Brazilian Journal

of Chemical ISSN 0104-6632

Printed in Brazil
Engineering
Vol. 20, No. 04, pp. 455 - 463, October - December 2003

FRICTION LOSSES IN VALVES AND FITTINGS

FOR POWER-LAW FLUIDS
M.A.Polizelli1, F.C.Menegalli2, V.R.N.Telis1 and J.Telis-Romero1*
1
Departamento de Engenharia e Tecnologia de Alimentos, Universidade Estadual Paulista,
Phone (55) (17) 221-2251, Fax (55) (17) 221-2299, 15054-000,
São José do Rio Preto - São Paulo, Brazil.
E-mail: javier@eta.ibilce.unesp.br
2
Departamento de Engenharia de Alimentos, Universidade Estadual de Campinas,
13083-970, Campinas - São Paulo, Brazil

(Received: October 5, 2002 ; Accepted: July 11, 2003)

Abstract - Data on pressure drop were obtained in stainless steel, sanitary fittings and valves during laminar and
turbulent flow of aqueous solutions of sucrose and xanthan gum, which were selected as model fluids. The
rheological properties of these solutions were determined and the power-law model provided the best fit for
experimental data. Friction losses were measured in fully and partially open butterfly and plug valves, bends and
unions. Values of loss coefficients (kf) were calculated and correlated as a function of the generalized Reynolds
number by the two-k method. The model adjustment was satisfactory and was better in the laminar flow range
(0.976 ≤ r2 ≤ 0.999) than in the turbulent flow range (0.774 ≤ r2 ≤ 0.989). In order to test the adequacy of the results
for predicting loss coefficients during flow of real fluids, experiments were conducted with coffee extract.
Comparison between experimental and predicted loss coefficients showed very good agreement.
Keywords: pressure drop; non-Newtonian fluids, friction factors, loss coefficients.

INTRODUCTION of piping systems often find it difficult to obtain the

necessary resistance coefficient values, since the
The design of piping and pumping systems for amount of available data in the literature is quite
chemical, pharmaceutical and food processing limited. The Crane Company (1982) published an
industries requires knowledge of the pressure drop extensive tabulation of loss coefficients for turbulent
due to flow in straight pipe segments and through flow of Newtonian fluids. For this same class
valves and fittings. Friction losses caused by the of fluids, flowing under laminar conditions, the
presence of valves and fittings usually results from classic reference is the Chemical Engineer’s
disturbances of the flow, which is forced to change Handbook (Perry and Chilton, 1986), while
direction abruptly to overcome path obstructions and Kittredge and Rowley (1957) have also published
to adapt itself to sudden or gradual changes in the some data. Other research has been conducted on
cross section or shape of the duct. non-Newtonian fluids, mainly for pseudoplastic
Evaluation of the friction loss in valves and fluids in laminar flow (Martínez-Padilla and Linares-
fittings involves determination of the appropriate García, 2001; Telis-Romero et al., 2000; Banerjee et
loss or resistance coefficient, k, which is calculated al., 1994; Das et al., 1991; Edwards et al., 1985;
from experimental measurement of the pressure drop Steffe et al., 1984). Turian et al. (1998) provided loss
in the fitting. coefficients for turbulent flows of concentrated non-
Engineers and technicians involved in the project Newtonian slurries, whereas Griskey and Green

