CONVECTIVE HEAT TRANSFER - Sahand University of …mech.sut.ac.ir/People/Courses/18/Chapter4.pdf · solution. convective heat transfer-chapter4 by: ... convective heat transfer-chapter4

CONVECTIVE HEAT TRANSFER

Mohammad GoharkhahDepartment of Mechanical Engineering, Sahand Unversity of Technology,

Tabriz, Iran

HEAT TRANSFER IN CHANNEL FLOW

CHAPTER 4

BASIC CONCEPTS

CONVECTIVE HEAT TRANSFER- CHAPTER4By: M. Goharkhah

SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING

BASIC CONCEPTS

Laminar vs. turbulent flow.Transition from laminar to turbulent flow takes place when the Reynolds numberreaches the transition value. For flow through tubes the experimentally determinedtransition Reynolds number is:

Entrance vs. fully developed region.(1)Entrance region (developing region).It extends from the inlet to the section where the boundary layer thickness reaches

the channel center.

(2) Fully developed region.This zone follows the entrance region. In general the lengths of the velocity andtemperature entrance regions are not identical.



Entrance vs. fully developed- Flow Field

Core velocity uc increases with axialdistance x (uc is not constant).

Pressure decreases with axial distance(dp/dx<0)

Velocity boundary layer thickness iswithin tube radius (δ<D/ 2 ).

Entrance Region (Developing Flow, 0 ≤ x ≤ Lh ) Fully Developed Flow Region (Lh ≤ x)

Streamlines are parallel vr = 0.

For two dimensional incompressibleflow the axial velocity u is invariant withaxial distance x. That is ∂u/∂ x=0.



Entrance vs. fully developed- Temperature Field

Entrance Region (Developing Temperature, 0 ≤ x ≤ Lt ).

Core temperature Tc is uniform equal toinlet temperature (Tc =Ti ).

Temperature boundary layer thicknessis within the tube’s radius (δ t<D/ 2).

Fully Developed Temperature Region (Lt ≤ x).

Fluid temperature varies radially andaxially. Thus ∂ T / ∂ x ≠ 0.

A dimensionless temperature φ(to bedefined later) is invariant with axial distancex. That is ∂φ / ∂ x =0.



Determination of Hydrodynamic and ThermalEntrance Lengths- Scale Analysis


Result of scale analysis for the velocityboundary layer thickness for external flow

Hydrodynamic Entrance Length

For the flow through a tube at the endof the entrance region with

the Reynolds number based ontube diameter D






Determination of Hydrodynamic and ThermalDetermination of Hydrodynamic and ThermalEntrance Lengths- Analytic and Numerical Solutions

channel flow area

channel perimeterLaminar Flow

Hydrodynamic Entrance Length

Thermal Entrance Length



Determination of Hydrodynamic and ThermalEntrance Lengths- Analytic and Numerical Solutions

For a rectangular channel of aspect ratio 2 at uniform surface temperature

scaling estimates the constant 0.22 to be unity.

For a rectangular channel of aspect ratio 2

scaling estimates the constant 0.29 to be unity.



Determination of Hydrodynamic and ThermalEntrance Lengths- Analytic and Numerical Solutions

Turbulent Flow



Determination of Hydrodynamic and ThermalEntrance Lengths- Integral Method

The integral momentum and energy equation

Since the core flow is inviscid, dP/dx isrelated to Uc (x) through the Bernoulliequation ρUc

2/2 + P = constant



Determination of Hydrodynamic and ThermalEntrance Lengths- Integral Method

mass conservation in the channel of half-width (from y = 0 to y = D/2)

Solve Eqs. 1 and 2 for δ(x) and Uc(x) by first assuming a boundary layer profile shape.Take u/Uc = 2y/δ − (y/δ)2

2

1

At the location X where the two boundary layers merge we set δ(X) = D/2,

ANALYTICAL SOLUTION OF THE FLOW FIELD



FULLY DEVELOPED FLOW

steady-statemass andmomentumconservation



FULLY DEVELOPED FLOW- Parallel Plate Channel

FULLY DEVELOPED FLOW- Parallel Plate Channel

The fully developed region is that section of theduct flow that is situated far enough from theentrance such that the scale of v is negligible

y momentum equation P is a function of x only

no-slip conditionsy is measured away from the centerline of the channel

X momentum equation

Conservation of Mass

Conservation of Momentum



FULLY DEVELOPED FLOW- Round Tube

The momentum equation for a duct of arbitrary cross section

the fully developed laminar flow in a round tube of radius r0

Hagen and Poiseuillesolution



HYDRAULIC DIAMETER AND PRESSURE DROPObjective

Calculation of the pressure drop in aduct with prescribed flow rate or thecalculation of the flow rate in a ductwith prescribed pressure drop

the momentum theorem in the longitudinal direction Perimeter of the

cross section

The calculation of P ispossible provided thatwe know the frictionfactor f.

