3.2 Chapter II. Human normal rectum and rectal tumor organoids. Calcitriol effect
3.2.5. Effect of calcitriol on normal colon, normal rectum and rectal tumor organoids
The purpose of this section is to provide the practical information necessary to predict pressure drop and power requirements for homo-geneous, non-time-dependent materials in fluid handling systems.
Rheological properties have a strong influence on the calculations and this information is needed to select proper pumps and related equipment when designing large scale tube (pipe) viscometers or commercial fluid handling systems (Steffe and Morgan, 1986; Steffe and Garcia, 1987).
Although rheological properties can only be evaluated from data taken in the laminar flow regime, the case of turbulent flow is also presented to provide a thorough analysis of pipeline design problems commonly encountered by food process engineers.
Mechanical Energy Balance. The mechanical energy balance for an incompressible fluid in a pipe may be written as
[2.105]
where , the summation of all friction losses, is
[2.106]
and subscripts 1 and 2 refer to two specific locations in the system. The friction losses include those from pipes of different diameters and a contribution from each individual valve, fitting, and similar parts.
is the work output per unit mass and the power requirement of the system is found by calculating the product of and the mass flow rate.
A negative value of indicates that work is being put into the system which is the normal function of a pump.
Rheological properties are required to evaluate the mechanical energy balance equation. Although there are many mathematical models available to describe flow behavior (Table 1.3), few can be con-sidered practical for making pressure drop calculations involving pipe flow. Most pumping problems involving fluid foods can be solved using the Newtonian, power law or Bingham plastic models. Over simplifi-cation, however, can cause significant calculation errors (Steffe, 1984).
Fanning Friction Factor. The Fanning friction factor ( ) is defined, from considerations in dimensional analysis, as the ratio of the wall shear stress in a pipe to the kinetic energy per unit volume:
(u2)2− (u1)2 α
+g(z2−z1) +P2−P1
ρ + ∑F+W=0
∑F
∑F= ∑2f(u)2L
D + ∑kf(u)2 2
W W
W
f
[2.107]
Substituting the definition of the shear stress at the wall of a pipe (Eq.
[2.2]) into Eq. [2.107] gives
[2.108]
where . Hence, the energy loss per unit mass (needed in the mechanical energy balance) may be expressed in terms of :
[2.109]
Some engineers calculate the friction losses with the Darcy friction factor which is equal to four times the Fanning friction factor. Pressure drop calculations may be adjusted for this difference. Final results are the same using either friction factor. Calculations in this text deal exclusively with the Fanning friction factor.
In laminar flow, values can be determined from the equations describing the relationship between pressure drop and flow rate for a particular fluid. Consider, for example, a Newtonian fluid ( ). Using Eq. [2.28], the volumetric average velocity for this material, in laminar tube flow, may be expressed as:
[2.110]
Simplification gives an expression for the pressure drop per unit length:
[2.111]
Substituting Eq. [2.111] into Eq. [2.108] yields the friction factor:
[2.112]
which is a common equation appropriate for predicting friction factors for Newtonian fluids when . Using the same approach, laminar flow friction factors for power law and Bingham plastic fluids may be calculated, respectively, from the following equations:
[2.113]
and
[2.114]
The above equations are appropriate when the laminar flow criteria, presented in Section 2.4, are satisfied. Eq. [2.114], plotted in Fig. 2.15, is an approximation for the Fanning friction factor based on the assumption that (Heywood, 1991a). This figure illustrates how the friction factor decreases with larger values of the Bingham Reynolds number, and increases with larger values of the Hedstrom number. A larger version of Fig. 2.15, more convenient for problem solving, is given in Appendix [6.17].
Figure 2.15. Fanning friction factors (from Eq. [2.114]) for Bingham plastic flu-ids in laminar flow at different values of the Hedstrom Number.
In turbulent flow, friction factors may be determined from empirical equations (Table 2.3) formulated from experimental data (Grovier and Aziz, 1972). The equations are only applicable to smooth pipes which include sanitary piping systems for food. It may be very difficult to accurately predict transition from laminar to turbulent flow in actual
f=16(6NRe,B+NHe) 6NRe,B2
(σo/σw)4 1
100 200 300 500 1,000 2,000 3,000 5,000 10,000
0.001 0.002 0.005 0.01 0.02 0.05 0.1 0.2 0.5 1
Fanning Friction Factor
100 1,000 5,000 10,000 25,000 50,000 100,000 200,000
NHe
NRe,B
Table 2.3. Fanning Friction Factor Equations for Turbulent Flow in Smooth Tubes
Fluid Fanning Friction Factor
Newtonian
processing systems and the equations given here are only intended for use in estimating the power requirements for pumping. Curves for power law fluids in turbulent flow are plotted in Fig. 2.16 (a larger version of the same plot is given in Appendix [6.18]). Newtonian fluids are represented by the curve with = 1.0.