*To whom correspondence should be addressed

456 M.A.Polizelli, F.C.Menegalli, V.R.N.Telis and J.Telis-Romero

(1971) presented data for dilatant fluids. The Fanning friction factor is defined as (Garcia
Except for the work of Martínez-Padilla and and Steffe, 1987)
Linares-García (2001) and Steffe et al. (1984),
experimental data on resistance coefficients have ∆PD
been collected using carbon steel valves and fittings. f= (4)
2ρv 2 L
Nevertheless, in the food, the pharmaceutical and
some chemical industries, fluids should be handled
where ∆P is the pressure drop observed in a length L
by means of sanitary piping components, which are
of straight tube.
made of stainless steel and often have distinct design
In the case of laminar flow, the friction factor can
patterns in order to assure hygienic cleanliness and
be obtained from a simple function of the
bacteriological safety.
generalized Reynolds number, which is identical to
Considering the lack of published data and the
the dimensionless form of the Hagen-Poiseuille
practical importance of their knowledge, the purpose
equation (Darby, 2001)
of this work was to obtain loss coefficients in the
laminar and turbulent flows of power-law fluids
through stainless steel, sanitary valves and fittings. 16
f= (5)
Reg
Friction Loss in Valves and Fittings
in which
In the flow of an incompressible fluid through a
horizontal section of uniform pipe with no work
Dn v(
2− n ) n
input/output, the mechanical energy balance can be ρ  4n 
Reg =   (6)
( n −1)
written as (Darby, 2001) 8 K  1 + 3n 

 P1 - P2  Equations (5) and (6) can be used for both


 ρ 
= ∑F (1) Newtonian and power-law fluids, since for
Newtonian fluids the behavior index, n, equals 1, and
the consistency index, K, equals the dynamic
where P is the flow pressure and ρ is the fluid viscosity, η, for Newtonian fluids. In this case the
density, while the subscripts indicate points 1 and 2, generalized Reynolds number (Equation 6) reduces
respectively. The term ∑ F accounts for the friction to Re = Dvρ η .
losses, which include losses in the straight pipe Friction factors for Newtonian fluids in turbulent
section and from expansions, contractions, valves flow can be calculated by the Nikuradse correlation,
and fittings in the system. These can be formulated as an empirical modification of the von Karman
equation given by
2fv 2 L kf v2
∑ ∑ F=
D
+ ∑ 2
(2) 1
(
= 4.0log Re f - 0.4 ) (7)
f
The friction losses in straight portions of the
piping system are based on the Fanning friction According to Darby (2001), Equation (7) is also
factor, f, and are represented by the first term at the known as the von Karman-Nikuradse equation and
right of Equation (2), where v is the average flow agrees well with experimental data for friction loss
velocity, D is the inside diameter of the tube and L over the range 5x103 < Re < 5x106.
the tube length. Pressure drops in valves and fittings An empirical equation that gives results similar to
are calculated with the loss coefficient, kf, as in the those of Equation (7) was proposed by Drew et al.
last term of the same equation. (apud Govier and Aziz, 1972) and has the advantage
Equations (1) and (2) can be combined to permit of being simpler, as is explicit in f:
the experimental determination of kf in the following
way: f = 0.00140 + 0.125Re −0.32 (8)

2  P1 − P2 2fLv 2  For the turbulent flow of power-law fluids, the

kf =  −  (3) still recommended correlation for the friction factor
v 2  ρ D 
is that obtained by Dodge and Metzner (1959):
Brazilian Journal of Chemical Engineering
 Friction Losses in Valves 457

