Steady State Assumption for Continuity Equation

Steady State Flow

In steady state flow within a tube (the blood's natural environment), platelet adhesion onto a surface will be reaction rate limited, that is the number of platelets adhering will be proportional to the affinity of the cells to the surface.

From: Encyclopedia of Biomedical Engineering , 2019

Fundamentals of Reservoir Oil Flow Analysis

Amanat U. Chaudhry , in Oil Well Testing Handbook, 2004

Steady-State Flow Equations and Their Practical Applications

In steady-state flow, there is no change anywhere with time, i.e., the right-hand sides of all the continuity and diffusivity equations are zero. Solutions for steady state are, however, useful for certain unsteady-state problems. The steady-state flow equations can be derived from integrating and evaluating the integration constants from the boundary conditions. The steady-state flow equation and Darcy's equations accounting for specific geometry arepresented here. The steady-state flow equations are based on the following assumptions:

1

Thickness is uniform, and permeability is constant.

2

Fluid is incompressible.

3

Flow across any circumference is a constant.

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Reservoir deliverability

Boyun Guo , in Well Productivity Handbook (Second Edition), 2019

3.2.2 Steady-state flow

Steady-state flow is defined as a flow condition under which the pressure at any point in the reservoir remains constant over time. This flow condition prevails when the pressure funnel shown in Fig. 3.1 has propagated to a constant-pressure boundary. The constant-pressure boundary might be the edge of an aquifer, or the region surrounding a water injection well. A sketch of this reservoir model is shown in Fig. 3.2, where p _e represents the pressure at the constant-pressure boundary. Under steady-state flow conditions due to a circular constant-pressure boundary at distance r _e from the wellbore centerline, assuming single-phase flow, the following theoretical relationship for an oil reservoir can be derived from Darcy's law:

Figure 3.2. A reservoir model illustrating a constant-pressure boundary.

(3.5) $q=\frac{kh\left(p_e-p_{wf}\right)}{141.2B_o{\mu }_o\left(\ln \frac{r_e}{r_w}+S\right)}$

where "ln" denotes 2.718-based natural logarithm (log _e ). The derivation of Eq. (3.5) is left to the reader as an exercise.

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Creep in Amorphous Metals

M.E. Kassner , K.K. Smith , in Fundamentals of Creep in Metals and Alloys (Third Edition), 2015

2.4 Primary and Transient Creep (Non–Steady-State Flow)

Steady-state flow has principally been discussed, so far. It has been presumed that STZs create free volume (leading to softening) and that recovery processes promote the annihilation of free volume (leading to hardening). Therefore, steady state has been regarded as a balance between free volume creation and annihilation. Other hardening effects such as chemical ordering have not been explicitly considered for steady state. It has been suggested that there can be a net free volume increase or decrease during deformation that precedes a steady state. Figure 142 from Lu et al. [1068] shows hardening at the onset of deformation that continues beyond the eventual steady state. The interpretation of this peak stress followed by softening to a steady state is unclear.

Figure 142. Effect of strain rate on the uniaxial stress-strain behavior of Vitreloy 1 at 643   K and strain rates of 1.0   ×   10⁻¹, 3.2   ×   10⁻², 5.0   ×   10⁻³ and 2.0   ×   10⁻⁴  s⁻¹ [1068].

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27th European Symposium on Computer Aided Process Engineering

Jakob D. Redlinger-Pohn , Stefan Radl , in Computer Aided Chemical Engineering, 2017

3.1 Fluid Flow Field

The steady-state flow field is presented in Figure 1. The lower panels show the velocity profile in streamwise direction and the velocity component in the channel cross section parallel to the channel width W. As expected, the stream wise velocity is higher at the outer bent. We find from our simulation data a lower speed of secondary motion for larger aspect ratio channels. We attribute this to the resulting lower curvature κ, which is expected to yield slower secondary motion. Interestingly, a shape change (from rectangular to trapezoidal) leads to faster secondary motion at the outer bent, but a slower secondary speed at the inner bent. Considering the secondary velocity along the channel height (normalized with the channel height at the sampling location) we find the change in the direction from outwards to inwards flow to take place at the same vertical position.

