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Heat transfer coefficient

From Wikipedia, the free encyclopedia

Quantity relating heat flux and temperature difference

Inthermodynamics, the heat transfer coefficient orfilm coefficient, orfilm effectiveness, is theproportionality constant between theheat flux and the thermodynamic driving force for theflow of heat (i.e., thetemperature difference,ΔT ). It is used to calculateheat transfer between components of a system; such as byconvection between a fluid and a solid. The heat transfer coefficient hasSI units inwatts per square meter perkelvin (W/(m²K)).

The total heat transfer rate for combinedmodes and system components is usually expressed in terms of anoverall heat transfer coefficient,thermal transmittance orU-value. The heat transfer coefficient is thereciprocal ofthermal insulance. This is used for building materials (R-value) and forclothing insulation.

There are numerous methods for calculating the heat transfer coefficient in different heat transfer modes, different fluids, flow regimes, and under differentthermohydraulic conditions. Often it can be estimated by dividing thethermal conductivity of theconvection fluid by alength scale. The heat transfer coefficient is often calculated from theNusselt number (adimensionless number). There are also online calculators available specifically forHeat-transfer fluid applications. Experimental assessment of the heat transfer coefficient poses some challenges especially when small fluxes are to be measured (e.g.< 0.2 W/cm²).^[1]^[2]

Definition

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The general definition of the heat transfer coefficient is:^[3]^: 19–21

h={\frac {q}{\Delta T}}

where:

q {\displaystyle q}

:heat flux (W/m²); i.e., thermal power per unitarea,

q=d{\dot {Q}}/dA

\Delta T

: difference in temperature (K) between the solid surface and surrounding fluid area

The heat transfer coefficient replaces the thermal conductivity within a generalization ofFourier's law postulated to also describe convection flows (including conduction). Upon reaching asteady state of flow, the heat transfer rate is:^[3]^: 11–21

{\dot {Q}}=hA(T_{2}-T_{1})

where (in SI units):

{\dot {Q}}

: Heat transfer rate (W)

h {\displaystyle h}

: Heat transfer coefficient (W/m²K)

A {\displaystyle A}

: surface area where the heat transfer takes place (m²)

T_{2}

: temperature of the surrounding fluid (K)

T_{1}

: temperature of the solid surface (K)

In much practical application, a heat transfer coefficient has a relatively constant value over its specified temperature range of usefulness.

Composition

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Convective heat transfer correlations

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Although convective heat transfer can be derived analytically throughdimensional analysis, exact analysis of the boundary layer, approximate integral analysis of the boundary layer and analogies between energy and momentum transfer, these analytic approaches may not offer practical solutions to all problems when there are no mathematical models applicable. Therefore, many correlations were developed by various authors to estimate the convective heat transfer coefficient in various cases including natural convection,forced convection for internal flow and forced convection for external flow. Theseempirical correlations are presented for their particular geometry and flow conditions. As the fluid properties are temperature dependent, they are evaluated at thefilm temperature $T_{f}$ , which is the average of the surface $T_{s}$ and the surrounding bulk temperature, ${{T}_{\infty }}$ .

{{T}_{f}}={\frac {{{T}_{s}}+{{T}_{\infty }}}{2}}

External flow, vertical plane

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Recommendations by Churchill and Chu provide the following correlation for natural convection adjacent to a vertical plane, both for laminar and turbulent flow.^[4]^[5]k is thethermal conductivity of the fluid,L is thecharacteristic length with respect to the direction of gravity, Ra_L is theRayleigh number with respect to this length and Pr is thePrandtl number (the Rayleigh number can be written as the product of the Grashof number and the Prandtl number).

h\ ={\frac {k}{L}}\left({0.825+{\frac {0.387\mathrm {Ra} _{L}^{1/6}}{\left(1+(0.492/\mathrm {Pr} )^{9/16}\right)^{8/27}}}}\right)^{2}\,\quad \mathrm {Ra} _{L}<10^{12}

For laminar flows, the following correlation is slightly more accurate. It is observed that a transition from a laminar to a turbulent boundary occurs when Ra_L exceeds around 10⁹.

h\ ={\frac {k}{L}}\left(0.68+{\frac {0.67\mathrm {Ra} _{L}^{1/4}}{\left(1+(0.492/\mathrm {Pr} )^{9/16}\right)^{4/9}}}\right)\,\quad \mathrm {1} 0^{-1}<\mathrm {Ra} _{L}<10^{9}

External flow, vertical cylinders

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For cylinders with their axes vertical, the expressions for plane surfaces can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter $D {\displaystyle D}$ . For fluids with Pr ≤ 0.72, the correlations for vertical plane walls can be used when^[6]

{\frac {D}{L}}\geq {\frac {35}{\mathrm {Gr} _{L}^{\frac {1}{4}}}}

where $\mathrm {Gr} _{L}$ is theGrashof number.

