How to Draw Line of Best Fit on Semilog Graph

Hydraulics

Malcolm J. Brandt BSc, FICE, FCIWEM, MIWater , ... Don D. Ratnayaka BSc, DIC, MSc, FIChemE, FCIWEM , in Twort's Water Supply (Seventh Edition), 2017

Weirs

Equation (14.14) can be re-arranged in the form: $(2/3)H_S^3=q^2/g$ . Hence, flow per unit width $q=(2/3)\sqrt{(2g/3)}{H}^{1.5}$ and for a rectangular channel of width, b:

(14.17) $Q=(2/3)\sqrt{(2g/3)}b{H}^{1.5}$

This is now in the form of a weir equation, which can be generalized as:

(14.18) $Q=C_{d}b{H}^{1.5}$

where C _d is a discharge coefficient .

C _d in Equation (14.18) has dimensions as it involves $\sqrt{g}$ . There are a number of other forms for this equation (Section 14.14), with the simplest involving a non-dimensional coefficient being:

(14.19) $Q={C\prime }_d\sqrt{g}b{H}^{1.5}$

In Equation (14.18) the discharge coefficient $C_d=(2/3)\sqrt{(2g/3)}=1.705$ in metric units and, in Equation (14.19), ${C\prime }_d=(2/3)\sqrt{2/3}=0.544$ in non-dimensional units. If the flow passes through critical depth over a weir crest then it might appear that C _d would always take that value. However, it is important to appreciate the inherent assumptions lying behind the theory that led to Equation (14.1) and all the subsequent equations derived from it:

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there is a uniform velocity distribution across the section so α=1.0 (Section 14.3);

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the streamlines are straight and parallel (i.e. there is no lateral pressure set up by curvature of the streamlines);

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the vertical pressure distribution is hydrostatic (i.e. the pressure is a linear function of depth);

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the effect of the longitudinal slope is negligible.

Provided these conditions are met, as is very nearly the case with a broad-crested weir such as illustrated in Figure 14.7(b), then the discharge coefficient C _d , is indeed about 1.705. However, many other crest profiles are used, ranging from simple plate or sharp-edged crests, to rounded tops of walls and triangular or ogee-profile crests. Their use depends on whether accurate flow measurement is required or whether they act merely as simple overflow hydraulic controls. Weir shapes used for flow measurement are discussed further in Section 14.14. The discharge coefficient can vary significantly from the basic broad-crested value of 1.705 and depends largely on the geometry of the crest, but it is also a function of the depth and velocity of the approach flow. The subject is wide-ranging and is beyond the scope of this book. As a rule of thumb, however, the discharge coefficient is likely to be greater than 1.705 if there is strong curvature to the flow, for example, over a half-round crest or an ogee crest. In the latter case, the profile of the crest is that of the underside of a free-falling jet of water, so, in theory, there should be little if any pressure on the solid surface. The pressure distribution through the depth of flow cannot, therefore, be hydrostatic and one way of considering the problem is that the back pressure on the flow over the crest is reduced, thus allowing an increased discharge and a corresponding increase in the discharge coefficient. Increases in C _d of 30–40% above the broad-crested weir value are possible.

Similarly, the discharge coefficient may be reduced if the weir crest is long (in the direction of flow), or very rough – as might be the case of flow over a grassed embankment. A value of 1.7 is however a good starting point for initial design or in the absence of more details of the weir shape.

It must be emphasized again that a weir only acts as a hydraulic control if it has free or 'modular' discharge; that is, the downstream water level is low enough to allow the flow to pass through critical depth. If the tailwater depth is high enough to affect the flow over the weir, the weir is said to be drowned. This condition is usually catered for in the weir equation by introducing a drowning factor f _d , which is a function of the height of the tailwater level above the crest. Referring to Figure 14.7(b) it would appear that, provided the tailwater level above the crest is not more than the critical depth, that is, two thirds of the upstream head above the crest, then critical depth flow will occur. Strictly, the tailwater level has a small effect even when it is at the level of the crest, particularly if lowered pressures can be generated below the nappe of the falling water (e.g. in an air pocket trapped behind the falling sheet of water), but this two thirds criterion is a useful rule of thumb. Although the crest shape does affect the drowning factor, in many instances where a weir is not being used for measurement it is reasonable to assume that the hydraulic control remains at the weir crest until the tailwater level rises to a depth greater than two thirds of the upstream head above the crest.

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Spring discharge hydrograph

Neven Kresic , Ognjen Bonacci , in Groundwater Hydrology of Springs, 2010

4.2 Equations of recession discharge

Analysis of the falling hydrograph limb shown in Figure 4-6, which corresponds to a period with no significant precipitation, is called the recession analysis. Knowing that the spring discharge is without disturbances caused by a rapid inflow of new water into the aquifer, the recession analysis provides good insight into the aquifer structure. By establishing an appropriate mathematical relationship between spring discharge and time, it is possible to predict the discharge rate after a given period without precipitation and to calculate the volume of discharged water. For these reasons, recession analysis has been a popular quantitative method in spring discharge analysis for a long time.

Figure 4-6. Spring discharge hydrograph with a recession period.

(From Kresic, 2007.)

The ideal recession conditions—a long period of several months without precipitation—are rare in moderate, humid climates. Consequently, frequent precipitation can cause various disturbances in the recession curve that may not be removed unambiguously during analysis. It is therefore desirable to analyze as many recession curves from different years as possible. Larger samples allow for derivation of an average recession curve as well as the envelope of minima (Figure 4-7), which enables a more accurate quantification of the expected long-term minimum discharge.

Figure 4-7. Three annual discharge hydrographs with main recession curves overlaid at the point of average discharge rate. The minima on the three overlaid curves are connected with the envelope, which represents a long-term recession curve.

(From Kresic, 2007.)

Two well-known mathematical formulas that describe the falling limb of hydrographs and the base flow were proposed by Boussinesq (1904) and Maillet (1905). Both equations give the dependence of the flow at specified time (Q_t ) on the flow at the beginning of recession (Q ₀). The Boussinesq equation is of hyperbolic form:

(4.3) $Q_t=\frac{Q_0}{{[1+\alpha (t-t_0)]}^2}$

where t is the time since the beginning of recession for which the flow rate is calculated; t ₀ is time at the beginning of recession usually (but not necessarily) set equal to 0.