Figure 1. Velocity distribution in the streamwise direction, as well as in the cross sectional plane for the rectangular channels (dashed line: AR-7p5, solid line: AR-3p3), and for the trapezoidal channel (dash-dotted line).

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Well Test Analysis: The Use of Advanced Interpretation Models

In Handbook of Petroleum Exploration and Production, 2002

1.2.1 Types of flow behavior

The different flow behaviors are usually classified in terms of rate of change of pressure with respect to time.

Steady state

During steady-state flow, the pressure does not change with time. This is observed for example when a constant pressure effect, such as resulting from a gas cap or some types of water drive, ensures a pressure maintenance in the producing formation.

Pseudo steady state

The pseudo steady state regime characterizes a closed system response. With a constant rate production, the drop of pressure becomes constant for each unit of time.

Transient state

Transient responses are observed before constant pressure or closed boundary effects are reached. The pressure variation with time is a function of the well geometry and the reservoir properties, such as permeability and heterogeneity.

Usually, well test interpretation focuses on the transient pressure response. Near wellbore conditions are seen first and later, when the drainage area expands, the pressure response is characteristic of the reservoir properties until boundary effects are seen at late time (then the flow regime changes to pseudo steady or steady state). In the following, several characteristic examples of well behavior are introduced, for illustration of typical well test responses.

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Advanced Forming Technologies

K.P. Rao , Y.V.R.K. Prasad , in Comprehensive Materials Processing, 2014

3.15.3.1 Kinetic Model

The steady-state flow stress in hot deformation may be related to the strain rate and temperature through an Arrhenius type of rate equation (7):

[1] $\dot{\varepsilon }=A{\sigma }^n\text{exp}\left[-\frac{Q}{RT}\right]$

Where σ  =   flow stress, , A  =   constant, n  =   stress exponent, Q  =   activation energy, R  =   gas constant, and T  =   temperature. In pure metals, this equation is valid over a wide range of temperature and strain rate, and the estimated apparent activation energy is close to that for self-diffusion. However, in alloys and commercial materials, the equation is valid over narrower ranges, and the apparent activation energy is generally higher than that for self-diffusion due to contributions from back stress generated by solid solution strengthening, second phase, or dispersion hardening. This complicates the evaluation of rate-controlling deformation mechanisms. Changes in the microstructure occurring in hot deformation, like grain size, are generally correlated with the help of a temperature-compensated strain rate or Zener-Hollomon parameter, Z, defined by

[2] $Z=\dot{\varepsilon }\exp \left[\frac{Q}{\mathit{RT}}\right]$

An inverse linear correlation of average grain diameter with Z on a log–log scale is generally observed in pure metals. The kinetic model does not help in optimization of workability of commercial materials or in the microstructural control or in the avoidance of defects in processing.

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Hydraulic Properties of Soils

Abdel-Mohsen Onsy Mohamed , Evan K. Paleologos , in Fundamentals of Geoenvironmental Engineering, 2018

13.5 Pumping Tests to Estimate Hydraulic Conductivity: Steady-State (Equilibrium) Radial Flow to a Well

Steady-state (equilibrium) radial flow to a well in unconfined and confined aquifers is described in Chapter 5, Section 5.10.

For steady-state flow, in a confined aquifer, to a well with a pumping rate Q, Thiem equation for two piezometers at distances r ₁ and r ₂ from the well, where drawdowns φ ₁ and φ ₂ are measured, respectively, yields for the transmissivity of the aquifer (Eq. (5.65) repeated here):

(13.13) $T=Kb=\frac{Q}{2\pi \left({\phi }_1-{\phi }_2\right)}ln\left(\frac{r_2}{r_1}\right)$

In practice, the drawdown may be estimated if it is possible to define a radius of influence, R, which represents the horizontal distance beyond which pumping of the well has little influence on the aquifer. Then, Eq. (5.64) gives

(13.14) $T=\frac{Q}{2\pi {\phi }_{\mathit{max}}}ln\left(\frac{R}{r_{\mathit{well}}}\right)\text{,}$

where r_well is the outer radius of the well screen and φ _max is the maximum drawdown that occurs at the well.

