11.4.1.2.1 Weymouth equation for horizontal flow

Eq. (11.97) takes the following form when the unit of scfh for gas flow rate is used:

$q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{1}{f_M}}\sqrt{\frac{(p_1^2-p_2^2)d^5}{{\gamma }_g\overline{T}\overline{z}L}},$ (11.98)

(11.98)

where $\sqrt{\frac{1}{f_M}}$ is called the “transmission factor.” The friction factor may be a function of flow rate and pipe roughness. If flow conditions are in the fully turbulent region, Eq. (11.89) degenerates to

$f_M=\frac{1}{{[1.14-2\log (e_D)]}^2},$ (11.99)

(11.99)

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

Therefore, use of Eq. (11.98) 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 as follows:

$f_M=\frac{0.032}{d^{1/3}}$ (11.100)

(11.100)

With this simplification, Eq. (11.98) reduces to

$q_h=\frac{18.062T_b}{p_b}\sqrt{\frac{(p_1^2-p_2^2)D^{16/3}}{{\gamma }_g\overline{T}\overline{z}L},}$ (11.101)

(11.101)

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 because 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 long-pipeline calculations.

The compressibility of the fluid can be considered constant and an average effective gas deviation factor may be used. When the two pressures p₁ and p₂ lie in a region where z is essentially linear with pressure, 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's with $\overline{z}=(z_1+z_2)/2$, where z₁ and z₂ are obtained at p₁ and p₂, respectively. On the other hand, should p₁ and p₂ 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:

$\overline{z}=\frac{\underset{p_1}{\overset{p_2}{\int }}zdp}{(p_1-p_2)},$ (11.102)

(11.102)

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

$\overline{p}=\frac{2}{3}\left(\frac{p_1^3-p_2^3}{p_1^2-p_2^2}\right).$ (11.103)

(11.103)

Regarding the assumption of horizontal pipeline, in actual practice, transmission lines seldom, if ever, are horizontal, so that factors are needed in Eq. (11.101) 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 11.5 For the following data given for a horizontal pipeline, predict gas flow rate in ft³/hr through the pipeline. Solve the problem using Eq. (11.101) with the trial-and-error method for friction factor and the Weymouth equation without the Reynolds number–dependent friction factor:

Solution The average pressure is

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

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

$\overline{z}=0.9188.$

With $\overline{p}$ 400 psia, T=540 °R and γ_g=0.70, Carr-Kobayashi-BurrowsViscosity.xls gives

$\mu =0.0099\mathrm{cp}.$

Relative roughness:

$e_D=0.0006/12.09=0.00005$

A. Trial-and-error calculation:
First trial:

$q_h=500,000\mathrm{scfh}$

$N_{\mathrm{Re}}=\frac{0.48(500,000)(0.7)}{(0.0099)(12.09)}=1,403,733$

$\frac{1}{\sqrt{f_M}}=1.14-2\log \left(0.00005+\frac{21.25}{{(1,403,733)}^{0.9}}\right)$

$f_M=0.01223$

$\begin{array}{ll}{q}_h\hfill & =\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01223}}\sqrt{\frac{({600}^2-{200}^2){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\hfill \\ \hfill & =1,148,450\mathrm{scfh}\hfill \end{array}$

Second trial:

$q_h=1,148,450\mathrm{cfh}$

$N_{\mathrm{Re}}=\frac{0.48(1,148,450)(0.7)}{(0.0099)(12.09)}=3,224,234$

$\frac{1}{\sqrt{f_M}}=1.14-2\log \left(0.00005+\frac{21.25}{{(3,224,234)}^{0.9}}\right)$

$f_M=0.01145$

$\begin{array}{ll}{q}_h\hfill & =\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01145}}\sqrt{\frac{({600}^2-{200}^2){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\hfill \\ \hfill & =1,186,759\mathrm{scfh}\hfill \end{array}$

Third trial:

$q_h=1,186,759\mathrm{scfh}$

$N_{\mathrm{Re}}=\frac{0.48(1,186,759)(0.7)}{(0.0099)(12.09)}=3,331,786$

$\frac{1}{\sqrt{f_M}}=1.14-2\log \left(0.00005+\frac{21.25}{{(3,331,786)}^{0.9}}\right)$

$f_M=0.01143$

$\begin{array}{ll}{q}_h\hfill & =\frac{3.23(520)}{14.7}\sqrt{\frac{1}{0.01143}}\sqrt{\frac{({600}^2-{200}^2){(12.09)}^5}{(0.7)(540)(0.9188)(200)}}\hfill \\ \hfill & =1,187,962\mathrm{scfh}\hfill \end{array}$

which is close to the assumed 1,186,759 scfh.