The Maillet equation, which is more commonly used, is an exponential function:

(4.4) $Q_t=Q_0\times {\text{e}}^{-\alpha (t-t_0)}$

The dimensionless parameter α in both equations represents the coefficient of discharge (or recession coefficient), which depends on the aquifer's transmissivity and specific yield. The Maillet equation, when plotted on a semilog diagram, is a straight line with the coefficient of discharge (α) being its slope:

(4.5) $\log Q_t=\log Q_0-0.4343\times \alpha \times \mathrm{\Delta }t$

$\mathrm{\Delta }t=t-t_0$

(4.6) $\alpha =\frac{\log Q_0-\log Q_t}{0.4343\times (t-t_0)}$

The introduction of the conversion factor (0.4343) is a convenience for expressing discharge in equation (4.6) in cubic meters per second and time in days. The dimension of α is day⁻¹.

Figure 4-8 is a semilog plot of time versus discharge rate for the recession period shown in Figure 4-6. The recorded daily discharges form three straight lines, which means that the recession curve can be approximated by three corresponding exponential functions with three coefficients of discharge (α). The three lines correspond to three microregimes of discharge during the recession. The coefficient of discharge for the first microregime, using equation (4.6), is (Kresic, 2007)

Figure 4-8. Semilog graph of time versus discharge for the recession period shown in Figure 4-7. The duration of the recession period is 54 days.

(Modified from Kresic, 2007.)

${\alpha }_1=\frac{\log Q_{01}-\log Q_{02}}{0.4343\times (t_{01}-t_{02})}$

${\alpha }_1=\frac{\log (3.55{\text{m}}^3/\text{s})-\log (2.25{\text{m}}^3/\text{s})}{0.4343\times 24.5\text{d}}=0.019$

The coefficient of discharge for the second microregime is

${\alpha }_2=\frac{\log (2.25{\text{m}}^3/\text{s})-\log (2.06{\text{m}}^3/\text{s})}{0.4343\times (44\text{d}-24.5\text{d})}=0.0045$

The third coefficient of discharge, or slope of the third straight line, is found by choosing a discharge rate anywhere on the line, including its extension if the actual line is short. In our case, the discharge rate after 60 days is 2.01 m³/s and α₃ is then

${\alpha }_3=\frac{\log (2.06{\text{m}}^3/\text{s})-\log (2.01{\text{m}}^3/\text{s})}{0.4343\times (60\text{d}-44\text{d})}=0.0015$

After determining the coefficients of discharge, the flow rate at any given time after the beginning of recession can be calculated using the appropriate Maillet equation. For example, the discharge of the spring 35 days after the recession started, when the second microregime is active, is calculated as

$Q_{35}=Q_{02}\times {\text{e}}^{-{\alpha }_2(35\text{d}-t_{02})}$

$Q_{35}=2.25{\text{m}}^3/\text{s}\times {\text{e}}^{-0.0045\times (35\text{d}-24.5\text{d})}=2.146{\text{m}}^3/\text{s}$

Note that the initial discharge rate for the second microregime is 2.25 m³/s and the corresponding time is 24.5 days.

Spring discharge three months (90 days) after the beginning of recession, assuming no precipitation for the entire period, may be predicted by using the characteristic values for the third microregime (see Figure 4-8), where Q ₀₃ is the initial discharge for that regime:

$Q_{90}=Q_{03}\times {\text{e}}^{-{\alpha }_3(90\text{d}-t_{03})}$

$Q_{90}=2.06{\text{m}}^3/\text{s}\times {\text{e}}^{-0.0015(90\text{d}-44\text{d})}=1.923{\text{m}}^3/\text{s}$

The coefficient of discharge (α) and the volume of free gravitational groundwater stored in the aquifer above spring level (i.e., groundwater that contributes to spring discharge) are inversely proportional:

(4.7) $\alpha =\frac{Q_t}{V_t}$

where Q_t is the discharge rate at time t and V_t is the volume of water stored in the aquifer above the level of discharge (spring level).

Equation (4.7) allows calculation of the volume of water accumulated in the aquifer at the beginning of recession as well as the volume discharged during a given period of time. The calculated remaining volume of groundwater always refers to the reserves stored above the current level of discharge. The draining of an aquifer with three microregimes of discharge (as in our case) and the corresponding volumes of the discharged water are shown in Figure 4-9. The total initial volume of groundwater stored in the aquifer (above the level of discharge) at the beginning of the recession period is the sum of the three volumes that correspond to three types of storage (effective porosity):

Figure 4-9. Schematic presentation of the recession with three microregimes of discharge, and the corresponding volumes of discharged water.

(From Kresic, 2007.)

(4.8) $V_0=V_1+V_2+V_3=\left[\frac{Q_1}{{\alpha }_1}+\frac{Q_2}{{\alpha }_2}+\frac{Q_3}{{\alpha }_3}\right]\times 86,400\text{s}[{\text{m}}^3]$

where discharge rates are given in cubic meters per second.

The volume of groundwater remaining in the aquifer at the end of the third microregime is the function of the discharge rate at time t* and the coefficient of discharge α₃:

(4.9) $V^{*}=\frac{Q^{*}}{{\alpha }_3}$

The difference between volumes V₀ and V* is the volume of all groundwater discharged during the period t* – t₀ . In our case, the volume of groundwater stored in the aquifer at the beginning of recession is

$V_0=\left[\frac{(Q_{01}-Q_{02})}{{\alpha }_1}+\frac{Q_{02}-Q_{03}}{{\alpha }_2}+\frac{Q_{03}}{{\alpha }_3}\right]\times 86,400\text{s}[{\text{m}}^3]$

$V_0=\left[\frac{(3.55{\text{m}}^3/\text{s}-2.55{\text{m}}^3/\text{s})}{0.019}+\frac{(2.55{\text{m}}^3/\text{s}-2.20{\text{m}}^3/\text{s})}{0.0045}+\frac{2.20{\text{m}}^3/\text{s}}{0.0015}\right]\times 86,400\text{s}$

