Laminar tube flow of sieved beef-cattle manure slurries

Agricultural Wastes 15 (1986) 35-49

Laminar Tube Flow of Sieved Beef-Cattle Manure Slurries Y. R. C h e n Roman L. Hruska US Meat Animal Research Center, Agricultural Research Service, US Department of Agriculture, Clay Center, Nebraska 68933, USA

ABSTRACT The rheological properties of beef-cattle manure slurries were studied using a tube viscometer. The previously proposed rheological model ( Chen, 1986) is used to describe the relationship between the shear stress and shear rate at the tube wall. The results are compared with those obtained with a rotational viscometer. A modified Reynolds number is defined and used to relate to the friction loss of the laminar flow in the tube. The Reynolds numbers where the laminar flow region terminated are also discussed.

INTRODUCTION The rheological properties of livestock manure slurries have great effects on the power and energy requirements when the slurry is pumped, mixed or aerated. The power and energy requirements for pumping and mixing slurries increase and oxygen transfer efficiency decreases as the slurry becomes more viscous (Hashimoto & Chen, 1975; Chen & Hashimoto, 1976; Chen, 1980, 1981). Understanding the rheological properties of manure slurry is necessary for rational designs of systems to handle and process the manure. The rheological properties of beef-cattle manure slurries have been studied using a rotational viscometer and reported previously (Chen, 1986). Beef-cattle manure slurry was found to be non-Newtonian pseudoplastic and the Power Law model could be used to describe the 35

36

Y. R. Chen

relationship between shear stress and shear rate only for slurries with TS < 4.5 ~o. For higher TS slurries the Power Law model could not be used. Different rheological models have been attempted to describe the relationship. It was also found that the cattle manure slurry exhibits negligible yield stress, and the Bingham Plastic, Herschel-Bulkley and Casson models were not applicable. The curvilinear relationship of shear stress and shear rate in the logarithmic plot for high TS slurries was due to the existence of the limiting viscosity. For beef-cattle manure slurry, a rheological model was proposed (Chen, 1986): z = tlo7 + K " ~ 'n''

(1)

where z is the shear stress; $ is the shear rate; r/o is the limiting viscosity; and K" and n" are rheological parameters. Nonlinear, least-squares regression results showed that the proposed model fitted very well to the rheological data of the slurries with TS > 4 . 5 ~ . Both t/0 and K" for sieved manure slurries increased as TS increased. The value of n" did not vary with TS or temperature. An average value of n" was found to be 0.307. This paper reports the results of the study of the rheological properties of sieved beef-cattle manure slurries using a tube viscometer. The previously proposed rheological model (Chen, 1986) is then used to describe the relationship between the shear stress and shear rate at the tube wall. The results are compared with those obtained with a rotational viscometer. A modified Reynolds number is defined and used to relate to the friction loss of the laminar flow in the tube.

MATERIAL The manure used in this study was 1-10 days old, and collected from steers confined in concrete-floored pens that were covered over the feed bunk area. The steers were fed a ration of 80 ~ corn, 18 ~ silage and 2 9/0 soybean meal. Water was added to the raw manure to form slurries and the coarse solids were removed from the slurries by passing the slurries through a vibrating sieve with a 10-mesh screen (opening 2 mm). The vibrating sieve was the same one used previously by Gilbertson & Nienaber (1978). After the coarse particles were removed with the vibrating sieve, the slurries were subjected to the tube and rotational viscometry tests. Results

[

Laminar tube .[tow of sieved beef-cattle manure slurries

37

of the rotational viscometry test have been reported separately (Chen, 1986).

METHODS

Apparatus Figure 1 shows a schematic diagram of the system for the rheological property measurements. A closed-loop piping system made of copper tubing of 1.3cm i.d. (nominal) was used as the tube viscometer. The storage tank, which was made of Plexiglas, had dimensions of 0.66 x 0.508 x 0.457m, equipped with a mixer to maintain the uniformity of the slurry, and was insulated with styrofoam. A constant head was maintained in the storage tank by overflowing the slurry into the overflow tank. The slurry in the overflow tank was constantly pumped back to the storage tank. The slurry was pumped into the piping system by a centrifugal pump powered by a 1 h.p., variable-speed DC motor. The slurry flowed through TEMPERATURESENSOR ~ CONCENTRICTUBE

DRAIN.~

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A schematic diagram of the capillary tube viscometer.

