Back to basics: Fundamentals of heat transfer to liquid flowing through a pipe

The post explains following

[1] Eddy transport in turbulent flow [2] Laminar and turbulent heat transfer correlations [3] Hydrodynamic and Thermal entrance’s role on fluid flow and temperature [4] Nusselt number [5] Prandtl number [6] Sieder-Tate equation [7] Dittus-Boelter correlation 

Heat transfer between the fluid and the pipe’s surroundings is an important aspect in heat exchangers, boilers, condensers, evaporators, and many other process equipment. The calculation of heat transfer coefficient, typically by convection or phase transition between a fluid and a solid is one of the major exercises we do in our heat transfer calculations. We classify the flow of a fluid in a straight circular pipe into either laminar or turbulent flow. We assume fully developed incompressible, Newtonian, steady flow conditions.

The first question is how flow develops in a pipe? Why does the velocity profile of laminar flow is parabolic and turbulent flow flat?

How does flow develop in a pipe?

A fluid travels a distance in a pipe before it becomes fully developed. The distance a fluid travels before becoming fully developed is called the entrance length

The area following the pipe entrance where effects originating from the interior wall of the pipe propagate into the flow as an expanding boundary layer. When the boundary layer expands to fill the entire pipe, the developing flow becomes a fully developed flow, where flow characteristics no longer change with increased distance along the pipe.

Why velocity profile of laminar flow is parabolic and flat for turbulent flow?

If the flow in a pipe is laminar, the velocity distribution at a cross-section will be parabolic in shape with the maximum velocity at the center being about twice the average velocity in the pipe. The velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid near the pipe.

In the case of turbulent flow, it gets a little flatter due to vigorous mixing in radial direction and eddy motion. surface.

How flow modifies heat transfer?

Hydrodynamic entrance length vs thermal entrance length on fluid flow and temperature

Hydrodynamic entrance length

The hydrodynamic entrance region refers to the area of a pipe where fluid entering a pipe develops a velocity profile due to viscous forces propagating from the interior wall of a pipe. This region is characterized by a non-uniform flow. The fluid enters a pipe at a uniform velocity, then fluid particles in the layer in contact with the surface of the pipe come to a complete stop due to the no-slip condition. Due to viscous forces within the fluid, the layer in contact with the pipe surface resists the motion of adjacent layers and slows adjacent layers of fluid down gradually, forming a velocity profile.  For the conservation of mass to hold true, the velocity of layers of the fluid in the center of the pipe increases to compensate for the reduced velocities of the layers of fluid near the pipe surface. This develops a velocity gradient across the cross section of the pipe.

The hydrodynamic entry length is the point at which the velocity profile is fully developed in the tube from the point of entry for the fluid. The thermal entry length is the point at which the temperature profile is fully developed from the point at which the tube wall is heated or cooled.

The hydrodynamic entry length is a function of Reynold's and Prandtl’s numbers. For turbulent flow, entry length is approximately independent of the Reynolds Number.

Thermal entrance length

The thermal entrance length describes the distance for incoming flow in a pipe to form a temperature profile of the stable shape. The shape of the fully developed temperature profile is determined by temperature and heat flux conditions along the inside wall of the pipe, as well as fluid properties.

The Prandtl number modifies the hydrodynamic entrance length to determine thermal entrance length. The thermal entrance length for a fluid with a Prandtl number greater than one will be longer than the hydrodynamic entrance length, and shorter if the Prandtl number is less than one.

To summarize, the turbulent flow hydrodynamic and thermal entrance lengths are much shorter than those for laminar flow due to velocity fluctuations that enhance the heat transfer convection and transfer of heat and momentum between the coolant particles.

The short entrance length in turbulent flow contributes significantly to an improved heat transfer compared to laminar flow

Concept of boundary layer

When the fluid just enters the pipe, the thickness of the boundary layer gradually increases from zero moving in the direction of fluid flow and eventually reaches the pipe center and fills the entire pipe. At the boundary layer, the shearing viscous forces are significant. This boundary layer is a hypothetical concept. It divides the flow in the pipe into two regions, boundary layer region - the region in which viscous effects and the velocity changes are significant and [2] the irrotational (core) flow region: The region in which viscous effects and velocity changes are negligible, also known as the inviscid core.

Reynolds Number

The Reynolds number is the ratio of inertial forces to viscous forces within a fluid that is subjected to relative internal movement due to different fluid velocities. A region where these forces change behavior is known as a boundary layer, such as the bounding surface in the interior of a pipe.

We define the Reynolds number as follows. Reynolds number, Re = DVρ/μ, D is the inside diameter of the pipe, V is the average velocity of the fluid, ρ is the density of the fluid, and μ is dynamic.

