Forced Convection Correlations for the Single-Phase Side ...

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Forced Convection Correlations for

the Single-Phase Side of Heat

Exchangers

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Laminar and turbulent forced convection correlations for single-phase fluids represent an

important class of heat transfer solutions for heat exchanger applications. When a viscous

fluid enters a duct, a boundary layer will form along the wall. The boundary layer

gradually fills the entire duct cross section and the flow is then said to be fully

developed. The distance at which the velocity becomes fully developed is called the

hydrodynamic or velocity entrance length (Lhe). Theoretically, the approach to the fully

developed velocity profile is asymptotic, and it is therefore impossible to describe a

definite location where the boundary layer completely fills the duct. But for all practical

purpose, the hydrodynamic entrance length is finite.


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If the walls of the duct are heated or cooled, then a thermal boundary layer will also develop along the duct wall.

At a certain point downstream, one can talk about the fully developed temperature profile where the thickness of

the thermal boundary layer is approximately equal to half the distance across the cross section. The distance at

which the temperature profile becomes fully developed is called the thermal entrance length (Lte).


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If heating or cooling starts from the inlet of the duct, then both the velocity and

the temperature profiles develop simultaneously. The associated heat transfer

problem is referred to as the combined hydrodynamic and thermal entry length

problem or the simultaneously developing region problem. Therefore, there are

four regimes in duct flows with heating/cooling, namely, hydrodynamically and

thermally fully developed, hydrodynamically fully developed but thermally

developing, thermally developed but hydrodynamically developing;

simultaneously developing; and the design correlations should be selected

accordingly.


The relative rates of development of the velocity and temperature profiles in the combined entrance region

depend on the fluid Prandtl number:

* For high Prandtl number fluids such as oils, even though the velocity and temperature profiles are

uniform at the tube entrance, the velocity profile is established much more rapidly than the temperature

profile.

* In contrast, for very low Prandtl number fluids such as liquid metals, the temperature profile is

established much more rapidly than the velocity profile.

* However, for Prandtl numbers around 1, such as gases, the temperature and velocity profiles develop

simultaneously at a similar rate along the duct, starting from uniform temperature and uniform velocity at

the duct entrance.

* For the limiting case of Pr → ∞, the velocity profile is developed before the temperature profile starts

developing.

* For the other limiting case of Pr → 0, the velocity profile never develops and remains uniform while the

temperature profile is developing.

* The idealized Pr → ∞ and Pr → 0 cases are good approximations for highly viscous fluids and liquid

metals (high thermal conductivity), respectively. 5


At a Reynolds number Re > 104, the flow is completely turbulent. Between the lower and

upper limits lies the transition zone from laminar to turbulent flow.

The critical Reynolds number in circular ducts is between 2100 and 2300.

hx is called the local heat transfer coefficient or film coefficient

Is defined based on the inner surface of the duct wall by using the convective boundary condition

k is the thermal conductivity of the fluid T is the temperature distribution in

the fluid, and Tw and Tb are the local wall and the fluid bulk temperatures,

respectively.

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The local Nusselt number is calculated from:

The local fluid bulk temperature Tbx, also referred to as the “mixing cup” or average fluid

temperature, is defined for incompressible flow as:

where um is the mean velocity of the fluid, Ac is the flow cross section, and u and T are,

respectively, the velocity and temperature profiles of the flow at position x along the duct

In design problems, it is necessary to calculate the total heat transfer rate over the total (entire)

length of a duct using a mean value of the heat transfer coefficient based on the mean value of the

Nusselt number defined as:

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Laminar Forced Convection

Laminar duct flow is generally encountered in compact heat exchangers, cryogenic

cooling systems, heating or cooling of heavy (highly viscous) fluids such as oils, and in

many other applications.

Different investigators performed extensive experimental and theoretical studies with

various fluids for numerous duct geometries and under different wall and entrance

conditions.

Laminar flow can be obtained for a specified mass velocity G = ρum for (1) small

hydraulic diameter Dh of the flow passage, or (2) high fluid viscosity μ. Flow passages

with small hydraulic diameter are encountered in compact heat exchangers since they

result in large surface area per unit volume of the exchanger. The internal flow of oils

and other liquids with high viscosity in non-compact heat exchangers is generally of a

laminar nature.