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Laminar tube .[tow o f sieved beef-cattle manure slurries

39

the viscometric section, which was an 8.33m long straight tube. The distance between the pressure taps was 6.096 m. The manometer taps had openings of 0.953 cm and were adjusted to the same high with a transit. The pressure drop between the manometer taps was measured with a differential pressure transducer (Model DP15-30, Validyne Engineering*), which was calibrated using a U-tube manometer installed in line. Pressure transmission devices (PTD) shown in Fig. 2 were fabricated and installed between each pressure tap and pressure transducer with slurry on one side and water on the other side. This prevented slurry from contaminating the pressure transducer. The membrane in each PTD was made of a thin rubber commonly used for surgical gloves. The same distances of deformation of the membrane were maintained for each PTD so that the pressure taken offby the thin rubber cancelled out on both sides of the differential pressure transducer. The length of the straight-pipe section before the pressure tap on the inflow end was 121 diameters of the pipe (1.54 m) so that the uniform flow profile was fully developed before the slurry flowed through the section between the pressure taps. After leaving the viscometric section, the slurry flowed through a heating section, which consists of a concentric-tube heat exchanger. The hot water, which was supplied by a hot-water heater, flowed through the annular space of the heat exchanger to maintain the slurry at the desired temperature. From the heating section, the slurry returned to the storage tank. The flow rate was measured by a magnetic-flow meter (Model MK3662PSO, Signet Scientific*). The magnetic-flow meter was calibrated for each slurry concentration against the weight tank's flow rate. Temperatures were measured at the storage tank and right after slurry left the viscometry section. All the readings (temperatures and voltages) were recorded with a data acquisition system (Autodata Nine, Acurex Autodata*).

Computational procedure For a capillary tube viscometer, the shear stress at the wall of the tube is given by: z = O AP/4L (2) * Mention of a trade name, proprietary product or specificequipmentdoes not constitute a guarantee or warranty by the US Department of Agriculture and does not imply its approval to the exclusion of other products that may be suitable.

40

Y. R. Chen

where D is the tube diameter in m; AP is the pressure drop between the pressure taps in Pa; and L is the distance between the pressure taps in m. The true shear rate at the wall of the tube was given by the Rabinowitsch-Mooney equation (Skelland, 1967): ~_3n'+l(~_)4n, and n' -

(3)

d[ln z]

(4)

d[ln(8V/D)]

where V is the flow speed and 8 V/D is called apparent shear rate. In general, n' is not a constant but varies with 8 V/D except for Power Law fluids. Equation (4) shows that the shear stress can be expressed as: ,[-3n'+ 1 {"8 V

"'

where K' and n' are rheological consistency and behavior indices, respectively. ' The measurements of a capillary tube viscometer are converted into a logarithmic plot of z versus 8 V/D, and n' is evaluated as the slope of the curve at a particular value of the shear stress. In this study, In(z) was expressed as a polynomial function of ln(8V/D) by a microcomputer program and n' was the computed derivative of ln(r) with respect to ln(8V/D) (see Chen, 1986). The value of n' at each shear stress was then used in eqn (3) for calculating the true shear rate. M O D I F I E D R E Y N O L D S NUMBER Many different Reynolds numbers, based on different rheological measurements of the liquids, have been defined for non-Newtonian liquids. However, the following generalized Reynolds number given by Metzner (1956), which used the rheological data from capillary tube viscometers, is most useful for scaling-up and predicting the pressure drops in pipes for time-independent, non-Newtonian liquids (Calderbank & Moo-Young, 1959):

,pVD{8V~'-" F 4n' .]"'

NRe ~ ~-S ~,-~-j where p is the liquid density.