Friction factor



HYDRAULIC DIAMETER AND PRESSURE DROPThe friction factors derived from the Hagen–Poiseuille solutions:

The product f ReDh is a number that depends only on the shape of the cross section. This number has been named the Poiseuille number, Po = f ReDh

The fact that f ReDh is a constant (of order close to 1) expresses the balance between the only two forces that are present, imposed pressure difference and fluid friction.

Poiseuille number





Calculation of the Friction Factor- Duct of rectangular cross section


In general, the friction factor f is obtained by solving the Poisson equation in the duct cross section of interest.

Fully developed laminar flow through a duct of rectangular cross section

Above Equation can be solved for u(y, z) by Fourier series.Here, we outline a more direct, approximate approach. To calculate f or τw, we needthe velocity distribution u(y, z): From the parallel-plate and round-tube solutionsdiscussed previously, we expect u(y, z) to be adequately represented by the expression

The problem reduces to calculating u0 (the centerline velocity)





Mean Velocity



The present f ReDh result coincides with the numerical result in the tall and flatcross-sectional shape limits because in those limits the profile shape assumption isexactly the Hagen–Poiseuille profile shape.Overall, the agreement between the current result and numerically derived resultsis better than 15 percent.

HEAT TRANSFER TO FULLY DEVELOPED DUCT FLOW





If the fluid as an ideal gas

The first law of thermodynamics

Mean Temperature

(dh = cp dTm)

(dh∼= c dTm) If the fluid is an incompressible liquid with negligible pressure changes



Fully Developed Temperature Profile

We must first determine the temperature field in the fluid T(x, r) by solving the energy equation subject to appropriate wall–temperature boundary conditions.

In the hydrodynamic fully developed region v = 0 and u = u(r)

ObjectiveCalculation of the convective heat transfer coefficient, h

How???

The energy equation for steady, ɵ-symmetric flow through a round tube

balance among three possible energy flows




Multiplying scales byD2/T and using thedefinition of heattransfer coefficienth = q’’/ΔT, we obtain:

The longitudinal conduction effect is negligible if:

from the convection–radial conduction balance, we learn that the Nusselt number is a constant of order 1




The flow is hydrodynamically fully developed; hence, the velocity profileu(r) is the same at any x along the duct.we assumed that the scale of ∂2T/∂r 2 is T/D2 ; in other words, the effectof thermal diffusion has had time to reach the centerline of the stream.This last assumption is not valid in a thermal entrance region XT near theduct entrance, where the proper scale of ∂2T/∂r 2 is T/δT

2, with δT <<D.

Assumptions



Definition of Fully Developed Temperature Profile

Definition of Fully Developed Temperature Profile



Uniform Wall Heat Flux

This means that the temperature everywherein the cross section varies linearly in x

Fully developed temperature

differentiate with respect to x



Uniform Wall Heat Flux

Fully developed temperature profile in a round tube with uniform heat flux



Integrating this equation twice and invoking one boundary condition (finite φ’ at r* = 0)

The mean temperature difference

Hagen–Poiseuillevelocity profile Energy equation

C2 is obtained from



Uniform Wall Temperature

A round tube with wall temperature T0 independent of xThe stream bulk temperature is T1 at some place x = x1 in the fully developed region

Eliminating q’’(x) and integrating the result from Tm = T1 at x = x1 yields




boundary conditions

Hagen–Poiseuillevelocity profile Energy equation

T = T0 − φ(T0 − Tm)




The radial profile φ will be a function of both r* and Nu, where Nu is theunknown in this problem.The additional condition for determining Nu uniquely is the definition of theheat transfer coefficient

The problem statement is now complete: The value of Nu must be such that the φ(r*, Nu) solution of Eqs. (*) and (+) satisfies the Nu definition (**).

boundary conditions

*

+

**




• The actual solution may be pursued in a number ofways, for example, by successively approximating(guessing) and improving the φ solution.

