Lesson 4 - Convection

Between the solid surface and the bulk flow outside it, a fluid boundary layer exists through which heat is transferred by conduction. Outside the boundary layer heat is transferred to the bulk of the fluid by convection. The film heat transfer coefficient is a measure of the ability of the laminar layer to transfer heat between the solid surface and the fluid. The film heat transfer coefficient (h) is measured in units of Wm^-2K^-1 and must be estimated to calculate heat transfer by convection.

Unlike thermal conductivity, the heat transfer coefficient is not an intrinsic property of the fluid, but depends on the flow patterns determined by fluid dynamics.

The correlation for a convective heat transfer coefficient is dependent on the geometric system at hand. For example, a rectangular plate, tube cylinder, and sphere all have different geometry and must be handled differently.

There are numerous methods for calculating the heat transfer coefficient for different fluids, flow regimes, and conditions. There are correlations for many different cases, including natural convection, forced convection for internal flow, and forced convection for external flow. These empirical correlations use dimensionless numbers which characterise the flow regime within a given geometry and given flow conditions. Let’s briefly look at these numbers before going any further.

The Nusselt number (Nu) is commonly used to calculate the convection heat transfer coefficient. The Nusselt number is:

Where

h is the convective heat transfer coefficient measured in Wm^-2K^-1 

L is the characteristic length measured in m

k is the thermal conductivity of the fluid measured in Wm^-2K^-1 

The characteristic length, L, depends on the geometry of the system associated with convective heat transfer. For example, in natural convection over a plate, the length of the plate would be the characteristic length. In a sphere, the diameter of the sphere is the characteristic length. Lastly, in flow in a pipe, the inner diameter of the pipe would be the characteristic length.

To determine the film heat transfer coefficient, we need a correlation for the Nusselt number.

For natural convection, the average Nusselt number is a function of the Reynold number and the Grashof number:

For forced convection, the Nusselt number is usually a function of the Reynolds number and the Prandtl number:

The dimensionless numbers mentioned above are important in fluid flow calculations. We will discuss the Reynolds number and the Pradtl number in the next section. The Grashof number will not be further discussed.

In fully developed laminar pipe flow with uniform temperature, the Nusselt number tends to be a constant value for long pipes. 

The Dittus-Boelter equation

Among the many correlations for forced convection in pipes with turbulent flow is the Dittus-Boelter equation. The Dittus-Boelter equation is an empirical correlation that has been developed to determine the heat transfer coefficient for heat transfer between a fluid and a solid surface in a pipe with turbulent flow.

The Dittus-Boelter equation is a relationship between multiple dimensionless numbers and is commonly written as:

Where            Nu is the Nusselt number

                       Re is the Reynolds number

                       Pr is the Prandtl number

                       n is 0.4 if fluid is being heated, and 0.3 if being cooled

The Reynolds number is a measure of the ratio of momentum of the fluid to the viscosity of the fluid (this was covered in detail in the Fluid Mechanics Unit). For fluid flow in a pipe, the Reynolds number is:

Where

            d is the internal pipe diameter measured in m

            u is the average velocity of the fluid measured in ms^-1

            ρ is the density of the fluid measured in kg m^-3

            μ is the dynamic viscosity of the fluid measured in Pa . s 

The Prandtl number is a measure of the ratio of the fluid momentum forces to the fluid thermal forces. The equation for the Prandtl number is:

Where

           C_p is the heat capacity of the fluid measured in J kg^-1 K^-1 

            μ is the viscosity of the fluid measured in Pa · s or N · s m-2

            κ is the thermal conductivity of the fluid measured in Wm^-1 K^-1

Recall that the purpose of developing and calculating these dimensionless numbers is to use them to calculate the film heat transfer coefficient for a convective heat transfer scenario.

Some typical film heat transfer coefficient values for brewery process fluids in phase change are shown in the table below.

Table: Film heat transfer coefficient ranges for brewery-related processes