And in fluids of Pr ≤ 6 when

{\frac {D}{L}}\geq {\frac {25.1}{\mathrm {Gr} _{L}^{\frac {1}{4}}}}

Under these circumstances, the error is limited to up to 5.5%.

External flow, horizontal plates

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W. H. McAdams suggested the following correlations for horizontal plates.^[7] The induced buoyancy will be different depending upon whether the hot surface is facing up or down.

For a hot surface facing up, or a cold surface facing down, for laminar flow:

h\ ={\frac {k0.54\mathrm {Ra} _{L}^{1/4}}{L}}\,\quad 10^{5}<\mathrm {Ra} _{L}<2\times 10^{7}

and for turbulent flow:

h\ ={\frac {k0.14\mathrm {Ra} _{L}^{1/3}}{L}}\,\quad 2\times 10^{7}<\mathrm {Ra} _{L}<3\times 10^{10}.

For a hot surface facing down, or a cold surface facing up, for laminar flow:

h\ ={\frac {k0.27\mathrm {Ra} _{L}^{1/4}}{L}}\,\quad 3\times 10^{5}<\mathrm {Ra} _{L}<3\times 10^{10}.

The characteristic length is the ratio of the plate surface area to perimeter. If the surface is inclined at an angleθ with the vertical then the equations for a vertical plate by Churchill and Chu may be used forθ up to 60°; if the boundary layer flow is laminar, the gravitational constantg is replaced withg cos θ when calculating the Ra term.

External flow, horizontal cylinder

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For cylinders of sufficient length and negligible end effects, Churchill and Chu has the following correlation for $10^{-5}<\mathrm {Ra} _{D}<10^{12}$ .

h\ ={\frac {k}{D}}\left({0.6+{\frac {0.387\mathrm {Ra} _{D}^{1/6}}{\left(1+(0.559/\mathrm {Pr} )^{9/16}\,\right)^{8/27}\,}}}\right)^{2}

External flow, spheres

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For spheres, T. Yuge has the following correlation for Pr≃1 and $1\leq \mathrm {Ra} _{D}\leq 10^{5}$ .^[8]

{\mathrm {Nu} }_{D}\ =2+0.43\mathrm {Ra} _{D}^{1/4}

Vertical rectangular enclosure

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For heat flow between two opposing vertical plates of rectangular enclosures, Catton recommends the following two correlations for smaller aspect ratios.^[9] The correlations are valid for any value of Prandtl number.

For $1<{\frac {H}{L}}<2$ :

h\ ={\frac {k}{L}}0.18\left({\frac {\mathrm {Pr} }{0.2+\mathrm {Pr} }}\mathrm {Ra} _{L}\right)^{0.29}\,\quad \mathrm {Ra} _{L}\mathrm {Pr} /(0.2+\mathrm {Pr} )>10^{3}

whereH is the internal height of the enclosure andL is the horizontal distance between the two sides of different temperatures.

For $2<{\frac {H}{L}}<10$ :

h\ ={\frac {k}{L}}0.22\left({\frac {\mathrm {Pr} }{0.2+\mathrm {Pr} }}\mathrm {Ra} _{L}\right)^{0.28}\left({\frac {H}{L}}\right)^{-1/4}\,\quad \mathrm {Ra} _{L}<10^{10}.

For vertical enclosures with larger aspect ratios, the following two correlations can be used.^[9] For 10 <H/L < 40:

h\ ={\frac {k}{L}}0.42\mathrm {Ra} _{L}^{1/4}\mathrm {Pr} ^{0.012}\left({\frac {H}{L}}\right)^{-0.3}\,\quad 1<\mathrm {Pr} <2\times 10^{4},\,\quad 10^{4}<\mathrm {Ra} _{L}<10^{7}.

For $1<{\frac {H}{L}}<40$ :

h\ ={\frac {k}{L}}0.46\mathrm {Ra} _{L}^{1/3}\,\quad 1<\mathrm {Pr} <20,\,\quad 10^{6}<\mathrm {Ra} _{L}<10^{9}.