1 4
= 0.75 log Reg .f (
f n
(
1− n 2 ) 0.4
− 1.2
n
) (9)
viscometer constant, while shear rate values were
obtained according to Krieger and Elrod (1953).
The performance of the viscometer was checked
using two fluids with well-known rheological
The above equations are applicable to smooth
pipes, which include sanitary piping systems for food properties: ethylene glycol and chlorobenzene, which
and pharmaceutical products (Steffe, 1996). have Newtonian behavior. Twenty-three repetitions
Under conditions of laminar flow, the pressure were accomplished to determine the rheological
drop coefficients for fittings and valves also change properties of each fluid at each of the working
as a function of the Reynolds number, with temperatures (-5, 10 and 70° C for ethylene glycol
increasing values of kf for a decreasing Reynolds and –22, 0 and 20° C for chlorobenzene).
number (Kittredge and Rowley, 1957; Steffe et al.,
1984; Telis-Romero et al., 2000). Some equations Pressure Drop Experiments
are traditionally used to correlate this dependence,
such as the potential equation for Newtonian fluids, The apparatus shown schematically in Figure 1
suggested for the first time by Kittredge and Rowley was used for measuring pressure drop in the
(1957): following stainless steel, sanitary fittings: 180°, 90°
and 45° bends; union; fully open and partially open
−B
k f = A ( Re ) (10) (10°, 20°, 40° and 60° opening angles) butterfly
valve; and fully open and half-open plug valve.
The equipment consists of a stainless steel piping
A less traditional approach is the two-k method
system connected to a stainless steel cylindrical tank
developed by Hooper (1981), which correlates the
pressure drop coefficient with the Reynolds number with a capacity of 100 liters (1). Circular tubes with
and the fitting diameter through the following three different external diameters (25.4 mm, 38.1
equation: mm and 50.8 mm) and a wall thickness of 2.87 mm
were used, and a positive displacement pump (KSB,
k f = k1 Re + k ∞ (1 + 1 D ) (11) model Triglav) (2) pumped the solutions. A butterfly
valve was used to regulate the flow (3) and flow rate
was measured by weighing fluid samples collected at
determined time intervals. Temperature transmitters
MATERIALS AND METHODS
(SMAR, model TT302) were used to measure
Model Fluids temperature (4). Differential pressure transmitters
(SMAR, model LD302) connected to pipes with
Solutions of xanthan gum (Star & Art, Brazil) and silicon tubes were installed throughout the
sucrose (commercial grade) were prepared at three equipment to measure static pressure. Flow pressure
different concentrations of xanthan gum (0.05, 0.15 varied from 0.209 kPa to 4.352 kPa. A HP data
and 0.25% w/w) and sucrose (10, 20, and 30% w/w). logger model 75.000-B, an interface HP-IB and an
The solutes were dissolved in distilled water with the HP-PC running a data acquisition and control
help of a mechanical agitator (Marconi, model program written in IBASIC monitored temperatures
MA59, Brazil). The mixing time was set at 20 and pressures (5).
minutes. In order to guarantee the complete Performance of the apparatus was checked using
hydration of the polymer, samples were kept at rest ethylene glycol, which was pumped through the
in a refrigerated chamber (10° C) during 24 hours. equipment at 31 different flow rates in the laminar
Densities of the model solutions were obtained by range and 25 flow rates in the turbulent domain. At
picnometry. each flow rate, pressure drop in a straight pipe
section of 0.80 m was measured with ten repetitions
Rheological Properties made at five-minute intervals.
The resistance coefficients in fittings and valves
Rheological measurements were taken using a were determined by measuring the pressure drop
Rheotest 2.1 viscometer (MLW, Germany) of the during the flow of model solutions through a specific
Searle type, equipped with a coaxial cylinder sensor fitting and a defined length of straight pipe.
(radii ratio, Rext/Rint = 1.04). A thermostatic bath was Measurements were obtained for each accessory
used to keep the working temperature at 32° C. The under 107 different conditions for laminar flow and
instrument was operated at 44 different speeds (from 49 conditions for turbulent flow. After adjustment of
0.028 to 243 rpm), which were changed stepwise the desired flow rate, the differential pressure data
with a selector switch. Shear stress values (σ) were were collected with ten repetitions made at five-
obtained by multiplying torque readings by the minute intervals.
Brazilian Journal of Chemical Engineering, Vol. 20, No. 04, pp. 455 - 463, October - December 2003
458 M.A.Polizelli, F.C.Menegalli, V.R.N.Telis and J.Telis-Romero

0.30 m 2.20 m 0.30 m 0.30 m

0.30 m
0.30 m 0.80 m 2.50 m

4
2.50 m 0.30 m

1
4
3
0.30 m
0.30 m 2.50 m
2

Figure 1: Schematic diagram of the experimental setup. Distances are in meters.

Table 1: Rheological properties of the standard fluids.