L3, ¥1

(6)

Laminar tube flow of sieved beef-cattle manure slurries

41

The correlation between the friction coefficient, which is defined by: (7)

f = O A P / ( 2 L p V 2)

and the generalized Reynolds number in the laminar flow region is given by: f = 16N~ 1

(8)

Equation (8) is universally applicable to all liquids in the laminar region, whether they are Newtonian or non-Newtonian. Dodge & Metzner (1959), working on purely viscous pseudoplastic fluids, found that different f-NRe correlations are needed for different n' values in the turbulent flow region for a smooth pipe. They showed that the friction coefficients for pseudoplastic fluids are generally lower than those for Newtonian fluids at the same generalized Reynolds number. The smaller the n' value, the lower the friction coefficient at the same NRe. The end of the laminar region is delayed compared to the Newtonian case which occurs at NRe = 2000. Similar conclusions were obtained by Chen & Hashimoto (1976) for the aerated dairy waste slurries flowing through a pipe. They used Power Law equations to describe the rheological properties of aerated dairy waste slurries. For the slurries which do not obey the Power Law, eqn (6) is not very practical to use, because n' varies with the shear stress. For the proposed rheological model (eqn (1)), a different Reynolds number was needed. Since eqn (1) shows that the shear stress is the sum of the shear stress contributed by the Newtonian resistances (t/o~) and the shear stress contributed by the non-Newtonian resistances (K"~n"), the reciprocal Reynolds number should be expressed as the sum of the reciprocals of the Reynolds number due to the Newtonian effect (N~¢) and that due to the Power Law effect (N~t¢): NRe = ~NRe +

(9)

where o

NRe

pVD -

-

(10)

t/o

and NR¢ - K" \ D ]

n E3n"4n+

1"

1

(1 l)

42

Y. R. Chen

RESULTS To determine the average diameter (D) of the copper tubing of the tube viscometer, water was first tested in the system at flow rates ranging from 1-492 to 883 cm 3 s- 1. The results of the water flow in the laminar range were then used to calibrate D as described by Chen & Hashimoto (1976). The result gave an averaged diameter of 1-367cm with a standard deviation of ___0.023 cm (4 data points). This value of D was subsequently used as the tubing diameter, instead of the value given by Bolz & Tuve (1973) (1.342cm i.d. for 18K gage copper tubing). Figure 3 gives the friction coefficient of water in the laminar and turbulent ranges from this experiment. It shows that the experimental friction coefficients agreed well with established relationships for transition and turbulent regions (solid line) given by Moody's diagram for smooth pipe (Perry & Chilton, 1973). The end of the laminar flow region occurred at a Reynolds number of 2000. The good agreement of the experimental results of water with previously published results in the transition and turbulent ranges indicated that the flow rate and pressure drop measurements were accurate.

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Plot of the friction coefficient against Reynolds number for water.

Laminar tube flow of sieved beef-cattle manure slurries

43

SIEVED CATTLE MANURE SLURRIES "-2.50% 44.6 * C • -2.82% 43.3 ° C 0-4.49% 45.9 *C

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Typical plots of pressure drop per unit length of tubing against the flow rate of the slurry at different TS.

Figure 4 shows the typical plots of pressure drop per unit length of the tubing against the flow rate at different TS. Figure 4 shows that as the flow rate increased, the pressure drop also gradually increased for each TS. However, when the flow rate was above a critical value, the pressure drop increased sharply, indicating that the flow had become turbulent. Figure 4 also indicates that the critical flow rate increased as TS increased. Figure 5 shows the rheological curves of sieved beef-cattle manure slurries of different TS at the temperature range between 16 and 35 °C in the laminar flow range (i.e. at flow rates less than the critical rate). Figure 5 shows that the rheological curve of slurries with TS greater than 2-84 cannot be represented by the Power Law. Table 1 summarizes the results of the fits of the Power Law model to the rheological data of the slurries with TS equal to 2.6% and 2.8%. They showed reasonably good correlations. Table 2 summarizes the results of the fit of eqn (1) to the rheological data of higher TS slurries. Table 2 indicates the excellent fit of the proposed rheological model (eqn (1)) to the flow curves of the sieved beef-cattle manure slurries in tube flow. The curves given by the parameters obtained through regression are also plotted in Fig. 5. Again, they all show a very good fit to the data. This indicates that the model not