$V_0=4.547\times {10}^6{\text{m}}^3+6.720\times {10}^6{\text{m}}^3+1.267\times {10}^8{\text{m}}^3=1.380\times {10}^8{\text{m}}^3$

The volume of water remaining in the aquifer above the spring level at the end of recession is

$V^{*}=\frac{2.03{\text{m}}^3/\text{s}}{0.0015}\times 86,400\text{s}=1.169\times {10}^8{\text{m}}^3$

which gives the following volume of water discharged at the spring for the duration of recession (54 days):

$V=V_0-V^{*}=1.380\times {10}^{-8}-1.169\times {10}^8{\text{m}}^3=21.1\times {10}^6{\text{m}}^3$

Recession periods of large perennial karstic springs or springs draining highly permeable fractured rock aquifers often have two or three microregimes of discharge, as in this example. However, the recession curve can vary greatly from spring to spring and its shape can have various physical meanings.

It has been argued that, for karst springs, the initial steep portion of the curve represents the turbulent drainage of large fractures and conduits (e.g., the first microregime in Figure 4-8), followed by a transitional portion of the curve, where the flow is less turbulent and reflects the contribution of smaller fractures and rock matrix (the second microregime), ending with the slowly decreasing curve, the so-called master recession curve, where the drainage of rock matrix and small fissures is dominant. This last portion of the curve is also the most important for predicting the future discharge in a complete absence of water inputs, that is, during prolonged droughts. However, as explained further, there may be other explanations for various breaks in the recession curve.

As discussed by Bonacci (1993), every break point in the recession line is caused by a change in a characteristic of the groundwater reservoir (aquifer). A common situation is represented in Figure 4-10. The break points in the recession curve result from the decrease in the spring drainage area and the decrease in the effective porosity of the aquifer. For example, in karsts, during high groundwater levels, the flow may be dominated by a more karstified epikarst zone, which has higher effective porosity than deeper portions of the aquifer.

Figure 4-10. Explanation of common causes of changes in the value of recession coefficients (α) caused by changes in the size of catchment area (A) and effective porosity (n_e ).

(Modified from Bonacci, 1993.)

Figure 4-11 presents some less common shapes of the recession curve and the changes in recession coefficients. The main difference between Figures 4-10 and 4-11 is that α₂ > α₁ in the latter; that is, the recession coefficient α increases with time. The reasons for this can be numerous. For example, Figure 4-11b gives a schematic presentation of a catchment that consists partly of limestone and partly of schists. The outflow through limestone is much faster than that through the schists, which causes a lag in the water inflow (in the period t_i – t_j ) from the schists, and thus a change (increase) in the slope of the recession curve in the Mikro Vuono Spring in Greece (Soulios, 1991). Figure 4-11c is a schematic representation of a temporarily flooded cave in a karst aquifer. The bottom of the main outlet of the cave is at height H. As long as the groundwater level is below that height, the discharge occurs very slowly or not at all. A sudden lowering of the groundwater level leads to water outflow from the flooded cave. Understandably, this is not particular to caves; it can occur in a wider aquifer area with locally higher effective porosity (Bonacci, 1993). An identical situation is presented in Figure 4-11d. In this case, a polje in a karst is temporarily flooded instead of an underground reservoir in a cave. The roles of the cave and the polje are identical.

Figure 4-11. Explanation of other causes for changes in the value of recession coefficients (α).

(From Bonacci, 1993.)

The initial portion of the recession curve reflecting rapid discharge may not correspond to the simple exponential expression of the Maillet's type and may be better explained by some other functions. Deviations from exponential dependence can be easily detected if the recorded data plotted on a semilog diagram do not form a straight line(s).

Often a good approximation of the rapid drainage at the beginning of recession is the hyperbolic relation of the Boussinesq type. Its general form is

(4.10) $Q_t=\frac{Q_0}{{(1+\alpha t)}^n}$

In many cases this function correctly describes the entire recession curve. On the basis of 100 analyzed recession curves of karstic springs in France, Drogue (1972) concludes that, among the six exponents studied, the best first approximations of exponent n are ½, , and 2.

The exact determination of exponent n and the discharge coefficient α for the function that best fits the measured data is performed graphically and by computation as follows ( Kresic, 2007):

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The minimum recorded discharge at the end of recession is noted (Q ₂=0.057 m³/s for the example in Figure 4-12).

Figure 4-12. Recession curve of a spring with discharges used to determine the parameters of hyperbolic function.

(Modified from Kresic, 2007.)

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Any discharge, Q ₁, on the recession curve, that is not the result of (possible) deviation due to recent rainfall is chosen in the section between Q ₂ and Q ₀.

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The value of α that satisfies the following equation

(4.11) $\frac{\log (Q_0/Q_1)}{\log (Q_0/Q_2)}=\frac{\log (1+\alpha t_1)}{\log (1+\alpha t_2)}$

is determined by trial and error, adopting an initial value for α (usually 0.5). The result can be graphically checked, as shown in Figure 4-13: The correct coefficient of discharge forms a straight line through the points defined by Q ₀, Q ₁, Q ₂, and the corresponding times, t ₀, t ₁, t ₂. The exact value of α for this example is 0.202:

$\frac{\log \left(\frac{0.240{\text{m}}^3/\text{s}}{0.105{\text{m}}^3/\text{s}}\right)}{\log \left(\frac{0.240{\text{m}}^3/\text{s}}{0.060{\text{m}}^3/\text{s}}\right)}=\frac{\log (1+0.202\times 15\text{d})}{\log (1+0.202\times 47\text{d})}$

$0.5963\approx 0.5929$

Figure 4-13. Graphical determination of discharge coefficient α for the example shown in Figure 4-12.

(From Kresic, 2007.)

