davidlunde curved pipe heat transfer

8/13/2019 DavidLunde Curved Pipe Heat Transfer

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ME 8342 - Convection

D. M. Lunde

Candidate M.S. Student

Single-Phase Convective Heat Transfer in a Pipewith Curvature

Abstract:

Heat transfer and heat exchanger devices that contain tubes many times contain tube bends

and fittings. The resulting streamwise curvature of fluid due to the presence of bends and fittings

creates secondary flows resulting in an overall increase in convective heat transfer. The physical

underpinnings that cause this effect are discussed and an attempt is made to tie together the previous

work in this area of study for the calculation of practical, engineering values. The discussion begins withidentifying curvature geometry, it then moves to the development of the governing momentum and

energy equations (both laminar and turbulent), and ends with a review of work done in solving these

equations and a summary of corresponding experimental work.

I. Introduction:Curved pipes are commonplace in nearly all piping networks. They are quite prevalent in

applications where heat transfer between fluid-fluid and fluid-solid mediums takes place. Some

common examples of curved pipes appear in large chillers and boilers, shell and tube heat exchangers,fin-tube heat exchangers, and chemical reactors for industrial processes [3]; they can be classified in the

following ways: helical coils, spirals, and tube bends (fittings). See Figures 1, 2 and 3 respectively.

Notice the radius of curvature Ris constant for a pipe bend and helical coil, but it continually varies from

Rminto Rmaxfor a spiral.

Figure 1: Helical Coil Geometry [3] Figure 2: Spiral Coil Geometry [3]


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Undergraduate heat transfer texts typically do not develop the necessary knowledge base to

successfully setup and solve practical, engineering problems associated with convective heat transfer in

curved pipe geometries. Yet, their ubiquitous use implies that one might find utility in understanding

the fundamental changes in the fluid flow and heat transfer processes associate with streamwise

curvature. In heat exchangers and piping networks the contribution to the overall heat transfer of a

device resulting from the streamwise curvature is great enough that many times it cannot be ignored

[4].

Figure 3: Tube Bend Geometry [3]

The geometry of a curved pipe naturally implies the use of one of two types of orthogonal

curvilinear coordinate systems: the cylindrical coordinate system (r, , z) orthe toroidal coordinate

system (r, ,) (where the radius of curvature of the pipe Ris held constant). Figures 4 and 5 show the

geometry of either system respectively.

Figure 4: Cylindrical Coordinates (r, , z)[3] Figure 5: Toroidal Coordinates (r, , )[3]


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Although the toirodal system can make more physical sense for the geometry under

consideration it is also far more esoteric, and for that reason all equations will be reported in cylindrical

coordinates. The interested reader is referred toSankariah and Rao(1972) and Tyagi and Sharma

(1975) for the momentum and energy equations in the toroidal system, respectively.

Curved pipes are subjected to some of the same complications as straight pipes. Entrance

effects, exit effects and validity of thermal boundary conditions must all be considered. Figure 6 displays

some of these areas of complications.

Figure 6: Areas of Analytical Complication for Curved Pipes

As fluid enters the bend it experiences a centrifugal force due to the streamwise curvature. This

force superimposed with the driving pressure gradient in the axial direction creates a secondary flow

transverse to the primary flow. The flow, therefore, becomes fully three dimensional while in the bend.

The momentum of the fluid entering the turn drives the fluid towards the outside of the tube. Thissqueezes the boundary layer, shrinking it and increasing the convective heat transfer coefficient. It

also causes a shift in the peak temperature and velocity to the outside of the bend [4]. The extra

centrifugal body force can be added to the momentum equation to account for its influence.

Complications arise, however, because the body force will vary with the fluids position in the pipe:


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Dean (1927) was the first to investigate the secondary flow in curved pipes. He introduced a non-

dimensional grouping to account for the curvature of the pipe:

In a helical coil there is also an influence from the coil pitch b (see Figure 1). An effective radius

is defined to account for this geometric parameter:

The Helical coil number is then defined:

II. Problem Formulation:To calculate values of engineering interests (such as friction factors and Nusselt numbers)

theoretically, the continuity, momentum, and energy equations must be solved. When solving the more

basic convection problems (such as laminar fluid flow and heat transfer on a flat plate or in a tube) one

can typically decouple the fluid flow and heat transfer problems without sacrificing greatly the accuracy

of the final solution. This technique greatly simplifies the solution to the governing differential

equations. When a new type of fluid flow/heat transfer situation arises, it is typical to solve the easier

or more tractable problems first. Laminar flows will be considered first and turbulent flows second.