µexp Standard Standard

Temperature µpred*
(mPa.s) Deviation Error
(°C) (mPa.s)
(mPa.s) (mPa.s)

-5 57.70 1.36 0.28 58.2

Ethylene glycol
10 34.22 1.38 0.29 34.0
CH2OHCH2OH
70 4.511 0.08 0.02 4.51
-22 1.632 0.003 0.001 1.63
Chlorobenzene
0 1.291 0.007 0.002 1.29
C6H5Cl
20 0.922 0.013 0.003 0.91
* Data from Perry and Chilton (1986).

RESULTS AND DISCUSSION experimental viscosities and their respective standard

deviations and standard errors (also included in
Rheological Properties Table 1 and calculated as shown by Telis-Romero et
al., 2002), it is possible to say that the values
The accuracy of the viscometer used for the obtained were very close to the data in the literature.
rheological measurements can be observed in Table Rheograms of the xanthan gum/sucrose solutions
1, which compares the experimental viscosity of were obtained at a fixed temperature of 32° C and
ethylene glycol and chlorobenzene with data shear rates in the range of 22 s-1 to 774 s-1. All the
published by Perry and Chilton (1986). Based on the solutions showed non-Newtonian behavior and the

Brazilian Journal of Chemical Engineering

 Friction Losses in Valves 459

power-law, Bingham and Herschel-Bulkley models power-law, also called the Ostwald de-Waele model,
(Equations 12, 13 and 14, respectively) were tested was selected to describe the rheological behavior of
in order to obtain the best fit to the rheological the model solutions. Table 2 contains the rheological
curves. In Equations 12 to 14, σ is the shear stress, γ parameters, K and n, as well as the RMS and r2
the shear rate, σ0 the yield stress, K the fluid values for Equation (12).
consistency index and n the flow behavior index Using the power-law model, statistical analysis of
(Steffe, 1996). The adequacy of the models was K and n dependence on solution composition showed
evaluated by the magnitude of the root mean square, that the consistency index (K) is significantly
RMS (Gabas et al., 2002), and the correlation affected (p < 0.05) by the xanthan gum and sucrose
coefficient (r2). weight fractions. K increased with the square of the
xanthan gum concentration, while increasing sucrose
concentration led K to increase linearly. The flow
σ = Kγ n (12)
behavior index was also significantly affected (p <
0.05) by both solutes. This parameter decreased
σ = σ0 + Kγ (13)
linearly with increasing xanthan gum and sucrose
concentration.
σ = σ0 + Kγ n (14)
Pressure Drop
Use of the Herschel-Bulkley model produced a
slightly better adjustment (0.37 ≤ RMS ≤ 1.75, r2 ≥ In order to evaluate the measurement of pressure
0.998) of the rheological curves than use of the drop in the system, experimental data obtained
power-law model (0.43 ≤ RMS ≤ 1.73, r2 ≥ 0.998). during flow of ethylene glycol were used. Pipe
This was already expected since the former is a dimensions, experimental density and measured
three-parameter model, while the latter is a two- pressure drop were substituted into Equation (4) to
parameter one. The Bingham model did not produce give the friction factor, f. This was then correlated
a satisfactory adjustment, with RMS values in the with the Reynolds number calculated by Equation
range of 7.23 to 11.31 and correlation coefficients (6) using the experimental rheological parameters, in
between 0.911 and 0.954. this case n=1 and K=η. These results are shown in
Even though the Herschel-Bulkley model Figure 2, which also includes predictions of Equation (5)
provided better statistical results, the yield stress for the laminar region and of Equation (7) for turbulent
values obtained were small (0.14 ≤ σ0 ≤ 2.80), conditions. The agreement between experimental and
including some negative values, which is predicted values is very satisfactory, indicating the
meaningless from a physical standpoint. Thus, the adequacy of the equipment and methodology used.

-1 Eq. (5)
10
Eq. (7)
f Fanning

-2
10

-3
10
2 3 4
10 10 10
Re
Figure 2: Experimental and predicted friction factors for ethylene glycol.