• It is more convenient to solve the problem numerically.the differential energy equation is first approximatedby finite differences and integrated from r* = 1 to r* = 0.To perform the integration at all, we must first guessthe value of Nu, which also gives us a guess for theinitial slope of the ensuing φ(r*) curve. The success ofthe Nu guess is judged by means of the first ofboundary conditions. the refined result is ultimately




(*) and (+)

(*) (*)

1

2



Uniform Wall Temperature1 2






Example

Air flows with a mean velocity of 2 m/s through a tube of diameter 1.0 cm. The meantemperature at a given section in the fully developed region is 35oC. The surface ofthe tube is maintained at a uniform temperature of 130oC. Determine the length ofthe tube section needed to raise the mean temperature to 105oC.

To compute L, it is necessary to determine the properties and the average h . Airproperties are determined at the mean temperature , defined as



The slight variation inthese values mimics thatof B = (πD2h/4)/Aduct,implying that it is causedby hydraulic diameternondimensionalization,that is, by the mismatchbetween hydraulicdiameter and effectivewall–stream distance. Thisbehavior again stressesthe importance of thenew dimensionless groupB = (πD2h/4)/Aduct.



the fully developed values of the friction factor and the Nusselt number in a duct with regular polygonal cross section

The thermally developed Nu value is considerably smaller in fully developed flow than in slug flow. The latter refers to the flow of a solid material (slug, rod), or a fluid with an extremely small Prandtl number (Pr→0), where the viscosity is so much smaller than the thermal diffusivity that the longitudinal velocity profile remains uniform over the cross section, u = U, like the velocity distribution of a solid slug.



HEAT TRANSFER TO DEVELOPING FLOWThermally Developing Hagen–Poiseuille Flow

the velocity profile is fully developed while the temperature profile is just being developed

Neglecting the effect of axial conduction (Pex >> 1),

Isothermal entering fluid, T = TIN for x < 0, where x is measured (positive) downstream from the location X



HEAT TRANSFER TO DEVELOPING FLOW

Graetz problem

The energy equation is linear and homogeneous. Separation of variables is achieved by assuming a product solution for ɵ*(r*, x*),

Sturm–Liouville type

Cn Constants determined by the x* = 0 condition

This problem was treated for the first time by Graetz




The average heat transfer coefficient is obtained by integrating the local heat transfer coefficient along a tube of length.




The overall Nusselt number for the thermal entrance region of a tube with isothermal wall is defined as

A simpler approach












HEAT TRANSFER TO DEVELOPING FLOW- Leveque solution

HEAT TRANSFER TO DEVELOPING FLOW- Leveque solution

• Simpler alternative to the Graetz series solution, which is known as the L´evˆeque solution



Thermally developing Hagen–Poiseuille flow in a round tube with uniform heat flux



Thermally developing Hagen–Poiseuille flow in a in a parallel-plate channel with isothermal surfaces

Thermally developing Hagen–Poiseuille flow in a in a parallel-plate channel with isothermal surfaces

• heat transfer to thermally developing Hagen–Poiseuille flow in a parallel-plate channel with isothermal surfaces are approximated by



Thermally developing Hagen–Poiseuille flow in a in a parallel-plate channel with uniform heat flux

Thermally developing Hagen–Poiseuille flow in a in a parallel-plate channel with uniform heat flux



Thermally and Hydraulically Developing Flow

The analytical expressions recommended for the local and overall Nusseltnumbers in the range 0.1 < Pr < 1000 in parallel-plate channels are:

The most realistic (and most difficult) tube flow problem consists of solvingthe following equation with the Hagen–Poiseuille profile 2(1 − r*

2) replacedby the actual x-dependent velocity profile present in the hydrodynamic entryregion. This, the finite-Pr problem, has been solved numerically by anumber of investigators.



Thermally and Hydraulically Developing Flow



A general expression for the entrance and fully developed regions

A general expression for the entrance and fully developed regions

• A closed-form expression for the local Nusselt number that covers boththe entrance and fully developed regions in a tube with uniform heatflux is

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CONVECTIVE HEAT TRANSFER - Sahand University of …mech.sut.ac.ir/People/Courses/18/Chapter4.pdf · solution. convective heat transfer-chapter4 by: ... convective heat transfer-chapter4