Fluids and Solids Handling
The G-Value for Agitator Design: De sign: Time to Retire It? Gregory T. Benz, P.E. Benz Technology International, Inc.
This commonly used number does not correctly address the influence of viscosity, and has no relationship to performance parameters such as fluid velocity and blend time. This article recommends actual measures of agitator performance that should be used instead.
C
amp’s G-value, or simply “G-value,” has been used as the standard way for specifying agitation systems in the water and wastewater treatment industry for many years. Engineers within the industry have instinctively known that the concept was inadequate or even wrong, but have been reluctant to challenge it because of its long tradition of use. Instead, they have attempted to “make it work” by supplementing the G-value with such requirements as minimum impeller diameter, maximum tip speed, and so on. Perhaps it is time to highlight the deficiencies of the Gvalue concept. Several others have gone on record pointing out the problems of G-value. For example: • “It is generally recognized that the velocity gradient or G-value concept is a gross, simplistic and totally inadequate parameter for design of rapid mixers” (1). • “Camp’s ‘G’ value is not intended for design or comparison of different impeller types, and has not been shown to accurately correlate mixing effectivenes effectivenesss for different mixing processes” (2). This article looks at what the G-value is and why it does not correctly address agitation design issues. It then gives examples of design procedures that are relevant to the water and wastewater treatment industry industry..
What is the G-value? The G-value, as defined by Camp and Stein in 1943 (3), is intended to represent the root mean square (rms) velocity gradient in a basin. Why this should be a relevant measure
of mixing performance, rather than such things as mean velocity,, blend time or other more direct measures, has velocity never been adequately explained. Additionally, Additionally, it is questionable whether the G-value even measures the true rms velocity gradient. The defining equation for the G-value is: G = (P / µV )0.5
(1)
Its units are normally chosen so that G is expressed as s–1. This equation implies that the required agitator power is directly proportional to the viscosity and liquid volume. In order to see why it is wrong to use the G-value to describe agitation, it is first necessary to briefly review some common agitator performance calculations.
Calculating agitator performance A discussion of design procedures procedures for the wide variety variety of agitation problems is beyond the scope of this article. However, as a first step, most water and wastewater applications can be classified as flow-v flow-velocit elocity-con y-contro trolled lled or blend-timecontrolled. Furthermore, virtually all are turbulent-flow applications, due to the low viscosity of water. Thus, it is possible to describe simplified, yet accurate, procedures for these types of situations. Viscosity has an effect on agitation, but not in a linear fashion. Instead, all attributes of agitation are functions of the impeller Reynolds number (4):
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Fluids and Solids Handling Table 1. Generic impeller characteristics (turbulent).
Nomenclature A B D G K N N P N Q N Re P Q T V V C Z
= tank cross sectional area, m 2 = blend time exponent on D / T , dimensionless = impeller diameter, m = root mean square velocity gradient, s –1 = blend time coefficient, dimensionless = shaft speed, s–1 or rpm = power number, P / ρ N 3 D5, dimensionless = pumping number, Q / ND3, dimensionless = impeller Reynolds number, D2 N ρ / µ, dimensionless = power, W or kg-m2 /s3 = impeller pumping rate, m3 /s = cylindrical tank diameter, m = liquid volume, m3 = characteristic velocity, m/s = liquid level, m
Greek Letters = blend time, s (or min) θ B = viscosity, kg/m-s (or cP) µ = fluid density, kg/m3 ρ
(2)
N Q = Q / ND 3
(3)
N P = P / ρ N 3 D 5
(4)
For fluid-motion-control applications, such as holding tanks, equalization basins, reagent make-up tanks, etc., where the retention time is long enough that the suspension of trace solids and the ability to blend reagents of higher or lower density or viscosity than water are limiting, rather than blend time, one measure of agitation intensity is the characteristic velocity, V C . This parameter is calculated on the basis of a cylindrical vessel of “square-batch” geometry, where the liquid level is the same as the tank diameter ( Z = T ), regardless of the actual geometry. The impeller pumping and power characteristics are measured based on this geometry; it is assumed that a given impeller will pump the same in a different geometry as long as the volume is the same. Characteristic velocity is defined as (4): V C = Q / A= 4Q / πT 2