Brazilian Journal of Chemical Engineering, Vol. 20, No. 04, pp. 455 - 463, October - December 2003
460 M.A.Polizelli, F.C.Menegalli, V.R.N.Telis and J.Telis-Romero

Experimental friction factors for ethylene glycol The loss coefficients, kf, for fittings and valves
were also submitted to nonlinear regression analysis, were obtained with Equation (3), using the
resulting in Equations (15) and (16) for laminar and experimental values of flow velocity, pressure loss,
turbulent flow, respectively. friction factors in the pipe and densities from Table
2. The two-k method proposed by Hooper (Equation
15.56 11) was adjusted to the results obtained by nonlinear
f= (15) regression, using the root mean square and
Re0.997
regression coefficient to evaluate adjustment quality.
0.089 The resultant parameters of Equation (11), k1 and
f = 0.00113 + (16) k∞, are shown in Tables 3 and 4, respectively for the
Re0.28 laminar and turbulent flow regimes. In these tables,
Equation (15) was adjusted in the range of 165 < the RMS and r2 values were also included. Based on
Re < 2,105 with a correlation coefficient, r2, of the values of RMS and regression coefficients, it can
0.999, and the parameters obtained were very similar be observed that agreement between the
to the theoretical values in Equation (5). Taking into experimental data and the adjusted model is better
account the turbulent region (Reynolds varying from for the laminar than for the turbulent flow, which had
9,428 to 25,141), Equation (16), which resulted in an already been verified for the experiments conducted
r2 value of 0.905, could be compared with the with ethylene glycol. It was also observed that
correlation proposed by Drew et al. (Equation 8). In fittings that cause a larger pressure drop and
this case, the parameters obtained in the present work therefore greater flow turbulence led to the poorest
were a little different from those in Equation (8), but of adjustment of Equation (11), indicating once again
a similar order of magnitude. These results also that the experimental procedure was more reliable in
confirm the suitability of the experimental apparatus. the absence of turbulence.
When measuring pressure drop in fittings and With the purpose of testing the validity of the
valves, inclusion of an additional loss in the parameters obtained, k1 and k∞, pressure drop in the
experimental value due to a straight piping section is same fittings and valves was measured during flow
unavoidable. This contribution should then be of coffee extract. Telis-Romero et al. (2001)
subtracted from the total in such a way that the presented the rheological properties for coffee
remaining value expresses only the fitting friction loss. extract as functions of temperature and
Friction factors corresponding to straight piping concentration, and in the present work, three
sections during flow of xanthan gum/sucrose solutions concentrations of the extract (36, 42 and 51°Brix)
were determined with Equation (4), substituting pipe were selected in order to guarantee pseudoplastic
dimensions, densities from Table 2 and experimental behavior at the experimental temperature (32° C).
values of pressure drop. The results, shown in Figure 3, Figure 4 illustrates the comparison between
were correlated with the generalized Reynolds number experimental resistance coefficients values, calculated
(Equation 6), also calculated with physical properties using Equation (3), and those predicted by the two-k
from Table 2 and experimental values of flow velocity. method (Equation 11) with parameters from Table 3
In addition to the experimental friction factors, Figure 3 and 4, respectively for the laminar and the turbulent
includes the predictions of Equation (5) for the laminar flow ranges for a 90° bend. The observed agreement
range and of the Dodge and Metzner correlation was very satisfactory, with only a few data points
(Equation 9) for the turbulent region. having deviations as large as ±20%.

Table 2: Composition, density and rheological parameters of the model solutions.

Composition (%) Density (kg/m3) K (Pa.sn) n r2 RMS(%)

Solution
Xanthan gum Sucrose
1 10 1047.1 0.137 0.469 0.998 1.73
2 0.05 20 1035.1 0.168 0.442 0.998 1.42
3 30 1028.2 0.187 0.432 0.998 1.31
4 10 1042.1 0.490 0.365 0.999 0.86
5 0.15 20 1035.0 0.555 0.349 0.998 1.08
6 30 1028.1 0.613 0.335 0.999 0.68
7 10 1042.1 0.649 0.279 0.999 0.43
8 0.25 20 1034.9 0.712 0.272 0.998 0.69
9 30 1028.0 0.823 0.253 0.998 0.52

Brazilian Journal of Chemical Engineering

 Friction Losses in Valves 461

-1
10 Eq. (5)
Eq. (9)

fFanning

-2
10
n = 0.9
n = 0.7
n = 0.6
n = 0.5
n = 0.4
n = 0.3

2 3 4 5
10 10 10 10

Reg
Figure 3: Experimental and predicted friction factors for xanthan gum/sucrose solutions.