B. Using the Weymouth equation:

$\begin{array}{ll}{q}_h\hfill & =\frac{18.062(520)}{14.7}\sqrt{\frac{({600}^2-{200}^2){(12.09)}^{16/3}}{(0.7)(540)(0.9188)(200)}}\hfill \\ \hfill & =1,076,035\mathrm{scfh}\hfill \end{array}$

Problems similar to this one can be quickly solved with the spreadsheet program PipeCapacity.xls.

11.4.1.2.2 Weymouth equation for nonhorizontal flow

Gas transmission pipelines 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

$\underset{1}{\overset{2}{\int }}wdP+\frac{g}{g_c}\mathrm{\Delta }Z+\underset{1}{\overset{2}{\int }}\frac{f_M{u}^2}{2g_{c}D}dL=0.$ (11.104)

(11.104)

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

$\frac{dP}{dz}=\frac{{\rho }_g}{144},$ (11.105)

(11.105)

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

$q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{(p_1^2-e^s{p}_2^2)d^5}{f_M{\gamma }_g\overline{T}\overline{z}L},}$ (11.106)

(11.106)

where

$e=2.718\mathrm{and}s=\frac{0.0375{\gamma }_g\mathrm{\Delta }z}{\overline{T}\overline{z}},$ (11.107)

(11.107)

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

$q_h=\frac{3.23T_b}{p_b}\sqrt{\frac{(p_1^2-e^s{p}_2^2)d^5}{f_M{\gamma }_g\overline{T}\overline{z}{L}_e},}$ (11.108)

(11.108)

where L_e is the effective length of the pipeline. For a uniform slope, L_e is defined as $L_e=\frac{(e^s-1)L}{s}$.

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

$\begin{array}{ll}{L}_e\hfill & =\frac{(e^{S_1}-1)}{S_1}{L}_1+\frac{e^{S_1}(e^{S_2}-1)}{S_2}{L}_2+\frac{e^{S_1+S_2}(e^{S_3}-1)}{S_3}{L}_3+\mathrm{.......}.+\sum _{i=1}^n\times \frac{e^{\sum _{j=1}^{i-1}{S}_j}(e^{S_i}-1)}{S_i}{L}_1,\hfill \end{array}$ (11.109)

(11.109)

where

$s_i=\frac{0.0375{\gamma }_g\mathrm{\Delta }{z}_i}{\overline{T}\overline{z}}.$ (11.110)

(11.110)

11.4.1.2.3 Panhandle-A equation for horizontal flow

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

$f_M=\frac{0.085}{N_{\mathrm{Re}}^{0.147}}$ (11.111)

(11.111)

The resultant pipeline flow equation is, thus,

$q=435.87\frac{d^{2.6182}}{{\gamma }_g^{0.4604}}{\left(\frac{T_b}{p_b}\right)}^{1.07881}{\left[\frac{(p_1^2-p_2^2)}{\overline{T}\overline{z}L}\right]}^{0.5394},$ (11.112)

(11.112)

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

11.4.1.2.4 Panhandle-B equation for horizontal flow (modified Panhandle)

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

$f_M=\frac{0.015}{N_{\mathrm{Re}}^{0.0392}},$ (11.113)

(11.113)

and it takes the following resultant form:

$q=737d^{2.530}{\left(\frac{T_b}{p_b}\right)}^{1.02}{\left[\frac{(p_1^2-p_2^2)}{\overline{T}\overline{z}L{\gamma }_g^{0.961}}\right]}^{0.510}$ (11.114)

(11.114)