For an unconfined aquifer in which the drawdown is a significant fraction of the saturated thickness, Eq. (5.69) is repeated here and gives the hydraulic conductivity as

(13.15) $K=\frac{Q}{\pi \left(h_1^2-h_2^2\right)}ln\left(\frac{r_1}{r_2}\right)\text{,}$

where h ₁ and h ₂ are the heads at distances r ₁ and r ₂, respectively. The transmissivity can be approximated from Eq. (13.15) as

(13.16) $T\approx K\frac{h_1+h_2}{2}$

If the drawdown φ is small compared with 2H (the original piezometric head, shown as h ₀ in Fig. 13.4), then Eq. (5.72) can be used to calculate K:

(13.17) $K\approx \frac{Q}{2\pi H\phi }ln\left(\frac{R}{r}\right)\text{,}$

where φ is the drawdown at a distance r from the well and R is again the radius of influence.

Eqs. (13.13)–(13.17) are applicable to steady-state or equilibrium conditions. Therefore, pumping needs to be continued at a uniform rate for a sufficient period of time to reach steady-state conditions, so that the drawdown, s or φ, at the point of measurement does not change with time. Other assumptions, inherent in the development of these equations include: (a) the observation wells are located close to the pumping well; (b) the aquifer is homogeneous and isotropic, and has uniform thickness and infinite areal extent; and (c) the well completely penetrates the thickness of the aquifer.

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FSI-based forced response analyses of a mistuned high pressure compressor blisk

T. Giersch , ... A. Kühhorn , in 10th International Conference on Vibrations in Rotating Machinery, 2012

4 RESULTS

The steady state flow computation represents part-speed and is fully subsonic. The rotor generates a total pressure ratio of 1.23. Determining aerodynamic damping values via an uncoupled approach, it has to be ensured that the flow-induced aerodynamic forces act linearly on the amplitude of the blade vibration. Figure 3 shows the aerodynamic damping at different vibration amplitudes and different nodal diameters, indicating that a linear behaviour is expected up to a 1   mm tip-displacement. Blade-mode shapes of blisks depend on the number of nodal diameters, however, the deviations from an isolated blade-mode commonly decrease for an increasing number of nodal diameters [17]. Since the method of AIC requires that blade-mode shapes remain largely unchanged, it is proved for the 1^st flap mode that blade-mode shapes taken from finite element results of CSM 4 and CSM 14, both fitted to the CFD-mesh to calculate the aerodynamic damping lead to similar results (figure 4). Since an EO 4 excitation will be enforced in the future rig-test, an aerodynamic damping matrix belonging to CSM 4 is chosen for further investigations.

Figure 3. Dependence of aerodynamic damping on nodal diameter (ND) and maximum displacement

Figure 4. Aerodynamic damping in dependence on IBPA

As stated in section 3, modal forces are evaluated via an integration of unsteady pressure distributions computed for the tuned system according to equation(2). Periodicity with respect to modal forces has been reached within the calculation of the unsteady rotor-stator-interaction in a half rotor revolution. A blade passing of upstream stator vanes has been discretised in 150 time-steps. Due to the low aerodynamic damping level the calculation of the coupled system requires multiple vibration-cycles even if unsteady forces reach periodicity very quickly. At least 107 rotor revolutions had to be calculated until displacement amplitudes converged in a satisfactory manner.

Considering both unsteady forces and aerodynamic damping in the ROMs the forced response due to an EO 4 excitation is computed, see figure 5. The strongly mistuned blade 1 exhibits a significantly lower vibration amplitude than all other blades (figure 5a). Bidirectional results indicate a good agreement to the SNM. Please note that the chosen exciting frequency at 717.12   Hz for the bidirectional CFD-approach merely represents an estimation of the resonance since it is not exactly known before this computation. That is why the maximum blade responses are predicted at slightly higher frequencies by both ROMs. Comparing EBM- and SNM-results, a perfect match of the tuned response has been found (figure 5b). However, deviations become apparent especially for blade 1 in case of the mistuned blisk (figure 5a) whereas the differences for the other blades decrease. The reason for this is that the operational deflection shape assigned to the blade 1 maximum frequency is characterised by a localisation at the same blade, which leads to stronger contribution of other NDs apart from ND 4 (figure 6). On the other hand, for these NDs the EBM frequency results strongly differ from those computed with an FEM approach in the nodal diameter map (figure 7). The amplification factor of the tuned response at maximum due to mistuning is 1.18 (SNM) and 1.27 (EBM), respectively.