For all four correlations, fluid properties are evaluated at the average temperature—as opposed to film temperature— $(T_{1}+T_{2})/2$ , where $T_{1}$ and $T_{2}$ are the temperatures of the vertical surfaces and $T_{1}>T_{2}$ .

Forced convection

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See main articleNusselt number andChurchill–Bernstein equation for forced convection over a horizontal cylinder.

Internal flow, laminar flow

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Sieder and Tate give the following correlation to account for entrance effects in laminar flow in tubes where $D {\displaystyle D}$ is the internal diameter, ${\mu }_{b}$ is the fluid viscosity at the bulk mean temperature, ${\mu }_{w}$ is the viscosity at the tube wall surface temperature.^[8]

\mathrm {Nu} _{D}={1.86}\cdot {{\left(\mathrm {Re} \cdot \mathrm {Pr} \right)}^{{}^{1}\!\!\diagup \!\!{}_{3}\;}}{{\left({\frac {D}{L}}\right)}^{{}^{1}\!\!\diagup \!\!{}_{3}\;}}{{\left({\frac {{\mu }_{b}}{{\mu }_{w}}}\right)}^{0.14}}

For fully developed laminar flow, the Nusselt number is constant and equal to 3.66. Mills combines the entrance effects and fully developed flow into one equation

\mathrm {Nu} _{D}=3.66+{\frac {0.065\cdot \mathrm {Re} \cdot \mathrm {Pr} \cdot {\frac {D}{L}}}{1+0.04\cdot \left(\mathrm {Re} \cdot \mathrm {Pr} \cdot {\frac {D}{L}}\right)^{2/3}}}

^[10]

Internal flow, turbulent flow

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Forced convection, external flow

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In analyzing the heat transfer associated with the flow past the exterior surface of a solid, the situation is complicated by phenomena such as boundary layer separation. Various authors have correlated charts and graphs for different geometries and flow conditions.For flow parallel to a plane surface, where $x {\displaystyle x}$ is the distance from the edge and $L {\displaystyle L}$ is the height of the boundary layer, a mean Nusselt number can be calculated using theColburn analogy.^[8]

Thom correlation

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There exist simple fluid-specific correlations for heat transfer coefficient in boiling. The Thom correlation is for the flow of boiling water (subcooled or saturated at pressures up to about 20 MPa) under conditions where the nucleate boiling contribution predominates over forced convection. This correlation is useful for rough estimation of expected temperature difference given the heat flux:^[13]

$\Delta T_{\rm {sat}}=22.5\cdot {q}^{0.5}\exp(-P/8.7)$

where:

\Delta T_{\rm {sat}}

is the wall temperature elevation above the saturation temperature, K

q is the heat flux, MW/m²

P is the pressure of water, MPa

This empirical correlation is specific to the units given.

Heat transfer coefficient of pipe wall

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The resistance to the flow of heat by the material of pipe wall can be expressed as a "heat transfer coefficient of the pipe wall". However, one needs to select if the heat flux is based on the pipe inner or the outer diameter.If theheat flux is based on the inner diameter of the pipe, and if the pipe wall is thin compared to this diameter, the curvature of the wall has a negligible effect on heat transfer. In this case, the pipe wall can be approximated as a flat plane, which simplifies calculations. This assumption allows the heat transfer coefficient for the pipe wall to be calculated as:

h_{\rm {wall}}={2k \over x}

where

k {\displaystyle k}

is the effectivethermal conductivity of the wall material

x {\displaystyle x}

is the difference between the outer and inner diameter.

However, when the wall thickness is significant enough that curvature cannot be ignored, the heat transfer coefficient needs to account for the cylindrical shape.^[14] Under this condition, the heat transfer coefficient can be more accurately calculated using :

h_{\rm {wall}}={2k \over {d_{\rm {i}}\ln(d_{\rm {o}}/d_{\rm {i}})}}

where

d_{i}

= inner diameter of the pipe [m]

d_{o}

= outer diameter of the pipe [m]

The thermal conductivity of the tube material usually depends on temperature; the mean thermal conductivity is often used.