(5)
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0.25 0.30 0.35 0.40 0.45 0.50
Hydrofoil (3-blade)
N P
N Q
N P
N Q
N P
N Q
3 3 3 3 3 3
0.7 0.7 0.7 0.7 0.7 0.7
1.37 1.37 1.37 1.37 1.37 1.37
0.88 0.80 0.74 0.68 0.64 0.60
0.33 0.32 0.31 0.29 0.28 0.27
0.57 0.55 0.54 0.53 0.52 0.51
To express the actual basin geometry on a square-batch basis, T in Eq. 5 is defined based on a square batch having the same volume as the actual basin (4): (6)
The impeller pumping rate can be derived from the pumping number (4):
In fully developed turbulent conditions (where the Reynolds number is above 10,000), all common measures of agitation performance become constant as a function of the Reynolds number. This means that viscosity has no effect when it is low enough to result in turbulent flow, which is nearly always the case in water. For example, the impeller pumping number, N Q , and the impeller power number, N p , are constant under turbulent conditions:
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D / T
Pitched (4-blade)
/ π)1/3 T = (4V
2 N Re = D N ρ / µ
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Radial (4-blade)
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Q = N Q ND 3
(7)
Thus: V C = 4 N Q ND 3 / πT 2
(8)
Note that the impeller pumping number, N Q , is a function of impeller type and the D / T ratio. Many agitator manufacturers are reluctant to disclose these figures, and when they do, the figures may not always be reliable. Table 1, therefore, gives some general guidance for generic impellers, as well as generic power numbers. Keep in mind that there are many combinations of impeller type and size that will give equal pumping, and that power and shaft speed will be different for these various combinations. Power is calculated from the definition of the power number (4): P = N Pρ N 3 D 5
(9)
The calculations are straightforward once the desired characteristic velocity is known. Table 2 compares typical values for various applications common to water and wastewater treatment plants. Table 2. Typical characteristic velocities ( VC ) for common applications.
Application Equalization Basin, Wastewater Equalization Basin, Water Flocculation Holding Tanks Rapid Mix
V C,
m/s
0.08 0.06 0.05 0.08 0.15+
Table 3. Blend time coefficients ( K ) and exponents ( B ) for various impellers.
Table 4. Effects of viscosity on power, pumping rate, G-value, and blend time.
Applications with short retention times or Impeller K B Type very fast critical reactions may also need to Radial 4.98 –2.32 Pitched 7.06 –2.20 be checked for blend Hydrofoil 16.9 –1.67 time. The normal definition of blend time, θ B, is the time for a disturbance or addition to be attenuated to within ±1% of the disturbance value. This is sometimes called 99% blend time. For most continuous-flow applications, the blend time is set to the retention time or less. However, for rapid mixing, the retention time is so low that it is acceptable to blend to only about ±20% attenuation; this is known as the 80% blend time. It is only 34% as long as the 99% blend time — because considerable blending occurs in the pipe downstream of the mixing chamber, it is not necessary to blend to the 99% level in the rapid mix chamber. In turbulent flow, the blend time is correlated by (4): θ B N = K ( D / T) B
(10)
Table 3 lists values of K and B for some common generic impellers. These values are based on a cylindrical vessel with a square-batch configuration and a single impeller. For a cylindrical vessel for which Z / T ≠ 1, a factor of ( Z / T) 0.44 should be applied. For other geometries, software such as that offered by Reyno, Inc. (www.ReynoInc.com) is recommended. Solids-suspension applications are far too complex to discuss here, as they incorporate many geometry effects as well as settling rate, and other factors. They also cannot be correlated as a function of the G-value. The same can be said for the dispersion of gases into liquids.
What’s wrong with the G-value? By analyzing the defining equation, one can see that there are several things wrong with the G- value: • it requires the agitator power to be proportional to viscosity • the implied scale-up is on a power/volume basis • no allowance is made for the impeller size • no allowance is made for the impeller type. These are all serious deficiencies in the concept, and will be explored in the subsequent sections of this article.
