Nusselt number

Physical key figure
Surname	Nusselt number
Formula symbol	${\displaystyle {\mathit {Nu}}}$
dimension	dimensionless
definition	${\displaystyle {\mathit {Nu}}={\frac {\alpha \cdot L}{\lambda }}}$
${\displaystyle \alpha }$ Heat transfer coefficient ${\displaystyle L}$ characteristic length ${\displaystyle \lambda }$ Thermal conductivity of the fluid
Named after	Wilhelm Nusselt
scope of application	Heat transfer

The Nusselt number (named after Wilhelm Nusselt ) is a dimensionless number from the similarity theory of heat transfer . It is used to describe the convective heat transfer between a solid surface and a flowing fluid . ${\displaystyle {\mathit {Nu}}}$

definition

The Nusselt number is defined as

${\displaystyle {\mathit {Nu}}={\frac {\alpha \cdot L}{\lambda }}}$.

This includes:

• the heat transfer coefficient , which describes the convective heat transfer between wall and fluid, ${\displaystyle \alpha }$
• the thermal conductivity of the fluid, ${\displaystyle \lambda }$
• as the characteristic length, a dimension which is decisive for the flow and which can be selected in different ways depending on the geometry of the system under consideration. Typical examples are the length of an area overflown in the direction of flow or the diameter of a pipe throughflow. ${\displaystyle L}$

meaning

Two physical systems are " geometrically similar " if all of their corresponding dimensions are in the same numerical relationship to one another (e.g. an original system and a scaled-down model system). They are also “physically similar” if their corresponding physical quantities are in constant relationships to each other (if, for example, all forces, all speeds etc. in the model system are smaller or larger by a certain factor than the corresponding forces and speeds in the original system). The similarity theory states that two geometrically similar systems are also physically similar if the dimensionless parameters that describe the two systems have the same numerical values ​​in both cases.

The Nusselt number is one of these dimensionless indicators. It has the same numerical value in the model system and in the original system if the model is geometrically similar to the original and is exposed to suitably selected conditions so that it is also physically similar to the original. If the Nusselt number is determined on the model, it is also known for the original. It can then be used, for example, to determine the heat transfer coefficient in the original system. ${\displaystyle \alpha ={\mathit {Nu}}\,{\tfrac {\lambda }{L}}}$

application

Model measurements

The possibility of transferring measured values ​​obtained on a model system to the original system if the Nusselt numbers of both systems are coordinated has already been mentioned.

In simple geometries, it is usually obvious which dimension is to be selected as the characteristic length. If there are several dimensions to choose from (in the case of the pipe through which there is flow, for example the radius or the diameter of the pipe can be chosen), the choice can be made as desired, but must be the same in all systems to be compared. The numerical value of the determined Nusselt number depends on the choice, but the equality of the Nusselt numbers of physically similar models is retained when the respective corresponding dimensions are used. If the numerical values ​​of the Nusselt numbers of a system are to be tabulated, then, in case of doubt, indicate which dimension was used.

Correlations

Measurements can also be suitably summarized and tabulated for frequently required systems. It is to be expected that the Nusselt number depends on the properties of the fluids used, such as their speed, density, viscosity, heat capacity, etc. As an analysis of the transport equations shows, these properties can also be combined into dimensionless groups of numbers, namely the Reynolds' Number and the Prandtl number . The local Nusselt number also depends on a coordinate that describes the considered location in the system. For the local Nusselt number, for a given geometry, there is therefore a functional dependence of the shape ${\displaystyle \textstyle {\mathit {Re}}}$ ${\displaystyle \textstyle {\mathit {Pr}}}$ ${\displaystyle x}$

${\displaystyle {\mathit {Nu}}=f(x,{\mathit {Re}},{\mathit {Pr}})}$

expected. If one considers instead the Nusselt number averaged over the system , the dependence on : ${\displaystyle {\mathit {\overline {Nu}}}}$ ${\displaystyle x}$

${\displaystyle {\mathit {\overline {Nu}}}=f({\mathit {Re}},{\mathit {Pr}})}$.