Kinetic Energy Evaluation. The kinetic energy term in the mechanical energy balance can be evaluated if the kinetic energy cor-rection factor ( ) is known. In turbulent flow of any fluid, . Expressions to compute values for various fluids in laminar flow are summarized in Table 2.4. These equations may be given in terms of , the flow behavior index, or which is defined as the ratio of the yield stress ( ) to the shear stress at the wall ( ). Equations provided for the Bingham plastic and Herschel-Bulkley cases are approximations.
1
Figure 2.16. Fanning friction factors (equation given in Table 2.3) for power law fluids in turbulent flow at different values of the flow behavior index.
An exact, but cumbersome, mathematical equation for the kinetic energy correction factor of a Herschel-Bulkley fluid has been published by Osorio and Steffe (1984). Values of , determined from this equation, are plotted in Fig. 2.17. This figure reveals some interesting features of the kinetic energy correction factor: values go to 2 as the yield stress approaches the wall shear stress for all values of ; values increase with decreasing values of . Overall, the numerical value of ranges from 0.74 to 2 for Herschel-Bulkley fluids (Osorio and Steffe, 1984). The minimum value of occurs at as approaches infinity. KE differences are usually small and often ignored in evaluating power requirements when selecting pumps.
2,000 4,000 6,000 8,000 10,000
0.002 0.004 0.006 0.008 0.01 0.012 0.014
Fanning Friction Factor
N
Re,PLn = 1.2 n = 1.0
n = 0.8
n = 0.6
n = 0.4 n = 0.3 n = 0.2
Power Law Fluids in Turbulent Flow
α
α
n α
n α
α=0.74 c=0 n
Table 2.4. Kinetic Energy Correction Factors for Laminar Flow in Tubes
Fluid , dimensionless
Newtonian 1.0
Power Law
+Bingham Plastic
+Herschel-Bulkley
for and for
+ Solution for the Bingham plastic material is within 2.5% of the true solution (Metzner, 1956). Errors in using the Herschel-Bulkley solution are less than 3% for 0.1 1.0 but as high as 14.2% for 0.0 0.1 (Briggs and Steffe, 1995).
Friction Losses: Valves, Fittings, and Similar Parts. Friction loss coefficients ( ) must be determined from experimental data. In general, published values are for the turbulent flow of water taken from Crane (1982). An adequate summary of these numbers may be found in Sakiadis (1984). Laminar flow data are much more limited. Some are available for various fluids: Newtonian (Kittredge and Rowley, 1957), shear-thinning (Banerjee et al., 1994; Lewicki and Skierkowski, 1988;
Steffe et al., 1984) and shear-thickening (Griskey and Green, 1971).
Overall, the quantity of engineering data required to predict pressure losses in valves and fittings for fluids, particularly non-Newtonian fluids, in laminar flow is insufficient.
Given this situation, a "rule of thumb" estimation procedure is needed. First some general observations should be made: a) The behavior of values for Newtonian and non-Newtonian fluids is similar (Metzner, 1961; Skelland, 1967), b) values decrease with increasing pipe diameter (Crane, 1982) -- they may drop as much as 30% in going from 3/4 to 4 inch (1.9 to 10.2 cm) pipe, c) values sharply increase with
α
σ = µγ˙
α =2(2n+1)(5n+3) 3(3n+1)2 σ =K(γ)˙n
α = 2 2−c σ = µpl˙γ + σo
α =exp(0.168 c−1.062 n c−0.954 n.5−0.115 c.5+0.831) 0.06≤n≤0.38
σ =K(γ)˙n+ σo
α =exp(0.849 c−0.296 n c−0.600 n.5−0.602 c.5+0.733) 0.38<n≤1.60
≤c≤ ≤c≤
kf
kf
kf
kf
Figure 2.17. Kinetic energy correction factors for the laminar flow of Herschel-Bulkley fluids (from Osorio and Steffe, 1984).
decreasing Reynolds numbers (Cheng, 1970; Kittredge and Rowley, 1957; Lewicki and Skierkowski, 1988; Steffe et al., 1984) in the laminar flow regime but are constant in the turbulent flow regime (Sakiadis, 1984) and show little change above (Kittredge and Rowley, 1957), d) Entrance pressure losses for power law fluids in laminar flow decrease with smaller values of the flow behavior index (Collins and Schowalter, 1963), e) Entrance losses for Bingham plastic fluids decrease with increasing values of the yield stress when the wall shear stress ( ) is constant (Michiyosi et al., 1966), f) Resistance to flow of non-Newtonian fluids in laminar flow, through similar valves, can be expected to be up to 133 percent higher than that observed for Newtonian fluids (Ury, 1966).