Agitator power proportional to viscosity This requirement grossly overstates the importance of the viscosity in agitator design. The power required to pump liquids in a pipe is not proportional to viscosity. In fact, viscosity has no effect on required pumping power as long as the pipe flow is fully turbulent. The same is true for agitators. Turbulence is measured by the Reynolds number (as defined by Eq. 2), which is conceptually the ratio of inertial
Basis: 10-m-dia. cylindrical vessel, 10-m liquid level, pitched-blade turbine with a 3-m-dia. impeller ( D = 3) and shaft speed of 20 rpm ( N = 20) µ,
cP
0.1 0.5 1 2 5 10 20 50 100 300
N Re
P , kW
Q , m3 /s
G , s–1
θB , min
3.00E+07 6,000,120 3,000,060 1,500,030 600,012 300,006 150,003 60,001 30,001 10,000
12.33 12.33 12.33 12.33 12.33 12.33 12.33 12.33 12.33 12.33
7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2 7.2
396.2 177.2 125.3 88.6 56 39.6 28 17.7 12.5 7.2
5 5 5 5 5 5 5 5 5 5
to viscous forces. Viscosity has virtually no effect on power draw, impeller pumping or blending performance unless the Reynolds number falls below 10,000, which almost never happens in water or wastewater applications. Designers in this industry have attempted to remedy this oversensitivity to viscosity by requiring the G-value to be calculated at some reference temperature, which will fix the viscosity. But, in practice, this means that the actual G-value varies with temperature, implying different mixing results. Anyone looking at the basin will not see any difference between the mixing in a tank with a 5°C water temperature and that in a 50°C tank, yet the G-value will be 66% higher at the warmer temperature because viscosity is 2.78 times as high at 5°C as at 50°C. Table 4 illustrates the lack of sensitivity to viscosity for a 10-m-dia. cylindrical basin with a 10-m liquid level, where the impeller size, type and shaft speed are fixed and viscosity is allowed to vary from 0.1 cP to 300 cP. This relatively high maximum viscosity is not chosen for its applicability to water treatment, where viscosities never get that high, but rather to emphasize the point that the results are insensitive to viscosity until it gets to 300 cP. The Reynolds number, power draw, impeller pumping rate, G-value and blend time are shown as a function of viscosity. Over this range of viscosity, the Reynolds number varies from 30 million to 10,000. The power, pumping rate and blend time all remain constant, indicating no change in real agitation performance. Yet the G-value varies from 396 to 7.2, a ratio of 55 to 1. Clearly, the G-value does not properly account for the effects of viscosity on agitation. Had the viscosity continued to increase, the Reynolds number would have dropped below 10,000, which would start to affect real agitation results. The correct way to account for the viscosity is to correlate power, pumping, blend time or other relevant parameters as a function of the Reynolds number. Article continues on p. 46 CEP
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Fluids and Solids Handling Table 5. Effects of impeller diameter on pumping rate and blend time at constant G-value.
Table 6. Effects of impeller diameter on power, G-value and blend time at constant impeller pumping rate.
Basis: 10-m-dia. cylindrical vessel, 10-m liquid level, pitchedblade turbine, viscosity µ = 1 cP, G-value = 100 s –1
Basis: 10-m-dia. cylindrical vessel, 10-m liquid level, pitchedblade turbine, viscosity µ = 1 cP, pumping rate Q = 7 m3 /s
D , m
N , rpm
N P
N Q
P , kW
Q , m3 /s
θB , min
2.5 3 3.5 4 4.5 5
23.34 17.23 13.32 10.66 8.76 7.35
1.37 1.37 1.37 1.37 1.37 1.37
0.88 0.8 0.74 0.68 0.64 0.6
7.88 7.88 7.87 7.87 7.87 7.87
5.3 6.2 7 7.7 8.5 9.2
6.4 5.8 5.3 5 4.7 4.4
Implied power/volume scale-up Although detailed scale-up procedures are beyond the scope of this article, it is valuable to understand that scale-up using equal G-value is a power/volume scale-up. Different process results scale up differently. For equal mean velocity and geometric similarity, it can be shown that the required scale-up rule corresponds to equal torque per volume, not equal power per volume. This rule also results in equal impeller tip speed. An equal power/volume scale-up would result in oversized equipment and waste power compared to the correct scale-up for velocity-controlled processes such as holding basins, flocculators and similar motioncontrolled applications. On the other hand, processes that require the same absolute blend time require a much larger agitator upon scale-up than a power/volume basis would imply. In fact, they require that the power/volume ratio increase in proportion to the volume raised to the 5/9 power. In such a case, scaling based on equal G-value would result in undersized equipment. The only situations where power/volume is commonly used for scale-up are mass-transfer-controlled applications, such as gas-liquid contacting. Even there, the G-value is weak because it overstates the viscosity effect. Thus, the G-value is not useful as a scale-up tool, and, in fact, leads to erroneous results for most common applications.