The functional relationship itself can only be derived from theory in very simple cases. As a rule, series of measurements are necessary to which a suitable function is adapted. Such empirical functions derived from measurements are also known as correlations or usage formulas. As experience shows, an approach of the form can often be used for the mean Nusselt number ${\displaystyle f({\mathit {Re}},{\mathit {Pr}})}$

${\displaystyle {\mathit {\overline {Nu}}}=C\ {\mathit {Re}}^{m}\ {\mathit {Pr}}^{1/3}}$

with suitable constants and can be used. ${\displaystyle C}$ ${\displaystyle m}$

The mean Nusselt number on a parallel flow plate of the length is, for example, for laminar flow and ${\displaystyle L}$ ${\displaystyle {\mathit {Pr}}\gtrsim 0{,}6}$

${\displaystyle {\mathit {\overline {Nu}}}_{L}={\frac {{\overline {\alpha }}L}{\lambda }}=0{,}664\ {\mathit {Re}}_{L}^{1/2}\ {\mathit {Pr}}^{1/3}}$

from which the mean heat transfer coefficient on the plate can be determined for numerous fluids and flow velocities . In this example, the formula symbols and also expressly state that these key figures are to be formed as a characteristic length in the present case . ${\displaystyle {\overline {\alpha }}}$ ${\displaystyle {\mathit {\overline {Nu}}}_{L}}$ ${\displaystyle {\mathit {Re}}_{L}}$ ${\displaystyle L}$

Calculation example

Wind flows across the width of a house facade from the side. The wind speed is the air temperature and the facade temperature . What is the mean convective heat transfer coefficient on the facade? ${\displaystyle L=4\ \mathrm {m} }$ ${\displaystyle v=2\ \mathrm {m/s} }$ ${\displaystyle \vartheta _{L}=22\ \mathrm {^{\circ }C} }$ ${\displaystyle \vartheta _{F}=18\ \mathrm {^{\circ }C} }$ ${\displaystyle {\overline {\alpha }}}$

The mean value of the fluid temperature and the surface temperature is generally used as representative of the temperature of the boundary layer between the fluid and the surface. The properties of the air are evaluated for 20 ° C. There are

characteristic length	${\displaystyle L}$	:	4th	m,		density	${\displaystyle \rho }$	:	1.188	kg / m³
Flow velocity	${\displaystyle v}$	:	2	m / s,		Thermal conductivity	${\displaystyle \lambda }$	:	0.02569	W / (m K)
specific heat capacity	${\displaystyle c_{p}}$	:	1007	J / (kg K),		kinematic viscosity	${\displaystyle \nu }$	:	153.5 · 10 −7	m² / s

This results in the Reynolds number

${\displaystyle {\mathit {Re_{L}}}={\frac {v\,L}{\nu }}\,=\,5{,}2\cdot 10^{5}}$

and for the Prandtl number

${\displaystyle {\mathit {Pr}}={\frac {\nu \,\rho \,c_{\mathrm {p} }}{\lambda }}=0{,}715}$.

The task describes a surface with a parallel flow, so the correlation presented in the previous section can be used. The mean Nusselt number follows from this

${\displaystyle {\mathit {\overline {Nu}}}_{L}\,=\,428}$

and from this the mean heat transfer coefficient

${\displaystyle {\overline {\alpha }}\,=\,{\mathit {\overline {Nu}}}_{L}\,{\frac {\lambda }{L}}\,=\,2{,}75\,\mathrm {\frac {W}{m^{2}K}} }$.

For a wider area or a higher wind speed, however, the Reynolds number increasingly exceeds the critical value at which turbulence begins, so that more complicated formulas for mixed laminar and turbulent flow conditions have to be used. Typical convective heat transfer coefficients are around 10 W / m²K under these conditions. ${\displaystyle {\mathit {Re}}_{c}\approx 5\cdot 10^{5}}$

Illustrative interpretations

Despite the non-illustrative definition of the Nusselt number as a dimensionless group of numbers, which appears in the similarity theory as a system number, clear interpretations are possible for some systems.

Formation of a boundary layer using the example of a flat plate with parallel flow. In the undisturbed area, the flow has a constant velocity profile u0. In the area of ​​influence of the plate, a profile forms in which the speed decreases near the plate and goes back to zero on the plate surface.