11.4.1.2.5 Clinedinst equation for horizontal flow

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

$\begin{array}{l}q=3973.0\frac{z_b{p}_b{p}_{pc}}{p_b}\times \sqrt{\frac{d^5}{\overline{T}{f}_{M}L{\gamma }_g}\left(\underset{0}{\overset{p_{r1}}{\int }}\frac{p_r}{z}d{p}_r\underset{0}{\overset{p_{r2}}{\int }}\frac{p_r}{z}d{p}_r\right)},\hfill \end{array}$ (11.115)

(11.115)

where

Based on Eqs. (2.29), (2.30), and (2.51), Guo and Ghalambor (2005) generated curves of the integral function ${\int }_0^{p_r}\frac{p_r}{z}d{p}_r$ for various gas-specific gravity values.

11.4.1.2.6 Pipeline efficiency

All pipeline flow equations were developed for perfectly clean lines filled with gas. In actual pipelines, water, condensates, sometimes crude oil accumulates in low spots in the line. There are often scales and even “junk” left in the line. The net result is that the flow rates calculated for the 100% efficient cases are often modified by multiplying them by an efficiency factor E. The efficiency factor expresses the actual flow capacity as a fraction of the theoretical flow rate. An efficiency factor ranging from 0.85 to 0.95 would represent a “clean” line. Table 11.1 presents typical values of efficiency factors.

11.4.2.1.1 Design procedure

Determination of pipeline wall thickness is based on the design internal pressure or the external hydrostatic pressure. Maximum longitudinal stresses and combined stresses are sometimes limited by applicable codes and must be checked for installation and operation. However, these criteria are not normally used for wall thickness determination. Increasing wall thickness can sometimes ensure hydrodynamic stability in lieu of other stabilization methods (such as weight coating). This is not normally economical, except in deepwater where the presence of concrete may interfere with the preferred installation method. We recommend the following procedure for designing pipeline wall thickness:

Note that in certain cases, it may be desirable to order a nonstandard wall. This can be done for large orders.

Pipelines are sized on the basis of the maximum expected stresses in the pipeline under operating conditions. The stress calculation methods are different for thin-wall and thick-wall pipes. A thin-wall pipe is defined as a pipe with D/t greater than or equal to 20. Fig. 11.11 shows stresses in a thin-wall pipe. A pipe with D/t less than 20 is considered a thick-wall pipe. Fig. 11.12 illustrates stresses in a thick-wall pipe.

11.4.2.1.2 Design for internal pressure

Three pipeline codes typically used for design are ASME B31.4 (ASME, 1989), ASME B31.8 (ASME, 1990), and DnV (1981). ASME B31.4 is for all oil lines in North America. ASME B31.8 is for all gas lines and two-phase flow pipelines in North America. DnV (1981) is for oil, gas, and two-phase flow pipelines in North Sea. All these codes can be used in other areas when no other code is available.

The nominal pipeline wall thickness (t_NOM) can be calculated as follows:

$t_{NOM}=\frac{P_{d}D}{2E_w\eta {\sigma }_y{F}_t}+t_a,$ (11.116)

(11.116)

where P_d is the design internal pressure defined as the difference between the internal pressure (P_i) and external pressure (P_e), D is nominal outside diameter, t_a is thickness allowance for corrosion, and is the specified minimum yield strength. Eq. (11.116) is valid for any consistent units.

Most codes allow credit for external pressure. This credit should be used whenever possible, although care should be exercised for oil export lines to account for head of fluid and for lines that traverse from deep to shallow water.

ASME B31.4 and DnV (1981) define P_i as the maximum allowable operating pressure (MAOP) under normal conditions, indicating that surge pressures up to 110% MAOP is acceptable. In some cases, P_i is defined as wellhead shut-in pressure (WSIP) for flowlines or specified by the operators.

In Eq. (11.116), the weld efficiency factor (E_w) is 1.0 for seamless, ERW, and DSAW pipes. The temperature de-rating factor (F_t) is equal to 1.0 for temperatures under 250°F. The usage factor (η) is defined in Tables 11.2 and 11.3 for oil and gas lines, respectively.

The under thickness due to manufacturing tolerance is taken into account in the design factor. There is no need to add any allowance for fabrication to the wall thickness calculated with Eq. (11.116).

