase in the flow rate as the down stream ressure is lowered, and ressure dro increases. Hence the mathematical model develoed is of great relevance as it is used to determine the gradual discharge of fluids through an ice. For examle, in the natural environment, where large ice lates are used to control onward flow in flood relief dams, the efficiency is deendent of the neumatic roortional control of the system. The flow characteristics of water, ethyl ethanol, benzene and butane are lowered as the different fluids asses through the ice late, which resulted to decrease in ressure.

12 Nomenclature d3 ie diameter (mm) d vena contracta diameter (mm) d ice diameter (mm) V ustream fluid velocity (m/s) V fluid velocity through the ice hole (m/s) Fluid ustream ressure, a (kg/m.s ) fluid down stream ressure a (kg/ms ) Fluid density, kg/m 3 Cross-sectional area of the ie (m ) Cross-sectional area of the ice late (m ) Volumetric flow rate (m 3 /s) V V Outut signal ressure (kg/m.s ) K c roortional gain o outut signal when there is no error E Error References. M.. l-marhoun, (003), The Coefficient of Isothermal Comressibility, SE aer 843. aer reared for resentation at the SE 3 th Middle East Oil show & Conference to be held in Bahrain 5-8 ril, ; Bergen, H Kiko, and T Weisenborn, (999), Research Note on hase Behaviour Guidelines, ET R0, SIERTS Rijswijk, Dec, S. Bradford,; Y.F Tadassa, and Y Jin, (006), Transort of Colihage in the resence and bsence of Manure Susension, J. Environ. ual,vol. 35, R. B. Byron; E. S. Warren and N. L. Edwin, (976), Transort henomena. uinn- Woodbrine-Inc ubsliher, New York London, C. V. Chrysikooulos;. Y. Hsuan; M. M. Fyrillas and K. Y. Lee, (003), Mass Transfer Coefficient and Concentration Boundary Layer Thickness for a Dissolving (Non-queous hase Liquid) NL ool in orous Media. Int. Journal of Hazardous Material. vol. 97, D. M. Consider, (9570, rocess Instruments and Controls, handbook New York McGraw-Hill,. 3-0.

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