Islamic Azad University Karaj Branch

Chapter 8 Convection: Internal Flow

Entrance Conditions •  Must distinguish between entrance and fully developed regions. •  Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at inlet of a

circular tube.

•  Velocity boundary layer develops on surface of tube and thickens with increasing x. •  Inviscid region of uniform velocity shrinks as boundary layer grows.

–  Does the centerline velocity change with increasing x? If so, how does it change? •  Subsequent to boundary layer merger at the centerline, the velocity profile becomes

parabolic and invariant with x. The flow is then said to be hydrodynamically fully developed.

–  How would the fully developed velocity profile differ for turbulent flow?

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Mean Velocity •  Velocity inside a tube varies over the cross section. For every differential

area dAc:

•  Overall rate of mass transfer through a tube with cross section Ac:

where um is the mean (average) velocity

!  Can determine average velocity at any axial location (along the x-direction), from knowledge of the velocity profile

(8.1)

(8.2)

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Velocity Profile in a pipe •  Recall from fluid methanics that for laminar flow of an incompressible,

constant property fluid in the fully developed region of a circular tube (pipe):

(8.3a)

(8.3b)

(8.3c)

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Thermal Considerations: Mean Temperature

•  We can write Newton’s law of cooling inside a tube, by considering a mean temperature, instead of T!

where Tm is the mean (average) temperature

(8.4)

•  For constant r and cp, Tm is defined:

(8.5)

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Fully Developed Conditions

•  For internal flows, the temperature, T(r), as well as the mean temperature, Tm generally vary in the x-direction, i.e.

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Fully Developed Conditions

•  A fully developed thermally region is possible, if one of two possible surface conditions exist : –  Uniform wall temperature (Ts=constant) –  Uniform heat flux (qx”=const)

•  Thermal Entry Length :

•  Although T(r) changes with x, the relative shape of the temperature profile remains the same: Flow is thermally fully developed.

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•  It can be proven that for fully developed conditions, the local convection coefficient is a constant, independent of x:

Fully Developed Conditions

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Mean temperature variation along a tube

We are still left with the problem of knowing how the mean temperature Tm(x), varies as a function of distance, so that we can use it in Newton’s law of cooling to estimate convection heat transfer.

Recall from Chapter 1, page 10 that by simplifying the energy balance for flow inside a control volume

where Tm,i and Tm,o are the mean temperatures of the inlet and outlet respectively

For flow inside a pipe: (8.6)

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Mean temperature variation along a tube

P=surface perimeter

For a differential control volume:

where P=surface perimeter =p D for circular tube, =width for flat plate

!  Integration of this equation will result in an expression for the variation of Tm as a function of x.

(8.7)

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Case 1: Constant Heat Flux

•  Integrating equation (8.7):

(8.8)

where P=surface perimeter =pD for circular tube, =width for flat plate

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Case 2: Constant Surface Temperature,Ts=constant

From eq.(8.7)

Integrating for the entire length of the tube:

where (8.10) (8.11)

(8.9)

As is the tube surface area, As=P.L=pDL, DTlm is the log-mean temperature difference

with Ts-Tm=DT:

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Case 3: Uniform External Temperature

"  Replace Ts by and by (the overall heat transfer coefficient, which includes contributions due to convection at the tube inner and outer surfaces, and due to conduction across the tube wall). Equations (8.9) and (8.10) become:

(8.11) (8.12)

Reminder from Chapter 3, p. 19 lecture notes

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Example (Problem 8.55) Water at a flow rate of 0.215 kg/s is cooled from 70°C to 30°C by passing it through a thin-walled tube of diameter D=50 mm and maintaining a coolant at 15°C in cross flow over the tube. What is the required tube length if the coolant is air and its velocity is V=20 m/s? The heat transfer coefficients are hi=680 W/m2.K for flow of water inside the tube and ho=83.5 W/m2.K for a cylinder in air cross flow of 20 m/s

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Summary (8.1-8.3) •  We discussed fully developed flow conditions for cases involving

internal flows, and we defined mean velocities and temperatures •  We wrote Newton’s law of cooling using the mean temperature,

instead of

•  Based on an overall energy balance, we obtained an alternative expression to calculate convection heat transfer as a function of mean temperatures at inlet and outlet.