11.4.2.1.3 Design for external pressure

Different practices can be found in the industry using different external pressure criteria. As a rule of thumb, or unless qualified thereafter, it is recommended to use propagation criterion for pipeline diameters under 16-in. and collapse criterion for pipeline diameters more than or equal to 16-in.

Propagation criterion

The propagation criterion is more conservative and should be used where optimization of the wall thickness is not required or for pipeline installation methods not compatible with the use of buckle arrestors such as reel and two methods. It is generally economical to design for propagation pressure for diameters less than 16-in. For greater diameters, the wall thickness penalty is too high. When a pipeline is designed based on the collapse criterion, buckle arrestors are recommended. The external pressure criterion should be based on nominal wall thickness, as the safety factors included below account for wall variations.

Although a large number of empirical relationships have been published, the recommended formula is the latest given by AGA.PRC (AGA, 1990):

$P_p-33S_y{\left(\frac{t_{NOM}}{D}\right)}^{2.46},$ (11.117)

(11.117)

which is valid for any consistent units. The nominal wall thickness should be determined such that P_p>1.3 P_e. The safety factor of 1.3 is recommended to account for uncertainty in the envelope of data points used to derive Eq. (11.117). It can be rewritten as

$t_{NOM}\ge D{\left(\frac{1.3P_p}{33S_y}\right)}^{\frac{1}{2.46}}.$ (11.118)

(11.118)

For the reel barge method, the preferred pipeline grade is below X-60. However, X-65 steel can be used if the ductility is kept high by selecting the proper steel chemistry and microalloying. For deepwater pipelines, D/t ratios of less than 30 are recommended. It has been noted that bending loads have no demonstrated influence on the propagation pressure.

Collapse criterion

The mode of collapse is a function of D/t ratio, pipeline imperfections, and load conditions. The theoretical background is not given in this book. An empirical general formulation that applies to all situations is provided. It corresponds to the transition mode of collapse under external pressure (P_e), axial tension (T_a), and bending strain (s_b), as detailed elsewhere (Murphey and Langner, 1985; AGA, 1990).

The nominal wall thickness should be determined such that

$\frac{1.3P_p}{P_C}+\frac{{\epsilon }_b}{{\epsilon }_B}\le g_p,$ (11.119)

(11.119)

where 1.3 is the recommended safety factor on collapse, B_B is the bending strain of buckling failure due to pure bending, and g is an imperfection parameter defined below.

The safety factor on collapse is calculated for D/t ratios along with the loads (P_e, ε_b, T_a) and initial pipeline out-of roundness (δ_o). The equations are

$P_C=\frac{P_{el}{P\prime }_y}{\sqrt{P_{el}^2+{P\prime }_y^2}},$ (11.120)

(11.120)

${P\prime }_y=P_y\left[\sqrt{1-0.75{\left(\frac{T_a}{T_y}\right)}^2}-\frac{T_a}{2T_y}\right],$ (11.121)

(11.121)

$P_{el}=\frac{2E}{1-v^2}{\left(\frac{t}{D}\right)}^3,$ (11.122)

(11.122)

$P_y=2S_y\left(\frac{t}{D}\right),$ (11.123)

(11.123)

$T_y=A{S}_y,$ (11.124)

(11.124)

where g_p is based on pipeline imperfections such as initial out-of roundness (δ_o), eccentricity (usually neglected), and residual stress (usually neglected). Hence,

$g_p=\sqrt{\frac{1+p^2}{p^2-\frac{1}{f_p^2}}},$ (11.125)

(11.125)

where

$p=\frac{{P\prime }_y}{P_{el}},$ (11.126)

(11.126)

$f_p=\sqrt{1+{\left({\delta }_o\frac{D}{t}\right)}^2}-{\delta }_o\frac{D}{t},$ (11.127)

(11.127)

${\epsilon }_B=\frac{t}{2D},$ (11.128)

(11.128)

and

${\delta }_o=\frac{D_{\max }-D_{\min }}{D_{\max }+D_{\min }}.$ (11.129)

(11.129)

When a pipeline is designed using the collapse criterion, a good knowledge of the loading conditions is required (T_a and ε_b). An upper conservative limit is necessary and must often be estimated.