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The exponent n is calculated by substituting the determined value for α into either of the following two equations:

(4.12) $n=\frac{\log (Q_0/Q_1)}{\log (1+\alpha t_1)}$

(4.13) $n=-\frac{\log (Q_1/Q_2)}{\log \left(\frac{1+\alpha t_1}{1+\alpha t_2}\right)}$

For this example, equation (4.12) gives

$n=\frac{\log \left(\frac{0.240{\text{m}}^3/\text{s}}{0.105{\text{m}}^3/\text{s}}\right)}{\log (1+0.202\times 15\text{d})}=0.593$

and the recession discharge equation is then

$Q_t=\frac{0.24{\text{m}}^3/\text{s}}{{(1+0.202\times t)}^{0.593}}$

The coefficient of discharge α and the exponent n have the following general relationship:

(4.14) $\alpha =\frac{\sqrt[\mathit{n}]{Q_0}-\sqrt[n]{{\mathit{Q}}_{\mathit{t}}}}{t\times \sqrt[n]{Q_t}}$

As in the case of the Maillet equation, the determined hyperbolic function can be used to calculate the volume of free gravitational water stored in the aquifer above the spring level. In general, this volume at any time t since the beginning of recession is

(4.15) $V_t=\frac{Q_0}{\alpha (n-1)}\left[1-\frac{1}{{(1+\alpha t)}^{n-1}}\right]\times 86,400\text{s}[{\text{m}}^3]$

4.2.1 Approximation with linear reservoirs

Analogous to the nonlinear nature of discharge in most natural hydrologic systems (Amorocho, 1964), the connection between the karst spring discharge (Q) during a recession period and the hydraulic head in the aquifer (H) can be approximated by the following equation (Castany, 1967; Avdagic, 1990; Bonacci, 1993):

(4.16) $Q=\lambda H^n$

where λ is a coefficient that characterizes the conductivity and effective porosity of the aquifer and n describes the type of the connection: When n = 1, the discharge is linear; and when n > 1, it is nonlinear. The same connection can be expressed in terms of the volume (V) of water discharged:

(4.17) $V=\lambda Q^n$

The equation of mass conservation (the continuity equation) is

(4.18) $I-Q=\frac{\mathit{dV}}{\mathit{dt}}$

where I is water input and t is time; it follows that

(4.19) $\frac{\mathit{dV}}{\mathit{dt}}=\lambda n{Q}^{n-1}\frac{dQ}{\mathit{dt}}$

Combining equations (4.18) and (4.19) and after integration, the discharge rate is

(4.20) $Q={\left[Q_0^{n-1}+\frac{1-n}{\lambda n}(t-t_0)\right]}^{1/(n-1)}$

Although the parameters λ and n can be identified from the discharge measurements (Q versus time), the procedure is somewhat complex and practitioners still favor approximation of aquifer (spring) discharge as outflow from a linear reservoir(s). Figure 4-14 illustrates this concept, where the volume of water (V) discharged between times t and t + dt is given as

Figure 4-14. Schematic of discharge from a linear reservoir.

(From Avdagic, 1990.)

(4.21) $\mathit{dV}=\mathit{Qdt}$

Using the reservoir (aquifer) active area (A) and the decrease in the hydraulic head (dH), this volume is

(4.22) $\mathit{dV}=\mathit{AdH}$

Since, for the linear reservoir, Q = λH, it follows that

(4.23) $\mathit{Qdt}=\lambda \mathit{Hdt}$

Combining equations (4.21) and (4.23) gives

(4.24) $\lambda \mathit{Hdt}=-\mathit{AdH}=-\mathit{dV}$

which, when integrated, results in

(4.25) $t=-\frac{A}{\lambda }\ln H+\mu$

Knowing that H = Q/λ, equation (4.25) becomes

(4.26) $Q={\text{e}}^{-\lambda /A(t-\mu )}$

The solution of equation (4.26) for the boundary conditions t = t ₀ and Q = Q ₀ is

(4.27) $Q=Q_0{\text{e}}^{-\lambda /A(t-t_0)}$

which is very similar to the Maillet equation (4.4). The coefficient of discharge α can be expressed with the active aquifer area (A) and the parameter dependent on the conductivity (effective porosity) of the aquifer (λ) as proposed by Castany (1967):

(4.28) $\alpha =\frac{\lambda }{A}$

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Hydraulics

Don D. Ratnayaka , ... K. Michael Johnson , in Water Supply (Sixth Edition), 2009

Broad-Crested Weir

For the broad-crested weir, illustrated in Figure 12.7(b) , the discharge coefficient, ${C^{″}}_d$ is indeed about $1/\sqrt{3}=0.577$ However, as indicated earlier it is easier to measure the actual water depth, h, above the weir crest, than the total energy head H (including the velocity of approach head), hence the equation is normally written as:

$Q=C_{v}0.577\left(2/3\right)\sqrt{\left(2g\right)}b{h}^{1.5}=C_{v}1.705b{h}^{1.5}$

where h is measured away from the local draw-down of the water surface as it passes over the crest, and C _v is a velocity coefficient to account for the approach velocity head. If the approach velocity is small then C _v can be taken as 1.0 and for other cases the effect of approach velocity head can be calculated and included in C _v .

The generalized weir equation can be written as $Q=C_D{C}_V{C}_S{C}_P{C}_{bl}\sqrt{g}b{h}^{1.5}$ where C _D is a discharge coefficient, C _v is the velocity coefficient, both as defined before, C _s is a shape coefficient depending on the lateral shape of the crest (e.g. shallow 'Vee'), C _p is a coefficient to take into account the height of the weir and C _bl is a boundary layer coefficient to take into account the growth of the boundary layer across the length of the weir crest in the direction of flow. For accurate measurement, all these factors need to be taken into account and are discussed in detail in Ackers (1978).

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Emphasizes how to apply techniques of process design and interpret results into mechanical equipment details

In Applied Process Design for Chemical & Petrochemical Plants, 1999

Nozzles and Orifices [3]

These piping items shown in Figures 2-17 and 2-18 are important pressure drop or head loss items in a system and must be accounted for to obtain the total system pressure loss. For liquids:

Figure 2-17. Flow coefficient "C" for nozzles. C based on the internal diameter of the upstream pipe.