Y. R. Chen

44

TABLE 1 Rheological Parameters of Sieved Cattle Manure Slurries Obtained by a Tube Viscometer (Power Law Model)

TS (%) 2.62 2.58 2.53 2.50 2.63 2.80 2.84 2.80 2-82 2-80

Temp. (°C) 19.9 ± 23.6 ± 35.4 + 44.7 ± 55.1 ± 65.1 ± 25.1 ± 33.9 ± 43.3 ± 54.1 ±

0.87 1-06 0-57 0.92 1-69 1-20 0-78 0.14 0.57 0-92

K' (Pa "s "°)

n' (dimension&ss)

CorreCtion coefficient

0.061333 0.035502 0.038878 0'021638 0'014258 0.018455 0.041903 0'017211 0-022600 0.012701

0.449614 0.584298 0.541895 0.605441 0.638501 0.582662 0.615279 0.740327 0.621442 0-723882

0-99592 0-98727 0.97896 0.99596 0.99374 0.99588 0.98617 0.98977 0.99819 0.98756

TABLE 2 Rheological Parameters of Sieved Cattle Manure Slurries Obtained by a Tube Viscometer (Previously Proposed Model)

TS (%) 4.79 4"50 4"84 4'49 7"46 7.45 7.51 7'64 7.42 7.43 7.41 7"55 8-13 8'35 7-87 7-91 8'51

Temp. (°C) 14'3 ± 1 . 2 7 14'2 ± 1 " 0 6 27"8 ± 0 " 0 7 45'9 ± 0.99 19.5 ± 0.57 20-5 ± 0.99 36"0 ± 0'64 35.1 ± 0'35 51.3 ± 0-42 50.1 ± 0.28 58'9 ± 2.05 59'7 ± 0.00 16"7 ± 1 " 0 6 27'6 ± 0 " 9 2 38'0 ± 0.64 48.9 ± 1'20 57-6 ± 0.35

To (Pa.s) 0"006698 0'007499 0"006091 0'002558 0'008857 0"006189 0"005532 0"007332 0"006219 0'005644 0'006244 0"006337 0.003462 0"004514 0'007206 0'004657 0"006077

K" (Pa.s") 0"211618 0'192809 0"241272 0.113522 0'633069 0'418607 0'596122 0'465566 0'560187 0'458091 0.522970 0'566486 0"282892 0"274785 0.450946 0'335649 0"241652

n"

0'382329 0'314344 0'279998 0.403234 0'360848 0'425950 0'352081 0'331450 0"289792 0.302787 0"243826 0"203066 0"557009 0"509225 0-413686 0-452203 0.407851

Correlation coefficient 0"99978 0"99937 0'99828 0"99845 0'99920 0"99947 0"99937 0'99966 0"99978 0'99966 0"99930 0"99963 0"99905 0"99967 0"99896 0-99928 0.99981

45

Laminar tube flow of sieved beef-cattle manure slurries I00

SIEVED

CATTLE MANURE SLURRIES

(TS, TEMP)

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( S -I)

Plots of shear stress and shear rate relationship of slurries in laminar flow. (Tube viscometer data.)

only fits the data from rotational viscometry very well (Chen, 1986), but that it is also quite useful for capillary tube viscometer results.

Comparing capillary and rotational viscometer results Figure 6 compares the results of the capillary and rotational viscometers for temperatures ranging from 14 to 57°C. The rotational viscometer's results were reported in a previous paper (Chen, 1986). It shows that the results of capillary tube and rotational viscometer tests agreed with each other rather well, except in the low shear rate range of the slurries with TS = 2.6 ~o and 2.8 ~o, where the shear stresses obtained by the rotational viscometer were lower than those for the capillary tube viscometer. This may be due to the fact that solids tend to settle in the rotational viscometer when the TS and shear rates are low.

