Nusselt number

Introduction

An understanding of convection boundary layers is necessary to understanding convective heat transfer between a surface and a fluid flowing past it. A thermal boundary layer develops if the fluid free stream temperature and the surface temperatures differ. A temperature profile exists due to the energy exchange resulting from this temperature difference.

The heat transfer rate can then be written as,

Q y = h A ( T s − T ∞ )

And because heat transfer at the surface is by conduction,

Q y = − k A ∂ ∂ y ( T − T s ) | y = 0

These two terms are equal; thus

− k A ∂ ∂ y ( T − T s ) | y = 0 = h A ( T s − T ∞ )

Rearranging,

h k = ∂ ( T s − T ) ∂ y | y = 0 ( T s − T ∞ )

Making it dimensionless by multiplying by representative length L,

h L k = ∂ ( T s − T ) ∂ y | y = 0 ( T s − T ∞ ) L

The right hand side is now the ratio of the temperature gradient at the surface to the reference temperature gradient. While the left hand side is similar to the Biot modulus. This becomes the ratio of conductive thermal resistance to the convective thermal resistance of the fluid, otherwise known as the Nusselt number, Nu.

N u = h L k

Derivation

The Nusselt number may be obtained by a non dimensional analysis of Fourier's law since it is equal to the dimensionless temperature gradient at the surface:

q = − k A ∇ T , where q is the heat transfer rate, k is the constant thermal conductivity and T the fluid temperature.

Indeed, if: ∇ ′ = − L ∇ , and T ′ = T h − T 0 T h − T c

we arrive at

− ∇ ′ T ′ = − L k A ( T h − T c ) q = h L k

then we define

N u L = h L k

so the equation becomes

N u L = − ∇ ′ T ′

By integrating over the surface of the body:

N u ¯ = − 1 S ′ ∫ S ′ N u d S ′ ,

where S ′ = S L 2

Empirical Correlations

Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and the Prandtl number, written as:

N u = f ( R a , P r )

Otherwise, for forced convection, the Nusselt number is generally a function of the Reynolds number and the Prandtl number, or

N u = f ( R e , P r )

Empirical correlations for a wide variety of geometries are available that express the Nusselt number in the aforementioned forms.

Free convection at a vertical wall

Cited as coming from Churchill and Chu:

N u ¯ L   = 0.68 + 0.67 R a L 1 / 4 [ 1 + ( 0.492 / P r ) 9 / 16 ] 4 / 9 R a L ≤ 10 9

Free convection from horizontal plates

If the characteristic length is defined

L   = A s P

where A s is the surface area of the plate and P is its perimeter.

Then for the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment

N u ¯ L   = 0.54 R a L 1 / 4 10 4 ≤ R a L ≤ 10 7 N u ¯ L   = 0.15 R a L 1 / 3 10 7 ≤ R a L ≤ 10 11

And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment

N u ¯ L   = 0.27 R a L 1 / 4 10 5 ≤ R a L ≤ 10 10

Forced convection on flat plate

Flat plate in laminar flow

The local Nusselt number for laminar flow over a flat plate is given by

N u x   = 0.332 R e x 1 / 2 P r 1 / 3 , ( P r > 0.6 )

The average Nusselt number for laminar flow over a flat plate is given by

N u ¯ x   = 2 ∗ 0.332 R e x 1 / 2 P r 1 / 3 , ( P r > 0.6 )

Gnielinski correlation

Gnielinski's correlation for turbulent flow in tubes:

N u D = ( f / 8 ) ( R e D − 1000 ) P r 1 + 12.7 ( f / 8 ) 1 / 2 ( P r 2 / 3 − 1 )

where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes from correlation developed by Petukhov:

f = ( 0.79 ln ⁡ ( R e D ) − 1.64 ) − 2

The Gnielinski Correlation is valid for:

0.5 ≤ P r ≤ 2000 3000 ≤ R e D ≤ 5 × 10 6

Dittus-Boelter equation

The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:

N u D = 0.023 R e D 4 / 5 P r n

where:

D is the inside diameter of the circular duct P r is the Prandtl number n = 0.4 for heating of the fluid, and n = 0.3 for cooling of the fluid.

The Dittus-Boelter equation is valid for

0.6 ≤ P r ≤ 160 R e D ≳ 10 000 L D ≳ 10

Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of 20 °C, viscosity 10.07×10⁻⁴ Pa·s and a heat transfer surface temperature of 40 °C (viscosity 6.96×10⁻⁴, a viscosity correction factor for ( μ / μ s ) can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of 100 °C (viscosity 2.82×10⁻⁴ Pa·s), making a significant difference to the Nusselt number and the heat transfer coefficient.

Sieder-Tate correlation

The Sieder-Tate correlation for turbulent flow is an implicit function, as it analyzes the system as a nonlinear boundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity ( μ and μ s ) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.

N u D = 0.027 R e D 4 / 5 P r 1 / 3 ( μ μ s ) 0.14

where:

μ is the fluid viscosity at the bulk fluid temperature μ s is the fluid viscosity at the heat-transfer boundary surface temperature

The Sieder-Tate correlation is valid for

0.7 ≤ P r ≤ 16 700 R e D ≥ 10 000 L D ≳ 10

Forced convection in fully developed laminar pipe flow

For fully developed internal laminar flow, the Nusselt numbers are constant-valued. The values depend on the hydraulic diameter.

For internal Flow:

N u = h D h k f

where:

D_h = Hydraulic diameter k_f = thermal conductivity of the fluid h = convective heat transfer coefficient

Convection with uniform surface heat flux for circular tubes

From Incropera & DeWitt,

N u D = 48 11 ≃ 4.36

Convection with uniform surface temperature for circular tubes

For the case of constant surface temperature,

N u D = 3.66