Laminar Flow

The laminar analysis holds for a range of Dean numbers, but the curvature must be small,

typically between De >20 and De < 1200 [5]. For Dean < 20, the velocity and temperature profiles are

not that different from a straight tube [6] (see Figure 7). As the radius of curvature decreases, the

likelihood of separation on the inside of the bend increases and the laminar analysis results will no

longer hold.

From [5] the continuity equation in cylindrical coordinates:

From [5] the laminar energy equation:


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The turbulent equations are computed in the normal way with a Prandtl mixing length model or a

model. The turbulent equations are always solved numerically.

III. Problem SolutionsLaminar Flow:

A number of techniques exist to solve the laminar momentum and energy equations. For fully

developed, hydrodynamic flow in a helical coil or tube bend integral techniques were used to solve the

momentum equations by Mori & Nakayama (1965) and Ito (1970), Fourier-series (product separation)

methods were used by McConalogue & Srivastava (1968), and Akiyama & Cheng (1971) predicted the

fully developed flow by finite-difference techniques. Patankar, Pratap & Spalding [5] also solved the

momentum and energy equations by a finite difference technique. Figure 7 shows their computed

velocity distribution for the two planes referenced in Figure 8.

Figure 7: Effect of Dean number on axial velocity profiles in (a) the plane AA and (b) the plane BB (see Figure 8). (i) De = 0

(straight tube). (ii) De = 60. (iii) De = 500. (iv) De = 1200 [5]

Figure 8: Plane Definitions for Velocity and Temperature Profiles [5]


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Patankar, Pratap & Spalding [5] also reported calculated temperature distributions. The results are seen

in Figure 9. Here, K is the Dean number and the plane AA is shown in Figure 8.

Figure 9: Effect of Dean number on the Temperature Profile along the Plane AA; Pr = 0.71

Notice there are oscillations in the temperature profile of Figure 9. Patankar, Pratap & Spalding

commented on this phenomenon finally concluding it to be the result of the secondary flow creating

small oscillation in the temperature profile. The calculations were run at different step sizes and it was

concluded that there were no instabilities in the solution technique used in the study [5]. The authorsthen compared their calculations of velocity in the development region of the bend to data taken by

Austin (1971). The results match well with the experiment and can be seen in Figure 10.

Figure 10 shows results from [5] of the development of the laminar velocity profile as it enters a

curved pipe. Different values of the bend angle are plotted versus the relative magnitude of the local

velocity over the cross-sectional mean velocity. Notice that the velocity peak is near the outside corner

of the pipe. The profile is nearly fully developed at a bend angle of 90 and is in good agreement with

the experimental data found by Austin (1971). The velocity profile must change in plane BB to account

for the movement of fluid to the outside of the pipe.


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Figure 10: Development of axial velocity at De = 198 and R/a = 29.1 in (a) the plane BB and (b) the plane AA (see Figure 8)

---------, predictions; * Austin (1971) [5]

For computing the average Nusselt value around a bend experimental work was performed by

Mori & Nakayama (1967), Dravid, Smith & Merrill (1971), Akiyama & Cheng (1972), Tarbell & Samuels

(1973), and Kalb & Seader (1974) [3]. Manlapaz-Churchill (1981) produced a curve fit based on the

available experimental data and came up with the following correlation for an isothermal pipe wall:

It is also interesting to note that the stream function for the fluid flow problem can be solved and

streamlines can be plotted to visualize the secondary flow. Sankaraiah & Rao [6] accomplished this and

plotted the streamlines in Figure 11 for De = 500.


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Figure 11: Streamlines for De = 500 [5]

Turbulent Flow

The validity of the solution of the turbulent equations rests heavily on the turbulence model

utilized. The Prandtl missing length model or the two equation could be used. Turbulent velocity

profiles in a bend are reported by Azzola & Humphrey (1984), Rowe (1970), and Weske (1948) [3].

Azzola and Humphrey noted that the flow distortion from the streamwise curvature in the bend persists

until up to five diameters [3]. How the distorted velocity and temperature profiles affect the Nusselt

number downstream of the bend is dealt with extensively in [2]. Kreith [4] used experimental results of

wall shear and velocity distribution obtained by Wattendorf (1934) and calculated Nusselt numbers

applicable for flow where , , and His results are

not for a pipe but for a rectangular duct with curvature. For the purpose of turbulent analysis the flow

in the curved duct/pipe is typically divided into the laminar sub layer, the buffer layer, and the turbulent

core. Using the experimental results Krieth was able to calculate how the eddy diffusivity varied across

the tube cross-section. His results and approximation (linear fit) are seen in Figure 12. The figures are

difficult to see because the requested copies were not made by the author. Kreiths paper was not in

circulation and library staff had to copy the article from storage. The axes are relabeled as best possible.


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The final Nusselt numbers obtained by Kreith are not repeated here because of their length and

formulation but can easily be found in [4].