The value of the Reynolds number permits us to determine whether the flow is laminar or turbulent

Flow in a pipe is considered laminar when the Reynolds number is below 2,300. In the range of Re 2,300 to Re 4,000, the flow is considered in transition from laminar to turbulent. Above Re = 4000, the flow is considered turbulent. Results for heat transfer in the transition window are difficult to predict, and it is generally avoided when someone is designing a heat exchanger.

Turbulent flow is inherently unsteady, being characterized by time-dependent fluctuations in the velocity and pressure, but we usually average over these fluctuations and define time-smoothed or time-average velocity and pressure; these time-smoothed entities can be steady or time-dependent. In this post, we have considered only steady conditions.

Heat transfer in laminar vs turbulent flow

The key difference is in the laminar flow it is the flow direction, and in the turbulent, the directions lie in a plane perpendicular to the tube axis. In the turbulent flow, the heat flow can be broken into radial and azimuthal components.

Azimuthal components of fluid flow

In simple words, the vector from an observer (origin) to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.

The important point is in the turbulent flow heat transfer takes place in the radial and azimuthal directions. This whole thing is commonly termed “eddy transport”.

Eddy transport

The eddy transport provides a much better transfer of energy across the flow at a given axial position in the turbulent flow than in laminar flow.

Wherein conduction the only mechanism that operates in the transverse directions made at right angles to the long axis of the body.

Another important point is the fluid is assumed to have a fully developed turbulent velocity profile throughout the length of the pipe. Local and fully developed Nusselt numbers are presented for fluids with Prandtl numbers ranging from 0.7 to 100 for Reynolds numbers between 50000 and 500000. 

Laminar heat transfer correlations

There is a number of dimensionless correlations which can predict heat transfer rates in laminar flow.

Nusselt number

Nu = φ (Re,Pr---), where Nu = hD/k = is the Nusselt number, φ is some function, and Pr = μ Cp/k = ν /α is the Prandtl number. Here, h is the heat transfer coefficient, k is the thermal conductivity of the fluid, and Cp is the specific heat of the fluid at constant pressure. The Prandtl number can also be written as the ratio of the kinematic viscosity ν to the thermal diffusivity of the fluid α. The heat transfer coefficient appearing in the Nusselt number is usually the average value over the heat transfer surface. You should assume this to be the case unless otherwise stated explicitly.

The physical significance of Prandtl number,

Pr = ν /α

= [Ability of fluid to transport momentum by molecular means] / [ Ability of that fluid to transport energy by molecular means].

Gases typically have Prandtl numbers in the range 0.7 - 1, while the Prandtl number for most liquids is much larger than unity. The Prandtl number for water ranges from 4-7, while that for oil might be of the order of 50-100. It is not uncommon to encounter Prandtl numbers for viscous liquids that are of the order of several thousand or even larger.

Prandtl number of turbulent vs laminar flow

Prandtl number for turbulent flow is about 0.7 while for water it is around 7.56 (At 18 °C)

Sieder and Tate’s correlation for the Nusselt number for laminar flow heat transfer

Nu= 1.86 x Re^1/3x Pr^1/3 x [D/L] ^1/3 x [μb/ μw] ^0.14, b stands for bulk and w stands for wall

 Turbulent flow heat transfer

The entrance length of laminar vs turbulent flows

The entrance lengths are much shorter for turbulent flows, LHS image, because of the additional transport mechanism explained earlier, across the cross-section. Thus, typical hydrodynamic entrance lengths [the point at which the velocity profile is fully developed] in turbulent flow are 10-15 tube diameters, and the thermal entrance lengths are even smaller. Therefore, for most engineering situations we use correlations L/ D ≥ 50, for fully developed conditions.

Dittus-Boelter correlation

The historic equation for use in turbulent pipe flow is the Dittus-Boelter Correlation

A common form is for fluids with Prandtl number in the approximate range of 0.7- 100, and tubes with L/ D > 50

Nu = 0.023 Re^0.8 x Pr^n, here, n = 0.4 if the fluid is being heated, that is, if the wall is at a higher temperature than the entering fluid, and n = 0.3 if the fluid is being cooled. All the physical properties used in the Dittus-Boelter correlation must be evaluated at the average bulk temperature of the fluid. This is the arithmetic average of the bulk average temperatures at the entrance and the exit. The usual recommendation is to use the Dittus-Boelter correlation for Re 10,000 >, but in practice, it is used even at lower Reynolds numbers so long as the flow is turbulent because it is a simple correlation to use.         Credit: Google