In the undisturbed area of ​​the fluid flow, the heat is transported both by conduction within the fluid and by convection in the fluid. The temperature of the fluid in this area can be viewed as constant in space and time. In the vicinity of the wall, however, the speed of the flowing fluid decreases due to friction and goes back to zero immediately at the wall ( sticking condition ). In the area in which the flow velocity gradually decreases from the undisturbed value to zero, the contribution of convection to heat transport also decreases, until only heat conduction remains as the only transport mechanism directly on the wall. The area in which the temperature gradually changes from the temperature of the undisturbed flow to the wall temperature is a boundary layer that represents a resistance for the heat transfer between the wall and the fluid because of the reduced heat transport. The heat transfer coefficient describes the strength of the heat flow density that flows through this boundary layer at a given temperature difference between the wall surface and the undisturbed fluid: ${\displaystyle \alpha }$ ${\displaystyle q_{\alpha }}$ ${\displaystyle \Delta T}$

${\displaystyle q_{\alpha }=\alpha \cdot \Delta T}$

Although both (limited) convection and heat conduction are usually involved as transport mechanisms in this heat flux density, they are usually referred to briefly as “convective”.

Comparison with pure heat conduction

The Nusselt number can be understood as the ratio of the real convective heat flux density flowing through the boundary layer to the imaginary heat flux density that would flow through a completely static fluid layer of thickness and thermal conductivity as a result of pure heat conduction . With ${\displaystyle q_{\alpha }}$ ${\displaystyle q_{\lambda }}$ ${\displaystyle L}$ ${\displaystyle \lambda }$

${\displaystyle q_{\lambda }={\frac {\lambda }{L}}\cdot \Delta T}$

is namely

${\displaystyle {\mathit {Nu}}={\frac {\alpha \cdot L}{\lambda }}={\frac {\alpha \cdot \Delta T}{{\frac {\lambda }{L}}\cdot \Delta T}}={\frac {q_{\alpha }}{q_{\lambda }}}}$.

In other words: the Nusselt number expresses the factor by which the heat transfer due to convection is stronger than if pure heat conduction would act.

This approach is particularly clear if the heat flow through an air layer is considered and the thickness of this air layer is chosen as the characteristic length . Then the real and the imaginary completely still air layer are identical and the contribution of convection to the heat transport in the real layer can be read directly from the Nusselt number: If the Nusselt number is greater than 1, there is convection in the air layer, which causes the Reinforced heat transport. ${\displaystyle L}$

Comparison with a stationary substitute layer

Imagine the boundary layer, in which there are different flow velocities, replaced by a completely static fluid layer of the thickness in which the heat can only be transported by conduction. If the substitute layer is exposed to the same temperature gradient, the following applies to the heat flux density flowing through it: ${\displaystyle d}$

${\displaystyle q_{\lambda }={\frac {\lambda }{d}}\cdot \Delta T}$.

If the same heat flow should flow through the substitute layer as through the real boundary layer,

${\displaystyle q_{\alpha }=\alpha \cdot \Delta T}$,

so must apply to: ${\displaystyle d}$

${\displaystyle d={\frac {\lambda }{\alpha }}}$,

and the Nusselt number can be written as

${\displaystyle {\mathit {Nu}}={\frac {\alpha \cdot L}{\lambda }}={\frac {L}{d}}}$.

It can also be understood as the ratio of the characteristic length to the thickness of the substitute layer . ${\displaystyle L}$ ${\displaystyle d}$

Dimensionless temperature gradient

The convective heat flow flowing through the boundary layer is given by

${\displaystyle q_{\alpha }=\alpha \cdot \Delta T=\alpha \cdot (T_{W}-T_{\infty })}$,

when the wall temperature is and the temperature of the undisturbed fluid. ${\displaystyle T_{W}}$ ${\displaystyle T_{\infty }}$

On the other hand, because of the adhesion condition, the fluid in the immediate vicinity of the wall is completely at rest, so that the transfer of heat from the wall to the fluid area adjacent to the wall occurs through pure heat conduction. In this boundary fluid area, the heat flow is driven solely by the temperature gradient directly on the wall : ${\displaystyle W}$

${\displaystyle q_{W}=\left.-\lambda \cdot {\frac {\mathrm {d} T}{\mathrm {d} x}}\right|_{W}}$

The two formulas must provide the same value for the heat flux density. Equating and multiplying both sides with a characteristic length yields: ${\displaystyle L}$