Table 3: Parameters for calculation of loss coefficients by

the two-k method (Equation 11) in laminar flow.
Fitting k1 k∞ RMS r2
Fully open butterfly valve 9.084 0.0240 1.54 0.998
10o open butterfly valve 14.83 0.0399 2.07 0.995
20o open butterfly valve 298.0 0.8018 2.07 0.995
40o open butterfly valve 1184.6 3.244 3.27 0.988
o
60 open butterfly valve 22579 59.63 4.72 0.976
Fully open plug valve 1022.9 0.2400 4.68 0.994
Half-open plug valve 1768.0 0.3964 7.77 0.984
o
45 bend 503.7 0.2486 1.56 0.999
90o bend 812.2 0.3955 3.11 0.997
o
180 bend 1001.5 0.7066 2.06 0.998
Union 24.86 0.0127 1.51 0.999

Table 4: Parameters for calculation of loss coefficients by the

two-k method (Equation 11) in turbulent flow.

Fitting k1 k∞ RMS r2
Fully open butterfly valve 118.7 0.1587 1.89 0.974
o
10 open butterfly valve 131.2 0.3862 4.67 0.859
o
20 open butterfly valve 250.5 1.136 7.40 0.796
40o open butterfly valve 1747.7 7.112 7.41 0.795
60o open butterfly valve 69778 88.37 8.53 0.636
Fully open plug valve 995.5 0.2402 5.27 0.783
Half-open plug valve 1937.7 0.4110 4.31 0.854
45o bend 465.1 0.2495 0.97 0.989
90o bend 798.9 0.3939 2.09 0.966
180o bend 1089.6 0.6622 7.30 0.774
Union 91.98 0.0805 2.04 0.970

Brazilian Journal of Chemical Engineering, Vol. 20, No. 04, pp. 455 - 463, October - December 2003
462 M.A.Polizelli, F.C.Menegalli, V.R.N.Telis and J.Telis-Romero

20
Laminar
Turbulent

15

kf (Experimental)
10

0
0 5 10 15 20
kf (Hooper)

Figure 4: Comparison of 90° bend loss coefficients using the two-k parameters obtained in this work and
experimental values obtained during flow of coffee extract. Dotted lines indicate ± 20% deviations.

CONCLUSIONS D Inside diameter of tube, m

f Fanning friction factor
The rheological behavior of the model solutions K Fluid consistency index, Pa.sn
could be described well by the power-law model, k∞, k1 Constants in Equation (11)
with K and n being significantly affected (p < 0.05) kf Loss or resistance coefficient
by the xanthan gum and sucrose fractions. The L Tube length, m
experimental apparatus used for measuring friction n Flow behavior index
losses during fluid flow through valves and fittings N Number of experiments
was shown to be satisfactory, and loss coefficients, P Pressure, Pa
kf, were calculated for fully and partially open r2 Correlation coefficient
butterfly and plug valves, bends and union. The two- Re Reynolds number
k method proposed by Hooper could be adjusted to Reg Generalized Reynolds number
the data obtained, resulting in good agreement RMS Root mean square
between predicted and experimental values.
Measurement of pressure drops in the same fittings
and valves carried out during flow of coffee extract ACKNOWLEDGMENTS
showed that the adjusted parameters were also
adequate to predict loss coefficients of real fluids. The authors wish to express their thanks to
FAPESP for its financial support (Proc. 01/02038-8).

NOMENCLATURE
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Brazilian Journal of Chemical Engineering, Vol. 20, No. 04, pp. 455 - 463, October - December 2003