Figure 5. Forced response due to EO 4 excitation: a) mistuned and b) tuned blisk

Figure 6. Operational deflection shapes (ODS) computed with EBM at blades' 1 and 2 maximum with Fourier transforms of ODSs

Figure 7. Nodal diameter map

In spite of the strong reduction of computational effort of the SNM/AIC, an excellent agreement with results of bidirectional calculation with respect to blade displacements at 717.25   Hz is achieved (figure 8). Though the EBM-approach still yields qualitatively good results, again higher deviations compared to the SNM-model (Blade 1: 2.4 %, other blades: 2 % on average) are obvious. Furthermore, figure 8 depicts a comparison of displacements calculated assuming a constant damping ratio of 0.81 % according to the -4 ND mode for all modes with those calculated via IBPA-dependent damping ratios. Due to the impact of mistuning on aerodynamic damping this leads to an underestimation of maximum displacements for both models the SNM (-8.7 % on average) and the EBM (-14.3 % on average). In addition the blade individual amplitude differences have been smoothed out for both methods due to the constant damping approach.

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Transportation

Dr. Boyun Guo , Dr. Ali Ghalambor , in Natural Gas Engineering Handbook (Second Edition), 2005

11.2.1.5 Theoretical Pipeline Equations

Designing a long-distance pipeline for transportation of natural gas requires knowledge of flow formulas for calculating capacity and pressure requirements. There are several equations in the petroleum industry for calculating the flow of gases in pipelines. In the early development of the natural gas transmission industry, pressures were low and the equations used for design purposes were simple and adequate. However, as pressure increased to meet higher capacity demands, more complicated equations were developed to meet the new requirements. Probably the most common pipeline flow equation is the Weymouth equation, which is generally preferred for smaller-diameter lines (D ≤ 15 in). The Panhandle equation and the Modified Panhandle equation are usually better for larger-sized transmission lines.

Based on the first law of thermal dynamics, the total pressure gradient is made up of three distinct components:

(11.17) $\frac{dp}{dL}=\frac{g}{g_c}\rho \sin \theta +\frac{f\rho u^2}{2g_{c}D}+\frac{\rho udu}{g_{c}dL}$

where $\frac{g}{g}\rho \sin \theta$ is the component due to elevation or potential energy change; $\frac{f\rho u^2}{2g_{c}D}$ is the component due to frictional losses; and $\frac{\rho udu}{g_{c}dL}$ is the component due to acceleration or kinetic energy change.

The elevation component is pipe angle dependent. It is zero for horizontal flow. The friction loss component applies to any type of flow at any pipe angle and causes a pressure drop in the direction of flow. The acceleration component causes a pressure drop in the direction of velocity increase in any flow condition in which velocity changes occurs. It is zero for constant-area, incompressible flow.

Equation (11.17) applies for any fluid in steady-state, one-dimensional flow for which ρ, f, and u can be defined. It is in differential equation form and would have to be integrated to yield pressure drop as a function of flow rate, pipe diameter, and fluid properties.

Consider steady-state flow of dry gas in a constant-diameter, horizontal pipeline. The mechanical energy equation, Equation (11.3), becomes

(11.18) $\frac{dp}{dL}=\frac{f\rho u^2}{2g_{c}D}=\frac{p{(MW)}_a}{zRT}\frac{f{u}^2}{2g_{c}D}$

which serves as a base for development of many pipeline equations. The difference in these equations originated from the methods used in handling the z-factor and friction factor. Integrating Equation (11.18) gives

(11.19) ${\displaystyle \int dp=\frac{{(MW)}_{a}f{u}^2}{2R{g}_{c}D}{\displaystyle \int \frac{p}{zT}dL}}$