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The well-known Nusselt–Graetz problem for heat transfer to an incompressible fluid with constant

properties flowing through a circular duct with a constant wall temperature boundary condition

(subscript T) and fully developed laminar velocity profile was solved numerically by several

investigators. The asymptotes of the mean Nusselt number for a circular duct of length L are

The superposition of two asymptotes for the mean Nusselt number derived by Gnielinski gives

sufficiently good results for most of the practical cases:

An empirical correlation has also been developed by Hausen for laminar flow in the thermal

entrance region of a circular duct at constant wall temperature and is given as:

Gnielinski

Hausen

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Hydro-dynamically developed and Thermally

developing Laminar Flow in Smooth Circular ducts


These equations may be used for the laminar flow of gases and liquids in the range of

Axial conduction effects must be considered if Peb d/L < 0.1. All physical properties are evaluated at the

fluid mean bulk temperature of Tb defined as:

where Ti and To are the bulk temperatures of the

fluid at the inlet and outlet of the duct, respectively.

Constant wall heat flux boundary condition:

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Simultaneously developing Laminar Flow in Smooth ducts

In the case of a short duct length, Nu values are represented by the asymptotic

equation of Pohlhausen for simultaneously developing flow over a flat plate; for a

circular duct, this equation becomes

For most engineering applications with short circular ducts (d/L > 0.1), it is

recommended that which ever of Equations gives the highest Nusselt number be

used.

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Laminar Flow through Concentric Annular Smooth ducts

Correlations for concentric annular ducts are very important in heat exchanger applications. The

simplest form of a two-fluid heat exchanger is a double-pipe heat exchanger made up of two

concentric circular tubes

Concentric tube annulus

The heat transfer coefficient in the annular duct depends on the ratio of the diameters Di/do because

of the shape of the velocity profile.

The hydraulic (equivalent) diameter approach is the simplest method to calculate the heat transfer

and the pressure drop in the annulus. In this approach, the hydraulic diameter of annulus Dh is

substituted instead of the tube diameter in internal flow correlations:

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This approximation is acceptable for heat transfer and pressure drop calculations.

The wetted perimeter for pressure drop calculation in the annulus is defined as:

and the heat transfer perimeter of the annulus can be calculated by

The net free-flow area of the annulus is given by

The hydraulic diameter based on the total wetted perimeter for pressure drop calculation is

which is hereafter called the equivalent diameter:

The only difference between Pw and Ph is Di,

which is the inner diameter of the shell

(outer tube) of the annulus.

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For the constant wall temperature boundary condition, Stephan8 has developed

a heat transfer correlation based on Equation.

The Nusselt number for hydrodynamically developed laminar flow in the

thermal entrance region of an isothermal annulus, the outer wall of which is

insulated, may be calculated by the following correlation:

where Nu∞ is the Nusselt number for fully developed flow, which is given by

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Effect of Variable Physical Properties

Two correction methods for constant-property correlations for the variable-property effect

have been employed: namely, the reference temperature method and the property

ratio method. In the former, a characteristic temperature is chosen at which the properties

appearing in non-dimensional groups are evaluated so that the constant-property results at

that temperature may be used to account for the variable-property behavior; in the latter,

all properties are taken at the bulk temperature and then all variable-property effects are

lumped into a function of the ratio of one property evaluated at the wall (surface)

temperature to that property evaluated at the average bulk temperature. Some correlations

may involve a modification or combination of these two methods.

When the previously mentioned correlations are applied to practical heat transfer problems

with large temperature differences between the wall and the fluid mean bulk temperatures,

the constant-property assumption could cause significant errors, since the transport

properties of most fluids vary with temperature, which influences the variation of velocity

and temperature through the boundary layer or over the flow cross section of a duct.

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For gases, the viscosity, thermal conductivity, and density vary with the absolute

temperature. Therefore, in the property ratio method, temperature corrections of the

following forms are found to be adequate in practical applications for the temperature-

dependent property effects in gases:

where Tb and Tw are the absolute bulk mean and

wall temperatures, respectively

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For liquids, the variation of viscosity with temperature is responsible for most of the property effects.

Therefore, the variable-property Nusselt numbers and friction factors in the property ratio method for

liquids are correlated by:

where μb and k are the viscosity and conductivity evaluated at the

bulk mean temperature, μw is the viscosity evaluated at the wall

temperature, and cp refers to the constant-property solution.