By permission, Crane Co. [3]. Crane reference [9] is to Fluid Meters, American Society of Mechanical Engineers, Part 1-6th Ed., 1971. Data used to construct charts. Chart not copied from A.S.M.E. reference. Copyright © 1971

Figure 2-18. Flow coefficient "C" for square edged orifices.

By permission, Crane Co. [3], Technical Paper 410 Engineering Div. (1976) and Fluid Meters, Their Theory and Application Part 1, 6th Ed., 1971, American Society of Mechanical Engineers and, Tuve, G. L. and Sprenkle, R. E., "Orifice Discharge Coefficients for Viscous Liquids," Instruments Nov. 1933, p. 201. Copyright © 1971

(2-46) $q={\text{C}}^{\prime }\text{A}\sqrt{2\text{g}\left(144\right)\left(\text{Δ}\text{P}\right)/\rho }={\text{C}}^{\prime }\text{A}{\left[2{\text{gh}}_{\text{L}}\right]}^{1/2}$

where q = cubic ft/sec of fluid at flowing conditions

C′ = flow coefficient for nozzles and orifices

(2-47) $\begin{array}{cc}{\text{C}}^{\prime }& ={\text{C}}_{\text{d}}/\sqrt{1-{\beta }^4},\text{corrected\hspace{0.17em}for\hspace{0.17em}velocity\hspace{0.17em}of}\hfill \\ & \text{approach}\hfill \end{array}$

Note:

C′ = C for Figures 2-17 and 2-18, corrected for velocity of approach.

C_d = discharge coefficient for nozzles and orifices

h_L = differential static head or pressure loss across flange taps when C or C values come from Figures 2-17 and 2-18, ft of fluid. Taps are located one diameter upstream and 0.5 diameter down from the device.

A = cross section area of orifice, nozzle or pipe, sq ft

h = static head loss, ft of fluid flowing

ΔP = differential static loss, lbs/sq in. of fluid flowing, under conditions of h_L above

β = ratio of small to large diameter orifices and nozzles and contractions or enlargements in pipes

For discharging incompressible fluids to atmosphere, take C values from Figures 2-17 or 2-18 if h_L or ΔP is taken as upstream head or gauge pressure.

For flow of compressible fluids use the net expansion factor Y (see later discussion) [3]:

(2-48) $\text{q}=\text{Y}{\text{C}}^{\prime }\text{A}{\left[2\text{g}\left(144\right)\left(\text{Δ}\text{P}\right)/\rho \right]}^{1/2}$

where

Y = net expansion factor for compressible flow through orifices, nozzles, and pipe.

C′ = flow coefficient from Figures 2-17 or 2-18. When discharging to atmosphere, P = inlet gauge pressure. (Also see critical flow discussion.)

For estimating purposes in usual piping systems, the values of pressure drop across an orifice or nozzle will range from 2 to 5 psi. For more exact system pressure drop calculations, the loss across these devices should be calculated using some size assumptions.

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Bit Hydraulics

Mark S. Ramsey P. E. , in Practical Wellbore Hydraulics and Hole Cleaning, 2019

2.8 Ongoing Continuous Hydraulics Optimization Calibration Example

For brevity, we assume the pumps have been calibrated and the driller has taken the data (OCHO steps 1 and 2). The collected data, with SPM converted to GPM, is shown below in the first two columns of Table 2.4.

Table 2.4. Example OCHO wellbore hydraulics calibration data

GPM (calculated from SPM and pump efficiency)	Standpipe pressure (measured)	ΔP BIT (calculated)	ΔP CIRC or the wasted energy line (calculated)
140 (SCR)	480	72	408
227	1200	191	1009
314	2200	367	1833

GPM, gallons per minute; SCR, slow circulation rate; SPM, strokes per minute.

Source: Courtesy: Texas Drilling Associates.

The right two columns represent OCHO step 3, where the bit pressure drop is calculated (column 3) and then subtracted from the standpipe pressure to yield P _CIRC (column 4), representing P _CIRC, the wasted pressure loss in the system, or everything except the bit nozzle pressure drop.

To calculate column three, the ΔP _BIT, the equation below is used.

(2.25) ${\Delta P}_{\mathrm{BIT}}=\frac{\mathrm{MW}\times Q^2}{12,042\times {C_D}^2{\times \mathrm{TNFA}}^2}$

The primary issue over recent years has been what the appropriate value of C _D should be (the coefficient that incorporates both the nozzle efficiency and the nozzle discharge coefficient into a single term C _D ).

If the pressure recovery effect is included, the bit pressure drop equation becomes

(2.26) ${\Delta P}_{\mathrm{BIT}}=\frac{\mathrm{MW}\times Q^2}{12,042{\times (1.03}^2)\times {\mathrm{TNFA}}^2}$

where ΔP _BIT is the pressure drop across the nozzles (psi); MW is the mud weight (ppg); Q is the flow rate (GPM); and TNFA is the total flow area of the nozzles, in square inches. Note that this TNFA term is squared in the equation.

This equation is slightly different from older and perhaps more familiar equations in that the nozzle coefficient of 0.95 has been replaced by a nozzle coefficient of 1.03 for the newer nozzles and bits. This is an attempt to quantify the pressure recovery effect observed from field measurements. This coefficient was also independently validated in controlled laboratory tests.^4,5 Note that in some companies, the 12,042 constant and the nozzle coefficient are combined into a single term for convenience.

(2.27) ${\Delta P}_{\mathrm{BIT}}=\frac{\mathrm{MW}\times Q^2}{12,775\times {\mathrm{TNFA}}^2}$

Further, note that API RP 13D committee members did not agree on an exact value of the nozzle discharge coefficient (and associated pressure recovery) but did recognize the problems facing hydraulics designers. The API RP 13D committee modified their recommended bit pressure drop equation to

(2.28) ${\Delta P}_{\mathrm{BIT}}=\frac{\mathrm{MW}\times Q^2}{12,042{{\times C}_V}^2{\times \mathrm{TNFA}}^2}$

The committee stopped short of endorsing a 1.03 value for the C _V (same as C _D in this text). In the words of the API 13D subcommittee,

"The discharge coefficient C_V, varies with the diametric ratio (output diameter/input diameter) and the fluid Reynolds numbers passing through the nozzles. There is significant evidence to update the long-standing C_V, value of 0.95 to 0.98, given the flow rates, drilling fluid densities, and nozzle ratios typical to oilfield operating conditions.

