1
Methods for Determining Convection Heat Transfer Coefficients
1.1. Introduction
In convection, heat transfer occurs between a fluid in motion and a neighboring surface. In reality, the fact that the fluid is in motion is the most specific aspect of convection. It logically follows that the convection heat transfer coefficient will depend intrinsically on the nature of the flow established. In fact, intuitively, we do not expect to have the same heat transfer in a fluid flowing in a laminar regime and a fluid in turbulent flow.
As such, when determining convection heat transfer, the first step is always to characterize the motion of the fluid.
1.2. Characterizing the motion of a fluid
Let us recall here that the flow of a fluid is characterized by a dimensionless number, known as the Reynolds number, Re (Bird, Stewart and Lightfoot, 1975; Knudsen and Katz, 1958; Landau and Lifshitz, 1989), defined by:
where:
d is the diameter of the pipe where the flow occurs
v is the flow velocity of the fluid
ρ is the density of the fluid
μ is the viscosity of the fluid
Two structurally different flow regimes can be encountered, depending on the Reynolds number (see Figure 1.1):
- – a laminar flow for Re ≤ 2,300: in this regime the fluid flow is orderly, moving in parallel layers;
- – a turbulent flow for Re > 4,000: in this regime the fluid flow is disorderly, but generally results in the fluid traveling in a clearly defined direction.
The interval corresponding to Reynolds numbers whereby 2,300 < Re < 4,000 corresponds to a transition zone between these two regimes.
Figure 1.1. Flow regimes
It should be noted that the limits between the different flow areas defined in Figure 1.1 are not as fixed, as they depend to a certain extent on the situations involved and the fluids used: whether the pipes are smooth or not, whether the fluids are viscous are not, etc. Consequently, it is not unusual to find that different authors consider different limits for the switching of the flow regimes.
Indeed, depending on the situation considered, the switch from the laminar zone to the transition zone can spread out from Re = 2,1000 to Re = 2,400. Likewise, the switch from the transition zone to the turbulent zone can span from Re = 2,400 to Re = 10,000. For the purposes of the engineering calculations of interest to us, however, we will retain the limits shown in Figure 1.1.
The flow regime of a fluid can be determined systematically by following the flowchart presented in Figure 1.2.
Figure 1.2. Determining flow regime
1.3. Transfer coefficients and flow regimes
In the various studies of interest to the engineer, we are called upon to determine the flux of energy exchanged by convection between a fluid (gas or liquid) in motion and a solid wall (pipe, furnace wall, etc.) that is in contact with said fluid.
It should be recalled that, whatever the flow regime (laminar or turbulent), the flux exchanged between the fluid and the wall is given by ϕ = h A (Δθ) where:
A is the transfer area between the fluid and the wall
Δθ is the difference between the average temperature of the fluid and that of the wall
h is the convection heat transfer coefficient
Intuitively, we can make the following observations regarding the influence of the flow regime on the convective flux (see Figure 1.3):
- – in laminar flow, the orderly movement of the fluid in parallel layers results in limited renewal of the layers of the fluid that are in contact with the solid wall;
- – in contrast, the disorderly motion of the fluid in turbulent flow results in higher rates of contact surface renewal.
The heat flux corresponding to turbulent flow, due to higher surface renewal rates, will be the larger in the laminar case:
These intuitive observations reflect the fact that energy transfer will be more effective in turbulent flow than in laminar flow, and also that the convection heat transfer coefficients resulting from turbulent flows will be much greater than those obtained for laminar flows.
Figure 1.3. Laminar and turbulent convective fluxes
Thus, determining the transfer coefficients is of great importance when calculating the fluxes transferred by convection. These coefficients are determined from correlations established based on experimentation and dimensional analysis.
1.4. Using dimensional analysis
As we have seen (Volume 1, Chapter 4), dimensional analysis is generally used when we wish to establish one or more equations (correlations) describing a given phenomenon or physical parameter, based on a series of experiments. It makes it possible to describe the evolution of a system through the variations of dimensionless numbers. The use of such numbers makes it possible to render the use of correlations that are general and independent of the system of units considered.
1.4.1. Dimensionless numbers used in convection
The different dimensionless numbers used in engineering calculations were defined in Chapter 4 of the first volume in this set. Let us recall in this section that the dimensionless groups (or numbers without dimensions) usually used to describe heat transfer by convection are:
- – The Nusselt number:
where:
h is the convection heat transfer coefficient, also known as the specific heat transfer coefficient
d is the inner or outer diameter of the pipe (depending on whether we are interested in the heat transfer between a fluid and the inner or outer wall of the pipe)
λ is the heat conductivity of the fluid
- – The Prandtl number:
where:
Cp is the specific heat of the fluid
μ is the viscosity of the fluid
λ is the heat conductivity of the fluid
It should be recalled that we can also write: 
.