Laminar Flow of Liquids

Desissler carried out a numerical analysis as described previously for laminar flow

through a circular duct at a constant heat flux boundary condition for liquid viscosity

variation with temperature given by:

and obtained n = 0.14 to be used with

Deissler9 also obtained m = –0.58 for heating and m = –0.50 for cooling of liquids, to be

used with

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A simple empirical correlation has been proposed by Seider and Tate to predict the mean Nusselt

number for laminar flow in a circular duct for the combined entry length with constant wall temperature

as

which is valid for smooth tubes for 0.48 < Prb < 16,700 and 0.0044 < (μb/μw) < 9.75. This correlation has

been recommended by Whitaker for values of

Below this limit, fully developed conditions will be established and

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Example 3.1

Determine the total heat transfer coefficient at 30 cm from the inlet of a heat exchanger where

engine oil flows through tubes with an inner diameter of 0.5 in. Oil flows with a velocity of 0.5

m/s and at a local bulk temperature of 30°C, while the local tube wall temperature is 60°C.

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Kuznestova conducted experiments with transformer oil and fuel oil in the range

of 400 < Reb < 1,900 and 170 < Prb < 640 and recommended


Test conducted an analytical and experimental study on the heat transfer and fluid

friction of laminar flow in a circular duct for liquids with temperature-dependent

viscosity. The analytical approach is a numerical solution of the continuity,

momentum, and energy equations. The experimental approach involves the use of a

hot-wire technique for determination of the velocity profiles. Test obtained the

following correlation for the local Nusselt number:

where

He also obtained the friction factor as

Equations should not be applied to extremely long ducts.

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Laminar Flow of gases

The first reasonably complete solution for laminar heat transfer of a gas flowing in

a tube with temperature-dependent properties was developed by Worsoe-Schmidt.

He solved the governing equations with a finite-difference technique for fully

developed and developing gas flow through a circular duct. Heating and cooling

with a constant surface temperature and heating with a constant heat flux were

considered. In his entrance region solution, the radial velocity was included. He

concluded that near the entrance, and also well downstream (thermally developed),

the results could be satisfactorily correlated

for heating 1 < (Tw/Tb) < 3 by n = 0, m = 1.00

And

for cooling 0.5 < (Tw/Tb) < 1 by n = 0, m = 0.81.

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Turbulent Forced Convection

Extensive experimental and theoretical efforts have been made to obtain the solutions for turbulent

forced convection heat transfer and flow friction problems in ducts because of their frequent

occurrence and application in heat transfer engineering. A compilation of such solutions and

correlations for circular and noncircular ducts has been put together by Bhatti and Shah.

There are a large number of correlations available in the literature for the fully developed

(hydrodynamically and thermally) turbulent flow of single- phase Newtonian fluids in smooth,

straight, circular ducts with constant and temperature-dependent physical properties. The objective

of this section is to highlight some of the existing correlations to be used in the design of heat

exchange equipment and to emphasize the conditions or limitations imposed on the applicability of

these correlations.

An example of the latter is the correlation given by Petukhov and Popov. Their theoretical

calculations for the case of fully developed turbulent flow with constant properties in a circular tube

with constant heat flux boundary conditions yielded a correlation, which was based on the three-

layer turbulent boundary layer model with constants adjusted to match the experimental data.

Petukhov also gave a simplified form of this correlation as

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Equation predicts the results in the range

and 0.5 < Prb < 200 with 5 to 6% error, and in the range 0.5 < Prb < 2,000 with 10% error.

Sleicher and Rouse correlated analytical and experimental results for the range

and obtained.

with

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Gnielinski recommended the following correlation for the average Nusselt

number, which is also applicable in the transition region where the Reynolds

numbers are between 2300 and 10000:

where

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Turbulent Flow in Smooth Straight Noncircular Ducts

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Therefore, the flow inside the annulus is turbulent. One of the correlations can be selected from Table

3.3. The correlation given by Petukhov and Kirillov (No. 3) is used here. It should be noted that, for the

annulus, the Nusselt number should be based on the equivalent diameter calculated from Equation 3.19:

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Effect of Variable Physical Properties in Turbulent Forced

Convection

Turbulent Liquid Flow in ducts

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Turbulent gas Flow in ducts

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Heat Transfer from Smooth-Tube Bundles

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Heat Transfer in Helical Coils and Spirals

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Nusselt numbers of Helical Coils—Laminar Flow

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Nusselt numbers for Spiral Coils—Laminar Flow

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Nusselt numbers for Helical Coils—Turbulent Flow

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Heat Transfer in Bends

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Heat Transfer in 90° Bends

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Heat Transfer in 180° Bends

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Forced Convection Correlations for the Single-Phase Side ...

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