Under high bending loads, care should be taken in estimating ε_b using an appropriate moment-curvature relationship. A Ramberg Osgood relationship can be used as

$K^{*}=M^{*}+A{M}^{*B},$ (11.130)

(11.130)

where K*=K/K_y and M*=M/M_y with K_y=2S_y/ED is the yield curvature and M_y=2IS_y/D is the yield moment. The coefficients A and B are calculated from the two data points on stress–strain curve generated during a tensile test.

11.4.2.1.4 Corrosion allowance

To account for corrosion when water is present in a fluid along with contaminants such as oxygen, hydrogen sulfide (H₂S), and carbon dioxide (CO₂), extra wall thickness is added. A review of standards, rules, and codes of practices (Hill and Warwick, 1986) shows that wall allowance is only one of several methods available to prevent corrosion, and it is often the least recommended.

For H₂S and CO₂ contaminants, corrosion is often localized (pitting) and the rate of corrosion allowance ineffective. Corrosion allowance is made to account for damage during fabrication, transportation, and storage. A value of $1/16$ in. may be appropriate. A thorough assessment of the internal corrosion mechanism and rate is necessary before any corrosion allowance is taken.

11.4.2.1.5 Check for hydrotest condition

The minimum hydrotest pressure for oil and gas lines is given in Tables 11.2 and 11.3, respectively, and is equal to 1.25 times the design pressure for pipelines. Codes do not require that the pipeline be designed for hydrotest conditions but sometimes give a tensile hoop stress limit 90% SMYS, which is always satisfied if credit has not been taken for external pressure. For cases where the wall thickness is based on P_d=P_i–P_e, codes recommend not to overstrain the pipe. Some of the codes are ASME B31.4 (Clause 437.4.1), ASME B31.8 (no limit on hoop stress during hydrotest), and DnV (Clause 8.8.4.3).

For design purposes, condition σ_h≤σ_y should be confirmed, and increasing wall thickness or reducing test pressure should be considered in other cases. For offshore pipelines connected to riser sections requiring P_h=1.4P_i, it is recommended to consider testing the riser separately (for prefabricated sections) or to determine the hydrotest pressure based on the actual internal pressure experienced by the pipeline section. It is important to note that most pressure testing of subsea pipelines is done with water, but on occasion, nitrogen or air has been used. For low D/t ratios (<20), the actual hoop stress in a pipeline tested from the surface is overestimated when using the thin wall equations provided in this chapter. Credit for this effect is allowed by DnV Clause 4.2.2.2 but is not normally taken into account.

Example Problem 11.6 Calculate the required wall thickness for the pipeline in Example Problem 11.4 assuming a seamless still pipe of X-60 grade and onshore gas field (external pressure P_e=14.65 psia).

Solution The wall thickness can be designed based on the hoop stress generated by the internal pressure Pi=2590 psia. The design pressure is

$P_d=P_i-P_e=2.590-14.65=2575.35\mathrm{psi}.$

The weld efficiency factor is E_w=1.0. The temperature de-rating factor F_t=1.0. Table 11.3 gives η=0.72. The yield stress is σ_y=60,000 psi. A corrosion allowance $1/16$ in. is considered. The nominal pipeline wall thickness can be calculated using Eq. (11.116) as

$t_{NOM}=\frac{(2574.3)(6)}{2(1.0)(0.72)(60,000)(1.0)}+\frac{1}{16}=0.2413\mathrm{in}.$

Considering that welding of wall thickness less than 0.3 in. requires special provisions, the minimum wall thickness is taken, 0.3 in.

11.4.2.2.1 Insulation materials

Polypropylene, polyethylene, and polyurethane are three base materials widely used in the petroleum industry for pipeline insulation. Their thermal conductivities are given in Table 11.4 (Carter et al., 2002). Depending on applications, these base materials are used in different forms, resulting in different overall conductivities. A three-layer polypropylene applied to pipe surface has a conductivity of 0.225 W/M-°C (0.13 btu/hr-ft-°F), while a four-layer polypropylene has a conductivity of 0.173 W/M-°C (0.10 btu/hr-ft-°F). Solid polypropylene has higher conductivity than polypropylene foam. Polymer syntactic polyurethane has a conductivity of 0.121 W/M-°C (0.07 btu/hr-ft-°F), while glass syntactic polyurethane has a conductivity of 0.156 W/M-°C (0.09 btu/hr-ft-°F). These materials have lower conductivities in dry conditions such as that in pipe-in-pipe (PIP) applications.

