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کلیات: جریان داخلی

• Liquid or gas flow through pipes or ducts

• forced to flow by a fan or pump through a tube

• Determination of the friction factor and convection coefficient sincethey are directly related to the pressure drop and heat transfer rate

• then used to determine the pumping power requirement and therequired tube length for the desired heat transfer

• In internal flow, the fluid is completely confined by the inner surfaces ofthe tube, and there is a limit on how much the boundary layer can grow

Chapter 8: Internal Forced Convection Dr. Afsaneh MojraChapter 8: Internal Forced Convection Dr. Afsaneh Mojra

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جریان داخلی

• For a fixed surface area, the circular tube gives the most heat transfer forthe least pressure drop

• flow sections of circular cross section are referred to as pipes (especiallywhen the fluid is a liquid)

• and the flow sections of noncircular cross section as ducts (especiallywhen the fluid is a gas)

• Small diameter pipes are usually referred to as tubes


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جریان داخلی

• rely on the experimental results and the empirical relations: error of 10percent (or more)

• friction between the fluid layers in a tube may cause a slight rise in fluidtemperature as a result of mechanical energy being converted to thermalenergy. But this frictional heating is too small

• flow of a fluid through a pipe or duct can be approximated to be one-dimensional: bulk average values over the cross section


Assumptions

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سرعت و دماي متوسط: جریان داخلی


Mean Velocity & Mean Temperature

The value of the mean velocity Vm in atube is determined from theconservation of mass principle

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• for incompressible flow in a circular tube


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conservation of energy

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fluid properties in internal flow are usually evaluated atthe bulk mean fluid temperature

= , + ,27/49

لایه اي و مغشوش: جریان داخلی

• For flow in a circular tube:

• For flow through noncircular tubes, the Reynolds number as well as theNusselt number and the friction factor are based on the hydraulicdiameter:


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• Under most practical conditions:

• laminar flow can be maintained at much higher Reynolds numbers invery smooth pipes


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• Flow regions:


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روابط: جریان داخلی


Hydrodynamic Entrance Region

The region from the tube inlet to the point at which the boundary layermerges at the centerline; Length: LhFlow: hydrodynamically developing flow

Hydrodynamically Fully Developed Region

beyond the entrance region in which the velocity profile is fully developedand remains unchanged.Velocity profile: parabolic in laminar flow and somewhat flatter inturbulent flow due to eddy motion in radial direction

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in fully developed region: friction factor & heat transfer coefficientremains constant in that region.

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• In laminar flow, the hydrodynamic and thermal entry lengths:

• The hydrodynamic entry length is much shorter in turbulent flow• In practice:


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Variation of local Nusselt number along a tube in turbulent flow for bothuniform surface temperature and uniform surface heat flux

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• The thermal conditions at the surface can usually be approximated withreasonable accuracy to be constant surface temperature ( = )or constant surface heat flux ( = )


General Thermal Analysis

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• mean fluid temperature:

• The surface temperature in the case of constant surface heat flux can bedetermined from

• In the fully developed region:h: constant &


Constant Surface Heat Flux ( = )

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&increase

linearly in theflow direction

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• steady-flow energy balance to a tube slice of thickness dx:

• p is the perimeter of the tube• We had:


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• in the fully developed region, we had:

• For a circular tube:


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in F.D. flow in a tube subjected to constant surface heat flux, thetemperature gradient is independent of x and thus the shape of thetemperature profile does not change along the tube

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• Newton’s law of cooling, the rate of heat transfer to or from a fluidflowing in a tube

• Tave is some appropriate average temperature difference between thefluid and the surface; Two Ways:

• Tb: bulk mean fluid temperatureChapter 8: Internal Forced Convection Dr. Afsaneh MojraChapter 8: Internal Forced Convection Dr. Afsaneh Mojra

Constant Surface Temperature (T = )

= ℎ ∆ ( )arithmetic mean temperature difference1st

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logarithmic mean temperature difference2nd

exact representation of the average temperature difference between thefluid and the surface.

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When Te differs from Ti by no more than 40 percent, the error in using thearithmetic mean temperature difference is less than 1 percent.But the error increases to undesirable levels when Te differs from Ti bygreater amounts.Therefore, we should always use the logarithmic mean temperaturedifference when determining the convection heat transfer in a tube whosesurface is maintained at a constant temperature Ts.

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Laminar Flow in Tubes

= −4 ( )(1 − )

= 2 = − 2 4 1 − = −8 ( )30/49



The maximum velocity occurs at the centerline

= 2Pressure Drop for Laminar Flow in Tubes

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• In practice, it is found convenient to express the pressure drop for alltypes of internal flows (laminar or turbulent flows, circular ornoncircular tubes, smooth or rough surfaces)

• f is the friction factor (also called the Darcy friction factor after Frenchengineer Henry Darcy, 1803–1858, who first studied experimentally theeffects of roughness on tube resistance).

• It should not be confused with the friction coefficient Cf which isdefined as = =


∆ =

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• Assumptions:– flow section is horizontal so that there are no hydrostatic or gravity effects,– flow section does not involve any work devices such as a pump or a turbine

since they change the fluid pressure, and– Cross sectional area of the flow section is constant and thus the mean flow

velocity is constant.


Poiseuille’s Law33/49


• For the ideal inviscid flow, the pressure drop is zero since there are noviscous effects


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Temperature Profile and the Nusselt Number

steady laminar flow of a fluid in a circulartube of radius R.

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• Boundary Conditions:– Constant Surface Heat Flux– Constant Surface Temperature


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• For fully developed flow in a circular pipe subjected to constant surfaceheat flux, we have


Constant Surface Heat Flux

− = ( )

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− = ( ) SolveODE

Applying B.C.s

= = ( ) & = =

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= .Circular tube, laminar ( = ):For fully developed laminar flow in a circular tube subjected to constantsurface heat flux, the Nusselt number is a constant. There is no dependenceon the Reynolds or the Prandtl numbers.

= = .

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Constant Surface Temperature

Circular tube, laminar (T = ): = = .thermal conductivity k should be evaluated at the bulk mean fluidtemperature, which is the arithmetic average of the mean fluidtemperatures at the inlet and the exit of the tube.For laminar flow, the effect of surface roughness on the friction factor andthe heat transfer coefficient is negligible

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Laminar Flow in Noncircular Tubes

Table 8–1: f (friction factor) & Nu for fully developed laminar flow in tubesof various cross sections

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• For a circular tube of length L subjected to constant surface temperature:

• average Nusselt number is larger at the entrance region, and itapproaches asymptotically to the fully developed value of 3.66 as L→∞.

• When the difference between the surface and the fluid temperatures islarge, it may be necessary to account for the variation of viscosity withtemperature.


Developing Laminar Flow in the Entrance Region

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• Non-circular Sections:

• isothermal parallel plates : twice the spacing of plates


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< (2300)flow is laminar

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much greater than the total length of the pipe.Assume thermally developing flow.

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+ = 0 → − + ℎ ∆ = 0

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≈ 0.05 ∗ 666 ∗ 0.3 = 9.99the laminar flow of oil is hydrodynamically developed.

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