If temperature is assumed to be constant at average value in a pipeline, $\overline{T}$ , and gas deviation factor, $\overline{z}$ , is evaluated at average temperature and average pressure, $\overline{p}$ , Equation (11.19) can be evaluated over a distance L between upstream pressure, p ₁, and downstream pressure, p ₂:

(11.20) $p_1^2-p_2^2=\frac{25{\gamma }_g{q}^2\overline{T}\overline{z}fL}{D^5}$

where

γ _g = gas gravity (air = 1)

q = gas flow rate, MMscfd (at 14.7 psia, 60 °F)

$\overline{T}$ = average temperature, °R

$\overline{z}$ = gas deviation factor at $\overline{T}$ and $\overline{p}$

$\overline{p}=(p_1+p_2)/2$

L = pipe length, ft

D = pipe internal diameter, in.

f = Moody friction factor

Equation (11.20) may be written in terms of flow rate measured at arbitrary base conditions (T_b and p_b ):

(11.21) $q=\frac{C{T}_b}{p_b}\sqrt{\frac{\left(p_1^2-p_2^2\right)D^5}{{\gamma }_g\overline{T}\overline{z}fL}}$

where C is a constant with a numerical value that depends on the units used in the pipeline equation. If L is in miles and q is in scfd, C = 77.54.

The use of Equation (11.21) involves an iterative procedure. The gas deviation factor depends on pressure and the friction factor depends on flow rate. This problem prompted several investigators to develop pipeline flow equations that are noniterative or explicit. This has involved substitutions for the friction factor f. The specific substitution used may be diameter-dependent only (Weymouth equation) or Reynolds number dependent only (Panhandle equations).

11.2.1.5.1 Weymouth Equation for Horizontal Flow

Equation (11.21) takes the following form when the unit of scfh for gas flow rate is used:

(11.22) $q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{1}{f}}\sqrt{\frac{\left(p_1^2-p_2^2\right)D^5}{{\gamma }_g\overline{T}\overline{z}fL}}$

where $\sqrt{\frac{1}{f}}$ is called transmission factor. The Moody friction factor may be a function of flow rate and pipe roughness. If flow conditions are in the fully turbulent region, Equation (11.16) degenerates to

(11.23) $f=\frac{1}{{\left[1.14-2\log (e_D)\right]}^2}$

where f depends only on the relative roughness, e_D . When flow conditions are not completely turbulent, f depends on the Reynolds number also:

(11.24) $N_{Re}\approx \frac{0.48q_h{\gamma }_g}{\mu D}$

Therefore, use of Equation (11.22) and Equation (11.24) requires a trial-and-error procedure to calculate q_h . To eliminate the trial-and-error procedure, Weymouth proposed that f vary as a function of diameter in inches as follows:

(11.25) $f=\frac{0.032}{D^{1/3}}$

With this simplification, Equation (11.22) reduces to

(11.26) $q_h=\frac{18.062T_b}{p_b}\sqrt{\frac{\left(p_1^2-p_2^2\right)D^{16/3}}{{\gamma }_g\overline{T}\overline{z}L}}$

which is the form of the Weymouth equation commonly used in the natural gas industry.

The use of the Weymouth equation for an existing transmission line or for the design of a new transmission line involves a few assumptions including no mechanical work, steady flow, isothermal flow, constant compressibility factor, horizontal flow, and no kinetic energy change. These assumptions can affect accuracy of calculation results.

In the study of an existing pipeline, the pressure-measuring stations should be placed so that no mechanical energy is added to the system between stations. No mechanical work is done on the fluid between the points at which the pressures are measured. Thus, the condition of no mechanical work can be fulfilled.

Steady flow in pipeline operation seldom, if ever, exists in actual practice because pulsations, liquid in the pipeline, and variations in input or output gas volumes cause deviations from steady-state conditions. Deviations from steady-state flow are the major cause of difficulties experienced in pipeline flow studies.