Because of their low thermal conductivities, more and more polyurethane foams are used in deepwater pipeline applications. Physical properties of polyurethane foams include density, compressive strength, thermal conductivity, closed-cell content, leachable halides, flammability, tensile strength, tensile modulus, and water absorption. Typical values of these properties are available elsewhere (Guo et al., 2005).

In steady-state flow conditions in an insulated pipeline segment, the heat flow through the pipe wall is given by

$Q_r=I{A}_r\mathrm{\Delta }T,$ (11.131)

(11.131)

where Q_r is heat-transfer rate; U is overall heat-transfer coefficient (OHTC) at the reference radius; A_r is area of the pipeline at the reference radius; ΔT is the difference in temperature between the pipeline product and the ambient temperature outside.

The OHTC, U, for a system is the sum of the thermal resistances and is given by (Holman, 1981):

$U=\frac{1}{A_r\left[\frac{1}{A_i{h}_i}+\sum _{m=1}^n\frac{1n(r_{m+1}/r_m)}{2\pi L{k}_m}+\frac{1}{A_o{h}_o}\right]},$ (11.132)

(11.132)

where h_i is film coefficient of pipeline inner surface; h_o is film coefficient of pipeline outer surface; A_i is area of pipeline inner surface; A_o is area of pipeline outer surface; r_m is radius of layer m; and k_m is thermal conductivity of layer m.

Similar equations exist for transient-heat flow, giving an instantaneous rate for heat flow. Typically required insulation performance, in terms of OHTC (U value) of steel pipelines in water, is summarized in Table 11.5.

Pipeline insulation comes in two main types: dry insulation and wet insulation. The dry insulations require an outer barrier to prevent water ingress (PIP). The most common types of this include the following:

Under certain conditions, PIP systems may be considered over conventional single-pipe systems. PIP insulation may be required to produce fluids from high-pressure/high-temperature (>150°C) reservoirs in deepwater (Carmichael et al., 1999). The annulus between pipes can be filled with different types of insulation materials such as foam, granular particles, gel, and inert gas or vacuum.

A pipeline-bundled system—a special configuration of PIP insulation—can be used to group individual flowlines together to form a bundle (McKelvie, 2000); heat-up lines can be included in the bundle, if necessary. The complete bundle may be transported to site and installed with a considerable cost savings relative to other methods. The extra steel required for the carrier pipe and spacers can sometimes be justified (Bai, 2001).

Wet-pipeline insulations are those materials that do not need an exterior steel barrier to prevent water ingress, or the water ingress is negligible and does not degrade the insulation properties. The most common types of this are as follows:

The main materials that have been used for deepwater insulations have been polyurethane and polypropylene based. Syntactic versions use plastic or glass matrix to improve insulation with greater depth capabilities. Insulation coatings with combinations of the two materials have also been used. Guo et al. (2005) gives the properties of these wet insulations. Because the insulation is buoyant, this effect must be compensated by the steel pipe weight to obtain lateral stability of the deepwater pipeline on the seabed.

11.4.2.2.2 Heat transfer models

Heat transfer across the insulation of pipelines presents a unique problem affecting flow efficiency. Although sophisticated computer packages are available for predicting fluid temperatures, their accuracies suffer from numerical treatments because long pipe segments have to be used to save computing time. This is especially true for transient fluid-flow analyses in which a very large number of numerical iterations are performed.