The f-NRe correlation for cattle manure slurries Figure 7(a) shows the correlation between f and NRe for the slurry with TS = 2-8~. It shows that in the laminar region the data follow the

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2.62 % 2 0 *C

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Laminar tube flow of sieved beef-cattle manure slurries

47

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48

Y. R. Chen

Newtonian relationship (f = 16NRel). However, the end of the laminar region occurred at a Reynolds number of about 1500, which was much lower than the case for Newtonian liquids (at a Reynolds number of 2000). Figures 7(b), (c) and (d) show the correlations between f and NRe for higher TS. In the laminar region the data closely follow the Newtonian relationship, f = N~¢1 (solid lines). The average value of f times NRe for all the data points in the laminar flow region (220 points) was found to be 16.23 with a standard deviation of + 1.31, which is no different from the value (16) for Newtonian liquids, indicating that the modified Reynolds number was useful for correlation with the friction coefficient. Also, in every case, laminar region terminated at a modified Reynolds number of about 2000, which agreed well with the results of Newtonian liquids.

SUMMARY AND CONCLUSIONS The tube flow (Poiseuille flow) of sieved cattle manure slurries was studied using a capillary tube viscometer. It was found that the Power Law model can only be used to describe the rheological properties of the slurries having low total solids concentrations (2.6 ~o and 2-8 ~o TS). For the slurries with higher TS, the model previously proposed for the Couette flow (eqn (1)) was successfully used to describe their shear stress and shear rate relationships. The slurry's shear stress and shear rate relationship obtained by the. tube viscometer agreed with the results of the rotational viscometer, except in the low shear rate range with TS equal to 2.6 ~ and 2.8 ~ where solids settling may have been a problem for rheological measurements. A modified Reynolds number was defined and used to correlate with the friction coefficient in the tube flow. In the laminar flow region the relationship between the friction coefficient and the modified Reynolds number was the same as in Newtonian liquids. The end of the laminar flow region was found to occur at a modified Reynolds number near 1500 for the slurries with TS equal to 2-6 ~o and 2.8 ~o where the Power Law model was used. However, for the slurries with higher TS (the proposed model was used to describe their rheological properties), the laminar region terminated at the modified Reynolds number near 2000, which agreed well with the results of Newtonian liquids.

Laminar tube flow of sieved beef-cattle manure slurries

49

ACKNOWLEDGEMENTS The author appreciates the technical assistance of Eldon Shetler, R a n d y Bradley and Miriam Eckblade.

REFERENCES Bolz, R. E. & Tuve, G. L. (1973). Handbook of tables for applied engineering science. CRC Press, Cleveland, Ohio, 1060 pp. Calderbank, P. H. & Moo-Young, M. D. (1959). The prediction of power consumption in the agitation of non-Newtonian fluids. Trans. Instn. Chem. Engrs., 37, 26-33. Chen, Y. R. (1980). Impeller power requirement in mixing livestock manure slurries. Trans. ASAE, 24, 187-92. Chen, Y. R. (1981). Scale-up of mechanical mixing for livestock slurries. Paper No. 81-4065, American Society of Agricultural Engineers, St Joseph, Michigan. Chen, Y. R. (1986). Rheological properties of sieved beef-cattle manure slurry: rheological model and effects of temperature and solids concentration. Agric. Wastes, 15, 17-33. Chen, Y. R. & Hashimoto, A. G. (1976). Pipeline transport of livestock waste slurries. Trans. ASAE, 19, 898-902, 906. Dodge, D. W. & Metzner, A. B. (1959). Turbulent flow of non-Newtonian systems. AIChE J., 5, 189-204. Gilbertson, C. B. & Nienaber, J. A. (1978). Separation of coarse solids from beef cattle manure. Trans. ASAE, 21, 1185-8. Hashimoto, A. G. &Chen, Y. R. (1975). Turbine-air aeration system for poultry wastes. In: Managing livestock wastes. American Society of Agricultural Engineers, St Joseph, Michigan, pp. 530-4. Metzner, A. B. (1956). Non-Newtonian technology. In: Advances in Chemical Engineering I (Thomas, B. D. & Hoopes, J. W., Jr (eds)). Academic Press, New York, pp. 111-42. Perry, R. Y. & Chilton, C. H. (1973). Chemical engineers' handbook, 5th edn. McGraw-Hill, New York, pp. 5-22. Skelland, A. H. (1967). Non-Newtonianflow on heat transfer. John Wiley, New York.