${\displaystyle \alpha L\cdot (T_{W}-T_{\infty })=\left.-\lambda L\cdot {\frac {\mathrm {d} T}{\mathrm {d} x}}\right|_{W}}$

and further

${\displaystyle {\frac {\alpha L}{\lambda }}=\left.-{\frac {\mathrm {d} {\frac {T}{(T_{w}-T_{\infty })}}}{\mathrm {d} {\frac {x}{L}}}}\right|_{W}}$,

so that the Nusselt number can be written as

${\displaystyle {\mathit {Nu}}={\frac {\alpha L}{\lambda }}=\left.-{\frac {\mathrm {d} \vartheta }{\mathrm {d} \xi }}\right|_{W}}$

with the dimensionless temperature and the dimensionless length . ${\displaystyle \vartheta ={\frac {T-T_{\infty }}{T_{W}-T_{\infty }}}}$ ${\displaystyle \xi ={\frac {x}{L}}}$

The Nusselt number can therefore be understood as the negative dimensionless fluid-side temperature gradient on the wall.

Rewrite

${\displaystyle {\mathit {Nu}}=\left.-{\frac {\mathrm {d} {\frac {T}{(T_{w}-T_{\infty })}}}{\mathrm {d} {\frac {x}{L}}}}\right|_{W}=-{\frac {\left.{\frac {\mathrm {d} T}{\mathrm {d} x}}\right|_{W}}{\frac {(T_{w}-T_{\infty })}{L}}}}$

shows that the Nusselt number also indicates how many times the temperature profile resulting from the convective heat transfer is steeper than a profile resulting from pure heat conduction in the fluid.

See also

The Biot number is formed formally similar to the Nusselt number. Unlike the Nusselt number, the thermal conductivity and characteristic length do not refer to the fluid, but to the solid body.

literature

• Merker, GP: Convective heat transfer. Springer, Berlin Heidelberg 1987, ISBN 978-3-540-16995-6
• Bergman Th. L., Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9

Remarks

1. ↑ Different fluids can be used in the original and in the model system, for example air in one system and water in the other.
2. ↑ This applies to systems with forced convection. For other systems, e.g. B. those with free convection, there are other key figures.
3. ↑ The thickness of the flow boundary layer is generally not identical to the thickness of the temperature boundary layer. The ratio of the two thicknesses is described by the Prandtl number .
4. ↑ This applies in any case if - as everywhere else in this article - stationary conditions are considered.
5. ↑ Note that . ${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \xi }}\left({\frac {T-T_{\infty }}{T_{W}-T_{\infty }}}\right)={\frac {\mathrm {d} }{\mathrm {d} \xi }}\left({\frac {T}{T_{W}-T_{\infty }}}\right)}$

Individual evidence

1. ↑ Baehr HD, Stephan K .: Heat and mass transfer. 5th edition, Springer, Berlin 2006, ISBN 978-3-540-32334-1 , p. 20.
2. ↑ ^{a b c} marker, GP: convective heat transfer. Springer, Berlin Heidelberg 1987, ISBN 978-3-540-16995-6 , p. 101.
3. ^ Bergman TL, Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9 , p. 398.
4. ^ ^{A b c} Bergman TL, Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9 , pp. 400f.
5. ^ Bergman TL, Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9 , p. 424.
6. ^ Bergman TL, Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9 , p. 442.
7. ↑ Baehr HD, Stephan K .: Heat and mass transfer. 5th edition, Springer, Berlin 2006, ISBN 978-3-540-32334-1 , p. 695.
8. ^ Bergman TL, Lavine AS, Incropera FP, Dewitt DP: Fundamentals of Heat and Mass Transfer. Seventh edition, John Wiley & Sons 2011, ISBN 978-0470-50197-9 , p. 390.
9. ↑ ^{a b c} marker, GP: convective heat transfer. Springer, Berlin Heidelberg 1987, ISBN 978-3-540-16995-6 , p. 113.
10. ↑ DIN EN 673: Glass in building - Determination of the heat transfer coefficient (U-value) - Calculation method , Beuth Verlag, Berlin 2011.
11. ↑ Merker, GP: Convective heat transfer. Springer, Berlin Heidelberg 1987, ISBN 978-3-540-16995-6 , p. 114.
12. ↑ Merker, GP: Convective heat transfer. Springer, Berlin Heidelberg 1987, ISBN 978-3-540-16995-6 , p. 3.