C_V=0.98

In field and laboratory test the flow not only through the nozzle but also the flow past the nozzle is considered when determining C_V. This results in a discharge coefficient of C_V=1.03 for roller cone bits. The design of the bit has in this case an impact on the discharge coefficient. Especially for PDC bits a single C_V has not been determined yet. The reported C_V ranges between 0.89 and 0.97. Therefore, a final recommendation for the discharge coefficient considering the flow past the nozzle cannot be made. Further tests are necessary to resolve the issue." [sic] ^14–16

Needless to say, additional research is needed in this area, particularly with respect to the pressure recovery effects and its relationship to mud weight, high shear rate viscosities, and drill bit/bit nozzle geometries.

OCHO steps 4–6 are shown graphically in Fig. 2.9, where the wasted energy line is plotted, a best fit straight line is drawn through the points, and where the slope of the line measured linearly (i.e., with a ruler, as is also shown in Fig. 2.9).

Note that if calculating slope numerically either with a calculator or in a spreadsheet, two cautions are in order.

First, the author strongly encourages some sort of visible plotting of the data be incorporated so that data integrity may be checked. Since the relationship between flow rate and pressure drop is an exponential one, identification of spurious or outlying data points is difficult if one is simply inspecting numerical data without a graphical plot.

Second, if determining the slope numerically, one must be careful to take the log of the values to compute the slope as opposed to the values themselves. Reviewing, the conventional slope calculation would be found in a Cartesian plot using

(2.29) $\mathrm{slope},u=\frac{\mathrm{rise}}{\mathrm{run}}$

(2.30) $u=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

However, for the exponential function, the slope is found by taking the log of each data pair value. If calculating based on the numeric values, the log of the values must be used.

(2.31) $u=\frac{\log y_2-\log y_1}{\log x_2-\log x_1}$

or

(2.32) $u=\frac{\log \left(\frac{y_2}{y_1}\right)}{\log \left(\frac{x_2}{x_1}\right)}$

For our example case, the slope was measured to be 1.62. This is considerably lower than the "standard" values used by many available computer software programs, which typically range from 1.8 to 1.9 or so. These larger values are artifacts of research conducted decades ago on drilling fluid systems of that day and do not accurately reflect modern drilling fluids in use today, generally speaking.

At this point, it must be reemphasized that the wasted pressure line constructed in Figs. 2.8 and 2.9 represents the entire circulation system except for the bit. It includes surface piping, all drill string components, and all return annuli. To flow at any flow rate across the x-axis (flow rate) will extract the corresponding amount of pressure (or energy) indicated on the y-axis (pressure).

Figure 2.8. Flow rate/pressure operating space.

Courtesy: Texas Drilling Associates.

Figure 2.9. Graphical measurement of slope (exponent u) on log–log data plot.

Courtesy: Texas Drilling Associates.

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CSI Tests on the Experimental Canal PAC-UPC

Enrique Bonet Gil , in Experimental Design and Verification of a Centralized Controller for Irrigation Canals, 2018

7.2 Initial and boundary conditions

We introduced the initial conditions from the water-level measurements at the checkpoints, the flow downstream of every pool (Table 7.5) and the position of gate 1. The upstream boundary condition is determined by gate 1 and the water level at the reservoir. The gate equation is represented for the next system of equations as:

[7.3] $\begin{array}{l}A\left(y_s\right)v_s=c_G\times a_G\times u\times \sqrt{L1\left(t\right)-y_s}\left(\mathit{submerged\; hydraulic\; jump}\right)\\ A\left(y_s\right)v_s=c_G\times a_G\times u\times \sqrt{L1\left(t\right)-\mathit{Cc}\times u}\left(\mathit{unsubmerged\; hydraulic\; jump}\right)\end{array}$

where y_S is the downstream water level of the reservoir; v_S is the velocity in that cross-section; L₁(t) is the water-level measurement upstream of the reservoir; Cc is the contraction coefficient; C_G is the discharge coefficient, which depends on different parameters, such as the upstream water level, gate opening, contraction coefficient and flow condition; a _G is the gate width and "u" is the gate opening.

Figure 7.7. Submerged hydraulic jump

The gate features are shown in Table 7.3.

Table 7.3. Features of the gate

Gate	Numerical node upstream	Numerical node downstream	Gate discharge coefficient	Contraction coefficient	L1 water level reservoir	Gate width (m)	Height of the gate opening (m)	Steep (m)
1	0	1	0.68	0.60	1.257	0.434	0.122	0.0

Finally, the downstream boundary condition is associated with weir 4, where the water level reaches a critical depth. The weir equation is represented by the following discharge equations of a sharp crested weir:

[7.4] $\begin{array}{l}A\left(y_{R1}\right)\times v_{R1}-c_W\times B\times {\left(H\right)}^{\frac{3}{2}}=0\\ c_W=\frac{2}{3}\times \sqrt{2g}\times c_d\end{array}$

where y _R1 and v _R1 are the water level and velocity at the sharp crested weir; H is the measured head above the crest, excluding the velocity head; C_d is the discharge coefficient and B is the weir width.

Figure 7.8. Sharp crested weir

The features of the sharp crested weir are shown in Table 7.4.

Table 7.4. Features of the weir

Weir	Numerical node	Weir discharge coefficient (CW)	Weir height (yW) (m)	Weir width (m)
4	341	0.5776	0.35	0.39

The steady state is the initial condition for the canal. The total flow is 110 l/s, and weirs 1, 2 and 3 are not operative. Table 7.5 shows the water-level measurements obtained manually and the flow rate at particular points at the initial time.

Table 7.5. Initial conditions (water level at particular points)

Checkpoint	Initial water level (m)	Flow (m3/s)
1 (L4)	0.758	0.110
2 (L6)	0.730	0.110
3 (L8)	0.689	0.110
4 (L10)	0.644	0.110
5 (L11)	0.604	0.110

The initial steady profile for the canal is shown in Figure 7.9.