The heat of compression is usually dissipated into the ground along a pipeline within a few miles downstream from the compressor station. Otherwise, the temperature of the gas is very near that of the containing pipe and, as pipelines usually are buried, the temperature of the flowing gas is not influenced appreciably by rapid changes in atmospheric temperature. Therefore, the gas flow can be considered isothermal at an average effective temperature without causing significant error in longpipeline calculations.

The compressibility of the fluid can be considered constant and an average effective gas deviation factor may be used. When the two pressures p ₁ and p ₂ lie in a region where z is essentially linear with pressure, then it is accurate enough to evaluate $\overline{z}$ at the average pressure $\overline{p}=(p_1+p_2)/2$ . One can also use the arithmetic average of the z with $\overline{z}=(z_1+z_2)/2$ where z ₁ and z ₂ are obtained at p ₁ and p ₂, respectively. On the other hand, should p ₁ and p ₂ lie in the range where z is not linear with pressure (double-hatched lines), the proper average would result from determining the area under the z-curve and dividing it by the difference in pressure:

(11.27) $\overline{z}=\frac{{\displaystyle {\int }_{p_1}^{p_2}zdp}}{(p_1-p_2)}$

where the numerator can be evaluated numerically. Also, $\overline{z}$ can be evaluated at an average pressure given by

(11.28) $\overline{p}=\frac{2}{3}\left(\frac{p_1^3-p_2^3}{p_1^2-p_2^2}\right)$

Regarding the assumption of horizontal pipeline, in actual practice, transmission lines seldom, if ever, are horizontal, so that factors are needed in Equation (11.26) to compensate for changes in elevation. With the trend to higher operating pressures in transmission lines, the need for these factors is greater than is generally realized. This issue of correction for change in elevation is addressed in the next section.

If the pipeline is long enough, the changes in the kinetic-energy term can be neglected. The assumption is justified for work with commercial transmission lines.

Example Problem 11.1

For the following data given for a horizontal pipeline, predict gas flow rate in cubic ft/hr through the pipeline.

$\begin{array}{c}D=12.09\text{ in}\\ L=200\text{ mi}\\ e=0.0006\text{ in}\\ T={80}^{\circ }\text{F}\\ {\gamma }_g=0.70\\ T_b={520}^{\circ }\text{R}\\ p_b=14.7\text{ psia}\\ p_1=600\text{ psia}\\ p_2=200\text{ psia}\end{array}$

Solution

The problem can be solved using Equation (11.22) with the trial-and-error method for friction factor and the Weymouth equation without the Reynolds number-dependent friction factor.

The average pressure is:

$\overline{p}=(200+600)/2=400\text{psia}$

With $\overline{p}=400\text{psia}$ , T = 540 °R, and γ _g = 0.70, Brill-Beggs-Z.xls gives:

$\overline{z}=0.9188$

With $\overline{p}=400\text{psia}$ , T = 540 °R, and γ _g = 0.70, Carr-Kobayashi-Burrows Viscosity.xls gives:

$\mu =0.0099\text{ cp}$

Relative roughness:

$e_D=0.0006/12.09=0.00005$

A.

Trial-and-Error Calculation:

First trial:

$\begin{array}{c}{q}_h=500,000\text{ cfh}\\ N_{Re}=\frac{0.48(500,000)(0.7)}{(0.0099)(12.09)}=1,403,733\\ \frac{1}{\sqrt{f}}=1.14-2\log \left(0.00005+\frac{21.25}{{(1,403,733)}^{0.9}}\right)\\ \text{undefined}f=0.01223\end{array}$

$\begin{array}{c}{q}_h=\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01223}}\sqrt{\frac{\left({600}^2-{200}^2\right){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\\ q_h=1,148,450\text{ cfh}\end{array}$

Second trial:

$\begin{array}{c}{q}_h=1,148,450\text{ cfh}\\ N_{Re}=\frac{0.48(1,148,450)(0.7)}{(0.0099)(12.09)}=3,224,234\\ \frac{1}{\sqrt{f}}=1.14-2\log \left(0.00005+\frac{21.25}{{(3,224,234)}^{0.9}}\right)\\ f=0.01145\\ q_h=\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01145}}\sqrt{\frac{\left({600}^2-{200}^2\right){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\\ q_h=1,186,759\text{ cfh}\end{array}$