•  We obtained relations to express the variation of Tm with length, for cases involving constant heat flux and constant wall temperature

(8.6)

(8.4)

(8.9) (8.8)

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Summary (8.1-8.3)

" In the rest of the chapter we will focus on obtaining values of the heat transfer coefficient h, needed to solve the above equations

•  We used these definitions, to obtain appropriate versions of Newton’s law of cooling, for internal flows, for cases involving constant wall temperature and constant surrounding fluid temperature

(8.10-8.12)

Heat Transfer Correlations for Internal Flow

Knowledge of heat transfer coefficient is needed for calculations shown in previous slides. "  Correlations exist for various problems involving internal flow, including laminar and

turbulent flow in circular and non-circular tubes and in annular flow.

"  For laminar flow we can derive h dependence theoretically

"  For turbulent flow we use empirical correlations

"  Recall from Chapters 6 and 7 general functional dependence

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Laminar Flow in Circular Tubes

•  For cases involving uniform heat flux:

•  For cases involving constant surface temperature:

(8.13)

(8.14)

1.  Fully Developed Region

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Laminar Flow in Circular Tubes

2. Entry Region: Velocity and Temperature are functions of x

•  Thermal entry length problem: Assumes the presence of fully developed velocity profile

•  Combined (thermal and velocity) entry length problem: Temperature and velocity profiles develop simultaneously

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Laminar Flow in Circular Tubes For constant surface temperature condition: •  Thermal Entry Length case or combined entry with Pr"5

•  Combined Entry Length case

All properties, except ms evaluated at average value of mean temperature

(8.15)

(8.16)

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Turbulent Flow in Circular Tubes •  For a smooth surface and fully turbulent conditions the

Dittus – Boelter equation may be used for small to moderate temperature differences Ts-Tm:

n=0.4 for heating (Ts>Tm) and 0.3 for cooling (Ts<Tm)

•  For large property variations, Sieder and Tate equation:

All properties, except ms evaluated at average value of mean temperature

(8.17)

(8.18)

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•  The Gnielinski correlation takes into account the friction factor:

Friction factors may be obtained from the Moody diagram.

(8.19)

•  For small Pr numbers 3x10-3# Pr #5x10-2 (i.e. liquid metals)

(8.20)

(8.21)

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Turbulent Flow in Circular Tubes

Example (Problem 8.55) Repeat Problem 8.55. This time the values of the heat transfer coefficients are not provided, therefore we need to estimate them. Water at a flow rate of 0.215 kg/s is cooled from 70°C to 30°C by passing it through a thin-walled tube of diameter D=50 mm and maintaining a coolant at 15°C in cross flow over the tube.

(a) What is the required tube length if the coolant is air and its velocity is V=20 m/s?

(b) What is the required tube length if the coolant is water is V=2 m/s?

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Non-Circular tubes

Use the concept of the hydraulic diameter:

where Ac is the flow cross-sectional area and P the wetted perimeter

!  See Table 8.1 textbook for typical values of Nusselt numbers for various cross sections

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Example (Problem 8.80) You have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. It is desirable to heat blood taken from the bank at 15°C to a physiological temperature of 37°C, at a flow rate of 200 ml/min. The blood passes through a rectangular cross-section tube, 6.4 mm by 1.6 mm, which is sandwiched between two plates held at a constant temperature of 40°C. Compute the length of the tubing required to achieve the desired outlet conditions at the specified flow rate. Assume the flow is fully developed and the blood has the same properties as water.

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Summary

•  Numerous correlations exist for the estimation of the heat transfer coefficient, for various flow situations involving laminar and turbulent flow.

•  Always make sure that conditions for which correlations are valid are applicable to your problem.

! Summary of correlations in Table 8.4 of textbook

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