Figure 7.9. Backwater profile for the canal from initial conditions (steady state). For a color version of the figure, please see www.iste.co.uk/bonet/canals.zip

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Choke Performance

Boyun Guo Ph.D. , ... Ali Ghalambor Ph.D. , in Petroleum Production Engineering, 2007

5.3 Single-Phase Liquid Flow

When the pressure drop across a choke is due to kinetic energy change, for single-phase liquid flow, the second term in the right-hand side of Eq. (5.1) can be rearranged as

(5.2) $q=C_{D}A\sqrt{\frac{2g_c\Delta P}{\rho }},$

where

q = flow rate, ft³/s

C_D = choke discharge coefficient

A = choke area, ft²

g_c = unit conversion factor, 32.17 lb_m-ft/lb_f-s²

ΔP = pressure drop, lb_f/ft²

ρ = fluid density, lb_m/ft³

If U.S. field units are used, Eq. (5.2) is expressed as

(5.3) $q=8074C_D{d}_2^2\sqrt{\frac{\Delta p}{\rho }},$

where

q = flow rate, bbl/d

d ₂ = choke diameter, in.

Δp = pressure drop, psi

The choke discharge coefficient C_D can be determined based on Reynolds number and choke/pipe diameter ratio (Figs. 5.2 and 5.3). The following correlation has been found to give reasonable accuracy for Reynolds numbers between 10⁴ and 10⁶ for nozzle-type chokes (Guo and Ghalambor, 2005):

Figure 5.2. Choke flow coefficient for nozzle-type chokes

(data used, with permission, from Crane, 1957).

Figure 5.3. Choke flow coefficient for orifice-type chokes

(data used, with permission, from Crane, 1957).

(5.4) $C_D=\frac{d_2}{d_1}+\frac{0.3167}{{\left(\frac{d_2}{d_1}\right)}^{0.6}}+0.025[\log (N_{\text{Re}})-4],$

where

d ₁ = upstream pipe diameter, in.

d ₂ = choke diameter, in.

N _Re = Reynolds number based on d₂

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EXPERIMENTAL TECHNIQUES

KAI SIREN , ... PETER V. NIELSEN , in Industrial Ventilation Design Guidebook, 2001

12.3.9.2 Orifice, Venturi, and Nozzle

The orifice, the venturi, and the nozzle are instruments for the measurement of duct or pipe flow rate. A constriction, throttling the flow, is placed in the duct, and the resulting differential pressure developed across the constriction is measured. It is the difference in the geometric shape that characterizes the three devices; see Fig. 12.22.

FIGURE 12.22. Schematic of three flow meters based on the obstruction principle.

The orifice plate is simple to manufacture and has a relatively low cost. It does, however, create a quite large permanent pressure loss when installed in the ductwork. The venturi is smoothly shaped with a low permanent pressure loss but requires more space and is more expensive. The nozzle is a compromise between the orifice and the venturi. All three devices are standardized flow meters with very detailed descriptions of their geometry, material, manufacturing, installation, and Use. ^38–42

Based on the measured pressure difference over the device, the throat diameter, and some other parameters, the flow rate can be determined. The equation for the volume flow rate is, in general, ³⁸

(12.30) $q_v=CE\in \frac{\pi d^2}{4}\sqrt{\frac{2\Delta p}{\rho }},$

where C is the coefficient of discharge, E is the velocity approach factor, ∈ is an expansion factor, d is the diameter of the throat of the device, Δp is the measured pressure differential, and Δ is the fluid density at the upstream pressure tapping. Correspondingly, the mass flow rate is

(12.31) $q_m=CE\in \frac{\pi d^2}{4}\sqrt{2\Delta p\rho }.$

The product of the discharge coefficient and the velocity approach factor, α = CE, is called the flow coefficient. For the orifice, the discharge coefficient is given by the equation ³⁸

(12.32) $\begin{array}{c}C=0.5959+0.0312{\beta }^{2.1}-0.184{\beta }^8+0.0029{\beta }^{2.5}{\left[\frac{{10}^6}{{\mathrm{Re}}_D}\right]}^{0.75}\\ +0.09L_1\frac{{\beta }^4}{1-{\beta }^4}-0.0337L_2{\beta }^3,\end{array}$

where β = d/D is the diameter ratio of the throat and the duct diameters, Re_D is the duct Reynolds number, L₁ = l₁/D is the distance of upstream pressure tapping from the upstream face of the orifice plate divided by the duct diameter, and L₂ = l₂/D is the distance of the downstream pressure tapping from the downstream face of the plate divided by the duct diameter. From the above it can be noticed that the calculation of the discharge coefficient is an iterative process, as the duct Reynolds number is included in the equation. The discharge coefficient for a nozzle is given by a slightly simpler equation:

(12.33) $\begin{array}{c}C=0.99+0.2262{\beta }^{4.1}+[0.000215-0.001125\beta \\ +0.00249{\beta }^{4.7}]{\left[\frac{{10}^6}{{\mathrm{Re}}_D}\right]}^{1.15}.\end{array}$

The discharge coefficient of the venturi tube depends on the convergent of the venturi, but is given as a constant value for a specified device. The range is from 0.984 for a rough-cast convergent to 0.995 for a machined convergent. ³⁸

The velocity approach factor is dependent on the diameter ratio only:

(12.34) $E=\frac{1}{\sqrt{1-{\beta }^4}}.$

The expansion factor e takes into account the compressibility effects of the fluid. It is close to unity in most industrial ventilation applications.

For the above equations to be valid, the measurement devices must be manufactured according to standards and also must be installed according to given specifications. There are strict requirements concerning the minimum upstream and downstream straight lengths. These lengths depend mainly on the diameter ratio of the device and the type of the nearest upstream fitting causing a disturbance in the incoming flow velocity profile. Table 12.6 provides typical values of the required straight lengths for orifice plates and nozzles.