I.e. 
, where: 
and 
.
- – The Péclet number:
where:
d is the inner or outer diameter of the pipe
v is the average velocity of the fluid
ρ is the density of the fluid
Cp is the specific heat of the fluid
λ is the heat conductivity of the fluid
We recall that: Pe = Re Pr.
- – The Grashof number:
, where:
d is the inner or outer diameter of the pipe
ρ is the density of the fluid
g is the acceleration of gravity
β is the volumetric expansion coefficient of the constant-pressure fluid
Δθ is the difference between the wall temperature and the average fluid temperature, or vice versa
- – The Stanton number:
, where:
h is the convection heat transfer coefficient
Cp is the specific heat of the fluid
v is the average velocity of the fluid
ρ is the density of the fluid
We also recall here that: 
.
- – The Graetz number:
, where:
W is the fluid mass flow rate
Cp is the specific heat of the fluid
λ is the heat conductivity of the fluid
L is the characteristic dimension of the area considered: length or width
- – The Elenbaas number:
where:
L is the length of the base comprising fins
Z is the distance between two fins
ρ is the density of the fluid
β is the volumetric expansion coefficient of the constant-pressure fluid
g is the acceleration of gravity
Δθ is the difference between the wall temperature of the fins and the temperature of the fluid at large
Cp is the specific heat of the fluid
μ is the viscosity of the fluid
1.4.2. Dimensional analysis applications in convection
1.4.2.1. Application in forced convection inside a tube
Consider a fluid flowing within a cylindrical pipe of diameter d and length L.
The specific heat transfer coefficient at the surface of the tube, h, is a function of:
- – the physical properties of the fluid, ρ, μ, Cp, λ;
- – its average velocity, v;
- – the characteristics of the pipe, d, L.
We can therefore write that the convection heat transfer coefficient is a function, f, of this set of parameters: h = f (ρ, μ, Cp, λ, v, d, L).
We assume that the function, f, can develop in a series of the form:
Thus, each term in the series must have the dimension of h, namely:
Yet:
Under these conditions, the dimensional equation of the series gives:
or:
This dimensional equation can be broken down as follows:
- – for dimension M: 1 = αi + βi + δi;
- – for dimension L: 0 = 2 γi + δi + εi + σi + ηi –3 αi – βi;
- – for dimension T: – 3 = – βi – 2 γi – 3 δi – εi;
- – for dimension θ: – 1 = – γi – δi.
We thus obtain the system of equations: 
.
The series giving h can thus be written in the form:
Or alternatively by grouping together the terms of power αi:
By proceeding in the same way with the terms of power γi, we obtain:
Lastly, by grouping together the terms of power σi, we arrive at:
Thus, dimensional analysis shows that the Nusselt number can be expressed as a function of the powers of Re, Pr and 
.
This makes it possible to orient the experiments to be conducted in order to determine the correlations that give the Nusselt number. Indeed, the correlations sought need to be of the form:
Parameters a, m, n and σ are to be determined from experiments during which we successively vary:
- – the Reynolds number, whilst keeping the other parameters constant: for example, for a given fluid and a set tube diameter, we can vary the fluid’s velocity by varying its flow rate;
- – the Prandtl number, whilst keeping the other parameters constant by changing the nature of the fluid;
- – the ratio
, whilst keeping the other parameters constant by taking, for example, different tube lengths.
Several researchers have worked on these types of experiments to determine the different parameters for flow situations in forced convection. This has led to correlations that often bear the names of the researchers who developed them. These correlations generally enable the calculation of the Nusselt number, and therefore the convection heat transfer coefficient, h, as a function of the Reynolds and Prandtl numbers.
This chapter presents the most important of these correlations, which are of great use when performing calculations in forced convection inside or outside of pipes.
1.4.2.2. Application in natural convection along a tube
We know that the phenomenon of natural convection must depend not only on the physical properties of the fluid considered, but also on the expansion coefficient, β, as well as of the acceleration of gravity, g, and the temperature difference, Δθ. We can thus write: h = f (β, g, Δθ, d, ρ, μ, Cp, λ).
