Dittus-Boelter Equation Calculation for Turbulent Flow in Tubes

The Dittus-Boelter equation is an empirical correlation used in heat transfer to calculate Convective Heat Transfer Coefficient for fluid flow inside pipes or ducts.

This equation works well for turbulent flow in smooth pipes, but it has limitations when used to rough tubes, which are frequently found in industrial settings, or when there are large temperature variations within the fluid.

Related: Heat Transfer through Conduction Calculator – Fourier’s law 

Dittus-Boelter Equation

Nu=0.023 ⋅ Re^0.8 ⋅ Prⁿ

where,

• Nu is the Nusselt number, which represents the ratio of convective to conductive heat transfer across the boundary layer.
• Re is the Reynolds number, which represents the ratio of inertial forces to viscous forces in the flow.
• Pr is the Prandtl number, which represents the ratio of momentum diffusivity to thermal diffusivity.
• n is an exponent dependent on the heating or cooling condition:
  • n=0.4 for heating (T_wall>T_fluid).
  • n=0.3 for cooling (T_wall<T_fluid​).

Note: The Dittus-Boelter Correlation is valid for turbulent flow (when Re ≥ 10,000) and Prandtl Number Range (0.7 ≤ Pr ≤ 160) and having L/D ≥ 10.

Related: Nusselt Number Calculator – Significance and Calculation

Also Read: LMTD Calculator with correction factor for Heat Exchanger Design

Dittus-Boelter Equation Calculator

This Dittus-Boelter Equation Calculator helps user to calculate the Nusselt number for fully developed turbulent flow in smooth circular tubes. Users can enter the Reynolds and Prandtl numbers and choosing the heat transfer method (heating or cooling).

Related: Convective Heat Transfer Calculator – Newton’s law of cooling

Related: Sieder-Tate Equation Solver

Dittus-Boelter Equation Assumptions

1. Fully Developed Flow (Hydrodynamically and Thermally):
   • Hydrodynamically : The velocity profile is fully formed and does not change along the tube.
   • Thermally: The temperature profile has reached a steady state, with temperature variations only in the radial direction, not along the length of the tube.
2. The Dittus-Boelter equation assumes that the fluid properties (such as viscosity, thermal conductivity, and specific heat) remain constant along the pipe.
3. The equations is used for small to moderate temperature differences.
4. This equation assumes that the fluid is in a single-phase state (liquid or gas)

Note: When the Dittus-Boelter equation isn’t suitable, other correlations are used to estimate the heat transfer coefficient under certain conditions.

The Sieder-Tate equation is used in case of large temperature differences which gives more accuracy since it takes into account the viscosity ratio between the bulk and wall temperatures.

\[Nu = 0.027 Re^{0.8} Pr^{0.33} \left(\frac{\mu}{\mu_w}\right)^{0.14}\]

where,

• μ is the fluid viscosity at bulk temperature
• μ_w is the fluid viscosity at wall temperature

The Gnielinski equation is used for turbulent flow in both smooth and rough pipes, as well as transitional flow, because it effectively considers surface roughness and friction effects.

\[Nu = \frac{\left(\frac{f}{8}\right)(Re – 1000)Pr}{1 + 12.7 \sqrt{\frac{f}{8}} \left(Pr^{0.66} – 1\right)}\]

where, f is the Darcy friction factor

These equations addresses specific limitations of the Dittus-Boelter correlation, and ensures precise heat transfer calculation.

Related: Thermal Boundary Layer Thickness (δT) for Flat Plate

Related: Heat and Mass Transfer – Analogy and Correlations for Chemical Engineers

Nusselt Number Correlations

Nusselt Number is correlated with other dimensionless numbers Reynolds number (Re) and the Prandtl number (Pr) , which characterize the heat transfer processes.

Here, we have summarized Nu correlations of flow over a flat plate, within pipes, and spheres within both laminar and turbulent regimes by their validity ranges based on both Reynolds and Prandtl numbers.

Also Read: Heat Transfer through Conduction Calculator – Fourier’s law 

Also Read: Hydraulic Diameter Calculator for Circular and Non-Circular cross-section

Example Problem on Dittus-Boelter

Hot liquid is flowing at a velocity of 2 m/s through a metallic pipe having an inner diameter of 3.5 cm and length 20 m. The temperature at the inlet of the pipe is 90^◦C. The heat transfer coefficient (in kW/m^2◦C) inside the tube is:

Following data is given for liquid at 90^◦C:

• Density = 950 kg/m³
• Specific heat = 4.23 kJ/kg.^◦C
• Viscosity = 2.55 × 10⁻⁴ kg/m.s
• Thermal conductivity = 0.685 W/m.^◦C

Solution:

Data given:

• Velocity of the liquid, v= 2 m/s
• Inner diameter of the pipe, D = 3.5 cm = 0.035 m
• Length of the pipe, L=20 m
• Inlet temperature of the liquid, T_inlet=90^∘C

From the data given we will be solving Reynolds Number and Prandtl Number using the formula:

For quick calculation you can use our reynolds number calculator and Prandtl Number Calculator

Re = (D * v * ρ) / μ
Re = (0.035 * 2 * 950) / (2.55 * 10^-4) = 260,784

Pr = (c_p * μ) / k
Pr = (4.23 * 1000 * 2.55 * 10^-4) / 0.685 = 1.575

Checking the validity of the conditions provided we can use the Dittus-Boelter Equation for solving this problem

Nu = 0.023 * Re^0.8 * Pr^0.33
Nu = 0.023 * (260784)^0.8 * (1.575)^0.33 = 575.2

Now, using the formula for Nusselt Number and then solving heat transfer coefficient we get,

Nu = (h * D) / k
h = (Nu * k) / D
h = (575.2 * 0.685) / 0.035 = 11,257 W/m^2°C

The heat transfer coefficient (in kW/m^2◦C) inside the tube is: 11.3

Must Read: 10 Mostly used Dimensionless Numbers in Chemical Engineering

Also Read: Critical Thickness of Insulation Calculator for Cylinder and Sphere

Resources

• Heat Transfer a Practical Approach – Book by Yunus A Çengel
• Industrial Chemical Process Design, 2nd Edition – Douglas L. Erwin, P.E
• “Heat Transfer Book” by David W. Hahn.
• “Introduction to Heat Transfer” by Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, and Adrienne S. Lavine.

Disclaimer: The content provided here is for educational purposes. While efforts ensure accuracy, results may not always reflect real-world scenarios. Verify results with other sources and consult professionals for critical applications. Contact us for any suggestions or corrections.