Combining convective heat transfer coefficients

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For two or more heat transfer processes acting in parallel, convective heat transfer coefficients simply add:

h=h_{1}+h_{2}+\cdots

For two or more heat transfer processes connected in series, convective heat transfer coefficients add inversely:^[15]

{1 \over h}={1 \over h_{1}}+{1 \over h_{2}}+\dots

For example, consider a pipe with a fluid flowing inside. The approximate rate of heat transfer between the bulk of the fluid inside the pipe and the pipe external surface is:^[16]

q=\left({1 \over {{1 \over h}+{t \over k}}}\right)\cdot A\cdot \Delta T

where

q {\displaystyle q}

= heat transfer rate (W)

h {\displaystyle h}

= convective heat transfer coefficient (W/(m²·K))

t {\displaystyle t}

= wall thickness (m)

k {\displaystyle k}

= wall thermal conductivity (W/m·K)

A {\displaystyle A}

= area (m²)

\Delta T

= difference in temperature (K)

Overall heat transfer coefficient

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Theoverall heat transfer coefficient $U {\displaystyle U}$ is a measure of the overall ability of a series of conductive and convective barriers to transfer heat. It is commonly applied to the calculation of heat transfer inheat exchangers, but can be applied equally well to other problems.

For the case of a heat exchanger, $U {\displaystyle U}$ can be used to determine the total heat transfer between the two streams in the heat exchanger by the following relationship:

q=UA\Delta T_{LM}

where:

q {\displaystyle q}

= heat transfer rate (W)

U {\displaystyle U}

= overall heat transfer coefficient (W/(m²·K))

A {\displaystyle A}

= heat transfer surface area (m²)

\Delta T_{LM}

=logarithmic mean temperature difference (K).

The overall heat transfer coefficient takes into account the individual heat transfer coefficients of each stream and the resistance of the pipe material. It can be calculated as the reciprocal of the sum of a series of thermal resistances (but more complex relationships exist, for example when heat transfer takes place by different routes in parallel):

{\frac {1}{UA}}=\sum {\frac {1}{hA}}+\sum R

where:

R = Resistance(s) to heat flow in pipe wall (K/W)

Other parameters are as above.^[17]

The heat transfer coefficient is the heat transferred per unit area per kelvin. Thusarea is included in the equation as it represents the area over which the transfer of heat takes place. The areas for each flow will be different as they represent the contact area for each fluid side.

Thethermal resistance due to the pipe wall (for thin walls) is calculated by the following relationship:

R={\frac {x}{kA}}

where

x {\displaystyle x}

= the wall thickness (m)

k {\displaystyle k}

= the thermal conductivity of the material (W/(m·K))

This represents the heat transfer by conduction in the pipe.

Thethermal conductivity is a characteristic of the particular material. Values of thermal conductivities for various materials are listed in thelist of thermal conductivities.

As mentioned earlier in the article theconvection heat transfer coefficient for each stream depends on the type of fluid, flow properties and temperature properties.

Some typical heat transfer coefficients include^{[citation needed]}:

Air -h = 10 to 100 W/(m²K)
Water -h = 500 to 10,000 W/(m²K).

Thermal resistance due to fouling deposits

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Often during their use, heat exchangers collect a layer of fouling on the surface which, in addition to potentially contaminating a stream, reduces the effectiveness of heat exchangers. In a fouled heat exchanger the buildup on the walls creates an additional layer of materials that heat must flow through. Due to this new layer, there is additional resistance within the heat exchanger and thus the overall heat transfer coefficient of the exchanger is reduced. The following relationship is used to solve for the heat transfer resistance with the additional fouling resistance:^[18]

{\frac {1}{U_{f}P}}

{\frac {1}{UP}}+{\frac {R_{fH}}{P_{H}}}+{\frac {R_{fC}}{P_{C}}}

where

U_{f}

= overall heat transfer coefficient for a fouled heat exchanger,

\textstyle {\rm {\frac {W}{m^{2}K}}}

P {\displaystyle P}

= perimeter of the heat exchanger, may be either the hot or cold side perimeter however, it must be the same perimeter on both sides of the equation,

{\rm {m}}

U {\displaystyle U}

= overall heat transfer coefficient for an unfouled heat exchanger,

\textstyle {\rm {\frac {W}{m^{2}K}}}

R_{fC}

= fouling resistance on the cold side of the heat exchanger,

\textstyle {\rm {\frac {m^{2}K}{W}}}

R_{fH}

= fouling resistance on the hot side of the heat exchanger,

\textstyle {\rm {\frac {m^{2}K}{W}}}

P_{C}

= perimeter of the cold side of the heat exchanger,

{\rm {m}}

P_{H}

= perimeter of the hot side of the heat exchanger,

{\rm {m}}

This equation uses the overall heat transfer coefficient of an unfouled heat exchanger and the fouling resistance to calculate the overall heat transfer coefficient of a fouled heat exchanger. The equation takes into account that the perimeter of the heat exchanger is different on the hot and cold sides. The perimeter used for the $P {\displaystyle P}$ does not matter as long as it is the same. The overall heat transfer coefficients will adjust to take into account that a different perimeter was used as the product $U P {\displaystyle UP}$ will remain the same.