Third trial:

$\begin{array}{c}{q}_h=1,186,759\text{ cfh}\\ N_{Re}=\frac{0.48(1,186,759)(0.7)}{(0.0099)(12.09)}=3,331,786\\ \frac{1}{\sqrt{f}}=1.14-2\log \left(0.00005+\frac{21.25}{{(3,331,786)}^{0.9}}\right)\\ f=0.01143\\ q_h=\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01143}}\sqrt{\frac{\left({600}^2-{200}^2\right){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\\ q_h=1,187,962\text{ cfh}\end{array}$

which is close to the assumed 1,186,759 cfh

B.

Using the Weymouth equation:

$\begin{array}{c}{q}_h=\frac{18.062(520)}{14.7}\sqrt{\frac{\left({600}^2-{200}^2\right){(12.09)}^{16/3}}{(0.7)(540)(0.9188)(200)}}\\ q_h=1,076,035\text{ cfh}\end{array}$

To speed up trial-and-error calculations, a spreadsheet program, PipeCapacity.xls, was developed. The solution given by the spreadsheet is shown in Table 11-1.

Table 11-1. Input Data and Results Given by PipeCapacity.xls( ^a )

Instructions: 1) Update input data; 2) Run Macro Solution and view results.
Input Data
Pipe ID:	12.09 in
Pipe roughness:	0.0006 in
Pipeline length:	200 mi
Average temperature:	80 °F
Base temperature:	60 °F
Base pressure:	14.7 psia
Inlet pressure:	600 psia
Outlet pressure:	200 psia
Gas properties:
Gas-specific gravity:	0.7 air = 1
Mole fraction of N2:	0
Mole fraction of CO2:	0
Mole fraction of H2S:	0
Calculated Parameter Values
Pseudocritical pressure:
Pseudocritical temperature:	389.14 °R
Uncorrected gas viscosity at 14.7 psia:	0.010079 cp
N2 correction for gas viscosity at 14.7 psia:	0.000000 cp
CO2 correction for gas viscosity at 14.7 psia:	0.000000 cp
H2S correction for gas viscosity at 14.7 psia:	0.000000 cp
Corrected gas viscosity at 14.7 psia (μ1):	0.010079 cp
Gas viscosity:	0.009899 cp
Pseudoreduced temperature:	1.34
Average pressure (psia):	400 psia
Pseudoreduced pressure:	0.599
Average z-factor:	0.9188
Pipe relative roughness:	0.000050
A) Use Reynolds number dependent friction factor:
Pipeline flow capacity:	1,188,000 cfh
Reynolds number:	3,335,270
Friction factor:	0.01143
Objective function:	0
B) Use the Weymouth equation:
Pipeline flow capacity:	1,076,035 cfh

a.

This program computes the capacity of gas pipelines (see Example Problem 11.1).

11.2.1.5.2 Weymouth Equation for Nonhorizontal Flow

Gas transmission lines are often nonhorizontal. Account should be taken of substantial pipeline elevation changes. Considering gas flow from point 1 to point 2 in a nonhorizontal pipe, the first law of thermal dynamics gives:

(11.29) ${\displaystyle \underset{1}{\overset{2}{\int }}vdp}+\frac{g}{g_c}\Delta z+{\displaystyle \underset{1}{\overset{2}{\int }}\frac{f{u}^2}{2g_{c}D}dL}=0$

Based on the pressure gradient due to the weight of gas column

(11.30) $\frac{dp}{dz}=\frac{{\rho }_g}{144}$

and real gas law, ${\rho }_g=\frac{p{(MW)}_a}{zRT}=\frac{29{\gamma }_{g}p}{zRT}$ , Weymouth (1912) developed the following equation:

(11.31) $q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{\left(p_1^2-{\text{e}}^s{p}_2^2\right)D^5}{f{\gamma }_g\overline{T}\overline{z}L}}$

where e = 2.718 and

(11.32) $s=\frac{0.0375{\gamma }_g\Delta z}{\overline{T}\overline{z}}$

and Δz is equal to outlet elevation minus inlet elevation (note that Δz is positive when outlet is higher than inlet). A general and more rigorous form of the Weymouth equation with compensation for elevation is