Ramey (1962) was among the first investigators who studied radial-heat transfer across a well casing with no insulation. He derived a mathematical heat-transfer model for an outer medium that is infinitely large. Miller (1980) analyzed heat transfer around a geothermal wellbore without insulation. Winterfeld (1989) and Almehaideb et al. (1989) considered temperature effect on pressure-transient analyses in well testing. Stone et al. (1989) developed a numerical simulator to couple fluid flow and heat flow in a wellbore and reservoir. More advanced studies on the wellbore heat-transfer problem were conducted by Hasan and Kabir (1994, 2002), Hasan et al. (1997, 1998), and Kabir et al. (1996). Although multilayers of materials have been considered in these studies, the external temperature gradient in the longitudinal direction has not been systematically taken into account. Traditionally, if the outer temperature changes with length, the pipe must be divided into segments, with assumed constant outer temperature in each segment, and numerical algorithms are required for heat-transfer computation. The accuracy of the computation depends on the number of segments used. Fine segments can be employed to ensure accuracy with computing time sacrificed.

Guo et al. (2006) presented three analytical heat-transfer solutions. They are the transient-flow solution for startup mode, steady-flow solution for normal operation mode, and transient-flow solution for flow rate change mode (shutting down is a special mode in which the flow rate changes to zero).

Temperature and heat transfer for steady fluid flow

The internal temperature profile under steady fluid-flow conditions is expressed as

$T=\frac{1}{{\alpha }^2}\left[\beta -\alpha \beta L-\alpha \gamma -e^{-\alpha (L+C)}\right],$ (11.133)

(11.133)

where the constant groups are defined as

$\alpha =\frac{2\pi Rk}{v\rho C_{p}sA},$ (11.134)

(11.134)

$\beta =\alpha G\cos (\theta ),$ (11.135)

(11.135)

$\gamma =-\alpha T_0,$ (11.136)

(11.136)

and

$C=-\frac{1}{\alpha }\ln (\beta -{\alpha }^2{T}_s-\alpha \gamma ),$ (11.137)

(11.137)

where T is temperature inside the pipe, L is longitudinal distance from the fluid entry point, R is inner radius of insulation layer, k is the thermal conductivity of the insulation material, v is the average flow velocity of fluid in the pipe, ρ is fluid density, C_p is heat capacity of fluid at constant pressure, s is thickness of the insulation layer, A is the inner cross-sectional area of pipe, G is principal thermal-gradient outside the insulation, θ is the angle between the principal thermal gradient and pipe orientation, T₀ is temperature of outer medium at the fluid entry location, and T_s is temperature of fluid at the fluid entry point.

The rate of heat transfer across the insulation layer over the whole length of the pipeline is expressed as

$q=-\frac{2\pi Rk}{S}\times \left(T_{0}L-\frac{G\cos (\theta )}{2}{L}^2-\frac{1}{{\alpha }^2}\left\{(\beta -\alpha \gamma )L-\frac{\alpha \beta }{2}{L}^2+\frac{1}{\alpha }\left[e^{-\alpha (L+C)}-e^{-\alpha C}\right]\right\}\right),$ (11.138)

(11.138)

where q is the rate of heat transfer (heat loss).

Transient temperature during startup

The internal temperature profile after starting up a fluid flow is expressed as follows:

$T=\frac{1}{{\alpha }^2}\left\{\beta -\alpha \beta L-\alpha \gamma -e^{-\alpha [L+f(L+vt)]}\right\},$ (11.139)

(11.139)

where the function f is given by

$f(L-vt)=-(L-vt)-\frac{1}{\alpha }\ln \left\{\beta -\alpha \beta (L-vt)-\alpha \gamma -{\alpha }^2[T_S-G\cos (\theta )(L-vt)]\right\}$ (11.140)

(11.140)

and t is time.

Transient temperature during flow rate change

Suppose that after increasing or decreasing the flow rate, the fluid has a new velocity v′ in the pipe. The internal temperature profile is expressed as follows:

$T=\frac{1}{{\alpha \prime }^2}\left\{\beta \prime -\alpha \prime \beta \prime L-\alpha \prime \gamma \prime -e^{-\alpha \prime [L+f(L-v\prime t)]}\right\},$ (11.141)

(11.141)

where

$\alpha \prime =\frac{2\pi Rk}{v\prime \rho C_{p}sA},$ (11.142)

(11.142)

$\beta \prime =\alpha \prime G\cos (\theta ),$ (11.143)

(11.143)