No allowance for impeller size The only agitation parameter in the G-value equation is the power draw. One could meet a specified G-value by using a 50-mm-dia. impeller turning at a very high shaft speed in a million-cubic-meter basin. Yet, intuitively, we know this would not work. Table 5 illustrates this for the same 10-m-dia. tank used in Table 4. The impeller diameter varies from 2.5 m to 5 m, and the shaft speed is chosen to maintain an equal G-value of 100 s–1. Equations 3 and 4 are rearranged to calculate the power and pumping rate from the power number and the pumping number. Although the larger impellers turn more slowly, they pump more and blend faster than the small ones. The 5-m-dia. impeller pumps 74% more than the 2.5m impeller, and blends 45% faster, yet they have the same 46
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D , m
N , rpm
N P
N Q
P , kW
G , s–1
θB , min
2.5 3 3.5 4 4.5 5
30.53 19.45 13.24 9.65 7.2 5.6
1.37 1.37 1.37 1.37 1.37 1.37
0.88 0.8 0.74 0.68 0.64 0.6
17.63 11.34 7.73 5.84 4.37 3.48
150 120 99 86 75 67
4.9 5.1 5.4 5.5 5.7 5.8
G-value. So, by failing to account for different impeller sizes, G-value fails to relate to real process performance. It has long been known that larger impellers can, in fact, save energy compared to smaller impellers. A large impeller can pump the same amount of liquid as a smaller impeller at a lower shaft speed and lower mean discharge pressure, thus drawing less power. Table 6 illustrates the required shaft speed and power for the same impellers and tank as Table 5, at a constant impeller discharge rate of 7 m3 /s. The power varies from more than 17 kW for the small impeller to 3.5 kW for the largest one. The G-value varies from 150 to 67. For flowvelocity-controlled applications, the 5-m impeller drawing 3.5 kW will perform as well as the 2.5-m impeller drawing 17.6 kW, with a power savings of 80%. Thus, G-value does not account for impeller size effects for flow-velocity-controlled applications. Notice that in Table 6, the blend time is not constant, even though the impeller pumping rate is. This is because blending is not simply a flow-controlled operation, but involves both flow and turbulence. The larger impellers produce less turbulence at a given flowrate, so their blend time is somewhat longer. In most cases, blend time is not a limiting factor, but for completeness, Table 7 shows the effects of varying impeller diameter at constant blend time. Based on a 5-min blend time, the required power varies from 16.4 kW to 5.4 kW, and the G-value varies from 144 to 83. So, the G-value does not account for impeller diameter effects on blend time either.
No allowance for impeller type There are many types of impellers on the market today. Some are proprietary, some are generic. Some have an axial discharge pattern, some have a radial pattern, some are mixed-flow. They have a wide range of performance characteristics, such as power numbers ranging from less than 0.2 to more than 5, pumping numbers ranging from less than 0.1 to about 1.0, and varying blend time characteristics. Yet, Gvalue accounts for none of these variations. Table 8 shows how the choice of impeller affects performance at constant G-value, using three different impeller
Table 7. Effects of impeller diameter at constant blend time.
Table 8. Effects of impeller type at constant G-value.
Basis: 10-m-dia. cylindrical vessel, 10-m liquid level, pitchedblade turbine, viscosity µ = 1 cP, blend time θB = 5 min
Basis: 10-m-dia. cylindrical vessel, 10-m liquid level, viscosity µ = 1 cP, G-value = 100 s -1
D , m
2.5 3 3.5 4 4.5 5
N , rpm
29.8 19.96 14.22 10.6 8.18 6.49
N P
1.37 1.37 1.37 1.37 1.37 1.37
N Q
0.88 0.8 0.74 0.68 0.64 0.6
P , kW
16.39 12.26 9.58 7.74 6.41 5.42
G , s–1
144 125 110 99 90 83
Q , m3 /s
6.8 7.2 7.5 7.7 8 8.1
types and two different impeller diameters. A 4-m-dia. hydrofoil turning at roughly the same speed as a 2.5-m radial turbine draws the same power and has the same G-value, but pumps more than three times as much and blends 75% faster. This comparison was chosen because these unequal types and sizes have similar torques, and would therefore require the same size gear drive and have a similar cost. At equal diameter, the hydrofoils turn at a faster speed and require less torque than pitched or radial turbines, and so would cost less. Yet, they pump more and blend faster at a constant G-value. So, G-value is useless as a means of accounting for variations in impeller type.