Figure 14: Circumferential Variation in Nusselt number in Fully Developed Flow Through a Coil [1]


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Figure 14 shows results from Iacovides & Lauder [1]. They solved the turbulent momentum and

heat transfer equations in the toroidal coordinate system with the two equation turbulent kinetic

energy and dissipation (k-) model. They used an elliptical equation solver. Their computations were

compared with experimental results from Sebam & McLaughlin (1963). Their results seem to be in good

agreement except at smaller curvatures the experimental results of Nusselt numbers are higher than

those predicted by the computation. Interestingly, their results show trends of Nusselt numbers on the

outer surface nearly two to three times the Nusselt numbers on the inner surface. They also reported

that the inside to outside wall Nusselt number ratio was 5:1 at a length 75from the bend inlet (Re =

4300, De = 1800, and Pr = 0.7) [1].

Schmidt (1967) computed the best fit average Nusselt numbers from various experimental and

theoretical works and found the following to have the largest application range:

Where is the equivalent Nusselt number for a straight run of pipe with the same Re and Pr.

IV. Concluding RemarksCurved pipes/bends are used in nearly all piping networks and across many industries. This

paper presented the fundamental physics behind heat transfer and fluid flow in pipes with streamwise

curvature. Typical geometry and nomenclature along with the appropriate curvilinear coordinate

systems used to represent curved pipes was introduced. The basic cylindrical coordinate forms of the

continuity, momentum, and thermal energy equations were formulated for both laminar and turbulent

flows. Due to the streamwise curvature, the concept of secondary flow was introduced and basic

physical reasoning for increased heat transfer in curved pipes was introduced. A review of solutions to

the fluid flow and heat transfer problems for both laminar and turbulent flow cases was presented.

Finally, basic average Nusselt number correlations for curved pipes were introduced. This paper was

primarily concerned with constant temperature surface boundary conditions. Other types of thermal

boundary conditions are presented in [3].


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V. Works Cited1. Iacovides, H., & Launder, B. (1984). The Computation of Momentum and Heat Transport in

Turbulent Flow Around Pipe Bends. First U.K. National Conference on Heat Transfer, (pp. 1097-

1114).

2. Kakac, S., Gogus, Y., & Ozgu, M. R. (1974). Investigation of the Effect of Tunrs on TurbulentForced Convection Heat Transfer in Pipes. In N. Afgan, & E. U. Schlunder, Heat Exchangers:

Design and theory Sourcebook(pp. 637-662). New York: McGraw-Hill.

3. Kakac, S., Shah, R. K., & Aung, W. (1987). Handbook of Single-Phase Convective Heat Transfer.New York: John Wiley & Sons.

4. Kreith, F. (1955). The influence of Curvature on Heat Transfer to Incompressible Fluids. Trans.ASME, 1247-1256.

5. Patankar, S. V., Pratap, V. S., & Spalding, D. B. (1974). Prediction of Laminar Flow and HeatTransfer in Helically Coiled Pipes.Journal of Fluid Mechanics, 539-551.

6. Sankaraiah, M., & Rao, Y. V. (1973). Analysis of Steady Laminar Flow of an IncompressibleNewtonian Fluid Through Curved Pipes of Small Curvature.Journal of Fluids Engineering, 75-80.


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Appendix A: Motivations for Study

The author is particularly interested in this topic because of ties to the local Minneapolis

Company, GARN. GARN was founded in 1984 and currently creates a line of wood-fired hydronic

heaters. This simply means that wood is burned in a combustor, hot combustion products pass through

a heat exchanger, and that heat is transferred to water as a thermal storage medium. See Figure 15.

Figure 15: Block Diagram of Wood Combustor

By burning the wood properly, the thermal efficiency of the combustion process is increased

dramatically, and pollutants from wood smoke are minimized. Hydronic heating has been around for

almost a century and is commonplace in most commercial and many residential buildings. Hot water

systems tie into conventional baseboard, radiant floor, and forced air systems.

New EPA regulations for wood-fired hydronic heaters require efficiency and pollutant standards

that surpass typical industrial boilers, so the GARN product line has undergone testing to see if it

complies with the regulations. It was found that heat transfer around the pipe bends in the heatexchanger could not be readily calculated and the deviations from heat transfer predictions were

significant. It was, therefore, realized that further investigations into the heat transfer in curved pipes

was needed. It is highly desirable in the design process to accurately estimate the heat transfer from

the hot gaseous products to the water storage tank in order to maximize the efficiency of the unit while

maintaining the simplest and most cost-effective heat exchanger. This paper will ultimately influence

the redesign of the GARN heat exchanger, and hopefully its success in passing the new EPA regulations.

davidlunde curved pipe heat transfer

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