If we proceed as above, we can admit that the function, f, can develop in a series of the form:
Thus, each term in the series must have the dimension of h, namely:
Yet:
Hence the dimensional equation:
This dimensional equation can be broken down as follows:
- – for dimension M: 1 = γi + δi + εi;
- – for dimension L: 0 = αi + βi – γi + δi – 3 εi + 2 ηi;
- – for dimension T: – 3 = – 2 αi – γi – 3 δi – 2 ηi;
- – for dimension θ: – 1 = – δi – ηi.
We thus obtain the system of equations: 
.
Thus, by grouping together the terms of power εi and those of power ηi, the series giving h can be written in the form:
Since 
, we can conclude that the Nusselt number will be a function of the Prandtl number and the Grashof number.
I.e. Nu = C Pra Grb.
Thus, by applying dimensional analysis to natural-convection problems, we are able to show that the Nusselt number can be expressed as a function of the Grashof and Prandtl numbers. Parameters C, a and b and are to be determined from experiments during which we successively vary:
- – the Prandtl number, whilst keeping the other parameters constant by changing the nature of the fluid;
- – the Grashof number, whilst keeping the other parameters constant, for example by varying the temperature difference, Δθ.
Several experiments of this type have been conducted and have enabled appropriate parameters to be determined. Chapter 5 of this volume presents the most significant of these correlations, which make it possible to calculate, in cases of natural convections, the Nusselt number, and therefore the convection heat transfer coefficient, h, as a function of the Grashof and Prandtl numbers.
1.5. Using correlations to calculate h
Let us recall that, in order to calculate convective fluxes, it is essential to know the convection heat transfer coefficient, h. This coefficient is calculated based on the physical data relating to the problem being considered: the nature of the fluid (ρ, Cp, λ, μp, etc.), the flow type (forced or natural), flow rate or velocity, pipe type (characteristic length: diameter or radius). Knowledge of these different parameters enables the dimensionless numbers to be calculated: Pr, then Re (in the case of forced convection) or Gr (in the case of natural convection). It thus enables the Nu to be easily calculated, and therefore h.
But, as we will see in the following sub-sections, it should be noted that the correlations that give Nu are each determined from experimentation, taking the general form of that corresponding to forced convection (Nu as a function of Re and Pr), or that corresponding to natural convection (Nu as a function of Gr and Pr). As a consequence, when using the different correlations, it is important to observe the type of convection considered for the establishment of the correlation by its initial author (see Figure 1.4).
Moreover, as specified by the various authors, the validity of each correlation is conditioned by observance of the conditions for which it has been established. Thus, an established correlation for a laminar flow within a cylindrical pipe will not be valid when estimating the transfer coefficient for a laminar flow on the outside of a pipe of the same type. Similarly, a correlation that has been established for a range of Reynolds numbers will not be valid within another range of Re and, needless to say, the same applies to all of the other parameters specified by each author for the use of their correlation. Ultimately, a correlation is only used appropriately if the conditions for its establishment are scrupulously observed.
Figure 1.4. Deetermining the convective flux
We therefore retain that, when calculating the convection heat transfer coefficient, h, great care needs to be taken, both in choosing which correlation to use and in its conditions of use.
In the following sections, the existing correlations used to determine the convection heat transfer coefficient are presented for:
- – flows in forced convection;
- – flows in natural convection.
1.5.1. Correlations for flows in forced convection
For forced convection, we distinguish between flows occurring inside and those occurring outside of the pipes. Two major classes of correlations are therefore to be mentioned for forced convection. Yet, it is obvious that a series of correlations established by conducting experiments on flows inside pipes can only be used for similar situations where we wish to estimate the transfer coefficient, h, for a fluid circulating inside a pipe. Conversely, the correlations established for forced convection on the outside of the pipes are strictly reserved for this type of situation.
It should also be noted that in each class of correlations, a distinction is made between situations presenting laminar flows and those with flows that are turbulent or in the transition area. Here too, it is essential to determine the appropriate correlation before starting any calculation. It should nevertheless be noted that engineering calculations mainly consider situations in laminar or turbulent regimes; it is rare for interest to be focused on heat transfer problems within the transition area.
1.5.2. Correlations for flows in natural convection
We have seen that natural convection is generated by gravitational forces and by differences in fluid densities. Thus, the expected correlations in this case are expressed as a function of the Grashof number instead of the Reynolds number.
In addition, it is necessary for natural convection to distinguish between situations where the fluid flows along plates and the cases where the flow is through horizontal or vertical tubes. The latter cases are important when calculating transfer coefficients in condensers.