What is the G-value useful for? It is this author’s opinion that the G-value has no legitimate use in designing or specifying agitators. So, what is the correct way to specify agitation performance? The best way is to be very specific about the task the agitator is expected to perform. This should be stated in purely physical terms. For example, agitators do not bring about chemical reactions; reactions are determined by composition and temperature only, which are not directly controlled by the agitator. Instead, specify volumes, the properties of each fluid being agitated, flowrates, retention times, descriptions of any solids present, and a clear statement of the desired physical process results.
Example We will calculate a design for the same tank used for the tables (T = 10 m, Z = 10 m), based on a 4-m-dia. hydrofoil impeller used to produce a characteristic velocity of 0.06 m/s. This involves calculating the shaft speed and power needed. At a D / T of 0.4, Table 1 gives us a pumping number of 0.53 and a power number of 0.29. Rearranging Eq. 8 to solve for shaft speed gives: N = V C πT 2 /4 N Q D3 = [(0.06 m/s)(π)(10 m)2)]/[(4)(0.53)(4m)3] = 0.139 s–1 = 8.34 rpm. Power is then calculated using Eq. 9: P = N Pρ N 3 D5 = (0.29)(1,000 kg/m 3)(0.139/s)3(4 m)5 = 798 kg-m 2 /s3 = 798 W = 0.798 kW. Motor power would need to be at least 10% more to allow for errors and mechanical transmission losses. Because motors come in standard sizes and reducers normally have
D ,
N , rpm
m
Impeller
2.5 2.5 2.5 4 4 4
Radial 17.96 Pitched 23.32 Hydrofoil 37.48 Radial 8.205 Pitched 10.655 Hydrofoil 17.88
N P
N Q
P , kW
3 1.37 0.33 3 1.37 0.29
0.7 0.88 0.57 0.7 0.68 0.53
7.86 7.86 7.86 7.86 7.86 7.86
θB , Q , min m3 /s
7 6 5 5 5 4
3.3 5.3 5.6 6.1 7.7 10.1
output speeds in accordance with American Gear Manufacturer’s Association (AGMA) standards, the actual design would have a standard motor size of 1.5 hp (1.1 kW) and a nominal shaft speed of 9 rpm. The impeller size would need to be adjusted for these conditions. Blend time is calculated by rearranging Eq. 10: θ B = (16.9)(4/10)–1.67 /(0.139 s–1) = 562 s = 9.36 min.
Recommendations It is time to end the practice of using the G-value in agitator specifications in water and wastewater treatment applications. Instead, specify in physical terms what the agitator must do in your process. Typical examples are characteristic velocity and blend time. Allow the equipment vendors to save you money by looking at equivalent alternatives that still achieve the required physical process results. CEP
Literature Cited 1.
2.
3.
4.
Amirtharajah, A., “Design of Rapid Mix Units,” in “Water Treatment for the Practicing Engineer,” Sanks, R. L., ed., Ann Arbor Science, Ann Arbor, MI (1978). “Camp’s Gt Values and In-Line Polymer Blending/Activation,” Fluid Dynamics, Inc., ww.dynablend.com/fdtech.html (viewed Mar. 2006). Camp, T. R., and P. C. Stein, “Velocity Gradients and Internal Work in Fluid Friction,” J. Bo ston Soc. C iv. Eng., 30 (4), pp. 219–237 (1943). Dickey, D. S., and J. G. Fenic , “Dimensional Analysis for Fluid Agitation Systems,” Chem. Eng., pp. 139–145 (Jan. 5, 1976).
GREGORY T. BENZ, P.E., is president of Benz Technology International, Inc.
(2305 Clarksville Rd., Clarksville, OH 45113; Phone: (937) 289-4504; Fax: (937) 289-3914; E-mail
[email protected]; Website: http://home.mindspring.com/~benztech/). He has over 30 years of experience in the design of agitation systems. Currently, his company offers general engineering and mixing consultation, including equipment specification and bid evaluation, as well as courses on agitation with CEU/PDH credits. Benz is also a course director at the Center for Professional Innovation and Education (CfPIE; www.CfPIE.com), and is a registered consulting expert with Intota (www.intota.com). He received his BSChE from the Univ. of Cincinnati in 1976, and has taken a course on fermentation biotechnology from the Center for Professional Advancement. He is a registered professional engineer in Ohio, and is a member of AIChE, Society for Industrial Microbiology (SIM), International Society for Pharmaceutical Engineering (ISPE) and the American Chamber of Commerce in Shanghai.
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