The Nusselt number is a dimensionless parameter that characterizes the relationship between the convective and conductive heat transfer rates across a boundary layer. Its value is influenced by various factors, including the type of flow – whether laminar or turbulent— as well as surface geometry.

This article discusses the Nusselt number in the context of laminar pipe flow, covering its calculation and its significance for heat transfer in low velocity conditions.

Table of Contents

## Nusselt Number For Laminar Flow In Pipe

The Nusselt number, denoted as *Nu* and named after German engineer Wilhelm Nusselt, is typically defined as the ratio between convective heat transfer and conductive heat transfer across a boundary layer of fluid. In this context, convective heat transfer accounts for the heat exchange driven by both fluid movement and conduction, whereas conductive heat transfer only considers heat conduction of a hypothetically immobile fluid under the same conditions. In other words, the Nusselt number quantifies the enhancement of heat transfer due to convection compared to conduction.

The general formula for the Nusselt number is:

Where:

- Nu
_{d}= Nusselt number [unitless] - h = convective heat transfer coefficient [W/m
^{2}-K] - L
_{c}= characteristic length of the flow [m] - k = thermal conductivity of the fluid [W/m-K]

In laminar pipe flows, the characteristic length usually refers to the diameter of the pipe. Knowing the Nusselt number is important in obtaining the heat transfer coefficient, which is then used to calculate the heat flux between the pipe wall and the fluid.

For a purely conductive heat transfer, the Nusselt number is equal to one. The larger the effect of convection, the larger the Nusselt number. Since convection only increases heat transfer, not reduce it, the Nusselt number cannot be less than one.

Typically, the Nusselt number falls within the range of 1 to 10 for laminar flows, and between 100 to 1000 for turbulent flows. In laminar pipe flows, the value of the Nusselt number depends on whether the flow is in the thermal entrance region or the thermally fully developed region.

As shown in the diagram above, the thermal entrance region is the initial portion of a pipe where the temperature distribution of the fluid still undergoes significant changes as it adjusts from its initial temperature to the temperature of the surrounding pipe walls. This means that the temperature profile of the fluid is not yet fully developed.

After a certain distance downstream from the pipe inlet, the temperature profile stabilizes, and the flow becomes thermally fully developed. In this region, the temperature distribution remains relatively constant along the pipe’s cross-section.

The Nusselt number is usually higher in the thermal entrance region than in the fully developed region. This is because in the thermal entrance region, the fluid experiences a greater temperature gradient near the wall, leading to increased convective heat transfer.

In this region, the Nusselt number decreases along the pipe length as the fluid temperature and heat transfer coefficient change. Once the flow becomes fully developed, the average Nusselt number maintains a nearly constant value, dependent only on the shape of the conduit and the boundary condition at the wall.

## Nusselt, Reynolds, And Prandtl Number Relationship In Laminar Pipe Flow

In general, the value of the Nusselt number is dependent on the Reynolds number and the Prandtl number, particularly in the thermal entrance region. In fact, in laminar pipe flow scenarios, the correlations used for predicting heat transfer rates are written in the following form:

Where:

- Re
_{d}= Reynolds number [unitless] - Pr = Prandtl number [unitless]

Reynolds and Prandtl numbers are both dimensionless quantities used to characterize different aspects of fluid flow.

The Reynolds number helps predict fluid flow patterns by measuring the ratio between inertial and viscous forces. It can be calculated using the following formula:

Where:

- D = pipe diameter [m]
- V = average fluid velocity [m/s]
- ρ = fluid density [kg/m
^{3}] - μ = dynamic viscosity of the fluid [Pa-s]

Laminar pipe flows typically occur at low Reynolds numbers below 2100.

On the other hand, the Prandtl number is defined as the ratio between the kinematic viscosity and thermal diffusivity of a fluid, as shown in the following equation:

Where:

- ν = kinematic viscosity [m
^{2}/s] - α = thermal diffusivity [m
^{2}/s] - c
_{p}= specific heat of the fluid [J/kg-K]

The Prandtl number is used to approximate the relative thicknesses of the momentum and thermal boundary layers. A Prandtl number smaller than one indicates that heat diffuses more rapidly than fluid velocity. Conversely, a Prandtl number larger than one implies that heat diffuses more slowly than fluid velocity.