(11.33) $q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{\left(p_1^2-{\text{e}}^s{p}_2^2\right)D^5}{f{\gamma }_g\overline{T}\overline{z}{L}_e}}$

where L_e is the effective length of the pipeline. For a uniform slope, L_e is defined as

(11.34) $L_e=\frac{\left(e^s-1\right)L}{s}$

For a nonuniform slope (where elevation change cannot be simplified to a single section of constant gradient), an approach in steps to any number of sections, n, will yield

(11.35) $L_{\text{e}}=\frac{\left({\text{e}}^{s_1}-1\right)}{s_I}{L}_I+\frac{{\text{e}}^{s_1}\left({\text{e}}^{s_2}-1\right)}{s_2}{L}_2+\frac{{\text{e}}^{s_{1+S_2}}\left({\text{e}}^{{\text{s}}_{\text{3}}}\text{-1}\right)}{s_3}{L}_3+\dots \dots \dots +{\displaystyle \sum _{i=1}^n\frac{{\text{e}}^{{\displaystyle \sum _{j=1}^{i-1}{s}_j}}({\text{e}}^{si}-1)}{s_i}{L}_i}$

where

(11.36) $s_i=\frac{0.0375{\gamma }_g\Delta z_i}{\overline{T}\overline{z}}$

11.2.1.5.3 Panhandle A Equation—Horizontal Flow

The Panhandle A pipeline flow equation assumes the following Reynolds number dependent friction factor:

(11.37) $f=\frac{0.085}{N_{Re}{}^{0.147}}$

The resultant pipeline flow equation is thus

(11.38) $q=435.87\frac{D^{2.6182}}{{\gamma }_g^{0.4604}}{\left(\frac{T_b}{p_b}\right)}^{1.07881}{\left[\frac{\left(p_1^2-p_2^2\right)}{\overline{T}\overline{z}}\right]}^{0.5394}$

where q is the gas flow rate in cfd measured at T_b and p_b , and other terms are the same as in the Weymouth equation.

11.2.1.5.4 Panhandle B Equation—Horizontal Flow (Modified Panhandle)

Panhandle B equation is the most widely used equation for long transmission and delivery lines. It assumes that f varies as:

(11.39) $f=\frac{0.015}{N_{Re}{}^{0.0392}}$

and it takes the following resultant form:

(11.40) $q=737D^{2.530}{\left(\frac{T_b}{p_b}\right)}^{1.02}{\left[\frac{\left(p_1^2-p_2^2\right)}{\overline{T}\overline{z}L{\gamma }_g^{0.961}}\right]}^{0.510}$

11.2.1.5.5 Clinedinst Equation—Horizontal Flow

The Clinedinst equation rigorously considers the deviation of natural gas from ideal gas through integration. It takes the following form:

(11.41) $q=3973.0\frac{z_b{T}_b{p}_{pc}}{p_b}\sqrt{\frac{D^5}{\overline{T}\text{undefined}fL{\gamma }_g}\left({\displaystyle \underset{0}{\overset{p_{r1}}{\int }}\frac{p_r}{z}d{p}_r-{\displaystyle \underset{0}{\overset{p_{r2}}{\int }}\frac{p_r}{z}d{p}_r}}\right)}$

where

q = volumetric flow rate, Mcfd

p_pc = pseudocritical pressure, psia

D = pipe internal diameter, in

L = pipe length, ft

p_r = pseudoreduced pressure

$\overline{T}$ = average flowing temperature, °R

γ _g = gas gravity, air = 1.0

z_b = gas deviation factor at T_b and p_b , normally accepted as 1.0 Based on Equation (2.5) for pseudocritical pressure (Wichert and Aziz 1972), the values of the integral function ${\displaystyle \underset{0}{\overset{p_r}{\int }}\frac{p_r}{z}d{p}_r}$ have been calculated for various gas-specific gravity values. The results are presented in Appendix B.

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Source: https://www.sciencedirect.com/topics/engineering/steady-state-flow

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