TABLE 12.6. Some Required Straight Lengths (LID) for Orifice Plates and Nozzles ³⁸

	Upstream side of the primary device			Downstream side
β	Single 90° bend or tee	Two or more 90° bends in different planes	Expander (0.5D to D over a length of I D to 2D)	All fittings
<0.2	10 (6)	34 (17)	16 (8)	4 (2)
0.4	14 (7)	36 (18)	16 (8)	6 (3)
0.6	18 (9)	48 (24)	22 (11)	7 (3.5)
0.8	46 (23)	80 (40)	54 (27)	8 (4)

If the lower values in the brackets are applied, an additional ±0.5 uncertainty (error on 5% risk level) has to be added arithmetically to the flow coefficient confidence limits. The use of flow straighteners is recommended in cases when a nonstandard type of upstream fitting disturbs the flow velocity profile.

Every measured quantity or component in the main equations, Eqs. (12.30) and (12.31), influence the accuracy of the final flow rate. Usually a brief description of the estimation of the confidence limits is included in each standard. The principles more or less follow those presented earlier in Treatment of Measurement Uncertainties. There are also more comprehensive error estimation procedures available. ^43–46 These usually include, beyond the estimation procedure itself, some basics and worked examples.

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Design of weirs and spillways

Hubert Chanson , in Hydraulics of Open Channel Flow (Second Edition), 2004

19.1.3 Discussion

Although a spillway is designed for specific conditions (i.e. design conditions: Q _des and H _des), it must operate safely and efficiently for a range of flow conditions.

Design engineers typically select the optimum spillway shape for the design flow conditions. They must then verify the safe operation of the spillway for a range of operating flow conditions (e.g. from 0.1 Q _des to Q _des) and for the emergency situations (i.e. Q > Q _des).

Table 19.2. Discharge coefficient for sharp-crested weirs (full-width weir in rectangular channel)

Reference (1)	Discharge coefficient C(2)	Range (3)	Remarks (4)
von Mises (1917)	$\frac{\pi }{\pi +2}$	$\frac{d_1-\Delta z}{\Delta z}\text{very}\text{large}$	Ideal fluid flow calculations of orifice flow
Henderson (1966)	$0.611+0.08\frac{d_1-\Delta z}{\Delta z}$	$\text{0}\le \frac{d_1-\Delta z}{\Delta z}<5$	Experimental work by Rehbock (1929)
	1.135	$\frac{d_1-\Delta z}{\Delta z}=10$
	$1.06{\left(1+\frac{\Delta z}{d_1-\Delta z}\right)}^{3/2}$	$20<\frac{d_1-\Delta z}{\Delta z}$
Bos (1976)	$0.602+0.075\frac{d_1-\Delta z}{\Delta z}$	$\begin{array}{l}{d}_1-\Delta z>0.03\text{m}\\ \frac{d_1-\Delta z}{\Delta z}<2\\ \Delta z>0.40\text{m}\end{array}$	Based on experiments performed at Georgia Institute of Technology
Chanson (1999)	1.0607	Δz = 0	Ideal flow at overfall

Table 19.3. Discharge correlations for sharp-crested weirs (full width in rectangular channel)

Reference(1)	Discharge per unit width q (m2/s) (2)	Comments (3)
Ackers et al. (1978)	$0.564\left(1+0.150\frac{d_1-\Delta z}{\Delta z}\right)\sqrt{g}{(d_1-\Delta z+0.0001)}^{3/2}$	Range of applications d 1 – Δz > 0.02 m Δz > 0.15 (d 1 – Δz)/Δz<2.2
Herschy (1995)	1.85(d 1 – Δz)3/2	Approximate correlation (±3%): (d 1 – Δz)/Δz<0.5

In the following sections, we present first the crest calculations, then the chute calculations followed by the energy dissipator calculations. Later the complete design procedure is described.

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Valves and Meters

Malcolm J. Brandt BSc, FICE, FCIWEM, MIWater , ... Don D. Ratnayaka BSc, DIC, MSc, FIChemE, FCIWEM , in Twort's Water Supply (Seventh Edition), 2017

18.20 Cavitation at Valves

Cavitation is the generation of pockets (cavities) of water vapour and their subsequent collapse (Section 15.12 ). A part open valve presents an orifice which produces a high velocity low-pressure jet. The onset of cavitation is marked by a fall-off in discharge coefficient as flow is increased, as downstream pressure is decreased or as the orifice is made smaller since any of these factors tends to depress the pressure in the orifice. The fall-off in discharge coefficient occurs because vapour pockets start to occupy the orifice opening. This effect can be used to determine when incipient cavitation occurs and is identified as the point where the curve of headloss coefficient against cavitation number begins to rise markedly. At this level there is minimal noise, vibration or risk of damage. Another method of assessing the onset and severity of cavitation is by taking noise and vibration measurements in the valve. Three stages of increasing cavitation are described in Control Valves (Borden, 1998): incipient, full and supercavitation. In this context incipient cavitation is characterized by the irregular occurrence of cavitation instances at vortices. As cavitation increases, there is constant production and collapse of cavities and noise is steady and reaches a maximum level, after which (in supercavitation) noise decreases somewhat due to the dampening effect of the volume of vapour present and as the collapse area is pushed downstream.

At small openings the internal shape of the valve (according to type) has little effect on the headloss coefficient. Therefore, in such circumstances critical cavitation conditions are likely to present themselves at very similar loss coefficients irrespective of valve type. The shape of the opening has a minor effect on the onset of cavitation. However, both the nature of the opening and the shape of valve internals significantly affect the risk of damage once cavitation is occurring. Valves in which collapse of vapour pockets occurs away from the valve body and other components should not be damaged by cavitation. The distance a jet travels before head recovers sufficiently to collapse vapour pockets depends on jet size. Therefore, porting the opening to produce numerous jets can help keep the area of collapse away from critical components. An added advantage of numerous small jets is that noise and vibration are reduced.

Some materials commonly used in valve construction (cast iron and brass) are not particularly resistant to cavitation damage but some bronzes are more resistant. Various stainless steels, when hardened, particularly Duplex types, provide superior resistance (Borden, 1998). In very extreme cases titanium may have to be used.

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How to Draw Line of Best Fit on Semilog Graph

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