Friction loss coefficients for many valves and fittings are summa-rized in Tables 2.5 and 2.6. values for the sudden contraction, or expansion, of a Newtonian fluid in turbulent flow, may be found in Crane (1982). The loss coefficient for a sudden contraction is calculated in terms of the small and large pipe diameters:
0 0.2 0.4 0.6 0.8 1
0.6 0.8 1 1.2 1.4 1.6 1.8 2
c = Yield Stress / Shear Stress at the Wall
n=0.1 n=0.3 n=0.5 n=0.7 n=0.9 n=2.0
Flow Behavior Index
NRe=500
δP R/(2L)
kf
Table 2.5. Friction Loss Coefficients ( Values) for the Laminar Flow of Newtonian Fluids through Valves and Fittings (from Sakiadis (1984) with Original Data from Kittredge and Rowley, 1957)
Type of Fitting or Valve =1000 500 100
90-deg. elbow, short radius 0.9 1.0 7.5
Tee, standard, along run 0.4 0.5 2.5
Branch to line 1.5 1.8 4.9
Gate valve 1.2 1.7 9.9
Globe valve, composition disk 11 12 20
Plug 12 14 19
Angle valve 8 8.5 11
Check valve, swing 4 4.5 17
[2.115]
Losses for a sudden enlargement, or an exit, are determined as [2.116]
The largest velocity which is the mean velocity in the smallest diameter pipe, should be used for both contractions and expansions in calculating the friction loss term ( ) found in Eq. [2.106].
After evaluating the available data for friction loss coefficients in laminar and turbulent flow, the following "rule-of-thumb" guidelines, conservative for shear-thinning fluids, are proposed for estimating values:
1. For Newtonian fluids in laminar or turbulent flow use the data of Kittredge and Rowley (1957) or Sakiadis (1984), respectively (Tables 2.5 and 2.6).
2. For non-Newtonian fluids above a Reynolds number ( ) of 500, use data for Newtonian fluids in turbulent flow (Table 2.6).
3. For non-Newtonian fluids in the Reynolds number range of use the following equation:
kf
Table 2.6. Friction Loss Coefficients for the Turbulent Flow of Newtonian Fluids through Valves and Fittings (from Sakiadis, 1984)
Type of Fitting of Valve
45-deg. elbow, standard 0.35
45-deg. elbow, long radius 0.2
90-deg. elbow, standard 0.75
Long radius 0.45
Square or miter 1.3
180-deg. bend, close return 1.5 Tee, standard, along run, branch blanked off 0.4 Used as elbow, entering run 1.0 Used as elbow, entering branch 1.0
Branching flow 1.0
Coupling 0.04
Union 0.04
Gate valve, open 0.17
3/4 open 0.9
1/2 open 4.5
1/4 open 24.0
Diaphragm valve, open 2.3
3/4 open 2.6
1/2 open 4.3
1/4 open 21.0
Globe valve, bevel seat, open 6.0
1/2 open 9.5
Composition seat, open 6.0
1/2 open 8.5
Plug disk, open 9.0
3/4 open 13.0
1/2 open 36.0
1/4 open 112.0
Angle valve, open 2.0
Y or blowoff valve, open 3.0
Plug cock =0 (fully open) 0.0
=5 0.05
=10 0.29
=20 1.56
=40 17.3
=60 206.0
Butterfly valve =0 (fully open) 0.0
=5 0.24
=10 0.52
=20 1.54
=40 10.8
=60 118.0
Check valve, swing 2.0
Disk 10.0
Ball 70.0
Foot valve 15.0
Water meter, disk 7.0
Piston 15.0
Rotary (star-shaped disk) 10.0
Turbine-wheel 6.0
[2.117]
where is depending on the type of fluid in question.
The constant, , is found for a particular valve or fitting (or related items like contractions and expansions) by multiplying the tur-bulent flow friction loss coefficient by 500:
[2.118]
Values of for many standard items may be calculated from the values provided in Table 2.6. The value of 20 was arbitrarily set as a lower limit of in Eq. [2.117] because very low values of the Reynolds number in that equation will generate unreasonably high values of the friction loss coefficient. Values of will cover most practical applications for fluid foods. Eq. [2.117] and [2.118] may also be used for Newtonian fluids when is used for .
The above guidelines are offered with caution and should only be used in the absence of actual experimental data. Many factors, such as a high extensional viscosity, may significantly influence values.
Generalized Pressure Drop Calculation. Metzner (1956) presents a generalized approach to relate flow rate and pressure drop for time-independent fluids in laminar flow. The governing equation is
[2.119]
where
[2.120]
and are easily determined from a plot of the experimental data.
There is a strong similarity with the above equation and those describing the flow of power law fluids in pipes. In fact, for a power law fluid,
[2.121]
With the general solution, may vary with the shear stress at the wall and must be evaluated at the particular value of in question.
Eq. [2.119] and [2.120] have practical value when considering direct scale-up from data taken with a small diameter tube or for cases where a well defined equation (power law, Bingham plastic or Herschel-Bulkley) is not applicable. A similar method is available for scale-up problems involving the turbulent flow of time-independent fluids (Lord et al., 1967).
Slip and time-dependent behavior may be a problem in predicting pressure loss in pipes. One solution is to incorporate these effects into the consistency coefficient. Houska et al. (1988) give an example of this technique for the transport of minced meat in pipes where incorpo-rates the property changes due to the aging of the meat and wall slip as a function of pipe diameter. An exercise in pipeline design is presented in Example Problem 2.12.6.