The fouling resistances can be calculated for a specific heat exchanger if the average thickness and thermal conductivity of the fouling are known. The product of the average thickness and thermal conductivity will result in the fouling resistance on a specific side of the heat exchanger.^[18]

R_{f}

{\frac {d_{f}}{k_{f}}}

where:

d_{f}

= average thickness of the fouling in a heat exchanger,

{\rm {m}}

k_{f}

= thermal conductivity of the fouling,

\textstyle {\rm {\frac {W}{mK}}}

References

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^Chiavazzo, Eliodoro; Ventola, Luigi; Calignano, Flaviana; Manfredi, Diego; Asinari, Pietro (2014)."A sensor for direct measurement of small convective heat fluxes: Validation and application to micro-structured surfaces"(PDF).Experimental Thermal and Fluid Science.55:42–53.Bibcode:2014ETFS...55...42C.doi:10.1016/j.expthermflusci.2014.02.010.
^Maddox, D.E.; Mudawar, I. (1989)."Single- and Two-Phase Convective Heat Transfer From Smooth and Enhanced Microelectronic Heat Sources in a Rectangular Channel".Journal of Heat Transfer.111 (4):1045–1052.doi:10.1115/1.3250766.
^^a ^bLienhard, John H. IV; Lienhard, John H., V (2024).A Heat Transfer Textbook (6th ed.). Cambridge, MA: Phlogiston Press.{{cite book}}: CS1 maint: multiple names: authors list (link)
^Churchill, Stuart W.; Chu, Humbert H.S. (November 1975). "Correlating equations for laminar and turbulent free convection from a vertical plate".International Journal of Heat and Mass Transfer.18 (11):1323–1329.Bibcode:1975IJHMT..18.1323C.doi:10.1016/0017-9310(75)90243-4.
^Sukhatme, S. P. (2005).A Textbook on Heat Transfer (Fourth ed.). Universities Press. pp. 257–258.ISBN 978-8173715440.
^Popiel, Czeslaw O. (2008)."Free Convection Heat Transfer from Vertical Slender Cylinders: A Review".Heat Transfer Engineering.29 (6):521–536.Bibcode:2008HTrEn..29..521P.doi:10.1080/01457630801891557.
^McAdams, William H. (1954).Heat Transmission (Third ed.). New York: McGraw-Hill. p. 180.
^^a ^b ^cJames R. Welty; Charles E. Wicks; Robert E. Wilson; Gregory L. Rorrer (2007).Fundamentals of Momentum, Heat and Mass transfer (5th ed.). John Wiley and Sons.ISBN 978-0470128688.
^^a ^bÇengel, Yunus.Heat and Mass Transfer (Second ed.). McGraw-Hill. p. 480.
^Subramanian, R. Shankar."Heat Transfer in Flow Through Conduits"(PDF).clarkson.edu.
^S. S. Kutateladze; V. M. Borishanskii (1966).A Concise Encyclopedia of Heat Transfer. Pergamon Press.
^F. Kreith, ed. (2000).The CRC Handbook of Thermal Engineering. CRC Press.
^W. Rohsenow; J. Hartnet; Y. Cho (1998).Handbook of Heat Transfer (3rd ed.). McGraw-Hill.
^Aggarwal, Nikita (27 October 2024)."Heat Conduction in Cylindrical Systems – Online Calculator & Python Code".ChemEnggCalc. Retrieved11 November 2024.
^This relationship is similar to theharmonic mean; however, it is not multiplied with the numbern of terms.
^"Heat transfer between the bulk of the fluid inside the pipe and the pipe external surface".Physics Stack Exchange. Dec 15, 2014. Retrieved15 December 2014.
^Coulson and Richardson, "Chemical Engineering", Volume 1, Elsevier, 2000
^^a ^bA.F. Mills (1999).Heat Transfer (second ed.). Prentice Hall, Inc.