$\gamma \prime =-\alpha \prime T_0,$ (11.144)

(11.144)

and the function f is given by

$\begin{array}{l}f(L-v\prime t)=-(L-v\prime t)-\frac{1}{\alpha \prime }\ln (\beta \prime -\alpha \prime \beta \prime (L-v\prime t)-\alpha \prime \gamma \prime -{\left({\displaystyle \frac{\alpha \prime }{\alpha }}\right)}^2\{\beta -\alpha \beta (L-v\prime t)-\alpha \gamma -e^{-\alpha [(L-v\prime t)C}\}).\hfill \\ \hfill & \hfill \end{array}$ (11.145)

(11.145)

Example Problem 11.7 A design case is shown in this example. Design base for a pipeline insulation is presented in Table 11.6. The design criterion is to ensure that the temperature at any point in the pipeline will not drop to less than 25°C, as required by flow assurance. Insulation materials considered for the project were polyethylene, polypropylene, and polyurethane.

Table 11.6

Base Data for Pipeline Insulation Design

Length of pipeline:	8047	M
Outer diameter of pipe:	0.2032	M
Wall thickness:	0.00635	M
Fluid density:	881	kg/M3
Fluid specific heat:	2012	J/kg-°C
Average external temperature:	10	°C
Fluid temperature at entry point:	28	°C
Fluid flow rate:	7950	M3/day

Solution A polyethylene layer of 0.0254 M (1 in.) was first considered as the insulation. Fig. 11.13 shows the temperature profiles calculated using Eqs. (11.133) and (11.139). It indicates that at approximately 40 minutes after startup, the transient-temperature profile in the pipeline will approach the steady-flow temperature profile. The temperature at the end of the pipeline will be slightly lower than 20°C under normal operating conditions. Obviously, this insulation option does not meet design criterion of 25°C in the pipeline.

Fig. 11.14 presents the steady-flow temperature profiles calculated using Eq. (11.133) with polyethylene layers of four thicknesses. It shows that even a polyethylene layer 0.0635-M (2.5-in.) thick will still not give a pipeline temperature higher than 25°C; therefore, polyethylene should not be considered in this project.

A polypropylene layer of 0.0254 M (1 in.) was then considered as the insulation. Fig. 11.15 illustrates the temperature profiles calculated using Eq. (11.133) and (11.139). It again indicates that at approximately 40 minutes after startup, the transient-temperature profile in the pipe will approach the steady-flow temperature profile. The temperature at the end of the pipeline will be approximately 22.5°C under normal operating conditions. Obviously, this insulation option, again, does not meet design criterion of 25°C in the pipeline.

Fig. 11.16 demonstrates the steady-flow temperature profiles calculated using Eq. (11.133) with polypropylene layers of four thicknesses. It shows that a polypropylene layer of 0.0508 M (2.0 in.) or thicker will give a pipeline temperature of higher than 25°C.

A polyurethane layer of 0.0254 M (1 in.) was also considered as the insulation. Fig. 11.17 shows the temperature profiles calculated using Eqs. (11.133) and (11.139). It indicates that the temperature at the end of pipeline will drop to slightly lower than 25°C under normal operating conditions. Fig. 11.18 presents the steady-flow temperature profiles calculated using Eq. (11.133) with polyurethane layers of four thicknesses. It shows that a polyurethane layer of 0.0381 M (1.5 in.) is required to keep pipeline temperatures higher than 25°C under normal operating conditions.

Therefore, either a polypropylene layer of 0.0508 M (2.0 in.) or a polyurethane layer of 0.0381 M (1.5 in.) should be chosen for insulation of the pipeline. Cost analyses can justify one of the options, which is beyond the scope of this example.

The total heat losses for all the steady-flow cases were calculated with Eq. (11.138). The results are summarized in Table 11.7. These data may be used for sizing heaters for the pipeline if heating of the product fluid is necessary.

Table 11.7

Calculated Total Heat Losses for the Insulated Pipelines (kW)

Material Name	Insulation Thickness (M) 0.0254	0.0381	0.0508	0.0635
Polyethylene	1430	1,011	781	636
Polypropylene	989	685	524	424
Polyurethane	562	383	290	234