Generally, the Nusselt number tends to rise with higher Reynolds and Prandtl numbers. This trend holds true in the thermal entrance region of laminar pipe flow. However, in the fully developed region, the Nusselt number converges to a constant value, becoming independent of both Reynolds and Prandtl numbers.

It is also worth noting that the length of the thermal entrance region is influenced by the Reynolds and Prandtl numbers. For a laminar pipe flow, the thermal entry length can be estimated using the following formula:

Where:

- L
_{t}= thermal entry length [m]

## Nusselt Number Calculation For Laminar Pipe Flow

Determining the Nusselt number involves considering several factors: the flow’s location within either the thermal entrance region or the thermally fully developed region, the prevailing flow regime, and the specific boundary conditions of the system. This section tackles Nusselt number calculation for laminar pipe flow under two kinds of boundary conditions: a constant pipe wall temperature and a constant wall heat flux.

### Calculating Nusselt Number For Laminar Pipe Flow At Constant Wall Temperature

To calculate the Nusselt number for a laminar pipe flow at constant wall temperature, the following empirical relation, developed by Hausen, can be used:

Where:

- Nu
_{d}= average Nusselt number across the pipe length [unitless] - L = pipe length [m]

The heat-transfer coefficient, obtained from the above equation, represents the average value across the entire pipe length. Notice that as the pipe length increases, the average Nusselt number decreases, approaching a constant value of 3.66. Hence, for a laminar pipe flow that is thermally fully developed at constant wall temperature, the average Nusselt number is approximately equal to 3.66.

An alternative and somewhat simpler empirical relation was proposed by Sieder and Tate as follows:

Where:

- μ
_{b}= bulk dynamic viscosity [Pa-s] - μ
_{w}= local dynamic viscosity at the pipe wall [Pa-s]

When applying the Sieder-Tate equation, it is necessary to calculate all the physical properties using the arithmetic average of the bulk temperatures of the fluid at the inlet and the outlet, except for the local dynamic viscosity at the pipe wall. However, this equation can only be used for the following condition:

This is because, at extremely large values of pipe length, the heat transfer coefficient obtained using the above equation would yield a zero value.

It is important to note that the calculation of laminar heat-transfer coefficients often becomes more complex due to the simultaneous influence of natural-convection and forced-convection effects. Furthermore, the empirical correlations above are specifically applicable to smooth tubes. Generally, there is a scarcity of correlations for laminar flow in the context of rough pipes.

### Calculating Nusselt Number For Laminar Pipe Flow At Constant Wall Heat Flux

In the case of uniform wall heat flux, the average fluid temperature increases linearly along the axial direction, such that:

In this case, the heat transfer coefficient can be calculated using the following formula:

Where:

- r = radial distance from the pipe center [m]
- r
_{o}= pipe radius [m] - T
_{w}= temperature at the pipe wall [K] - T
_{b}= bulk temperature [K]

This formula assumes that the pressure is uniform at any cross section, and that the temperature and velocity fields are independent of each other. That is, the temperature gradient does not affect the calculation of the velocity profile.

For a thermally fully developed flow, the above formula can be simplified as follows:

From this heat transfer coefficient formula, the Nusselt number can be obtained as follows:

Hence, for a thermally fully developed laminar pipe flow at constant heat flux, the Nusselt number is approximately equal to 4.364.

## Nusselt Number vs Graetz Number For Laminar Pipe Flow

Kays, Sellars, Tribus, and Klein have conducted calculations of the local and average Nusselt numbers for the laminar entrance regions of circular pipes, assuming a fully developed velocity profile. The results are presented in terms of the inverse Graetz number.

The Graetz number is a dimensionless quantity defined as:

Where:

- Gz = Graetz number [unitless]
- x = axial distance of the flow from the pipe inlet [m]

These data are shown in the diagram below.

As shown in the graph, the Nusselt numbers reach their peak at the inlet and subsequently decrease nonlinearly along the length of the pipe. This decline continues until they stabilize at a constant value upon entering the thermally fully developed region.